Edges Of Graphene Nanoribbons Research Articles

Optimizing Edge Smoothness in Graphene Nanoribbons for Enhanced Transistor Performance: A Lithography Approach

Graphene nanoribbons (GNRs) emerge as quintessential candidates for advanced transistor applications, attributed to their distinctive electronic attributes. Their tunable band gap, coupled with exceptional electron mobility, renders GNRs exceedingly conducive for transistor engineering. Moreover, GNRs' inherent spin-polarized edge states present intriguing prospects for Quantum Information Science (QIS) endeavors, leveraging electron spin for information storage or processing. A pivotal determinant to the performance of GNR-based single electron transistors is the roughness or irregularity of edges. Edge imperfections can introduce scattering events, impeding electron transport and subsequently diminish the device functionality. Attaining optimal edge structure and roughness in GNRs is crucial for mitigating such scattering phenomena, thereby enhancing charge carrier mobility and stability. Here, we employ sophisticated lithography modalities, including electron beam and probe lithography, to meticulously tailor GNR width and edge configurations. We pedantically explore the ramifications of edge roughness on their electronic characteristics and resultant device performance. Through this rigorous analysis, our goal is to elucidate strategies for optimizing edge smoothness in GNRs, thereby unlocking superior electronic performance for transistor and QIS applications, marking a significant stride in the realm of nanoscale semiconductor technology.

(Invited) Computational Design of Low-D Graphene-Based Systems in View of Energy Applications

Graphene-based nanostructures, such as 1D graphene nanoribbons [1-4], and graphene-related 2D materials, such as C3N [5-7], offer extraordinary versatility as next-generation semiconductors for nanoelectronics, optoelectronics and energy devices (e.g. batteries) [8,9], due to their tunable properties, including charge-carrier mobility, optical absorption, excitonic features, electronic bandgap and mechanical properties which are uniquely defined by their chemical structure.We here adopt predictive first-principles computational schemes - based on both density-functional theory and many-body perturbation theory - to acquire a comprehensive understanding of the electronic, optical and vibrational (Raman and IR) excitations of such low-D graphene-based materials, as a function of their structure, functionalization, distortion, and different reaction stage in their fabrication. Our results, often combined to experimental investigations, allow not only to fingerprint their precise structures, but also to physically interpret and predict their spectral features also in view of a material design perspective. N. Cavani, M. De Corato, A. Ruini, D. Prezzi, E. Molinari, A. Lodi Rizzini, A. Rosi, R. Biagi, V. Corradini, X.Y. Wang, X. L. Feng, A. Narita, K. Mullen, V. De Renzi, “Vibrational signature of the graphene nanoribbon edge structure from high-resolution electron energy-loss spectroscopy”, Nanoscale 12, 19681-19688 (2020)D. Rizzo, D. Prezzi, A. Ruini, V. Nagyte, A. Keerthi, A. Narita, U. Beser, F.G. Xu, Y,Y, Mai, X.L. Feng, K. Mullen, E. Molinari, C. Casiraghi, “Multiwavelength Raman spectroscopy of ultranarrow nanoribbons made by solution-mediated bottom-up approach”, Phys. Rev. B 100, 045406 (2019)B. Hu, P. Xie, M.De Corato, A. Ruini, S. Zhao, F. Meggendorfer, L.A. Straaso, L. Rondin, P. Simon, J. Li, J.J. Finley, M.R. Hansen, J.S. Lauret, E. Molinari, X.L. Feng, J.V. Barth, C.A. Palma, D. Prezzi, K. Mullen, A. Narita, “Bandgap Engineering of Graphene Nanoribbons by Control over Structural Distortion”, J. Am. Chem. Soc.140, 7803-7809 (2018)R. Denk, M. Hohage, P. Zeppenfeld, J. M. Cai, C. A. Pignedoli, H. Sode, R. Fasel, X. L. Feng, K. Mullen, S. D. Wang, D. Prezzi, A Ferretti, A. Ruini, E. Molinari, P. Ruffieux, “Exciton-dominated optical response of ultra-narrow graphene nanoribbons”, Nature Communications 5, 4253 (2014).M. Zanfrognini, M. Bonacci, F. Paleari, E. Molinari, A. Ruini, A. Ferretti, M. J. Caldas, D. Varsano, “Quenching of low-energy optical absorption in bilayer polytypes”, Physical Review Materials 7 (6), 064006 (2023)M. Zanfrognini, N. Spallanzani, M. Bonacci, E. Molinari, A. Ruini, M.J. Caldas, A. Ferretti, D. Varsano, “Effect of uniaxial strain on the excitonic properties of monolayer: A symmetry-based analysis”, Physical Review B 107 (4), 045430 (2023)M. Bonacci, M. Zanfrognini, E. Molinari, A. Ruini, M. J Caldas, A. Ferretti, D. Varsano, “Excitonic effects in graphene-like C3N”, Phys. Rev. Mat. 6, 034009 (2022)M. A. Salvador, R. Maji, F. Rossella, E. Degoli, A. Ruini, R. Magri, “Structural and Dynamic Characterization of Li–Ionic Liquid Electrolyte Solutions for Application in Li-Ion Batteries: A Molecular Dynamics Approach”, Batteries 9 (4), 234 (2023)R. Maji, M. A. Salvador, A. Ruini, R. Magri, E. Degoli, “A first-principles study of self-healing binders for next-generation Si-based lithium-ion batteries”, Materials Today Chemistry 29, 101474 (2023)

