en Français IX I warmly thank Christophe Coillot for kindly showing me around the lab when I first arrived, and for his support and example in being a sporty and green person. Special thanks to Dilek Cakiroglu

Introduction XVII

1 Theory 1 Afin d'atteindre cet objectif, ou plus généralement, de pouvoir développer des applications (opto-)électroniques réalistes à base de MoS 2 , une compréhension plus fine de ses propriétés électroniques est nécessaire. A ce jour, différents régimes de transport ont été reportés dans le régime isolant dans la littérature [START_REF] Ghatak | Nature of Electronic States in Atomically Thin MoS 2 Field-Effect Transistors[END_REF][START_REF] Jariwala | Band-like transport in high mobility unencapsulated single-layer MoS 2 transistors[END_REF][START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Papadopoulos | Efros-Shklovskii variable range hopping and nonlinear transport in 1T/1T ′ -MoS 2[END_REF]. De même, une description précise de la transition vers un régime métallique [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Wang | Superconductor-insulator transition in space charge doped one unit cell Bi 2.1 Sr 1.9 CaCu 2 O 8+x[END_REF][START_REF] Nihar | Metal to Insulator Quantum-Phase Transition in Few-Layered ReS 2[END_REF] et des mécanismes de diffusion dans ce régime [START_REF] Kaasbjerg | Electron and hole transport in disordered monolayer MoS 2 : Atomic vacancy induced short-range and Coulomb disorder scattering[END_REF][START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF][START_REF] Zhu | Electronic transport and device prospects of monolayer molybdenum disulphide grown by chemical vapour deposition[END_REF][START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF] sont encore nécessaires. De plus, même pour des problèmes en apparence plus basiques, tels qu'à l'interface de contact avec un métal, il n'y a pas d'étude complète des mécanismes en jeu selon le dopage et la température [START_REF] Pan | Reexamination of the Schottky Barrier Heights in Monolayer MoS 2 Field-Effect Transistors[END_REF][START_REF] Ahmed | Carrier transport at the metal-MoS 2 interface[END_REF][START_REF] Kim | Fermi Level Pinning at Electrical Metal Contacts of Monolayer Molybdenum Dichalcogenides[END_REF]. Cette thèse présente nos efforts dans le développement d'un procédé de fabrication contrôlé de composants à base de MoS 2 CVD.

De nombreuses approches tentant de rendre les contacts plus ohmiques, et de les doper le plus efficacement possible ont été étudiés. Après avoir observé que le recuit in-situ était le plus efficace, nous avons étudié en détail le transport dans les régimes isolants et métalliques et la transition entre ces deux régimes, ainsi que le transport à l'interface or/MoS 2 .

Ce manuscrit est divisé en 4 chapitres principaux:

Dans le chapitre 1, Théorie, nous décrivons les bases théoriques et d'état de l'art nécessaires à la compréhension des chapitres suivants. Dans sa première partie, les TMD sont présentés en nous concentrant sur MoS 2 et ses structures de bandes cristallographiques et électroniques. Nous présentons ensuite la motivation derrière le projet: les jonctions Josephson accordables par grille, et quelques-unes de ses utilités possibles. Dans une seconde partie, le modèle classique de Drude de la conduction électrique dans les métaux est présenté.

Le débat de longue date sur la transition métal-isolant est présenté dans une troisième partie, ainsi que les modèles de transport électronique les plus 'populaires' dans le régime isolant. La dernière partie est consacrée aux modèles de transport électronique à l'interface métal-MoS 2 .

Dans le chapitre 2, Techniques Expérimentales, le processus de croissance de MoS Ainsi, de manière générale, la densité électronique s'est avérée être une variable difficile à contrôler en raison des impuretés sur les échantillons, de la faible densité électronique intrinsèque, et de la grande bande interdite de la monocouche MoS 2 . La présence d'une grande densité d'états (DoS) dans la bande interdite liée aux impuretés implique qu'une grande tension de grille est nécessaire pour remplir ces états et déplacer le niveau de Fermi vers la bande de conduction. La densité électronique n'a pas pu être augmentée suffisamment pour observer la supraconductivité, mais a permis l'observation de la transition métal-isolant.

Introduction

The discovery of graphene by Andre Geim et Novoselov in 2004 awoke a huge interest by the scientific community. Graphene is a bi-dimensional material composed only by carbon atoms arranged in a hexagonal way like a honeycomb. It is considered as one of the most promising materials due to its unique properties: stronger than steel, lighter than aluminium, better conductor than copper, flexible and almost transparent. Since this discovery that led to a Nobel Prize, there has been a growing interest in two dimensional (2D) systems for applied and fundamental physics research.

Among them, Transition Metal Dichalcogenides (TMDs) represent a family of 2D materials with unique properties. The TMDs have the form MX 2 where M is the transition metal and X 2 are two chalcogen atoms. They can also exist in monolayer form and have a honeycomb structure like graphene. They can exhibit a wide range of band gaps (0-2 eV), and strong spin orbit coupling strengths (150-460 meV), making the, suitable for spintronics and valleytronics. Bulk (or 3D) TMDs have been known and used for a very long time, but most of the older applications (i.e. as solid lubricants) were related to their mechanical properties determined by the presence of van der Waals bonding between the layers. It is only recently, following the success of graphene, that TMDs moved to the forefront of solid state research, with most of the research realized on mono and few-layer structures. Of all TMDs, MoS 2 has been so far the most studied one. Monolayer MoS 2 has a large direct band gap at the K-points of 1.8-1.9 eV, which qualifies it for numerous photonic applications, and it has excellent mechanical flexibility, which makes it a compelling semiconducting material for flexible electronics.

One of the most interesting features of this material is that it can behave as an insulator, a metal, or even a superconductor, depending on its carrier density. Given those characteristics, the project "Josephson Junctions by Electrostatic tuning of Dichalcogenides" (JJEDi)

was born. This project aims to make a Josephson junction with the use of TMDs, by con-trolling the electron density with three different electrostatic gates. To reach this goal, or to develop MoS 2 potential for realistic (opto)-electronic applications, a better understanding of its electronics properties is still needed. To this day, many different transport regimes in the insulating phase have have been reported. Similarly, a precise description of the transition to a metallic state regime and of scattering in this last one are still needed. Moreover, even problems which could appear as basic, such as the contact interface with the metal, still lacks of a comprehensive study of the mechanisms involved as a function of doping and temperature.

This thesis work aimed at developing a controlled fabrication of CVD MoS 2 -based transistors, with the most ohmic contacts possible, at studying the most efficient gating techniques.

We then studied the transport in the insulating & metallic regime, as well as at the MoS 2metal interface. The thesis is outlined as follow. In Chapter 1 are described some of the features of monolayer MoS 2 , and the classical and quantum electronic transport mechanism that take place within the material. The transport mechanisms at the metal-MoS 2 interface is also briefly discussed. In Chapter 2 is introduced the growth process of MoS 2 and the fabrication method for the different devices. It is also shown how to extract intrinsic quantities of the MoS 2 , together with the experimental setup used to characterize these devices.

Finally, some typical preliminary results are presented, which illustrate the technical difficulties encountered. In Chapter 3, a detailed electronic transport of 3 different Hall bars is presented, focusing on the 4-probe conductivity data extracted after thermally annealing the samples. Different transport mechanism are observed depending on the temperature and electron density, and a discussion on the metal-insulator transition is presented. Finally, Chapter 4 is devoted to the analysis of the 2-probe data, focusing at the metal-MoS 2 electronic transport. The models of thermionic emission of electrons over the Schottky barrier and the tunneling across it are presented.

Chapter 1 Theory

We will describe in the first part of this Chapter what are TMDs, focusing on MoS 2 and its crystallographic and electronic band structures. We will then present the motivation behind the project: gate tunable Josephson junctions, and some of its possible utilities.

On the second part is presented the classical Drude model of electric conduction in metals.

The long-standing debate of the metal-insulator transition is presented in the third part of the Chapter, together with the most 'popular' electronic transport models in the insulator regime. The last part is devoted to the electronic transport models at the metal-MoS 2 interface.

Transition Metal Dichalcogenides

Transition metal dichalcogenides (TMDs) are a class of materials of type MX 2 in which M is a transition metal of groups 4-10 and X is a chalcogen atom. There are about sixty in number, with two-thirds forming layered structures [START_REF] Kolobov | In: Two-Dimensional Transition-Metal Dichalcogenides[END_REF]. In Fig. 1.1a the periodic table of elements is shown with the chalcogen atoms highlighted in orange, the transition metals that form layered structures highlighted in different colours, and those metals that form layered structures with some chalcogens but not with others are colour-framed, e.g. while NiTe 2 has a layered structure NiS 2 possesses a three-dimensional structure. For those that form layered materials the stability of the crystal is ensured by weak van der Waals (vdW) interactions between layers, as schematically shown in Fig. 1.1b. Adapted from [START_REF] Kolobov | In: Two-Dimensional Transition-Metal Dichalcogenides[END_REF].

Crystal structure

Layered TMDs are usually found in three polymorphs called 1T, 2H and 3R. Here the numbers stand for the number of layers in the unit cell and the letters indicate symmetry (Ttrigonal, H-hexagonal, and R-rhombohedral). These polymorphs are schematically shown in Fig. 1.2 [START_REF] Kolobov | In: Two-Dimensional Transition-Metal Dichalcogenides[END_REF]. Each layer is formed of transition metal atoms sandwiched between two layers of chalcogen atoms, to which they are covalently bonded. The unit cell is defined with the c-axis perpendicular to the layers while the parameter a represents the minimal in-plane distance between two chalcogen atoms.

Monolayer MoS 2

Molybdenum disulfide (MoS 2 ) is certainly the most known and the most studied of all TMDs. The most remarkable properties of MoS 2 monolayers are their large direct band gap (1.8-1.9 eV [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Shang | Electronic structures of defects and magnetic impurities in MoS 2 monolayers[END_REF][START_REF] Fai | Atomically Thin MoS 2 : A New Direct-Gap Semiconductor[END_REF][START_REF] Shi | Superconductivity Series in Transition Metal Dichalcogenides by Ionic Gating[END_REF]), which qualify them for numerous photonic applications, and their excellent mechanical flexibility which makes them a compelling semiconducting material for flexible electronics [START_REF] Li | Large-scale flexible and transparent electronics based on monolayer molybdenum disulfide field-effect transistors[END_REF][START_REF] Singh | Flexible Molybdenum Disulfide (MoS 2 ) Atomic Layers for Wearable Electronics and Optoelectronics[END_REF].

The most stable structure of MoS 2 is the 2H. When viewed from the top, a monolayer wiched between two layers of sulfur atoms (yellow). From [START_REF]Theory & Applications[END_REF].

Brillouin zone

The two Bravais primitive lattice vectors are (Fig. 1.4a):

⃗ R 1 = (a, 0, 0) (1.1) ⃗ R 2 = a 2 , √ 3a 2 , 0 , (1.2
Γ = (0,0), K = ( 2π 3a , -2π √ 3a ), M = ( π a , -π √ 3a
) and the Q point, which is located half way between the Γ and K points [START_REF] Ridolfi | A tight-binding model for MoS 2 monolayers[END_REF][START_REF] Piatti | Multi-Valley Superconductivity in Ion-Gated MoS 2 Layers[END_REF]. The reciprocal lattice basis vectors are: are the symmetry high points. Adapted from [START_REF] Ridolfi | A tight-binding model for MoS 2 monolayers[END_REF].

⃗ K 1 = 4π √ 3 √ 3a 2 , - 1 2 , 0 (1.3)
⃗ K 2 = 4π √ 3a (0, 1, 0) (1.4) (a) (b)

Band structure

First-principles band structure calculations for bulk and a few layer MoS 2 show that the minimum of the conduction band (CBM) is located at the Q point, between the K and Γ points in the Brillouin zone, and the maximum of the valence band (VBM) is located at the Γ point [START_REF] Kobayashi | Electronic structure and scanning-tunnelingmicroscopy image of molybdenum dichalcogenide surfaces[END_REF]. In more recent studies [START_REF] Splendiani | Emerging Photoluminescence in Monolayer MoS 2[END_REF], using density functional theory (DFT) calculations, it has been shown that the direct band gap at the K point of the Brillouin zone barely changes as a function of the number of layers, while the indirect band gap increases monotonically as the number of layers decreases. In the case of a monolayer, the band gap becomes direct and located at the K points corresponding to the corners of the hexagonal Brillouin zone, as shown in Fig. 1.5. The energy gap between the CBM and VBM for bulk MoS 2 is 1.2 eV while it is 1.9 eV for monolayer [START_REF] Wang | Superconductor-insulator transition in space charge doped one unit cell Bi 2.1 Sr 1.9 CaCu 2 O 8+x[END_REF]. The Q point is important in electronic transport even for monolayer MoS 2 , since its energy is close to the CBM, with an energy difference of 0.1-0.3 eV depending on the position of the Fermi energy [START_REF] Ridolfi | A tight-binding model for MoS 2 monolayers[END_REF][START_REF] Piatti | Multi-Valley Superconductivity in Ion-Gated MoS 2 Layers[END_REF].

Spin orbit coupling

A tight binding model [START_REF] Liu | Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides[END_REF] reveals that the Bloch states near the band edge for the of the band structure at the band edges located at the K points. From [START_REF] Xiao | Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group-VI Dichalcogenides[END_REF].

Ising superconductivity

Superconductivity is the ability of certain materials to conduct electric current without energy losses and expel magnetic fields. It was first observed in 1911 by H. K. Onnes, a

Dutch physicist [START_REF] Van Delft | The discovery of superconductivity[END_REF], conducting experiments with mercury at liquid helium temperature.

He observed an abrupt drop on the resistance of mercury when cooling it down. It wasn't until 1957 when John Bardeen, Leon Cooper, and John Robert Schrieffer developed a model explaining this phenomena, describing superconductivity as a microscopic effect caused by a condensation of Cooper pairs [START_REF] Bardeen | Microscopic Theory of Superconductivity[END_REF]. The Nobel Prize in Physics 1972 was awarded jointly to John Bardeen, Leon Neil Cooper and John Robert Schrieffer "for their jointly developed theory of superconductivity, usually called the BCS-theory" [START_REF]The Nobel Prize in Physics[END_REF].

Nowadays exist a group of superconductors called "unconventional superconductors", which do not follow the BCS theory. In several unconventional superconductors the transition temperature (the temperature at which the abrupt drop on resistivity occurs) has a maximum as a function of external parameters such as chemical doping or pressure. In 2012 [START_REF] Ye | Superconducting Dome in a Gate-Tuned Band Insulator[END_REF] it was observed for the first time that few layers of MoS 2 undergo a superconducting transition at high electron density, with a critical temperature as high as 10.8 K. In After gated MoS 2 was found to be superconducting, two experimental groups independently discovered that the in-plane upper critical fields of the material were several times higher than the bulk material and six times higher than the Pauli limits [START_REF] Lu | Evidence for two-dimensional Ising superconductivity in gated MoS 2[END_REF][START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF]. Due to the SOC and broken in-plane mirror symmetry, the electron spin is pinned in the out-of-plane directions and is generally called "Ising SOC" to distinguish it from the Rashba SOC, which pins electron spins in the in-plane directions [START_REF] Benjamin | Ising superconductivity and Majorana fermions in transitionmetal dichalcogenides[END_REF]. Owing to the strong pinning of electron spins in the out-of-plane directions by Ising SOC, external in-plane magnetic fields are much less effective in aligning electron spins. As a result, the resistance to in-plane magnetic fields in MoS 2 superconductors is enhanced. In Fig. 1.8 are reproduced the results of applying a perpendicular and parallel magnetic field to a few layers MoS 2 transistor [START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF]. This special type of superconductors are referred to as Ising superconductors, and has been studied in the formation of Majorana states [START_REF] Benjamin | Ising superconductivity and Majorana fermions in transitionmetal dichalcogenides[END_REF], spintronic and valleytronic physics [START_REF] Xiao | Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group-VI Dichalcogenides[END_REF]. From [START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF].

Original thesis objective: gate tunable Josephson junction

A Josephson Junction (JJ) is a device that consists of two or more superconductors coupled by a weak link, as shown in the simplified sketch in Fig 1 .9a. The weak link can be a thin insulating barrier, a short section of non-superconducting metal, or a physical constriction that weakens the superconductivity at the region of contact. The device is named after Brian Josephson, who predicted in 1962 that pairs of superconducting electrons (Cooper pairs) could tunnel through the barrier from one superconductor to another, predicting also the form of the current and voltage relations for the junction [START_REF] Josephson | Possible new effects in superconductive tunnelling[END_REF]. A supercurrent will exist until a critical current is reached, after which the JJ behaves as a resistor, as shown in the I-V diagram in Fig. 1.9b. The Nobel Prize in Physics 1973 was given to Leo Esaki and Ivar

Giaever "for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors", and to Brian David Josephson "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier" [START_REF]The Nobel Prize in Physics[END_REF]. Josephson junctions have important applications in quantum-mechanical circuits, such as superconducting quantum interference device (SQUID) for low magnetic field measurements [START_REF] Sochnikov | dc SQUID Design with Femtotesla Sensitivity for Quantum-Ready Readouts[END_REF], long lifetime superconducting qubits [START_REF] Kono | Breaking the trade-off between fast control and long lifetime of a superconducting qubit[END_REF][START_REF] Rigetti | Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms[END_REF], and rapid single flux quantum for high-performance digital electronics [START_REF] Dimov | The asynchronous rapid single-flux quantum electronics -a promising alternative for the development of high-performance digital circuits[END_REF]. The superconducting state of the MoS 2 will be reached with the use of the ionic liquid gating technique. Ionic liquid can push the carrier density on TMDs as high as ∼ 1×10 14 cm -2

[1, [START_REF] Shi | Superconductivity Series in Transition Metal Dichalcogenides by Ionic Gating[END_REF][START_REF] Lu | Evidence for two-dimensional Ising superconductivity in gated MoS 2[END_REF][START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF][START_REF] Costanzo | Gate-induced superconductivity in atomically thin MoS 2 crystals[END_REF][START_REF] Yoshida | Electrostatic and electrochemical tuning of superconductivity in two-dimensional NbSe 2 crystals[END_REF][START_REF] Sanghyun | Electrostatically Induced Superconductivity at the Surface of WS 2[END_REF][START_REF] Zeng | Gate-Induced Interfacial Superconductivity in 1T-SnSe 2[END_REF], which allows the superconducting state to manifest. Therefore, the whole MoS 2 sample will be in a superconducting state except in the region surrounding the local gate. This local gate will tune the carrier density in a small region around it, and is expected to modify and even destroy the superconducting state, making the weak link in the JJ. A back gate, made of highly doped Si, will act as a fine control on the electron density.

In Fig. 1.11 is shown a simplified sketch of the goal device. JJ made with superconductors exhibiting strong spin-orbit coupling are of prime interest since they could host topological states called Majorana fermions [START_REF] Benjamin | Ising superconductivity and Majorana fermions in transitionmetal dichalcogenides[END_REF][START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF][START_REF] Wang | Topological superconducting phases from inversion symmetry breaking order in spin-orbit-coupled systems[END_REF][START_REF] Chu | Spin-orbit-coupled quantum wires and Majorana fermions on zigzag edges of monolayer transition-metal dichalcogenides[END_REF][START_REF] Zhang | Observation of topological superconductivity on the surface of an iron-based superconductor[END_REF]. These states and their manipulation are expected to lead to applications in quantum computing and other condensed matter areas of physics. First hints in one-dimensional and bidimensional materials have been already reported [START_REF] Mourik | Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices[END_REF][START_REF] Zhang | Observation of topological superconductivity on the surface of an iron-based superconductor[END_REF], but they require hybrid materials where superconductivity is induced by proximity into conductors with strong spin-orbit coupling, or by placing magnetic materials in proximity with normal superconductors. In Fig. A tunnable JJ could also find uses in the area of metrology. In 1963, Shapiro [START_REF] Shapiro | Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations[END_REF] showed that when a JJ is under microwave excitation the critical current is reduced and voltage steps appears. Those steps, now called Shapiro steps, are directly proportional to the excitation frequency, and are of particular interest for the volt standard [START_REF]A Primary Voltage Standard for the Whole World[END_REF], since these voltage steps are linked to frequency only through fundamental constants as V DC = nhf /2e, where n is an integer. In a more recent study performed by Moon et al. [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF] on multilayer MoS 2 , is revealed a complex temperature and doping dependence of the conductivity, as shown in Fig. 1.13b.

It is shown that monolayer MoS 2 transistor devices are commonly found in the literature, having different shapes depending on the utility. A more complete review can be found in [START_REF] Wang | Evolution Application of Two-Dimensional MoS2-Based Field-Effect Transistors[END_REF].

Classical electronic transport

Electrical transport involves the motion of charges under the influence of electric and/or magnetic fields. In non-ballistic systems, the charge carriers are, on one hand, accelerated by the driving fields and, on the other hand, scattered when colliding onto the medium's defects, phonons or between themselves. These opposite interactions lead to a constant mean velocity of the charge carriers. The classical treatment of conduction relies on the Drude model, which introduces a mean scattering time to take into account the diffusion processes.

This model will be presented with and without magnetic field in the following Sections.

Drude model of conduction

The Drude model was proposed in 1900 by Paul Drude to explain the transport of electrons in metals [START_REF] Drude | Zur Elektronentheorie der Metalle[END_REF]. The model treats electrons as classical balls, free and independents.

The electron-electron interactions are neglected and the interactions with the ions and with the impurities are described by a drag force which is proportional to the electron's velocity.

Effect of electric field

The displacement of an electron of charge -e and of mass m under an electric field ⃗ E is described by the following equation (from Newton's second law):

∂ ⃗ P ∂t = ∂ ⃗ P e ∂t + ∂ ⃗ P d ∂t = -e ⃗ E - m⃗ v τ , (1.5)
where ⃗ P is the total momentum, ∂ ⃗ Pe ∂t is the force due to the electric field and ∂ ⃗ P d ∂t is the "drag force" due to collisions, and τ is the average collision time.

In steady state, the velocity has the following expression:

⃗ v = - eτ m ⃗ E (1.6)
The current density in the material is given by:

⃗ j = -ne⃗ v = ne 2 τ m ⃗ E ≡ σ 0 ⃗ E, (1.7)
where σ 0 is the conductivity of the material.

The mobility is defined as

⃗ v = µ ⃗ E: µ = q τ m (1.8)

Effect of electric and magnetic field

The force ⃗ F acting on a particle of electric charge q with velocity ⃗ v, due to an external electric field ⃗ E and magnetic field ⃗ B, is given by the commonly known Lorentz force [START_REF] Lorentz | Ergebnisse und Probleme der Elektronentheorie[END_REF]:

⃗ F = q( ⃗ E + ⃗ v × ⃗ B) (1.9)
The electron's velocity in the material is then given by:

⃗ v = - eτ m ( ⃗ E + ⃗ v × ⃗ B), (1.10)
then, the current density is given by the following expression:

⃗ j = -ne⃗ v = ne 2 τ m ⃗ E - 1 ne ⃗ j × ⃗ B (1.11)
This expression can be expressed in terms of the electric field in a matrix form, in 2D

ones get:

  E x E y   =   1 σ 0 ωcτ σ 0 -ωcτ σ 0 1 σ 0     j x j y   , (1.12)
where ω c = eB/m is the cyclotron frequency of the electron.

