We study the edge magnetization of bilayer graphene ribbons with zigzag edges. The presence of flat edge-state bands at the Fermi energy of undoped bilayer, which gives rise to a strong peak in the density of states, makes bilayer ribbons magnetic at the edges even for very small on-site electronic repulsion. Working with the Hubbard model in the Hartree Fock approximation we show that the magnetic structure in bilayer ribbons with zigzag edges is ferromagnetic along the edge, involving sites of the two layers, and antiferromagnetic between opposite edges. It is also shown that this magnetic structure is a consequence of the nature of the edge states present in bilayer ribbons with zigzag edges. Analogously to the monolayer case, edge site magnetization as large as $m \approx0.2 μ_{B}$ (per lattice site) even at small on-site Hubbard repulsion $U \approx 0.3 {eV}$ is realized in nanometer wide bilayer ribbons.
As the three-dimensional analogs of graphene, Weyl semimetals display signatures of chiral anomaly which arises from charge pumping between the lowest chiral Landau levels of the Weyl nodes in the presence of parallel electric and magnetic fields. In this work, we study the pseudo chiral anomaly and its transport signatures in graphene ribbon with zigzag edges. Here "pseudo" refers to the case where the inverse of width of zigzag graphene ribbon plays the same role as magnetic field in three-dimensional Weyl semimetals. The valley chiral bands in zigzag graphene ribbons can be introduced by ed…
Interlayer Conductance in the Armchair Nanotube -- Zigzag Graphene Ribbon Parallel Contact: Theoretical Proposal of Detection of Wavefunction Growing from the Edge to the Center in the Graphene Ribbon
We theoretically investigate effect of a transverse electric field generated by side gates and a vertical electric field generated by top, back gates on energy bands and transport properties of zigzag bilayer graphene ribbons (Bernal stacking). Using atomistic Tight Binding calculations and Green function formalism we demonstrate that bandgap is opened when either field is applied and even enlarged under simultaneous influence of the two fields. Interestingly, although vertical electric fields are widely used to control bandgap in bilayer graphene, here we show that transverse fields exhibit m…
We explore the electronic structure, orbital character and topological aspect of a monolayer MoS$_2$ nanoribbon using tight-binding (TB) and low-energy (${\bm k}\cdot{\bm p} $) models. We obtain a mid-gap edge mode in the zigzag ribbon of monolayer MoS$_2$, which can be traced back to the topological properties of the bulk band structure. Monolayer MoS$_2$ can be considered as a valley Hall insulator. The boundary conditions at armchair edges mix the valleys on the edges, and a gap is induced in the edge modes. The spin-orbit coupling in the valence band reduces the hybridization of the bulk s…
We report a thorough study of the reducibility of edge correlation effects in graphene to much-simplified effective models for the edge states. The latter have been used before in specially tailored geometries. By a systematic investigation of corrections due to the bulk states in second order perturbation theory, we show that the reduction to pure edge state models is well-justified in general geometries. The framework of reduced models discussed here enables the study of non-mean-field correlation physics for system sizes far beyond the reach of conventional methods, such as, e.g., quantum M…
On the basis of first principles calculations, we report energy estimated to cut a graphene sheet into nanoribbons of armchair and zigzag configurations. Our calculations show that the energy required to cut a graphene sheet into zigzag configuration is higher than that of armchair configuration by an order of 0.174 eV. Thus, a control over the threshold energy might be helpful in designing an experiment for cutting a graphene sheet into smooth edged armchair or zigzag configurations.
We investigate the effects of Rashba spin-orbit interactions on the electronic band-structure and corresponding wave-functions of graphene. By exactly solving a tight-binding model Hamiltonian we obtain the expected splitting of the bands -due to the SU(2) spin symmetry breaking- that is accompanied by the appearance of additional Dirac points. These points are originated by valence-conduction band crossings. By introducing a convenient gauge transformation we study a model for zigzag nanoribbons with RSO interactions. We show that the RSO interactions lifts the quasi-degeneracy of the edge ba…
Zigzag graphene nanoribbons can have magnetic ground states with ferromagnetic, antiferromagnetic, or canted configurations, depending on carrier density. We show that an electric field directed across the ribbon alters the magnetic state, favoring antiferromagnetic configurations. This property can be used to prepare ribbons with a prescribed spin-orientation on a given edge.
