Effect of edge vacancies on localized states in a semi-infinite zigzag graphene sheet

The effect of vacancies on the robustness of zero-energy edge electronic states in zigzag-type graphene layer is studied at different concentrations and distributions of defects. All calculations are performed by using the Green’s function method and the tight-binding approximation. It is found that the arrangement of defects plays a crucial role in the destruction of the edge states. We have specified a critical distance between edge vacancies when their mutual influence becomes significant and affects markedly the density of electronic states at graphene edge.     

Spin-filtered edge states in graphene

Spin–orbit coupling changes graphene, in principle, into a two-dimensional topological insulator, also known as quantum spin Hall insulator. One of the expected consequences is the existence of spin-filtered edge states that carry dissipationless spin currents and undergo no backscattering in the presence of non-magnetic disorder, leading to quantization of conductance. Whereas, due to the small size of spin–orbit coupling in graphene, the experimental observation of these remarkable predictions is unlikely, the theoretical understanding of these spin-filtered states is shedding light on the electronic properties of edge states in other two-dimensional quantum spin Hall insulators. Here we review the effect of a variety of perturbations, like curvature, disorder, edge reconstruction, edge crystallographic orientation, and Coulomb interactions on the electronic properties of these spin filtered states.     

Edge states of graphene bilayer strip

The electronic structure of the zig-zag bilayer strip is analyzed. The electronic spectraof the bilayer strip is computed. The dependence of the edge state band flatness on thebilayer width is found. The density of states at the Fermi level is analytically computed.It is shown that it has the singularity which depends on the width of the bilayer strip.There is also asymmetry in the density of states below and above the Fermi energy.     

Terahertz radiation driven chiral edge currents in graphene

We observe photocurrents induced in single-layer graphene samples by illumination of the graphene edges with circularly polarized terahertz radiation at normal incidence. The photocurrent flows along the sample edges and forms a vortex. Its winding direction reverses by switching the light helicity from left to right handed. We demonstrate that the photocurrent stems from the sample edges, which reduce the spatial symmetry and result in an asymmetric scattering of carriers driven by the radiation electric field. The developed theory based on Boltzmann's kinetic equation is in a good agreement with the experiment. We show that the edge photocurrents can be applied for determination of the conductivity type and the momentum scattering time of the charge carriers in the graphene edge vicinity.     

Acoustic analog of monolayer graphene and edge states

Acoustic analog of monolayer graphene has been designed by using silicone rubber spheres of honeycomb lattices embedded in water. The dispersion of the structure has been studied theoretically using the rigorous multiple-scattering method. The energy spectra with the Dirac point have been verified and zigzag edge states have been found in ribbons of the structure, which are analogous to the electronic ones in graphene nanoribbons. The guided modes along the zigzag edge excited by a point source have been numerically demonstrated. The open cavity and “Z” type edge waveguide with 60° corners have also been realized by using such edge states.     

Spin-filtered edge states and quantum Hall effect in graphene

Electron edge states in graphene in the quantum Hall effect regime can carry both charge and spin. We show that spin splitting of the zeroth Landau level gives rise to counterpropagating modes with opposite spin polarization. These chiral spin modes lead to a rich variety of spin current states, depending on the spin-flip rate. A method to control the latter locally is proposed. We estimate Zeeman spin splitting enhanced by exchange, and obtain a spin gap of a few hundred Kelvin.     

Edge states versus diffusion in disordered graphene flakes

We study the localization properties of the wavefunctions in graphene flakes with short range disorder, via the numerical calculation of the inverse participation ratio (IPR) and its scaling which provides the fractal dimension D ₂. We show that the edge states which exist at the Dirac point of ballistic graphene (no disorder) with zig-zag edges survive in the presence of weak disorder with wavefunctions localized at the boundaries of the flakes. We argue, that there is a strong interplay between the underlying destructive interference mechanism of the honeycomb lattice of graphene leading to edge states and the diffusive interference mechanism introduced by the short-range disorder. This interplay results in a highly abnormal behavior, wavefunctions are becoming progressively less localized as the disorder is increased, indicated by the decrease of the average ?IPR? and the increase of D ₂. We verify, that this abnormal behavior is absent for graphene flakes with armchair edges which do not provide edge states.     

Band structure and edge states of star-like zigzag graphene nanoribbons

Connecting three zigzag graphene nanoribbons(ZGNRs) together through the sp~3 hybrid bonds forms a star-like ZGNR(S-ZGNR). Its band structure shows that there are four edge states at k = 0.5, in which the three electrons distribute at three outside edge sites, and the last electron is shared equally(50%) by two sites near the central site. The lowest conductance step in the valley is 2, two times higher than that of monolayer ZGNR(M-ZGNR). Furthermore, in one quasithree-dimensional hexagonal lattice built, both of the Dirac points and the zero-energy states appear in the band structure along the z-axis for the fixed zero k-point in the x-y plane. In addition, it is an insulator in the x-y plane due to band gap 4 eV, however, for any k-point in the x-y plane the zero-energy states always exist at k_z = 0.5.     

Floquet bands and photon-induced topological edge states of graphene nanoribbons

Floquet theorem is widely used in the light-driven systems. But many 2 D-materials models under the radiation are investigated with the high-frequency approximation, which may not be suitable for the practical experiment. In this work,we employ the non-perturbative Floquet method to strictly investigate the photo-induced topological phase transitions and edge states properties of graphene nanoribbons under the light irradiation of different frequencies(including both low and high frequencies). By analyzing the Floquet energy bands of ribbon and bulk graphene, we find the cause of the phase transitions and its relation with edge states. Besides, we also find the size effect of the graphene nanoribbon on the band gap and edge states in the presence of the light.     

