Low-Energy Electronic Properties of Graphene
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- Institutt for fysikk 
We investigate the electronic structure of two-dimensional (2D) mate- rials with and without spin-orbit coupling. Initially, we use the tight- binding model to derive the electronic structure of graphene with stag- gered sublattice potential. Subsequently, we introduce group theory as a powerful tool to understand how symmetries of different points in the Brillouin zone affect the electronic band structure of graphene. Additionally, the merging of the bands at different points is interpreted in the framework of the group theory. We present the symmetry classification of the electron energy bands in gapless graphene, and we derive the low-energy Hamiltonian of gapless graphene by means of group theory arguments near the edges of the Brillouin zone. Then we introduce the spin-orbit coupling, and investigate the effect of this interaction on the band structure of graphene. By introducing the double groups, we calculate the reduction in symmetries and the en- ergy splittings when intrinsic spin-orbit effects are taken into account. Finally, we derive an effective Hamiltonian of graphene in presence of spin-orbit interaction.