We combine density-functional theory and the nonequilibrium Green's function method to study the thermal conductance of graphene nanoribbons with armchair and zigzag edges. Zigzag ribbons have higher thermal conductance than armchair ribbons of comparable widths due to an anisotropy in phonon dispersion for graphene nanoribbons: low-frequency bands in zigzag ribbons are more dispersive than those in armchair ribbons. The edges with and without hydrogen-passivation modify the atomic structure and ultimately influence the phonon thermal transport differently for the two ribbon types.
We explore electron transport properties in honeycomb lattice ribbons with zigzag edges coupled to two semi-infinite one-dimensional metallic electrodes. The calculations are based on the tight-binding model and the Green's function method, which numerically compute the conductance-energy and current-voltage characteristics as functions of the lengths and widths of the ribbons. Our numerical results predict that for such a ribbon an energy gap always appears in the conductance spectrum across the energy E=0. With the increase of the size of the ribbon, the gap gradually decreases but it never …
Using the tight-binding approach we study the electronic transport in a $\mathrm{MoS}_2$ zigzag ribbon with a spatially varying potential profile. Considering a ribbon with a smooth potential step in the Fermi energy regime where the transport is dominated by the edge modes, we find that the conductance exhibits sharp resonances due to the resonant transport through a n-p-n junction effectively created in the structure. We show that in a gated wire the current carried on the wire edges can be blocked despite the metallic band structure of the ribbon. For the Fermi energies corresponding to $\m…
Zigzag nanoribbons of monolayer graphene-like two-dimensional materials host spontaneous edge magnetism at the zigzag terminations, whose configuration controls the band gap. In this article, the edge magnetism of zigzag nanoribbons of bilayer hexagonal crystals are studied. The specific models of bilayer graphene and bilayer silicene are both studied. As the gated voltage increases, the total energy level, magnetic structure and band structures of the ground state and the first quasi-stable excited state are tuned. For certain region of the gated voltage, the band gaps of spin up and down are…
We study the conductance of a biased bilayer graphene flake with monolayer nanoribbon contacts. We find that the transmission through the bilayer ribbon strongly depends on the applied bias between the two layers and on the relative position of the monolayer contacts. Besides the opening of an energy gap on the bilayer, the bias allows to tune the electronic density on the bilayer flake, making possible the control of the electronic transmission by an external parameter.
We study the conductance of graphene nanoribbons with long-range disorder. Due to the absence of intervalley scattering from the disorder potential, time-reversal symmetry (TRS) can be effectively broken even without a magnetic field, depending on the type of ribbon edge. Even though armchair edges generally mix valleys, we show that metallic armchair nanoribbons possess a hidden pseudovalley structure and effectively broken TRS. In contrast, semiconducting armchair nanoribbons inevitably mix valleys and restore TRS. As a result, in strong disorder metallic armchair ribbons exhibit a perfectly…
Boundary conditions for the two-dimensional fermions in ribbons of the hexagonal lattice are studied in the dice model whose energy spectrum in infinite system consists of three bands with one completely flat band of zero energy. Like in graphene the regular lattice terminations are of the armchair and zigzag types. However, there are four possible zigzag edge terminations in contrast to graphene where only one type of zigzag termination is possible. Determining the boundary conditions for these lattice terminations, the energy spectra of pseudospin-1 fermions in dice model ribbons with zigzag…
The field screening effect on the electronic and field-emission properties of zigzag graphene nanoribbons (ZGNRs) has been studied using first-principles calculations. We have systematically investigated the effect of inter-ribbon distance and ribbon width on the work function, field enhancement factor, band gap and edge magnetism of zigzag graphene nanoribbons (ZGNRs). It is found that the work function of ZGNRs increases rapidly as the inter-ribbon distance Dx increases, which is caused by the positive dipole at the edge of the ribbon. For a given Dx, the work function of ZGNRs decreases as …
We investigate the effects of external electric fields on the electronic properties of bilayer armchair graphene nano-ribbons. Using atomistic simulations with Tight Binding calculations and the Non-equilibrium Green function formalism, we demonstrate that (i) in semi-metallic structures, vertical fields impact more effectively than transverse fields in terms of opening larger bandgap, showing a contrary phenomenon compared to that demonstrated in previous studies in bilayer zigzag graphene nano-ribbons; (ii) in some semiconducting structures, if transverse fields just show usual effects as in…
We study the interaction energy between two graphene nanoribbons by first principles calculations, including van der Waals interactions and spin polarization. For ultranarrow zigzag nanoribbons, the direct stacking is even more stable than Bernal, competing in energy for wider ribbons. This behavior is due to the magnetic interaction between edge states. We relate the reduction of the magnetization in zigzag nanoribbons with increasing ribbon width to the structural changes produced by the magnetic interaction, and show that when deposited on a substrate, zigzag bilayer ribbons remain magnetic…