Since ⃗ E = ρ ⃗ j, and using the expression 1.8 for the mobility, we obtain the elements of the resistivity tensor ρ:

ρ xx = ρ yy = 1 σ 0 ⇒ ρ xx = ρ yy = 1 neµ (1.13)
ρ xy = -ρ yx = ω c τ σ 0 ⇒ ρ xy = -ρ yx = B ne (1.14)
According to the previous equations, the longitudinal resistivity is independent of magnetic field, but the transverse resistivity (often called Hall resistivity) varies linearly with magnetic field.

Hall effect

If a conductor in which a current is flowing is placed in a region with a magnetic field, the carriers (electrons or holes) will experience the Lorentz force and will deviate from their main trajectory. If the conductor has a finite width, the carriers will start to accumulate at the edges and, eventually, generate an electric field as shown in Figure 1.14. The steady state is reached when the electric force cancels the magnetic force, and the electrons then travel without any deviation. This induced electric field is responsible for the Hall voltage V H , and was discovered by Edwin Hall in 1879 [START_REF] Hall | On a New Action of the Magnet on Electric Currents[END_REF].

For 2D materials, resistances are related to resistivities by the following equations: and the resistivities are related to the conductivities by the following equations:

R xx = V x I x = L W E x j x = L W ρ xx (1.15) R xy = V y I x = W W E y j x = ρ xy (1.16)
ρ xx = σ xx σ 2 xx + σ 2 xy ⇐⇒ σ xx = ρ xx ρ 2 xx + ρ 2 xy
(1.17)

ρ xy = - σ xy σ 2 xx + σ 2 xy ⇐⇒ σ xy = - ρ xy ρ 2 xx + ρ 2 xy (1.18)
The Hall resistivity, with the help of expression 1.14, is often written in the following form:

R H = R xy = B ne = K H B, (1.19)
where K H = 1/ne is the Hall coefficient. By measuring the Hall resistivity, the charge density can be determined with the use of Eq. 1.14, and by measuring the longitudinal resistivity, the mobility of the material can be calculated with the use of Eq. 1.13.

Mesoscopic transport: the metal-insulator transition

The metallic and insulating state of a material are defined by the temperature dependence of its conductivity at low temperatures: if the conductivity decreases as temperature tends to zero the material then falls into the insulator category, but if the opposite occurs it is then considered a metal. A variety of materials can show both states, transitioning from one to another (metal-insulator transition) depending on external parameters others than temperature. The study of the metal-insulator transition (MIT) has been an important field of research in solid state physics for more than half a century, the reason being that the MIT raised unsolved fundamental question. One of the first explanation for the MIT phenomena was given by Anderson in the late 50's [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF] and Mott in the early 60's [START_REF] Mott | The transition to the metallic state[END_REF].

In Vleck "for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems" [START_REF]The Nobel Prize in Physics[END_REF]. Those models are referred today as "Mott transition" and "Anderson transition" and will be presented in the following Sections. Good reviews can be found in [START_REF] Shklovskii | Electronic Properties of Doped Semiconductors[END_REF][START_REF] Nevill | Metal-Insulator Transitions[END_REF][START_REF] Evers | Anderson transitions[END_REF].

Mott transition

The model developed by Mott starts from a cubic crystalline array of hydrogen-like (oneelectron) atoms with a lattice parameter d [START_REF] Mott | Metal-Insulator Transition[END_REF], as shown in Fig. 1.15 . The system can be a metal or an insulator depending on d: for large values of d the array is an insulator, but for small values of d the array will be a metal. When decreasing d, the electron density increases and starts to screen the potential they feel from the nucleus of the atom. This potential can be expressed by the Yukawa potential V (r) = (e/r)exp(-r/λ T F ), where λ -2 T F = 4m * e 2 (3n/π) 1/3 /ℏ 2 is the Thomas-Fermi length. At some critical d or, equivalently, a critical

Anderson transition

In 1958 Anderson published his paper on "the absence of diffusion in certain random lattices" [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF], in which Anderson assumed distributed sites in three-dimensional space that can be occupied by any particle. If the site j is occupied, it will have an energy E j , which is a stochastic variable distributed over a band of energies completely random, with a probability distribution P (E)dE. This band of energy can be characterized by a width W , called impurity band, and schematically shown in Fig. 1.17. This model also assumes that between two sites, j and k, there is an interaction matrix element V jk (r jk ), which transfers the particle from one site to the next. Anderson showed that if V o /W (V o being the average value of the interactions) is lower than a certain critical value, the solutions of the Schrodinger equation for any energy in the band are no longer extended Bloch states, but are localized in space.

Mobility edge

Mott [START_REF] Mott | The electrical properties of liquid mercury[END_REF] first pointed out that, if V 0 /W lies above the critical value, localized states will nonetheless exist for energies near the extremities of a band, and that an energy E c must separate energies where states are localized from energies where they are extended (see Fig. 1.18); E c was later called the "mobility edge" [START_REF] Morrel | Simple Band Model for Amorphous Semiconducting Alloys[END_REF]. If E F > E c , where E F is the Fermi energy, then the system is "metallic", and the conductivity σ tends to a finite value as T → 0. If on the other hand E F < E c , then electrons can only move by thermally activated hopping and σ goes to zero as T → 0. For E F near the mobility edge on the metallic side, Mott found a minimum metallic conductivity value of σ min = 0.03e 2 /ℏa [START_REF] Mott | The conductivity near a mobility edge[END_REF], where a is the mean distance between defects at E F .

Ioffe-Regel criteria

Ioffe and Regel [START_REF] Ioffe | Progress in Semiconductors[END_REF] were the first to point out that values of the mean free path l such that k F l < 1/2π are impossible because, in metals, this mean l ∼ d where d is the distance between atoms, thus, setting a minimum value for the mean free path. Using the distance

MESOSCOPIC TRANSPORT: THE METAL-INSULATOR TRANSITION19

between impurities instead of the distance between atoms, Ioffe and Regel found a minimum for the conductivity:

σ IR ∼ e 2 /3ℏd, (1.20)
being one order of magnitude higher than σ min found by Mott. This idea of minimum conductivity was later abandoned when came out Abrahams work on scaling theory of localization.

The Ioffe-Regel criterion can be used in two-dimensional systems, even if the localization theory demonstrates that all these systems are insulators at zero temperature. The Ioffe-

Regel criteria can still be used because it separates the strongly localized regime from the weakly localized one. This idea has been largely developed by Das Sarma and Hwand [START_REF] Sarma | Two-dimensional metal-insulator transition as a strong localization induced crossover phenomenon[END_REF].

Some materials undergo a MIT as a function of external parameters such as doping, where decreasing 2D carrier density drives the system from being a 2D metal at high density to being a 2D insulator at lower density, as the Fermi level moves through the mobility edge. Das Sarma treats the metallic state as a weakly localized high-density state, and the insulating state as a low-density strongly localized state, viewing the MIT as a transition from weak to strong localization. This transition is then expected to happen around k F l = 1, where k F is the wavevector at E F , and l is the mean free path of the carriers. Since both k F and l are doping dependent, the transition then occurs at a critical electron density n c , with n > n c being a metal and n < n c being an insulator.

One can obtain the conductivity at the transition using the Ioffe-Regel criteria k F l = 1, where l = v F τ ; v F = p F /m = ℏk F /m is the Fermi velocity; τ is the transport relaxation time; k F = (4πn/g s g v ) 1/2 for a 2D electron gas, with g s and g v being, respectively, the spin and valley degeneracy factor. Substituting all this on the Drude formula for the electrical conductivity presented in Eq. 1.7, Das Sarma got the following condition for the critical conductivity σ c = σ(n c ) at the transition [START_REF] Sarma | Two-dimensional metal-insulator transition as a strong localization induced crossover phenomenon[END_REF]:

σ c = g s g v 2 e 2 h (1.21)
Values close to this critical conductivity at the transition have been observed numerous times on MoS 2 as well as other TMDs [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Nihar | Metal to Insulator Quantum-Phase Transition in Few-Layered ReS 2[END_REF][START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF][START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF][START_REF] Britton | Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS 2[END_REF][START_REF] Radisavljevic | Mobility engineering and a metal-insulator transition in monolayer MoS 2[END_REF][START_REF] Ovchinnikov | Electrical Transport Properties of Single-Layer WS 2[END_REF].

Scaling theory of localization

In order to account more generally for the dimensionality of the system, a scaling theory of localization was introduced by Abrahams et al. in 1979 [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF], based on concepts of Thouless.

Thouless [START_REF] Thouless | Maximum Metallic Resistance in Thin Wires[END_REF] imagined a system where the electrons move in a wire of length L which have a uniform potential but with enough irregularities to give the conductivity of the bulk metal a finite value at zero temperature. The time that takes an electron to travel from end to where dE/dN is the average spacing between energy levels, with N being the number of energy levels available. According to Thouless, the individual energy levels are sensitive to the boundary conditions applied at the ends of the wire, and can be shifted by an amount of order ℏ/τ , which can be found using the uncertainty relation between time and energy:

end is τ = L 2 /D,
τ δE = ℏ. Generalizing the sample size to an hypercube of size L d Abrahams got:

δE dE(L)/dN = 2ℏ e 2 σL d-2 = g(L), (1.23)
where g(L) is the generalized dimensionless conductance which is called "Thouless number".

Abrahams then imagined combining many cubes of size L d and computed the new Thouless number:

g(bL) = f (b, (L)), (1.24)
which can be written as:

dlng(L) dlnL = β(g(L)), (1.25)
where β(g(L)) is a scaling function. Abrahams then studied the limits of this function: for large g, where macroscopic theory of conduction is correct giving G(L) = σL d-2 , Abrahams got:

lim g→∞ β(g) = d -2 (1.26)
In the case of small g, localization occurs and therefore g falls off exponentially, giving

g(L) = g 0 exp(-L/ξ): lim g→0 β(g) = lng -lng 0 , (1.27)
where g 0 is dimensionless ratio of order unity and ξ is the localization length. From the asymptotics of Eq. 1.26 and Eq. 1.27, Abrahams sketched the universal curve shown in Fig. 1.19 for β(g) in d = 1,2 and 3 dimensions. In light of this results, it is shown that lower-dimension systems (d < 3) do not undergo a metal-insulator transition because the conductance always decreases with system size. Note that this theory does not involve electron-electron interactions nor weak localization.

Hopping transport

As we have seen previously, disorder induces a bandtail of localized states at the edges of the bands. We present here what happens to the d.c. conductivity when the Fermi energy lies in the range of energies where states are localized. Two mechanisms for conduction are then possible:

(i) Excitation of electrons to the bottom of the conduction band, or mobility edge E c .

The contribution to the conductivity is (neglecting interaction with phonons) [START_REF] Nevill | Metal-Insulator Transitions[END_REF]:

σ = σ min exp - ϵ kT , (1.28)
where σ min = 0.03e 2 /ℏa is the value of the conductivity at E = E c , ϵ = E c -E F and a is the average distance between defects. This kind of conduction dominates at high temperature or at small ϵ, and is often called thermally activated transport.

(ii) Thermally activated hopping of electrons between states near the Fermi energy. In this process, an electron hops from a filled state i to another empty state j. The two states must be close to the Fermi energy, so their energy difference is not too high. The probability that this process occurs is the product of several factors [START_REF] Mott | Conduction in glasses containing transition metal ions[END_REF]:

1. The Boltzmann factor exp(-ϵ ji /kT ), where ϵ ji is the difference between the energies of the two states;s 2. A phonon rate ν ph which depends on the strength of the interaction with phonons;

3. A factor which depends on the overlap of the wavefunctions for the two states. If the localization is very strong, as for hydrogenoid impurities or structural defects, this factor varies as exp(-2r ji /ξ), where r ji is the distance between the two states, and ξ is the spatial extension (the localization length) of these wavefunctions.

The hopping probability is then:

ν ph exp - 2r ji ξ - ϵ ji kT (1.29)
In a first approximation, Mott proposed one can assume that the total conductivity is well approximated by Eq. 1.29:

σ ∼ exp - 2r ji ξ - ϵ ji kT (1.30)

Variable range hopping

Let us assume the the electron hops at a distance r, having πr 2 ϵDoS available sites to hop to, where DoS is the density of states at E F . Assuming at least one site available, this gives the minimal value for ϵ (the average spacing between energies):

ϵ = 1 πr 2 DoS , (1.31)
and the conductivity is then given by:

σ ∼ exp - 2r ξ - 1 πr 2 DoSkT (1.32)
The derivation of the term in the exponential with respect to r gives the optimal hopping distance. One obtains:

σ ∼ exp - T 3 T 1/3 , (1.33)

MESOSCOPIC TRANSPORT: THE METAL-INSULATOR TRANSITION23

where:

T 3 = 27 πξ 2 DoSk B (1.34) r = ξ πDoSk B T 1/3 = ξ 3 T 3 T 1/3 (1.35) ϵ = k 2 B T 2 πDoSξ 2 1/3 = 1 3 k B T 3 T 2 1/3 (1.36)
Ambegaokar et al. [START_REF] Vinay Ambegaokar | Hopping Conductivity in Disordered Systems[END_REF] and Pollak [START_REF] Pollak | A percolation treatment of dc hopping conduction[END_REF] were the first to apply percolation theory to the hopping problem. They found a value of T 3 differing slightly from the value given in Eq. 1.34.

In two dimensions, percolation theory gives:

T 3 = 13.8 ξ 2 DoS(E F )k B (1.37)

Nearest neighbour hopping

If the localization length is too large, which is the case near the Anderson transition, one can neglect the first term in the exponential on Eq. 1.32, the result is:

σ ∼ exp - T 1 T , (1.38)
where T 1 = (k B a 2 DoS) -1 . This kind of transport is known as nearest neighbour hopping (NNH).

Efros-Schklovskii

The previous discussion neglects Coulomb interaction which may exist between the two sites if they correspond, for instance, to two donors. This discussion also assumed a constant density of states at the Fermi level being equal to DoS 0 . Efros and Shklovskii re-derived this law for electrons with Coulomb interactions [START_REF] Efros | Coulomb gap and low temperature conductivity of disordered systems[END_REF]. The effect of the Coulomb interaction is to empty the density of states near the Fermi energy. In this case, the DoS varies with energy below a critical energy ∆ from the Fermi level and vanishes at the Fermi level.

Consider the equilibrium case at temperature T = 0 K, where all states below the Fermi level are occupied and those above it are empty. Consider now two states i and j, which in the ground state are occupied and vacant respectively. The transfer of an electron from state i to state j should increase the energy of the system:

ϵ ji = ϵ j -ϵ i - e 2 κr ij > 0, (1.39)
where κ is the dielectric of the system. We consider now a small energy interval ε centered at the Fermi level. According to 1.39, any two donors in this band with energies on opposite sides of the Fermi level must be separated by a distance r ij not less than e 2 /κε. Therefore the donor concentration n(ε) in a band of width ε cannot exceed ε 2 κ 2 /e 4 , and the density of states DoS(ε) = dn(ε)/dε must vanish when ε → 0. In the 2D case, the density of states is: Thus the density of state in 2 dimensions is then [START_REF] Boris | Electronic properties of doped semiconductors[END_REF]:

DoS(ε) = 2 π |ε|κ 2 e 4 (
DoS(ε) = 2 π κ 2 e 4 |ϵ| if ε ≪ ∆ (1.42) DoS(ε) = DoS 0 if ε ≫ ∆ (1.43)
The gap size should be compared with the optimum band ϵ in Eq. 1.31. When the optimum band is much larger than ∆, the influence of the Coulomb gap can be safely neglected. At the opposite, when ϵ ≪ ∆, the Coulomb gap must be taken into account. Eq. 1.30 then becomes:

σ ∼ exp - 2r ξ - U rk B T , (1.44)
where U = e 2 /κ. Following the same approach as when deriving Eq. 1.38, we find the optimal hopping distance:

σ ∼ exp - T 2 T 1/2 , (1.45)
where:

T 2 = 8e 2 κξk B (1.46) r = e 2 ξ 2k B T 1/2 = ξ 4 T 2 T 1/2
(1.47)

A slightly different expression for T 2 has been found by numerical methods. It was found that k B T 2 = βe 2 /κξ, where β is a numerical coefficient which depends on the dimensionality of the system, and has been calculated by numerical methods [START_REF] Nguen | Two-Dimensional Hopping Conduction in a Magnetic Field[END_REF]: β = 6.2 in two dimensions.

MESOSCOPIC TRANSPORT: THE METAL-INSULATOR TRANSITION25

Hopping conduction in strong electric fields

In the low-temperature and strong electric field regime, an additional term, representing the energy that an electron gains with the electric field, appears in Eq. 1.30. The conduction is then driven by the electric field with a negligible thermal dependence. In the presence of the ES Coulomb gap, the field-driven conductivity in the insulating phase is described as [START_REF] Van | Hopping conduction in strong electric fields and directed percolation[END_REF][START_REF] Zavaritskaya | Hopping conductivity of germanium bicrystals in strong electric fields at T < 1 K[END_REF][START_REF] Nenashev | Hopping conduction in strong electric fields: Negative differential conductivity[END_REF]:

σ ∼ exp - E ES E 1/2 , (1.48)
where E is the electric field and

E ES = k B T 2 2eξ
, with T 2 given by Eq. 1.46. This electric-fielddependence conductivity leads to non-linear current-voltage curves, and it has been observed in monolayer graphene [START_REF] Liu | Variable range hopping and nonlinear transport in monolayer epitaxial graphene grown on SiC[END_REF] and few layers of MoS 2 [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF].

Percolation method

In the solution presented above, we neglected the fact that there is not only two impurities, but a network of sites on which the electrons can hop. Percolation theory applied to this problem gives a more rigorous treatment. In this approach, the hopping problem is transposed to the problem of percolation through a network of random spheres (in 3D) or discs (in 2D). Webman et al. [START_REF] Webman | Numerical simulation of electrical conductivity in microscopically inhomogeneous materials[END_REF] studied the MIT as a percolation problem by numerical simulations of a cubic resistor network, where the material is viewed as consisting of a random submicroscopic mixture of metallic and nonmetallic regions, with the parameter C being the fraction of the total volume which is metallic at the Fermi energy. If C is large then continuous metallic paths extend throughout the material, so that electrons at the Fermi More recently, Das Sarma et al. [START_REF] Sarma | Two-Dimensional Metal-Insulator Transition as a Percolation Transition in a High-Mobility Electron System[END_REF][START_REF] Tracy | Observation of percolation-induced two-dimensional metal-insulator transition in a Si MOSFET[END_REF][START_REF] Sarma | Two-dimensional metal-insulator transition as a potential fluctuation driven semiclassical transport phenomenon[END_REF] suggested that as the carrier density is lowered in a 2D system, screening becomes progressively weaker and strongly nonlinear, and eventually leads to a highly inhomogeneous 2D system as the electron gas becomes unable to screen the disorder potential. This gives rise to a random "hill-and-valley" potential landscape, as shown in Fig. 1.20, with the 2D carriers repelled from the potential hills and accumulating at the potential valleys, in contrast to the high carrier density homogeneous situation. Once these potential hills are numerous enough to prevent percolating conducting paths to span the 2D system, an effective 2D MIT transition takes place with the system being an effective metal (insulator) for n > (<) n c , where n c is the critical percolation density. Das Sarma et al. were able to fit the low-temperature conductivity of GaAs/AlGaAs heterojunctions [START_REF] Sarma | Two-Dimensional Metal-Insulator Transition as a Percolation Transition in a High-Mobility Electron System[END_REF] and Si MOSFETs [START_REF] Tracy | Observation of percolation-induced two-dimensional metal-insulator transition in a Si MOSFET[END_REF] with the expression σ(n) = A(n -n c ) δ , the results are summarized on the table 1.1. From [START_REF] Sarma | Two-dimensional metal-insulator transition as a potential fluctuation driven semiclassical transport phenomenon[END_REF].

Sample δ n c Si 1.20 -1.24 1.2 -2.0 ×10 11 cm -2
GaAs/AlGaAs 1.4 -1.5 1.8 -2.8 ×10 9 cm -2

Table 1.1: Percolation fitting results from [START_REF] Sarma | Two-Dimensional Metal-Insulator Transition as a Percolation Transition in a High-Mobility Electron System[END_REF][START_REF] Tracy | Observation of percolation-induced two-dimensional metal-insulator transition in a Si MOSFET[END_REF].

All these transport mechanism will be used in Chapter 3 to describe the gate and temperature dependent electronic transport in monolayer MoS 2 . manifests itself as a potential energy barrier that leads to rectifying behavior between the metal and the n-type semiconductor, i.e., the flow of electrons from the semiconductor to the metal is easier than conduction in the opposite direction [START_REF] Tung | The physics and chemistry of the Schottky barrier height[END_REF]. A metal-semiconductor junction that conducts current in both directions without rectification is called an ohmic contact. A decrease in barrier height and width will improve the charge injection, which in turn will have significant impact in fabricating high-performance electronic devices.

Electric contacts are commonly of Schottky type on MoS 2 [13-15, 95, 96]. At the metal-MoS 2 contact (where a barrier is formed) the charge injection occurs either (i) as a result of thermionic emission over the top of the barrier due to the transfer of thermal energy or (ii) as a result of quantum mechanical tunneling of carriers across the barrier width. The tunneling behavior can be direct or Fowler-Nordheim (FN) type, depending on the shape and width of the barrier. The direct tunneling and the F-N tunneling are determined by the nature of the interfacial barrier, that is, the former occurs when the barrier is trapezoidal (wide) and the latter occurs when the barrier is triangular (thin). Those mechanism are shown in Fig. 1.21b. Besides the SB, an interface barrier is also created at the interface, as schematically shown in Fig. 1.21a. The shape of this barrier is characterized for being rectangular, with the width depending on the distance between the metal and the semiconductor, or more precisely, the quantum overlapping between the electronics states at the metal with those at the semiconductor. Fowler-Nordheim (F-N) tunneling. From [START_REF] Biddut | Thermionic Emission and Tunneling at Carbon Nanotube-Organic Semiconductor Interface[END_REF].

Thermionic emission

In vacuum, thermionic emission is commonly known as the liberation of electrons from an electrode because of its thermal energy. This occurs because the thermal energy given to the charge carrier overcomes the work function of the material. In the case of contacted semiconductors, the electrons emitted by thermionic emission overcome the SB. Some of the earliest work on electron emission in vacuum from hot bodies were carried out by Richardson, naming the subject as "thermionics". In 1901 [START_REF] Richardson | On the Negative Radiation from Hot Platinum[END_REF], Richardson was able to show that each unit area of a platinum surface emitted a definite number of electrons per unit time, and that it was governed by the well-known law:

J = AT 2 exp(-ω/k B T ), (1.49)
where T is the temperature of the emitter, ω is its work function, k B is the Boltzmann constant and A is a constant. Originally, Richardson derived the Eq. 1.49 with an exponent on T of 1/2 instead of 2, and the Boltzmann constant wasn't present. The equation was later modified by Saul Dushman in 1923 [START_REF] Dushman | Electron Emission from Metals as a Function of Temperature[END_REF]. The Nobel Prize in Physics 1928 was awarded to Owen Willans Richardson "for his work on the thermionic phenomenon and especially for the discovery of the law named after him" [START_REF]The Nobel Prize in Physics[END_REF].

In semiconductors, the Eq. 1.49 takes the form:

J = AT 2 exp(-qΦ B /k B T )exp(q∆Φ img /k B T ), (1.50)
where Φ B is the Schottky barrier height, A is the Richardson constant and Φ img is the barrier lowering due to the image force. The Eq. 1.50 is usually used on 3D materials, but it was shown by Anwar et al. [START_REF] Anwar | Effects of electron confinement on thermionic emission current in a modulation doped heterostructure[END_REF] that due to electron confinement and reduced DoS on 2D materials the equation for thermionic emission takes the form:

J = A * T 3/2 exp(-qΦ B /k B T )exp(-qV /ηk B T ), (1.51)
where A * = q(8πk 3 B m * ) 1/2 /h 2 and η is the ideality factor.

