Nanoestruturas de grafeno e o problema do confinamento de partículas de Dirac na descrição do contínuo - Souza, José Fernando Oliveira de (2014)
In this work, we investigate in parallel physical and mathematical aspects inherent to the problem of confinement of massless Dirac fermions in graphene nanostructures. In a low energy approach, we propose models to describe confining systems in graphene and study how the choice of boundary conditions of the problem - or, equivalently, of domains of the Dirac operator - affects the physical properties of such systems. In this scenario, we concentrate essentially on the study of the physical behavior of graphene nanorings and nanoribbons in response to aspects such as topology, edge and interface geometry and interactions with external fields. At the same time, a rigorous investigation concerning formal aspects of the problem and the way that they manifest themselves physically is also performed. In light of the theory of linear operators on Hilbert spaces, we analyze the role played by the notion of self-adjointness in the problem and establish sets of boundary conditions physically acceptable in graphene, which mathematically corresponds to the definition of self-adjoint extensions of the Dirac Hamiltonian from the continuum description. Sets proposed in the treatment of some studied configurations are approached in this context. In addition, we present a particular study in which we examine the influence of topological defects on the physics of massive fermions in graphene in the presence of Coulomb and uniform magnetic fields.