Field-effect transistors (FET) based on graphene as channel has extraordinaryproperties in terms of charge mobility, charge carrier density etc. However, there aremany challenges to graphene based FET due to the fact graphene is a monolayer ofatoms in 2-dimentional space that is strongly influenced by the operating conditions.One issue is that the Dirac point, or K-point, shifts to higher gate voltage whengraphene is exposed to atmosphere. In this study graphene field-effect transistors(GFET) based on commercially available CVD graphene are electrically characterizedthrough field effect gated measurements. The Dirac point is initially unobservable andlocated at higher gate voltages (>+42 V), indicating high p-doping in graphene.Different treatments are tried to enhance the properties of GFET devices, such astransconductance, mobility and a decrease of the Dirac point to lower voltages, thatincludes current annealing, vacuum annealing, hot plate annealing, ionized water bathand UV-ozone cleaning. Vacuum annealing and annealing on a hot plate affect thegated response; they might have decreased the overall p-doping, but also introducedDirac points and non-linear features. These are thought to be explained by localp-doping of the graphene under the electrodes. Thus the Dirac point of CVDgraphene is still at higher gate voltages. Finally, the charge carrier mobility decreasedin all treatments except current – and hot plate annealing, and it is also observed that charge carrier mobilities after fabrication are lower than the manufacturer estimatesfor raw graphene on SiO2/Si substrate.

The connection discovered by M. Sato, T. Miwa and M. Jimbo (SMJ) between the monodromy-preserving deformation theory of the two-dimensional Euclidean Dirac operator and quantum fields is rigorously established for the case of nonreal S¹ monodromy parameters. This connection involves the expression of the associated n-point functions in terms of solutions to deformation equations which arise as necessary conditions for the monodromy exhibited by a class of multivalued solutions of the Euclidean Dirac equation to be preserved under perturbations of branch points. Our approach utilizes recent results involving infinite-dimensional group representations. A lattice version of the n-point function is introduced as a section of a determinant bundle defined over an infinite dimensional Grassmannian. A trivialization for this bundle is singled out so that the corresponding n-point functions behave like Ising correlations in the massive scaling regime. Then the SMJ n-point functions are recovered as the scaled functions. A parallel scaling analysis is carried out with lattice analogues of the Euclidean Dirac wave functions which scale to square-integrable multivalued solutions of the Euclidean Dirac equation and the connection between the SMJ deformation theory and the n-point functions is rigorously established in terms of local Fourier expansion coefficients of these wave functions. These results are presented in detail for two-point functions with the same monodromy associated to each site.

This thesis presents a study of two-dimensional electron systems in GaAs-(Al,Ga)As heterojunctions and quasi-two-dimensional electron and hole systems in graphite within the quantum Hall effect regime of low temperature and high magnetic field. This thesis covers three main sets of experimental work as well as details of the experimental methods (chapter 2) used and the background theory behind the observed results (chapter 1). The first experimental results presented in this thesis in chapter 3 focus on contactless measurement of the equilibrium magnetisation of sample A2268, a ten layer multiple quantum well sample. Fitting the shape of dHvA oscillations at various temperatures to different models for the density of states, various properties of the system can be estimated, such as the shape of the disorder-broadened density of states and the presence of a background density of states between the Landau levels. Chapter 4 focuses on measurements of the decay of induced circulating currents in the quasi-dissipationless quantum Hall regime in two samples, V0049 and T73. The induced current is measured via contactless measurement of the associated magnetic moment. The magnitude of the induced current is found to be affected by the sweep rate of the magnetic field and also the distance of approach. The decay of the induced currents is observed at several temperatures and for different magnetic field sweep rates and distances of approach. Decays are observed for up to several days at time, far longer than previously possible. Information about the rate of decay can be used to build a picture of the decay mechanisms present in the quantum Hall regime. The presence of a power-law decay regime indicates many decay mechanisms contribute to the decay of a circulating current in the quasi-dissipationless quantum Hall regime. Chapter 5 focuses on both contactless magnetometry and transport experiments carried out on a graphite sample. The experiments aim to confirm or dispute recent claims of Dirac fermions in graphite. Experiments are carried out at temperatures in the range 30 mK to ~4 K and at two different angles to the applied magnetic field. Phase analysis of both Shubnikov de Haas and de Haas van Alphen oscillations is used to distinguish between normal and Dirac fermions. Observation of quantum Hall effect displays the presence of a half-integer quantum Hall staircase similar to that observed in graphene.

