We develop a general theoretical framework based on $Z$-classification to count the number of topological bound states at a junction of chiral-symmetric one-dimensional systems. The formulation applies to general multiway junctions composed of an arbitrary number of channels and an arbitrary joint structure. By using the formula, we calculate the zero-energy bound states in various types of two-way and three-way junctions of semiconducting graphene nanoribbons. We then consider periodic two-dimensional networks of graphene nanoribbons, and show that the topological junction states form isolated energy bands inside the bulk energy gap, which can be viewed as a two-dimensional crystal of the effective atoms. Depending on the $Z$ number of a single junction, we have a different set of effective atomic orbitals, resulting in various types of nanoscale metamaterials, which are often accompanied by flat bands. The system would provide an ideal platform for quantum simulator to emulate a strongly-interacting fermion system on various types of lattices.
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…
We investigate transport properties of the junctions in which the graphene nanoribbon with the zigzag shaped edges consisting of the $N$ legs is sandwiched by the two normal metals by means of recursive Green's function method. The conductance and the transmission probabilities are found to have the remarkable properties depending on the parity of $N$. The singular behaviors close to E=0 with $E$ being the Fermi energy are demonstrated. The channel filtering is shown to occur in the case with $N=$ even.
We demonstrate that the phenomenon of current confinement along graphene n-p junctions at high magnetic fields can be used to form an Aharonov-Bohm interferometer. The interference system exploits a closed n-p junction that can be induced by a floating gate within the sample, and coupling of the junction currents with the edge currents in the quantum Hall regime. Operation of the device requires current splitting at the edge and the n-p junction contacts which is found for armchair ribbons at low Fermi energy.
We investigate the electron transport and thermoelectric property of twisted bilayer graphene nanoribbon junction (TBGNRJ) in $0^o$, $21.8^o$, $38.2^o$ and $60^o$ rotation angles by first principles calculation with Landauer-Buttiker and Boltzmann theories. It is found that TBGNRJs exhibit negative differential resistance (NDR) in $21.8^o$ and $38.2^o$ rotation angles under $\pm$ 0.2 V bias voltage. More importantly, three peak ZT values of 2.0, 2.7 and 6.1 can be achieved in the $21.8^o$ rotation angle at 300K. The outstanding ZT values of TBGNRJs are interpreted as the combination of the red…
A thermoelectric voltage is induced in a junction, constituted of two dissimilar materials under a temperature gradient. Similarly, a thermosize voltage is expected to be induced in a junction made by the same material but having different sizes, so-called thermosize junction. This is a consequence of dissimilarity in Seebeck coefficients due to differences in classical and/or quantum size effects in the same materials with different sizes. The studies on thermosize effects in literature are mainly based on semi-classical models under relaxation time approximation or even simpler local equilib…
Graphene nanoribbons with armchair edges (AGNRs) have bandgaps that can be flexibly tuned via the ribbon width. A junction made of a narrower AGNR sandwiched between two wider AGNR leads was recently reported to possess two perfect transmission channels close to the Fermi level. Here, we report that by using a bias voltage to drive these transmission channels into the gap of the wider AGNR lead, we can obtain a negative differential resistance (NDR) effect. Owing to the intrinsic properties of the AGNR junctions, the on-set bias reaches as low as ~ 0.2 V and the valley current almost vanishes.…
Stimulated by recent advances in isolating graphene, we discovered that quantum dot can be trapped in Z-shaped graphene nanoribbon junciton. The topological structure of the junction can confine electronic states completely. By varying junction length, we can alter the spatial confinement and the number of discrete levels within the junction. In addition, quantum dot can be realized regardless of substrate induced static disorder or irregular edges of the junction. This device can be used to easily design quantum dot devices. This platform can also be used to design zero-dimensional functional…
To address the robustness of the transport gap induced by locally strained regions in graphene nanostructures, the effect of disorder and smoothness of the interface region is investigated within the Landauer-Büttiker formalism. The electronic conductance across strained junctions and barriers in graphene nanoribbons is calculated numerically, with and without various types of disorder, and comparing smooth and sharp strain junctions. A smooth strain barrier in graphene is seen to be generically as efficient in suppressing transport at low densities as a sharp one, and the critical density (or…
We investigate the thermoelectric performance of armchair graphene nanoribbon (AGNR), bilayer GNRs junction (BGNRJ) and BGNRJ with holes (BGNRJ-H) by the first principles calculation with non-equilibrium Green function. It is found that the BGNRJ-H exhibits high ZT values of 9.65 and 5.55 at 300K. The reason of these significantly larger ZT values than previously observed has been calculated due to reduced thermal conductivity and enhanced electrical conductivity. The low thermal conductance comes from the van der waals (vdW) interaction between two graphene layers. The increased electrical co…
