Abstract
The physical realization of Chern insulators is of fundamental and practical interest, as they are predicted to host the quantum anomalous Hall (QAH) effect and topologically protected chiral edge states which can carry dissipationless current. Current realizations of the QAH state often require complex heterostructures and subKelvin temperatures, making the discovery of intrinsic, high temperature QAH systems of significant interest. In this work we show that timereversal symmetry breaking Weyl semimetals, being essentially stacks of Chern insulators with interlayer coupling, may provide a new platform for the higher temperature realization of robust chiral edge states. We present combined scanning tunneling spectroscopy and theoretical investigations of the magnetic Weyl semimetal, Co_{3}Sn_{2}S_{2}. Using modeling and numerical simulations we find that depending on the strength of the interlayer coupling, chiral edge states can be localized on partially exposed kagome planes on the surfaces of a Weyl semimetal. Correspondingly, our dI/dV maps on the kagome Co_{3}Sn terraces show topological states confined to the edges which display linear dispersion. This work provides a new paradigm for realizing chiral edge modes and provides a pathway for the realization of higher temperature QAH effect in magnetic Weyl systems in the twodimensional limit.
Introduction
The quantized Hall conductance of the quantum Hall effect is a striking example of the macroscopic consequences of quantum phenomena^{1}. In the quantum Hall effect, large magnetic fields generate Landau levels in a twodimensional (2D) material. The Landau levels acquire a nonzero topological index, resulting in chiral edge currents that are a manifestation of the quantized Hall response. Haldane’s conception of the Chern insulator^{2}, or quantum anomalous Hall insulator^{3}, takes this idea a step further. A Chern insulator is a 2D material that exhibits the quantum Hall effect in the absence of an external magnetic field. The distinctive features of Chern insulators are their quantized Hall conductance, and topologically protected chiral edge states^{4}, which travel in unidirectional channels (see Fig. 1a). The transport signatures of the quantum anomalous Hall effect (QAHE) were originally reported in a 2D magnetically doped topological thin film^{5}, and advancements in QAHE signatures in magnetically doped topological insulators^{6,7,8,9,10,11,12,13} have recently been extended to intrinsic magnetic topological insulators^{14} and twisted bilayer graphene^{15}. While these results are breakthroughs in studying QAHE, future progress in studying chiral edge states is limited by the low Curie temperatures (<30 K) of these material systems and the complex heterostructures often necessary to realize chiral edge states.
Interestingly, Chern insulators are also related to a variety of higherdimensional topological systems, not least of which are the magnetic Weyl semimetals (WSMs). One can in fact model magnetic WSMs as layers of 2D Chern insulators that are coupled in the stacking direction^{16,17}. Thus, an unexplored route to 2D Chern insulators is to identify a threedimensional (3D) layered magnetic Weyl semimetal fitting this description. Recent developments in candidate magnetic WSMs^{18,19} now provide a promising alternative arena for the study of chiral edge states. In the spirit of the coupledlayer model presented by Balents and Burkov^{16}, we can show that stepped terraces on the surface of a WSM can harbor chiral edge states localized on the steps. First, we note that Weyl semimetals are an intermediate critical phase between a trivial insulator and a magnetic weak topological insulator, the latter of which is adiabatically connected to a decoupled stack of Chern insulators^{20,21,22}. Importantly, the two gapped phases and the intermediate gapless phase can be reached by starting with decoupled layers of Chern insulators and increasing the strength of the interlayer coupling. To illustrate, take a bilayer of Chern insulators both having the same nonzero Chern number. If the coupling between the two layers is increased, the system will eventually undergo a transition where the strong tunneling creates a trivial phase of the bilayer with a vanishing Chern number. Interestingly, if we strip off a part of one of the layers as depicted in the schematic in Fig. 1b, the exposed singlelayer region will revert to a nontrivial Chern insulator since it was the interlayer coupling that drove it to be trivial. Thus, both the end of the single layer and the singlestep defect itself will harbor a chiral edge state (see Supplementary Note 1 for more details). The existence of the edge state on the end of the single layer is obvious, but the edge state on the step defect appears only because the remaining bilayer region with strong tunneling is a trivial insulator. Thus, this process serves to expose a region of the Chern insulator despite the full bilayer system being trivial. The concept of exposing topological subsystems when the combined system is trivial was explored in two recent papers^{23,24} in the context of a bulk topological proximity effect, and embedded topological insulators, respectively.
