Bilayer graphene has been a subject of intense study in recent years. We extend a structural phase field crystal method to include an external potential from adjacent layer(s), which is generated by the corresponding phase field and changes over time. Moreover, multiple layers can be added into the structure. Using the thickness of the boundaries between different stacking variants of the bilayer structure as the key parameter, we quantify the strength of the adjacent layer potential by comparing with atomistic simulation results. We then test the multiple graphene structures, including bilayers, triple layers, up to 6 layers. We find that besides the initial conditions, the way of new layers added into the structure will also affect the layout of the atomic configuration. We believe tour results can help understanding the mechanism of graphene structure consists of more than one layer.
In this paper we review thermal properties of graphene and multilayer graphene and discuss the optothermal technique developed for the thermal conductivity measurements. We also outline different theoretical approaches used for the description of phonon transport in graphene and provide comparison with available experimental thermal conductivity data.
We calculate the static polarizability of multilayer graphene and study the effect of stacking arrangement, carrier density, and onsite energy difference on graphene screening properties. At low densities, the energy spectrum of multilayer graphene is described by a set of chiral two-dimensional electron systems and the associated chiral nature determines the screening properties of multilayer graphene showing very different behavior depending on whether the chirality index is even or odd. As density increases, the energy spectrum follows that of the monolayer graphene and thus the polarizabil…
We study the electronic structure of multilayer graphene using a $π$-orbital continuum model with nearest-neighbor intralayer and interlayer tunneling. Using degenerate state perturbation theory, we show that the low-energy electronic structure of arbitrarily stacked graphene multilayers consists of chiral pseudospin doublets with a conserved chirality sum.
We study the optical transmittance of multilayer graphene films up to 65 layers thick. By combing large-scale tight-binding simulation and optical measurement on CVD multilayer graphene, the optical transmission through graphene films in the visible region is found to be solely determined by the number of graphene layers. We argue that the optical transmittance measurement is more reliable in the determination of the number of layers than the commonly used Raman Spectroscopy. Moreover, optical transmittance measurement can be applied also to other 2D materials with weak van der Waals interlaye…
We address the intrinsic polarisation and screening of external electric field in a broad range of ordered and twisted configurations of multilayer graphene, using an ab initio approach combining density functional theory and the Wannier function formalism. We show that multilayer graphene is intrinsically polarized due to the crystal field effect, an effect that is often neglected in tight-binding models of twisted bilayer graphene and similar systems. This intrinsic polarization of the order of up to few tens of meVs has different out-of-plane alignments in ordered and twisted graphene multi…
We discuss the chiral decomposition of non-symmetric stacking structures. It is shown that the low-energy electronic structure of Bernal stacked graphene multilayer deposited on h-BN consists of chiral pseudospin doublets. N-layer graphene stocks on h-BN layer have N/2 effective bilayer graphene systems and one effective h-BN layer if N is even or (N-1)/2 effective graphene bilayers plus one graphene monolayer modified by h-BN layer if N is odd. We present the decomposition procedure and we derive the recurrence relations for the effective parameters characterizing the chiral subsystems. The e…
By limiting the carbon segregation at the copper surface defects, a pulsed chemical vapor deposition method for single layer graphene growth is shown to inhibit the formation of few-layer regions, leading to a fully single-layered graphene homogeneous at the centimeter scale. Graphene field-effect devices obtained after transfer of pulsed grown graphene on oxidized silicon exhibit mobilities above 5000 cm^2.V^-1.s^-1.