It is known that the parity of reflection amplitude can either be even or odd under the mirror operation. Up to now, all the parities of reflection amplitude in the one-mode energy region are even under the mirror operation. In this paper, we give an example of odd parity for Andreev reflection (AR) in a three-terminal graphene-supercondutor hybrid systems. We found that the parity is even for the Andreev retroreflection (ARR) and odd for specular Andreev reflection (SAR). We attribute this remarkable phenomenon to the distinct topology of the band structure of graphene and the specular Andree…
Electron transport in small graphene nanoribbons is studied by microwave emulation experiments and tight-binding calculations. In particular, it is investigated under which conditions a transport gap can be observed. Our experiments provide evidence that armchair ribbons of width $3m+2$ with integer $m$ are metallic and otherwise semiconducting, whereas zigzag ribbons are metallic independent of their width. The contact geometry, defining to which atoms at the ribbon edges the source and drain leads are attached, has strong effects on the transport. If leads are attached only to the inner atom…
Zigzag edges of neutral armchair-oriented Graphene Nano-Ribbons show states strongly localized at those edges. They behave as free radicals that can capture electrons during processing, increasing ribbon's stability. Thus, charging and its consequences should be investigated.Total energy calculations of finite ribbons using spin polarized Density Functional Theory (DFT) show that ribbon's charging is feasible. Energies for Pariser-Parr-Pople (PPP) model Hamiltonian are compatible with DFT allowing the study of larger systems. Results for neutral ribbons indicate: i) the fundamental gap of spin…
We study the electronic properties of electrons in flat and curved zigzag graphene ribbons using a tight-binding model within the Slater Koster approximation. We find that curvature dramatically enhances the action of spin orbit effects in graphene ribbons and has a strong effect on the spin orientation of the edge states: whereas spins are normal to the surface in the case of flat ribbons, this is no longer the case in the case of curved ribbons. We find that for the edge states, the spin density lies always in the plane perpendicular to the ribbon axis, and deviate strongly from the normal t…
We construct a phenomenological scattering theory for the triangular network of valley Hall states that arises in twisted bilayer graphene under interlayer bias. Crucially, our network model includes scattering between different valley Hall states within the same valley and spin. We show that in the absence of forward scattering, symmetries reduce the network model to a single parameter that interpolates between a nested Fermi surface and flatbands, which can be understood in terms of one-dimensional chiral zigzag modes and closed triangular orbits, respectively. We demonstrate how unitarity a…
We use quantum Monte Carlo method to study the indirect coupling between two magnetic impurities on the zigzag edge of graphene ribbon, with respect to the chemical potential $μ$. We find that the spin-spin correlation between two adatoms located on the nearest sites in the zigzag edge are drastically suppressed around the zero-energy. As we switch the system away from half-filling, the antiferromagnetic correlation is first enhanced and then decreased. If the two adatoms are adsorbed on the sites belonging to the same sublattice, we find similar behavior of spin-spin correlation except for a …
Magnetic order emerging in otherwise non-magnetic materials as carbon is a paradigmatic example of a novel type of s-p electron magnetism predicted to be of exceptional high-temperature stability. It has been demonstrated that atomic scale structural defects of graphene can host unpaired spins. However, it is still unclear under which conditions long-range magnetic order can emerge from such defect-bound magnetic moments. Here we propose that in contrast to random defect distributions, atomic scale engineering of graphene edges with specific crystallographic orientation, comprising edge atoms …
We calculate the band structure and the conductance of periodic corrugated graphene nanoribbons within the framework of the tight-binding $p$-orbital model. We consider corrugated structures based on host ribbons with armchair and zigzag edges and three different types of corrugations (armchair edges, zigzag edges as well as a rectangular corrugation). We demonstrate that for armchair host ribbons, depending on the type of corrugation, a band gap or low-velocity minibands appear near the charge neutrality point. For higher energies the allowed Bloch state bands become separated by mini-stopban…
We study the properties of edge states in in-plane heterostructures made of adjacent zigzag graphene and BN ribbons. While in pure zigzag graphene nanoribbons, gapless edge states are nearly flat and cannot contribute significantly to the conduction, at BN/Graphene interfaces the properties of these states are significantly modified. They are still strongly localized at the zigzag edges of graphene but they exhibit a high group velocity up to 4.3x10^5 m/s at the B/C interface and even 7.4x10^5 m/s at the N-C interface. For a given wave vector the velocities of N/C and B/C hybrid interface stat…
Interacting disordered zigzag graphene nanoribbons have fractional charges, are quasi-one-dimensional, and display an exponentially small gap. Our numerical computations showed that the topological entanglement entropy of these systems has a small finite but universal value, independent of the strength of the interaction and the disorder. The result that was obtained for the topological entanglement entropy shows that the disorder-free phase is critical and becomes unstable in the presence of disorder.
