Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior (original) (raw)
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Physical Review B, 2009
We investigate the way that the degenerate manifold of midgap edge states in quasicircular graphene quantum dots with zigzag boundaries supports, under free-magnetic-field conditions, strongly correlated many-body behavior analogous to the fractional quantum Hall effect (FQHE), familiar from the case of semiconductor heterostructures in high magnetic fields. Systematic exactdiagonalization (EXD) numerical studies are presented for the first time for 5 ≤ N ≤ 8 fully spin-polarized electrons and for total angular momenta in the range of N (N − 1)/2 ≤ L ≤ 150. We present a derivation of a rotating-electron-molecule (REM) type wave function based on the methodology introduced earlier [C. Yannouleas and U. Landman, Phys. Rev. B 66, 115315 (2002)] in the context of the FQHE in two-dimensional semiconductor quantum dots. The EXD wave functions are compared with FQHE trial functions of the Laughlin and the derived REM types. It is found that a variational extension of the REM offers a better description for all fractional fillings compared with that of the Laughlin functions (including total energies and overlaps), a fact that reflects the strong azimuthal localization of the edge electrons. In contrast with the multiring arrangements of electrons in circular semiconductor quantum dots, the graphene REMs exhibit in all instances a single (0, N) polygonal-ring molecular (crystalline) structure, with all the electrons localized on the edge. Disruptions in the zigzag boundary condition along the circular edge act effectively as impurities that pin the electron molecule, yielding single-particle densities with broken rotational symmetry that portray directly the azimuthal localization of the edge electrons.
Patterns of the Aharonov-Bohm oscillations in graphene nanorings
Using extensive tight-binding calculations, we investigate (including the spin) the Aharonov-Bohm (AB) effect in monolayer and bilayer trigonal and hexagonal graphene rings with zigzag boundary conditions. Unlike the previous literature, we demonstrate the universality of integer (hc/e) and half-integer (hc/2e) values for the period of the AB oscillations as a function of the magnetic flux, in consonance with the case of mesoscopic metal rings. Odd-even (in the number of Dirac electrons, N ) sawtooth-type patterns relating to the halving of the period have also been found; they are more numerous for a monolayer hexagonal ring, compared to the cases of a trigonal and a bilayer hexagonal ring. Additional more complicated patterns are also present, depending on the shape of the graphene ring. Overall, the AB patterns repeat themselves as a function of N with periods proportional to the number of the sides of the rings.
Electronic structure of triangular, hexagonal and round graphene flakes near the Fermi level
New Journal of Physics, 2008
The electronic shell structure of triangular, hexagonal and round graphene quantum dots (flakes) near the Fermi level has been studied using a tight-binding method. The results show that close to the Fermi level the shell structure of a triangular flake is that of free massless particles, and that triangles with an armchair edge show an additional sequence of levels ("ghost states"). These levels result from the graphene band structure and the plane wave solution of the wave equation, and they are absent for triangles with an zigzag edge. All zigzag triangles exhibit a prominent edge state at F , and few low-energy conduction electron states occur both in triangular and hexagonal flakes due to symmetry reasons. Armchair triangles can be used as building blocks for other types of flakes that support the ghost states. Edge roughness has only a small effect on the level structure of the triangular flakes, but the effect is considerably enhanced in the other types of flakes. In round flakes, the states near the Fermi level depend strongly on the flake radius, and they are always localized on the zigzag parts of the edge.
Electronic and magnetic excitations in graphene and magnetic nano-ribbons
The discovery of graphene-a 2D material with superior physical properties-in 2004 was important for the intensive global research to find alternatives to three-dimensional (3D) semiconductor materials in industry. At the same time there have been exciting advances for 2D magnetic materials on the nanometer scale. The superior properties of graphene are mainly attributed to its crystal structure and its relatively short-range interactions. These properties show that natural and artificial 2D materials are promising for new applications. In this thesis we have carried out a comprehensive investigation of the effects of the 2D lattice structures, the roles of nearest neighbor (NN) and next nearest neighbor (NNN) interactions and the formation of coupled bilayer systems in both the electronic and the magnetic geometries (chosen specifically to be nano-ribbons or stripes). In the case of honeycomb lattices (which occur in graphene and can be produced artificially by growing nanodot arrays for ferromagnetic structures) the effects of different edges of the zigzag and armchair types are studied with emphasis on the localized modes that may occur. Impurity sites in the form of one or more lines of impurities introduced substitutionally are considered from the perspectives that they give additional localized mode effects and they change the spatial quantization (for example, as studied via the density of states for the modes). The theoretical methods employed throughout the thesis are based on the second quantization forms of both the tightbinding Hamiltonian for electronic excitations and the Heisenberg exchange Hamiltonian for the ferromagnetic excitations (or spin waves). The translational symmetry along the length of the ribbon or stripe is utilized to make a wave-vector Fourier transform in this longitudinal direction, while the finite number of rows in the transverse direction are treated within a matrix formulation.