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…
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…
Based on first-principles calculations we predict that periodically repeated junctions of armchair graphene nanoribbons of different widths form superlattice structures. In these superlattice heterostructures the width and the energy gap are modulated in real space and specific states are confined in certain segments. Orientation of constituent nanoribbons, their width and length, the symmetry of the junction are the structural parameters to engineer electronic properties of these quantum structures. Not only the size modulation, but also composition modulation, such as periodically repeated, …
Topological surface states of three-dimensional topological insulator nanoribbons and their distinct magnetoconductance properties are promising for topoelectronic applications and topological quantum computation. A crucial building block for nanoribbon-based circuits are three-terminal junctions. While the transport of topological surface states on a planar boundary is not directly affected by an in-plane magnetic field, the orbital effect cannot be neglected when the surface states are confined to the boundary of a nanoribbon geometry. Here, we report on the magnetotransport properties of su…
We study the electron transport through a graphene nanoribbon-superconductor junction. Both zigzag and armchair edge graphene nanoribbons are considered, and the effects of the magnetic field and disorder on the transport property are investigated. By using the tight-binding model and the non-equilibrium Green's function method, the expressions of the current, conductance, normal tunneling coefficient, and Andreev reflection coefficient are obtained. For a clean system and at zero magnetic field, the linear conductance increases approximatively in a linear fashion with the on-site energy. In t…
Fano Resonances in Mismatched C$_3$N Nanoribbon Junctions
Technische Universität Dresden（德累斯顿工业大学）；Université Paris-Saclay, Centre de Nanosciences et de Nanotechnologies（巴黎萨克雷大学，纳米科学中心）；Universidad del País Vasco UPV/EHU（巴斯克地区大学）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯克，巴斯克科学基金会）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）
We report anomalous enhancement of the critical current at low temperatures in gate-tunable Josephson junctions made from topological insulator BiSbTeSe$_2$ nanoribbons with superconducting Nb electrodes. In contrast to conventional junctions, as a function of the decreasing temperature $T$, the increasing critical current $I_c$ exhibits a sharp upturn at a temperature $T_*$ around 20$\%$ of the junction critical temperatures for several different samples and various gate voltages. The $I_c$ vs. $T$ demonstrates a short junction behavior for $T>T_*$, but crosses over to a long junction behavio…
Quantized Conductance through Surface States in High Quality Three-Dimensional Dirac Semimetal Cd$_3$As$_2$ Nanowire/Nanoribbon p-n Junctions
DGIST Research Institute, DGIST（大邱科学技术院研究学院）；Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）；Department of Physics, KTH Royal Institute of Technology（皇家理工学院物理系）；Industrial AX Innovation Research Institute and Sensor Technology Support Center, DGIST（大邱科学技术院工业AX创新研究所及传感器技术支持中心）；Department of Advanced Materials Chemistry, Korea University（韩国大学先进材料化学系）；Department of Physics and Chemistry, DGIST（大邱科学技术院物理化学系）；Department of Interdisciplinary Engineering, DGIST（大邱科学技术院交叉工程学系）；Institute of next generation semiconductor technology (INST), DGIST（大邱科学技术院下一代半导体技术研究所）
We study Andreev reflection in graphene nanoribbon/superconductor hybrid junctions. By using a tight-binding approach and the scattering formalism we show that finite-size effects lead to notable differences with respect to the bulk graphene case. At subgap voltages, conservation of pseudoparity, a quantum number characterizing the ribbon states, yields either a suppression of Andreev reflection when the ribbon has an even number of sites in the transverse direction or perfect Andreev reflection when the ribbon has an odd number of sites. In the former case the suppression of Andreev reflectio…
We analyze the thermal fluctuations of a narrow graphene nanoribbon. Using a continuum, membrane-like model we study the height-height correlation functions and the destabilization modes corresponding to two different boundaries conditions: ribbons which are fixed or free on the edges. For the first situation, the thermal spectrum has a gap and the correlations along the ribbon decay exponentially. Thermal fluctuations produce only local perturbations of the flat situation. However, the long range crystalline order is not distorted. For free edges, the situation changes as thermal excitations …
Thermoelectric junctions are often made of components of different materials characterized by distinct transport properties. Single material junctions, with the same type of charge carriers, have also been considered to investigate various classical and quantum effects on the thermoelectric properties of nanostructured materials. We here introduce the concept of defect-induced thermoelectric voltage, namely, {\it thermodefect voltage}, in graphene nanoribbon (GNR) junctions under a temperature gradient. Our thermodefect junction is formed by two GNRs with identical properties except the existe…
Using the partition-free time-dependent Landauer-B{ü}ttiker formalism for transient current correlations, we study the traversal times taken for electrons to cross graphene nanoribbon (GNR) molecular junctions. We demonstrate electron traversal signatures that vary with disorder and orientation of the GNR. These findings can be related to operational frequencies of GNR-based devices and their consequent rational design.