Rhombohedral stacked multilayer graphene displays the occurrence of a magnetic surface state at low temperatures. Recent angular resolved photoemission experiments demonstrate the robustness of the magnetic state in long sequences of ABC graphene. Here, by using first-principles calculations, we show that field-effect doping of these graphene multilayers induces a perfect half-metallic behaviour with 100% of spin current polarization already at dopings attainable in conventional field effect transistors with solid state dielectrics. Our work demonstrates the realisability of a new kind of spin…
We report the synthesis of AB-stacked multilayer graphene via ambient pressure chemical vapor deposition on Cu foils, and demonstrate a method to construct suspended multilayer graphene devices. In four-terminal geometry, such devices were characterized by hot carrier transport at temperatures down to 240 mK and in magnetic fields up to 14 T. The differential conductance (dI/dV) shows a characteristic dip at longitudinal voltage bias V=0 at low temperatures, indicating the presence of hot electron effect due to a weak electron-phonon coupling. Under magnetic fields, the magnitude of the dI/dV …
Edges naturally exist in a single-layer graphene sample. Similarly, individual graphene layers in a multilayer graphene sample contribute their own edges. We study the Raman spectrum at the edge of a graphene layer laid on n layer graphene (nLG). We found that the D band observed from the edge of the top graphene layer exhibits an identical line shape to that of disordered (n+1)LG induced by ion-implantation. Based on the spectral features of the D and 2D bands, we identified two types of alignment configurations at the edges of mechanically-exfoliated bilayer and trilayer graphenes, whose edg…
With the motivation of improving the performance and reliability of aggressively scaled nano-patterned graphene field-effect transistors, we present the first systematic experimental study on charge and current distribution in multilayer graphene field-effect transistors. We find a very particular thickness dependence for Ion, Ioff, and the Ion/Ioff ratio, and propose a resistor network model including screening and interlayer coupling to explain the experimental findings. In particular, our model does not invoke modification of the linear energy-band structure of graphene for the multilayer c…
We present a simple analytical approach to study anharmonic effects in single layer, bilayer, and multilayer graphene. The coupling between in plane and out of plane modes leads to negative Grüneisen coefficients and negative thermal expansion. The value of the thermal expansion coefficient depends on the coupling to the substrate. The bending rigidity in bilayer graphene shows a crossover between a long wavelength regime where its value is determined by the in plane elastic properties and a short wavelength regime where its value approaches twice that of a single layer.
We suggest a new class of hyperbolic metamaterials for THz frequencies based on multilayer graphene structures. We calculate the dielectric permittivity tensor of the effective nonlocal medium with a periodic stack of graphene layers and demonstrate that tuning from elliptic to hyperbolic dispersion can be achieved with an external gate voltage. We reveal that such graphene structures can demonstrate a giant Purcell effect that can be used for boosting the THz emission in semiconductor devices. Tunability of these structures can be enhanced further with an external magnetic field which leads t…
We use a tight-binding model and the random-phase approximation to study the Coulomb excitations in simple-hexagonal-stacking multilayer graphene and discuss the field effects. The calculation results include the energy bands, the response functions, and the plasmon dispersions. A perpendicular electric field is predicted to induce significant charge transfer and thus capable of manipulating the energy, intensity, and the number of plasmon modes. This could be further validated by inelastic light scattering or electron-energy-loss spectroscopy.
The presence of twist angles between layers of two-dimensional materials has a profound impact on their physical properties. Turbostratic multilayer graphene is a system containing a distribution of rotational stacking faults, and these interfaces also have variable twist angles. In this work, we examine the influence of turbostratic single-layer graphene content on the in-plane thermal conductivity of a defect free multilayer graphene system with low defect density. Detailed Raman mode analysis is used to quantify the content of turbostratic single-layer graphene in the system while complemen…
We present a theoretical study on the electron transmission through the AB-BA stacking boundary in multilayer graphene. Using the tight-binding model and the transfer matrix method, we calculate the electron transmission probability through the boundary as a function of electron Fermi energy in multilayers from bilayer to five-layer. We find that the transmission is strongly suppressed particularly near the band touching point, suggesting that the electronic conductivity in general multilayer graphenes is significantly interfered by stacking fault. The conductivity suppression by stacking faul…
We demonstrate electrical spin injection into multilayer graphene (MLG) in a lateral spin valve device from a highly spin-polarized Co2FeSi (CFS) Huesler electrode. Exfoliated MLG was transferred onto pre-patterned epitaxial CFS wires grown on an Si(111) substrate by a polymer-based transfer method. This method enabled us to fabricate multiple single-crystal CFS electrodes in contact with MLG. Electrical spin injection from CFS to MLG was detected through non-local magnetoresistance (MR) measurement. A non-local spin signal of 430 ohm was observed; this is the largest value among all reported …
