In this study, based on the self-energy method and the total energy calculation, the indirect exchange coupling between two semi-infinite ferromagnetic strips (FM electrodes) separated by metallic graphene nanoribbons (GNRs) is investigated. In order to form a FM/GNR/FM junction, a graphitic region of finite length is coupled to the FM electrodes along graphitic zigzag or armchair interfaces of width $N$. The numerical results show that, the exchange coupling strength which can be obtained from the difference between the total energies of electrons in the ferromagnetic and antiferromagnetic couplings, has an oscillatory behavior, and depends on the Fermi energy and the length of the central region.
High-quality BN-Graphene-BN nanoribbon capacitors with double side-gates of graphene are experimentally realized. Graphene electronic properties can be significantly modulated by the double side-gates. The modulation effects are very obvious and followed the metallic electrode behavior of numerical simulations, while the theoretically predicted negative quantum capacitance was not observed, possibility due to the over-estimated or weakened interactions between the graphene nanoribbon and side-gate electrodes.
It is shown that apart from well-known factors, like temperature, substrate, and edge reconstruction effects, also the presence of external contacts is destructive for the formation of magnetic moments at the edges of graphene nanoribbons. The edge magnetism gradually decreases when graphene/electrode interfaces become more and more transparent for electrons. In addition to the graphene/electrode coupling strength, also the aspect ratio parameter, i.e. a width/length ratio of the graphene nanoribbon, is crucial for the suppression of edge magnetism. The present theory uses a tight-binding meth…
The magnetic and spin transport properties of a carbon chain between two armchair graphene nanoribbon (AGNR) electrodes were studied using tight-binding Hamiltonian, mean-field Hubbard model and Landauer-Butikker formalism. The results showed that only odd-numbered carbon chains show intrinsic magnetic moments in chain-graphene junctions. It was also found that the electronic, magnetic and spin transport properties of carbon chain-graphene junctions strongly depend on the position and the length of the carbon chains between AGNR electrodes. Interestingly, we found a fully spin-polarized transm…
Atomically precise graphene nanoribbons are a promising emerging class of designer quantum materials with electronic properties that are tunable by chemical design. However, many challenges remain in the device integration of these materials, especially regarding contacting strategies. We report on the device integration of uniaxially aligned and non-aligned 9-atom wide armchair graphene nanoribbons (9-AGNRs) in a field-effect transistor geometry using electron beam lithography-defined graphene electrodes. This approach yields controlled electrode geometries and enables higher fabrication thro…
We study all-carbon-hydrogen molecular transistors where zigzag graphene nanoribbons play the role of three metallic electrodes connected to a ring-shaped 18-annulene molecule. Using the nonequilibrium Green function formalism combined with density functional theory, recently extended to multiterminal devices, we show that the proposed nanostructures exhibit exponentially small transmission when the source and drain electrodes are attached in a configuration that ensures destructive interference of electron paths around the ring. The third electrode, functioning either as an attached infinite-…
Thermoelectric properties of finite graphene nanoribbons (GNRs) coupled to metallic electrodes are theoretically studied in the framework of tight-binding model and Green's function approach. When the zigzag sides are coupled to the electrodes, the electron transport through the localized edge states can occur only if the channel length between electrodes is smaller than the decay length of these localized zigzag edge states. When the armchair edges are coupled to the electrodes, there is an interesting thermoelectric behavior associated with the mid-gap states when the GNR is in the semicondu…
Based on first-principles approaches, we study the ballistic phonon transport properties of finite monatomic carbon chains stretched between graphene nanoribbons, an $sp$-$sp^2$ hybrid carbon nanostructure that has recently seen significant experimental advances in its synthesis. We find that the lattice thermal conductance anomalously increases with tensile strain for the even-numbered carbon chains that adopt the alternating bond-length polyyne configuration. On the other hand, in the odd-numbered carbon chain cases, which assume the equal bond-length cumulene configuration, phonon conductan…
Graphene nanoribbons (GNRs) are a novel and intriguing class of materials in the field of nanoelectronics, since their properties, solely defined by their width and edge type, are controllable with high precision directly from synthesis. Here we study the correlation between the GNR structure and the corresponding device electrical properties. We investigated a series of field effect devices consisting of a film of armchair GNRs with different structures (namely width and/or length) as the transistor channel, contacted with narrowly spaced graphene sheets as the source-drain electrodes. By ana…
