Uso de geometria no estudo de Nanocones duplos de carbono e mantos de invisibilidade

Abstract

In this thesis we study the influence of the geometry on the electronic properties of double carbon nanocones. We also propose an invisibility cloak detecting strategy by exploring the gometry of the cloak built using the theory of transformation optics. For the study of double carbon nanocones, we use the continuous model based on a Dirac Hamiltonian for massless fermions where topological defects are described through non-abelian gauge fields. For this we demonstrate how the Dirac equation can be obtained from the coupling of spin to the relativistic analogue of kinetic energy and demonstrate how the electronic properties of graphene can be modeled by an effective Dirac equation for massless fermions. We then develop the geometric approach that describes two cocnes in a single structure, by extending the radial coordinate to the whole set of real numbers. For a better understanding of the characteristics of this type of surface, we demonstrate the dynamics of a particle on this surface in classical, quantum and quantum-relativistic regimes. We then solve the effective Dirac equation for a free particle on double carbon nanocone surfaces. We show thet for some combinations of different nanocones, the local density of states near the apex of the cones is not zero in the Fermin energy and presents a strong dependence on angular momentum. We also get Landau levels for charged particles under the influence of an azimuthal magnetic field and do a detailed analysis of the energy spectrum considering the combinations of quantum numbers. We show how the magnetic field breaks the "summetry" between the cones by introducing states of different classes in each cone. The geometry of surface is also explored in the study of metamterials through the theory of transformation optics. We show how it is possible to manipulate in the laboratory the electric permittivity (e) and the magnetic permeability (u) of a dielectric and, in this way, obtain control over the refractive index of the medium. This approach allows us to model the geometry that reproduces a desired effect and determine the parameters of the dielectric that reproduces the desired geometry. More specifically, we explore how to construct a material capable of completely camouflaging an object within it, the so-called invisibility cloak. At the same time, we show how the invisibility cloak can be built, we propose a strategy to detect the presence of this cloak through the difference of phase generated by the curvature resulting from its building.