Physical Review B, 2011
We study the energy spectrum and electronic properties of graphene in a periodic magnetic field of zero average with a symmetry of triangular lattice. The periodic field leads to formation of a set of minibands separated by gaps, which can be manipulated by external field. The Berry phase, related to the motion of electrons in k space, and the corresponding Chern numbers characterizing topology of the energy bands are calculated analytically and numerically. In this connection, we discuss the anomalous Hall effect in the insulating state, when the Fermi level is located in the minigap. The results of calculations show that in the model of gapless Dirac spectrum of graphene the anomalous Hall effect can be treated as a sum of fractional quantum numbers, related to the nonequivalent Dirac points.
Physical Review B, 2009
By combining analytic and numerical methods, edge states on a finite width graphene ribbon in a magnetic field are studied in the framework of low-energy effective theory that takes into account the possibility of quantum Hall ferromagnetism (QHF) gaps and dynamically generated Dirac-like masses. The analysis is done for graphene ribbons with both zigzag and armchair edges. The characteristic features of the spectrum of the edge states in both these cases are described. In particular, the conditions for the existence of the gapless edge states are established. Implications of these results for the interpretation of recent experiments are discussed.
Physical Review B, 2018
We study narrow zigzag graphene nanoribbons (ZGNRs), employing density functional theory (DFT) simulations and the tight-binding (TB) method. The main result of these calculations is the braiding of the conduction and valence bands, generating Dirac cones for non-commensurate wave vectors k. Employing a TB Hamiltonian, we show that the braiding is generated by the thirdneighbor hopping (N3). We calculate the band structure, the density of states and the conductance, new conductance channels are opened, and the conductance at the Fermi energy assumes integer multiples of the quantum conductance unit Go = 2e 2 /h. We also investigate the satisfaction of the Stoner criterion by these ZGNRs. We calculate the magnetic properties of the fundamental state employing LSDA (spin-unrestricted DFT) and we confirm that ZGNRs with N = (2, 3) do not satisfy the Stoner criterion and as such the magnetic order could not be developed at their edges. These results are confirmed by both tight-binding and LSDA calculations.
Electronic spectra, topological states, and impurity effects in graphene nanoribbons
Low Temperature Physics
We consider the finite ribbons of graphene with two principal orientations, zigzag and armchair, of their edges to study in detail impurity effects on their edge states. An alternative to the known description of quasiparticle states in terms of transversal standing waves is proposed in the recurrence relations for their spectra vs discrete numbers of atomic chains in the ribbon, permitting to simplify the Green function approach to the disorder effects in these systems. The derived analysis shows the microscopic mechanisms of perturbation by different types of impurities on low energy states and clarifies how the stability of topological states in zigzag systems to disorder is related to the discrete amplitudes of these states across the ribbon. An opposite possibility for Mott localization under local impurity perturbations is found for armchair type nanoribbons but at special values of their width.
Revealing Hofstadter spectrum for graphene in a periodic potential
Physical Review B, 2014
We calculate the energy bands for graphene monolayers when electrons move through a periodic electrostatic potential in the presence of a uniform perpendicular magnetic field. We clearly demonstrate the quantum fractal nature of the energy bands at reasonably low magnetic fields. We present results for the energy bands as functions of both wave number and magnetic flux through the unit cells of the resulting moiré superlattice. The effects due to pseudo-spin coupling and Landau orbit mixing by a strong scattering potential have been exhibited. At low magnetic fields when the Landau orbits are much larger than the period of the modulation, the Landau levels are only slightly broadened. This feature is also observed at extremely high magnetic fields. The density of states has been calculated and shows a remarkable self-similarity like the energy bands. We estimate that for modulation period of 10 nm the region where the Hofstadter butterfly is revealed at B ≤ 2 T .
Energy Spectrum and Quantum Hall Effect in Twisted Bilayer Graphene
Arxiv preprint arXiv:1202.4365, 2012
We investigate the electronic structure and the quantum Hall effect in twisted bilayer graphenes with various rotation angles in the presence of magnetic field. Using a low-energy approximation, which incorporates the rigorous interlayer interaction, we computed the energy spectrum and the quantized Hall conductivity in a wide range of magnetic field from the semi-classical regime to the fractal spectrum regime. In weak magnetic fields, the low-energy conduction band is quantized into electronlike and holelike Landau levels at energies below and above the van Hove singularity, respectively, and the Hall conductivity sharply drops from positive to negative when the Fermi energy goes through the transition point. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadter's butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy. The typical electron density and magnetic field amplitude characterizing the spectrum monotonically decrease as the rotation angle is reduced, indicating that the rich electronic structure may be observed in a moderate condition.