Magnetochiral Spin-Polarized Tunneling in a Paramagnetic State

Crystallographic chirality can mediate various optical and electrical magnetochiral effects. Since these effects have been studied in bulk optical, transport or non-local probe setups, investigation with a local probe is necessarily the next step towards further understanding of the intriguing phenomena closer to the quantum regime. We observed a spin-polarized scanning tunneling microscopy (SP-STM) contrast in the chiral domains of Co1/3NbS2 in a paramagnetic state, which is unexpected in the conventional SP-STM mechanism. This spin-polarized tunneling, depending on the local structural chirality, is argued to be an inverse magnetochiral effect due to a dynamic coupling between tunneling electrons and chirality. In addition, using the standard STM, we also find magnetochiral nonreciprocal tunneling in the presence of external magnetic fields, considered as the inverse process. Our results demonstrate a new application of SP-STM in detecting the dynamic interaction of tunneling electrons with broken crystallographic symmetries. Magnetochiral effects can arise from the broken mirror symmetry of chiral materials interacting with various moving entities such as photon1, phonon2, or electron3. Lately, chirality-induced spin selection, where flow of photoelectrons gets spin-polarization through magnetochiral interaction, has been studied intensively on chiral molecules4–7. In crystalline materials, where crystallographic chirality interplays with various exotic magnetic orders such as skyrmion8, polar magnetism9, multiferroicity10, and chiral topological spin texture11,12, a similar effect has been found that electric conduction through a chiral crystal, e.g. trigonal Te13–15, can directly induce magnetic moment. Interestingly, the THz-range Faraday effect of chiral Te in the presence of electric current was reported four decades ago, and can be simply explained in terms of the induced magnetization by a current16. These effects can also be understood based on the symmetry operational similarity (SOS) between a magnetic moment and a chiral structure placed in a quasi-equilibrium electric conduction17,18. Although the chirality-induced spin selection has a classical analogy of the magnetic-field generation by flowing electric current through a conducting coil, the quantum mechanical understanding is not intuitive and there is still much room for improvement4,5. Specifically, there still lacks any detailed investigation of the magnetochiral interaction at the atomic level where quantum mechanical tweak often acts in a counter-intuitive way. Moreover, whether the magnetochiral interaction can be associated with local probing techniques such as STM is a crucial question19 that might provide superior spatial resolution compared to the existing bulk optical or transport techniques in chirality studies20–22. However, it is highly unclear how the accumulated effect in bulk contributes to the surface measurement with an atomic spatialresolution, or even whether the spin-polarization induced by the magnetochiral interaction is a quantity that can be detected with SP-STM. SP-STM is a powerful tool to visualize various magnetic orders in, e.g. ferromagnets23, antiferromagnets24, helimagnets25, and skyrmion sytems26, at the atomic scale. In addition, considering various interactions that magnetism can have with other order parameters, its use may be not limited to visualizing magnetic orders. In this paper, we report our discovery of a novel spin-polarization mechanism in the quantum tunneling due to the magnetochiral interaction of electrons with a chiral structure and demonstrate the detection of the atomic chiral environment with SP-STM. As an exemplary material, we chose Co1/3NbS2, one of the chiral intercalated transition metal dichalcogenides, where we found, for the first time, the numerous structural chiral domains outlined by a network of topological vortex/antivortex27. The existence of micron-scale structural chiral domain boundaries makes this system unique to detect the change of magnetochiral interaction across the domain boundaries. Moreover, Cr1/3NbS2 (isostructural to Co1/3NbS2) has been reported to show magnetochiral interactions in a bulk transport setup, i.e. chirality-induced spin transport19 and anomalous nonreciprocal electrical transport28. Specifically, Co1/3NbS2, compared with Cr1/3NbS2, is advantageous since it shows paramagnetic magnetism at temperatures above the antiferromagnetic transition temperature of ~26 K29, thus we can eliminate the influence of magnetic ordering at the liquid nitrogen temperature and test solely the intrinsic magnetochiral effect. Furthermore, a study of the magnetochiral interaction in chiral van der Waals materials could suggest a novel mechanism that controls spin degrees of freedom with electrical means for two-dimensional spintronics30. Structural Chiral Domains and Topological Vortices