Based on atomistic simulations, the nonlinear elastic properties of monolayer graphene nanoribbons under quasistatic uniaxial tension are predicted, emphasizing the effect of edge structures (armchair and zigzag, without and with hydrogen passivation). The results of atomistic simulations are interpreted within a theoretical framework of thermodynamics, which enables determination of the nonlinear functions for the strain-dependent edge energy and the hydrogen adsorption energy, for both zigzag and armchair edges. Due to the edge effects, the initial Young's modulus of graphene nanoribbons und…
The rules that govern spin exchange interaction in pristine graphene nanostructures are constrained by the bipartite character of the lattice, so that the sign of the exchange is determined by whether magnetic moments are on the same sublattice or else. The synthesis of graphene ribbons with perfect zigzag edges and a fluoranthene group with a pentagon ring, a defect that breaks the bipartite nature of the honeycomb lattice, has been recently demonstrated. Here we address how the electronic and spin properties of these structures are modified by such defects, both for indirect exchange interac…
We study the electronic structure of chiral and achiral graphene nanoribbons with symmetric edges, including curvature and spin-orbit effects. Curved ribbons show spin-split bands, whereas flat ribbons present spin-degenerate bands. We show that this effect is due to the breaking of spatial inversion symmetry in curved graphene nanoribbons, while flat ribbons with symmetric edges possess an inversion center, regardless of their having chiral or achiral edges. We find an enhanced edge-edge coupling and a substantial gap in narrow chiral nanoribbons, which is not present in zigzag ribbons of sim…
We investigate transport in the network of valley Hall states that emerges in minimally twisted bilayer graphene under interlayer bias. To this aim, we construct a scattering theory that captures the network physics. In the absence of forward scattering, symmetries constrain the network model to a single parameter that interpolates between one-dimensional chiral zigzag modes and pseudo-Landau levels. Moreover, we show how the coupling of zigzag modes affects magnetotransport. In particular, we find that scattering between parallel zigzag channels gives rise to Aharonov-Bohm oscillations that a…
Finite-length armchair graphene nanoribbons can behave as one dimensional topological materials, that may show edge states in their zigzag-terminated edges, depending on their width and termination. We show here a full solution of Tight-Binding graphene rectangles of any length and width that can be seen as either finite-length armchair or zigzag ribbons. We find exact analytical expressions for both bulk and edge eigen-states and eigen-energies. We write down exact expressions for the Coulomb interactions among edge states and introduce a Hubbard-dimer model to analyse the emergence and featu…
We investigate the diffusive electron-transport properties of charge-doped graphene ribbons and nanoribbons with imperfect edges. We consider different regimes of edge scattering, ranging from wide graphene ribbons with (partially) diffusive edge scattering to ribbons with large width variations and nanoribbons with atomistic edge roughness. For the latter, we introduce an approach based on pseudopotentials, allowing for an atomistic treatment of the band structure and the scattering potential, on the self-consistent solution of the Boltzmann transport equation within the relaxation-time appro…
We construct a two-channel scattering model for the triangular network of valley Hall states in interlayer-biased minimally twisted bilayer graphene from symmetry arguments and investigate electronic transport in a four-terminal setup. In the absence of forward scattering, a single phenomenological parameter tunes the network between a triplet of chiral zigzag modes and pseudo-Landau levels. Moreover, the chiral zigzag modes give rise to robust Aharonov-Bohm resonances in the longitudinal conductance in the presence of a perpendicular magnetic field or an in-plane electric field. Interestingly…
