Abstract
This thesis discusses correlation effects in graphene and bilayer graphene. The discovery of graphene was awarded 2010 Nobel Prize in Physics. Graphene is one of the most intriguing topics around the world. Its flexibilities make it a very promising material in device physics. From theoretical point of view, graphene connects condensed matter physics to quantum field theory, and is an excellent candidate for model studies. Furthermore, it stimulates researches in low-dimensional electron systems. Bilayer grapheme is an interesting variants of graphene, and is one of the new directions in developing low-dimensional systems.
Due to honeycomb lattice symmetry, the low-energy effective Hamiltonian of a graphene is described by gapless Dirac equation σ•p around K(K’) point. In this thesis, symmetry of Dirac equation is reviewed. In graphene, there are four copies of gapless Dirac equations. In addition, spin-orbit couplings are also discussed by using point-group techniques. Some typical models on honeycomb lattice are reviewed, including Haldane model and Kane-Mele model. We investigate the attractive-U Kane-Mele-Hubbard model by using a mean-field theory, and find strong superconducting correlations along the edge, analogous to edge magnetism in positive U case. We investigate mesoscopic spin Hall effect on the surface of a three-dimensional topological insulator using McMillan Green's function techniques, and discuss the robustness of edge states and stabilities against interactions in topological insulator.
Bilayer graphene is also investigated. Our study follows the recent experiments and theoretical proposals. As suggested by previous works, quantum spin Hall state and layer antiferromagnetic state are two most possible candidates of the ground state. We propose by tiny doping, a half-metallic state can be realized based on layer antiferromagnetic state. The responses to in-plane and perpendicular magnetic fields are also reviewed.