Localized spin-orbit polaron in magnetic Weyl semimetal Co3Sn2S2

The kagome lattice Co3Sn2S2 exhibits the quintessential topological phenomena of a magnetic Weyl semimetal such as the chiral anomaly and Fermi-arc surface states. Probing its magnetic properties is crucial for understanding this correlated topological state. Here, using spin-polarized scanning tunneling microscopy/spectroscopy (STM/S) and non-contact atomic force microscopy (nc-AFM) combined with first-principle calculations, we report the discovery of localized spin-orbit polarons (SOPs) with three-fold rotation symmetry nucleated around single S-vacancies in Co3Sn2S2. The SOPs carry a magnetic moment and a large diamagnetic orbital magnetization of a possible topological origin associated relating to the diamagnetic circulating current around the S-vacancy. Appreciable magneto-elastic coupling of the SOP is detected by nc-AFM and STM. Our findings suggest that the SOPs can enhance magnetism and more robust time-reversal-symmetry-breaking topological phenomena. Controlled engineering of the SOPs may pave the way toward practical applications in functional quantum devices.

Co3Sn2S2 has a layered structure which consists of two hexagonal planes of S and Sn as well as a Co3Sn kagome layer sandwiched between them (Fig. 1a-b).As previous literature reported, the Curie temperature of such compound is at 177 K and the spontaneous magnetic moment is about 0.3 μB/Co 8,14,15 .Single crystals of Co3Sn2S2 were cleaved in-situ at 6 K under ultrahigh-vacuum and immediately transferred to the STM head.Weak bonds between S and Sn atoms offer a cleave plane and lead to S-terminated (top in Fig. 1b) and Sn-terminated surface (middle in Fig. 1b).In topographic STM images, S-terminated surfaces show a hexagonal lattice with some randomly distributed vacancies (Fig. 1d), while Sn-terminated surfaces show a similar hexagonal lattice with adatoms and clusters (Fig. 1f).The dI/dV spectra on these two surfaces measured using a tungsten tip (Extended Data Fig. 1) are in good agreement with previous reports 10,16,17 .
In order to study the magnetism of Co3Sn2S2, we carried out spin-polarized measurements on S and Sn terminated surfaces with a ferromagnetic (FM) Ni tip, as schematically shown in Fig. 1c.The spinpolarization of the Ni tip was calibrated on Co/Cu(111) (Extended Data Fig. 2).External magnetic fields of +0.6 T (-0.6 T) were applied to magnetize the Ni tip to be in the spin up (spin down) states.This tip was then used to measure the spin-dependent dI/dV spectra at 0 T. In the defect-free regions of the S terminated surface, we did not observe spin-polarized contrast, indicating that the intact S-terminated surface is non-magnetic (Fig. 1e).On the Sn surface, however, the dI/dV spectrum taken with the spin-up tip shows much stronger intensity than that taken with the spin-down tip (Fig. 1g), indicating the presence of an FM order in the Sn layer.
To understand the distinct response of the two surfaces with different atomic terminations to spinpolarized tunneling, we carried out DFT calculations of the electronic structures of Co3Sn2S2.The spinresolved, projected density of states (PDOS) on the S-terminated surface (Extended Data Fig. 1) shows that the two spin channels are almost degenerate.The net magnetic moment in the S layer is nearly zero.
For the Sn-terminated surface, however, the calculated PDOS shows significant spin-polarization (Extended Data Fig. 1) with a net magnetic moment of 0.04 µB.These results are consistent with our STM/S measurements.Moreover, the DFT calculations show that the FM moment in the Sn layer is antiparallel to that (0.3 µB) of a Co atom in the Co3Sn layer, which show the same magnetic order as that in the bulk material (Extended Data Method part).The STM/S together with the DFT results reveals the layer-resolved magnetic structure: a dominant FM ordered Co3Sn layer, a weaker FM Sn layer with opposite magnetization, and a nonmagnetic S layer.Thus, the FM Co3Sn2S2 is in fact a weak ferrimagnet.