Effects of nanocutting environments on the electronic structure of armchair-type graphene nanoribbons: the first-principles study

In order to investigate the effect of nanocutting environment on the electronic structure of armchair-type graphene nanoribbons, this paper adopts a first-principle computational approach to study the effect of different substrates and solutions, such as on the motion of electrons in the middle and outer orbitals of graphene nanoribbons, by observing the energy band structure, the value of the band gap, and the density of the split-wave states. The results show that the adsorption of Si and C atoms at the edge of the nanoribbon leads to a decrease in the band gap value. The adsorption of Al and O atoms at the edges of graphene nanoribbons leads to a decrease in the nanoribbon band gap value to 0 eV. Different substrate atoms mainly affect the p-orbital electron motion in the nanobelt. Bare-edge graphene nanoribbons are indirect bandgap structures, and graphene nanoribbons with H, O and OH atoms adsorbed at the edges of the nanoribbons are direct bandgap structures. Edge O-isation leads to a nanobelt band gap of 0, which exhibits metallic properties. The edge H-isation nanoribbon band gap is higher than the bare edge nanoribbon band gap. Nanoribbon edge OH-isation reduces the nanoribbon band gap value. Nanoribbon edge adsorption of atoms in solution affects p-orbital electron motion. The formation energy of five-ring defects and seven-ring defects is low, and the defects are easier to form. The edges containing defects all reduce the band gap values of graphene nanoribbons. The defects mainly affect the p-orbital electron motion, leading to differences in the band gap values. The bandgap decreases with increasing nanobelt width, and the bandgap value conforms to 3 N+2<3 N<3 N+1, with regular fluctuations in the curve with period 3. The larger the band gap, the smaller the curvature of the curve at the extremes, and the sparser the curve. In this paper, the electronic structures of different edge structures are analysed from a quantum mechanical point of view, and the synthesis of these results will provide theoretical guidance for obtaining high-quality semiconductor nanoribbons by mechanochemical nanocutting.

