The Landau level spectrum of graphene superlattices is studied using a tight-binding approach. We consider non-interacting particles moving on a hexagonal lattice with an additional one-dimensional superlattice made up of periodic square potential ba
rriers, which are oriented along the zig-zag or along the arm-chair directions of graphene. In the presence of a perpendicular magnetic field, such systems can be described by a set of one-dimensional tight-binding equations, the Harper equations. The qualitative behavior of the energy spectrum with respect to the strength of the superlattice potential depends on the relation between the superlattice period and the magnetic length. When the potential barriers are oriented along the arm-chair direction of graphene, we find for strong magnetic fields that the zeroth Landau level of graphene splits into two well separated sublevels, if the width of the barriers is smaller than the magnetic length. In this situation, which persists even in the presence of disorder, a plateau with zero Hall conductivity can be observed around the Dirac point. This Landau level splitting is a true lattice effect that cannot be obtained from the generally used continuum Dirac-fermion model.
The electronic states of an electrostatically confined cylindrical graphene quantum dot and the electric transport through this device are studied theoretically within the continuum Dirac-equation approximation and compared with numerical results obt
ained from a tight-binding lattice description. A spectral gap, which may originate from strain effects, additional adsorbed atoms or substrate-induced sublattice-symmetry breaking, allows for bound and scattering states. As long as the diameter of the dot is much larger than the lattice constant, the results of the continuum and the lattice model are in very good agreement. We also investigate the influence of a sloping dot-potential step, of on-site disorder along the sample edges, of uncorrelated short-range disorder potentials in the bulk, and of random magnetic-fluxes that mimic ripple-disorder. The quantum dots spectral and transport properties depend crucially on the specific type of disorder. In general, the peaks in the density of bound states are broadened but remain sharp only in the case of edge disorder.
The electronic properties of graphene zig-zag nanoribbons with electrostatic potentials along the edges are investigated. Using the Dirac-fermion approach, we calculate the energy spectrum of an infinitely long nanoribbon of finite width $w$, termina
ted by Dirichlet boundary conditions in the transverse direction. We show that a structured external potential that acts within the edge regions of the ribbon, can induce a spectral gap and thus switches the nanoribbon from metallic to insulating behavior. The basic mechanism of this effect is the selective influence of the external potentials on the spinorial wavefunctions that are topological in nature and localized along the boundary of the graphene nanoribbon. Within this single particle description, the maximal obtainable energy gap is $E_{rm max}propto pihbar v_{rm F}/w$, i.e., $approx 0.12$,eV for $w=$15,nm. The stability of the spectral gap against edge disorder and the effect of disorder on the two-terminal conductance is studied numerically within a tight-binding lattice model. We find that the energy gap persists as long as the applied external effective potential is larger than $simeq 0.55times W$, where $W$ is a measure of the disorder strength. We argue that there is a transport gap due to localization effects even in the absence of a spectral gap.
We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with coupling J in one lattice direction and couplings J along the other two directions. For J/J > 1, th
is model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J < 1/2 and a large manifold of canted spin states for J/J > 1/2. To include quantum effects self-consistently, we investigate the Sp(N) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter kappa that measures the importance of quantum effects. Our numerical calculations reveal that in the kappa-J/J plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J >> 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.
In the search for spin-1/2 kagome antiferromagnets, the mineral volborthite has recently been the subject of experimental studies [Hiroi et al.,2001]. It has been suggested that the magnetic properties of this material are described by a spin-1/2 Hei
senberg model on the kagome lattice with spatially anisotropic exchange couplings. We report on investigations of the Sp(N) symmetric generalisation of this model in the large N limit. We obtain a detailed description of the dependence of possible ground states on the anisotropy and on the spin length S. A fairly rich phase diagram with a ferrimagnetic phase, incommensurate phases with and without long range order and a decoupled chain phase emerges.
