Edge Spin Research Articles

Tunable topological phases in nanographene-based spin-1/2 alternating-exchange Heisenberg chains.

Unlocking the potential of topological order in many-body spin systems has been a key goal in quantum materials research. Despite extensive efforts, the quest for a versatile platform enabling site-selective spin manipulation, essential for tuning and probing diverse topological phases, has persisted. Here we utilize on-surface synthesis to construct spin-1/2 alternating-exchange Heisenberg chains by covalently linking Clar's goblets-nanographenes each hosting two antiferromagnetically coupled spins. Using scanning tunnelling microscopy, we exert atomic-scale control over chain lengths, parities and exchange-coupling terminations, and probe their magnetic response via inelastic tunnelling spectroscopy. Our investigation confirms the gapped nature of bulk excitations in the chains, known as triplons. Their dispersion relation is extracted from the spatial variation of tunnelling spectral amplitudes. Depending on the parity and termination of chains, we observe varying numbers of in-gap spin-1/2 edge excitations, reflecting the degeneracy of distinct topological ground states in the thermodynamic limit. By monitoring interactions between these edge spins, we identify the exponential decay of spin correlations. Our findings present a phase-controlled many-body platform, opening avenues toward spin-based quantum devices.

Existence of long-range magnetic order in Heisenberg spin nanoribbons with edge modification

Long-range magnetic order appears on a side decorated Heisenberg spin nanoribbon at nonzero temperature, although no spontaneous magnetization exists in a one- or two-dimensional isotropic Heisenberg model at any nonzero temperature according to the Mermin–Wagner theorem. By use of the spin Green’s function method, we calculated the magnetizations of Heisenberg nanoribbons decorated by side spins with single-ion anisotropy and found that the system exhibits a nonzero transition temperature, whether the decorated edge spins of the system link together or separate from each other. When the width of the nanoribbon achieves infinite limit, the transition temperatures of the system tend to the same finite constant eventually whether one edge or both edges are decorated by side spins in the nanoribbon. The results reveal that the magnetism of a low-dimensional spin system is different from that of a three-dimensional spin system. When the single-ion anisotropy of edge spins in a Heisenberg spin nanoribbon can be modulated by an electric field experimentally, various useful long-range magnetic orders of the system can be obtained. This work can provide a detailed theoretical basis for designing and fabricating next-generation low-dimensional magnetic random-access memory.

Chiral symmetry breaking and topological charge of zigzag graphene nanoribbons

Interacting quasi-one-dimensional zigzag graphene nanoribbons display gapped edge excitations. Although the self-consistent Hartree–Fock fields break chiral symmetry, our work demonstrates that zigzag graphene nanoribbons maintain their status as short-range entangled symmetry-protected topological insulators. The relevant symmetry involves combined mirror and time-reversal operations. In undoped ribbons displaying edge ferromagnetism, the band gap edge states with a topological charge form on the zigzag edges. An analysis of the anomalous continuity equation elucidates that this topological charge is induced by the gap term. In low-doped zigzag ribbons, where the ground state exhibits edge spin density waves, this topological charge appears as a nearly zero-energy edge mode. Our system is outside the conventional classification for topological insulators.

Broken edge spin symmetry induces a spin-polarized current in graphene nanoribbon

Zigzag graphene nanoribbons (ZGNRs) are known to possess spin moments at the hydrogen-terminated edge carbon atoms; thus, spin-polarized electron transmission is expected, while the current is longitudinally passed through the ZGNRs. However, in pristine ZGNRs, spin-polarized transmission is not observed due to symmetric anti-parallel distributions of the spin densities between the edges. Here, the hypothesis is that any physical or chemical process that breaks such anti-parallel spin symmetry can induce spin-polarized transmission in ZGNRs. In this work, we have established this proof-of-concept by depositing the trimethylenemethane (TMM) radical on 6ZGNRH and investigating the quantum transport properties by employing density functional theory in conjunction with the nonequilibrium Green’s function method. Although TMM has a high magnetic moment ( 2μB ), it does not induce magnetization in 6ZGNRH when TMM is physisorbed. However, during the chemisorption of TMM, it forms the π−π bond with the 6ZGNRH in a particular geometric configuration, where the p z orbitals of carbon atoms of TMM have maximum overlap with the p z orbitals of carbon atoms of 6ZGNRH. The chemisorption of TMM transfers the spin moment to 6ZGNRH, which breaks the edge spin symmetry of pristine 6ZGNRH. The adsorption of the TMM radical results in transmission dips in the transmission spectra due to interference between localized states of TMM and 6ZGNRH states. This induces spin-polarized transmission with 60% spin-filtering efficiency at zero bias, which can further be enhanced up to 92% by applying a bias voltage of 1.0 V.