Tunneling

In 1963 Simmons derived an expression for the electric tunnel effect between similar electrodes separated by a thin insulating film. The purpose of this film is to introduce a potential barrier between the electrodes, impeding the flow of electrons between them [START_REF] Simmons | Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film[END_REF].

Starting from the equilibrium conditions (the top of the energy gap of the insulator being positioned above the Fermi level of the electrodes), the electronic current can flow through the insulating region between the two electrodes if: (a) the electrons in the electrodes have enough thermal energy to surmount the potential barrier and flow in the conduction band (already discussed in the previous Section), or (b) the barrier is thin enough to permit its penetration by the electric tunnel effect. With this, Simmons was able to come up with the expression of the current flow through a generalized barrier:

J = J 0 φe -A φ1/2 -( φ + eV )e -A( φ+eV ) 1/2 , (1.52)
where

J 0 = e/2πh(βd) 2 , A = (4πβd/h)(2m * ) 1/2
, with e, h and d being the electron charge, Planck constant and width of the barrier respectively. β is a correction factor, defined in the appendix of Simmons work's, and φ is the mean barrier height above Fermi level.

Two regions can be studied: low and high source-drain voltage (V sd ). In Fig. 1.21b is shown the effect of V sd on the barrier. In the low V sd limit (V ≃ 0) the expression 1.52 takes the form shown in Eq. 1.53, while in the high V sd limit (V > ϕ/e) it takes the form shown in Eq. 1.54, which is called Fowler-Nordheim (F-N) tunneling, after their contribution to electronic emission under high electric fields [START_REF] Howard | Electron emission in intense electric fields[END_REF].

J = 3 2d (2m * ϕ B ) 1/2 e h 2 V sd e -2d ℏ (2m * ϕ B ) 1/2
(1.53)

J = 2.2e 3 8πhϕ B d 2 V 2 sd e -4d 3eℏV (2m * ϕ 3 B ) 1/2 (1.54)
Expression 1.53 is obtained assuming a rectangular SB shape, while expression 1.54 is obtained assuming a triangular SB shape. The work done by Simmons does not take into account the interfacial barrier that is sometimes created at the semiconductor-metal interface, however, those expressions can still work given a sufficiently thin interfacial layer.

Conclusion

In this Chapter we have reviewed the basic properties of TMDs, in particular those of monolayer MoS 2 . The commonly known electronic transport models, through the material and at the semiconductor-metal interface, both classical and mesoscopic, has been also presented. Those models will be used in the next Chapters, but before presenting the main results, we will describe the devices and the experimental setup used to characterize them.

Chapter 2 Experimental techniques

In this Chapter, we first introduce a brief description of transistors and their working principle. Then, the growth process of MoS 2 and the fabrication methods for the different devices is presented. The general experimental setups used to characterize these devices is also presented. Finally, some typical preliminary results are shown, which illustrate the technical difficulties encountered. We detail how we made changes on the fabrication steps and the type of measurements we could perform to overcome these difficulties.

Device working principle

Here we present the transistor configuration used, the different shapes of the devices, and how we can extract intrinsic information of monolayer MoS 2 .

Field effect transistor

A transistor is a semiconductor device used to amplify or switch electrical signals and power. It is composed of a semiconductor material, usually with at least three terminals for electrical connection to an electronic circuit, called source, drain, and gate. A voltage or current applied to the gate electrode controls the current injected between the source and drain terminals. Because the controlled (output) power can be higher than the controlling (input) power, a transistor can amplify a signal. A field effect transistor (FET) is a type of transistor that uses electric fields to modulate the current flow between the source and drain terminals. The Nobel Prize in Physics 1956 was awarded jointly to William Bradford Shockley, John Bardeen and Walter Houser Brattain "for their researches on semiconductors and their discovery of the transistor effect" [START_REF]The Nobel Prize in Physics[END_REF]. In Fig. 2.1 is shown a sketch of a FET and its working principle. The ability to control the chemical potential of a material using electric fields is crucial in semiconductor electronics, and is a major degree of freedom in many experiments in solid state physics. Usually, in a FET, the gate is made up of a metal, but it is possible to use liquids and even light for gating. The electron density below the gate can hence be tuned with an electric field, and is usually approximated to a parallel plate capacitor: ∆n = C∆V g ;

where n, C and V g are the electron density, capacitance and gate voltage respectively. Those types of transistors will be discussed in the following.

Ionic liquid gating

In conventional field-effect gating techniques using a solid gate dielectric, SiO 2 or high-κ dielectric, the charge carrier density accumulated is rather small, of the order of 1×10 13 cm -2 , a density that is far from what is expected to induce the desired electronic phase changes (See Section 1.1.3). A different route for carrier doping that has gained popularity in 2D materials is electrochemical doping with an electric field using ionic liquids. An ionic liquid is a salt containing separable and mobile cations and anions [START_REF] Vaquero | Ionic-Liquid Gating in Two-Dimensional TMDs: The Operation Principles and Spectroscopic Capabilities[END_REF]. Under an externally applied voltage, the ions in the liquid will separate and a layer of cations or anions accumulates near the sample surface, forming an electric double layer (EDL) that is essentially a capacitor with a gap of a few nanometers. When a voltage is applied to an ionic liquid, a huge electric field of the order of 10 MV/cm is generated at the solid/liquid interface due to the formation of an EDL. Owing to this extremely high electric field, EDL gating has the unique capability to push the charge density accumulation as high as a few 10 14 cm -2 , with some electrolytes reaching 1×10 15 cm -2 , determined by Hall effect measurements. This is well beyond the limitations of any solid gate dielectric [START_REF] Ye | Superconducting Dome in a Gate-Tuned Band Insulator[END_REF][START_REF] Shi | Superconductivity Series in Transition Metal Dichalcogenides by Ionic Gating[END_REF][START_REF] Lu | Evidence for two-dimensional Ising superconductivity in gated MoS 2[END_REF][START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF][START_REF] Costanzo | Gate-induced superconductivity in atomically thin MoS 2 crystals[END_REF][START_REF] Yoshida | Electrostatic and electrochemical tuning of superconductivity in two-dimensional NbSe 2 crystals[END_REF][START_REF] Sanghyun | Electrostatically Induced Superconductivity at the Surface of WS 2[END_REF][START_REF] Zeng | Gate-Induced Interfacial Superconductivity in 1T-SnSe 2[END_REF].

Photogating

Photogating is a way of modulating the device channel conductance with light-induced gate voltage. Photogating can be ascribed to the prolonged excess carrier lifetime induced by defects and impurities. If one type of the photogenerated carriers is trapped and they have a certain spatial distribution, they can produce an additional electric field similar to the gate voltage, modulating then the channel conductance [START_REF] Fang | Photogating in Low Dimensional Photodetectors[END_REF]. Photogating have been applied in TMDs and topological insulators as a tool to modify the sample's conductance [START_REF] Wu | Dynamically controllable polarity modulation of MoTe 2 field-effect transistors through ultraviolet light and electrostatic activation[END_REF][START_REF] Yu Yeats | Persistent optical gating of a topological insulator[END_REF].

Devices geometries

Different devices were fabricated for specific purposes. The most common used and widely known is the Hall bar, but TLM and vdP geometries were also fabricated. In the following, we give a bit more details on these devices and their purpose.

Hall bar devices

One of the most common geometry used in electronics research is a Hall bar. This type of devices allows extracting the conductivity of the material by 4-probe measurements. As a consequence, if a magnetic field is applied perpendicular to the Hall bar plane, mobility and carrier density can be estimated via the Hall effect (see Section 1.2.2). A sketch of a Hall bar is shown in Fig. 2.2, and the resistivity can be obtained with the use of the following expression:

ρ xx = V x I sd w l (2.1)

The van der Pauw method

The van der Pauw method is a technique used to measure the resistivity and the Hall coefficient of thin films with four contacts. Its power lies in its ability to measure the properties of a sample of any arbitrary shape, as long as the sample is homogeneous, approximately two-dimensional, does not contain any (geometrical) holes, and the electrodes are small and placed on its perimeter, as shown in Fig. 2.3 [START_REF] Van Der Pauw | A METHOD OF MEASURING SPECIFIC RESISTIVITY AND HALL EFFECT OF DISCS OF ARBITRARY SHAPE[END_REF]. By applying a current i M N between contacts M and N , measuring the potential difference V P -V O and defining R M N,OP = V P -V O i M N , the resistivity of the sample can be obtained using the following expression:

R M N,OP + R N O,P M 2 f, (2.2)
where d is the thickness of the sample, f is a factor which depends on the ratio

R M N,OP /R N O,P M
and is given by the following equation:

R M N,OP -R N O,P M R M N,OP -R N O,P M = f cosh -1 exp(ln 2/f ) 2 (2.
3)

The Hall coefficient (R H ) can be also determined using this method. If we measure R M O,N P at zero magnetic field, and if ∆R M O,N P is the change on resistance when turning on the magnetic field, the Hall coefficient can be calculated using the following expression: the intercept of the line with the y-axis is two times the contact resistance. Thus, the sheet resistance as well as the contact resistance can be determined from this technique.

R H = d B ∆R M O,N P (2.

DC measurements

All the measurements were performed using DC power sources. Two different Keithleys source-measure units (SMUs), the 2400 and 2612B models, were used as power sources. Each SMU instrument is both, a stable DC power source and a 6. The most common technique used for measuring the sample resistance is the 4-probe method. This technique involves using four probes to make electrical contacts with the material, as shown in Fig 2 .6. The applied current I enters and leaves the sample via the outer probes, and flows through the sample. The voltage is then measured between the inner probes. Voltmeters with higher electrical impedance than the sample will prevent them from affecting the circuit being measured, meaning that no current will flow through the inner two probes. Therefore, the wire resistances R W 2 and R W 3 , and the contact resistances R C2 and R C3 do not contribute to the measurement. Any measured decrease in voltage (∆V )

will therefore arise entirely from the sample resistance R S2 [START_REF] Ossila | Sheet Resistance Measurements of Thin Films[END_REF]. with some of the flakes merging with others forming a random big non-uniform flake. Large, isolated monolayer flakes were commonly available, which has been used in fabricating the MoS 2 transistors.

Design of the devices

All the designs were done with the open source KLayout software [114]. KLayout is an viewer and editor of graphic data system files (GDS files). A GDS file contains information about the layout of a circuit, layers, geometric shapes, and text labels. GDS files are used as a standard format between integrated circuit design applications. The created GDS files were then transferred to the electron beam lithography system of the Centrale de Technologie de Montpellier (CTM), where all the devices have been fabricated. Ultraviolet lithography could no be used since the flakes are at random positions, it would have been required to make the physical mask for each growth sample.

Electron beam lithography (EBL)

The process of shaping MoS 2 flakes into any desired form is carried out using electron beam lithography (EBL). The process is summarised in Fig. 2.10. The typical process is the following: first, two thin film of PMMA are uniformly deposited on the substrate by spin coating and separately baked at 170 • C for 5 min. Then, part of the resist is chemically modified by exposing it to an electron beam (EB). Finally, the sample is put in a solvent called "developer", which consist of a mix of Methyl isobutyl ketone (MIBK) and isopropanol (IPA). The MIBK/IPA ratio used is 1:3, and it will remove the exposed or unexposed parts of the resist if positive or negative development is used:

❼ Positive resist:
The resist is exposed with the EB where the underlying material is to be removed. In these resists, exposure to the EB changes the chemical structure of the resist so that it becomes more soluble in the developer. The exposed resist is then washed away by the developer solution.

❼ Negative resist: When exposed to the EB, the negative resist becomes crosslinked/polymerized and less soluble in the developer. Therefore, the negative resist remains on the surface of the substrate where it is exposed, and the developer solution removes only the unexposed areas. We have used this method to minimize the exposure of the MoS 2 to the EB.

Plasma etching

In order to shape the MoS 2 flakes into any form, CHF 3 plasma is used. The plasma is generated by radio frequency (RF) at a pressure of 10 mTorr with a power of 20 W. The gases flowing into the plasma are: 10 sccm of argon, 20 sccm of oxygen and 60 sccm of CHF 3 .

With this low power plasma, the time required to etch a few layers is on the order of a few minutes. Below we show one of the plasma etching tests on PMMA and SiO 2 , which was performed before making the real devices. In Fig. 2.11 is shown a PMMA Hall bar made by EBL using the design shown in Fig. 2.9a. After 5 min of plasma etching, the lateral size reduction is around 200-300 nm. The thickness of the PMMA is also reduced by ∼ 100 nm,

as measured with a profilometer. After etching, the remaining PMMA can be easily removed using acetone. The SiO 2 is also etched with this kind of plasma at a rate of 2-3 nm/min, as measured with the reflectometry technique.

Description of the lithography process

The process of shaping the MoS 2 consists of three levels (S1, S2 and S3), each of them divided into several sub-levels (a,b,c...). Positive EBL is used during the lithography process. All the levels are explained in the following, and shown in Fig. 2.12.

S1 Markers level.

S1a Resist: A layer of electron-sensitive resist PMMA4 is spin coated on the sample's surface. This layer is then backed at 170 • C for 5 min. Two layers of resist are used, each of them separately baked, just to have a thick ∼ 400 nm layer of resist.

S1b EBL: This level consist of irradiating small areas of the resist with a cross pattern.

These pattern will be used in the alignment of the next levels, and to precisely locate the MoS 2 flakes.

S1c Development: It is carried out using MIKB/IPA 1:3 for 30 s. This chemical mix dissolves the exposed resist leaving behind the unexposed areas. In this way, we have resist everywhere except on the exposed parts.

S1d Metallization: Metal is deposited on the sample by EB sputtering. The contacts are mainly gold (Au) but a small layer of titanium (Ti) metal is used to have a better adhesion to the SiO 2 surface. On this step, the thickness of the metals deposited is not important as long as they are visible optically with a microscope and with a scanning electron microscope, since the markers are used only for alignment.

S1e Lift-off: The excess of metal is removed using hot acetone, leaving the exposed areas untouched, and hence, the crosses are transferred to the sample. In Fig. 2.12a is shown the result of markers level.

S2 Plasma etching level.

S2a Resist: Again, two layers of PMMA4 are separately spin coated and baked.

S2b EBL: The area needed to be etched away is exposed since only positive resist is used.

The large pads for bonding the devices and the paths connecting them are also exposed in this step. This is done to avoid any shortcut within contacts or with other devices when there are many MoS 2 flakes.

S2c Development: As before, MIKB/IPA 1:3 is used for the development. After developing, we have resist everywhere except around the desired pattern. In Fig. 2.12b is shown the the exposed area to be etched.

S2c Plasma etching: The sample is then exposed to CHF 3 plasma, already presented in Section 2.2.4.

S3 Contacts level.

S3a Resist: As usual, two layers of PMMA4 are separately spin coated and baked.

S3b EBL: The exposed areas of the sample are where the metal for electrical contacts is desired.

S3c Development: As before, MIKB/IPA 1:3 is used for the development. After developing, the pads and paths ready for metallization. In Fig. 2.12c is shown the exposed area where the metal will be deposited.

S3d Metallization: As with the markers, Ti/Au is deposited on the sample by EB sputtering. In this step, the thickness of the metals deposited are important, since they are contacting the MoS 2 . Different thickness has been tried, they will be described at the end of the current Section.

S3e Lift-off: The excess of metal is removed using hot acetone. This is the final step of the fabrication process, Fig. 2.12d shows a fabricated device.

Fabricated devices

The

(d) (e) (f) (g) (h) (i) (j) (k) (l)

Experimental setups

Before showing the measurement results, we present here the setup used for the different characterizations. Some of those has been performed on MoS 2 flakes prior and after fabricating the devices.

Raman spectroscopy

Raman spectroscopy is a spectroscopic technique used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be The MoS 2 has two characteristic Raman peaks or modes called E 1 2g and A 1g and they exhibit sensitive thickness dependence. The E 1 2g is an in-plane vibrational mode and A 1g is an out-of-plane vibrational mode. The distance in frequency between those peaks increases with the number of layers in MoS 2 , which provides a convenient and reliable mean for determining layer thickness with atomic-level precision [START_REF] Lee | Anomalous Lattice Vibrations of Single-and Few-Layer MoS 2[END_REF]. Raman spectroscopy has been used to evaluate the thickness of the flakes before the EBL process using an Horiba commercial micro-Raman. A home-made system from the group has been used for more accurate studies.

Reflectometry

Reflectometry uses the reflection of light at surfaces and interfaces to measure characteristics such as film thickness and index of refraction. The main components of a reflectometer are a broadband white light source, focused onto the sample's surface via an microscope objective (see Fig. 2.23b). The reflected light is collected by the same objective and sent to a spectrometer. The working principle is described in the following [START_REF]Lurie Nanofabrication Facility Wiki[END_REF]:

❼ A source of stable intensity light is used to illuminate a spot on the sample. ❼ Some light is reflected from the top surface of the sample and some is transmitted into it. There is another reflection when the transmitted light reaches the bottom interface of the sample. The light from each of these reflections recombines to generate varying levels of constructive or destructive interference depending on the wavelength of light and the optical distance the light travels (see Fig. 2

.23a).

❼ A model is used to fit the reflected intensity versus wavelength data to extract the sample thickness. This model uses the Fresnel coefficients, the optical indices of thin film and of the substrate. These fixed values are stored beforehand.

❼ This technique gives access to the SiO 2 thickness through the interference pattern, as well as the MoS 2 thickness through its absorption coefficient. It can only solve for layer thicknesses, from one layer up to a few layers in the case of MoS 2 .

Ball bonding

Ball bonding is a common type of wire bonding used to make electrical connections between small devices and the sample holder. This process is carried out using the microscopic bonding machine shown in Fig. 2.24. Its working principle is described in the following.

There is a small capillary through which the wire is delivered. A high-voltage electric discharge is applied to the wire, melting its tip and forming a ball because of the surface tension.

Finally, the capillary is lowered to the surface of the sample, which is typically heated until 100 • C, and ultrasound is applied to the wire in contact with the metallic pad of the sample.

The combined heat, pressure, and ultrasound create a weld between the ball and the contact pad. The wire is them brought to the sample holder and the process is repeated, this time with higher strength, forming the second bond and cutting the wire. This process is resumed in Fig. 2.25a.

A series of small crosses or markers were used to test the bonding parameters prior fabricating the devices. The markers were done by EBL and are made of Ti/Au 10/75 nm.

The ball bonding creates a circular gold contact on the surface with 150 µm of diameter, and does not breaks the 300 nm of SiO 2 , which was tested by applying a voltage between the marker and the back-gate. The bonding parameter used are shown in Fig. 2.25b, where Bond1 and Bond2 are, respectively, the bonding done on the device, and the one done on the sample holder.

Probe Station

After finishing the fabrication process in the clean room, the first step is to check which devices are working properly, i.e, they are conducting, and the back-gate is not leaking. For this step a probe station, shown in Fig. 2.27, is used. The probe station is used to electrically test a device before and after bonding wires on it. It utilizes micrometer screws to allow precise positioning of thin metal needles on the device's surface, with the force applied to the contacts being a controllable parameter. The material of the needles used were copper and tungsten, having a tip diameter of 150 µm and 7 µm respectively. These needles are connected to coaxial cables.

Doping control

The doping was controlled using mainly the back-gate of the devices, which consists of highly doped silicon. It was electrically connected by pasting the sample with conductive epoxy. The voltage was applied with a SMU in the range -100 V -100 V, which is close to the maximum that 300 nm of SiO 2 can hold. The current leakage was also systematically monitored to prevent breakdowns.

Two other ways of controlling doping were also tested: the ionic liquid Diethylmethyl

He free cryostat

A Janis helium-free cryostat, shown in Fig. 2.29, with a temperature range of 10 K -800 K was used to test the effect of thermal annealing on the samples. Most of the electrical measurements were performed on this setup, at high and low temperatures. The high temperatures are reached under low (∼ 10 -3 mTorr) primary pump vacuum. This setup uses helium gas in a closed loop to lower the temperature, meaning efficient recycling of the helium gas. With this setup, it is also possible to apply a magnetic field up to 1 T, using water cooled electro magnets. Distances are given in millimeters.

Device Performance

In this Section we present a few preliminary results that show the difficulties encountered.

Before and after transport measurement, device thickness and/or homogeneity has been checked by optical means. All the fabricated devices has been tested in the probe station, bonding the ones that worked. All of the working devices exhibited high, non-linear 2probe resistance and low doping, even at high back gate voltages. We will present here the various attempt to improve in-situ those devices, with the thermal annealing being the most favorable given our setup.

Reflectometry and Raman spectra

The typical reflectometry fit window is shown in Fig. 2.31a, where the ratio between the reference light and the reflected light from the sample is plotted against their wavelength.

The region of interest is between 450 nm and 900 nm. The reflected light, both from the MoS 2 flake (red dots) and from the SiO 2 (white dots) are fitted, and the number of layers of the MoS 2 flake is then obtained. In A more thorough and detailed mapping of two devices, namely J339 and J12B, shown in Fig. 2.33a and 2.33b respectively, has been done after they were electrically tested (the electrical measurement result of those device will be shown in the next Chapters). These Raman measurements were realized with a laser of optical wavelength of 532 nm and a spot size ∼ 1 µm, allowing a complete spatial mapping of the device. In Fig. 2.33c is shown a color map plot of the frequency difference δ = ω(A 1g ) -ω(E 1 2g ) measured on device J339. The energy difference δ is very homogeneous over the entire MoS 2 surface. It is centered around 20 cm -1 , with a full width at half maximum of less than 0.8 cm -1 . Such a peaked frequency dispersion is the signature of a homogeneous MoS 2 monolayer [START_REF] Lee | Anomalous Lattice Vibrations of Single-and Few-Layer MoS 2[END_REF].

Similar results were obtained on device J12B, where, however, a few additional MoS 2 bilayer patches were also evidenced covering less than 1% of the total surface. These patches were also visible with optical and SEM images, confirming this low coverage. Fig. 2.33d and 2.33e show some typical Raman spectra obtained in devices J339 and J12B. The spectrum of an additional sample, obtained by exfoliation of a MoS 2 monolayer, is also shown for comparison. Fig. 2.33d focuses on the two A 1g and E 1 2g modes, whereas Fig. 2.33e evidences the presence of additional modes at lower frequencies. Among these modes, the most prominent is the LA(M ) peak around 227 cm -1 . Its intensity is proportional to the density of defects n D . Following Mignuzzi et al. [START_REF] Mignuzzi | Effect of disorder on Raman scattering of single-layer MoS 2[END_REF], the average inter-defect distance L D can be evaluated from the ratio R of the integrated intensities of the LA(M ) and A 1g mode. For device J339, R ∼ 0.044 gives a L D = 3-4 nm, meaning a density of defects of n D = (2-3.5)×10 12 cm -2 , calculated as n D = (πL 2 D ) -1 , assuming a random defect distribution. Device J339 hosts a typically large number of defects, whereas, by comparison, device J12B appears to be less defective, being below the detection limit of this approach, with R < 0.005, L D > 10 nm and n D ≤ 3 × 10 11 cm -2 . Those values will be compared to the density of defects obtained in transport measurements in the next Chapter.

