Introdução à teoria de grupos e aplicação no grafeno

Abstract

Group theory is a very important mathematical tool in materials physics. The strength and beauty of this method lie in its simple algebraic structure, capable of summarizing a series of complex symmetry operations. It provides eigenvalues and selection rules for Hamiltonians, helping in the description of electronic energy levels in the occupied and unoccupied states. When one studies the electronic properties of materials one deals with complex calculations as well as interpretations of experimental results. The point groups deal with the symmetries of the crystalline lattices, and from these symmetry operations it is possible to simplify the resolution of the Schrödinger equation. Graphene is only a monolayer of the graphite-forming structure containing only two carbon atoms per unit cell. To understand the character table of graphene, one needs to present the basic elements of group theory. We will present an introduction to the reader of elementary concepts, which will be necessary for a better understanding of the symmetry group. In this work we also aim to make the concepts of group theory clearer, thus easier to comprehend the graphene character table. Through group theory, it is possible to find operations that leave the structure under study invariant. Since graphene is a material that has lattice symmetry, we can classify its unit cell into one of the groups. And from the operations of symmetry, it is possible to find the Hamiltonian, through different methods. We will present a simple calculation using the Hückel method to obtain an approximation for the graphene dispersion relation and to better understand its physical properties.