Coexistence of Quantum-Spin-Hall and Quantum-Hall-Topological-Insulating States in Graphene/hBN on SrTiO3 Substrate.

SrTiO3 (STO) substrate, a perovskite oxide material known for its high dielectric constant (ɛ), facilitates the observation of various (high-temperature) quantum phenomena. A quantum Hall topological insulating (QHTI) state, comprising two copies of QH states with antiparallel two ferromagnetic edge-spin overlap protected by the U(1) axial rotation symmetry of spin polarization, has recently been achieved in low magnetic field (B) even as high as ≈100 K in a monolayer graphene/thin hexagonal boron nitride (hBN) spacer placed on an STO substrate, thanks to the high ɛ of STO. Despite the use of the heavy STO substrate, however, proximity-induced quantum spin Hall (QSH) states in 2D TI phases, featuring a topologically protected helical edge spin phase within time-reversal-symmetry, is not confirmed. Here, with the use of a monolayer hBN spacer, it is revealed the coexistence of QSH (at B = 0T) and QHTI (at B ≠ 0) states in the same single graphene sample placed on an STO, with a crossover regime between the two at low B. It is also classified that the different symmetries of the two nontrivial helical edge spin phases in the two states lead to different interaction with electron-puddle quantum dots, caused by a local surface pocket of the STO, in the crossover regime, resulting in a spin dephasing only for the QHTI state. The results obtained using STO substrates open the doors to investigations of novel QH spin states with different symmetries and their correlations with quantum phenomena. This exploration holds value for potential applications in spintronic devices.

Quasi-fractionalization of edge spin in chirality-assisted cluster-based Haldane state on triangular spin tube

Fractionalization of quantum degrees of freedom holds the key to finding new phenomena in physics, e.g., the quark model in hadron physics and the spin-charge separation in strongly-correlated electron systems. A typical example of the fractionalization in quantum spin systems is the spin-1 Haldane state, whose intriguing characteristics are well described by fractionalized virtual spins, delivering two individual spin-1/2 degrees of freedom as edge states. Here we theoretically propose an exotic extension of the Haldane state to the pseudo spin-1 model consisting of the mixture of real spin and spin chirality, resulting in quasi-fractionalization of spin-1/2 magnetization, i.e., an approximately-1/4 spin. Existence of the edge state is confirmed both analytically and numerically in a triangular spin tube, combining a low-energy perturbation theory and variational matrix-product state method. Our study not only proposes an unconventional quantum spin object but paves a way to chop the elementary quantities further.

Light-induced pure spin current in carbon hexagonal-connected zigzag graphene nanoribbons via magnetic field modulation.

Pure spin current, exhibiting no Joule heat and self-powered characteristics, has recently attracted intensive attention. Here, through first-principles calculations and symmetry analysis, we propose a new method to generate photoelectric pure spin current in carbon hexagonal connected three zigzag graphene nanoribbons (ZGNRs) via magnetic field modulation. Specifically, a device with centro-symmetry is designed, which consists of three ZGNRs using two carbon hexagons as connectors ('2-C6'). When the edge spin states of the three ZGNRs from left to right are modulated to AFM-AFM-AFM or FM-AFM-FM by magnetic fields, excellent pure spin currents are obtained which are independent of the photon energy and the angle of the linearly polarized light. However, when the edge spin states are FM-FM-FM orderly, the photocurrent is nearly zero and can be neglected. Analysis show that the first two spin magnetic structures own the spatial inversion antisymmetric spin density which is the origin of stable pure spin currents, while the FM-FM-FM structure owns Cs symmetric spin density, leading to the nearly zero photocurrent. Our findings provide a scheme for obtaining pure spin currents by changing the spin states of the graphene nanoribbons via magnetic field modulation, which is of great importance for the design of spintronic devices.