Probe Station

Just after the fabrication, the devices were systematically characterized electrically with a probe station. The transport property has been characterized by connecting the sourcedrain electrodes to a SMU, which supplies the source-drain voltage (V sd ) and measures the source-drain current (I sd ). A second SMU applies a back-gate voltage (V g ) and measures the leakage current (I g ).

Two tests were carried out: 2-probe current-voltage measurements (IV) and transfer measurements. With IV measurements, V g is fixed while V sd is varied, measuring at the same time the source-drain current. In the case of transfer measurements, V sd is fixed while V g is varied, measuring at the same time the source-drain current. Fig. 2.34 shows some typical characteristic behaviors, obtained with device J114. The device has a high resistance, in the order of GΩs, non-ohmic contacts, and a large (∼ 10 V) back-gate hysteresis.

The flakes that were not shaped into Hall bars had lower electrical resistances, probably because of lower contact resistance, as can be seen in Fig. 2. [START_REF] Saito | Superconductivity protected by spin-valley locking in ion-gated MoS 2[END_REF], where the results obtained with the J16A device are shown. This motivated us to change the design of the contacts, and to maximize the size of the interface between MoS 2 and metal. The transfer curves shows a ∼ 10 V hysteresis for all pair of contacts.

All fabricated devices showed non-linear IVs, because of non-ohmic/Schottky contacts (discussed on Chapter 4), and big (10-20 V) back-gate hysteresis under ambient conditions.

Those topics will be discussed in the following Sections.

Current annealing

Passing a high current density through the samples have been shown to be an effective way for improving device's quality [START_REF] Islam | Quality enhancement of low temperature metal organic chemical vapor deposited MoS 2 : an experimental and computational investigation[END_REF]. In the lattice, the electron scattering is mostly elastic in nature, however, the scattering is high at the defects where the electrons lose their momentum, developing atomic scale force known as the "electron wind force" [START_REF] Conrad | Electroplasticity-the effect of electricity on the mechanical properties of metals[END_REF].

The electron wind force could make migrate the vacancies, enhancing the sample's mobil-

V g = 30 V → -30 V → 30 V.
ity. Additionally, a high current density could locally increase the temperature, desorb the molecules attached to the sample's surface, and reduce the amount of those defects.

A moderate current density of 3.4 A/m was passed through a few devices, all of them giving similar results. Below we present the data for a single device, namely J113, as shown in Fig. 2.36a. The IVs measurements were performed in the following cycle: V sd = 0 V → 16 V → -10 V → 0 V, and has been performed under primary vacuum at T = 130 K.

The sample shows instability when V sd > 5 V, but becomes stable as the annealing takes effect. This is evident at negative V sd , once the current annealing at positive V sd takes effect, as can be seen in Fig. 2.36b. Passing a higher current damaged the contacts, as shown in Fig. 2.36c. Therefore, this approach for quality enhancement was abandoned.

Ionic liquid gating

One of the first ionic liquid (IL) drops was placed on top of device J124. As the droplet was quite big, it covered and disabled other devices around, as can be seen in Fig. 2.37a.

After placing the IL drop, the sample was electrically tested at room temperature, performing IVs and transfer curves with the back-gate before performing them with the IL gate. The results at room temperature are shown in Fig. 2.37b and Fig. 2.37c for IV and transfer curves respectively. Although the temperature was not stable during the measurements, the behaviour was usual, with non-linear IVs, and a bit of gate hysteresis. The cycle of the transfer measurements with the back-gate is the following:

V bg = 0 V → 30 V → -30 V → 0 V.
The contacts used for the measurements are shown in Fig. 2.38c and are labeled as 'S' and 'D'.

After knowing that the back-gate was working, the IL gate (ILG) was tested, but nothing could be measured, as shown in Fig. 2.38. It turned out that the sample was destroyed during the previous measurements because of the high V sd which was also applied to the IL, resulting in a degradation of the device, as shown in Fig. 2.38c.

Another device, J126, was tested with the IL trying to overcome some of the mistakes made on the previous device. A small "spoon" was made by using a piece of 25 µm diameter gold wire, and twisting the end of it forming a loop. With this handmade tool it was possible to take a small drop of IL and to place it on top of the desired device. In Fig. 2.39a is shown a picture of J126 with the IL on top.

As with the previous device, the back-gate was tested prior to the ILG, and a high

V sd was applied. The IVs and transfer measurement results are shown in Fig. 2.39b and Fig. 2.39c respectively. This time, the operating temperature was kept stable at T = 275 K.

The cycle of the transfer measurements with the back-gate is the following:

V bg = 0 V → 80 V → -80 V → 0 V.
Luckily, after the back-gate test, the device was still operating and we could effectively control the electron density with the ILG, as show in Fig. 2.40a.

Unfortunately, the device stopped working after the transfer measurement, and the IV curves made afterwards were meaningless, as can be seen in Fig. 2.40b. As with J124 (Fig. 2.38c), J126 (Fig. 2.40c) also appears to be dissolved in the IL.

Initially, it was thought that a too high ILG voltage rate and too high V sd were the causes of this dissolution. We reduced the ILG voltage rate and reduced the maximal V sd on subsequent devices but still the MoS 2 seemed to be dissolved by IL.

Following the work of Lezama et al. [START_REF] Gutiérrez-Lezama | Ionic gate spectroscopy of 2D semiconductors[END_REF], the ILG was trained by sweeping it slowly over a small range of voltages, increasing this range gradually. A few devices were tested with this approach and all of them showed similar results. We present the data for one of them, called J6AF, as shown in Fig. 2.41. Unfortunately, we could not perform 4-probe measurements on this device because the metallization of contacts at the end of the fabrication process was not uniform, as can be seen on Fig. 2.41b, making difficult the bonding of the contacts pads. The next discussion will then be on the results obtained with 2-probe measurements only. The notation for back-gate voltage and ionic liquid gate voltage are, respectively, V bg and V tg . During operation, the ILG leakage needs to be kept negligibly small by sweeping the ILG voltage slowly (the leakage increases with the sweeping rate). A rapid increase of leakage upon increasing ILG voltage may be indicative of chemical reactions that can cause device degradation and should, therefore, be avoided. Environment conditions like temperature and pressure also play a role on the ILG performance, with the ILG leakage increasing with temperature.

In Fig. 2.42 is shown the "ILG training" at room temperature and high vacuum (∼ 10 -6 mBarr):

transfer measurements with the ILG, increasing voltage range gradually. A small V sd voltage of 10 mV was applied in order to know if the training was taking place. A color bar has been included to indicate the time during the measurements, since, as mentioned before, the sweep is performed over a small range of voltages which is increased over time. The ILG sweep range starts with an amplitude of 0.5 V and is increased in steps of 0.2 V after completing a full cycle. For instance, the ILG cycle at the beginning of the training is the following: V tg = 0 V → 0.5 V → -0.5 V → 0.7 V → -0.7 V → 0.9 V..., and so on. The ILG goes from positive to negative and then back to positive with the added voltage step.

As shown in Fig. 2.42a, the IL effectively gates the sample. The ILG voltage at which a noticeable current is measured is lower as the training takes place, starting from V tg ∼ 1 V at the beginning of the training and decreasing down to V tg ∼ 0.5 V at the end. Also, the slope of the I sd -V tg curves becomes steeper as the training takes place, meaning the ILG efficiency is increased with the training.

A maximum in the current is reached around V tg ∼ 2 V, attributed to the Q valley which also starts to be filled once the Fermi level is high enough [START_REF] Piatti | Multi-Valley Superconductivity in Ion-Gated MoS 2 Layers[END_REF][START_REF] Vaquero | Ionic-Liquid Gating in Two-Dimensional TMDs: The Operation Principles and Spectroscopic Capabilities[END_REF]. As discussed in Section 1.1.2, monolayer MoS 2 has a direct gap at the K-points, but there is a local minimum at the Q-point with an indirect gap. The electron density at the Q-valley is estimated to be 1 × 10 13 cm -2 [START_REF] Piatti | Multi-Valley Superconductivity in Ion-Gated MoS 2 Layers[END_REF]. Fig. 2.42b shows the 2-probe conductance, defined as I sd /V sd and measured between contacts 5-B. A wide region of low conductance (G < 10 -7 S, between V tg : -2.5 V --0.5 V) is observed, and the hole conduction band is not reached, indicating that the efficiency of the ILG was not good enough, or that a extremely large impurity band is present on this device. In Fig. 2.42c is shown the ILG leakage, which increases with the sweeping range and shows peaks in both, positive and negative V tg , which may be signs of occurring irreversible chemical reactions [START_REF] Ulissi | Low-Polarization Lithium-Oxygen Battery Using [DEME][TFSI] Ionic Liquid Electrolyte[END_REF].

After the ILG training, we stopped the ILG voltage at 2.5 V, close to the peak in conductance, and measured the current over time. In Fig. 2.42d, we can see that the conductance decreases over time, which we attribute to a decrease in electron density, possibly due to the ions in the IL still rearranging themselves.

Then, an ILG voltage of V tg = 3 V (past the peak on conductance) was set and the device was cooled down to liquid He temperature and electrically tested. Transfer and IV measurements were performed, this time on a different pair of contacts: 3-8, as shown in surpassed by the noise.

After the previous IVs and transfer measurements, the device was warmed up to room temperature in order to apply a different V tg , training the IL before that, and cooled down to He liquid temperature to be electrically tested again. This process was repeated a couple of times, resulting in a higher resistance of the device each time, regardless of the V tg applied. A total of 3 cooling downs at different ILG voltages has been performed on this device; the first a V tg = 3 V, the second a V tg = 6 V and the third a V tg = 2 V. The extracted conductance during each cooling down is shown in Fig. 2.44a, in a Efros-Shklovskii plot. In some regions of temperatures, the data follows very well the Efros-Shklovskii VRH transport mechanism, as seen by the linearity of the data in that plot, although, as mentioned before, we can not be conclusive with 2-probe data. This transport mechanism will be studied in more detail in the next Chapter.

During the first cooling down (blue curve), the conductance was significantly higher than the other two since the device degraded with each cycle of cooling down, warming up and ILG training. After the third cool down, the device stopped working and no I sd could be measured at low temperature. It turned out the device got damaged because of unwanted chemical reactions caused by the IL, as shown in Fig. 2.44b and Fig. 2.44c. We attribute this chemical reaction to the temperature at which the IL's training was performed (room temperature). A low temperature of slightly above 220 K (the freezing point of the IL), is recommended for operating the ILG in order to avoid chemical reactions.

Ionic liquid gating is a promising approach if high electron densities are desired. However, controlling it requires technique and very specific conditions. We could reach the Q-point, which is estimated to happen at a density of 1 × 10 13 cm -2 , but with consequent cycles of warming ups and cooling downs the device got damaged. The LED gating showed not reproducibility, making it hard to be used as a tool for efficient doping control. As a consequence, this way of controlling doping was discontinued.

Thermal annealing

Vacuum thermal annealing is commonly used to improve the contact quality and achieve enhanced device performance [START_REF] Yang | Multilayer MoS 2 transistors enabled by a facile dry-transfer technique and thermal annealing[END_REF]. In our case, we performed annealing with the Janis cryostat shown in Fig. 2.29 under primary vacuum, without further exposure to air. This method was the only one that allowed us to obtain reproducible results and several complete data sets as a function of temperature. A detailed analysis of these data will be presented in the next Chapters. Below we simply present this method, and the first results obtained.

J11B was the first device measured at high temperature. For this device, the measurements were performed in order of increasing temperature from 300 K up to 600 K, with a noticeable change in the device performance above T = 500 K. The annealing times and temperatures are: 87.6 min at 500 K, 37.5 min at 550 K and 37.5 min at 600 K. In Fig. 2.47 are compared the IV and transfer data obtained at T = 300 K before and after annealing.

It is clear from the results that the total resistance decreases, as seen by the higher current at the same V sd . Also, the doping increases, as seen by the change in threshold voltage V T h on the transfer curves. Following [START_REF] Ortiz-Conde | Revisiting MOSFET threshold voltage extraction methods[END_REF], a linear fit on the transfer curve was performed in order to determine V T h , as shown by the dashed line in the insert of Fig. 2.47c. This gives a value of V T h ∼ 40 V before annealing, but an unknown value for after annealing. The annealing changed the V T h to negative values on the gate voltage, meaning that the device got n doped and a negative gate voltage is needed in order to deplete it. After the thermal annealing, the contact resistance is negligible compared to the MoS 2 resistance, as can be seen from the 2-probe and 4-probe results at T = 500 K, shown in Similar results were obtained for several other devices. In Fig. 2.50 are shown the annealing results for devices J31A and J339. The estimated change of V T h with annealing is ∼ 120 V, or equivalently, an electron density increase of around n = 8 × 10 12 cm -2 , estimated using the capacitive model:

n = C g (V g -V T h )/e
, where e is the electron charge and C g = 1.04 × 10 -8 F/cm -2 is the geometric capacitance for 300 nm thickness of SiO 2 .

This increase in doping with annealing has been already documented [START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF][START_REF] Britton | Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS 2[END_REF][START_REF] Yang | Multilayer MoS 2 transistors enabled by a facile dry-transfer technique and thermal annealing[END_REF], and a possible explanation could be that the annealing is desorbing molecules from the MoS 2 surface, exposing the (sulphur) vacancies which in turn acts as electron donors [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Kaasbjerg | Electron and hole transport in disordered monolayer MoS 2 : Atomic vacancy induced short-range and Coulomb disorder scattering[END_REF]. The fact that there is a noticeable difference in doping when the sample is on ambient air or in vacuum, as shown by the black and blue curves in Fig. 2.50, supports this explanation of desorbing molecules from the surface [START_REF] Kwon | Electrical characteristics of multilayer MoS 2 transistors at real operating temperatures with different ambient conditions[END_REF]. The insert shows the device with the contacts used marked. The scale bar for both devices is 25 µm.

Different annealing temperatures and consecutive annealings have also been studied. A temperature of T = 500 K is not sufficient (but induces already noticeable changes) while consecutive annealings at T = 600 K negatively affect the device performance. This study was performed in several devices, but here we show the results for a single one, named J348. Transfer measurements, shown in Fig. 2.51, were performed at T = 400 K after annealing the device at different temperatures: at T = 520 K for 30 min (blue curve), at T = 600 K for 30 min (orange curve), and finally, at T = 600 K for 20 min (green curve), in that order. The device's conductivity changes by almost 1 order of magnitude after being annealed at T = 600 K compared to the effect of annealing at T = 520 K, indicating that temperatures close to 500 K are not sufficient for a proper thermal annealing. However, consecutive annealing at high temperature are detrimental for the devices. After a second annealing at T = 600 K, the conductivity dropped by almost 1 order of magnitude (green curve) compered to the first annealing at T = 600 K (orange curve). Of all devices that were thermally annealed, all of them had varying degrees of doping change, with some of them breaking during the annealing. The magnitude of the impact that the annealing will have on the device, such as the change in doping and the device's resistance to the annealing (meaning the device doesn't brake during the annealing) can not be known a priori, since it is strongly device-dependent.

Hysteresis

Hysteresis is a bistability in the operational transistor current. It appears as a difference in the source-drain current values observed during forward and backward sweeps of the gate voltage. Hysteresis is useful in nonvolatile memory devices but it is generally avoided in standard integrated circuits [START_REF] He | Thermally Assisted Nonvolatile Memory in Monolayer MoS 2 Transistors[END_REF][START_REF] Kaushik | Reversible hysteresis inversion in MoS 2 field effect transistors[END_REF][START_REF] Shu | The intrinsic origin of hysteresis in MoS 2 field effect transistors[END_REF]. The hysteresis behavior is schematically depicted in All of our devices showed anti-clockwise hysteresis at temperatures above T = 300 K.

Figure 2.53 shows the characteristic transfer data of four devices. Each device was annealed at T = 600 K for about 3.8 min, 10.6 min, 20 min and 15 min for devices J339, J338, J12Bbot and J12B-top respectively. After the thermal annealing, the transfer measurements were performed starting at T = 500 K down to T = 300 K, where the hysteresis starts to be negligible for almost all devices. The sweeping of the gate voltage V g is from 0 V up to the maximum V g , then to the minimum V g an finally back to 0 V. For instance, for device J339, shown in Fig. 2.53e, the transfer cycle is the following: 0 V → 80 V → -20 V → 0 V. As we can see from the results, the magnitude and behavior of the hysteresis is sample dependent:

for devices from the sample Jjedi03, the hysteresis decreases with decreasing temperature, as shown in Fig. 2.53e and 2.53f, while for devices from sample Jjedi01 the opposite occurs, but only above T = 300 K, as shown in Fig. 2.53g and 2.53h. This behavior can be explained by considering V T h higher than the V g range used, with the later increased as the temperature was lowered.

All devices showed counterclockwise BSC hysteresis, possibly due to oxide traps close to MoS 2 -SiO 2 interface [START_REF] Kaushik | Reversible hysteresis inversion in MoS 2 field effect transistors[END_REF][START_REF] Egginger | Current versus gate voltage hysteresis in organic field effect transistors[END_REF]. Some other authors have reported a transition from clockwise to anti-clockwise as temperature increases [START_REF] Kaushik | Reversible hysteresis inversion in MoS 2 field effect transistors[END_REF]. However, the origin of the increase of hysteresis when decreasing the temperature, as observed on devices from sample Jjedi01, is unclear.

The typical hysteresis for device J339 at T = 400 K is ∆V T ∼ 10 -20 V, which allows us to estimate a lower value for J339's doping inhomogeneity of N T ≳ 10 12 cm -2 , using a simple capacitive model: N T > ∆N T ⩾ ∆V T * C ox /e. This value will be compared to the density of defects calculated in the next Chapter.

Conclusion

In this Chapter we have introduced the fabrication process of the different devices, the setup used to characterized them, and a few preliminary measurements that helped us to understand the electronic transport mechanism on MoS Chapter 3

Electronic transport through the MoS 2

This Chapter is devoted to the low-temperature electronic transport measurements performed on devices J339, J338 and J12B. The three devices are shown in Fig. 3.1, with a label on the contacts that were used in 2-probe and 4-probe measurements.

Those measurements were performed on the Janis He-free cryostat presented in Section were extremely resistive with resistances of the order of GΩ, and electron conduction merely accessible at high positive gate voltages. To reduce the electric resistance, the devices were annealed at 600 K in primary vacuum (∼ 10 -3 mbar) before the measurements. Devices J339, J338 and J12B were annealed during 4 min., 11 min. and 15 min. respectively. The measurements were performed starting at 500 K, measured at fixed T , and progressively cooling down to 20 K without any further exposure to ambient air. Devices J339 and J338 have been measured in primary vacuum down to 300 K, and at low pressure He gas afterwards; whereas device J12B has been measured in primary vacuum down to 108 K, and at low pressure He gas afterwards.

For each measurement, V sd and V g were sourced, while the current I sd , the 4-probe voltage V x = V 12 , and the gate leakage I g were recorded. Two set of measurements has been performed: sweeping V sd at fixed V g (IV measurements), and sweeping V g at fixed V sd (transfer measurements).

2-and 4-probe I-V

The in-situ thermal annealing considerably doped the devices with n-type carriers, shifting the threshold voltage V T by ∼ 120 V towards negative values for device J339, as already discussed in Section 2.4.6. This increase in doping with thermal annealing has been already documented [START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF][START_REF] Britton | Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS 2[END_REF][START_REF] Yang | Multilayer MoS 2 transistors enabled by a facile dry-transfer technique and thermal annealing[END_REF], and it is thought to be caused by the desorbtion of molecules in the vicinity of MoS 2 and, in particular, near sulfur vacancies which will then act as electron donors [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Kaasbjerg | Electron and hole transport in disordered monolayer MoS 2 : Atomic vacancy induced short-range and Coulomb disorder scattering[END_REF]. This results in a large increase of both carrier density and conductivity. The gate dependent conductivity curves exhibited hysteresis at temperatures above ∼ 350 K, as discussed in Section 2.4.7. At T ≤ 300 K, the amplitude of the hysteresis is negligible for the three devices. The measured I sd -V x curves are straight lines, which ensures that the properties of the devices respectively. V g = 100 V for J339 and J338 devices, while V g = 120 V for device J12B.

MoS 2 can be extracted from the 4-probe measurements. Thus, in the following, we use only the 4-probe data, and we restrict the analysis to the domain in which the gate hysteresis is negligible (T ≤ 300 K), to ensure that the resistivity of the MoS 2 is extracted correctly.

Conductivity: Metallic and insulating behavior

Given the linearity of the 4-probe data shown earlier, the conductivity is then confidently extracted as σ = I sd /V x at fixed V sd while sweeping V g , i.e., using the transfer data. conductivity is vanishingly small at low V g for devices J339 and J338, and steadily increases with V g . In the case of J339 (see Fig. 3.3a), for V g < 45 V, the conductivity decreases with T , indicating a clear insulating behavior. At the opposite, for V g > 45 V, the conductivity decreases with increasing temperatures, indicating a metallic behavior. The crossover between the two regimes can be associated to a crossing point between the σ(V g ) curves taken at low temperatures. In the temperature range T = 20 -120 K, the crossing point marking this transition is remarkably temperature-independent and is located at

V c = V g ≃ 45 V
, where V c is the gate voltage at the crossover. At higher temperatures, the crossing point shifts slowly from V g ≃ 45 V at T = 20 -120 K down to V g ≃ 0 V at T = 300 K.

For comparison, Fig. 3.3c shows the conductivity of device J338, the onset of the apparent crossover is visible at V g ≃ 120 V. As in device J339, σ c ≃ 1.4e 2 /h = 55 µS at the transition, and the apparent crossover shifts progressively to lower gate voltages when T increases. We

recall that e 2 /h ≃ 39 µS.

Device J12B, shown in Fig. 3.3e, was considerably more doped than J339 and J338 and only the metallic region could be observed, with σ in the range 100-700 µS. For this device, the measurements were performed under low vacuum down to T = 108 K. At lower T , the chamber was filled with He at low pressure. This had a negative effect on the device performance, as shown in Fig. 3.3f, where the asterisk ( * ) marks the temperature at which the chamber was filled. Below this point, the conductivity is much more noisy and with an apparent thermal-dependence suppression. This phenomena will be discussed on appendix B. For this device, the data for temperatures above 108 K will be considered for further analysis. This apparent two-dimensional metal-insulator transition (2D MIT) was first observed in MoS 2 monolayer by Radisavlevic and Kis [START_REF] Radisavljevic | Mobility engineering and a metal-insulator transition in monolayer MoS 2[END_REF], and later by Baugher et al. [START_REF] Britton | Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS 2[END_REF], Schmidt et al. [START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF], Yu et al. [START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF] and more recently by Moon et al. [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF]. The experimental ubiquity of this MIT, systematically observed with a critical conductivity around e 2 /h in various samples, has not been elucidated yet. It was suggested that this phenomena originates from percolation [START_REF] Chen | Electrically tunable physical properties of two-dimensional materials[END_REF], as already observed in two dimensional electron gases [START_REF] Sarma | Two-Dimensional Metal-Insulator Transition as a Percolation Transition in a High-Mobility Electron System[END_REF][START_REF] Sarma | Two-dimensional metal-insulator transition as a potential fluctuation driven semiclassical transport phenomenon[END_REF], or from a genuine 2D MIT induced by interaction [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Byoung | Quantum critical scaling for finite-temperature Mott-like metal-insulator crossover in few-layered MoS 2[END_REF][START_REF] Byoung | Metal-insulator transition in two-dimensional transition metal dichalcogenides[END_REF]. It should also be noted that the term "MIT" is often used ambiguously. In some papers, it is used to indicate the presence of a genuine phase transition. In other works [START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF][START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF], it is purely phenomenological, and refers to a transition between metallic and insulator as a function of doping. In our case, as a real phase transition can not be demonstrated, we refer to the MIT as "apparent MIT".