Formation of Metal Atom Chains at the Edges of Graphene Nanoribbons Supported by Graphene

Formation of Metal Atom Chains at the Edges of Graphene Nanoribbons Supported by Graphene

Nanoscale terahertz conductivity and ultrafast dynamics of terahertz plasmons in periodic arrays of epitaxial graphene nanoribbons

The dynamics of plasmons in nanoribbons of (hydrogen intercalated) quasi-freestanding single-layer graphene is studied by terahertz spectroscopy both in the steady state and upon photoexcitation by an ultrashort near infrared laser pulse. The use of two-dimensional frequency-domain analysis of the optical pump–terahertz (THz) probe signals allows us to determine the evolution of carrier temperature and plasmon characteristics with ∼100 fs time resolution. Namely, we find that the carrier temperature decreases from more than 5000 K to the lattice temperature within about 7 ps and that during this evolution the carrier mobility remains practically constant. The time-resolved THz conductivity spectra suggest that graphene nanoribbons contain defects which act as low potential barriers causing a weak localization of charges; the potential barriers are overcome upon photoexcitation. Furthermore, the edges of graphene nanoribbons are found to slightly enhance the scattering of carriers. The results are supported by complementary measurements using THz scanning near-field microscopy, which confirm a high uniformity of the THz conductivity across the sample and demonstrate high enough sensitivity to resolve even the impact of nanometric terrace steps on SiC substrate under the graphene monolayer. Published by the American Physical Society 2024

Micromechanical effects of substrate hardness on graphene nano-cutting quality

In the nanocutting process of graphene, the mechanism on the atomic scale of the effect of substrate hardness on the edge quality of graphene nanoribbons has not been thoroughly investigated. In this paper, we construct a diamond probe (radius of 5 Å) to cut graphene (190 Å × 113 Å) models with different substrates (silicon/sapphire/silicon carbide), molecular dynamics simulations were used to study the effect of substrate hardness and cutting depth on the atomic structure of graphene nanoribbon edges by observing the atomic morphology, the number of C-C bonds, the radial distribution function and friction. The results show that during the nanoindentation stage, when graphene is in the critical damage state, the substrate with high hardness has high deformation strength, the substrate deformation is small, and graphene undergoes deformation is also small, but the force Fz on the probe, the critical graphene stress σz and the equivalent stress are all large, and the number of C atoms with CN=3 is large; At the nano-cutting stage, the breakage of C-C bonds and the generation of edge defects are caused by local stress concentration, and the graphene stress σxx along the x-direction has a clear directionality. With the same cutting depth (critical damage depth of silicon substrate), silicon substrate hardness is small, the graphene nanoribbon obtained has less number of circular defects generated at the edge, less total number of defective atoms, more number of CN=3 atoms, large peak of radial distribution function of graphene after cutting, and graphene integrity is better. The hardness of the silicon carbide substrate is large, and the number of C-C bond breaks in graphene during the cutting process is high. Each substrate is cut at its own critical damage depth, and the substrate hardness has no significant effect on the number of C-C bond breaks, the number of circular defects, the total number of defective atoms, and the number of CN=3 atoms in graphene. When the silicon substrate was used, the stable force of the probe is small, the edge structure of graphene has little impact, the edge effect obtained is good. The cutting depth and substrate hardness have little effect on the graphene temperature during the nano-cutting process. In this paper, the atomic structure of graphene edges after cutting is analysed and investigated on the atomic scale, and the above results are combined to provide theoretical guidance for obtaining efficient and high-quality and low-damage graphene nanoribbon cutting.

Mechanical behaviour and micro-mechanism of atomically precise edge graphene nanoribbons mechanical cutting guided by hydrogen atom adsorption

The preparation of graphene nanoribbons with atomically precise edges is a major challenge. In this paper, we proposed a mechanical cutting method for graphene nanoribbons guided by hydrogen atom adsorption and analysed the mechanical behaviour and microscopic cutting mechanism of graphene with hydrogen atom adsorption using reactive molecular dynamics simulations. The results showed that hydrogen atom adsorption weakened the mechanical properties and improved the cutting performance of graphene. Hydrogen atom adsorption showed a substantial guiding effect on graphene cutting, increasing the regular edge ratio from 12.21 % to 91.89 %, and graphene nanoribbons with atomically precise edges were prepared using this method. We determined that the cutting mechanism of graphene without hydrogen atom adsorption was a tensile-breaking mechanism. In contrast, graphene with hydrogen atom adsorption exhibited a shear-breaking mechanism. The cutting performance of the armchair-H graphene was lower than that of the zigzag-H graphene owing to the unique hexagonal distribution of carbon atoms. This study provides new insights into the cutting mechanism of graphene guided by hydrogen atom adsorption and establishes the method for preparing atomically precise edge graphene nanoribbons by mechanical cutting.

