Camadas de heteroestruturas hexagonais planas modeladas por grafos quânticos

Abstract

We started this work by reviewing an application of periodic quantum graph theory to model monolayer hexagonal materials with δ_a and δ_b parameters associated with the different types of atoms located in their vertices. We verified that materials of this nature have gaps in their spectral bands and express the size of this opening according to these parameters. In the Chapters 2 and 3, extend this modeling to equal bilayers, stacked in type AA and AA′, and in the Chapters 4, we studied heterostructures with two mixed layers and the “sandwich” hexagonal boron graphene-nitride: a single graphene sheet between two layers of hBN, and a single hBN sheet between two layers of graphene. In each of these configurations, we use the Schrödinger operator with its respective boundary conditions and we introduced a weak t_0 interaction parameter between the connections of different layers. We analyzed initially the dispersion relationship obtained in these models regarding the existence of conical or parabolic touches and confirmed, in rigorous models, known results in the physical literature, namely: hBN bilayers do not have Dirac cones, but, in AA stacking we identified the presence of parabolic touches. In the case of mixed bilayers, our study allows us to conclude that the inclusion of an hBN layer over a graphene layer can induce a gap in the graphene sheet and we express the width of this gap according to the parameters t_0 e δ_a. In the study of “sandwiches”, hBN-graphene-hBN and graphene-hBN-graphene, for certain particular values of the parameters, we found that the inclusion of a single graphene sheet between two sheets of hBN does not eliminate the gap of the hBN, but induces a reduction in the width of the spectral gap in an order of magnitude; by on the other hand, in the case graphene-hBN-graphene, the graphene cone at the origin prevails in this sandwich, but it also caused gaps in the other Dirac cones of the graphene. Such results can be justified by the fact that, in these heterostructures, carbon atoms have interacted with other inequivalent hBN, nitrogen, and boron atoms, causing a reduction or increase of the gaps. Finally, in the last chapter, we consider hexagonal quantum graphs and we adapt our proposal to include a magnetic field in the hBN sheet. We demonstrate that if the magnetic flux is constant in the hexagonal network and is a rational multiple of 2π, then there will be values of thisflux such that, for certain boundary conditions at the vertices (modeling the hBN), the conical touches in the operator scattering relation will cease to exist and we guarantee the existence of gaps.