As introduced in Section 1.3, the scaling theory predicts that a non-interacting 2D electron gas (2DEG) is always an insulator, and cannot have a genuine 2D MIT. However, it could be induced by another phenomenon, for example the crossing between weak and strong localization [START_REF] Sarma | Two-dimensional metal-insulator transition as a strong localization induced crossover phenomenon[END_REF]. In the strong localization regime, the conductivity falls exponentially with temperature because the carriers become exponentially localized. In the weak localization regime, the metallic behaviour comes from the temperature-dependence of phonons and screened impurities scattering. The Ioffe-Regel criteria for the wave localization in disordered media could be used to study such a crossover. It stipulates that an Anderson phase crossover from weak to strong localization [START_REF] Sarma | Two-dimensional metal-insulator transition as a strong localization induced crossover phenomenon[END_REF] occurs in a 2DEG when k F l e ≃ 1. Here,

k F = √ 2πn 2D
is the wavevector at the Fermi energy, n 2D is the electron concentration, l e = ℏk F σ/n 2D e 2 is the mean free path of the electrons and e is the electron charge. In two dimensions, neglecting screening, the Ioffe-Regel criteria gives a conductivity σ c at the crossover independent of the charge concentration:

σ c,IR = g s g v 2 e 2 h , (3.1)
where g s and g v are the spin and valley degeneracies respectively. For MoS 2 , g s = 2 and

g v = 2, thus σ c,IR = 2e 2 /h ≃ 80 µS.
As the system is confined in two dimensions, Coulomb interactions between electrons become important. To tell if our system have important electron-electron interactions we look at the parameter r s , which is the ratio between potential (Coulomb, E C ) and kinetic (Fermi, E F ) energies:

r s = E C E F = g v a * B √ πn 2D = m * e 2 g v /4πϵ r ϵ 0 ℏ 2 √ πn 2D , (3.2)
where a * B = (4πϵ r ϵ 0 )/(m * e 2 ) is the effective Bohr radius, m * = 0.45 is the effective electron mass for monolayer MoS 2 [START_REF] Huynh | Tuning the Electronic Properties, Effective Mass and Carrier Mobility of MoS 2 Monolayer by Strain Engineering: First-Principle Calculations[END_REF][START_REF] Shi | Quasiparticle band structures and optical properties of strained monolayer MoS 2 and WS 2[END_REF], and ϵ r is the relative dielectric constant of the system, which we evaluate by the Keldysh formula [START_REF] Keldysh | Coulomb interaction in thin semiconductor and semimetal films[END_REF]:

ϵ r = (ϵ r (SiO 2 ) + ϵ r (air))/2 + dq(ϵ r (MoS 2 ) -1)/2, (3.3)
where d is the MoS 2 thickness, and q is the in-plane reciprocal lattice vector of the screening particles. In the insulator regime, all electrons are localized and the last term can be safely neglected, thus, ϵ r = (3.9 + 1)/2 ≃ 2.5.

The carrier density at any V g can be obtained using the capacitive model:

n 2D = C g (V g - V T h )/e
, where e is the electron charge, C g = 1.04×10 -8 F/cm -2 is the geometric capacitance, and V T h is the threshold voltage, above which conduction starts to take place through the conduction band. Following [START_REF] Ortiz-Conde | Revisiting MOSFET threshold voltage extraction methods[END_REF], V T h is extracted by finding the intercept with the V g axis of the linear extrapolation of the σ -V g curves at high V g , as illustrated in Fig. 3.4a.

At T = 20 K we get V T h ≃ 18 V and V T h ≃ 87 V for device J339 and J338 respectively.

For J338, V T h is probably underestimated because the conductivity curves are not perfectly linear, even at the highest V g . It is worth mentioning that V T h should be extracted at the lowest temperature possible in order to precisely locate the bottom of the conduction band.

When T is increased, in-gap states contribute to the conductance through hopping and thermal activation, and artificially decrease the threshold voltage, see Fig. 3.4b. The change of V T h occurs because of trapping/detrapping electrons from the impurities or disorder [START_REF] Giannazzo | Impact of contact resistance on the electrical properties of MoS 2 transistors at practical operating temperatures[END_REF].

Disorder states are occupied or unoccupied depending if the FL is above or below their energy.

Decreasing temperature traps electrons since they no longer have the thermal energy to be activated and participate in the conduction. Hence, electron trapping results in a positive shift of V T h with decreasing temperature.

As the crossover voltage takes place at V c -V T h ≃ 27 V for devices J339 and J338, the crossover charge density is n c ≃ 27 × C g /e ≃ 1.7 × 10 12 cm -2 for both devices. Using Eq. 3.2, with the obtained n c , ones get r s ≃ 20. Even if we use the value ϵ r = 7.3 calculated for monolayer MoS 2 [START_REF] Salmani-Jelodar | Single layer MoS 2 band structure and transport[END_REF], we still get r s ≃ 8. In both cases, such a high interaction parameter (r s ≫ 1) points toward systems where electron-electron interactions cannot be neglected to describe transport mechanisms [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Radisavljevic | Mobility engineering and a metal-insulator transition in monolayer MoS 2[END_REF]. Thus, the MIT might be a real transition coming from e -e interactions. With this value of n c , we can estimate the mean free path at the crossover:

l c = ℏ √ 2π/
√ n c e 2 ≃ 3 nm. This value is close to the average defect distance in device J339 deduced from the Raman analysis presented in Section 2.4.1. It is also similar to the substrate-induced doping inhomogeneity estimated from the width of the hysteresis at high temperature, see Section 2.4.7.

Before studying the transition itself, we will look into more details at the models describing these insulating and metallic regimes.

Transport mechanisms in the insulating regime

In the insulating regime, the carrier transport on MoS 2 can take place via thermal activation at relatively high temperatures and via variable-range hopping (VRH) at low but finite temperatures [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Xue | Mott variable-range hopping transport in a MoS 2 nanoflake[END_REF]. Whichever the transport mechanism, it can be modelled by Eq. 3.4

σ = C p exp - T p T 1 p , (3.4)
where C p is a constant of proportionality and T p is the characteristic temperature. The exponent p depends on the transport regime: p = 1 when the transport is dominated by thermal activation, p = d + 1 for Mott VRH (d is the dimension of the system, d = 2 in our case) and p = 2 for Efros-Shklovskii (ES) VRH. Those transport mechanism were presented in Section 1.3, and will be discussed in the following.

Thermally activated transport

The Arrhenius plot (p = 1) of conductivity at different gate voltages is shown in Fig. 3.5

for devices J339 and J338. The fit is performed on the temperature range 220-150 K and 300-220 K for J339 and J338 respectively. Unfortunately, for both samples, we could fit only three temperature points. The higher the gate voltage, the closer is the Assuming that E a is the energy difference between E F and E c , we get the relation (valid at T = 0 but commonly used at T ̸ = 0) [START_REF] Xue | Mott variable-range hopping transport in a MoS 2 nanoflake[END_REF][START_REF] Luryi | Quantum capacitance devices[END_REF]:

- dE a dV g = dE F dV g = eC g C g + C q , (3.5)
where C q = e 2 DoS(E F ) is the quantum capacitance and DoS(E) is the density of states at the energy E. From the Arrhenius fit, one gets E a (V g ), which in turns allows us to reconstruct the DoS with the help of Eq. 3.5. In Fig. 3.6 is shown the extracted E a from the Arrhenius fit, and the computed DoS from Eq. 3.5. The activation energy depends exponentially on V g for negative values of V g (see Fig. 3.6a), and E a → 0 when V g → V T h , reinforcing the idea that this point indeed correspond to E c . We assume that E a = E c -E F , and that the increasing of gate voltage moves E F closer to E c , eventually vanishing E a .

We find that DoS(V g = V T h ) ∼ 3 × 10 14 eV -1 cm -2 (shown by the dashed lines in Fig.

3.6b

), which is the expected DoS value for pristine or ideal MoS 2 with a parabolic band and an effective mass m * = 0.45m 0 . This result further confirms that, above T = 100 K, the temperature dependence of the conductivity indeed corresponds to the thermal activation from localized states to the bottom of the conduction band.

Our range of gate voltages for device J339 was not enough to look deep below the bottom of conduction band E c , reaching a maximum of 30 meV below E c . However, the situation is different for device J338, where E a goes up to 240 meV. We notice that in the range -80 --60 meV the DoS is not properly computed because of a small bump on the activation energy for J338 (the DoS extraction involves numerical differentiation, a bump introduces diverging points). We also notice that the DoS does not fall quickly but instead seems to reach a plateau.

One of the main results of this analysis is that below E c , the DoS decreases exponentially on an energy scale s ∼ 25 meV:

DoS(E -E c ) =DoS(E = E c ) × exp[(E -E c )/s]+ DoS 0 ,
where DoS 0 ≃ 10 13 eV -1 cm -2 is an additional constant contribution. This exponential decay, together with the constant term, are indicated in Fig. 3.6b by a red solid line. It is consistent with spatial doping fluctuations encountered in 2D materials on SiO 2 [START_REF] Ghatak | Nature of Electronic States in Atomically Thin MoS 2 Field-Effect Transistors[END_REF]. This residual DoS 0 is in the same order of magnitude than the density of defects found by Raman analysis (see Section 2.4.1). Moreover, in Section 2.4.7, we attributed the hysteresis observed at T > 350 K to traps in SiO 2 . The energy scale s ∼ 25 meV is roughly quantitatively consistent with this explanation. Indeed, DoS(V g = V T h ) × s ≃ 7.5 × 10 12 cm -2 , is in good agreement with the residual DoS 0 found.

Variable Range Hopping

At lower temperature, there is a departure from the thermal activation model, as seen in Fig. 3.5, where the data can no longer be fitted by straight lines. Usually, in disordered semiconductors, σ(T ) can be very well modeled in terms of variable range hopping (VRH), which have two expressions: p = 2 or p = 3 in Eq. 3.4 [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Byoung | Quantum critical scaling for finite-temperature Mott-like metal-insulator crossover in few-layered MoS 2[END_REF][START_REF] Xue | Mott variable-range hopping transport in a MoS 2 nanoflake[END_REF]. As detailed in Section 1.3.6, the value of p depends on the exact form of the VRH mechanism. When electronelectron interactions can be neglected, the exponent is given by the Mott's law and p = 3 in two dimensions. However, when screening is poor, Coulomb interaction must be included.

This corresponds to the Efros-Schklovskii (ES) VRH regime and p = 2 in any dimensions.

Previous studies in semiconductors unveiled two possible distinct behaviors for the prefactor C p : a constant prefactor C p = e 2 /h, and a temperature-dependent prefactor C p = AT -m .

For monolayer graphene, a temperature-dependent prefactor has been reported [START_REF] Bennaceur | Unveiling quantum Hall transport by Efros-Shklovskii to Mott variable-range hopping transition in graphene[END_REF]. For GaAs/AlGaAs heterostructures, both cases have been reported: a temperature dependent factor, with m = 1 in the quantum Hall regime [START_REF] Furlan | Electronic transport and the localization length in the quantum Hall effect[END_REF] or m = 0.8 with or without magnetic field [START_REF] Van Keuls | Screening of the Coulomb interaction in two-dimensional variable-range hopping[END_REF], and a constant prefactor with the value e 2 /h [START_REF] Shlimak | Hopping conductivity in gated δdoped GaAs: universality of prefactor[END_REF][START_REF] Khondaker | Two-dimensional hopping conductivity in a δ-doped GaAs/Al x Ga 1-x As heterostructure[END_REF]. This remarkable universal prefactor e 2 /h has also been observed in Si metal-oxide semiconductor field-effect transistors [START_REF] Mason | Experimental evidence for a Coulomb gap in two dimensions[END_REF], and in insulating Si:B [START_REF] Sarachik | Scaling of the conductivity of insulating Si:B: A temperatureindependent hopping prefactor[END_REF]. In these experiments, the temperature invariance of the prefactor implied that the hopping in these systems is not mediated by phonons, as originally assured in the VRH model [START_REF] Efros | Electron-Electron Interactions in Disordered Systems[END_REF][START_REF] Nevill | Metal-Insulator Transitions[END_REF]. The possibility of variable-range hopping involving electron-electron interactions, instead of the usual electron-phonon mechanism, has been considered theoretically by a number of authors [START_REF] Fleishman | Elementary Excitations in the Fermi Glass[END_REF][START_REF] Aleiner | 22nd International Conference on the Physics of Semiconductors[END_REF][START_REF] Berkovits | Statistics of energy spectra of a strongly disordered system of interacting electrons[END_REF][START_REF] Kozub | Fluctuation-stimulated variable-range hopping[END_REF].

Interestingly, the importance of the prefactor C p and its temperature dependence have not been studied in monolayer MoS 2 . In most experimental studies of MoS 2 thin films, a T -independent prefactor has been used [START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF][START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Papadopoulos | Efros-Shklovskii variable range hopping and nonlinear transport in 1T/1T ′ -MoS 2[END_REF][START_REF] Xue | Mott variable-range hopping transport in a MoS 2 nanoflake[END_REF], while in some others,

C p ∝ T -0.8 [2, 3,
154] has been used to fit the conductivity, without further justification.

Furthermore, Mott [START_REF] Ghatak | Nature of Electronic States in Atomically Thin MoS 2 Field-Effect Transistors[END_REF][START_REF] Jariwala | Band-like transport in high mobility unencapsulated single-layer MoS 2 transistors[END_REF][START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF] and ES [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Papadopoulos | Efros-Shklovskii variable range hopping and nonlinear transport in 1T/1T ′ -MoS 2[END_REF] expressions for VRH are both commonly used in monolayer MoS 2 , without a clear distinction between them. Xue et al. [START_REF] Xue | Mott variable-range hopping transport in a MoS 2 nanoflake[END_REF] was able to discriminate between the two VRH expressions by looking at the change of magnetoresistance with magnetic field. In some other materials, a transition from ES to Mott has been reported.

For AlGaAs/GaAs heterostructures, this transition can be magnetic field dependent [START_REF] Van Keuls | Screening of the Coulomb interaction in two-dimensional variable-range hopping[END_REF],

or doping and temperature-dependent [START_REF] Shlimak | Hopping conductivity in gated δdoped GaAs: universality of prefactor[END_REF]. For graphene in the quantum hall regime, the transition occurs depending on the filling factor [START_REF] Bennaceur | Unveiling quantum Hall transport by Efros-Shklovskii to Mott variable-range hopping transition in graphene[END_REF]. This motivates us to have a closer look into the different VRH mechanisms.

To identify which VRH mechanism is at play, we introduce the parameter

W = T -1 ∂ ln ρ(T )/∂T -1 .
If Eq. 3.4 is satisfied, and if the temperature dependence of C p is weak or non-existent, one finds:

ln W = - 1 p ln T + const., (3.6)
and the slope of W on a double logarithmic scale (ln(W ) vs. ln(T )) gives directly the exponent 1/p of Eq. 3.4. Fig. 3.7 shows the plot of W vs T in a log-log scale at different gate voltages, and three dashed lines show the slope for the three different transport regimes.

For device J339, shown in Fig. 3.7a, the value of p is very close to 2 for gate voltages less than -20 V and temperatures less than 100 K. By contrast, p = 3 would give a too moderate slope, and the steeper slope for p = 1 indeed starts to appear at T > 100 K. In the case of device J338, shown in Fig. 3.7b, it seems not to follow any regime to the letter, with p between 2 and 1. A fit on the W vs T plots has been performed in order to extract p as a function of gate voltage, the results are shown in Fig. 3.8. As mentioned before, the value of p for device J339 is close to 2, while p ∼ 1.3 in the case of device J338.

As the index p = 2 has been determined for J339 device, we can plot the logarithm of the conductivity in units of e 2 /h as a function of T -1/2 , as shown in Fig. Nonetheless, from the fits we extracted the temperature coefficient T 2 for both devices. The universality of the prefactor C p is better seen in Fig. 3.10, where the conductivity in units of e 2 /h is plotted as a function of (T /T 2 ) -1/2 . For device J339, all the data collapse onto the same straight line. The extrapolation to T -1 = 0 of the the linear fit confirms that the value of the prefactor is very close to e 2 /h. The small residual discrepancy can be easily explained, e.g. by a slightly incorrect geometric factor w/l when computing the conductivity with the inverse of Eq. 2.1. However, for device J338, the data is highly scattered, and the extrapolation gives a value of C p = 3.2 ± 1.9 e 2 /h.

For ES VRH, the localization length is related to the characteristic temperature T 2 by:

ξ = C ES e 2 4πϵ r ϵ 0 T 2 , (3.7)
where the prefactor C ES ≃ 6.2 in two dimensions [START_REF] Nguen | Two-Dimensional Hopping Conduction in a Magnetic Field[END_REF] (see Section 1.3.6). The fitting results from the ES (Fig. 3.9) and Mott (Fig. 3.12) models show that just the quality of the fitting is not sufficient to discriminate which VRH transport is at play. VRH regime with a p value of 3/2. This new model fits very well the conductivity data of J338, and the corresponding characteristic temperature and localization length are extracted and discussed.

Effective mobility, effective doping and metallic regime

In this Section, a discussion on the mobility, doping and electronic transport in the conduction band is given. Since we could not perform Hall measurements to extract those parameters, we resort to numerical derivation of the conductivity.

FET & effective mobilities

In 2012, Kaasbjerg et al. [START_REF] Kaasbjerg | Phonon-limited mobility in n-type single-layer MoS 2 from first principles[END_REF] showed that, from first principles, the phonon-limited mo- attenuating the Coulomb scattering on charged impurities [START_REF] Jena | Enhancement of Carrier Mobility in Semiconductor Nanostructures by Dielectric Engineering[END_REF], and it has been successfully implemented in the literature [START_REF] Radisavljevic | Single-layer MoS 2 transistors[END_REF][START_REF] Radisavljevic | Mobility engineering and a metal-insulator transition in monolayer MoS 2[END_REF].

From the conductivity data presented in Fig. 3.3, we extracted the field-effect mobility (µ FET ) using the expression

µ FET = [∆σ/∆V g ] × [1/C],
where the linear fit [∆σ/∆V g ] is performed in the linear region of the σ vs V g curves at high gate voltage. Figure 3.15 shows the temperature dependence of the extracted FET mobility in a log-log scale, from where we observe a monotonous increase of the mobility as the temperature is decreased, with a saturation at low temperatures. The fitting on the conductivity curves for device J338 is not quite accurate since, as mentioned before, the transfer curves at low temperature are not linear, and because of this we might be underestimating the mobility. The extracted FET mobility, shown in Fig. 3.15, hardly shows linearity against temperature in a log-log plot. Nonetheless, a linear fit was performed on the high temperature region and a value for the damping factor was extracted for each device. We find large differences between devices, with J338 having the lowest value of damping factor (γ = 1.1) while J12B having the highest (γ = 2.4).

In order to take a closer look at the damping factor, we introduce the "effective mobility" (µ eff ) defined as µ eff = dσ/CdV g , and numerically computed over the entire range of gate voltages and temperatures. The gate-dependent µ eff at different temperatures is shown in gate voltage, indicating screening of charged impurities; and for high positive gate voltages, the transport is metallic and reveals an upper limit of mobility, becoming almost doping independent. This saturation of mobility with temperature as well as with charge carrier density is a signature of short-range scattering limiting the device performance [START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF][START_REF] Schmidt | Transport Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition[END_REF].

The mobility of device J338 (see Fig. The damping factor γ is then extracted from the effective mobility and is shown in Fig.

Capacitive & effective doping

With the conductivity data and the effective mobility defined in the previous Section, we now define the effective doping as n eff = σ/(eµ eff ). Since we could not perform Hall measurements to extract the doping, our only options were to compute the doping using the capacitive model (n C ) and the effective doping defined before.

In the extraction of n C , the temperature dependent threshold voltage, shown in Fig. 3.4b, has been used. In Fig. 3.17 is shown the similarities between n eff (solid lines) and n C (dashed lines) for device J339 in the metallic regime. As can be seen in the figure, the two models gives similar doping results, with noticeable deviation at low gate voltage, close to the MIT.

This deviation occurs because there is still a large part of VRH conduction at play, given the relatively high thermal broadening.

Metallic conduction

In the following, we focus on the metallic part of the conductivity data, at temperatures above 90 K and gate voltages above V c . There, VRH hopping is not the main electronic transport mechanism, and it is possible to show that the data are reasonably well fitted by the usual single-particle models, when phonon scattering and impurity scattering are considered in a single electron picture. More precisely, the theoretical conductivity is simulated by considering electron scattering with phonons and charged point defects with a full energyand momentum-dependent Boltzmann transport equation. Electron-phonon interactions are computed ab initio using density functional perturbation theory, and charge screening is included for both types of scattering. This work has been done in collaboration with Thibault Sohier at Laboratoire Charles Coulomb.

In Fig. 3.18 is shown the fit on conductivity and effective mobility. The experimental data is shown by thin solid lines with error bars, while the fit is shown by thick solid lines.

The conductivity for each device is obtained at four different doping values, with the error bars showing the conductivity between a 10% error on doping.

It appears from Fig. 3.18d that the combination of these two scatterings give a fair fit of the mobility versus carrier concentration for J339 and J12B devices, except at low doping n < 2 × 10 12 cm -2 . J338 is not well described by this model because it does not reach the metallic regime. These observations are consistent with the presence of a impurity DoS and thermal broadening which makes the actual band carrier concentration more poorly defined.

For a finer analysis, Fig. 3.18 shows the fitted experimental conductivities for the three devices. The individual adjustment of the defect density for each device is enough to obtain fair fits for the 3 devices, confirming the validity of this approach. The defect density is n D = 1.0×10 12 , 1.5×10 12 , 1.2×10 12 cm -2 for devices J339, J338 and J12B respectively. These values cannot be directly related to the defect densities observed by Raman spectroscopy, since the Raman spectra was obtained after the electrical measurements under ambient air and may have absorbed impurities. Nevertheless, one can note that the agreement is fairly good for device J339, while the discrepancy in J12B could be explained either by a less good approximation of doping, or the uncertainties in the position in energy of the local band minimum at the Q-point, which introduces newer scattering paths at dopings above

∼ 10 13 cm -2 .
This model of phonons and charged point defect scattering describe well the conductivity in the conduction band, at sufficiently high doping ant temperatures. However, the transition from an insulator regime where VRH transport dominate, to a metallic regime where phonons and impurity scattering transport dominates, is still an open issue as we will see in the next Section.

A genuine MIT?

From these previous analysis, it still appears reasonable to investigate the existence of a possible scaling function, i.e., a function that collapses all the conductivity curves where σ c is the conductivity at the limit x → x c , f is the scaling function, x is the tuning parameter which in our case is the gate voltage, t(T ) = T -1/zν is the scaling factor with z and ν being the dynamical and correlation length exponents respectively [START_REF] Matthew | Quantum phase transitions in disordered two-dimensional superconductors[END_REF]. The critical conductivity together with the scaling factor encode the nature of the MIT transition [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Wang | Superconductor-insulator transition in space charge doped one unit cell Bi 2.1 Sr 1.9 CaCu 2 O 8+x[END_REF][START_REF] Nihar | Metal to Insulator Quantum-Phase Transition in Few-Layered ReS 2[END_REF]. Furthermore, ξ is expected to diverge close to the transition as ξ ∼ |n 2D -n c | -ν . Fig. 3.19a shows the results of the scaling on conductivity for device J339. We were able to collapse all the conductivity curves for temperatures below 90 K, where the critical voltage V c and conductivity σ c are independent of the temperature, into two branches: the upper and lower branches containing the conductivity data above and below the MIT respectively.

