Une décennie après la découverte du graphème par A.K. Geim et K.S. Novoselov ,beaucoup de ses propriétés fondamentales ont été intensément étudiées. En effet, lespromesses du graphème a été etenues et ont conduit a une recherche fructueuse dansdes domaines aussi varies que l'optique, la mécanique, la chimie ou l'électronique. Legraphème mérite ainsi d'être appelée le matériau miracle puisque c'est un très conducteurde la chaleur et de l'électricité, il est l'un des matériau les plus solides tout en étantléger et optiquement transparent. Son succès est en partie dû à la technique de micro clivage,ou exfoliation qui est facile et accessible à tout laboratoire sans équipementlourd. Cela a permis d'étudier les propriétés du graphème avec un large éventail de techniquesexpérimentales. Après l'immense succès du graphème, d'autres matériaux bidimensionnels(2D) ont été obtenus avec la même technique, conduisant à une physiqueparticulièrement riche. / A decade after the discovery of grapheme by A.K. Geim and K.S. Novoselov , manyof its fundamental properties have been intensely studied. Indeed, the early promise ofgrapheme has been kept, leading to a fruitful research in many felds as diverse as optics,mechanics, chemistry or electronics, and grapheme still deserves the name of wondermaterial as among its attributes there is an exceptionally good conduction of heat andelectricity, it is one of the strongest known material while being light and opticallytransparent. Its success was partly due to the easy micromechanical cleavage or scotchtape exfoliation technique accessible to any laboratory without heavy equipment, thatallowed to study the properties of grapheme with a wide range of experimental techniques.Following the success of grapheme, other two-dimensional (2D) materials were obtainedwith the same technique, leading to a particularly rich physics.

The following dissertation presents a comprehensive study of the interaction between Dirac fermion quasiparticles (DFQs) and surface phonons on the surfaces of the topological insulators Bi2Se3 and Bi2Te3. Inelastic helium atom surface scattering (HASS) spectroscopy and time of flight (TOF) techniques were used to measure the surface phonon dispersion of these materials along the two high-symmetry directions of the surface Brillouin zone (SBZ). Two anomalies common to both materials are exhibited in the experimental data. First, there is an absence of Rayleigh acoustic waves on the surface of these materials, pointing to weak coupling between the surface charge density and the surface acoustic phonon modes and potential applications for soundproofing technologies. Secondly, both materials exhibit an out-of-plane polarized optical phonon mode beginning at the SBZ center and dispersing to lower energy with increasing wave vector along both high-symmetry directions of the SBZ. This trend terminates in a V-shaped minimum at a wave vector corresponding to 2kF for each material, after which the dispersion resumes its upward trend. This phenomenon constitutes a strong Kohn anomaly and can be attributed to the interaction between the surface phonons and DFQs. To quantify the coupling between the optical phonons experiencing strong renormalization and the DFQs at the surface, a phenomenological model was constructed based within the random phase approximation. Fitting the theoretical model to the experimental data allowed for the extraction of the matrix elements of the coupling Hamiltonian and the modifications to the surface phonon propagator encoded in the phonon self energy. This allowed, for the first time, calculation of phonon mode-specific quasiparticle-phonon coupling λⱱ(q) from experimental data. Additionally, an averaged coupling parameter was determined for both materials yielding ¯λ^Te ≈ 2 and ¯λ^Se ≈ 0.7. These values are significantly higher than those of typical metals, underscoring the strong coupling between optical surface phonons and DFQs in topological insulators. In an effort to connect experimental results obtained from phonon and photoemission spectroscopies, a computational process for taking coupling information from the phonon perspective and translating it to the DFQ perspective was derived. The procedure involves using information obtained from HASS measurements (namely the coupling matrix elements and optical phonon dispersion) as input to a Matsubara Green function formalism, from which one can obtain the real and imaginary parts of the DFQ self energy. With these at hand it is possible to calculate the DFQ spectral function and density of states, allowing for comparison with photoemission and scanning tunneling spectroscopies. The results set the necessary energy resolution and extraction methodology for calculating ¯λ from the DFQ perspective. Additionally, determining ¯λ from the calculated spectral functions yields results identical to those obtained from HASS, proving the self-consistency of the approach.