We theoretically investigate the quantum transports in a junction between a superconductor and a silicene nanoribbon, under the effect of a magnetic exchange field. We find that for a narrow nanoribbon of silicene, remarkable crossed Andreev reflection (with a fraction $>50\%$) can be induced in the energy window of the elastic cotunneling, by destroying some symmetries of the system. Since the energy responses of electrons to the exchange field are opposite for opposite spins, these transport channels can be well spin polarized. Moreover, due to the helicity conservation of the topological ed…
We study crossed Andreev reflection (CAR) of electrons or holes in normal metal-superconductor-normal metal junctions and highlight some very strong effects of the underlying lattice. In particular, we demonstrate that for sharp interfaces and under certain, albeit generic, symmetry conditions, the CAR probability exactly vanishes for an even number of atoms in the superconducting region. This even-odd effect applies notably to NSN junctions made of graphene nano-ribbons with armchair edges and for zigzag edges with somewhat more restrictive conditions. We analyze its robustness towards smooth…
Graphene nanoribbons (GNRs) have recently been shown by Cao, Zhao, and Louie [Cao, T.; Zhao, F.; Louie, S. G. Phys. Rev. Lett. 2017, 119, 076401] to possess distinct topological phases in general, characterized by a Z2 invariant. Cove-edged and chevron GNRs moreover are chemically and structurally diverse, quasi-one-dimensional (1D) nanostructures whose structure and electronic properties can be rationally controlled by bottom-up synthesis from precursor molecules. We derive the value of the topological invariant of the different types of cove-edged and chevron GNRs, and we investigate the ele…
Spin-dependent electronic transport properties of Fe-terminated zig-zag graphene nanoribbons (zGNR) have been studied using first-principles transport simulations. The spin configuration of proposed zGNR junction can be controlled with external magnetic field, and the tunneling junction show MR>1000 at small bias and is a perfect spin-filter by applying uniform external magnetic filed at small bias.
In graphene nanoribbon junctions, the nearly perfect transmission occurs in some junctions while the zero conductance dips due to anti-resonance appear in others. We have classified the appearance of zero conductance dips for all combinations of ribbon and junction edge structures. These transport properties do not attribute to the whole junction structure but the partial corner edge structure, which indicates that one can control the electric current simply by cutting a part of nanoribbon edge. The ribbon width is expected to be narrower than 10 nm in order to observe the zero conductance dip…
Junctions composed of two crossed graphene nanoribbons (GNRs) have been theoretically proposed as electron beam splitters where incoming electron waves in one GNR can be split coherently into propagating waves in \emph{two} outgoing terminals with nearly equal amplitude and zero back-scattering. Here we scrutinize this effect for devices composed of narrow zigzag GNRs taking explicitly into account the role of Coulomb repulsion that leads to spin-polarized edge states within mean-field theory. We show that the beam-splitting effect survives the opening of the well-known correlation gap and, mo…
Thanks to their highly tunable band gaps, graphene nanoribbons (GNRs) with atomically precise edges are emerging as mechanically and chemically robust candidates for nanoscale light emitting devices of modulable emission color. While their optical properties have been addressed theoretically in depth, only few experimental studies exist, limited to ensemble measurements and without any attempt to integrate them in an electronic-like circuit. Here we report on the electroluminescence of individual GNRs suspended between the tip of a scanning tunneling microscope (STM) and a Au(111) substrate, c…
We report on the fabrication of Josephson junctions using the topological crystalline insulator Pb$_{0.5}$Sn$_{0.5}$Te as the weak link. The properties of these junctions are characterized and compared to those fabricated with weak links of PbTe, a similar material yet topologically trivial. Most striking is the difference in the AC Josephson effect: junctions made with Pb$_{0.5}$Sn$_{0.5}$Te exhibit rich subharmonic structure consistent with a skewed current-phase relation. This structure is absent in junctions fabricated from PbTe. A discussion is given on the origin of this effect as an ind…
Stacking two-dimensional layered materials such as graphene and transitional metal dichalcogenides with nonzero interlayer twist angles has recently become attractive because of the emergence of novel physical properties. Stacking of one-dimensional nanomaterials offers the lateral stacking offset as an additional parameter for modulating the resulting material properties. Here, we report that the edge states of twisted bilayer zigzag graphene nanoribbons (TBZGNRs) can be tuned with both the twist angle and the stacking offset. Strong edge state variations in the stacking region are first reve…
