Theoretical And Numerical Studies Of Two-Dimensional Graphene Affected By An In-Plane Magnetic Field

This thesis deals with theoretical and numerical study of a two-dimensional graphene lattice. In our study we mainly focused on the derivation of graphene band structure in the presence of an applied in-plane magnetic field. We briefly present an overview of various carbon allotropes, to which graphene belongs, and then introduce graphene with a discussion of its geometry and electrical properties.We also introduce the nearest-neighbor tight binding model that we used in the calculation of the band structure of graphene as well as presenting an overview of the basic concepts that describe electron states in solids. Here we deal with how the electron wave function changes under the action of the operator of translation from a lattice cell to the next one which is vital in the determination of electron energy eigenvalues in graphene. We also discuss the tight-binding (TB) technique, also known as the Linear Combination of Atomic Orbitals (LCAO) method, for the study of lattice structures and then discuss the calculation of the band structure of graphene sheet and the band structures of graphene nanoribbons in the absence of a magnetic field using the nearest-neighbor tight-binding (TB) technique. When we compared the band structure of an infinite graphene sheet to that of the ribbon's first Brillouin zone we observed that he most striking difference between them affects the highest occupied and the lowest unoccupied band.We also proceeded on to the investigation of the effect of an in-plane magnetic field applied to graphene by carrying out a detailed derivation of the hopping integrals in the presence of the magnetic field. These hopping integrals were then used in the calculation of the electron energy eigenvalues. We considered a magnetic field applied in the x-direction inside the plane of the graphene lattice and investigated how varying this magnetic field affects the hopping parameters between the carbon atoms that make up the lattice. Our results showed that the value of hopping parameter t₁, along the c-c bond perpendicular to thedirection of the applied magnetic field, increase more rapidly than the other two hopping parameters t₂ and t₃ along the other two c-c bonds between a carbon atom and its three nearestneighbors. We also found that the values of hopping parameters t₂ and t₃ for a given value of the magnetic field are the same. We also observed the opening of a band gap in the band structure of graphene at the Dirac points due to the in-plane magnetic field. An analysis of the band gap for various values of the magnetic field showed an increase in the band gap with increase in the magnetic field. We could therefore deduce that an applied in-plane magnetic field can be used to tune the band structure of graphene.