While a single bilayer of Chern insulators is not sufficient to realize the intermediate WSM phase between the topological and trivial regimes, we extend this analysis to the case of many layers (as shown schematically in Fig. 1c) to model the properties of a WSM (Supplementary Note 1). We find that steplocalized chiral edge states continue to exist in WSMs in a wide swath of the topological phase diagram that is parameterized by the Cherninsulator gap of decoupled QAH planes and interlayer tunneling (see Supplementary Note 1). Indeed, we find that whenever the surface terrace exhibits the localized chiral modes on the steps, the system is in a WSM phase. Thus, while not every Weyl semimetal will harbor steplocalized chiral modes, a large fraction do. One can therefore be optimistic that, given a magnetic WSM, there is a large probability that terraces will exhibit localized QAH regions and manifest localized chiral modes.
To investigate these theoretical predictions, we study the magnetic Weyl semimetal Co_{3}Sn_{2}S_{2}. There is substantial prior evidence for the topological WSM nature of this material, including a large anomalous Hall effect^{25,26}, signatures of Fermi arc states in STM^{27,28,29}, and flat band diamagnetism caused by Berry curvature^{30}. Importantly, the compound is predicted to host the QAHE in the 2D limit^{31}, and models of a single, magnetic Co_{3}Sn kagome layer predict a nonzero Chern number^{30}. Therefore, this material appears to fit the model of a WSM constructed from stacked and coupled Chern insulators. In addition, the material’s high Curie temperature of 170 K^{32} makes Co_{3}Sn_{2}S_{2} an ideal candidate to test the predictions of terraces within the coupled Chern insulator model. If edge states are observed, then it would provide strong experimental evidence that the 2D limit of this material may host the QAHE at elevated temperatures.
Results
Topography and spectroscopy of Ssurfaces and Snsurfaces
We use scanning tunneling microscopy and spectroscopy (STM/S) at 4 K to investigate the possibility of realizing chiral edge states in Co_{3}Sn_{2}S_{2}. Co_{3}Sn_{2}S_{2} is a layered material consisting of a kagome Co_{3}Sn plane inbetween two hexagonal S layers, and all sandwiched in between two hexagonal Sn layers (Figs. 1d to 1f). The hexagonal lattice constant is a = 5.3 Å, while three stackings of SnSCo_{3}SnSSn layers each translated by (1/3, 1/3, 1/3) construct a full unit cell, giving a lattice constant c = 13.2 Å^{19}. The material is a halfmetallic ferromagnet, with magnetic properties derived from the moments of the Co atoms aligned along the caxis^{18,32}.
Co_{3}Sn_{2}S_{2} bulk single crystals cleaved along the (001) direction most often expose two distinct surfaces^{28,29,30} that are both hexagonal in nature. This suggests that the main cleavage plane is between the SSn/SnS layers revealing either the Sn or S layer. A third possible termination is a honeycomblike kagome Co_{3}Sn plane which is rare. The two surfaces most seen in STM can be distinguished topographically (one consisting of vacancies and the other adatoms) and also spectroscopically (as shown in Figs. 1g and 1h). On the surface with adatoms, we observe a sharp peak in the spectra near −12 meV associated with a diamagnetic flat band^{30}, along with another large peak near −300 meV. On the surface with vacancies, we see a depression in the density of states from −300 meV to 0 meV, with two broad peaks occurring at 50 meV and 200 meV, respectively.