Low-frequency noise with the spectral density S(f)~1/f^g (f is the frequency and g~1) is a ubiquitous phenomenon, which hampers operation of many devices and circuits. A long-standing question of particular importance for electronics is whether 1/f noise is generated on the surface of electrical conductors or inside their volumes. Using high-quality graphene multilayers we were able to directly address this fundamental problem of the noise origin. Unlike the thickness of metal or semiconductor films, the thickness of graphene multilayers can be continuously and uniformly varied all the way dow…
There has been a lot of excitement around the observation of superconductivity in twisted bilayer graphene, associated to flat bands close to the Fermi level. Such correlated electronic states also occur in multilayer rhombohedral stacked graphene (RG), which has been receiving increasing attention in the last years. In both natural and artificial samples however, multilayer stacked Bernal graphene (BG) occurs more frequently, making it desirable to determine what is their relative stability and under which conditions RG might be favored. Here, we study the energetics of BG and RG in bulk and …
We report the existence of zero energy surface states localized at zigzag edges of $N$-layer graphene. Working within the tight-binding approximation, and using the simplest nearest-neighbor model, we derive the analytic solution for the wavefunctions of these peculiar surface states. It is shown that zero energy edge states in multilayer graphene can be divided into three families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene; (ii) states with finite amplitude over the two last, or the two first layers of the stack, equivalent to surface states …
Most experimental studies to date of multilayer epitaxial graphene on C-face SiC have indicated that the electronic states of different layers are decoupled as a consequence of rotational stacking. We have measured the third order nonlinear tensor in epitaxial graphene as a novel approach to probe interlayer electronic coupling, by studying THz emission from coherently controlled photocurrents as a function of the optical pump and THz beam polarizations. We find that the polarization dependence of the coherently controlled THz emission expected from perfectly uncoupled layers, i.e. a single gr…
We calculate the conductivity of arbitrarily stacked multilayer graphene sheets within a relaxation time approximation, considering both short-range and long-range impurities. We theoretically investigate the feasibility of identifying the stacking order of these multilayers using transport measurements. For relatively clean samples, the conductivities of the various stacking configurations depend on the carrier density as a power-law for over two decades. This dependence arises from a low density decomposition of the multilayer band structure into a sum of chiral Hamiltonians. For dirty sampl…
We synthesize large-area graphene via atmospheric-pressure (AP) chemical vapor deposition (CVD) on copper, and transfer to SiO2 wafers. In contrast to low-pressure (LP) CVD on copper, optical contrast and atomic force microscopy measurements show AP-CVD graphene contains significant multi-layer areas. Raman spectroscopy always shows a single Lorentzian 2D peak, however systematic differences are observed in the 2D peak energy, width, and intensity for single- and multi-layer regions. We conclude that graphene multi-layers grown by AP-CVD on Cu are rotationally disordered.
In this work, we study structural and vibrational properties of multilayer graphene using density-functional theory (DFT) with van der Waals (vdW) functionals. Initially, we analyze how different vdW functionals compare by evaluating the lattice parameters, elastic constants and vibrational frequencies of low energy optical modes of graphite. Our results indicate that the vdW-DF1-optB88 functional has the best overall performance on the description of vibrational properties. Next, we use this functional to study the influence of the vdW interactions on the structural and vibrational properties…
Scanning tunneling microscopy (STM) images are obtained for the first time on few layer and twisted multilayer epitaxial graphene states synthesized on n+ 6H-SiC a-plane non-polar surface. The twisted graphene is determined to have a rotation angle of 5.4° between the top two layers, by comparing moiré patterns from stick and ball models of bilayer graphene to experimentally obtained images. Furthermore, the experimental moiré pattern shows dynamic behavior, continuously shuffling between two stable surface arrangements one bond length apart. The moiré pattern shifts by more than 1 nm, making …
When electrons are confined in a two dimensional (2D) system, typical quantum mechanical phenomena such as Landau quantization can be detected. Graphene systems, including the single atomic layer and few-layer stacked crystals, are ideal 2D materials for studying a variety of quantum mechanical problems. In this article, we review the experimental progress in the unusual Landau quantized behaviors of Dirac fermions in monolayer and multilayer graphene by using scanning tunneling microscopy(STM) and scanning tunneling spectroscopy(STS). Through STS measurement of the strong magnetic fields, dis…
We demonstrate the existence of a new type of zero energy state associated to vacancies in multilayer graphene that has a finite amplitude over the layer with a vacancy and adjacent layers, and the peculiarity of being quasi-localized in the former and totally delocalized in the adjacent ones. In a bilayer, when a gap is induced in the system by applying a perpendicular electric field, these states become truly localized with a normalizable wavefunction. A transition from a localized to an extended state can be tuned by the external gate for experimentally accessible values of parameters.