The collapse has been performed using the parameter t = T -1/zν with zν = 4.16. Fig. 3.19b

shows the results of the scaling on ξ for the same device, giving a value of ν = 2.96. This leads to a value of the dynamical exponent of z = 1.4, which is is supposed to be equal to 1 in the case of strongly interacting electrons [START_REF] Matthew | Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition[END_REF]. Although the presented scaling theory has been developed for superconductors [START_REF] Matthew | Quantum phase transitions in disordered two-dimensional superconductors[END_REF], it has been applied in the literature on different sample materials, with some of then showing superconductor-insulator transitions [START_REF] Wang | Superconductor-insulator transition in space charge doped one unit cell Bi 2.1 Sr 1.9 CaCu 2 O 8+x[END_REF][START_REF] Aubin | Magnetic-field-induced quantum superconductor-insulator transition in Nb 0.15 Si 0.85[END_REF] and some others showing metal-insulator transitions [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF][START_REF] Nihar | Metal to Insulator Quantum-Phase Transition in Few-Layered ReS 2[END_REF][START_REF] Lin | Critical Behavior of a Strongly Disordered 2D Electron System: The Cases of Long-Range and Screened Coulomb Interactions[END_REF]. Different values of the critical exponent zν can be found, usually in the range 0.6 -4, with zν values being low the more symmetric are the two scaled conductivity branches. It has been suggested that low values of the critical exponent correspond to low disordered samples [START_REF] Byoung | Soft Coulomb gap and asymmetric scaling towards metalinsulator quantum criticality in multilayer MoS 2[END_REF].

While we can successfully apply the scaling theory suggesting a quantum phase transition, we can't rule out the possibility of this being a classical phonon-related phenomena.

Following Das Sarma 2D screening work [START_REF] Sarma | Screening and transport in 2D semiconductor systems at low temperatures[END_REF], three characteristic temperatures, namely

Fermi T F , Bloch-Grüneisen T BG and Dingle T D temperatures suffice to tell if phonons still plays a role in the observed MIT, or if the metallic temperature dependence in the conductivity arises purely from electronic effects. The Fermi temperature T F defines the intrinsic quantum temperature scale for the 2D electrons, and when T F is very large there cannot be any temperature dependence in the metallic conductivity at low temperatures arising from intrinsic electronic effects. The Bloch-Grüneisen temperature T BG defines the characteristic temperature scale for phonon scattering effects to become important in the 2D metallic conductivity. The Dingle temperature T D acts as a lower cutoff, suppressing the temperature dependence in conductivity for T < T D . This is because the strong temperature dependence of carrier screening is cutoff for T < T D by impurity disorder effects parametrized by the Dingle temperature. Thus, we end up with the condition shown in Eq. 3.12 for intrinsic 2D electronic effects, with T BG > T F guaranteeing that phonon scattering would not play a role in the 2D MIT physics, and T D < T guaranteeing that disorder is not restraining the temperature dependence of conductivity [START_REF] Sarma | Screening and transport in 2D semiconductor systems at low temperatures[END_REF]. Those characteristic temperatures are shown in Eqs. 3.9, 3.10 and 3.11.

k B T F = ℏ 2 k 2 F 2m * = ℏ 2 2m * 4πn g s g v (3.9) k B T BG = 2ℏk F v ph = 2ℏv ph 4πn g s g v 1/2 (3.10) k B T D = ℏ 2 e m * µ (3.11) T D < T < T F < T BG (3.12)
T BG depends on the phonon velocity v ph which, considering acoustic phonons for MoS 2 [START_REF] Kaasbjerg | Phonon-limited mobility in n-type single-layer MoS 2 from first principles[END_REF], is equal to v ph = 6.7×10 3 m/s. As T F and T BG depend on the electron density n, we use the n obtained by the capacitive model (n = C g (V g -V T h )/e) for any V g > V T h , where V T h = 18 V is the threshold voltage obtained at T = 20 K (explained in Section 3.2). In the case of T D , the mobility used is µ eff (T = 20 K), which depends on the electron density. We end up then with the three characteristic temperatures being doping dependent that, for simplicity, is translated to a gate voltage dependence. We then interpolate the conductivity data for temperatures between (20 K, 220 K), and find the value of the interpolated σ(V g , T s )

for any V g > V T h and T s , where T s is any of the characteristic temperatures. With this we were able to show, in the conductivity plot, the three characteristic temperatures, as shown in Fig. 3.20.

The Bloch-Grüneisen temperature (green dashed line) is on the order of the lowest experimental temperature at the critical point T BG (n c ) ≃ 24 K, meaning that phonon plays a role in almost the entire temperature region. The Dingle temperature (cyan dashed line) is also high, meaning that the temperature dependence of conductivity is being suppressed by disorder. The black solid line with dots is the conductivity σ c at the critical voltage

V c : σ c = σ(V g = V c
). For T ≤ 90 K, σ c is temperature-independent and equal to ∼ 1.4e 2 /h.

The condition shown in Eq. 3.12 is not met by device J339, therefore the possibility of the MIT being a classical phonon-related phenomena cannot be ruled out. The existence of a 2D MIT should be crosschecked with additional measurements in other materials [START_REF] Wang | Superconductor-insulator transition in space charge doped one unit cell Bi 2.1 Sr 1.9 CaCu 2 O 8+x[END_REF][START_REF] Nihar | Metal to Insulator Quantum-Phase Transition in Few-Layered ReS 2[END_REF], at temperatures lower than T BG , and as a function of the source-drain electric field [START_REF] Byoung | Quantum critical scaling for finite-temperature Mott-like metal-insulator crossover in few-layered MoS 2[END_REF]. The MIT can be observed intuitively by the combination of (i) the increase in mobility and (ii) positive V T h shift with decreasing the temperature, as already pointed out by Nan Fang and Kosuke Nagashio [START_REF] Fang | Band tail interface states and quantum capacitance in a monolayer molybdenum disulfide field-effect-transistor[END_REF]. The mobility increases with decreasing temperature due to suppression of phonon scattering, while the change of V T h occurs because of trapping/detrapping electrons from the impurities or disorder. Thus, while we can successfully apply the scaling theory suggesting a quantum phase transition, we can't rule out the possibility of this being a classical phonon-related phenomena.

Conclusion

We have performed IVs and transfer measurements on three different devices after thermally annealing them at T = 600 K. The increase of doping with the annealing pushed the Fermi level deeper into the conduction band, after which it was varied with the back gate, resulting in the observation of both, the metallic and insulator regimes. results obtained during the 2-probe test of contacts F and A. Some combinations of contacts included a floating contact in-between, and they are marked with an asterisk symbol ( * ) in the labels, i.e., the 2-probe measurements performed with contacts DA have an asterisk because between contacts D and A there is the contact F, which was floating when measuring DA contacts.

The measurements were performed at different back-gate voltages and a temperature of T = 350 K, after annealing the device at T = 600 K for 42 min. A linear fit is performed on the IV curves, for V sd values between -0.1 V< V sd <0.1 V, in order to extract the resistance of each pair of contacts. The extracted resistance is then plotted against the distance between the contacts, and the result is shown in Fig. 4.2c. In theory, this method should give a linear dependence R L = ρ × (L/w) + 2R c , and then the contact resistance is extracted from the y-intercept, extrapolating to zero distance. Our TLM devices showed strongly non-linear IVs, and large inhomogeneities on the contacts resistance. Thus, as the inhomogeneities dominated over the total resistance, it was impossible to extract a mean contact resistance value. A 4-probe measurement was also performed giving non-linear results, as shown in Fig. 4.2d, which could be a sign of VRH transport. In this case, four different contacts combination has been used for the 4-probe measurements. For instance, the black curve shows the results obtained using B and H as 2-probe contacts with G and C as the 4-probe contacts.

Non linear IVs

Non-linear 2-probe IVs were the usual results in all devices just after fabrication, with some devices showing non-linearity on the 4-probe measurements. This non-linear 4-probe behavior could be a sign of variable range hopping conduction, or an impedance mismatch issue with the voltmeter used. Here we present the results of using a high input impedance preamplifier (HIP). The device used is J11B, which was already presented in the thermal annealing Section 2.4.6, although the data shown here were taken before the annealing. J11B By contrast, the data obtained with the HIP are linear at high temperature, as shown in Fig. 4.4d, but lose their linearity at low temperature, as shown in Fig. 4.4c. However, the non-linearity on the 4-probe data could also be a sign of VRH transport.

The non-linearity of IV curves might come from an impedance mismatch or because VRH transport is at play. Nonetheless, having an input impedance higher than 10 GΩ is desirable when measuring MoS 2 transistors at low doping & temperatures.

Contact resistance

This Section is devoted to the analysis of the 2-probe electronic transport data of devices J339, J338 and J12B. The three devices were presented in the previous Chapter, where the 4-probe data has been studied. The 2-probe set of data presented here has been obtained at the same time as the 4-probe data. The three devices are shown again in Fig. 4.5, with a label on the contacts that were used in 2-probe and 4-probe measurements.

To extract the contact resistance (R c ), the devices were connected as shown in For device J339 the contact resistance could not be reliably extracted at low doping (V g < 0 V), where the contact resistance drops quickly to zero, which is absurd. This could be caused by an incorrect measurement of the length L, since the source and drain contacts have a saw-tooth shape, as can be seen in Fig. 4.5a. This saw-tooth shape also increases The contact resistance also varies with the temperature, increasing when T is lowered in devices J339 and J338 but having the opposite effect in device J12B. This behavior could be caused by the differences in the contacts between J1 and J3 devices: for J3 devices, the contacts are made of Ti/Au 2/150 nm; while for J1 devices, they are made of Ti/Au 15/75 nm. As a consequence, the contacts from J1 devices behave more like Ti contacts, while those from J3 sample behave more like a Au contacts. This phenomenon could also be related to electron density. Device J339 shows a R c that decreases with T , but in a small region at V g > 70 V and T < 120 K; while J12B shows a R c that increases when decreasing T , in the region V g < 0 V and T < 108 K. Thus, at low doping, R c increases when decreasing T , but the opposite occurs at high doping. This could be related to a transition from thermionic emission at low doping to "ohmic" contact or tunneling dominated transport at high doping, as we will discuss in the next Sections.

Electronic transport and Schottky barrier

Electric contacts are commonly of Schottky type on MoS 2 [13-15, 95, 96, 164]. At the metal-MoS 2 contact (where a barrier is formed) the charge injection occurs either (i) as a result of thermionic emission over the top of the barrier due to the transfer of thermal energy or (ii) as a result of quantum mechanical tunneling of carriers across the barrier being the exception where it decreases for V g < 0 V. Decreasing the temperature resulted in an increase on R c for devices J339 and J338 but with the opposite effect for device J12B.

width, as schematically shown in Fig. 4.8c. In the following Sections, those charge injections mechanism will be discussed.

Schottky barrier

All of our devices exhibited Schottky contacts, especially at low temperature and gate voltages, when the thermal energy is small, and the Fermi level is far away from the conduction band. Here, we present the data for a single device, called J35A, which showed a relationship between the area of MoS 2 below the contact and its resistance. shown the S and D contacts as two back-to-back connected Schottky diodes, but this configuration can be treated as a single Schottky diode [START_REF] Osvald | Back-to-back connected asymmetric Schottky diodes with series resistance as a single diode[END_REF], where the transport is dominated by the blocking diode at high V sd . As can be seen in Fig. 4.7a, there is significant difference of MoS 2 area below S and D contacts, and this has an effect on the contact resistance as we will show in the following.

The 2-probe and 4-probe IV results are shown in Fig. 4.7c and 4.7d respectively. It is clear from the results that the device conducts better for positive V sd bias than for negative bias, given the higher current magnitude reached under positive bias than negative bias.

With positive V sd the current flows from D1 to D2 and the resistance is dominated by D2.

But with negative V sd the resistance is dominated by D1, which has a higher resistance. The small contacted MoS 2 area in contact S manifest itself as a higher resistance for electrons when going from the metal contact to the MoS 2 , showing that the contacted area is important when fabricating low resistance contacts.

Thermionic emission

To understand the nature of the electron transmission at the Schottky barrier (SB), we investigate the two-probe data as a function of the inverse of temperature at various gate voltages, in the range 300 K < T < 150 K. Electrical transport across a Schottky contact into a 2D material can be described by the 2D thermionic emission equation [START_REF] Anwar | Effects of electron confinement on thermionic emission current in a modulation doped heterostructure[END_REF]:

I sd = AA * 2D T 3/2 exp - e k B T ϕ B - V sd η (4.1)
where A is contact area of junction, A * 2D is the two-dimensional equivalent Richardson constant, e is the magnitude of the electron charge, ϕ B is the Schottky barrier height, k B is the Boltzmann constant and η is the ideality factor. was omitted for simplicity. A V sd value of ±0.5 V has been used on devices J339 and J338, but V sd = ±0.01 V for device J12B (the data's average between positive and negative V sd was used, with the error bars being the difference).

For devices J339 and J12B, the fit was not performed since the data does not show any linearity and the current decreases as the temperature increases, being opposite to what happens with thermionic emission. This happens because those devices are so highly doped that the Fermi level lies deep in the conduction band, and the transport is of a tunneling nature, as will be discussed in the next Section. However, for device J338, as the temperature increases so does the measured current. Thus, for this device, the SB height ϕ B , could be extracted for gate voltages below 60 V, and is shown in semiconductor and a metal [START_REF] Kim | Fermi Level Pinning at Electrical Metal Contacts of Monolayer Molybdenum Dichalcogenides[END_REF]. For V g > V F B , the change in SB height is no longer linear because the carriers can tunnel through the SB, and their transport is no longer described by thermionic emission alone, and tunneling becomes more and more important. This situation is shown in Fig. 4.8c, and will be subject of the next Section. In order to avoid making assumptions on the ideality factor η and the effect of V sd , the Schottky barrier has also been extracted using the I sd (V sd ) measurements. The results are shown in Fig. 4.10. The fit is done using the same equation 4.1 and the slope obtained is then plotted against V sd , with the interception with y-axis being proportional to the SB [START_REF] Chen | Control of Schottky Barriers in Single Layer MoS 2 Transistors with Ferromagnetic Contacts[END_REF][START_REF] Kim | Origins of genuine Ohmic van der Waals contact between indium and MoS 2[END_REF]. As before, devices J339 and J12B show the opposite behavior to what happens with thermionic emission because of their high doping. However, for device J338 at V g = 50 V, a SB ϕ B ∼ 7 ± 2 meV is obtained. This value is consistent with the one obtained with transfer measurements, ϕ B ∼ 7.1 ± 0.7 meV, as shown by the black dashed lines in Fig. 4.9d.

Such small SB could come from the fact that the devices are highly doped, with the Fermi level close to the bottom or deep into the conduction band. This comes from the fact that the fitting is done on a temperature range not high enough, underestimating the SB due to the weak thermionic emission at low temperatures [START_REF] Kim | Fermi Level Pinning at Electrical Metal Contacts of Monolayer Molybdenum Dichalcogenides[END_REF]. In such case, the thermionic emission model does not applies very well because electron tunneling across the barrier becomes important, as we will see in the next Section. The image force due to the high V sd used could also affect the SB value extracted with the transfer data [START_REF] Vaknin | Schottky Barrier Height and Image Force Lowering in Monolayer MoS 2 Field Effect Transistors[END_REF].

Tunneling

The tunneling behavior can be of two different types: direct or Fowler-Nordheim (FN) depending on the shape and width of the barrier. The direct tunneling and the F-N tunneling are determined by the nature of the interfacial barrier, that is, the former occurs when the barrier is trapezoidal (wide) and the latter occurs when the barrier is triangular (thin), as already discussed in Section 1.4.

The equations governing both type of transport can be rewritten as equation 4. In the following, (I sd /V 2 sd ) vs 1/V sd plots are presented in semi-log and log-log scales. If the plot in semi-log scale shows linearity, then F-N tunneling is expected to occur, whereas when the plot in log-log scale shows linearity, direct tunneling is thought to occur. A high V g has been used in order to push the Fermi level deep into the conduction band, bending it, at least, as shown in Fig. 4.8c. In this scenario, tunneling transport is more likely to occur.

In Fig. 4.11, the tunneling plots for the three devices are shown. The F-N plots for the three devices are not linear, as can be seen in Fig. 4.11a, 4.11c and 4.11e, meaning that the V sd used is not high enough to bend the barrier [START_REF] Vaknin | Schottky Barrier Height and Image Force Lowering in Monolayer MoS 2 Field Effect Transistors[END_REF], or F-N tunneling simply does not occurs. However, the direct plots show linearity in almost the entire range of V sd for devices J339 and J12B, as shown in Fig. 4.11b and 4.11f, but partially for J338 device, as can be seen in Fig. 4.11d. This deviation from direct tunneling on device J338 could be a transition to F-N tunneling caused by high enough V sd [START_REF] Ahmed | Carrier transport at the metal-MoS 2 interface[END_REF][START_REF] Biddut | Thermionic Emission and Tunneling at Carbon Nanotube-Organic Semiconductor Interface[END_REF][START_REF] Beebe | Transition from Direct Tunneling to Field Emission in Metal-Molecule-Metal Junctions[END_REF].

The linearity shown in all Direct tunneling plots from devices J339 and J12B, regardless of the temperature, happens because the contacts are of ohmic-like nature. In those devices, the Fermi level is deep in the conduction band and the SB does not affect the electronic transport. Moreover, they show similar Direct tunneling behaviour, considering their different metalcontact thickness. This shows that there is no difference between 2/150 nm and 15/75 nm Ti/Au at this range of doping and temperatures. The apparent transition to F-N tunneling on device J338 disappears when increasing temperature where thermionic emission occurs, since tunneling current is dominant mainly in the low temperature regime.

Conclusion

The results presented in this Chapter suggest that the charge carrier injection from the metal to the MoS 2 is dominated by direct tunneling at low temperature, and thermionic emission at high temperature. The extracted Schottky barrier for device J338 is being underestimated for two reasons: the electrons are tunneling through the barrier, and the temperature range used in its extraction is not the adequate. However, a flat band SB ϕ F B = 84.6 meV has been extracted, and is commonly found in the literature. Ohmic contacts can be realized at sufficiently high doping as in the case of device J12B, but also depending on the shape of the contacts, as in the case of J339. A contact resistance as low as ∼ 82.5 kΩ • µm has been achieved, which is commonly reported in the literature.

The electron conduction in the insulating side can be well described by variable range hoping models. It is found that the prefactor of the VRH models is a constant, which points to a hopping mechanism which is not assisted by phonons, contrary to the common belief.

The hopping mechanism is then thought to be assisted by electron-electron interaction, which is demonstrated to be high in our samples, as commonly found in 2D materials. There are The conductivity in the metallic regime seems to be of a classical nature. A model of phonons and charged point defect scattering describe very well the conductivity in the conduction band, a sufficiently high doping and temperatures. Regarding the transition itself, a scaling on the conductivity and the localization length has been performed, suggesting that this phenomena is a transition of second order, and a quantum phase transition is occurring.

However, we could not rule out the possibility of this transition being a classical phenomena, or even just an artifact. Classically, the MIT can be observed intuitively by the combination of (i) the increase in mobility and (ii) positive threshold voltage shift with decreasing the temperature. If a correction of doping or threshold voltage on the conductivity is performed, the MIT is no longer visible, suggesting that the MIT could even be an artifact.

The analysis of the 2-probe data showed that, after the thermal annealing at T = 600 K, the contact resistance drops significantly, reducing also the Schottky barrier. The shape of the contacts also affects the Schottky barrier, with the "saw-tooth" contact shape having a low Schottky barrier. Altogether, the contacts become ohmic and the intrinsic quantities such as mobility and sheet resistance can be reliably extracted. The transport at the metal-MoS 2 interface is then dominated by thermionic emission at high temperatures, and by Direct tunneling at low temperatures.

In this PhD study, we have made progress in the fabrication process of MoS 2 transistors with ohmic contacts, and in the understanding of the electronic transport at different temperatures and electron density. This progress constitutes an important step towards future studies on MoS 2 , in particular for the Jjedi project. The goal of the project was to build a tunable Josephson junction, with important sub-goals as making the proper design, fabricating working transistors, knowing the operation limits of those transistors, and how stable they are over time. Although the tunable Josephson junction could not be build, important progress has been done in the fabrication of functioning devices, and in the understanding of its electronic properties.

Figure 1 :

 1 Figure 1: Dôme supraconducteur en fonction de la densité électronique observée dans MoS 2 . Les états de l'isolant et du métal sont également observés. De [1].

Figure 2 :

 2 Figure 2: Schéma simplifié du dispositif d'une jonction Josephson. Le liquide ionique, le CNT et le Si hautement dopé sont, respectivement, les grilles supérieure, locale et arrière.

2

 2 et les méthodes de fabrication des différents dispositifs sont présentés. Le MoS 2 est cru par CVD, et de grandes surfaces avec des feuillets monocouches, comme le montre la Fig. 3a, ont été obtenues. La majorité des composants fabriqués sont des barres de Hall, sans pouvoir construire la jonction Josephson souhaitée. Certains autres dispositifs ont également été fabriqués pour extraire des informations sur le MoS 2 , comme la méthode de van der Pauw et la méthode "transfer length" (TLM). Les montages expérimentaux utilisés pour caractériser ces dispositifs sont également présentés. Les monocouches de MoS 2 sont identifiées grâce à des mesures de spectroscopie Raman et de réflectométrie. Les couches sélectionnées sont ensuite gravées selon la forme désirée et connectées via des couches de Ti/Au à l'aide d'un système de lithographie électronique, comme le montre la Fig 3b. Quelques résultats préliminaires typiques sont présentés, ceux-ci illustrent les difficultés techniques rencontrées. En particulier, lors des premières mesures, différentes techniques ont été utilisées pour contrôler la densité électronique des transistors MoS 2 (photodopage avec des LED et grille ionique en particulier), avec plus ou moins de succès. Le recuit thermique des échantillons (sous vide à T = 600 K) s'est avéré être l'une des meilleures façons d'augmenter la qualité des échantillons. L'ensemble complet des données présentées dans les deux chapitres suivants a été obtenu juste après ce type de recuit (sans exposition intermédiaire à l'air).

Figure 3 :

 3 Figure 3: Mise en forme d'un monocouche MoS 2 en une barre Hall par lithographie électronique. (a) Vue d'ensemble de feuillets de MoS 2 monocouches. Le triangle rouge indique la monocouche utilisée pour la fabrication. (b) Barre de Hall fabriquée avec des contacts Ti/Au.

Figure 4 :

 4 Figure 4: (a) Dépendance à la température de la conductivité par rapport à T -1/2 (courbe ES). Seul un jeu de données réduit est indiqué pour plus de clarté, représenté par des points reliés par de fines lignes pleines. Les lignes épaisses sont les résultats d'ajustement avec le modèle ES, d'où T 2 (la température caractéristique de modèle ES) est extraite. Les données ci-dessous T = 120 K et V g = 20 V ont été ajustées. (b) Conductivité sans dimension σ en fonction de (T /T 2 ) -1/2 . L'ajustement linéaire est indiqué par une ligne cyan, qui est extrapolée à T -1 = 0, et interceptée avec l'axe y. Cette intersection donne une valeur pour le préfacteur C p proche de la conductance quantique e 2 /h.

