Uma revisão sobre a diagonalização do Grafeno e os pontos de Dirac: formalismo de segunda quantização no modelo Tight-Binding

Abstract

Graphene, a single atomic layer of carbon, stands out for its exceptional electronic, mechanical, and thermal properties, making it promising for various technological appli- cations. To understand these properties, it is essential to analyze its crystal structure and associated mathematical aspects. Graphene can be described as two interpenetrating triangular Bravais sublattices, with its reciprocal lattice linked to wave vectors k that exhibit the periodicity of the structure. Particularly, the K and K points, known as Dirac points, are of great importance due to the unique electronic phenomena occurring in these re- gions. The second quantization formalism, a crucial tool in quantum physics, is applied to the tight-binding model to describe the interactions between carbon atoms and diago- nalize the system, facilitating the understanding of graphene’s energy bands. This work reviews the diagonalization process of graphene, exploring the second quantization for- malism within the tight-binding model and providing a detailed analysis of Dirac points. The main objective was to show the structure of graphene and the mathematical part about the allowed energies and the Dirac points. In this context, the present work aims to serve as a guide for those who wish to begin their studies on graphene.