Bottom-up synthesized graphene nanoribbons (GNRs) are quantum materials that can be structured with atomic precision, providing unprecedented control over their physical properties. Accessing the intrinsic functionality of GNRs for quantum technology applications requires the manipulation of single charges, spins, or photons at the level of an individual GNR. However, experimentally, contacting individual GNRs remains challenging due to their nanometer-sized width and length as well as their high density on the metallic growth substrate. Here, we demonstrate the contacting and electrical chara…
In this work we numerically calculate the electric current through three kinds of DNA sequences (telomeric, λ-DNA, and p53-DNA) described by different heuristic models. A bias voltage is applied between two zig-zag edged graphene contacts attached to the DNA segments, while a gate terminal modulates the conductance of the molecule. The calculation of current is performed by integrating the transmission function (calculated using the lattice Green's function) over the range of energies allowed by the chemical potentials. We show that a telomeric DNA sequence, when treated as a quantum wire in t…
We overview nonequilibrium Green function combined with density functional theory (NEGF-DFT) modeling of independent electron and phonon transport in nanojunctions with applications focused on a new class of thermoelectric devices where a single molecule is attached to two metallic zigzag graphene nanoribbons (ZGNRs) via highly transparent contacts. Such contacts make possible injection of evanescent wavefunctions from ZGNRs, so that their overlap within the molecular region generates a peak in the electronic transmission. Additionally, the spatial symmetry properties of the transverse propaga…
The effects of tensile strain on the current-voltage (I-V) characteristics of hydrogenated-edge armchair graphene nanoribbons (HAGNRs) are investigated by using DFT theory. The strain is introduced in two different ways related to the two types of systems studied in this work: in-plane strained systems (A), and out-of-plane strained systems due to bending (B). These two kinds of strain lead to make a distinction among three cases: in-plane strained systems with strained electrodes (A1) and with unstrained electrodes (A2), and out-of-plane homogeneously strained systems with unstrained, fixed e…
We present numerical simulations of the capacitive coupling between graphene nanoribbons of various widths and gate electrodes in different configurations. We compare the influence of lateral metallic or graphene side gate structures on the overall back gate capacitive coupling. Most interestingly, we find a complex interplay between quantum capacitance effects in the graphene nanoribbon and the lateral graphene side gates, giving rise to an unconventional negative quantum capacitance. The emerging non-linear capacitive couplings are investigated in detail. The experimentally relevant relative…
With nonequilibrium Green's function approach combined with density functional theory, we perform an ab initio calculation to investigate transport properties of graphene nanoribbon junctions self-consistently. Tight-binding approximation is applied to model the zigzag graphene nanoribbon (ZGNR) electrodes, and its validity is confirmed by comparison with GAUSSIAN03 PBC calculation of the same system. The origin of abnormal jump points usually appearing in the transmission spectrum is explained with the detailed tight-binding ZGNR band structure. Transport property of an edge defect ZGNR junct…
The coherent spin-polarized electron transport through a zigzag-edge graphene flake (ZGF), sandwiched between two semi-infinite armchair graphene nanoribbons, is investigated by means of Landauer-Buttiker formalism. To study the edge magnetism of the ZGF, we use the half-filled Hubbard model within the Hartree-Fock approximation. The results show that the junction acts as a spin filter with high degree of spin polarization in the absence of magnetic electrodes and external fields. By applying a gate voltage the spin-filtering efficiency of this device can be effectively controlled and the spin…
The effects of Coulomb interactions on the electronic properties of bilayer graphene nanoribbons (BGNs) covered by a gate electrode are studied theoretically. The electron density distribution and the potential profile are calculated self-consistently within the Hartree approximation. A comparison to their single-particle counterparts reveals the effects of interactions and screening. Due to the finite width of the nanoribbon in combination with electronic repulsion, the gate-induced electrons tend to accumulate along the BGN edges where the potential assumes a sharp triangular shape. This has…