Orbital Hanle Magnetoresistance in a 3d Transition Metal.

The Hanle magnetoresistance is a telltale signature of spin precession in nonmagnetic conductors, in which strong spin-orbit coupling generates edge spin accumulation via the spin Hall effect. Here, we report the existence of a large Hanle magnetoresistance in single layers of Mn with weak spin-orbit coupling, which we attribute to the orbital Hall effect. The simultaneous observation of a sizable Hanle magnetoresistance and vanishing small spin Hall magnetoresistance in BiYIG/Mn bilayers corroborates the orbital origin of both effects. We estimate an orbital Hall angle of 0.016, an orbital relaxation time of 2ps and diffusion length of the order of 2nm in disordered Mn. Our findings indicate that current-induced orbital moments are responsible for magnetoresistance effects comparable to or even larger than those determined by spin moments, and provide a tool to investigate nonequilibrium orbital transport phenomena.

MoRe Electrodes with 10 nm Nanogaps for Electrical Contact to Atomically Precise Graphene Nanoribbons.

Atomically precise graphene nanoribbons (GNRs) are predicted to exhibit exceptional edge-related properties, such as localized edge states, spin polarization, and half-metallicity. However, the absence of low-resistance nanoscale electrical contacts to the GNRs hinders harnessing their properties in field-effect transistors. In this paper, we make electrical contact with nine-atom-wide armchair GNRs using superconducting alloy MoRe as well as Pd (as a reference), which are two of the metals providing low-resistance contacts to carbon nanotubes. We take a step toward contacting a single GNR by fabricating electrodes with needlelike geometry, with about 20 nm tip diameter and 10 nm separation. To preserve the nanoscale geometry of the contacts, we develop a PMMA-assisted technique to transfer the GNRs onto the prepatterned electrodes. Our device characterizations as a function of bias voltage and temperature show thermally activated gate-tunable conductance in GNR-MoRe-based transistors.

Magnetic edge states and edge current in honeycomb zigzag nanoribbons by Kane–Mele–Hubbard model

We investigate spin-dependent edge states of honeycomb zigzag nanoribbons with a focus on electron–electron interaction as the origin of the initial edge magnetism which is assumed to be anti-ferromagnetism order as a stable form. We study the effect of intrinsic spin–orbit interaction (SOI) and also applied in-plane transverse electric field on low energy spectrum and edge spin density. Tight-binding method in the framework of the self-consistent Kane–Mele–Hubbard model has been utilized for exploring electronic properties. Edge currents and electric dipole moments are also calculated. Results show electric field and SOI can dramatically affect the edge states and energy gaps. A transition from a magnetic edge state to a non-magnetic one occurs at critical values of SOI and electric field. Furthermore, the system goes into the topological insulator phase at sufficiently large SOI strengths. The results of edge spin density, electric dipole moment, and spin currents indicate that one can reach the favorite spin-dependent transport by manipulating the electric field and SOI in a zigzag nanoribbon. The outcome may pave the way to further control of spin-based transport in future devices.

Spin-dependent polarization and quantum Hall conductivity in decorated graphene: influence of locally induced spin–orbit-couplings and impurities

We study spin transport through graphene-like substrates in the presence of one or several, locally induced spin–orbit coupling (SOC) terms resulting from periodically placed strips, on their top and decorated with a random distribution of impurities. Intrinsic SOC, Rashba SOC and/or pseudo-spin-inversion-asymmetry coupling are considered. A systematic investigation of the spin conductance identifies the main SOC terms which lead to its energy dependence as well as the extent to which the impurity concentration and each SOC term can affect or tune it, In addition, the spin current flow is considered in the presence of different SOC impurities and their related group symmetry such C 6v , C 3v , D 6h and D 3h . Further, we show that the quantum spin-Hall effect (QSHE) related to the spin edge states depends only on the spin character when the PIA and ISO terms are not sublattice resolved, and on both the spin and sublattice character when they are. In addition, we show that the RSO term plays a major role in obtaining edge states that are either protected on both edges or only on one edge against backscattering. This Rashba term creates an anticrosing gap that affects the symmetry in the edge localizations and leads to half-topological states. The results can facilitate the experimental choice of appropriately decorated strips to (i) develop spin-transistor devices by tuning the Fermi energy, (ii) control the robustness of the QSHE against backscattering even in the presence of on-site sublattice asymmetry induced by a transverse electric field or functionalizations, and (iii) provide a strong theoretical support for spintronic quantum devices.