A rigorous proof for non-existence of edge state in the semi-infinite armchair edged graphene

Within the transfer matrix method, we study the conditions for the existence of the edge states in the semi-infinite armchair edged graphene. We discuss zero-energy and non-zero-energy edge states, respectively, and show the non-existence of the edge states in the model analytically and rigorously.     

Interplay between edge states and simple bulk defects in graphene nanoribbons

We study the interplay between the edge states and a single impurity in a zigzag graphene nanoribbon. We use tight-binding exact diagonalization techniques, as well as density functional theory calculations to obtain the eigenvalue spectrum, the eigenfunctions, as well as the dependence of the local density of states (LDOS) on energy and position. We study the strictly zero-energy eigenfunctions using symmetry considerations, as well as tight-binding techniques. Moreover, we note that roughly half of the unperturbed eigenstates in the spectrum of the finite-size ribbon hybridize with the impurity state, and the corresponding eigenvalues are shifted with respect to their unperturbed values. The maximum shift and hybridization occur for a state whose energy is inverse proportional to the impurity potential, and give rise to an impurity peak in the DOS spectrum. We find that the interference between the impurity and the edge gives rise to peculiar modifications of the LDOS of the nanoribbon, in particular to oscillations of the edge LDOS. These effects depend on the size of the system, and decay with the distance between the edge and the impurity.     

Edge states at the interface between monolayer and bilayer graphene

The electronic properties for monolayer-bilayer hybrid graphene with zigzag interface are studied by both the Dirac equation and numerical calculation in zero field and in a magnetic field. Basically there are two types of zigzag interface dependent on the way of lattice stacking at the edge. Our study shows they have different locations of the localized edge states. Accordingly, the energy-momentum dispersion and local density of states behave quit differently along the interface near the Fermi energy EF=0.     

Structure, stability, edge states, and aromaticity of graphene ribbons

We determine the stability, the geometry, the electronic, and magnetic structure of hydrogen-terminated graphene-nanoribbon edges as a function of the hydrogen content of the environment by means of density functional theory. Antiferromagnetic zigzag ribbons are stable only at extremely low ultravacuum pressures. Under more standard conditions, the most stable structures are the mono- and dihydrogenated armchair edges and a zigzag edge reconstruction with one di- and two monohydrogenated sites. At high hydrogen concentration "bulk" graphene is not stable and spontaneously breaks to form ribbons, in analogy to the spontaneous breaking of graphene into small-width nanoribbons observed experimentally in solution. The stability and the existence of exotic edge electronic states and/or magnetism is rationalized in terms of simple concepts from organic chemistry (Clar's rule).     

Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.     

Effect of winding edge currents

We discuss persistent currents for particles with internal degrees of freedom. The currents arise because of winding properties essential for the chaotic motion of the particles in a confined geometry. The currents do not change the particle concentrations or thermodynamics, similar to the skipping orbits in a magnetic field.     

Spin currents in graphene under tension

Based on the Tight-Binding model, we have asymmetric massless Dirac fermions as the carriers in graphene under tension. Because of uniaxial strain, the velocities of Dirac fermions depend on their directions. This work studies the effect of the uniaxial strain on the spin transport through a single magnetic barrier of the strained graphene system. The result shows that graphene has a great potential for applications in nano-mechanical spintronic devices. This is because of strain in graphene can induce the spin-dependent pseudo-potentials at the barrier to control the spin currents of the junction.     

Spin-polarized edge and magnetoresistance in graphene flake

We examine spin-polarized edge and magnetoresistance (MR) in Fe/graphene flake (GF)/Fe junctions. For simulating various edge designs, we use the tight-binding approximation, mean-field scheme for the Hubbard model and the Landauer-Büttiker formalism. The Fe electrodes strongly affect electronic states and magnetic properties of the GF and induce magnetism in the edge atoms even in case of armchair interfaces. Also, the edge magnetic moments of the zigzag interface rotate and couple antiferromagnetically with the Fe electrodes. The conductivity of the junctions strongly depend on the relative magnetic orientation of the Fe electrodes, so, the junctions show high MR ratios. Moreover, edge geometry and localized edge state in the GF alter the MR ratios and produce large (small) variation in the MR at low (high) bias voltages.     

Surface states/modes in one-dimensional semi-infinite crystals

Analytical expressions on how the eigenvalue of an existing surface state/mode in one-dimensional semi-infinite periodic systems depends on the boundary location and the boundary condition are obtained, by using a Sturm-Liouville theory approach. The obtained equations are verified by numerical calculations. A direct consequence of the results obtained is that the termination of the periodicity at a boundary τ in a semi-infinite periodic system does not always cause a surface state/mode in a specific band gap.     

Self-passivating edge reconstructions of graphene

Planar reconstruction patterns at the zigzag and armchair edges of graphene were investigated with density-functional theory. It was unexpectedly found that the zigzag edge is metastable and a planar reconstruction spontaneously takes place at room temperature. The reconstruction changes electronic structure and self-passivates the edge with respect to adsorption of atomic hydrogen from a molecular atmosphere.