Motivated by the recent synthesis of single layer TiSe2 , we used state-of-the-art density functional theory calculations, to investigate the structural and electronic properties of zigzag and armchair- edged nanoribbons of this material. Our analysis reveals that, differing from ribbons of other ultra-thin materials such as graphene, TiSe2 nanoribbons have some distinctive properties. The electronic band gap of the nanoribbons decreases exponentially with the width and vanishes for ribbons wider than 20 Angstroms. For ultranarrow zigzag-edged nanoribbons we find odd-even oscillations in the b…
We theoretically investigate the one-color injection currents and shift currents in zigzag graphene nanoribbons with applying a static electric field across the ribbon, which breaks the inversion symmetry to generate nonzero second order optical responses by dipole interaction. These two types of currents can be separately excited by specific light polarization, circularly polarized lights for injection currents and linearly polarized lights for shift currents. Based on a tight binding model formed by carbon 2p$_z$ orbitals, we numerically calculate the spectra of injection coefficients and sh…
A graphene nanoribbon is a good candidate for a $(1+1)$ Chern-Simons topological insulator since it obeys particle-hole symmetry. We show that in a finite semiconducting armchair ribbon, which has two zigzag edges and two armchair edges, a $(1+1)$ Chern-Simons topological insulator is indeed realized as the length of the armchair edges becomes large in comparison to that of the zigzag edges. But only a quasi-topological insulator is formed in a metallic armchair ribbon with a pseudogap. In such systems a zigzag edge acts like a domain wall, through which the polarization changes from $0$ to $e…
We present an analytical description of pi electrons of a finite size bilayer graphene within a framework of the tight-binding model. The bilayered structures considered here are characterized by a rectangular geometry and have a finite size in one or both directions with armchair- and zigzag-shaped edges. We provide an exact analytical description of the spectrum of pi electrons in the zigzag and armchair bilayer graphene nanoribbons and nanotubes. We analyze the dispersion relations, the density of states, and the conductance quantization.
We investigate higher-order plasmons in graphene nanoribbons, and present how electronic edge states and wavefunction fine structure influence the graphene plasmons. Based on nearest-neighbor tight-binding calculations, we find that a standing-wave model based on nonlocal bulk plasmon dispersion is surprisingly accurate for armchair ribbons of widths even down to a few nanometers, and we determine the corresponding phase shift upon edge reflection and an effective ribbon width. Wider zigzag ribbons exhibit a similar phase shift, whereas the standing-wave model describes few-nanometer zigzag ri…
Graphene on a substrate will suffer an inversion-symmetry-breaking (ISB) lattice potential. Taking electron-electron interaction into account, we study in this paper the possibility of half-metallicity and noncollinear (NC) magnetic phase for graphene zigzag nanoribbons without inversion symmetry. At half-filling it is found that half-metallic(HM) state can be achieved at an intermediate value of the ISB potential due to its competition with the electron-electron interaction. Away from half-filling, the phase diagrams of doping versus ISB potential for different ribbon width are given, where t…
We investigate the effect of vacancy defects on the electronic and magnetic properties of zigzag graphene nanoribbons (zGNRs) by making use of the Green's function formalism in combination with the tight-binding Hamiltonian. The evolution of the indirect exchange coupling, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, including single, double, and multiple 5-8-5 divacancy defects is explained. Our numerical calculations show that the changes in the electronic structure and the exchange coupling of zGNRs depend significantly on the location of the divacancy defects with respect to …
We study in this paper the edge polarizations and their consequences for a biased Bernal stacked bilayer graphene nanoribbon with zigzag termination. The magnetic states are classified according to the interlayer and intralayer couplings between the edge polarizations, and the magnetic phase diagram of doping versus bias voltage is given. Coplanar magnetic phase is found and the variation of its magnetic structure with the bias voltage is investigated. For all the magnetic states, we also discuss the possibility of the half-metallicity, and for a ribbon with perfect zigzag edges we predict sev…