We next study the properties of localized excitations by focusing on a region with S vacancies on the Sterminated surface (Fig. 2).An atomic topographic image shows two single-S vacancies labeled as A and B, as well as a vacancy-dimer C (Fig. 2a).A typical dI/dV spectrum taken off of vacancies resembles closely that taken in a clean region on the S-surface far away, exhibiting an energy gap of 300meV (black curve in Fig. 2b) 10,16 .In particular, the dI/dV map of the broad hump around +50 meV (Fig. 2c), which is believed to originate from the topological surface states of the magnetic Weyl semimetal 10,16 , shows extended states with weakened intensity at the S-vacancies.Similarly, the peak near the top of the valence band at -350 meV has a spatial distribution seen in the dI/dV map (Fig. 2d) with suppressed density of states around the S-vacancy sites.Remarkably, the dI/dV spectrum taken at the site of vacancy A (red circle in Fig. 2a) reveals a series of approximately equal-spaced spectral peaks just above the valence band top (orange curve in Fig. 2b), indicative of S-vacancy induced in-gap bound states.To elucidate the spatial structure of the bound states, we measured dI/dV maps (Fig. 2e, 2f, and 2g) at -322, -300, and -280 meV, corresponding to the three discernable peaks.The bound states are localized and have a flower petal shaped pattern with a three-fold rotation symmetry around the single-S vacancies A and B. The electrons scatter from the S-vacancy potential and form a bound state corresponding to the sharp primary peak ~-280 meV much like a localized polaron, and the higher order peaks can be understood as due to the polaron shake-off process.These features at the single-S vacancies are highly reproducible in different regions and on different samples (Extended Data Fig. 3).From the statistical analysis, we determine the average spacing between the spectral peaks to be ~16meV (Extended Data Fig. 3).
To investigate the magnetic properties of the bound states, we switch to spin-polarized Ni tip and obtain the dI/dV spectra on a single-S vacancy D (Fig. 3a).The strong magnetic contrast over the energy range from -350 to -270 meV shows that the bound states are magnetic with a spin-down majority.The spin flip operation of the Ni tip is shown in Fig. 3b, where we zoom in to one of the two sub-peaks associated with the primary bound state, possibly due to the exchange field induced energy splitting (Extended Data Fig. 4).A spin down-tip was initially prepared, which gave a pronounced bound state peak around -283 meV (left panel in Fig. 3b).The polarization of the tip was then flipped to spin-up by an external magnetic field of +0.6 T. The intensity of the peak at -283 meV was reduced while the peak position did not change (middle panel in Fig. 3b).After flipping the tip back to spin-down, the peak intensity increases back to the initial level (right panel in Fig. 3b).These demonstrate the excellent reproducibility of the spin flip operation 18,19 , and that the bound states are magnetic polarons introduced by the S vacancies.Interestingly, the magnetization of all bound magnetic polarons is opposite to the FM moment in the Sn layer, but in the same direction as the ordered moment in the Co3-Sn layer.Indeed, from the high resolution dI/dV map acquired with the W tip around vacancy D (Fig. 3c), the three-fold symmetric spatial profile of the bound magnetic polaron can be traced out and superimposed onto the atomic structure projected to the S-surface (Fig. 3d), revealing its overall correlation with the underlying Co atoms.These experimental findings are qualitatively in agreement with and supported by our DFT calculations (Extended Data Fig. 5).