The crystal structure of Co1/3NbS2 consists of 2H-NbS2 matrix and intercalated Co ions occupying one of three Nb-atop sites, A, B, and C. The Co ions are arranged in such a way that the intercalation site changes alternately along the z direction, e.g. AB-type stacking. In Fig. 1a, we draw the AB-stacked and BA-stacked structures with opposite chiralities. The planar displacements of S ions determine the chirality of the structures as shown with red or blue triangular arrows. In relation with the fictitious cleaving planes (the horizontal black lines), the top-most NbS2 layer has upper and lower Co ion lattices, and the first and second characters in a stacking sequence denote the intercalation site of the upper and lower Co lattices, respectively. Throughout this paper, we will use Upper (Lower) Co to refer the top-most (subsurface) Co lattice. There exist six combinations of stacking sequence, the chirality of which can be identified from the permutation sequence (i.e. [AB, BC, CA] and [BA, CB, AC] have the opposite chiralities). The structure belongs to one of the Sohncke space groups (P6322) that can bear the opposite chiralities within one space group31. The chiral P6322 structure is obtained from the high-temperature P63/mmc prototype through Co ordering and S displacements in Co1/3NbS2, accompanied by the cell tripling. Ref. 27 reported the first discovery that in chiral Fe1/3TaS2, the six-types of domains are interlocked around a topological vortex/antivortex core, and adjacent domains always show the opposite chirality27. Our superlattice dark-field transmission electron microscopy (DF-TEM) image of Co1/3NbS2 manifests clear micron-scale bright and dark regions, corresponding to opposite-chirality (i.e., left-handed and right-handed) chiral domains, as shown in Fig. 1b. The DF-TEM image also shows that similar with Fe1/3TaS2, Co1/3NbS2 displays topological vortex domains: depending on the sign of the vorticity, a topological defect is either a topological vortex or antivortex, and these vortices and antivortices tend to be paired in a few-micrometers scale, which can be manipulated through the cooling rate across Tc. (see Supplementary Information, SI Fig. S1) Fig. 1c represents the atomic model of the chiral domains around a vortex core. Note that there are two types of domain boundaries, solid and dashed lines, where we have a change of intercalation sites only in Upper or Lower Co, respectively. We will similarly refer them as Upper and Lower domain boundaries. Then, it is evident that a cleaved surface only exposes three Upper domain boundaries while rest three Lower domain boundaries exist underneath the top-most NbS2 layer, which is an important structural aspect in STM images. In Fig. 1d, we show the magnetic susceptibility revealing antiferromagnetic ordering below 26 K, which is consistent with the previous result32. Since all the following experiments are performed at the liquid nitrogen temperature, we expect no spin-polarized signal from ordered Co magnetic moments, which is in the paramagnetic state. Chiral Structural Domains in STM/SP-STM Co1/3NbS2 is cleaved at a sample stage cooled by liquid nitrogen to minimize the effect of thermal diffusion. (SI note 1) Because Co ions reside in the van der Waals gap, they can remain on either side of exposed surfaces, so two types of surface, e.g. Co-/NbS2-types, can be produced. Fig. 2a shows the STM topographies, the simulated images, and FFTs of the two types. The primary difference is the predominant periodicity shown in the images as well as in FFT. The Co-type surface is dominated by �√3 × √3�R30° super-structure peak (red circles) while NbS2-type one reveals more obvious (1 × 1) peak (white circles) of the host unit cell. Although both types show mixed traces of two periodicities, the type of surface can be identified from the stronger peaks in FFT (especially at low biases). The coexistence of the two types becomes more obvious when it comes to a boundary between the two types as shown in Fig. 2b. In order to compare STM and SP-STM on chiral domains, we have used two Pt/Ir-alloy tips that are treated on non-magnetic Cu(111) and magnetic Cr(001) surface, respectively. The SP-STM tip is coated with Cr from the Cr(001) surface, and its magnetic sensitivity is verified by the layer-bylayer contrast of Cr(001) antiferromagnetic order33. (SI note 2) It is known that the magnetization direction of antiferromagnetic Cr tips tends to be randomly oriented, so it shows both in-plane and out-of-plane sensitivity34. Throughout this paper, we assume the dominant out-of-plane sensitivity along the chiral axis, which is the expected induced magnetization direction by symmetry17. (It is also consistent with the observed two levels of contrast, which will be discussed later.) Fig. 2c and 2d show the topographies of topological vortices in STM and SP-STM on Co-type surfaces. It is immediately noticeable that STM shows only “three” Upper