We report the fabrication and characterization of superconducting quantum interference devices (SQUIDs) made of Sb-doped Bi2Se3 topological insulator (TI) nanoribbon (NR) contacted with PbIn superconducting electrodes. When an external magnetic field was applied along the NR axis, the TI NR exhibited periodic magneto-conductance oscillations, the so-called Aharonov-Bohm oscillations, owing to one-dimensional subbands. Below the superconducting transition temperature of PbIn electrodes, we observed supercurrent flow through TI NR-based SQUID. The critical current periodically modulates with a m…
The nonequilibrium time-dependent fluctuations of charge current have recently emerged as a sensitive experimental tool to probe ballistic transport through evanescent wave functions introduced into clean wide and short graphene strips by the attached metallic electrodes. We demonstrate that such "pseudo-diffusive" shot noise can be substantially modified in zigzag graphene nanoribbon (ZGNR) due to the topology of its edges responsible for localized states that facilitate ferromagnetic ordering along the edge when Coulomb interaction is taken into account. Thus, the shot noise enhancement of u…
Carbon-based nanostructures and graphene, in particular, evoke a lot of interest as new promising materials for nanoelectronics and spintronics. One of the most important issue in this context is the impact of external electrodes on electronic properties of graphene nanoribbons (GNR). The present theoretical method is based on the tight-binding model and a modified recursive procedure for Green's functions. The results show that within the ballistic transport regime, the so called end-contacted geometry (of minimal GNR/electrode interface area), is usually more advantageous for practical appli…
We report on the fabrication and electrical transport properties of superconducting quantum interference devices (SQUIDs) made from a (Bi_{1-x}Sb_x)_2Se_3 topological insulator (TI) nanoribbon (NR) connected with Pb0.5In0.5 superconducting electrodes. Below the transition temperature of the superconducting Pb0.5In0.5 electrodes, periodic oscillations of the critical current are observed in the TI NR SQUID under a magnetic field applied perpendicular to the plane owing to flux quantization. Also the output voltage modulates as a function of the external magnetic field. Moreover, the SQUID the S…
We investigate electronic transport in lithographically patterned graphene ribbon structures where the lateral confinement of charge carriers creates an energy gap near the charge neutrality point. Individual graphene layers are contacted with metal electrodes and patterned into ribbons of varying widths and different crystallographic orientations. The temperature dependent conductance measurements show larger energy gaps opening for narrower ribbons. The sizes of these energy gaps are investigated by measuring the conductance in the non-linear response regime at low temperatures. We find that…
Phase change memory (PCM) devices are known to reduce in power consumption as the bit volume and contact area of their electrodes are scaled down. Here, we demonstrate two types of low-power PCM devices with lateral graphene ribbon electrodes: one in which the graphene is patterned into narrow nanoribbons and the other where the phase change material is patterned into nanoribbons. The sharp graphene "edge" contacts enable switching with threshold voltages as low as ~3 V, low programming currents (<1 μA SET, <10 μA RESET) and ON/OFF ratios >100. Large-scale fabrication with graphene grown by ch…
We demonstrate that giant current and high spin rectification ratios can be achieved in atomic carbon chain devices connected between two symmetric ferromagnetic zigzag-graphene-nanoribbon electrodes. The spin dependent transport simulation is carried out by density functional theory combined with the non-equilibrium Green's function method. It is found that the transverse symmetries of the electronic wave functions in the nanoribbons and the carbon chain are critical to the spin transport modes. In the parallel magnetization configuration of two electrodes, pure spin current is observed in bo…
We have elaborately studied the electronic structure of 555-777 divacancy (DV) defected armchair edged graphene nanoribbon (AGNR) and transport properties of AGNR based two-terminal device constructed with one defected electrode and one N doped electrode, by using density functional theory and non-equilibrium Green's function based approach. The introduction of 555-777 DV defect into AGNRs, results in a shifting of the π and π* bands towards the higher energy value which indicates a shifting of the Fermi level towards the lower energy. Formation of a potential barrier, very similar to that of …
Coherent spin-dependent transport through a junction containing of Normal/Ferromagnetic/Normal bilayer graphene nanoribbon with zigzag edges is investigated by using Landauer formalism. In a more realistic set-up, the exchange field is induced by two ferromagnetic insulator strips deposited on the ribbon edges while a perpendicular electric field is applied by the top gated electrodes. Our results show that, for antiparallel configuration, a band gap is opened giving rise a semiconducting behavior, while for parallel configuration, the band structure has no band gap. As a result, a giant magne…