In the context of graphene-based composite applications, a complete understanding of charge conduction in multilayer reduced graphene oxides (rGO) is highly desirable. However, these rGO compounds are characterized by multiple and different sources of disorder depending on the chemical method used for their synthesis. Most importantly the precise role of interlayer interaction in promoting or jeopardizing electronic flow remains unclear. Here, thanks to the development of a multiscale computational approach combining first-principles calculations with large scale transport simulations, the tra…
We calculate the temperature dependent long-range magnetic coupling in the presence of dilute concentrations of random magnetic impurities in chiral multilayer two-dimensional semimetals, i.e., undoped intrinsic multilayer graphene. Assuming a carrier-mediated indirect RKKY exchange interaction among the well-separated magnetic impurities with the itinerant carriers mediating the magnetic interaction between the impurities, we investigate the magnetic properties of intrinsic multilayer graphene using an effective chiral Hamiltonian model. We find that due to the enhanced density of states in t…
Carbon nanomaterials continue to amaze scientists due to their exceptional physical properties. Recently there have been theoretical predictions and first reports on graphene multilayers, where, due to the rotation of the stacked layers, outstanding electronic properties are retained while the susceptibility to degradation and mechanical stress is strongly reduced due to the multilayer nature. Here we show that fully turbostratic multilayer graphitic microstructures combine the high charge carrier mobilities necessary for advanced electronic and spintronic devices with the robustness of graphi…
Circular polarization resolved magneto-infrared studies of multilayer epitaxial graphene (MEG) are performed using tunable quantum cascade lasers in high magnetic fields up to 17.5 T. Landau level (LL) transitions in the monolayer and bilayer graphene inclusions of MEG are resolved, and considerable electron-hole asymmetry is observed in the extracted electronic band structure. For monolayer graphene, a four-fold splitting of the $n=0$ to $n=1$ LL transition is evidenced and attributed to the lifting of the valley and spin degeneracy of the zeroth LL and the broken electron-hole symmetry. The …
Active sites and catalytic mechanism of nitrogen-doped graphene in oxygen reduction reaction (ORR) have been extensively studied but are still inconclusive, partly due to the lack of an experimental method that can detect the active sites. It is proposed in this report that the active sites on nitrogen-doped graphene can be determined via the examination of its chemical composition change before and after ORR. Synchrotron-based X-ray photoelectron spectroscopy analyses of three nitrogen-doped multilayer graphene samples reveal that oxygen reduction intermediate OH(ads) which should chemically …
We propose a mechanism for the quenching of the Shubnikov de Haas oscillations and the quantum Hall effect observed in epitaxial graphene. Experimental data show that the scattering time of the conduction electron is magnetic field dependent and of the order of the cyclotron orbit period, \textit{i.e.} can be much smaller than the zero field scattering time. Our scenario involves the extraordinary graphene $n=0$ Landau level of the uncharged layers that produces a high density of states at the Fermi level. We find that the coupling between this $n=0$ Landau level and the conducting states of t…
We report on spatial measurements of the superconducting proximity effect in epitaxial graphene induced by a graphene-superconductor interface. Superconducting aluminum films were grown on epitaxial multilayer graphene on SiC. The aluminum films were discontinuous with networks of trenches in the film morphology reaching down to exposed graphene terraces. Scanning tunneling spectra measured on the graphene terraces show a clear decay of the superconducting energy gap with increasing separation from the graphene-aluminum edges. The spectra were well described by Bardeen-Cooper-Schrieffer (BCS) …
We investigate the optical properties of bromine intercalated highly orientated pyrolytic graphite (Br-HOPG) and provide a novel interpretation of the data. We observe new absorption features below 620 meV which are absent in the absorption spectrum of graphite. Comparing our results with those of theoretical studies on graphite, single and bilayer graphene as well as recent optical studies of multilayer graphene, we conclude that Br-HOPG contains the signatures of ultrapure bilayer, single layer graphene, and graphite. The observed supermetallic conductivity of Br-HOPG is identified with the …