domain boundaries depicted by aligned Co deficiencies, while SP-STM shows “domain contrast” exhibiting “six” domains and six boundaries for one vortex. Considering the fact that Co lattices exhibit localized orbitals well separated by the delocalized electrons of NbS2, the appearance of only three Upper domain boundaries in STM seems reasonable. The observed Co-deficient Upper domain boundaries are consistent with the fine atomic-structure model of domain boundaries around a topological vortex core. (SI note 3) On the other hand, the significant domain contrast in SP-STM is obviously not from magnetic order at this paramagnetic temperature, and we attribute it to magnetochiral interaction as will be discussed later. Note that the domain contrast becomes clearer in a larger size scan as it averages out the Co deficiencies. (Fig. 2d inset) To examine the Co lattice shift, we obtained atomic images near boundaries. The STM image in Fig. 3a shows lined-up Co deficiencies that form an Upper domain boundary (gray sold line). As the intercalation sites, A, B, and C, have a relative translational shift that corresponds to 1/3 of the unit cell length (along both of the a and b directions), Co lattice manifests a 2ππ 3 ⁄ phase shift upon changing intercalation site. The shift is evident in the expanded image in Fig. 3d, which is consistent with the atomic model of an Upper domain boundary given in Fig. 3g. Note that this atomic-scale configuration of a domain boundary is also consistent with the extinction rule for the antiphase boundaries in super-lattice DF-TEM image27. (SI note 3) To our best knowledge, this is the first atomic observation of topological vortex domain boundaries in chiral 1/3 intercalated transition metal dichalcogenides that can reveal the topological nature of vortex structures. Regarding the topological nature, we emphasize that the S displacement vectors, in modulus to unit cell translation, exhibits a 2ππ 3 ⁄ rotation between adjacent domains, and thus enclosing a topological vortex core results in an overall 4π phase shift, i.e. the topological charge of ±236. (SI note 4) The completion of full rotation in the order parameter space (i.e. the presence of topological charge) is closely related to the robustness of the defect structure (i.e. topological protection), especially, to the prohibited domain boundary switch. (SI Fig. S4) Domain boundaries obtained with SP-STM show more complicated and non-trivial features. The main difference is that it shows contrast-change corresponding to the chirality (i.e. domain contrast), which is shown as depressed height in the middle domain of Fig. 3b. As a result, both the Upper and the Lower domain boundaries are traced by gray solid and dashed lines, respectively. The Co lattice shift of the Upper domain boundary (Fig. 3e) follows the same manner as in the case of STM and is consistent with Fig. 3g again. However, in case of the Lower domain boundary, there is no shift as shown in Fig. 3f, and it matches with the model of Fig. 3h. Instead of phase shift, the domain boundary exhibits a step-like topographic change as depicted in the height profile in Fig. 3c. Note that the alternating contrast shows a direct correlation with the chirality and is consistent with the expected current-induced magnetization along the electric current direction in transport setup19 and within SOS concept17. Emphasize that instead of phase shift, the change of atomic “chiral” environment makes the spin-polarized signal only appearing in SP-STM, not in STM. Inverse Magnetochiral Effect The spectroscopic signature of the magnetochiral interaction has been checked by obtaining tunneling spectra across a domain boundary in SP-STM. Fig. 4a shows a series of spin-polarized tunneling spectra across a Lower domain boundary. While the modulation by atomic corrugation is visible throughout the spectra, there is a definitive change of spectral intensity across the boundary. To examine the change in more detail, the spectral intensities from two representative biases are depicted in Fig. 4b. The changes of spectral intensity that follow step-like guides are obvious, and the step-like feature is a result of adding the opposite magnetochiral interaction to the base tunneling spectra (gray lines). Therefore, the domain contrast in topography is due to the enhanced/suppressed tunneling probability upon chirality change. Next, we check the dependence on the electric field direction. Although it can be seen briefly in the inversion of the step-like guides in Fig. 4b, it is depicted fully by comparing the spectra obtained from the two points (red and blue stars in Fig. 4a). The two spectra in Fig. 4c indicate zero bias as the crossing point where the magnetochiral interaction vanishes. We can further eliminate the “non-magnetochiral portion” by getting polarization PP = (GG↑ − GG↓) (GG↑ + GG↓) ⁄ where GG↑(GG↓) depicts the tunneling conductance of parallel(antiparallel) spin-polarized