To further investigate the nature of the bound magnetic polaron, we measure the magnetic field response of the spectral peaks in dI/dV using a normal W tip. The magnetic field is applied perpendicular to the sample surface, ranging from -6 T to 6 T. As shown in Fig. 4a and b, when the amplitude of the magnetic field increases, the peak positions shift linearly toward the higher energy side, independent of the direction of the magnetic field.Such an anomalous Zeeman response is reproducible on different vacancies (Extended Data Fig. 6), and indicates the important orbital contribution to the magnetic moment of the polaron.Because of this, we will refer to the polaron as a bound spin-orbit polaron (SOP).By fitting the two peak positions as a function of the magnetic field (Fig. 4c), we obtained a slope of 75 µeV/T = 1.35 µB for the effective moment of the SOP.We have also measured the magnetic field dependence of the near-zero-energy peak on the Sn-terminated surface, and observed a similar anomalous shift (Extended Data Fig. 7), which was attributed to the Berry phase induced orbital magnetization in the kagome flat band 13 .It is remarkable that the bound SOP nucleated at the S vacancies in Co3Sn2S2 presents a localized analog of the anomalous Zeeman shift, highlighting the rich and novel behaviors of the correlated and topological d-electrons in this kagome-lattice Weyl semimetal.
Finally, we study the magnetic-lattice coupling of the bound SOP.The lattice distortion around the S vacancies (Fig. 4d) shows significant local atomic displacements (see also Extended Data Fig. 8).To quantify the lattice distortion, we measure the average nearest atom distance around the S vacancy and determine the local atomic displacement ratio as its percentage changes from the average nearest atom distance far away from the vacancy.The local atomic displacement ratio is ~6% under zero magnetic field.
An external magnetic field is then applied along the c-axis.We find the local atomic displacement ratio decreases significantly with increasing magnetic field strength (Extended Data Fig. 9), and about onethird of the displacement ratio can be manipulated by a magnetic field up to 6 T.These observations further support the SOP nature of the S vacancy-induced bound states and the strong magnetic-lattice coupling in Co3Sn2S2.
We have revealed the intriguing and diverse magnetism in Co3Sn2S2 with both itinerant and localized contributions.The main findings are summarized in the schematic illustration (Fig. 4e).The itinerant FM order is weakly ferrimagnetic with oppositely polarized magnetic moments in the Co3Sn and the Sn layers, while the S surface is nonmagnetic.Such layer-dependent magnetism may be relevant for the recently observed diversity in the Fermi arc topological surface states 10 .The missing S atom causes strong scattering and hybridization between the p-electrons of S and the d-electrons of the underlying Co, giving rise to a localized bound SOP with antiparallel (parallel) magnetization direction as compared to that of the Sn (Co3Sn) surface.Bound magnetic polarons play an important role in dilute magnetic semiconductors [20][21][22] , where they form around the doped magnetic impurity ions and order magnetically through the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions.The bound SOP at the vacancies in the nonmagnetic layer discovered here is new and open a direction for manipulating the magnetic order in the Weyl semimetal Co3Sn2S2.While it is currently unclear the extent to which S vacancies exist in as grown stoichiometric materials, S deficient Co3Sn2S2-δ crystals have been synthesized 23 and should allow controlled and systematic study of the magnetism as a function of the density of the S vacancies.The bound SOP should emerge and enhance the saturated ordered moment as observed recently at small S deficiency 23 .We propose that increasing the SOP density will raise the Curie temperature and enable topological properties, such as the giant anomalous Hall and Nernst transport, to be observed at higher temperatures.

Single crystal growth of Co3Sn2S2
The single crystals of Co3Sn2S2 were grown by flux method with Sn/Pb mixed flux.The starting materials of Co (99.95% Alfa), Sn (99.999%Alfa), S (99.999% Alfa) and Pb (99.999%Alfa) were mixed in molar

Density functional theory
Quantum mechanical calculations based on density functional theory (DFT) were performed by using the Vienna Ab initio Simulation Package (VASP) 24,25 .The projector augmented wave (PAW) 26 method was employed, and the Perdew-Burke-Ernzerhof (PBE) 27 type of exchange correlation functional was used.
The slab models containing six Co-Sn layers and extra S or Sn layers were used to simulate the Sterminated and Sn-terminated surfaces.S surfaces with a single S vacancy were simulated by a 4×4 supercell.In structural relaxations, the atoms in the two middle Co3Sn layers were fixed, while atoms in other layers were totally relaxed.The vacuum layers of the slab models are larger than 15 Å.