Stochastic strong zero modes and their dynamical manifestations.

Strong zero modes (SZMs) are conserved operators localized at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyze analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbor transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general nondiagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.

Two-particle topological Thouless spin pump

We show that two particles interacting via spin exchange exhibit topological features found in one-dimensional single particle lattice models. This is accomplished by absorbing all of the spatial degrees of freedom of the lattices into the spin degrees of freedom of the two particles. Comparing the spin system with the Su-Schrieffer-Heeger model, we show the existence of topologically protected edge spin states and establish the bulk-edge correspondence. Modifying the spin system with a chiral symmetry breaking term results in it resembling the Rice-Mele model and can therefore act as a Thouless spin pump of one of the particles when periodically and adiabatically driven. By using the spin states as a synthetic spatial dimension, we show two particles are enough to simulate well known topological properties in condensed matter physics.

The exact solution of the edge-modified graphene nanoribbon-like structure Ising-Heisenberg spin (1,1/2) system

A two-fold rotational symmetry ferromagnetic Ising-Heisenberg model modified by edge spin ( S = 1 / 2 ) with antiferromagnetic Heisenberg exchange interaction, whose shape is graphene nanoribbon-like has been constructed, and the host Ising spin ( S = 1 ). By use of the star-triangle transformation and the transfer matrix method, the edge states, thermodynamics and magnetization behavior of the system display various properties. The results perform that the unique edge-spin states and the host-spin states of the system are influenced with each other. And there are eleven different ground state phases, which are presented as the values of magnetic susceptibility multiplied by temperature at ultra-low temperature in the parameter space of single-ion anisotropy and antiferromagnetic Heisenberg interaction. Because of a large number of excited states close to the ground state, the specific heat curve shows multi-peak phenomena, which are analyzed by corresponding to the entropy of the system. The magnetization process of the system is complex, due to the direct correlation of the edge-spin states and the host-spin states of system, in which many different magnetization plateaus appear. The rigorous results of the number and width of magnetization plateaus and the saturation of the magnetization are written. This study offers unique opportunities for future technological applications such as proposed designs of graphene-based quantum spintronic devices as well as magnetic ordering.

Evolution of local edge state braiding and spin topological transport characterization of Te-doped monolayer 1T'-MoS2.

We conducted first-principles calculations to investigate the dynamic braiding of local edge states and the spin topological transport mechanism in a strong topological MoS1.75Te0.25 matrix. The presence of type-II Van Hove singularity in the middle of the X-S path indicates a strong cohesive interaction and a paring condensation mechanism within the matrix. The surface state data of MoS1.75Te0.25 clearly demonstrate the characteristic features of strong regular loop braiding in spin transport. The spin Hall conductivity of the matrix was determined from the anisotropic characteristics of the spin Berry curvature. The phase transition of the spin Hall conductivity was evidenced by the positive sign of local spin polarization strength, primarily contributed by the dz2 orbital of Mo atoms, and the negative sign of spin polarization strength, mainly contributed by the p-px orbitals of S atoms. Moreover, the inclusion of Te selectively tuned the spin transport efficiency of the dz2 and px orbitals. Comprehensive braiding and readout of edge states can be achieved using an artificially designed MoS1.75Te0.25 spintronic device. This 2D fractional braiding holds significant potential for applications in topological quantum computation.