Previous STM studies have arrived at different conclusions on the chemical identification of these two common hexagonal surfaces^{28,29,30}. To reconcile this issue, we utilize the symmetry of the local density of states signatures of defects to identify the termination layer. This method has been used successfully for the chemical identification of surface lattices of other layered materials^{33}. The density of state signatures of defects centered at positions that do not correspond to the positions of the top layer atoms can be attributed to defects in the layer below (DLB). Such signatures are present in both commonly observed hexagonal surfaces. However, DLBs existing on the surface with vacancies have a reduced symmetry (a triangle with one bright vertex) than the symmetry of DLBs on the surface with adatoms (a clover with equally bright vertices). The reduced symmetry of DLBs seen on the surface with vacancies is consistent with the layer below being the Co_{3}Sn kagome plane, which has the same reduced symmetry compared to hexagonal layers. This allows for the identification of the surface with vacancies as being the S plane, and the surface with the adatoms being the Sn plane (see Supplementary Note 2) which is consistent with recent STM studies^{34}.
Edge modes on Co_{3}Sn step edges
To look for edge modes, we investigate large Co_{3}Sn terraces terminated by a step edge (Fig. 2a). The Co_{3}Sn surface can be distinguished spectroscopically from the S and Sn planes. From Fig. 2b, the spectra seen on the Co_{3}Sn surface have two sharp peaks in the density of states at 0 meV and +60 meV which are not seen in the spectra on either the S and Sn surfaces and are consistent with spectra previously reported on the Co_{3}Sn surface^{28,29}. Point spectra taken on the edge of this surface reveals a broad accumulation density of states that is not present in spectra taken away from the edge. The broad accumulation of density of states at the edge over these energies is unique to the Co_{3}Sn surface (see Supplementary Fig. 3). Taking spectra along the line profile shown in Fig. 2a reveals that this enhanced density of states is localized to the edge, with an extent of approximately 1.5 nm transverse to the step edge (Fig. 2c and Supplementary Fig. 4). The highly localized nature of these states suggests that they can be attributed to an edge mode existing on an exposed Co_{3}Sn plane. Interestingly, despite the presence of impurities on the edge which act as scattering centers, we find no sign of quantization in the dI/dV spectra of the edge mode (see Supplementary Fig. 4). This however changes markedly when we investigate narrow terraces, as we show next.
Linearly dispersing edge states on narrow Co3Sn terraces
To further explore the nature of these edge states, we study narrow terraces of the kagome Co_{3}Sn planes. Incomplete cleavage between Sn and S layers occasionally exposes small terraces near step edges on the S surface (Fig. 3a and Supplementary Fig. 5 and 6). These terraces are approximately 130 pm below the S surface (Fig. 3b). As shown in the extended data (Supplementary Fig. 7), the closest plane to S is the Co_{3}Sn plane at a height of 156 pm below the Sn surface with the next plane being 312 pm below. The thin terrace can thus be unambiguously identified as the Co_{3}Sn plane. Taking a dI/dV map of one of these terraces reveals striking features, highlighted by quantum welllike bound states at various energies shown in the DOS images in Fig. 3c, f. Note that these states are responsible for obscuring the atomic resolution on this terrace. The number of nodes increases with energy, indicating a positive dispersion. The bound state energies for each distinct quantum welllike state can be identified from peaks in the spectra at locations where the density of states is maximum (see Fig. 3g). In the terrace of length 5.2 nm, shown in Fig. 3a, we observe a sequence of four quantum well like states ranging from n = 1 to n = 4, while in the other two terraces, with lengths of 5 nm and 6.1 nm as shown in Supplementary Fig. 3, we observe a sequence of four and five quantum well like states, respectively. The notation for the nth state is the same as in the classic quantum well, where n indicates a bound state with n + 1 nodes and n maxima in the local density of states. Converting the linear energy dependence on the bound state wavelength into a dispersion velocity we find a value of approximately 5 × 10^{4} m/s. Quantum well like one dimensional (1D) states have been observed previously with STM in systems such as Au/Cu adatom chains and semiconductor terraces^{35,36,37,38,39}, however all of these studies show a quadratic dependence of the bound state/subband energy on the number of nodes, as expected from the conventional quantum well states originating from freeelectronlike quadratic dispersion. Unlike these examples, the dispersion seen on all the terraces we observe is linear (as plotted in Fig. 3h).