The wavefunctions are expanded in plane waves with a kinetic energy cutoff of 400 eV.For pristine Sterminated and Sn-terminated surfaces of Co3Sn2S2, the k-points sampling is 8×8×1, generated by Monkhorst-Pack grids with the origin at the Γ-point.The structures were relaxed until the energy and residual force on each atom were smaller than 10 -6 eV and 0.001 eV/Å, respectively.For S-terminated surface with single vacancy, the k-points sampling is with only the Γ-point.The structures were relaxed until the energy and residual force on each atom were smaller than 10 -4 eV and 0.01 eV/Å, respectively.
With these parameters, the optimized lattice constant of bulk Co3Sn2S2 is 5.37 Å and 13.15 Å for a and c directions, respectively.The magnetic moment is -0.35 μB, 0.02 μB, 0.03 μB, and 0 μB for Co, Sn in Co3Sn plan, Sn in Sn layer, and S, respectively.distributed single S vacancies (Scanning setting: bias: V s =-400 mV, setpoint I t =100 pA).e, dI/dV spectra at intact S surface region using up-polarized tip (red curve) and down-polarized tip (blue curve), showing no polarization contrast on intact S surface (V s =-400 mV, I t =100 pA, Modulation V mod = 0.5 mV).f, atomicresolution STM image of the Sn-terminated surface (V s =-400 mV, I t =100 pA).g, Spin-polarized dI/dV spectra at Sn surface, showing a more massive intensity of spin-up contribution (V s =-400 mV, I t =100 pA, V mod = 0.5 mV).
ratio of Co : S : Sn : Pb = 12 : 8 : 35 : 45.The mixtures were placed in Al2O3 crucibles sealed in quartz tubes.The quartz tubes were slowly heated to 673 K over 6 h and kept there over 6 h to avoid the heavy loss of sulfur.The quartz tubes were further heated to 1323 K over 6 h and left there for 6 h.Then the melt was cooled down slowly to 973 K over 70 h.At 973 K, the flux was removed by rapid decanting and subsequent spinning in a centrifuge.The hexagonal-plate single crystals with diameters of 2 ~ 5 mm were obtained.The compositions and phase structure of the crystals were checked by energy-dispersive x-ray spectroscopy and x-ray diffraction, respectively.Scanning tunneling microscope and spectroscopyThe samples used in the experiments were cleaved in situ at 6 K and immediately transferred to an STM head.Experiments were performed in an ultrahigh vacuum (1×10 -10 mbar) ULT-STM systems (40 mK) equipped with 9-2-2 T magnetic field.All the scanning parameter (setpoint voltage and current) of the STM topographic images are listed in the captions of the figures.Unless otherwise noted, the differential conductance (dI/dV) spectra were acquired by a standard lock-in amplifier at a frequency of 973.1 Hz.Non-magnetic tungsten tip was fabricated via electrochemical etching and calibrated on a clean Au(111) surface prepared by repeated cycles of sputtering with argon ions and annealing at 500 ℃.Ferromagnetic Ni tips were applied in the spin-polarized STM measurement.The Ni tips were fabricated via electrochemical etching of Ni wires in a constant-current mode28 .To calibrate the spin-polarization of Ni tip, the prepared Ni tips have been applied to resolve magnetic-state-dependent contrast of Co islands grown on a Cu(111) surface in SP-STM experiments (details are shown in Extended Data Fig.2).

. 5 |
The calculated electronic structure of S-terminated surfaces with a S vacancy (VS).a, The band structures of the pristine S surface (in a 4 × 4 supercell).b, The band structure of S surfaces with a VS.By comparing (a) and (b), the bound state was recognized and labeled by the blue arrow.c, Spin resolved DOS projected on the S vacancy.It is clearly shown that the S vacancy is spin polarized, which is in agreement with the experimental observation.d, Spin resolved DOS projected on the three Co atoms under the S vacancy.Comparing (c) and (d), we can conclude that the magnetic moment of VS has the same direction as that of the Co3Sn plane.