We report on the observation of gate-tunable proximity-induced superconductivity and multiple Andreev reflections (MAR) in a bulk-insulating BiSbTeSe$_2$ topological insulator nanoribbon (TINR) Josephson junction (JJ) with superconducting Nb contacts. We observe a gate-tunable critical current ($I_C$) for gate voltages ($V_g$) above the charge neutrality point ($V_{CNP}$), with $I_C$ as large as 430 nA. We also observe MAR peaks in the differential conductance ($dI/dV$) versus DC voltage ($V_{dc}$) across the junction corresponding to sub-harmonic peaks (at $V_{dc} = V_n = 2Δ_{Nb}/en$, where $…
We study the transport properties of a twisted bilayer graphene flake contacted by two monolayer nanoribbons which act as leads. We analyze the conductance in terms of the spectra of the bilayer nanoribbon and the monolayer contacts. The low-energy transport properties are governed by the edge states with AB stacking. Remarkably, the electronic conductance in this energy region does not depend much on the relative position of the leads, in contrast with that of bilayer flakes with more symmetric stackings. We attribute this feature to the localization of these low-energy states in the AB edge …
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 study the transport of charge carriers through finite graphene structures. The use of numerical exact kernel polynomial and Green function techniques allows us to treat actual sized samples beyond the Dirac-cone approximation. Particularly we investigate disordered nanoribbons, normal-conductor/graphene interfaces and normal-conductor/graphene/normal-conductor junctions with a focus on the behavior of the local density of states, single-particle spectral function, optical conductivity and conductance. We demonstrate that the contacts and bulk disorder will have a major impact on the electro…
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 linear conductance spectrum of a metallic graphene junction formed by interconnecting two gapless graphene nanoribbons is calculated. A strong conductance suppression appears in the vicinity of the Dirac point. We found that such a conductance suppression arises from the antiresonance effect associated with the edge state localized at the zigzag-edged shoulder of the junction. The conductance valley due to the antiresonance is rather robust in the presence of the impurity and vacancy scattering. And the center of the conductance valley can be readily tuned by an electric field exerted on t…
Usually, graphene is used in its horizontal directions to design novel concept devices. Here, we report a single electron tunneling diode based on quantum tunneling through a vertical graphene two-barrier junction. The junction is formed by positioning a scanning tunnelling microscopy (STM) tip above a graphene nanoribbon that was deposited on a graphite surface. Because of the asymmetry of the two-barrier junction, the electrons can unidirectional transfer from the tip to the graphene nanoribbon but not from the graphene to the tip. This result opens intriguing opportunities for designing new…
We theoretically demonstrate that the high-critical-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO) is a natural candidate for the recently envisioned classical space-time crystal. BSCCO intrinsically forms a stack of Josephson junctions. Under a periodic parametric modulation of the Josephson critical current density, the Josephson currents develop coupled space-time crystalline order, breaking the continuous translational symmetry in both space and time. The modulation frequency and amplitude span a (nonequilibrium) phase diagram for a so-defined spatiotemporal order paramet…
Superconducting proximity junctions made of topological insulator (TI) nanoribbons (NRs) provide a useful platform for studying topological superconductivity. We report on the fabrication and measurement of Josephson junctions (JJs) using Sb doped Bi$_2$Se$_3$ NRs in contact with Al electrodes. Aharonov$-$Bohm and Altshuler$-$Aronov$-$Spivak oscillations of the axial magneto-conductance of TI NR were observed, indicating the existence of metallic surface states along the circumference of the TI NR. We observed the supercurrent in the TI NR JJ and subharmonic gap structures of the differential …
Knowledge of the topology of the electronic ground state of materials has led to deep insights to novel phenomena such as the integer quantum Hall effect and fermion-number fractionalization, as well as other properties of matter. Joining two insulators of different topological classes produces fascinating boundary states in the band gap. Another exciting recent development is the bottom-up synthesis (from molecular precursors) of graphene nanoribbons (GNRs) with atomic precision control of their edge and width. Here we connect these two fields, and show for the first time that semiconducting …
We demonstrate snake-like motion of graphene nanoribbons atop graphene and hexagonal boron nitride (h-BN) substrates using fully-atomistic non-equilibrium molecular dynamics simulations. The sliding dynamics of the edge-pulled nanoribbons is found to be determined by the interplay between in-plane ribbon elasticity and interfacial lattice mismatch. This results in an unusual dependence of the friction-force on the ribbon's length, exhibiting an initial linear rise that levels-off above a junction dependent threshold value dictated by the pre-slip stress distribution within the slider. As part …