Fermi-Level Engineering of Nitrogen Core-Doped Armchair Graphene Nanoribbons.

Substitutional heteroatom doping of bottom-up engineered 1D graphene nanoribbons (GNRs) is a versatile tool for realizing low-dimensional functional materials for nanoelectronics and sensing. Previous efforts have largely relied on replacing C-H groups lining the edges of GNRs with trigonal planar N atoms. This type of atomically precise doping, however, only results in a modest realignment of the valence band (VB) and conduction band (CB) energies. Here, we report the design, bottom-up synthesis, and spectroscopic characterization of nitrogen core-doped 5-atom-wide armchair GNRs (N2-5-AGNRs) that yield much greater energy-level shifting of the GNR electronic structure. Here, the substitution of C atoms with N atoms along the backbone of the GNR introduces a single surplus π-electron per dopant that populates the electronic states associated with previously unoccupied bands. First-principles DFT-LDA calculations confirm that a sizable shift in Fermi energy (∼1.0 eV) is accompanied by a broad reconfiguration of the band structure, including the opening of a new band gap and the transition from a direct to an indirect semiconducting band gap. Scanning tunneling spectroscopy (STS) lift-off charge transport experiments corroborate the theoretical results and reveal the relationship among substitutional heteroatom doping, Fermi-level shifting, electronic band structure, and topological engineering for this new N-doped GNR.

Theoretical insights into the methane catalytic decomposition on graphene nanoribbons edges.

Catalytic methane decomposition (CMD) is receiving much attention as a promising application for hydrogen production. Due to the high energy required for breaking the C-H bonds of methane, the choice of catalyst is crucial to the viability of this process. However, atomistic insights for the CMD mechanism on carbon-based materials are still limited. Here, we investigate the viability of CMD under reaction conditions on the zigzag (12-ZGNR) and armchair (AGRN) edges of graphene nanoribbons employing dispersion-corrected density functional theory (DFT). First, we investigated the desorption of H and H2 at 1200K on the passivated 12-ZGNR and 12-AGNR edges. The diffusion of hydrogen atom on the passivated edges is the rate determinant step for the most favourable H2 desorption pathway, with a activation free energy of 4.17eV and 3.45eV on 12-ZGNR and 12-AGNR, respectively. The most favourable H2 desorption occurs on the 12-AGNR edges with a free energy barrier of 1.56eV, reflecting the availability of bare carbon active sites on the catalytic application. The direct dissociative chemisorption of CH4 is the preferred pathway on the non-passivated 12-ZGNR edges, with an activation free energy of 0.56eV. We also present the reaction steps for the complete catalytic dehydrogenation of methane on 12-ZGNR and 12-AGNR edges, proposing a mechanism in which the solid carbon formed on the edges act as new active sites. The active sites on the 12-AGNR edges show more propensity to be regenerated due lower free energy barrier of 2.71eV for the H2 desorption from the newly grown active site. Comparison is made between the results obtained here and experimental and computational data available in the literature. We provide fundamental insights for the engineering of carbon-based catalysts for the CMD, showing that the bare carbon edges of graphene nanoribbons have performance comparable to commonly used metallic and bi-metallic catalysts for methane decomposition.

Evaluation of Hysteresis Response in Achiral Edges of Graphene Nanoribbons on Semi-Insulating SiC

Hysteresis response of epitaxially grown graphene nanoribbons devices on semi-insulating 4H-SiC in the armchair and zigzag directions is evaluated and studied. The influence of the orientation of fabrication and dimensions of graphene nanoribbons on the hysteresis effect reveals the metallic and semiconducting nature graphene nanoribbons. The hysteresis response of armchair based graphene nanoribbon side gate and top gated devices implies the influence of gate field electric strength and the contribution of surface traps, adsorbents, and initial defects on graphene as the primary sources of hysteresis. Additionally, passivation with AlOx and top gate modulation decreased the hysteresis and improved the current-voltage characteristics.