We study thermoelectric devices where a single 18-annulene molecule is connected to metallic zigzag graphene nanoribbons (ZGNR) via highly transparent contacts that allow for injection of evanescent wave functions from ZGNRs into the molecular ring. Their overlap generates a peak in the electronic transmission, while ZGNRs additionally suppress hole-like contributions to the thermopower. Thus optimized thermopower, together with suppression of phonon transport through ZGNR-molecule-ZGNR structure, yield the thermoelectric figure of merit ZT ~ 0.5 at room temperature and 0.5 < ZT < 2.5 below li…
We calculate the local current density in pristine armchair graphene nanoribbons (AGNRs) with varying width, $N_\mathrm{C}$, employing a density-functional-theory-based ab initio transport formalism. We observe very pronounced current patterns (streamlines) with threefold periodicity in $N_\mathrm{C}$. They arise as a consequence of quantum confinement in the transverse flow direction. Neighboring streamlines are separated by stripes of almost vanishing flow. As a consequence, the response of the current to functionalizing adsorbates is very sensitive to their placement: adsorbates located wit…
We investigate chiral graphene nanoribbons using projective quantum Monte Carlo simulations within the local Hubbard model description and study the effects of electron-electron interactions on the electronic and magnetic properties at the ribbon edges. Static and dynamical properties are analyzed for nanoribbons of varying width and edge chirality, and compared to a self-consistent Hartee-Fock mean-field approximation. Our results show that for chiral ribbons of sufficient width, the spin correlations exhibit exceedingly long correlation lengths, even between zigzag segments that are well sep…
We calculate quantum transport for metal-graphene nanoribbon heterojunctions within the atomistic self-consistent Schrödinger/Poisson scheme. Attention is paid on both the chemical aspects of the interface bonding as well the one-dimensional electrostatics along the ribbon length. Band-bending and doping effects strongly influence the transport properties, giving rise to conductance asymmetries and a selective suppression of the subband formation. Junction electrostatics and p-type characteristics drive the conduction mechanism in the case of high work function Au, Pd and Pt electrodes, while …
Quantum-dot states in graphene nanoribbons (GNR) were calculated using density-functional theory, considering the effect of the electric field of gate electrodes. The field is parallel to the GNR plane and was generated by an inhomogeneous charge sheet placed atop the ribbon. Varying the electric field allowed to observe the development of the GNR states and the formation of localized, quantum-dot-like states in the band gap. The calculation has been performed for armchair GNRs and for armchair ribbons with a zigzag section. For the armchair GNR a static dielectric constant of ε approx. 4 coul…
In this work we develop a time-dependent extension of the Landauer-Büttiker approach to study transient dynamics in time-dependent quantum transport through molecular junctions. A key feature of the approach is that it provides a closed integral expression for the time-dependence of the density matrix of the molecular junction after switch-on of a bias or gate potential which can be evaluated without the necessity of propagating individual single-particle orbitals. This allows for the study of time-dependent transport in large molecular systems coupled to wide band leads. As an application of …
We report on the fabrication and transport characterization of atomically-precise single molecule devices consisting of a magnetic porphyrin covalently wired by graphene nanoribbon electrodes. The tip of a scanning tunneling microscope was utilized to contact the end of a GNR-porphyrin-GNR hybrid system and create a molecular bridge between tip and sample for transport measurements. Electrons tunneling through the suspended molecular heterostructure excited the spin multiplet of the magnetic porphyrin. The detachment of certain spin-centers from the surface shifted their spin-carrying orbitals…
In graphene nanoribbons (GNRs), the lateral confinement of charge carriers opens a band gap, the key feature to enable novel graphene-based electronics. Successful synthesis of GNRs has triggered efforts to realize field-effect transistors (FETs) based on single ribbons. Despite great progress, reliable and reproducible fabrication of single-ribbon FETs is still a challenge that impedes applications and the understanding of the charge transport. Here, we present reproducible fabrication of armchair GNR-FETs based on a network of nanoribbons and analyze the charge transport mechanism using nine…
