Browsing by Author "Conca, C."
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Publication A class of solutions for the graphene hamiltonian operator(2022) Conca, C.; San Martín, J.; Solano, VivianaThe 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.Item A mathematical basis for the graphene(2020) Conca, C.; San Martín, J.; Solano, VivianaWe present a new basis of representation for the graphene honeycomb structure that facilitates the solution of the eigenvalue problem by reducing it to one dimension. We define spaces in these geometrical basis that allow us to solve the Hamiltonian in the edges of the lattice. We conclude that it is enough to analyze a one-dimensional problem in a set of coupled ordinary second-order differential equations to obtain the behavior of the solutions in the whole graphene structureItem On the graphene Hamiltonian operator(2020) Conca, C.; Orive, R.; San Martín, J.; Solano, VivianaWe solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805–826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.