We consider two one-dimensional harmonic chains coupled along their length via linear springs. Casting the elastic wave equation for this system in a Dirac-like form reveals a directional representation. The elastic band structure, in a spectral representation, is constituted of two branches corresponding to symmetric and antisymmetric modes. In the directional representation, the antisymmetric states of the elastic waves possess a plane wave orbital part and a 4x1 spinor part. Two of the components of the spinor part of the wave function relate to the amplitude of the forward component of waves propagating in both chains. The other two components relate to the amplitude of the backward component of waves. The 4x1 spinorial state of the two coupled chains is supported by the tensor product Hilbert space of two identical subsystems composed of a non-interacting chain with linear springs coupled to a rigid substrate. The 4x1 spinor of the coupled system is shown to be in general not separable into the tensor product of the two 2x1 spinors of the uncoupled subsystems in the directional representation. (C) 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

In J.D. Jackson's Classical Electrodynamics textbook, the analysis of Dirac's charge quantization condition in the presence of a magnetic monopole has a mathematical omission and an all-too-brief physical argument that might mislead some students. This paper presents a detailed derivation of Jackson's main result, explains the significance of the missing term, and highlights the close connection between Jackson's findings and Dirac's original argument. (C) 2016 Sharif University of Technology. All rights reserved.

This thesis is devoted to quantum confinement effects in low-dimensional Dirac materials. We propose a variety of schemes in which massless Dirac fermions, which are notoriously diffcult to manipulate, can be trapped in a bound state. Primarily we appeal for the use of external electromagnetic fields. As a consequence of this endeavor, we find several interesting condensed matter analogues to effects from relativistic quantum mechanics, as well as entirely new effects and a possible novel state of matter. For example, in our study of the effective Coulomb interaction in one dimension, we demonstrate how atomic collapse may arise in carbon nanotubes or graphene nanoribbons, and describe the critical importance of the size of the band gap. Meanwhile, inspired by groundbreaking experiments investigating the effects of strain, we propose how to confine the elusive charge carriers in so-called velocity barriers, which arise due to a spatially inhomogeneous Fermi velocity triggered by a strained lattice. We also present a new and beautiful quasi-exactly solvable model of quantum mechanics, showing the possibilities for confinement in magnetic quantum dots are not as stringent as previously thought. We also reveal that Klein tunnelling is not as pernicious as widely believed, as we show bound states can arise from purely electrostatic means at the Dirac point energy. Finally, we show from an analytical solution to the quasi-relativistic two-body problem, how an exotic same-particle paring can occur and speculate on its implications if found in the laboratory.

Konforme Potenzen des Dirac Operators einer semi Riemannschen Spin-Mannigfaltigkeit werden untersucht. Wir präsentieren einen neuen Beweis, basierend auf dem Traktor Kalkül, für die Existenz von konformen ungeraden Potenzen des Dirac Operators auf semi Riemannschen Spin-Mannigfaltigkeiten. Desweiteren konstruieren wir eine neue Familie von konform kovarianten linearen Differentialoperatoren auf dem standard spin Traktor Bündel. Weiterhin verallgemeinern wir den Existenzbeweis für konforme ungerade Potenzen des Dirac Operators auf semi Riemannsche Spin-Mannigfaltigkeiten. Da die Existenzbeweise konstruktive sind, erhalten wir explizite Formeln für die konforme dritte und fünfte Potenz des Dirac Operators. Basierend auf den expliziten Formeln zeigen wir, dass die konforme dritte und fünfte Potenz des Dirac Operators formal selbstadjungiert (anti selbstadjungiert) bezüglich des L2-Skalarproduktes auf dem Spinorbündel ist. Abschliessend präsentieren wir neue Strukturen der konformen ersten, dritten und fünften Potenz des Dirac Operators: Es existieren lineare Differentialoperatoren auf dem Spinorbündel der Ordnung kleiner gleich eins, so dass die konforme erste, dritte und fünfte Potenz des Dirac Operators ein Polynom in jenen Operatoren ist. / Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin manifolds using the tractor machinery. We will also present a new family of conformally covariant linear differential operators on the standard spin tractor bundle. Furthermore, we generalize the known existence proof of conformal power of the Dirac operator on Riemannian spin manifolds to semi Riemannian spin manifolds. Both proofs concering the existence of conformal odd powers of the Dirac operator are constructive, hence we also derive an explicit formula for a conformal third- and fifth power of the Dirac operator. Due to explicit formulas, we show that the conformal third- and fifth power of the Dirac operator is formally self-adjoint (anti self-adjoint), with respect to the L2-scalar product on the spinor bundle. Finally, we present a new structure of the conformal first-, third- and fifth power of the Dirac operator: There exist linear differential operators on the spinor bundle of order less or equal one, such that the conformal first-, third- and fifth power of the Dirac operator is a polynomial in these operators.