tunneling (Fig. 4c inset), which denotes zero bias as its crossing point. In other words, the effect is exactly antisymmetric about the zero bias. Generally, in any magnetic ordered states, spin polarization of tunneling current reflects the difference between opposite spin states, and the zero-crossing point in polarization appears at a somewhat arbitrary energy level, as can be found in many examples in the comprehensive review by Wiesendanger37. The coincidence of the crossing point of spin polarization with zero bias is a crucial characteristic, which is remarkably different from the conventional mechanism for SPSTM contrasts. We try to understand our observation based on the comparison with the previously known two magnetochiral phenomena: (A) current-induced magnetization in bulk chiral crystal13,19 and (B) spin-polarization in photoelectron transmission of chiral molecules5,6. First, our result has a similarity with (A) in the sense that the material has chirality in bulk structure but is different from (A) since our measurement detects atomic local regions. The observation of spin-polarized current in chiral Cr1/3NbS2 with bulk transport setups19 was claimed to be associated with the nonlocal measurement of diffused spins, which is distinct from our results with local tunneling current. This remarkable difference makes it difficult to employ the theories relevant to (A)14,15,38,39 directly. The local-probing nature is in line with (B), dealing transmitted photoelectrons interacting with an individual chiral molecule. However, a perturbative approach5 in (B) utilizing a small spin-orbit coupling of light atoms composing chiral molecules cannot be applied to Co1/3NbS2, where dorbital bands of Co/Nb dominate the electronic structure near the Fermi energy40. Moreover, the nature of tunneling electrons and photoelectrons6 should be very much different. As a result, we argue that a new theoretical framework is required for the unprecedented atomic observation of the magnetochiral interaction, which we will refer as the inverse magnetochiral effect. (the name is used in contrast to the various magnetochiral effects exhibiting nonreciprocal change of quantities induced by magnetic field1,3) We pay particular attention to the common characteristics among our result and the cases in (A) and (B). Firstly, the induced spin-polarization should be reversible by changing chirality. Secondly, since the effect is caused by interaction between nonequilibrium electric carrier and chiral structure, the effect should disappear when electric current vanishes and reverses when current flows backwards5,14,15,38. We emphasize that both of these are already shown in the tunneling spectra analysis by the reversal upon chirality change and the zerobias crossing point. Thus, we argue that all these magnetochiral phenomena have something in common at the fundamental level. Magnetochiral Nonreciprocal Tunneling Lastly, we test the cross-coupling between tunneling electrons and magnetism. In, for example, linear magnetoelectric systems, the magnetoelectric coupling depends on the strength of linearly cross-coupled term (i.e. EE ∙ HH) in free energy41, and gives a mutual relationship that induces one by another. A similar relationship can be assumed for the inverse magnetochiral effect. In other words, “electric tunneling induced by magnetic field” and “magnetization induced by tunneling electron” are all possible in chiral systems. However, any permanent electric current induced by magnetic field in an isolated system is thermodynamically prohibited, unless it is, for example superconducting. Instead, when there is a current flow by an external electromotive force, the magnetochiral interaction is in the form of enhancement/suppression of electric current. This phenomenon has already been reported in single wall carbon nanotube42, chiral MnSi3, or helically deformed Bi crystal43 as known as electrical magnetochiral anisotropy. The chirality-dependent resistance-change under magnetic field has an analogue with tunneling probability change in the presence of external magnetic field in a STM setup although it is not straightforward to match tunneling probability and resistance. We will refer this effect as magnetochiral nonreciprocal tunneling. Note that it does not require any magnetic probe to see this effect. The STM image in Fig. 4d demonstrates the emergence of domain contrast upon application of a magnetic field, which confirms the existence of the magnetochiral nonreciprocal tunneling effect. Although, from the viewpoint of SOS, it was predicted that the magnetic field applied to a chiral structure could exhibit nonreciprocity17, it has never been associated with electric tunneling or scanning probe microscopy. In conclusion, we have demonstrated that the inverse magnetochiral effect can be exploited to observe chiral domains and domain boundaries using SP-STM. The inverse magnetochiral effect is