Abstract

The graphene is a substance with carbon atoms arranged in a honeycomb lattice. The dynamics of the electrons in the structure is governed by the Hamilton equations of the system in the form of its associated spectral problem: HΨ = λΨ, with the additional condition that the eigenfunction Ψ must satisfy the so-called Kirchhoff’s conditions. In this paper, we study a class of solutions (λ,Ψ) that, in addition to meeting these conditions, are periodic in one of the two main directions of the lattice, and satisfy a pseudo-periodicity type like condition in the other direction. Our main results lead to an adequate characterization of the dispersion relationships of the honeycomb lattice, providing a precise description of the regions of stability and instability of the eigenfunctions in terms of λ. As a consequence, a tool is thus obtained for a better understanding of the propagation properties and the behavior of the wave function of electrons in a hexagonal lattice, a key issue in graphene-based technologies.