Effectuating tunable valley selection via multiterminal monolayer graphene devices

Valleytronics using two-dimensional materials opens unprecedented opportunities for information processing using a valley polarizer as a basic building block. Various methodologies, such as strain engineering, the inclusion of line defects, and the application of static magnetic fields, have been widely explored for creating valley polarization. However, these methods suffer from low transmission or lack of polarization directionality. To overcome the above limitations, we propose an all-electrical valley polarizer using zigzag edge graphene nanoribbons in a multiterminal device geometry. The proposed device can be gate-tuned to operate along two independent regimes: (i) a terminal-specific valley filter that utilizes band-structure engineering, and (ii) a parity-specific valley filter that exploits the parity selection rule in zigzag edge graphene. We show that the device exhibits intriguing physics in the multimode regime of operation affecting the valley polarization; we investigate various factors influencing polarization in wide-device geometries, such as optical analogs of graphene Dirac fermions, angle-selective transmission via $p\ensuremath{-}n$ junctions, and the localization of edge states. Furthermore, we evaluate the performance of the proposed device structures in the presence of Anderson short-range disorder at the edges and the bulk, and we find it to be resilient to edge disorder even for a higher disorder strength. The device geometry is optimized to achieve maximum valley polarization, thereby paving the way for a physics-based tunable valleytronic device.

Role of symmetry in quantum blocking of Andreev reflection in graphene nanoribbons side-terminated by superconductors

Andreev reflection can be forbidden by means of quantum interference when superconductors are attached at the side edges of graphene nanoribbons (GNRs). The blocking is restricted to single-mode nanoribbons having symmetric zigzag edges and is destroyed by the application of a magnetic field. These characteristics are shown to be the effects of the wavefunction parity on the Andreev retro and specular reflections. Not only the mirror symmetry of the GNRs but also symmetric coupling of the superconductors is required for the quantum blocking. The quasi-flat-band states around the Dirac point energy induced for armchair nanoribbons by adding carbon atoms at the nanoribbon edges do not cause the quantum blocking due to the lack of the mirror symmetry. Furthermore, the phase modulation by the superconductors is shown to be able to convert the quasi-flat dispersion for the edge states of zigzag nanoribbons to a quasi-vertical dispersion.

Magnetism in Nonplanar Zigzag Edge Termini of Graphene Nanoribbons.

Graphene nanoribbons (GNRs) and nanographenes synthesized by on-surface reaction using tailor-made molecular precursors offer an ideal playground for a study of magnetism towards nano-spintronics. Although the zigzag edge of GNRs has been known to host magnetism, underlying metal substrates usually veil the edge-induced Kondo effect. Here, we report the on-surface synthesis of unprecedented, π-extended 7-armchair GNRs using 7-bromo-12-(10-bromoanthracen-9-yl)tetraphene as the precursor. Characterizations by scanning tunneling microscopy/spectroscopy revealed unique rearrangement reactions leading to pentagon- or pentagon/heptagon-incorporated, nonplanar zigzag termini, which demonstrated Kondo resonance even on bare Au(111). Density functional theory calculations indicate that the nonplanar structure significantly reduces the interaction between the zigzag terminus and the Au(111), leading to a recovery of the spin localization of the zigzag edge. Such distortion of planar GNR structures offers a degree of freedom to control the magnetism on metal substrates.