Figure 5 :Figure 6 :

 56 Figure 5: Mise à l'échelle MIT. (a) Mise à l'échelle sur la conductivité. (b) Mise à l'échelle sur la longueur de localisation.

Figure 1 . 1 :

 11 Figure 1.1: (a) The transition metals and the three chalcogen elements that crystallise into layered structures are highlighted in the Periodic Table. The transition metals that crystallise into layered structures with some chalcogens but not with others are color-framed. (b) Three-dimensional representation of a typical layered MX 2 structure, with the metal atoms shown in green and the chalcogen atoms shown in orange.

Figure 1 . 2 :

 12 Figure 1.2: Schematics of the structural polytypes of TMDs. From left to right: 1T (tetragonal symmetry, one layer per unit cell, octahedral coordination of the metal), 2H (hexagonal symmetry, two layers per unit cell, trigonal prismatic coordination) and 3R (rhombohedral symmetry, three layers per unit cell, trigonal prismatic coordination). The yellow-filled triangles highlight the spatial position of the chalcogen atoms within one layer. For the 1T and 2H polytypes, top views are additionally shown. From [16].

Figure 1 . 3 :

 13 Figure 1.3: The crystal structure of monolayer MoS 2 showing a layer of molybdenum atoms (blue) sand-

) where a = 3 . 2

 32 16 Å is the lattice constant. The S atoms are located in planes 1.56 Å above and below the Mo plane. This yields a distance between neighboring Mo and S atoms of d = 2.40 Å. The angle between the Mo-S bond and the Mo plane is θ = 40.6 • [23]. The MoS Brillouin zone is hexagonal, the most important symmetry points and symmetry lines are indicated in Fig. 1.4b, namely,

Figure 1 . 4 :

 14 Figure 1.4: (a) Top view of the MoS 2 lattice structure. Dark (light) circles represent Mo (S) atoms. Notice that in this view two S atoms sit on top of each other. The unit cell is shown in the highlighted hexagon. The lattice constant in the Mo plane is a. The two Bravais lattice vectors ( ⃗ R 1 and ⃗ R 2 ) are indicated. Brillouin zone for the MoS 2 lattice. ⃗ K 1 and ⃗ K 2 are the reciprocal lattice basis vectors, and Γ, K, K', M, Q and Q'

Figure 1 . 5 :

 15 Figure 1.5: Density functional theory results of band structures of (a) bulk MoS 2 , (b) quadrilayer MoS 2 , (c) bilayer MoS 2 , and (d) monolayer MoS 2 . The solid arrows indicate the lowest energy transitions. Bulk and few layers MoS 2 are characterized by an indirect bandgap while monolayer is direct occurring at the K point. Adapted from [26].

Figure 1 . 6 :

 16 Figure 1.6: (a) Orbital projected band structures for monolayer MoS 2 . Contributions from Mo d-orbitals: blue dots for d xy and d x 2 -y 2 , red open circles for d z 2 , and green open diamonds for d xz and d yz . The size of the symbols is proportional to the population in the corresponding state. From [28]. (b) Schematic drawing

Fig. 1 .

 1 7 is reproduced the results of the first observed superconductivity transition in MoS 2 [1], as a function of temperature and electron density.

Figure 1 . 7 :

 17 Figure 1.7: Superconducting dome as a function of electron density observed in MoS 2 . The insulator and metal states are also observed. From [1].

Figure 1 . 8 :

 18 Figure 1.8: Sheet resistance of few layers MoS 2 as a function of temperature under the application of perpendicular magnetic fields (a), and parallel magnetic fields (b). The magnetic field varies in 1 T step.

Figure 1 . 9 :

 19 Figure 1.9: (a) Simplified sketch of a JJ. (b) I-V diagram of a JJ. After reaching a critical current, a voltage appears and the JJ behaves as a resistor.

Figure 1 . 10 :

 110 Figure 1.10: Goal device: tunable JJ. Adapted from [43].

Figure 1 . 11 :

 111 Figure 1.11: Simplified sketch of the goal device. The ionic liquid, the CNT and the highly doped Si will act as the top, local and back gates respectively.

Figure 1 . 12 :

 112 Figure 1.12: (a) Conductance dI/dV versus magnetic field and bias voltage of a topological JJ (InSb nanowires partially contacted by superconductors and non-superconductors). The green dashed oval corresponds to the onset of a Majorana fermion, the superconducting gap is given by the green dashed lines, and yellow lines underlines other types of sub-bandgap states, called bound Andreev states. From [48]. (b) Characteristic current versus bias voltage of a Josephson junction with microwave excitation. The steps formed are called Shapiro steps. Adapted from [54].

Figure 1 . 13 :

 113 Figure 1.13: (a) Electric field effect in single-atomic-sheet crystals at T = 300 K. The inset shows a typical devices used for such measurements. Scale bar: 5 µm. From [55]. (b) Room-temperature electric field effect of monolayer MoS 2 . Backgate voltage is applied to the substrate and the top gate is disconnected. The top left inset shows the current-voltage curves at different backgate voltages, whereas the lower right inset shows a optical image of the device used. Scale bar: 10 µm. Adapted from [56]. (c) Conductivity σ vs. backgate bias V BG for various temperatures. The inset shows a optical image of the device. From [5].

1. 3 .Figure 1 . 14 :

 3114 Figure 1.14: Hall effect measurement setup for electrons.

  value for the density n c , the MIT occurs. Mott estimated this critical value by comparing the Thomas-Fermi length to the effecrive Bohr radius: λ T F ∼ a * B , where a * B = 4πε 0 ε r ℏ 2 /m * e 2 , finding n 1/3 c a * B ≃0.25. Besides its simplicity, this criteria is obeyed by a large number of semiconducting materials, as shown in Fig. 1.16.

Figure 1 .

 1 Figure 1.15: A crystalline array of hydrogen-like atoms. From [67].

1. 3 .Figure 1 . 16 :

 3116 Figure 1.16: Critical density n c at the MIT, and atomic Bohr radius a * B , in a variety of semiconducting materials. The straight line follows the criteria developed by Mott. From [68].

Figure 1 . 17 :

 117 Figure 1.17: Random impurities states below the conduction band E c used by Anderson. The impurity band have a width W . States below the Fermi energy E F are all occupied at T = 0 K, while at finite T they can hop between states that fall in the thermal energy band.

Figure 1 . 18 :

 118 Figure 1.18: Density of states (DoS) in an Anderson band with the two mobility edges E C and E ′ C

1. 3 .Figure 1 . 19 :

 3119 Figure 1.19: Plot of β(g(L)) vs ln(g) for d=1,2 and 3 dimensions.

1 . 40 )= π 2 e 4 κ 2

 1402 Expression 1.40 is valid provided the energy ε is so small that the distance r ij = e 2 /κε much exceeds the mean separation between donors or, equivalently, when DoS(ε) ≪ DoS 0 , where DoS 0 is the constant density of states used by Mott. The width of the Coulomb gap ∆ is given by the equation DoS(∆) = DoS 0 : ∆ DoS 0 (1.41)

  energy are in extended states, whereas when C decreases to some critical value C * , called the percolation threshold, extended metallic paths cease to exist, and the electrons become localized. Webman et al. were able to fit the conductivity data of metal-ammonia solutions and of alkali-tungsten bronzes (which undergo a MIT) with the expression σ(C) = A(C -C * ) γ , finding values for C = 0.13 -0.25 and γ = 1.5 -1.75.

Figure 1 . 20 :

 120 Figure 1.20: Schematic diagram of the potential landscape for a disorder strength of s = 0.05 meV at different Fermi levels: (a) E F = 0.01 meV. (b) E F = 0.05 meV. (c) E F = 0.1 meV. (d) E F = 0.2 meV.

1. 4 .

 4 METAL-SEMICONDUCTOR INTERFACE: THE SCHOTTKY BARRIER27 1.4 Metal-semiconductor interface: the Schottky barrier As the monolayer MoS 2 is contacted with metal in order to perform electrical measurements, one needs to understand what happens at the semiconductor-MoS 2 interface. The electrical current flowing across such interface is usually non-linear against the applied bias voltage, as a result of the discontinuity of the energy of the electronic states responsible for conduction in the two materials. In Fig. 1.21, a band diagram of the interface is sketched to show this discontinuity. Delocalized electronic states around the Fermi level are responsible for electrical conduction in the metal, drawn on the left, but these states are not coupled to any delocalized electronic states in the semiconductor drawn on the right. The lowestlying states for n-type semiconductor that can communicate with electrons in the metal are now at an energy Φ B above the Fermi level, as shown in the figure. For electronic transport across the metal-semiconductor interface, this energy offset, known as Schottky barrier (SB),

Figure 1 . 21 :

 121 Figure 1.21: (a) Band diagram for n-type semiconductor Schottky barrier (SB). (b) Schematic diagram of a SB at different temperatures and voltages. The circles represent the charge carriers, and the arrow indicates the charge carrier injection processes: (I) thermionic emission, (II) direct tunneling, and (III)

Figure 2 . 1 :

 21 Figure 2.1: Sketch of a FET and its working principle. A negative-channel metal-oxide semiconductor employs a positive secondary voltage to switch a thin layer of p-type semiconductor material below the gate into n-type. From [105].

Figure 2 . 2 :

 22 Figure 2.2: Sketch of a Hall bar showing the current and voltages that can be measured.

Figure 2 . 3 :

 23 Figure 2.3: A flat lamella of arbitrary shape, with four contacts M,N,O and P placed on the periphery.

4 )

 4 Fig. 2.4. Voltages are applied to pairs of adjacent contacts, measuring at the same time the current, and hence determining the resistance between them. The current flows from the first probe into the metal contact, across the metal-semiconductor junction, through the sheet of semiconductor, across the metal-semiconductor junction again (except this time in the other direction) into the second contact, and from there into the second probe and into the external circuit. The resistance measured is the sum of the two probes, the two contacts, and the sheet resistance of the semiconductor between the contacts. When several such measurements are made between pairs of contacts that are separated by different distances, a plot of resistance versus contact separation can be obtained, being linear if the contact resistance is homogeneous. The slope of the line corresponds to the sheet resistance, and

  5-digit multimeter. The 2400 model, shown in Fig. 2.5a, have a voltage range of 100 µV to 200 V and a current range of 10 pA to 1 A, while the model 2612B, shown in Fig. 2.5b, have similar voltage range but a current range of 1 pA to 10 A. A Hewlett Packard voltmeter is added to measure voltages

Figure 2 . 4 :

 24 Figure 2.4: Simplified sketch of a TLM. The MoS 2 , shown in gray color, has a rectangular shape and is contacted on top with metal contacts, shown in yellow color. The distance between the contacts increases with the addition of new contacts.

Figure 2 . 5 :

 25 Figure 2.5: Power sources and voltmeters. (a) Keithley 2400 used as source-drain voltage power supply and source-drain current measurement. A second one was used for the back-gate. (b) Keithley 2612B used during ionic liquid measurements, the channel a was used as source-drain voltage input and channel b for the ionic liquid voltage. (c) Hewlett Packard voltmeter for measuring voltages in a 4-probe configuration.

Figure 2 . 6 :

 26 Figure 2.6: (a) Schematic diagram of a four-probe circuit. From [111].

Figure 2 . 7 :

 27 Figure 2.7: (a) Two routes for MX 2 synthesis. (b) Typical setup of a CVD systems. Adapted from [112].

Figure 2 . 8 :

 28 Figure 2.8: Usual triangular shape of two different CVD grown MoS 2 samples: Jjedi01 (a-c) and Jjedi06 (d). (a) Large field view. (b) close look at the triangular monolayer MoS 2 flakes. (c) Agglomeration of flakes into a big non-uniform structure. The white center is a bulk MoS 2 structure. (d) Triangular monolayer MoS 2 flakes.

Figure 2 . 9 :

 29 Figure 2.9: Masks of the different devices. The MoS 2 is represented by pink color and the metal contacts by blue color. (a) A simple Hall Bar (HB). (b) Transmission length method (TLM). (c) van der Pauw (vdP) square. (d) A second generation of Hall bars.

Figure 2 . 10 :

 210 Figure 2.10: Illustration of a EBL process: resist deposition by spin coating; electron beam exposure; development carried out with MIBK/IPA with a ratio of 1:3; metallization; and lift-off. Positive resist has been used during the full fabrication process.

Figure 2 . 11 :

 211 Figure 2.11: (a) and (b) PMMA Hall bar before and after plasma etching. A close analysis reveals a lateral reduction of a 200-300 nm on the PMMA layer. (c) and (d) profilometer results before and after plasma etching showing ∼ 100 nm of thickness reduction on the PMMA.

Figure 2 . 12 :

 212 Figure 2.12: Shaping a MoS 2 flake into a Hall bar by EBL. The device is named J339 (the name follows a notation that will be explained later). (a) Step S1e: markers. There are four crosses forming a square of length 860 µm, indicated by black arrows. The red triangle shows the flake used for the EBL. The insert shows a closer look at the marker. (b) Step S2c plasma-etching. (c) Step S3c: contacts. (d) Step S3e: metallization of contacts.

  Fig. 2.13.A closer look into a few WFs for the different samples are shown in Fig.2.17, 2.18, 2.19, and 2.20, for samples Jjedi01, Jjedi03, Jjedi04, and Jjedi06 respectively. Close to 200 devices were fabricated. Fig.2.21 shows a few of them.

Figure 2 . 13 :

 213 Figure 2.13: Jjedi01. The red circle marks the device J162. A total of 28 HBs and 7 vdP shapes were fabricated on this sample. This sample was cut into 4 parts on column 7 and row 5, indicated by the yellow lines.

Figure 2 . 14 :

 214 Figure 2.14: Jjedi03. A total of 58 HBs and 15 TLM shapes were fabricated on this sample. This sample was cut into two parts on column 6, indicated by the yellow line.

Figure 2 . 15 :

 215 Figure 2.15: Jjedi04. A total of 47 HBs, 7 TLM and 1 vdP shapes were fabricated on this sample.

Figure 2 . 16 :Figure 2 . 17 :Figure 2 . 18 :

 216217218 Figure 2.16: Jjedi06. A total of 30 HBs and 4 vdP shapes were fabricated on this sample.

Figure 2 . 19 :Figure 2 . 20 :

 219220 Figure 2.19: Two different WFs of sample Jjedi04. (a) J462 with a single TLM. (b) J423 also with a single TLM.

Figure 2 . 21 :

 221 Figure 2.21: Some of the fabricated devices: (a) HB J11B. (b) HB J339. (c) HB J338. (d) TLM J335. (e) TLM J462. (f) vdP J17B. (g) HB J16A. (h) HB J12B. (i) TLM J349. (j) HB J6DC. (k) HB J69G. (l) vdP J6FD.

Figure 2 . 22 :

 222 Fig. 2.22b. This setup uses a laser of known wavelength which is focused onto the sample surface. The scattered light is then collected by the microscope objective and sent to a spectrometer.

Figure 2 .

 2 Figure 2.23: (a) Illustration of reflectometry principle. (b) The microscope Nikon eclipse LV100 used for the reflectometry measurements.

Figure 2 . 24 :Figure 2 . 25 :Figure 2 . 26 :

 224225226 Figure 2.24: Ball bonding machine.. (a) Full view of the bonding machine. The labels are: 1) binoculars to look at the sample; 2) interface to view and change bonding parameters such as ultrasonic time and force; 3) handle to lower the capillary with the gold wire to make the bond; 4) Sample holder. (b) Close view at the capillary. The labels are: 5) light source; 6) high voltage discharge wire; 7) capillary; 8) surface of the sample holder that can be heated up to 100 • C.

  (2methoxyethyl)ammonium bis(trifluoromethylsulfonyl)imide, shown in Fig. 2.28a, and photogating with an UV light emitting diode (LED), shown in Fig 2.28b.

Figure 2 . 27 :Figure 2 . 28 :

 227228 Figure 2.27: Probe station Karl Suss PM5. (a) Full view of the probe station. Five controllers are visible (2 on the left and 3 on the right). They are used to control 5 different metal needles. (b) Close view of 2 needles electrically contacting a 5×5 mm sample.

Figure 2 . 29 :

 229 Figure 2.29: The high-temperature Janis Helium-free cryostat used for most of the measurements done with a controlled temperature.

Figure 2 . 30 :

 230 Figure 2.30: Liquid helium-4 cryostat. (a) Optical picture. (b) Diagram indicating the main parts.

Fig. 2 .Figure 2 . 31 :Figure 2 . 32 :

 2231232 Figure 2.31: (a) Typical reflectometry fit window. The vertical blue and yellow lines correspond to the limit of the fitting in wave length lambda, while the vertical red line shows the maximum of the reflected light. The red and white dots correspond to the measured reflected light from the MoS 2 flake and the SiO 2 respectively, which are being fitted by the blue and green line respectively. (b) Number of layers given by reflectometry on different spots whitin the flake S9. The spots are indicated by a red dot, and the corresponding number of layers is written just above it.

Figure 2 . 33 :Figure 2 . 34 :Figure 2 . 35 :

 233234235 Figure 2.33: (a) and (b) Optical pictures of J339 and J12B respectively. (c) Histogram of the frequency difference δ = ω(A 1g ) -ω(E 1 2G ), extracted from the map analysis shown in the inset (device J339). (d) Three Raman spectra: from device J339, device J12B, and from an exfoliated monolayer MoS 2 reference. The spectra show the E 1 2g mode at ≃ 380 cm -1 and the A 1g mode at ≃ 405 cm -1 . (e) Raman spectra as in panel d but at lower frequencies, showing the LA mode centered around 240 cm -1 .

Figure 2 . 36 :

 236 Figure 2.36: Current annealing of J113. (a) Device J113, scale bar = 50 µm. (b) IV curves at high V sd for the different pairs of contacts indicated by color lines in panel (a). (c) Optical picture after current annealing, evidencing the contact burned.

Figure 2 . 37 :Figure 2 . 38 :

 237238 Figure 2.37: Test of the back-gate with IL on top of J124. (a) J124 covered with the IL. (b) IV curves at different V bg . (c) Transfer curves at different V sd . Positive and negative values has been used to look at the symmetry of the curves.

Figure 2 . 39 :Figure 2 . 40 :

 239240 Figure 2.39: Test of the back-gate with IL on top of J126. (a) J126 covered with a small drop of IL. (b) Transfer curves at different V sd . Positive and negative values has been used to look at the symmetry of the curves. (c) IV curves at different V bg .

Figure 2 . 41 :

 241 Figure 2.41: Device J6AF with IL drop on top. (a) Large area image. The biggest contact pad is the ILG pad. (b) Closer look at the Hall bar, the contacts used are labelled.

Fig. 2 .Figure 2 . 42 :

 2242 Fig. 2.41b. The results are shown in Fig. 2.43. It is worth noting that the conductance increases when decreasing the back-gate voltage (reducing the electron density) which is apparent since we are past the maximum in conductance shown in Fig. 2.42b. The IV curves, shown in Fig. 2.43b, are strongly non-linear, which could be a contact issue (high Schottky barrier) or VRH transport mechanism taking place, which can not be discriminated by 2probe measurements. Unfortunately, no Hall effect could be detected because the electron density and mobility of the device gives a theoretical Hall voltage value on the order of µV,

Figure 2 . 43 :

 243 Figure 2.43: Measurements performed at T = 4.2 K on contacts 3-8. V tg = 3 V. (a) Transfer curves with the back-gate at different V sd . (b) IV curves at different back-gate voltages.

Figure 2 . 44 :

 244 Figure 2.44: (a) Cooling downs of device J6AF at different V tg in a Efros-Shklovskii plot. (b) and (c) Optical pictures of the device after measurements.

2. 4 . 5 PhotogatingFigure 2 . 45 :

 45245 Figure 2.45: Photogating of device J35A. (a) Schematic of the UV illumination and measurements. Scale bar = 25 µm. (b) 4-probe conductivity as a function of time during UV illumination.

Figure 2 . 46 :

 246 Figure 2.46: IV (a) and transfer (b) measurements for different LED conditions: LED OFF (blue curves), LED ON at low power (Orange curve), and LED ON at high power (green curve).

Figure 2 . 47 :

 247 Figure 2.47: J11B before and after thermal annealing. (a) Device J11B showing the contacts used for the measurements. Scale bar = 25 µm. (b) and (c) IV and transfer results before (in black color) and after (blue color) thermal annealing at T = 600 K.

Fig. 2 .

 2 Fig. 2.48, and at T = 300 K, shown in Fig. 2.49. V sd is 4 times V x , with the distance between contacts used in the 4-probe configuration being roughly 1/4 the distance between sourcedrain contacts, meaning there is a negligible (compared to the MoS 2 resistance) voltage drop at the contacts.

Figure 2 . 48 :Figure 2 . 49 :

 248249 Figure 2.48: J11B IV plots at T = 500 K. (a) 2-probe results (b) 4-probe results.

Figure 2 . 50 :

 250 Figure 2.50: Change in doping with annealing for different devices. (a) Device J31A. (b) Device J339.

Figure 2 . 51 :

 251 Figure 2.51: Effect of different annealing temperature and consecutive high temperature annealing. Device J348, scale bar = 25 µm. All the measurements were performed at T = 400 K after performing the annealing.The temperatures and times of the thermal annealing are: T = 520 K for 30 min (blue curve), T = 600 K for 30 min (orange curve), and T = 600 K for 20 min (green curve), measured in that order.

Fig. 2 .Figure 2 . 52 :

 2252 Fig. 2.52, where it shows on cyclic transfer curves, I sd vs. V g , with I sd depending on the sweep direction of V g . Depending on the microscopic effect, the hysteresis of the full transfer cycle can result in clockwise or anti-clockwise direction. In Fig. 2.52a and Fig. 2.52b are shown both directions for n-doped semiconductors. For n-type FETs, a clockwise hysteresis is

Figure 2 . 53 :

 253 Figure 2.53: (a), (b), (c) and (d) Devices J339, J338, J12B (bot) and J12B (top) respectively. Scale bar = 25 µm. (e), (f), (g) and (h) Transfer curves showing hysteresis for the aforementioned devices. The black arrows shows the back-gate sweeping direction. The hysteresis is noticeable above T = 300 K.

2 .

 2 Several of those measurements resulted in damages of the samples (current annealing, ionic liquid gating) while some others gives us hints on what to change on the device itself and the type of measurements we could perform without damaging the sample. The ionic liquid gating is a delicate technique that requires a fine control over the temperature, the sweeping voltage rate and the current leakage. The temperature should be as low as possible while keeping mobile ions, which is around 220 K. Hall effect measurements has been tried but the mobility of the samples was not the adequate to measure any significant signal. Thermally annealing the devices resulted in a big change of doping, more than what the SiO 2 back gate could change, pushing the Fermi level close to the conduction band. With this method, the electronic transport has been successfully studied, as will be shown in the next Chapters.

2. 3 . 6 .

 36 The DC two-terminal and four-terminal current-voltage (I -V ) characteristics were recorded as a function of the back gate voltage, and as function of the source-drain voltage at different temperatures. As mentioned in Section 2.4.2, after fabrication, all measured devices

Figure 3 . 1 :

 31 Figure 3.1: Optical pictures of J339 (a), J338 (b), and J12B (c). Scale bar = 25 µm. S and D stand for source and drain respectively. The contacts 1 and 2 were used to measure voltage in a 4-probe configuration.

Figure 3 .