Graphene can be used to create circuits that are almost superconducting, potentially speeding electronic components by as much as 1000 times [1]. Such blazing speed might also help produce ever-tinier computing devices with more power than your clunky laptop [2]. Graphite is a polymorph of the element carbon [3]. Graphite is made up of tiny sheets of graphene. Graphene sheets stack to form graphite with an interplanar spacing of 0.335 nm, which means that a stack of 3 million sheets would be only one millimeter thick. [1] This nano scale 2 dimensional sheet is graphene. Novoselov and Geim's discovery is now the stuff of scientific legend, with the two men being awarded the Nobel Prize in 2010 [4]. In 2004, two Russian-born scientists at the University of Manchester stuck Scotch tape to a chunk of graphite, then repeatedly peeled it back until they had the tiniest layer possible [2]. Graphene has exploded on the scene over the past couple of years. "Six years ago, it didn't exist at all, and next year we know that Samsung is planning to release their first mobile-phone screens made of graphene." - Dr Kostya Novoselov [4]. It is a lattice of hexagons, each vertex tipped with a carbon atom. At the molecular level, it looks like chicken wire [4]. There are two common lattice formations of graphene, armchair and zigzag. The most studied edges, zigzag and armchair, have drastically different electronic properties. Zigzag edges can sustain edge surface states and resonances that are not present in the armchair case Rycerz et al., 2007 [5]. This research focused on the armchair graphene nanoribbon formation (acGNR). Graphene has several notable properties that make it worthy of research. The first of which is its remarkable strength. Graphene has a record breaking strength of 200 times greater than steel, with a tensile strength of 130GPa [1]. Graphene has a Young's modulus of 1000, compared to just that of 150 for silicon [1]. To put it into perspective, if you had a sheet of graphene as thick as a piece of cellophane, it would support the weight of a car. [2] If paper were as stiff as graphene, you could hold a 100-yard-long sheet of it at one end without its breaking or bending. [2] Another one of graphene's attractive properties is its electronic band gap, or rather, its lack thereof. Graphene is a Zero Gap Semiconductor. So it has high electron mobility at room temperature. It's a Superconductor. Electron transfer is 100 times faster than Silicon [1]. With zero a band gap, in the massless Dirac Fermion structure, the graphene ribbon is virtually lossless, making it a perfect semiconductor. Even in the massive Dirac Fermion structure, the band gap is 64meV [6]. This research began, as discussed in Chapter 2, with an armchair graphene nanoribbon unit cell of N=8. There were 16 electron approximation locations (ψ) provided per unit cell that spanned varying Fermi energy levels. Due to the atomic scales of the nanoribbon, the carbon atoms are separated by 1.42Å. The unit vector is given as, ~a = dbx, where d = 3αcc and αcc = 1.42°A is the carbon bond length [5]. Because of the close proximity of the carbon atoms, the 16 electron approximations could be combined or summed with their opposing lattice neighbors. Using single line approximation allowed us to reduce the 16 points down to 8. These approximations were then converted into charge densities (ρ). Poisson's equation, discussed in Chapter 3, was expanded into the 3 dimensional space, allowing us to convert ρ into voltage potentials (φ). Even though graphene is 2 dimensional; it can be used nicely in 3 dimensional computations without the presence of a substrate, due to the electric field lines and voltage potential characteristics produced being 3 dimensional. Subsequently it was found that small graphene sheets do not need to rest on substrates but can be freely suspended from a scaffolding; furthermore, bilayer and multilayer sheets can be prepared and characterized.

This thesis describes two topological phases of matter, the Weyl semimetal and the line node semimetal, that are related to but distinct from topological insulator phases. These new topological phases are semimetallic, having electronic energy bands that touch at discrete points or along a continuous curve in momentum space. These states are achieved by breaking time-reversal symmetry near a transition between an ordinary insulator and a topological insulator, using a model based on alternating layers of topological and ordinary insulators, which can be tuned close to the transition by choosing the thicknesses of the layers. The semimetallic phases are topologically protected, with corresponding topological surface states, but the protection is due to separation of the band-touching points in momentum space and discrete symmetries, rather than being protected by an energy gap as in topological insulators. The chiral surface states of the Weyl semimetal give it a non-zero Hall conductivity, while the surface states of the line node semimetal have a flat energy dispersion in the region bounded by the line node. Some transport properties are derived, with a particular emphasis on the behaviour of the conductivity as a function of the impurity concentrations and the temperature.