Linear indium atom chains at graphene edges

The presence of metal atoms at the edges of graphene nanoribbons (GNRs) opens new possibilities toward tailoring their physical properties. We present here formation and high-resolution characterization of indium (In) chains on the edges of graphene-supported GNRs. The GNRs are formed when adsorbed hydrocarbon contamination crystallizes via laser heating into small ribbon-like patches of a second graphitic layer on a continuous graphene monolayer and onto which In is subsequently physical vapor deposited. Using aberration-corrected scanning transmission electron microscopy (STEM), we find that this leads to the preferential decoration of the edges of the overlying GNRs with multiple In atoms along their graphitic edges. Electron-beam irradiation during STEM induces migration of In atoms along the edges of the GNRs and triggers the formation of longer In atom chains during imaging. Density functional theory (DFT) calculations of GNRs similar to our experimentally observed structures indicate that both bare zigzag (ZZ) GNRs as well as In-terminated ZZ-GNRs have metallic character, whereas in contrast, In termination induces metallicity for otherwise semiconducting armchair (AC) GNRs. Our findings provide insights into the creation and properties of long linear metal atom chains at graphitic edges.

Graphene-edge-supported iron dual-atom for oxygen reduction electrocatalysts.

Pyrolyzed Fe-N-C-based catalysts, particularly FeN4, are reported to show enhanced catalytic activity for some chemical reactions, particularly for the oxygen reduction reaction (ORR). Here, we present a computational study to investigate another pyrolyzed Fe-N-C-based catalyst, i.e. Fe2N6, adsorbed on graphene with special emphasis on the edges of graphene nanoribbons (both zig-zag and armchair configurations) as a candidate for Fe dual-atom catalysts (Fe-DACs). Utilizing density functional theory calculations along with microkinetic simulations, we investigate the influence of graphitic edges on the stability and ORR activity of Fe-DAC active sites. Our findings indicate that the presence of graphitic edges, particularly the zig-zag configuration, significantly lowers the formation energy of Fe-DAC active sites, making them more likely to form at the edges. Furthermore, several Fe-DAC active sites at graphitic edges exhibit exceptional ORR performance, surpassing the commonly employed FeN4 active site in SAC systems and even exceeding the benchmark Pt(111) surface. Notably, the (Fe2N6)o@z1 active site demonstrates outstanding performance in both associative and dissociative mechanisms. These results highlight the role of graphitic nanopores in enhancing the catalytic behavior of Fe-DAC active sites, providing valuable insights for designing efficient non-precious metal catalysts for ORR applications.

Simulation-Based Recommendations for Digital Circuits Design Using Schottky-Barrier-Type GNRFET

The use of graphene nano-ribbon field-effect transistors (GNRFETs) in the nanoscale circuits design is challenging because there are several adjustable parameters that need to be selected carefully. In this paper, we evaluate the impact of changes in different GNRFETs’ parameters including channel length (Lch), oxide thickness (Tox), line-edge roughness (Pr), number of dimer lines (N), supply voltage, and temperature on the performance of inverter, Flip-Flop, and SRAM circuits. Performance analysis in terms of noise margin (NM), delay, average power, and energy-delay-product (EDP) show that those adjustable parameters of the GNRFETs are of a significant role in attaining low-power or high-performance requirements. To achieve low-power designs, GNRFETs with perfectly smooth edges and higher Tox should be used. For low-power designs with a smaller oxide thickness, it is suggested to select 9 for N with very low Pr. Otherwise, the priority would be choosing 13 for N. To attain high-performance designs with low delay, it is recommended to use the GNRFETs without Pr. If the edges of graphene nano-ribbons are not smooth, choosing (3p+1, ) periodic for N provides a potential improvement in circuit delay. Moreover, selecting 18 for N and smaller Tox provides a potential improvement in circuit delay.

Manipulation of Spin Polarization in Boron-Substituted Graphene Nanoribbons.

The design of magnetic topological states due to spin polarization in an extended π carbon system has great potential in spintronics application. Although magnetic zigzag edges in graphene nanoribbons (GNRs) have been investigated earlier, real-space observation and manipulation of spin polarization in a heteroatom substituted system remains challenging. Here, we investigate a zero-bias peak at a boron site embedded at the center of an armchair-type GNR on a AuSiX/Au(111) surface with a combination of low-temperature scanning tunneling microscopy/spectroscopy and density functional theory calculations. After the tip-induced removal of a Si atom connected to two adjacent boron atoms, a clear Kondo resonance peak appeared and was further split by an applied magnetic field of 12 T. This magnetic state can be relayed along the longitudinal axis of the GNR by sequential removal of Si atoms.

