Propriedades eletrônicas da matéria topológica: heteroestruturas e efeitos da rotação - Lima, Jonas Romero Fonseca de (2014)
In this thesis we study the electronic properties of several systems of condensed matter physics using two different continuum models, the effective mass theory and the effective Dirac Hamiltonian. In several systems, there is an effective mass depending on position. Some models for the kinetic energy operator were proposed to describe these systems, but there is no definition of which one is the most appropriate. It is one of the oldest open questions in solid state physics. We propose a new model, where we consider all permutations among the operators and show that it satisfies the fundamental requirements of quantum mechanics. We use this model to obtain the minibands structure of a heterostructure composed by two different materials and compare our model with other models previously proposed. We also get the Schrödinger equation for a particle constrained to a curved surface with position dependent mass. We follow the da Costa approach, where there is a geometric potential. We show that the position dependent mass does not affect the geometric potential, contributing only to the kinetic part. We use this equation to study the electronic transport in a junction of two cylinders with different radii, with the effective mass varying with the cylinder radius. Using the effective Dirac Hamiltonian, we consider a graphene sheet on a periodic substrate heterostructure composed by two different materials. Each material induces a specific energy gap and Fermi velocity in the graphene, so the Dirac Hamiltonian has a gap (mass) term and a Fermi velocity depending on position. We write this operator taking into account that it has to be Hermitian and we obtain the minigaps induced by the substrate in the electronic structure of graphene. Motivated by experimental results, we study the effects of rotation on the electronic structure of carbon nanotubes, fullerene C60 and topological insulators, using an effective Dirac operator. In the carbon nanotube and C60 cases, the rotation adds a shift in the energy levels and a break in spin degeneracy. In the topological insulator case, the rotation adds only a shift in the energy.