Numerical simulation of hybridized chiral edge states
To interpret the STM results using the layered WSM model (Supplementary Note 1) we first perform numerical simulations of a WSM having a surface terrace with a partially exposed Chern insulator plane as shown in Fig. 4a top panel. Spacer layers between Chern insulator layers, in this case, the S and Sn layers, effectively change the coupling in the stacking direction. Identifying S/Sn as spacer layers is consistent with earlier work that shows that the band structure near the Fermi energy mainly arises from the Co3d bands^{26}, as well as work showing that the Kagome monolayer alone is a Chern insulator^{30}. Remarkably, we find that the chiral modes that we argued are present in the strongly coupled bilayer are recovered in the WSM regime, as seen in Fig. 4a bottom panel. These modes counterpropagate and are localized at edges of the partially exposed Chern insulator planes and decay exponentially along the surface and as a powerlaw into the bulk.
After confirming the existence of the chiral terrace modes within the coupled Chern insulator model, we further simulate the DOS signatures of the two (counterpropagating) chiral edge states in a quantum well using a simple onedimensional lattice model for linearly dispersing, counterpropagating modes in a potential well. We use a onedimensional Hamiltonian of the form \(H\,=\,\mathop{\sum}\nolimits_{n}\big(\frac{i}{2}{c}_{n+1,{{\alpha }}}^{\dagger }{c}_{n,{{\beta }}}{{{\sigma }}}_{{\rm{\alpha }}{\rm{\beta }}}^{z}\,\,\frac{i}{2}{c}_{n,{{\alpha }}}^{\dagger }{c}_{n+1,{\rm{\beta }}}{{\rm{\sigma }}}_{{\rm{\alpha }}{{\beta }}}^{z}\,+\,m{c}_{n,{{\alpha }}}^{\dagger }{c}_{n,{{\beta }}}{{{\sigma }}}_{{{\alpha }}{{\beta }}}^{x}+V\left(n\right){c}_{n,{{\alpha }}}^{\dagger }{c}_{n,{{\beta }}}{{{\delta }}}_{{{\alpha }}{{\beta }}}\big)\). Here the operator \({c}_{n,{\rm{\alpha }}}^{\dagger }\) creates an electron on site n in orbital α, the parameter m represents the evanescent hybridization between the counterpropagating modes,\(V\)(n)is a discretized finite square well potential, and all energy scales are in units of the tunneling strength which we have set to unity. See Fig. 4a, b and Supplementary Fig. 8 for a schematic illustration of the model setup. In the absence of the square well potential, this Hamiltonian yields an energy spectrum \(E=\pm \sqrt{{{\sin }}{\left({{\bf{k}}}_{a}\right)}^{2}+{m}^{2}}\) which, for vanishing hybridization (m = 0) and long wavelengths (k_{a}«1), recovers the linear dispersion \(E=\pm \left{\bf{k}}\right\) (see the leftmost panel in Fig. 4c and Supplementary Fig. 8). Turning on the hybridization term m opens a gap, lifting the degeneracy at zero momentum as shown in the middle panel of Fig. 4c. The length of the terrace is modeled as a finite square well, V(n), with a width that is a small fraction of the total system size. Adding a finite potential well with a nonzero mixing term creates eigenstates within the hybridizationinduced energy gap, as illustrated in the rightmost panel in Fig. 4c. The electron density of the eigenstates confined to the potential well resembles the bound states observed in our experiment (see Fig. 4d). In addition, for a wide range of m, the energies of the confined states are linearly dependent on the confinement quantum number n (see Fig. 4e and Supplementary Fig. 8). Also, if m becomes too large, the dispersion relation begins to resemble a typical quadratic band and the bound state energies cross over to a quadratic dependence on the confinement quantum number. If m is too small the counterpropagating modes will not effectively form a bound state, e.g., if the plateau is very wide so that the evanescent coupling between the opposed edge modes is small, the chiral edge mode will hit the potential wall and turn to continue its circulation around the plateau boundary instead of forming a coherent bound state. Physically, this indicates that when the terrace width is small enough to hybridize chiral edge states, we should expect quantum welllike states to develop. Notably, the transverse extent of the edge state observed on the Co_{3}Sn surface is of the same order (~1.5 nm) as the width of terraces containing quantum welllike bound states (Fig. 2 and Supplementary Fig. 5). This suggests that the edge states are within the intermediate evanescent mixing regime necessary to observe linearly dispersing bound states.