The edge reconstruction of zigzag graphene nanoribbons to a stable line of alternatively fused seven and five membered rings with hydrogen passivation has been studied within density functional theory with both localized and extended basis approximations. Reconstruction of both edges results in a nonmagnetic metallic ground state, whereas the one edge reconstruction stabilizes the system in a ferromagnetic metallic ground state. The reconstructed edge suppresses the local spin density of atoms and contributes finite density of states at Fermi energy. Our study paves a new way to fabricate the …
This contribution reports on comparative studies on giant magnetoresistance (GMR) in carbon nanotubes (CNTs) and graphene nanoribbons of similar aspect ratios (i.e perimeter/length and width/length ratios, for the former and the latter, respectively). The problem is solved at zero temperature in the ballistic transport regime, by means of the Green's functions technique within the tight-binding model and with the so-called wide band approximation for electrodes. The GMR effect in graphene is comparable to that of CNTs, it depends strongly on the chirality and only slightly on the aspect ratio.…
Atomically precise graphene nanoribbons (GNRs) are increasingly attracting interest due to their largely modifiable electronic properties, which can be tailored by controlling their width and edge structure during chemical synthesis. In recent years, the exploitation of GNR properties for electronic devices has focused on GNR integration into field-effect-transistor (FET) geometries. However, such FET devices have limited electrostatic tunability due to the presence of a single gate. Here, we report on the device integration of 9-atom wide armchair graphene nanoribbons (9-AGNRs) into a multi-g…
We present electrical transport measurements of a van-der-Waals heterostructure consisting of a graphene nanoribbon separated by a thin boron nitride layer from a micron-sized graphene sheet. The interplay between the two layers is discussed in terms of screening or, alternatively, quantum capacitance. The ribbon can be tuned into the transport gap by applying gate voltages. Multiple sites of localized charge leading to Coulomb blockade are observed in agreement with previous experiments. Due to the strong capacitive coupling between the ribbon and the graphene top layer sheet, the evolution o…
Recently, porphyrin units have been attached to graphene nanoribbons (Por-GNR) enabling a multitude of possible structures. Here we report first principles calculations of two prototypical, experimentally feasible, Por-GNR hybrids, one of which displays a small band gap relevant for its use as electrode in a device. Embedding a Fe atom in the porphyrin causes spin polarization with a spin ground state $S=1$. We employ density functional theory and nonequilibrium Green's function transport calculations to examine a 2-terminal setup involving one Fe-Por-GNR between two metal-free, small band gap…
Transport and thermoelectric properties of finite textured graphene nanoribbons (t-GNRs) connected to electrodes with various coupling strengths are theoretically studied in the framework of the tight-binding model and Green's function approach. Due to quantum constriction induced by the indented edges, such t-GNRs behave like serially-coupled graphene quantum dots (SGQDs). These types of SGQDs can be formed by tailoring zigzag GNRs (ZGNRs) or armchair GNRs (AGNRs). Their bandwidths and gaps can be engineered by varying the size of the quantum dot and the neck width at indented edges. Effects …
We will present brief overview on the electronic and transport properties of graphene nanoribbons focusing on the effect of edge shapes and impurity scattering. The low-energy electronic states of graphene have two non-equivalent massless Dirac spectrum. The relative distance between these two Dirac points in the momentum space and edge states due to the existence of the zigzag type graphene edges are decisive to the electronic and transport properties of graphene nanoribbons. In graphene nanoribbons with zigzag edges, two valleys related to each Dirac spectrum are well separated in momentum s…
The recent fabrication of graphene nanoribbon (GNR) field-effect transistors poses a challenge for first-principles modeling of carbon nanoelectronics due to many thousand atoms present in the device. The state of the art quantum transport algorithms, based on the nonequilibrium Green function formalism combined with the density functional theory (NEGF-DFT), were originally developed to calculate self-consistent electron density in equilibrium and at finite bias voltage (as a prerequisite to obtain conductance or current-voltage characteristics, respectively) for small molecules attached to me…
The low-energy spectrum of graphene nanoribbons with armchair edges (armchair nanoribbons) is described as the superposition of two non-equivalent Dirac points of graphene. In spite of the lack of well-separated two valley structures, the single-channel transport subjected to long-ranged impurities is nearly perfectly conducting, where the backward scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wavefunction. For multi-channel energy regime, however, the conventional exponential decay of the averaged conductance occurs. Since the int…