We investigate spin transport in diffusive graphene nanoribbons with both clean and rough zigzag edges, and long-range potential fluctuations. The long-range fields along the ribbon edges cause the local doping to come close to the charge neutrality point forming $p$-$n$ junctions with localized magnetic moments, similar to the predicted magnetic edge of clean zigzag graphene nanoribbons. The resulting random edge magnetization polarizes charge currents and causes sample-to-sample fluctuations of the spin currents obeying universal predictions. We show furthermore that, although the average sp…
By first principle calculations, a dual spin filter effect under finite bias voltages is demonstrated in an antiferromagnetic junction of symmetric zigzag graphene nanoribbon (ZGNR). Unlike conventional spin filter devices using half metallic materials, the up- and down-spin electrons are unidirectionally filtered in the counter direction of the bias voltage, making the junction a dual spin filter. On the contrary, asymmetric ZGNRs do not exhibit such a spin filter effect. By analyzing Wannier functions and a tight-binding model, we clarify that an interplay between the spin polarized band str…
Center for Quantum Information Physics, New York University（纽约大学量子信息物理中心）；Department of Materials Science and Engineering, The Ohio State University（俄亥俄州立大学材料科学与工程系）
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 co…
The electron transport between two zigzag graphene nanoribbons (ZGNRs) connected by carbon atomic chains has been investigated by the nonequilibrium Green's function method combined with the density functional theory. The symmetry of the orbitals in the carbon chain selects critically the modes and energies of the transporting electrons. The electron transport near the Fermi energy can be well-manipulated by the position and the number of carbon chains contacting the nanoribbons. In symmetric ZGNRs connected by a central carbon chain, a square conductance step appears at the Fermi energy becau…
Efforts to modulate the electronic properties of atomically thin crystalline nanoribbons requires precise control over their morphology. Here, we perform atomistic simulations on freestanding graphene nanoribbons (GNRs) to first identify the minimal shapes, and then employ a core-edge framework based on classical plate theory to quantify the width dependence in more general systems. The elastic edge-edge interactions force ultra-narrow ribbons to be flat, which then bifurcate to twisted and bent shapes at critical widths that vary inversely with edge stress. Compressive edge stresses results i…
We study the conductance through two types of graphene nanostructures: nanoribbon junctions in which the width changes from wide to narrow, and curved nanoribbons. In the wide-narrow structures, substantial reflection occurs from the wide-narrow interface, in contrast to the behavior of the much studied electron gas waveguides. In the curved nanoribbons, the conductance is very sensitive to details such as whether regions of a semiconducting armchair nanoribbon are included in the curved structure -- such regions strongly suppress the conductance. Surprisingly, this suppression is not due to t…
We consider an interface between two strong time-reversal invariant topological insulators having surface states with opposite spin chirality, or equivalently, opposite mirror Chern number. We show that such an interface supports gapless modes that are protected by mirror symmetry. The interface states are investigated with a continuum model for the Bi2Se3 class of topological insulators that takes into account terms up to third order in the crystal momentum, which ensures that the model has the correct symmetry. The model parameters are obtained from ab initio calculations. Finally, we consid…
Interplay between structure and function in atomically thin crystalline nanoribbons is sensitive to their conformations yet the ability to prescribe them is a formidable challenge. Here, we report a novel paradigm for controlled nucleation and growth of scrolled and folded shapes in finite-length nanoribbons. All-atom computations on graphene nanoribbons (GNRs) and experiments on macroscale magnetic thin films reveal that decreasing the end distance of torsionally constrained ribbons below their contour length leads to formation of these shapes. The energy partitioning between twisted and bent…
We have investigated structure, electronic, and magnetic properties of metal-terminated zigzag graphene nanoribbons (M-ZGNRs) by first-principles calculations. Two families of metal terminations are studied: (1) 3d-transition metals (TMs) Fe, Co, and Ni and (2) noble metals (NMs) Cu, Ag, and Au. All systems have spin-polarized edge states with antiferromagnetic (AFM) ordering between two edges, except Co-ZGNRs and Ni-ZGNRs which exhibit negligibly small energy differences between AFM and ferromagnetic states with the given ribbon width. In the AFM state the TM terminations transform semiconduc…