reversible readily by changing chirality or the electron tunneling direction, which is in a clear distinction with the conventional SP-STM contrasts from different orientations of ordered local spins. Thus, SP-STM can, now, be utilized to reveal atomic-scale chiral structures, in addition to atomic spin structures. Moreover, the magnetochiral nonreciprocal tunneling, an inverse phenomenon, gives rise to domain contrast upon application of external magnetic field without using a magnetized probe. Our observations provide a new paradigm to unveil quantum-level interrelationship between electric tunneling and crystallographic chirality. We also would like to remark that the novel spin-polarization mechanism using a chiral van der Waals material can be an innovative spin-polarizing technique in two-dimensional spintronics, where spin-injection has been relied on integration of ferromagnetic materials or circularly polarized light so far30. Methods Crystal growth Single crystals of Co1/3NbS2 were grown by a chemical vapor transport reaction method in the presence of iodine as a transport agent. About 0.2 g mixture of cobalt powder (Alfa Aesar, 99.99%), niobium powder (Alfa Aesar, 99.9%), and sulfur piece (Alfa Aesar, 99.999%) with a molar ratio Co:Nb:S=0.7:1:2 were sealed in a quartz tube with 100 mg of iodine (Alfa Aesar, 99.99%) under vacuum. Then, the quartz tube was placed under an optimum temperature gradient in a two-zone tube furnace. The hot and cold zones were kept at 9500C and 8000C, respectively. After about 4 weeks, the tube furnace was turned off and the quartz tube was cooled to room temperature naturally. Crystals in hexagonal shape were collected both at the hot and cold end of the quartz tube. X-ray diffraction data on ground powder specimens were taken at room temperature with Cu Kα(λ = 0.15418 nm) radiation in a Malvern Panalytical X’Pert 3 powder diffractometer. Transmission Electron Microscopy Specimens for TEM experiments were prepared by Ar-ion milling and studied using a JEOL2010F TEM. To unveil the chiral domains, DF-TEM images were taken by selecting g±= ±222 along the [101] zone axis. All images are raw data. Scanning Tunneling Microscopy STM measurements were performed using a Unisoku ultra-high vacuum SPM system (USM-1500) equipped with a cleaving stage capable of cryogenic cooling. Cu(111) and Cr(001) samples cleaned by repeated cycles of sputtering and annealing have been used as reference samples as well as a tip treatment base. Two Pt/Ir alloy tips were prepared by electron bombardment heating in ultra-high vacuum, and treated separately on Cu(111) and Cr(001) surface to be used for normal STM and SP-STM, respectively. The treatment was done by tip shaping mechanism implemented in the controller (Nanonis) typically with indentation into the surface with a negative bias to the tip and pulling out with a few nm/s of speed. Several repetitions of the procedure result in Cr transfer to the tip. The magnetization of Cr-coated tips are checked by tunneling spectroscopy across a step edge with the alternating contrast of antiferromagnetic ordering of Cr(001) as described in SI Note 2. Co1/3NbS2 sample is glued to a sample plate by silver epoxy (Epotek H20E) and a metal post is attached to the top with the same epoxy. Then the sample is cleaved at 80 K in the cleaving stage after precooling of 1 hour. All the STM and SP-STM measurements are acquired at 78 K. The tunneling spectroscopy measurement is obtained by modulation of bias and demodulation of tunneling current using lock-in technique. STM Simulation Density functional theory Calculations were performed using GPAW package44 within the PAW formalism implemented in Atomic Simulation Environment45 (ASE). We used PBE exchangecorrelation functional46 with SCF convergence criteria of 4.0 × 10−8 eV2 el. ⁄ (for eigenstates), 5.0 × 10−4 eV (energy), and 10-4 electrons (electron density). Gamma-centered k-point mesh of (2 × 2 × 2) was used to sample the k-space, and 0.2 Å real space grid was used for wavefunction expansion in the finite-difference scheme. The surface of Co1/3NbS2 was modeled with four slabs of 2H-NbS2 with embedded intercalants in between. The top-most Co lattice was added/removed according to the type of surface that is being simulated with 10 Å of vacuum added on top. The STM simulations were performed using the local density of states obtained from the density functional theory calculations following the Tersoff-Hamann approach47 as implemented in the GPAW package. Author contributions S.C. initiated the study; S.L. performed STM and simulation; F.H. performed the TEM; S.P. and K.W. grew the crystals; J.K. performed susceptibility measurement; S.L. and S.C. analyzed the data and wrote the paper. Acknowledgement This work was supported by the center for Quantum Materials Synthesis (cQMS), funded by the Gordon and Betty Moore Foundation’s EPiQS initiative through grant GBMF6402, and by Rutgers University.