 3 Figure3.2 shows the 2-probe and 4-probe curves at different temperatures and fixed gate voltage (V g ) for J339, J338 and J12B devices. Here, I sd and V sd correspond to the current and voltage between the source and the drain, and V x is the local voltage measured in the four probe configuration. As we can see from the figure, the four-probe curves are linear, while the two-probe curves are not. This is especially obvious for device J338. The non-linearity is induced in the vicinity of the contacts by Schottky barriers, as will be discussed in the next Chapter. As the I sd current is limited by the blocking Schottky contact, the superlinear behavior (in the 2-probe data) may be due to Schottky contacts as will be discussed on the next Chapter.

Figure 3 . 2 :

 32 Figure 3.2: Two-probe ((a), (c) and (e)) and four-probe ((b), (d) and (f)) curves for J339, J338 and J12B

Fig- ure 3

 3 .3 shows the conductivity σ of the three devices as a function of backgate voltage at different temperatures, and as a function of temperature at different backgate voltages. The

Figure 3 . 3 :

 33 Figure 3.3: Four-probe conductivity of devices J339 (a and b), J338 (c and d), and J12B (e and f), as a function of gate voltage and temperature. In panel (a) is shown the the critical conductivity σ c ≃ 55 µS, and the critical gate voltage V c = 45 V. For the data of device J12B shown in panel (f), the asterisk marks the temperature below which the sample's chamber was filled with Helium at low pressure.

Figure 3 . 4 :

 34 Figure 3.4: Threshold voltage (V T h ) of devices J339 (black color) and J338 (blue color). (a) V T h extraction at T = 20 K. (b) V T h as a function of temperature.

Figure 3 . 5 :

 35 Figure 3.5: (a) and (b) Arrhenius plot of conductivity for J339 and J338 respectively. The data is shown by dots joined by thin solid lines, and the fitting results by thermally activated model are shown by solid thick lines. The dashed lines are extrapolations of the fit to show the deviation at lower temperatures.

Figure 3 . 6 :

 36 Figure 3.6: (a) Extracted activation energy as a function of gate voltage. The bottom of the conduction band is reached at V g = V T h , where the conduction band stats to being populated and there are not more thermally activated electrons. (b) Computed density of states (DoS) as a function of activation energy. The DoS reaches the expected value for pristine or ideal MoS 2 at the bottom of the conduction band.

3 . 9 .

 39 The experimental data are very well fitted by straight lines over large domains of gate voltages and temperatures. J338 device has been also fitted with p = 2 for comparison. The fit is extrapolated to T -1 = 0, as shown by the dashed lines. In the case of J339 device, the extrapolated fit almost converges at T -1 = 0, giving a value of C p = 0.8 ± 0.2 e 2 /h. In the case of J338 device, it gives C p = 3.3 ± 1.5 e 2 /h. The discrepancy of the C p value between J339 and J338 devices comes from the fact that the ES VRH model does not describe well the transport mechanism on J338, as shown by the extracted p value, and the poor ES fit.

Figure 3 . 7 :

 37 Figure 3.7: Parameter W = T -1 ∂ ln ρ(T )/∂T -1 on a double logarithmic plot. The thick solid lines are the linear fits, and the dashed lines correspond to different transport mechanisms, each with its corresponding p value (Eq. 3.4). Thermally activated transport (TA) with p = 1, Efros-Shklovskii (ES) VRH with p = 2, and Mott VRH with p = 3. (a) and (b) J339 and J338 respectively.

Figure 3 . 8 :

 38 Figure 3.8: Extracted p from the linear fit performed on the W plots shown in Fig. 3.7.

Figure 3 . 9 :

 39 Figure 3.9: Temperature dependence of the conductivity versus T -1/2 (ES plot). Only a reduced data set is indicated for clarity, represented by points joined with thin solid lines. The thick lines are the fitting results using the ES model. (a) Device J339. The data below T = 120 K and V g = 20 V have been fitted. (b) Device J338. The fitting temperature region varies depending on the gate voltage. The fit is extrapolated to T -1 = 0 to extract the C p coefficient.

Figure 3 .

 3 11 shows the T 2 temperature and localization length as extracted from Eq. 3.7, as a function of gate voltage. One can see that T 2 → 0 when approaching E c , while ξ increases to large values signaling delocalization. Device J339 has very large values of ξ at V g values close to V T h . If ϵ r is constant as defined by the Keldysh formula shown in Eq. 3.3 (ϵ r ∼ 2.5), then ξ ∼ 1.7 µm

Figure 3 . 10 : 2 ( 3 Figure 3 . 11 :

 31023311 Figure 3.10: Dimensionless conductivity σ as a function of (T /T 2 ) -1/2 . The linear fit is indicated by a cyan line, which is extrapolated to T -1 = 0 and intercepted with the y-axis. This intersection gives a value close to the quantum conductance e 2 /h for device J339 (a), while the data is highly scattered for device J338 (b).

Figure 3 . 12 :Figure 3 . 13 :

 312313 Figure 3.12: Temperature dependence of the conductivity versus T -1/3 (Mott plot). Only a reduced data set is indicated for clarity, represented by points joined with thin solid lines. The thick lines are the fitting results using the Mott model. (a) Device J339. The data below T = 180 K and V g = 20 V have been fitted. (b) Device J338. The fitting temperature region varies depending on the gate voltage.

2 e 4 κ 2

 242 Therefore, the VRH type must be crosschecked by other means, e.g., the parameter W , which was shown in Fig.3.7. We introduce another way of discriminating between those models, by looking at the energy gap ∆ = π DoS in the ES model (Eq. 1.41), and compare it to the average spacing between energies ϵ = 1 3 k B (T 3 T 2 ) 1/3 in the Mott model (Eq. 1.36).As the ES model was deduced assuming ∆ ≫ ϵ, while the Mott model was deduced assuming ∆ ≪ ϵ, we then look at the dimensionless energy ratio ∆/ϵ. The results are shown in Fig. 3.14. ∆/ϵ ≫ 1 means that the electronic transport mechanism is governed by the ES model, whereas ∆/ϵ ≪ 1 means it is governed by the Mott model. The result shows that the electronic transport for device J339 is effectively dominated by ES VRH, whereas for device J338 it is neither Mott nor ES. Those VRH models were obtained under the assumption of the gap ∆ being much greater or much smaller than the Mott average energy spacing ϵ, and device J338 does not meet any of those assumptions, except for V g > 60 V where ∆/ϵ > 10. It is evident that neither Mott nor ES model fits well the conductivity data of J338, and since the ES/Mott ratio gives values around 1, it means that the assumptions made to derive Mott and ES models are not valid. A revision on the derivation of the VRH models has been done by the doctorant on Appendix A, where no assumptions on the energy ∆ and ϵ are made, and both terms are included in the transport equation. The result is an intermediate

Figure 3 . 14 :

 314 Figure 3.14: Dimensionless energy ratio for device J339 (black color) and J338 (blue color) at T = 180 K.

bility in pristine monolayer MoS 2

 2 is weakly dependent on the carrier density, and depends on temperature as µ ∼ T -γ for T > 100 K, where the damping factor γ ∼ 1.7. They also predicted a theoretical value at room temperature of ∼ 410 cm 2 V -1 s -1 . Those studies were performed taking into account only electron-phonon interactions, without considering any other type of electron scattering. A few years later, in 2019, Kaasbjerg et al.[START_REF] Kaasbjerg | Electron and hole transport in disordered monolayer MoS 2 : Atomic vacancy induced short-range and Coulomb disorder scattering[END_REF] addressed the problem again but this time on disordered monolayer MoS 2 , with impurities such as sulphur vacancies acting as both, short-range and long-range scatterers. Their result are quite different from their first work, with the Coulomb scattering being the main scattering mechanism. The dominance of Coulomb disorder scattering gives rise to a strong temperature and density dependence of the mobility, which arise from the temperature and density dependent screening of the Coulomb potential[START_REF] Sarma | Screening and transport in 2D semiconductor systems at low temperatures[END_REF]. Kaasbjerg et al. also showed that the damping factor does not have a fixed value but depends on temperature and carrier density, as commonly found in experiments[3, 10-12, 74, 75]. Thus, having multiple scattering mechanisms modifies the damping factor. It has been suggested that a change in the dielectric environment by using high-κ dielectrics on top of the samples can improve the mobility by

Fig. 3 . 16 .

 316 Fig. 3.16. It shows low values at large negative gate voltages (low doping) and increases as the electron density increases, reaching a plateau at high gate voltages in the case of J339 (see Fig. 3.16a). We can distinguish three different transport regimes for device J339: gate voltages below -40 V, µ eff slowly increases with doping, and the transport is dominated by variable range hopping; at gate voltages between -40 V and 40 V, µ eff increases quickly with

Figure 3 . 15 :

 315 Figure 3.15: Extracted FET mobility (µ FET ) from the linear fit of conductivity at high gate voltage. The high temperature region was fitted and the damping factor was then extracted.

  3.16b) does not reach a saturation, indicating poor screening due to low (high) doping (density of defects), showing only the VRH and screening regions. In the case of device J112B (see Fig.3.16c), only the metallic region is observed, with µ eff being almost doping-independent.

3 .

 3 16d for device J339. It decreases over the entire range of temperature (in agreement with Kaasbjerg et al.[START_REF] Kaasbjerg | Electron and hole transport in disordered monolayer MoS 2 : Atomic vacancy induced short-range and Coulomb disorder scattering[END_REF]), and there is little change with V g for values above the critical voltageV c .The low mobilities usually found in the literature are because those devices have low electron density, falling in the VRH region. The mobility should then be obtained once the metallic regime is reached, where the electrons are conducting because of the conduction band, and not because the impurity band.

Figure 3 . 16 :

 316 Figure 3.16: (a), (b) and (c) Gate dependent of effective mobility for devices J339, J338 and J12B respectively. For device J12B, the µ eff is computed only above T = 108 K, since at lower temperatures the data is very noisy. (d) Damping factor of device J339 extracted from the effective mobility (µ eff ) as γ = dlogµ eff /dlogT , for gate voltages above V c (in the metallic regime).

Figure 3 . 17 :

 317 Figure 3.17: Electron densities extracted from two different ways for device J339. Solid lines correspond to n eff while dashed lines correspond to n C .

Figure 3 . 18 :

 318 Figure 3.18: (a), (b) and (c) Conductivity versus temperature at different carrier concentrations for devices J339, J338 and J12B respectively. The experimental data are shown as colored dots and the best theoretical fits are indicated by solid lines. The best fit parameters are n D = 1.0 × 10 12 , 1.5 × 10 12 , 1.2 × 10 12 cm -2 for devices J339, J338 and J12B respectively. (d) Effective mobility of the three devices, J339 (black dots), J338 (red dots), and J12B (blue dots), at T = 220 K. The theoretical mobility is shown for phonon scattering only (cyan curve), impurity scattering only (orange curve), and both type of scatterings (green curve).

Figure 3 . 19 :

 319 Figure 3.19: MIT of device J339. (a) Scaling on conductivity. (b) Scaling on localization length.

Figure 3 . 20 :

 320 Figure 3.20: Conductivity with the three characteristic temperatures shown by colored dashed lines. The black solid line with dots is the curve used in the scaling analisys shown in Fig. 3.19a.

  The insulator regime has been studied as a function of temperature and gate voltages, showing TA and VRH transport mechanisms, with a possible existence of an intermediate VRH transport mechanism. The metallic regime has been modeled with electrons experiencing phonons and impurity scattering. The long-debated MIT phenomena has been studied in terms of a QPT, although the analysis could be performed only on a single device, with the conditions not being the optimal to be conclusive. We were able to model the conductivity of monolayer MoS 2 in the metallic regime, as well as in the insulating regime. A small overview summarizing the transport mechanism found is shown in the Fig 3.21. The conductivity depends on temperature and electron density. At low doping and low temperature, all the data enclosed by the red box, VRH takes place. At low doping and high temperature, all the data enclosed by the green box, thermally activated transport takes place. At high doping and high temperature, all the data enclosed by the black box, electrons are delocalized and band-like conduction takes place. A quantum phase transition might be occurring at temperatures below 100 K, corresponding to all the data enclosed by the cyan color box.

Figure 3 . 21 : 4 1

 32141 Figure 3.21: Conductivity of device J339. Colored boxes are enclosing different regions with their particular electronic transport mechanism.

Figure 4 . 1 :

 41 Figure 4.1: Schematic of the 2-and 4-probe measurements. (a) side view: S and D stand for Source and Drain respectively and are the outer electrodes contacting the MoS 2 (in gray color). (b) top view showing the width w, external length L and internal length l of a Hall bar shape.

Figure 4 . 2 :

 42 Figure 4.2: TLM technique on device J31A. The measurements were performed at T = 350 K after annealing at T = 600 K for about 42 min. (a) and (b) IV and transfer curves respectively. The colored curves represent different pair of contacts. The insert of (b) shows the TLM device with a label for each contact, the scale bar is 25 µm. The distance between the contacts is on the labels. (c) extracted contact resistance from fitting the IV curves. (d) 4-Probe measurements for a few contacts showing a non-linear behaviour.

Figure 4 . 3 :

 43 Figure 4.3: J11B before (a) and after (b) metallization. Scale bar = 50 µm. S and D labels indicate the Source and Drain contacts respectively, while V x1 and V x2 labels indicate the 2 pair of contacts used for DC 4-probe measurements. V x1 was measured with the help of the HIP.

Figure 4 .

 4 Figure 4.4 shows the 2-probe and 4-probe IVs data obtained at different gate voltages and temperatures. As we can see on Fig. 4.4a and 4.4b, the 2-probe IVs curves are not linear (non-ohmic contacts) caused by barriers between the metal contact and MoS 2 . The non linearity even persist in the 4-probe measurements, especially at low temperatures, as shown in Fig. 4.4c and 4.4e. The data obtained in the 4-probe configuration with (Fig. 4.4c and 4.4d) and without (Fig. 4.4e and 4.4f) the HIP are noticeably different. All the V x2 curves are shifted (not centered at zero) with the shift being temperature and doping dependent, and making no sense at low gate voltages, as can be seen in Fig. 4.4f. This happens because at low

Fig. 4 .

 4 1a, with the outer contacts (S and D) serving as the current source -drain, and two inner contacts as the voltage probes. The contact resistance was then computed from the resistivity as R c = w(V sd /I sd -ρ(L/w)), where L and w are the full sample length and width, respectively (see Fig. 4.1b), and ρ = (V x /I sd )(w/l), with l being the length between the inner contacts in the 4-probe configuration. It is usually given in terms of the width the transistor channel, in units of Ω • µm, to take into account the sample width. The extracted contact resistance as a function of gate voltage at different temperatures is shown in Fig. 4.6 for the three devices. As can be seen from the figures, R c decreases as V g increases. This happens because the Fermi level is increasing, decreasing the effective Schottky barrier. This phenomena will be discussed in the next Section. A low R c of ∼ 90 kΩ • µm, ∼ 2 MΩ • µm and ∼ 115 kΩ • µm, has been be achieved at high densities for devices J339, J338 and J12B respectively.

Figure 4 . 4 :

 44 Figure 4.4: IVs curves for device J11B at (a,c,e) different temperatures for fixed V g , and (b,d,f) different V g for fixed temperature. V x1 is measured with a HIP of 10 TΩ input impedance and a Hewlett Packard Voltmeter, while V x2 is measured directly with another Hewlett Packard Voltmeter with 10 GΩ input impedance.

Figure 4 . 5 :

 45 Figure 4.5: Optical pictures of J339 (a), J338 (b), and J12B (c). Scale bar = 25 µm. S and D stand for source and drain respectively. The contacts 1 and 2 were used to measure voltage in a 4-probe configuration.

Figure 4 . 6 :

 46 Figure 4.6: Contact resistance as a function of gate voltage at different temperatures for devices J339 (a), J338 (b), and J12B (c). The contact resistance R c increases when decreasing V g for all devices, with J339

Fig. 4 .

 4 Fig. 4.7b. In Fig. 4.7b are shown the source (S) and drain (D) contacts. The high of the SMU was connected to contact S and the low was connected to contact D. In Fig. 4.7a are

Figure 4 . 7 :

 47 Figure 4.7: Device J35A before (a) and after (b) metallization. Scale bar 25 µm. (c) and (d) 2-probe and 4-probe IV results respectively. Negative bias have a higher resistance than positive bias, and this behaviour could be related to the area of MoS 2 below the contact.

Fig. 4 .

 4 9d.If carrier transport is dominated by thermionic emission, for low doping ϕ(V g ) varies in a linear fashion until the flat band voltage (V F B ) is reached. This situation is schematically shown in Fig.4.8, and increasing V g corresponds to a continuous change from the situation shown in Fig.4.8a to the one shown in Fig.4.8b. Thus, the SB height obtained at the endpoint of the linearity (at V g = V F B ), ϕ F B , indicates the true SB height between a 2D

Figure 4 . 8 :

 48 Figure 4.8: Energy band bending when changing the Fermi energy E F . The electron injection into the band is indicated by the blue arrows. (a), (b) and (c) correspond to V g < V F B , V g = V F B and V g > V F B , respectively.

Figure 4 . 9 :

 49 Figure 4.9: 2D Richardson plots for device J339 (a), J338 (b) and J12B (c). The dots represent the data which are joined by thin solid lines as a guide for the eye. The thick solid lines are the linear fits, performed only on device J338. (d) Extracted Schottky barrier Φ B for device J338 as a function of V g . V F B is the gate voltage at which the flat band condition is reached.

Figure 4 . 10 :

 410 Figure 4.10: 2D Richardson plots at fixed V g for device J339 (a), J338 (b) and J12B (c). The dots represent the data which are joined by thin solid lines as a guide for the eye. The thick solid lines are the linear fits, performed only on device J338. (d) Slopes extracted from (b) as a function of V sd . The Schottky barrier (Φ B ) is derived from the y-intercept, S 0 .

  2 for direct tunneling, and equation 4.3 for FN tunneling: I ∝ V sd expϕ, m * , e and ℏ are the SB width, SB high, effective mass, electron charge, and the reduced Planck constant, respectively.

Figure 4 . 11 :

 411 Figure 4.11: I sd /V sd plotted vs. the inverse of the drain bias (1/V sd ) for different temperatures and fixed gate voltage: (a), (b) and (c) Direct-like plots and (d), (e) and (f) FN-like plots for devices J339, J338 and J12B respectively.

  two commonly known VRH models: Efros-Shklovskii and Mott models, which have different power law in the temperature-dependent conductivity. An intermediate VRH mechanism seems to occur on device J338, where neither the Efros-Shklovskii nor Mott models can describe the conductivity. An attempt to derive a complete model, working on the ideas already developed by Efros and Shklovskii, is presented in the appendix. The new derived model describe very well the conductivity of J338, suggesting that this transport mechanism could be real.

  the Mott model the transition occurs because of electron screening of their host atom's potential at high enough electron density, while in the Anderson model electron localization occurs if the disorder in the lattice is sufficiently large. The Nobel Prize in Physics 1977 was awarded jointly to Philip Warren Anderson, Sir Nevill Francis Mott and John Hasbrouck Van

bandes interdites (de 0 à ∼ 2 eV), combinés à de forts couplages spin-orbite. De tous les TMD, MoS 2 reste le plus étudié à ce jour. Les monocouches de MoS 2 ont une large bande interdite (∼ 1.9 eV) directe située aux points K du réseau réciproque. Ce 'gap' les qualifie pour de nombreuses applications photoniques, et leur excellente flexibilité mécanique en fait un matériau semiconducteur prometteur pour l'électronique flexible.En 2012, il a été observé pour la première fois qu'un contrôle électrostatique du dopage de quelques couches de MoS 2 induisait une transition de la phase isolante vers une phase métallique, jusqu'à atteindre un phase supraconductrice à haute densité électronique, avec une température critique pouvant atteindre 10.8 K. Sur la Fig.1sont reproduits les résultats de la première observation de cette transition dans MoS 2 , où la résistance est tracée en fonction de la température et de la densité électronique. Suite à cette découverte, deux groupes expérimentaux ont indépendamment mis en évidence que les champs critiques dans le plan du matériau monocouche étaient plusieurs fois supérieurs à ceux de plusieurs couches, et six fois supérieurs aux limites de Pauli. En raison du couplage spin-orbite (SOC) et de l'absence de centre d'inversion, les spins des électrons sont orientés dans la direction perpendiculaire au plan 2D. Ce type de couplage est généralement appelé "Ising SOC" pour le distinguer du Rashba SOC, qui maintient les spins

(a) (b)

Acknowledgements

I want to express to my supervisors Benoit Joault and Sébastien Nanot all my gratitude for welcoming me to the laboratoire Charles Coulomb, introducing me to this field of research, and supporting me during this three short years. Thank you for all the unconditional support.

Bibliography

IV

Conclusion & Outlook

The discovery of graphene by Novoselov and Geim in 2004 has opened a new field of research on 2D materials. The properties of these materials range from zero-gap, like in graphene, to finite gaps, of 1-2 eV in semiconducting TMDs. Of all TMDs, MoS 2 has been the most studied because of its abundance, and since it has been used as dry lubricant since the 60s.

Nowadays, monolayer MoS 2 is studied for its electronic and optical properties. One of the most important electronic property is that it can transition from an insulating to a metallic state, even reaching a superconducting state, depending on its electron density. However, at the fundamental level, these interesting transitions are not fully understood yet. Within this thesis we studied the transport mechanism of MoS 2 in the insulating and metallic regimes, hopefully shedding some light into this transition that still remains debated for 2D materials.

We also discussed some of the features of metal-MoS 2 contact, and the electronic transport occurring at this interface. Different techniques for controlling the electron density and improving the quality of the sample have been explored, with the in-situ thermal annealing at T = 600 K being the most favorable, and the only reliable method with which complete data-sets could be obtained.

Our MoS

The mobility of our MoS 2 devices after thermal annealing is one of the largest reported for CVD grown MoS 2 , reaching values ∼ 200 cm 2 V -1 s -1 . The thermal annealing also increased the electron density of the samples, pushing the Fermi level deep into the conduction band where the metal-insulator transition (MIT) could be observed.

Appendix A Variable range hopping

A more complete picture of variable range hopping (VRH) must include the average energy spacing and the coulomb interaction. Once both terms are included, a different VRH model can be derived. The result is a value of p = 3/2 in Eq. A.1.

The derivation is shown below, starting with the conductivity shown in Eq. A.2:

The term -2r ξ correspond to a tunneling probability between impurities; the term -1 πDoSk B T corresponds to the average energy between impurities states, with a density of impurities depending on energy given by DoS(E F ); the last term, -e 2 rκk B T , corresponds to the Coulomb interaction between states.

The optimal hopping distance is then obtained by finding the maximum of the previous expression, i.e. differentiating against r and equating to zero:

which can be rewritten as:

where A = 2/ξ, B = 2/πDoSk B T and C = e 2 /κk B T . The Eq. A.4 was solved with the use of Wolfram Alpha [START_REF] Lin | WolfrAmalpha computational intelligence[END_REF], and it have only one real root:

At sufficient low DoS (< 1×10 17 eV -1 m -2 ) and low ξ (< 1×10 -9 m), the term 27A 4 B 2 ≫ 4A 3 C 3 , hence the term 4A 3 C 3 on Eq. A.5 can be omitted and the expression is reduced to:

Using the same condition as before (low DoS and low ξ), the term 3

the Eq. A.7 can be further reduced to:

We have then found the optimal hopping distance r. This value of r is now reintroduced into the exponent of Eq. A.2:

(A.12)

At sufficient low T , the term 3 3 √ k B T in Eq. A.13 can be omitted, reducing the expression to:

We now define the characteristic temperature T 3/2 and localization length ξ:

Finally, Eq. A.2 can be rewritten as:

We then proceed to fit the conductivity data of J338 with the Eq. A.