Topological insulators (TI) are a new phase of matter with very exotic electronic properties on their surface. As a direct consequence of the topological order, the surface electrons of TI form bands that cross the Fermi surface odd number of times and are guaranteed to be metallic. They also have a linear energy-momentum dispersion relationship that satisfies the Dirac equation and are therefore called Dirac fermions. The surface Dirac fermions of TI are spin-polarized with the direction of the spin locked to momentum and are immune from certain scatterings. These unique properties of surface electrons provide a platform for utilizing TI in future spin-based electronics and quantum computation. The surface bands of 3D TI can be directly mapped by angle-resolved photoemission spectroscopy (ARPES) and the spin polarization can be determined by spin-resolved ARPES. These types of experiments are the first to establish the 3D topological order, which demonstrates the power of ARPES in probing the surface of strongly spin-orbit coupled materials. Extensive investigation of TI has ranged from understanding the fundamental electronic and lattice structure of various TI compounds to building TI-based devices in search of more exotic particles such as Majorana fermions and magnetic monopoles. Surface-sensitive techniques that can efficiently disentangle the charge and spin degrees of freedom have been crucially important in tackling the multi-faceted problems of TI. In this thesis, I show that laser-based ARPES in combination with a time-of-flight spectrometer is a powerful tool to study the spin structure and charge dynamics of the Dirac fermions on the surface of TI. Chapter 1 gives a brief introduction of TI. Chapter 2 describes the basic principles behind ARPES and time-resolved ARPES (TrARPES). Chapter 3 provides a detailed account of the experimental setup to perform laser-based ARPES and TrARPES. In Chapters 4 and 5, how these two techniques are effectively applied to investigate two unique electronic properties of TI is elaborated. Through these studies, I have obtained a complete mapping of the spin texture of several prototypical topological insulators and have uncovered the cooling mechanism governing the hot surface Dirac fermions. / Physics

This thesis describes two topological phases of matter, the Weyl semimetal and the line node semimetal, that are related to but distinct from topological insulator phases. These new topological phases are semimetallic, having electronic energy bands that touch at discrete points or along a continuous curve in momentum space. These states are achieved by breaking time-reversal symmetry near a transition between an ordinary insulator and a topological insulator, using a model based on alternating layers of topological and ordinary insulators, which can be tuned close to the transition by choosing the thicknesses of the layers. The semimetallic phases are topologically protected, with corresponding topological surface states, but the protection is due to separation of the band-touching points in momentum space and discrete symmetries, rather than being protected by an energy gap as in topological insulators. The chiral surface states of the Weyl semimetal give it a non-zero Hall conductivity, while the surface states of the line node semimetal have a flat energy dispersion in the region bounded by the line node. Some transport properties are derived, with a particular emphasis on the behaviour of the conductivity as a function of the impurity concentrations and the temperature.

Dans cette thèse, nous étudions l’effet du désordre magnétique sur les propriétés de transport électronique du graphène et des isolants topologiques 2D de type HgTe. Le graphène et les isolants topologiques sont des matériaux dont les excitations électroniques sont assimilées à des fermions de Dirac sans masse. L’influence des impuretés magnétiques sur les propriétés de transport du graphène est étudiée dans le régime de forts champs électriques. En conséquence de la production de paires électron-trou, la réponse devient non linéaire et dépend de la polarisation magnétique. Nous étudions une transition entre un isolant topologique bi-dimensionnel conducteur, caractérisé par une conductance G = 2 (en quantum de conductance) et un isolant de Chern avec G = 1, induite par des impuretés magnétiques polarisées. / In this thesis, we study the effect of a magnetic disorder on the electronic transport properties of graphene and HgTe-type 2D topological insulators. Graphene and topological insulators are materials whose electronic excitations are treated as massless Dirac fermions.The influence of magnetic impurities on the transport properties of graphene is investigated in the regime of strong applied electric fields. As a result of electron-hole pair creation, the response becomes nonlinear and dependent on the magnetic polarization.We investigate a transition between a two-dimensional topological insulator conduction state, characterized by a conductance G = 2 (in conductance quantum) and a Chern insulator with G = 1, induced by polarized magnetic impurities.

The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for<br>studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this<br>frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.<br>Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between<br>hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the<br>topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported. <br>