Magnetochiral effects can arise from the broken mirror symmetry of chiral materials interacting with various moving entities such as photon 1 , phonon 2 , or electron 3 . Lately, chirality-induced spin selection, where flow of photoelectrons gets spin-polarization through magnetochiral interaction, has been studied intensively on chiral molecules [4][5][6][7] . In crystalline materials, where crystallographic chirality interplays with various exotic magnetic orders such as skyrmion 8 , polar magnetism 9 , multiferroicity 10 , and chiral topological spin texture 11,12 , a similar effect has been found that electric conduction through a chiral crystal, e.g. trigonal Te [13][14][15] , can directly induce magnetic moment. Interestingly, the THz-range Faraday effect of chiral Te in the presence of electric current was reported four decades ago, and can be simply explained in terms of the induced magnetization by a current 16 . These effects can also be understood based on the symmetry operational similarity (SOS) between a magnetic moment and a chiral structure placed in a quasi-equilibrium electric conduction 17,18 . Although the chirality-induced spin selection has a classical analogy of the magnetic-field generation by flowing electric current through a conducting coil, the quantum mechanical understanding is not intuitive and there is still much room for improvement 4,5 .
Specifically, there still lacks any detailed investigation of the magnetochiral interaction at the atomic level where quantum mechanical tweak often acts in a counter-intuitive way. Moreover, whether the magnetochiral interaction can be associated with local probing techniques such as STM is a crucial question 19 that might provide superior spatial resolution compared to the existing bulk optical or transport techniques in chirality studies [20][21][22] . However, it is highly unclear how the accumulated effect in bulk contributes to the surface measurement with an atomic spatialresolution, or even whether the spin-polarization induced by the magnetochiral interaction is a quantity that can be detected with SP-STM.
SP-STM is a powerful tool to visualize various magnetic orders in, e.g. ferromagnets 23 , antiferromagnets 24 , helimagnets 25 , and skyrmion sytems 26 , at the atomic scale. In addition, considering various interactions that magnetism can have with other order parameters, its use may be not limited to visualizing magnetic orders. In this paper, we report our discovery of a novel spin-polarization mechanism in the quantum tunneling due to the magnetochiral interaction of electrons with a chiral structure and demonstrate the detection of the atomic chiral environment with SP-STM. As an exemplary material, we chose Co1/3NbS2, one of the chiral intercalated transition metal dichalcogenides, where we found, for the first time, the numerous structural chiral domains outlined by a network of topological vortex/antivortex 27 . The existence of micron-scale structural chiral domain boundaries makes this system unique to detect the change of magnetochiral interaction across the domain boundaries. Moreover, Cr1/3NbS2 (isostructural to Co1/3NbS2) has been reported to show magnetochiral interactions in a bulk transport setup, i.e. chirality-induced spin transport 19 and anomalous nonreciprocal electrical transport 28 . Specifically, Co1/3NbS2, compared with Cr1/3NbS2, is advantageous since it shows paramagnetic magnetism at temperatures above the antiferromagnetic transition temperature of ~26 K 29 , thus we can eliminate the influence of magnetic ordering at the liquid nitrogen temperature and test solely the intrinsic magnetochiral effect. Furthermore, a study of the magnetochiral interaction in chiral van der Waals materials could suggest a novel mechanism that controls spin degrees of freedom with electrical means for two-dimensional spintronics 30 .

Structural Chiral Domains and Topological Vortices
The crystal structure of Co1/3NbS2 consists of 2H-NbS2 matrix and intercalated Co ions occupying one of three Nb-atop sites, A, B, and C. The Co ions are arranged in such a way that the intercalation site changes alternately along the z direction, e.g. AB-type stacking. In Fig. 1a, we draw the AB-stacked and BA-stacked structures with opposite chiralities. The planar displacements of S ions determine the chirality of the structures as shown with red or blue triangular arrows. In relation with the fictitious cleaving planes (the horizontal black lines), the top-most NbS2 layer has upper and lower Co ion lattices, and the first and second characters in a stacking sequence denote the intercalation site of the upper and lower Co lattices, respectively.