We report a method to promote photoluminescence emission in graphene materials by enhancing carrier scattering instead of directly modifying band structure in multilayer reduced graphene oxide (rGO) nanospheres. We intentionally curl graphene layers to form nanospheres by reducing graphene oxide with spherical polymer templates to manipulate the carrier scattering. These nanospheres produce hot-carrier luminescence with more than ten-fold improvement of emission efficiency as compared to planar nanosheets. With increasing excitation power, hot-carrier luminescence from nanospheres exhibits abn…
We investigate the competition between acoustic-phonon-mediated superconductivity and the long-range Coulomb interaction in moiréless graphene multilayers, specifically, Bernal bilayer graphene, rhombohedral trilayer graphene, and ABCA-stacked tetralayer graphene. In these graphene multilayers, the acoustic phonons can realize, through electron-phonon coupling, both spin-singlet and spin-triplet pairings, and the intra-sublattice pairings ($s$-wave spin-singlet and $f$-wave spin-triplet) are the dominant channels. Our theory naturally explains the distinct recent experimental findings in Berna…
Graphene sustains transverse out-of-plane mechanical vibrations (flexural phonons). At the nanometer scale, these appear as travelling ripples, or cavities, if excited in counter-phase in alternate multilayers. In this work we explore by means of classical molecular dynamics simulations the possibility of using these moving nano-cavities to actively transport hydrogen. We find that the gas can be efficiently transported for hundreds of nanometers in the wave propagation direction, before the phonons damp down. Therefore, this effect could be used to move and pump gases through multilayers grap…
The intrinsic resistance peak (ridge) structures were recently found to appear in the carrier density dependence plot of the resistance of the AB-stacked multilayer graphene with even numbers of layers.The ridges are due to topological changes in the Fermi surface. Here, these structures were studied in AB-stacked multilayer graphene with odd numbers of layers (5 and 7 layers) by performing experiments using encapsulated high-quality graphene samples equipped with top and bottom gate electrodes.The intrinsic resistance peaks that appeared on maps plotted with respect to the carrier density and…
The Seebeck coefficient in multilayer graphene is investigated within the density-functional theory, using the semiclassical Boltzmann equations and interpolating the bands in a maximally-localized Wannier functions basis set. We compare various graphene stackings (AA, AB and ABC) both free-standing and deposited on a $4H$-SiC(0001) C-terminated substrate. We find that the presence of the SiC substrate can significantly affect the thermopower properties of graphene layers, depending on the stacking, providing a promising way to tailor efficient graphene-based devices.
The unusual transport properties of graphene are the direct consequence of a peculiar bandstructure near the Dirac point. We determine the shape of the pi bands and their characteristic splitting, and the transition from a pure 2D to quasi-2D behavior for 1 to 4 layers of graphene by angle-resolved photoemission. By exploiting the sensitivity of the pi bands to the electronic potential, we derive the layer-dependent carrier concentration, screening length and strength of interlayer interaction by comparison with tight binding calculations, yielding a comprehensive description of multilayer gra…
We investigate electrical conduction and breakdown behavior of 3D nano-carbon-stacked multilayer graphene (s-MLG) system with complete interlayer decoupling. The s-MLG is prepared by transferring and stacking large-area CVD-grown graphene monolayers, followed by wire patterning and plasma etching. Raman spectroscopy was used to confirm the layer number. The D-band peak indicates low defect level in the samples. Electrical current stressing induced doping is performed to shift the charge-neutrality Dirac point and decrease the graphene/metal contact resistance, improving the overall electrical …
The mechanical response of single and multiple graphene sheets under uniaxial compressive loads was studied with molecular dynamics simulations, using different semi-empirical force fields at different boundary conditions or constrains. Compressive stress-strain curves were obtained and the critical stress/strain values were derived. For single layer graphenes, the critical stress/strain for buckling was found to scale to the inverse length square. For multilayer graphenes the critical buckling stress also decreased with increasing length, though at a slower rate than expected from elastic buc…
We show that the origin of the universal optical conductivity in a normal $N$-layer graphene multilayer is an emergent chiral symmetry which guarantees that $σ(ω)=Nσ_{uni}$ in both low and high frequency limits. [$σ_{uni}=(π/2) e^2/h$]. We use this physics to relate intermediate frequency conductivity trends to qualitative characteristics of the multilayer stacking sequence.