Thermophoresis of carbon nanoparticles (nanoribbons and nanotubes) on a flat multilayer substrate of hexagonal boron nitride (h-BN)

Using the method of molecular dynamics and a 2D chain model, it is shown that thermophoresis of carbon nanoparticles (nanoribbons and nanotubes) on a flat multilayer substrate (on a flat surface of a hexagonal boron nitride crystal) has high efficiency. Placing a nanoparticle on a flat surface of a substrate involved in heat transfer leads to its movement in the direction of the heat flow. The heat flow along the substrate leads to the formation of constant forces acting on the nanoparticle nodes (thermophoresis forces). The main effect of the force is exerted on the edges of graphene nanoribbons, exactly where the main interaction of the nanoribbon with the bending phonons of the substrate occurs. These phonons have a long free path, so the effective transfer of nanoparticles using thermophoresis can occur at sufficiently large distances. The motion of carbon nanoparticles under the action of a heat flow has the form of particle motion in a viscous medium under the action of a constant force. Over time, the nanoparticles always enter the mode of movement at a constant speed. The velocity of the stationary motion is almost the same for all sizes and types of carbon nanoparticles, which is explained by the fact that the thermophoresis force and effective friction have the same source --- the interaction of the nanoparticle with the bending thermal vibrations of the substrate layers. Keywords: nanoribbons, nanotubes, flat multilayer substrates, thermophoresis, 2D model of the multilayer substrate.

Edge state in AB-stacked bilayer graphene and its correspondence with the Su-Schrieffer-Heeger ladder

We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the Chern number of the corresponding two-dimensional (2D) bulk Hamiltonian is zero. The existence and topological features of edge states when two layers ended with the same or different edge terminations (zigzag, bearded, armchair) are discussed. The edge states (non-dispersive bands near the Fermi level) are states localized at the edge of graphene nanoribbon that only exists in certain range of momentum $k_y$. Their existence near the Fermi level are protected by the chiral symmetry with topology well described by coupled Su-Schrieffer-Heeger (SSH) chains model, i.e., SSH ladder, based on the bulk-edge correspondence of one-dimensional (1D) systems. These zero-energy edge states can exist in the whole $k_y$ region when two layers have zigzag and bearded edges, respectively. Winding number calculation shows a topological phase transition between two distinct non-trivial topological phases when crossing the Dirac points. Interestingly, we find the stacking configuration of BLG ribbon is important since they can lead to unexpected edge states without protection from the chiral symmetry both near the Fermi level in armchair-armchair case and in the gap within bulk bands that are away from Fermi level in the general case. The influence of interlayer next nearest neighbor (NNN) interaction and interlayer bias are also discussed to fit the realistic graphene materials, which suggest the robust topological features of edge states in BLG systems.

Электронные свойства графеновых нанолент, допированных пирролоподобным азотом

Doping of graphene nanoribbons with various chemical elements leads to a change in their band structure, which significantly expands the range of applications of these objects in modern electronic devices. In this work, the authors investigate graphene nanoribbons of the «armchair» and «zigzag» types with different concentrations of pyrrole-like nitrogen at the edges. The SCC-DFTB method was used to establish the most energetically favorable configurations of pyrrole-like nitrogen at each edge of graphene nanoribbons. The relationship between the energy gaps of graphene nanoribbons and the content of the considered functional nitrogen-containing groups in them was determined. Calculations have shown that, by incorporating into the atomic lattice, pyrrole-like nitrogen at the «zigzag» edge transfers a greater amount of charge to nearby carbon atoms, which makes such nanoribbons more chemically active in comparison with «armchair» type nanoribbons. Nitrogen doped «zigzag» graphene nanoribbons may be a promising chemoresistive element of nanosensors; however, these conclusions require further calculations.