Throughout this paper, we will use Upper (Lower) Co to refer the top-most (subsurface) Co lattice. layer, which is an important structural aspect in STM images. In Fig. 1d, we show the magnetic susceptibility revealing antiferromagnetic ordering below 26 K, which is consistent with the previous result 32 . Since all the following experiments are performed at the liquid nitrogen temperature, we expect no spin-polarized signal from ordered Co magnetic moments, which is in the paramagnetic state.

Chiral Structural Domains in STM/SP-STM
Co1/3NbS2 is cleaved at a sample stage cooled by liquid nitrogen to minimize the effect of thermal diffusion. (SI note 1) Because Co ions reside in the van der Waals gap, they can remain on either side of exposed surfaces, so two types of surface, e.g. Co-/NbS2-types, can be produced. Fig. 2a shows the STM topographies, the simulated images, and FFTs of the two types. The primary difference is the predominant periodicity shown in the images as well as in FFT. The Co-type surface is dominated by �√3 × √3�R30° super-structure peak (red circles) while NbS2-type one reveals more obvious (1 × 1) peak (white circles) of the host unit cell. Although both types show mixed traces of two periodicities, the type of surface can be identified from the stronger peaks in FFT (especially at low biases). The coexistence of the two types becomes more obvious when it comes to a boundary between the two types as shown in Fig. 2b.
In order to compare STM and SP-STM on chiral domains, we have used two Pt/Ir-alloy tips that are treated on non-magnetic Cu(111) and magnetic Cr(001) surface, respectively. The SP-STM tip is coated with Cr from the Cr(001) surface, and its magnetic sensitivity is verified by the layer-bylayer contrast of Cr(001) antiferromagnetic order 33 . (SI note 2) It is known that the magnetization direction of antiferromagnetic Cr tips tends to be randomly oriented, so it shows both in-plane and out-of-plane sensitivity 34 . Throughout this paper, we assume the dominant out-of-plane sensitivity along the chiral axis, which is the expected induced magnetization direction by symmetry 17 . (It is also consistent with the observed two levels of contrast, which will be discussed later. from magnetic order at this paramagnetic temperature, and we attribute it to magnetochiral interaction as will be discussed later. Note that the domain contrast becomes clearer in a larger size scan as it averages out the Co deficiencies. (Fig. 2d inset) To examine the Co lattice shift, we obtained atomic images near boundaries. The STM image in

Inverse Magnetochiral Effect
The spectroscopic signature of the magnetochiral interaction has been checked by obtaining tunneling spectra across a domain boundary in SP-STM. Fig. 4a shows a series of spin-polarized tunneling spectra across a Lower domain boundary. While the modulation by atomic corrugation is visible throughout the spectra, there is a definitive change of spectral intensity across the boundary. To examine the change in more detail, the spectral intensities from two representative biases are depicted in Fig. 4b. The changes of spectral intensity that follow step-like guides are obvious, and the step-like feature is a result of adding the opposite magnetochiral interaction to the base tunneling spectra (gray lines). Therefore, the domain contrast in topography is due to the enhanced/suppressed tunneling probability upon chirality change. Next, we check the dependence on the electric field direction. Although it can be seen briefly in the inversion of the step-like guides in Fig. 4b, it is depicted fully by comparing the spectra obtained from the two points (red and blue stars in Fig. 4a). The two spectra in Fig. 4c indicate zero bias as the crossing point where the magnetochiral interaction vanishes. We can further eliminate the "non-magnetochiral portion" by where ↑ ( ↓ ) depicts the tunneling conductance of parallel(antiparallel) spin-polarized tunneling (Fig. 4c inset), which denotes zero bias as its crossing point. In other words, the effect is exactly antisymmetric about the zero bias. Generally, in any magnetic ordered states, spin polarization of tunneling current reflects the difference between opposite spin states, and the zero-crossing point in polarization appears at a somewhat arbitrary energy level, as can be found in many examples in the comprehensive review by Wiesendanger 37 . The coincidence of the crossing point of spin polarization with zero bias is a crucial characteristic, which is remarkably different from the conventional mechanism for SP-STM contrasts.