The growing fields of spintronics and nanotechnology have created increased interest in developing the means to manipulate the spin of electrons. One such method arises from the combination of the spin-orbit interaction and the broken inversion symmetry that arises at surfaces and interfaces, and has prompted many recent investigations on metallic surfaces. A method by which surface states, in the absence of spin orbit effects, have been successfully investigated is the Green function embedding scheme of Inglesfield. This has been integrated into a self consistent FLAPW density functional framework based on the scalar relativistic K¨olling Harmon equation. Since the spin of the electron is a direct effect of special relativity, calculations involving the spin orbit interaction are best performed using solutions of the Dirac equation. This work describes the extension of Green’s function embedding to include the Dirac equation and how fully relativistic FLAPW surface electronic structure calculations are implemented. The general procedure used in performing a surface calculation in the scalar relativistic case is closely followed. A bulk transfer matrix is defined and used to generate the complex band structure and an embedding potential. This embedding potential is then used to produce a self consistent surface potential, leading to a Green’s function from which surface state dispersions and splittings are calculated. The bulk embedding potential can also be employed in defining channel functions and these provide a natural framework in which to explore transport properties. A relativistic version of a well known expression for the ballistic conductance across a device is derived in this context. Differences between the relativistic and nonrelativistic methods are discussed in detail. To test the validity of the scheme, a fully relativistic calculation of the extensively studied spin orbit split L-gap surface state on Au(111) is performed, which agrees well with experiment and previous calculations. Contributions to the splitting from different angular momentum channels are also provided. The main advantages of the relativistic embedding method are the full inclusion of the spin orbit interaction to all orders, the true semi infinite nature of the technique, allowing the full complex bands of the bulk crystal to be represented and the fact that a only small number of surface layers is needed in comparison to other existing methods.

We study theoretically graphene-like plasmonic metamaterials, in particular a honeycomb structured array of identical metallic nanoparticles, and examine the collective plasmonic modes that arise due to the near-field dipolar coupling between the localised surface plasmons of each individual nanoparticle. An analysis of the band structure of these eigenmodes reveals a phenomenal tunability granted by the polarisation of the dipole moments associated with the localised surface plasmons. As a function of the dipole orientation we uncover a rich phase diagram of gapped and gapless phases, where remarkably every gapless phase is characterised by the existence of collective plasmons that behave as massless chiral Dirac particles, in analogy to electrons in graphene. We consider lattices beyond the perfect honeycomb structure in two ways. Firstly, we break the inversion symmetry which leads to collective plasmons described as massive chiral modes with an energy dependent Berry phase. Secondly, we break the three-fold rotational symmetry and investigate generic bipartite lattices. In this scenario we progressively shift one sublattice away from the original honeycomb arrangement and observe a sequence of topological phase transitions in the phase diagram, as well as the merging and annihilation of Dirac points in the dispersions. After examining the purely plasmonic response we wish to address the true eigenmodes responsible for transporting electromagnetic radiation. For this reason we examine plasmon polaritons that arise from the strong light-matter coupling between the collective plasmons in a honeycomb array of metallic nanoparticles and the fundamental photonic mode of an enclosing cavity. Here we identify that the Dirac point remains robust and fixed in momentum space, irrespective of the light-matter coupling strength. Moreover, we demonstrate a qualitative modification of the polariton properties through modulation of the photonic environment, including order-of-magnitude renormalisation of the group velocity and the intriguing ability to invert the chirality of Dirac polaritons.

Orientador: Edmundo Capelas de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-09-25T13:12:35Z (GMT). No. of bitstreams: 1 Gomes_Denilson_D.pdf: 2902377 bytes, checksum: 93ed340e89d6a94739429631eed2a4d4 (MD5) Previous issue date: 2002 / Resumo: Neste trabalho são apresentadas e discutidas as chamadas equações de Klein-Gordon e Dirac generalizadas, associadas ao grupo de Fantappié-de Sitter - isometrias do espaço-tempo de Robertson- Walker. A equação de Klein-Gordon generalizada é obtida a partir do operador de Casimir de segunda ordem associada ao grupo de Fantappié-de Sitter. Por sua vez, a equação de Dirac generalizada é obtida fatorando o operador de Casimir de segunda ordem num produto de dois operadores de primeira ordem. A solução destas duas equações é obtida por separação de variáveis. Também é discuta a imersão do espaço-tempo de Robertson-Walker, desprovido de matéria e radiação, num espaço pseudo-euclidiano, tanto no caso de curvatura positiva como no caso de curvatura negativa. Apresentam-se ainda, os geradores da álgebra de Lie do grupo de Fantappié-de Sitter e seus respectivos operadores diferenciais / Abstract: We consider and discuss the so-called Klein-Gordon and Dirac generalized wave equations, related to Fantappié-de Sitter group - Robertson- Walker space-time isometries. The generalized Klein-Gordon wave equation is obtained by means of the second order Casimir invariant operator related to the Fantappié-de Sitter group. The generalized Dirac wave equation is obtained by writing the second order Casimir invariant operator as the product of two first order operators. The solution oí these equations is obtained by variable separation. We also discuss the Robertson- Walker space-time, without matter and radiation, embedded in a pseudo-euclidian space in both cases: positive and negative curvatures. We present the Lie algebra generator related to the Fantappié-de Sitter group and its differential operators / Doutorado / Doutor em Matemática Aplicada