One-dimensional electron systems on graphene edges
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In this dissertation several aspects on one-dimensional edge states in grapheme are studied. First, a background in the history and development of graphitic forms is presented. Then some novel features found in two-dimensional bulk graphene are presented. Here, some focus is given to the chiral nature of the Dirac equation and the symmetries found in the grahene. Magnetism and interactions in graphene is also briefly discussed. Finally, the graphene nanoribbon with its two typical edges: armchair and zigzag is introduced. Gaps due to finite-size effects are studied. Next, the problem of determining the zigzag ground state is presented. Later, we develop this in an attempt to add the Coulomb interaction to the zigzag flat-band states. These nanoribbons can be stimulated with a tight-binding code on a lattice model in which many different effects can be added, including an A/B sublattice asymmetry, spin-orbit coupling and external fields. The lowest Landau level solutions in the different ribbon orientations is of particular current interest. This is done in the context of understanding new physics and developing novel applications of graphene nanoribbon devices. Adding spin-orbit to a graphene ribbon Hamiltonian leads to current carrying electronic states localized on the sample edges. These states can appear on both zigzag and armchair edges in the semi-finite limit and differ qualitatively in dispersion and spin-polarization from the well known zigzag edge states that occur in models that do not include spin-orbit coupling. We investigate the properties of these states both analytically and numerically using lattice and continuum models with intrinsic and Rashba spin-orbit coupling and spin-independent gap producing terms. A brief discussion of the Berry curvature and topological numbers of graphene with spin-orbit coupling also follows.