Eliminando o problema de duplicamento de férmions em nanofitas de grafeno: modelos e condições de contorno via teoria de grupos

Abstract

In 2004, N. Novoselov and A. Geim (Nobel 2010) have isolated a single layer of graphene on a substrate. Thus yielding novel research fields in two-dimensional materials, and materials that are characterized by Dirac cones in their electronic structure. The lateral confinement of a monolayer introduces the graphene nanoribbons, which can be of the type zigzag or armchair, whose names refer to the shape of the atomic arrangement at their borders.

In this dissertation, we investigate the electronic structure and boundary conditions of graphene nanoribbons under novel perspectives. The effective model of graphene is given by a Dirac Hamiltonian, linear in momentum k, which introduces subtleties. First, the hard wall confinement is not given by trivial boundary conditions (ψ = 0 at the Edges). Instead, one usually applies the Brey and Fertig (BF) boundary conditions, which requires an analysis of the atomistic terminations of each boundary. Second, whenever numerical simulations are necessary, the discretization of the operator k = −i∂x, via the finite differences method, yield the Fermion doubling problem for k-linear Hamiltonians, which introduces spurious numerical solutions near the Fermi energy. Moreover, the BF conditions are not compatible with the finite differences method. Here, in this dissertation, we revisit these issues, which are seemingly unrelated, to find out that they are closely related, allowing us to propose an unique solution.

Our results are based on the demonstration that the BF boundary conditions are equiv- alent to those introduced by McCann and Fal’ko (MF), which are valid for all k-linear Hamiltonians. We will show that the MF conditions can be established using group theory, being defined by the broken symmetries caused by the confinement. The group theory analysis also allow us to show that the k-quadratic corrections of the Hamiltonian are given by the same matrices that define the MF conditions. We show here that these quadratic corrections, known as Wilson’s mass, not only solves the Fermion doubling problem, but also modify the boundary conditions, allowing for the trivial ψ = 0 condition at the edges.

Our proposal for a new approach for Hamiltonians dominated by k-linear terms is not yet established in the literature. However, it is of broad interest, since our results are valid beyond graphene, and can be applied to all k-linear models, like topological insulators, Weyl fermions, etc.