We try to understand our observation based on the comparison with the previously known two vanishes and reverses when current flows backwards 5,14,15,38 . We emphasize that both of these are already shown in the tunneling spectra analysis by the reversal upon chirality change and the zerobias crossing point. Thus, we argue that all these magnetochiral phenomena have something in common at the fundamental level.

Magnetochiral Nonreciprocal Tunneling
Lastly, we test the cross-coupling between tunneling electrons and magnetism. In, for example, linear magnetoelectric systems, the magnetoelectric coupling depends on the strength of linearly cross-coupled term (i.e. • ) in free energy 41 , and gives a mutual relationship that induces one by another. A similar relationship can be assumed for the inverse magnetochiral effect. In other words, "electric tunneling induced by magnetic field" and "magnetization induced by tunneling electron" are all possible in chiral systems. However, any permanent electric current induced by magnetic field in an isolated system is thermodynamically prohibited, unless it is, for example superconducting. Instead, when there is a current flow by an external electromotive force, the magnetochiral interaction is in the form of enhancement/suppression of electric current. This phenomenon has already been reported in single wall carbon nanotube 42 , chiral MnSi 3 , or helically deformed Bi crystal 43 as known as electrical magnetochiral anisotropy. The chirality-dependent resistance-change under magnetic field has an analogue with tunneling probability change in the presence of external magnetic field in a STM setup although it is not straightforward to match tunneling probability and resistance. We will refer this effect as magnetochiral nonreciprocal tunneling. Note that it does not require any magnetic probe to see this effect. The STM image in Fig. 4d demonstrates the emergence of domain contrast upon application of a magnetic field, which confirms the existence of the magnetochiral nonreciprocal tunneling effect. Although, from the viewpoint of SOS, it was predicted that the magnetic field applied to a chiral structure could exhibit nonreciprocity 17 , it has never been associated with electric tunneling or scanning probe microscopy.
In conclusion, we have demonstrated that the inverse magnetochiral effect can be exploited to observe chiral domains and domain boundaries using SP-STM. The inverse magnetochiral effect is reversible readily by changing chirality or the electron tunneling direction, which is in a clear distinction with the conventional SP-STM contrasts from different orientations of ordered local spins. Thus, SP-STM can, now, be utilized to reveal atomic-scale chiral structures, in addition to atomic spin structures. Moreover, the magnetochiral nonreciprocal tunneling, an inverse phenomenon, gives rise to domain contrast upon application of external magnetic field without using a magnetized probe. Our observations provide a new paradigm to unveil quantum-level interrelationship between electric tunneling and crystallographic chirality. We also would like to remark that the novel spin-polarization mechanism using a chiral van der Waals material can be an innovative spin-polarizing technique in two-dimensional spintronics, where spin-injection has been relied on integration of ferromagnetic materials or circularly polarized light so far 30 .

Crystal growth
Single crystals of Co1/3NbS2 were grown by a chemical vapor transport reaction method in the presence of iodine as a transport agent. About 0.2 g mixture of cobalt powder (Alfa Aesar, 99.99%), niobium powder (Alfa Aesar, 99.9%), and sulfur piece (Alfa Aesar, 99.999%) with a molar ratio Co:Nb:S=0.7:1:2 were sealed in a quartz tube with 100 mg of iodine (Alfa Aesar, 99.99%) under vacuum. Then, the quartz tube was placed under an optimum temperature gradient in a two-zone tube furnace. The hot and cold zones were kept at 950 ⁰ C and 800 ⁰ C, respectively. After about 4 weeks, the tube furnace was turned off and the quartz tube was cooled to room temperature naturally. Crystals in hexagonal shape were collected both at the hot and cold end of the quartz tube. X-ray diffraction data on ground powder specimens were taken at room temperature with Cu K α (λ = 0.15418 nm) radiation in a Malvern Panalytical X'Pert 3 powder diffractometer.

Transmission Electron Microscopy
Specimens for TEM experiments were prepared by Ar-ion milling and studied using a JEOL-2010F TEM. To unveil the chiral domains, DF-TEM images were taken by selecting g±= ±222 along the [101] zone axis. All images are raw data.

Scanning Tunneling Microscopy
STM measurements were performed using a Unisoku ultra-high

STM Simulation
Density functional theory Calculations were performed using GPAW package 44