Discussion
We have found that our model can reproduce all the features seen in our experiment, providing a clear, selfconsistent explanation for the existence of linearly dispersing quantum welllike bound states, composed of hybridized chiral edge states on an exposed kagome Co_{3}Sn terrace. While the chiral nature of these states cannot be directly probed with STM, it follows naturally from topological edge states within a material with broken timereversal symmetry. It is worth noting that similar interference patterns of topological edge states have been observed by STM in Bi single crystals^{40,41}. While Bi edge state interference patterns are due to intraband scattering of topological edge states, our result originates from chiral edge states with the same quantum numbers that couple because of their close spatial proximity.
This observation of chiral edge modes within a bulk magnetic Weyl semimetal provides evidence for the physical realization of the model presented by Balents and Burkov^{16}. The modification of this original model to terrace geometries, combined with our experimental observations, theorizes a new paradigm for studying chiral edge states in a wide range of magnetic Weyl semimetals via local probes without requiring thin film growth or heterostructures. Most importantly, a reductionist approach to this model suggests that a material fitting this description will, in the 2D limit, be a Chern insulator hosting the QAHE. Co_{3}Sn_{2}S_{2}’s high Curie temperature^{32} which persists into the 2D limit^{42}, existing theoretical calculations in the 2D limit^{31}, and our observation of linearly dispersing chiral edge modes make the 2D limit of Co_{3}Sn_{2}S_{2} a strong candidate for the observation of intrinsic QAHE at elevated temperatures.
Data availability
The data in this work will be made available at reasonable request. Correspondence and requests for materials should be addressed to V.M. or H.B.
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Acknowledgements
The research at the University of Illinois at UrbanaChampaign was primarily supported by NSF through Materials Research Science and Engineering Center DMR1720633. V.M. acknowledges partial support from the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9465. C.F. acknowledges support by the European Research Council (ERC) Advanced Grant No. 742068 (TOPMAT), Deutsche Forschungsgemeinschaft (DFG) under SFB 1143 (Project No. 247310070), and WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, project no. 39085490). H.B. acknowledges support by the European Research Council (ERC) starting grant No. 678702 and German Science Foundation (DFG) by project A02 of the CRCTR 183 “entangled states of matter.
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S.H., L.J., H.B., C.F., and V.M. conceived the experiments; S.H., Z.W., N.M., R.B., P.N., and N.A. carried out STM studies; P.V., E.L., and C.S. made the single crystal samples; S.H. and T.H. carried out the theoretical calculations; and S.H., L.J., V.M., H.B., and T.H. wrote the paper with input from all authors.
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Howard, S., Jiao, L., Wang, Z. et al. Evidence for onedimensional chiral edge states in a magnetic Weyl semimetal Co_{3}Sn_{2}S_{2}. Nat Commun 12, 4269 (2021). https://doi.org/10.1038/s41467021245613
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DOI: https://doi.org/10.1038/s41467021245613
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