The electronic structure and conductance of substitutionally edge-doped zigzag silicene nanoribbons (ZSiNRs) are investigated using the nonequilibrium Green's function method combined with the density functional theory. Two-probe systems of ZSiNRs in both ferromagnetic and antiferromagnetic states are considered. Doping effects of elements from groups III and V, in a parallel or antiparallel magnetic configuration of the two electrodes, are discussed. Switching on and off the external magnetic field, we may convert the metallic ferromagnetic ZSiNRs into insulating antiferromagnetic ZSiNRs. In …
We investigate the production of spin-polarized currents in corrugated graphene nanoribbons. Such corrugations are modeled as multiple regions with Rashba spin-orbit interactions, where concave and convex curvatures are treated as Rashba regions with opposite signs. Numerical examples for different separated Rashba-zone geometries calculated within the tight-binding approximation are provided. Remarkably, the spin-polarized current in a system with several Rashba areas can be enhanced with respect to the case with a single Rashba part of the same total area. The enhancement is larger for confi…
In this paper, we study the conductance of the graphene nanoribbons(GNRs) in the presence of the Stone-Wales(S-W) reconstruction, using the transfer matrix method. The ribbon is connected with semi-infinite quantum wires as the leads. The S-W reconstruction occurs on the edges and the interfaces between the electrodes and ribbon. When the reconstruction occurs on the edges, the conductance is suppressed considerably if the gate voltage $V_g$ takes intermediate values around $|V_{g}|\sim t_0$($t_0$ being the hopping amplitude of grahene) in both positive and negative energy regions. In contrast…
Graphene nanoribbons (GNRs) have attracted a strong interest from researchers worldwide, as they constitute an emerging class of quantum-designed materials. The major challenges towards their exploitation in electronic applications include reliable contacting, complicated by their small size ($<$50 nm), as well as the preservation of their physical properties upon device integration. In this combined experimental and theoretical study, we report on the quantum dot (QD) behavior of atomically precise GNRs integrated in a device geometry. The devices consist of a film of aligned 5-atoms wide GNR…
Liquid-phase-exfoliation is a technique capable of producing large quantities of two-dimensional material in suspension. Despite many efforts in the optimization of the exfoliation process itself not much has been done towards the integration of liquid-phase-exfoliated materials in working solid-state devices. In this article, we use dielectrophoresis to direct the assembly of liquid-phase-exfoliated TiS3 nanoribbons between two gold electrodes to produce photodetectors working in the visible. Through electrical and optical measurements we characterize the responsivity of the device and we fin…
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…
Impact of Valley Degeneracy on Thermoelectric Properties of Zigzag Graphene Nanoribbons with Staggered Sublattice Potentials and Transverse Electric Fields
Excitonic spectra of armchair graphene nanoribbons (AGNRs) obtained from a full many-body exact diagonalization of the Hubbard model are reported for both longitudinally and transversely polarized photons, thus providing a complete survey of low-energy may-body optical excitations in these systems. The resulting one-photon allowed eigenstates turn out to be well separated in energy from each other but both couple to the same set of two-photon allowed states. The magnitude of the calculated optical oscillator strengths for perpendicular polarization suggest that these optical features can be in…
The concept of a novel graphene P-I-N junction switching device with a nanoribbon is proposed, and its basic operation is demonstrated in an experiment. The concept aims to optimize the operation scheme for graphene transistors toward a superior on-off property. The device has two bulk graphene regions where the carrier type is electrostatically controlled by a top gate, and these two regions are separated by a nanoribbon which works as an insulator. As a result, the device forms a (P or N)-I-(P or N) junction structure. The off state is obtained by lifting the band of the bulk graphene of the…
We present an analytical investigation of the quasi-Coulomb impurity states in a narrow gapped armchair graphene nanoribbon (GNR) in the presence of a uniform external electric field directed parallel to the ribbon axis. The effect of the ribbon confinement is taken to be much greater than that of the impurity electric field, which in turn considerably exceeds the external electric field. Under these conditions we employ the adiabatic approximation assuming that the motion parallel ("slow") and perpendicular ("fast") to the ribbon axis are separated adiabatically. In the approximation of the i…