Tunable On‐Chip Terahertz Isolator Based on Nonreciprocal Transverse Edge Spin State of Asymmetric Magneto‐Plasmonic Waveguide

AbstractTerahertz (THz) isolators are crucial to one‐way transmission in THz application systems. However, it is still challenging to implement THz isolators with integrated structure and tunable characteristics. Herein, a THz isolator based on a magneto‐plasmonic waveguide (MPWG) has been demonstrated. In this InSb‐air‐metal asymmetric waveguide, the plasmonic band gap is excited and thermally tuned by surface plasmons. Moreover, under weak external magnetic fields (EMFs), magneto‐plasmons excite nonreciprocal transverse edge spin states, which break the time‐reversal symmetry of the MPWG and lead to a pair of isolation bandgaps that only allow one‐way transmission. The experimental results of both broadband time‐domain spectrum system and single‐frequency system confirm this isolator can achieve more than 20 dB isolation and only 4 dB insertion loss. More importantly, its isolation can be actively modulated by EMFs from 0 to 0.23 T, and also realize broadband tuning of over 1 THz frequency range by controlling the temperature from 80 to 280 K. This topological edge band structure of transverse spin states and its active manipulation mechanism are helpful to deepen the understanding of nonreciprocal plasmon propagation, and this broadband tunable THz on‐chip isolator is significant to urgently apply in strong THz sources and ultra‐sensitive detectors.

Full-electrostatic management of spin-valley transmission through topological edge states of the hexagonal materials

All-electrostatic generation and control of both spin and valley polarizations provides a highly promising route towards much smaller and energy-efficient nanoscale devices. However, the requirement of broken time reversal symmetry is a serious limitation in this procedure. Here, we show that the chiral spin-resolved topological edge states in each valley of the two-dimensional hexagonal materials, aid to get round this restriction. By topological analysis of the spin and valley dependent edge states, we demonstrate that bulk-to-edge transport in these materials allows all-electrostatic generation and control of fully polarized spin-valley data, preserved during a long enough time and distance. We also obtain an analytic relation for bulk-to-edge transmission and show that the perfect transmission is insensible to the applied electric fields which enables even more convenient implementation of this scheme in spin- and valley-based devices.

Time-reversal symmetry broken quantum spin Hall phase in the van der Waals heterostructure ZrTe5/Cr2Ge2Te6

The topological phases with time-reversal symmetry (TRS) breaking have always attracted intense studies due to their potential applications to spintronics. The previous studies mainly focused on the exploration of quantum anomalous Hall effects, but another typical TRS-broken two-dimensional topological phase, i.e., the TRS-broken quantum spin Hall (QSH) effect, has rarely been proposed in realistic materials. Here, based on first-principles calculations and topology analysis, we show that the van der Waals heterostructure ZrTe5/Cr2Ge2Te6 can realize the robust TRS-broken QSH state. We utilize the topological edge states and spin Hall conductance, which can be measured in experiments directly, to confirm the appearance of TRS-broken QSH phase. Moreover, we uncover that the built-in electric field is essential to realize such topological state and an external electric field can effectively tune the nontrivial band gap. Our findings not only provide a reliable candidate to obtain the TRS-broken QSH phase with a high tunability but also can facilitate further applications to topological quantum transport.

New disordered anyon phase of doped graphene zigzag nanoribbon

We investigate interacting disordered zigzag nanoribbons at low doping, using the Hubbard model to treat electron interactions within the density matrix renormalization group and Hartree-Fock method. Extra electrons that are inserted into an interacting disordered zigzag nanoribbon divide into anyons. Furthermore, the fractional charges form a new disordered anyon phase with a highly distorted edge spin density wave, containing numerous localized magnetic moments residing on the zigzag edges, thereby displaying spin-charge separation and a strong non-local correlation between the opposite zigzag edges. We make the following new predictions, which can be experimentally tested: (1) In the low doping case and weak disorder regime, the soft gap in the tunneling density of states is replaced by a sharp peak at the midgap energy with two accompanying peaks. The e^-/2 fractional charges that reside on the boundary of the zigzag edges are responsible for these peaks. (2) We find that the midgap peak disappears as the doping concentration increases. The presence of e^-/2 fractional charges will be strongly supported by the detection of these peaks. Doped zigzag ribbons may also exhibit unusual transport, magnetic, and inter-edge tunneling properties.

Probing Majorana modes via local spin dynamics

We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here, the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for p-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap whilst retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the inter-dimer coupling to generate a finite size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.