A single graphene layer placed between two parallel Ni(111) surfaces screens the strong attractive force and results in a significant reduction of adhesion and sliding friction. When two graphene layers are inserted, each graphene is attached to one of the metal surfaces with a significant binding and reduces the adhesion further. In the sliding motion of these surfaces the transition from stick-slip to continuous sliding is attained, whereby non-equilibrium phonon generation through sudden processes is suppressed. The adhesion and corrugation strength continues to decrease upon insertion of t…
We propose a generalized Peierls substitution method in conjunction with the tight-binding model to explore the magnetic quantization and quantum Hall effect in twisted multilayer graphene under a magnetic field. The Bloch-basis tight-binding Hamiltonian is constructed for large twist angle while a simplified tight-binding model is employed for the magic angle. We investigate extensively the band structures, Landau levels (LLs), and quantum Hall conductivity (QHC) of twisted bilayer graphene and twisted double bilayer graphene, as well as their dependence on the twist angle. Comparison between…
We present a formalism and numerical results for the energy loss of a charged particle scattered at an arbitrary angle from epitaxially grown multilayer graphene (MLG). It is compared with that of free-standing graphene layers. Specifically, we investigated the effect of the substrate induced energy gap on one of the layers. The gap yields collective plasma oscillations whose characteristics are qualitatively and quantitatively different from those produced by Dirac fermions in gapless graphene. The range of wave numbers for undamped self-sustaining plasmons is increased as the gap is increase…
Plasmon resonance in nanopatterned single layer graphene nanoribbon (SL-GNR), double layer graphene nanoribbon (DL-GNR) and triple layer graphene nanoribbon (TL-GNR) structures is studied both experimentally and by numerical simulations. We use 'realistic' graphene samples in our experiments to identify the key bottle necks in both experiments and theoretical models. The existence of electrical tunable plasmons in such stacked multilayer GNRs was first experimentally verified by infrared microscopy. We find that the strength of the plasmonic resonance increases in DL-GNR when compared to SL-GN…
We systematically calculate thermopower of biased and unbiased multilayer grphene systems. The effect of screening to a bias field perpendicular to the graphene planes is taken into account self-consistently under the Hartree approximation. The model including nearest neighbor hopping and the more complete Slonczewski-Weiss-McClure (SWMcC) model are both considered for a comparison. The effect of impurity scattering is studied for monolayer and unbiased bilayer graphene and is treated in terms of the self-consistent Born approximation. For a monolayer graphene, only when the effect of impurity…
The intrinsic channel properties of monolayer and multilayer graphene were systematically investigated as a function of layer number by the exclusion of contact resistance using four-probe measurements. We show that the continuous change in normalized sheet resistivity from graphite to a bilayer graphene is governed by one unique property, i.e., the band overlap, which markedly increases from 1 meV for a bilayer graphene to 11 meV for eight layers and eventually reaches 40 meV for graphite. The monolayer graphene, however, showed a deviation in temperature dependence due to a peculiar linear d…
The electronic band structure of as-grown and doped graphene grown on the carbon face of SiC is studied by high-resolution angle-resolved photoemission spectroscopy, where we observe both rotations between adjacent layers and AB-stacking. The band structure of quasi-freestanding AB- bilayers is directly compared with bilayer graphene grown on the Si-face of SiC to study the impact of the substrate on the electronic properties of epitaxial graphene. Our results show that the C-face films are nearly freestanding from an electronic point of view, due to the rotations between graphene layers.
We use ultra-high vacuum chemical vapor deposition to grow polycrystalline silicon carbide (SiC) on c-plane sapphire wafers which are then annealed between 1250 and 1450°C in vacuum to create epitaxial multilayer graphene (MLG). Despite the surface roughness and small domain size of the polycrystalline SiC, a conformal MLG film is formed. By planarizing the SiC prior to graphene growth, a reduction of the Raman defect band is observed in the final MLG. The graphene formed on polished SiC films also demonstrates significantly more ordered layer-by-layer growth and increased carrier mobility for…
Using first-principles density-functional theory, we study the electronic structure of multilayer graphene nanoribbons as a function of the ribbon width and the external electric field, applied perpendicular to the ribbon layers. We consider two types of edges (armchair and zigzag), each with two edge alignments (referred to as alpha- and beta-alignments). We show that, as in monolayer and bilayer armchair nanoribbons, multilayer armchair nanoribbons exhibit three classes of energy gaps which decrease with increasing width. Nonmagnetic multilayer zigzag nanoribbons have band structures that ar…
