H. A. Fertig

We study the electronic states of narrow graphene ribbons (``nanoribbons'') with zigzag and armchair edges. The finite width of these systems breaks the spectrum into an infinite set of bands, which we demonstrate can be quantitatively understood using the Dirac equation with appropriate boundary conditions. For the zigzag nanoribbon we demonstrate that the boundary condition allows a particlelike and a holelike band with evanescent wave functions confined to the surfaces, which continuously turn into the well-known zero energy surface states as the width gets large. For armchair edges, we show that the boundary condition leads to admixing of valley states, and the band structure is metallic when the width of the sample in lattice constant units has the form $3M+1$, with $M$ an integer, and insulating otherwise. A comparison of the wave functions and energies from tight-binding calculations and solutions of the Dirac equations yields quantitative agreement for all but the narrowest ribbons.

Graphene subject to a spatially uniform, circularly polarized electric field supports a Floquet spectrum with properties akin to those of a topological insulator. The transport properties of this system, however, are complicated by the nonequilibrium occupations of the Floquet states. We address this by considering transport in a two-terminal ribbon geometry for which the leads have well-defined chemical potentials, with an irradiated central scattering region. We demonstrate the presence of edge states, which for infinite mass boundary conditions may be associated with only one of the two valleys. At low frequencies, the bulk dc conductivity near zero energy is shown to be dominated by a series of states with very narrow anticrossings, leading to superdiffusive behavior. For very long ribbons, a ballistic regime emerges in which edge state transport dominates.

We investigate the excitation spectrum of two- and three-layer electron systems in a strong perpendicular magnetic field with \ensuremath{\nu}=(1/2 and (1/3, respectively, in each layer. For layer separation z=0 the dispersion relations \ensuremath{\omega}(k) vanish as ${k}^{2}$ for k\ensuremath{\rightarrow}0, as one expects for Goldstone modes. For z>0, \ensuremath{\omega}(k) behaves as an acoustic mode, vanishing linearly for small k. For large values of k one finds that the dispersion relations have the form \ensuremath{\Delta}(z)-${e}^{2}$/\ensuremath{\kappa}${\mathrm{kl}}_{0}^{2}$, where ${l}_{0}$ is the magnetic length and \ensuremath{\kappa} the dielectric constant of the medium. At ${\mathrm{kl}}_{0}$ of order unity, the dispersion relations develop a dip as z is increased. These become soft modes at certain critical values of z, indicating that the system undergoes a phase transition as the layer spacing is increased.

We study edges states of graphene ribbons in the quantized Hall regime, and show that they can be described within a continuum model (the Dirac equation) when appropriate boundary conditions are adopted. The two simplest terminations, zigzag and armchair edges, are studied in detail. For zigzag edges, we find that the lowest-Landau-level states terminate in two types of edge states, dispersionless and current-carrying surface states. The latter involve components on different sublattices that may be separated by distances far greater than the magnetic length. For armchair edges, the boundary conditions are met by admixing states from different valleys, and we show that this leads to a single set of edges states for the lowest Landau level and two sets for all higher Landau levels. In both cases, the resulting Hall conductance step for the lowest Landau level is half that between higher Landau levels, as observed in experiment.

We study RKKY interactions between local magnetic moments for both doped and undoped graphene. In the former case interactions for moments located on definite sublattices fall off as 1/R2, whereas for those placed at interstitial sites they decay as 1/R3. The interactions are primarily (anti)ferromagnetic for moments on (opposite) equivalent sublattices, suggesting that at low temperature dilute magnetic moments embedded in graphene can order into a state analogous to that of a dilute antiferromagnet. In the undoped case we find no net magnetic moment in the ground state, and demonstrate numerically this effect for ribbons, suggesting the possibility of an unusual spin-transfer device.