## Abstract

A kagome lattice (KL) is a two-dimensional atomic network comprising hexagons interspersed with triangles, which provides a fascinating platform for studying competing quantum ground states. The KL contains three atoms in a unit cell, and their degrees of freedom combine to yield Dirac bands and a flat band. Despite many studies to understand the flat band in KL, exploring the pseudospin of Dirac bands in KL has been scarce. In this paper, we suggest pseudospin-polarized scanning tunneling microscopy that is analogous to spin-polarized scanning tunneling microscopy. Using a pseudospin-polarized tip, we possibly observed the pseudospin texture of kagome metal FeSn in real space. Based on a simple tight-binding calculation, we further simulated the pseudospin texture of KL, confirming the geometric origin of pseudospin. This work potentially deepens our understanding of the lattice symmetry-preserving tunneling process in Dirac materials.

## Introduction

Over the last several years, kagome lattices (KLs) of 3d-transition metals have emerged as a new class of materials for studying topological, frustrated, and correlated electronic ground states^{1,2,3,4,5,6,7,8}. The tight-binding model reveals the presence of dispersionless flat bands and linear Dirac bands in KLs, which suggests that the band topology and many-body interactions are intertwined in this class of materials^{7,8,9,10,11,12,13,14}. Recently, the angle-resolved photoemission spectroscopy (ARPES) study measured such flat bands in two-dimensional (2D) kagome metal FeSn and CoSn^{7,8,9}. However, an in-depth study of pseudospin that can impose a non-trivial Berry’s phase to Dirac bands has been lacking in KLs.

Pseudospin is a sublattice degree of freedom that has a mathematical analogy to real spin in describing Dirac bands^{15,16,17,18}. The pseudospin operator \({\hat{S}}_{z}\) is diagonalized in natural sublattice basis of a honeycomb lattice, and its eigenstates *m*_{s} = ±1/2 denote the pseudospin states^{19}. In contrast, \({\hat{S}}_{z}\) is not diagonalized in sublattice basis of KL, and the pseudospin states can be revealed by superpositions between the sublattices. Although it is demonstrated that the electron back scattering is suppressed in graphene due to the pseudospin conservation^{17,20}, it has been challenging to visualize the pseudospin as a measurable quantity in real space. Here, we have employed scanning tunneling microscopy (STM) to reveal spin and pseudospin textures in kagome metal FeSn. We suggest that the STM tip can be both spin- and pseudospin-polarized when it is terminated with FeSn, thus realizing spin-polarized scanning tunneling microscopy (SPSTM) and pseudospin-polarized scanning tunneling microscopy (pSPSTM). Combined with the density functional theory (DFT) and tight-binding calculations, our STM work reveals the potential mechanism of pseudospin-dependent tunneling in KLs.

## Results and discussion

The FeSn crystal used in the experiment is shown in Fig. 1a. It is made of repeatedly stacked Fe_{3}Sn and Sn_{2} layers along the c-axis (Fig. 1b)^{7,21}. Scanning transmission electron microscopy (STEM) images resolve the atomic configuration of the FeSn (Fig. 1c, d). The Fe_{3}Sn layers contain KLs formed by the Fe atoms, and the Sn_{2} layers serve as buffer layers to separate the Fe_{3}Sn layers. Figure 1e depicts the KL in a Fe_{3}Sn layer. A Star-of-David pattern is featured by a red star as a kagome motif. In KLs, the electrons are self-localized owing to the destructive interference around the hexagonal structures, which leads to the formation of a flat band^{7,22}. When the flat band is located near the Fermi energy in 3d-transition kagome metals, the spin-degeneracy is lifted through the Stoner transition^{23}. Consistently, the Fe_{3}Sn layers are ferromagnetic in FeSn, whereas they are coupled antiferromagnetically due to the indirect exchange interaction via the Sn_{2} layers^{7,21,24}.

Figure 1f shows an STM image of the FeSn measured with the bias voltage (V_{bias}) of −50 mV and the tunneling current (*I*_{t}) of 100 pA. Two different surfaces are identified, denoted as Fe_{3}Sn and Sn_{2}. While the Sn_{2} surface is free of defects, many vacant islands are observed on the Fe_{3}Sn surfaces. This is because the Fe_{3}Sn surface is thermodynamically unstable compared to the Sn_{2} surface, as confirmed by earlier DFT calculations^{21}. A zoomed-in image of the Fe_{3}Sn surface is presented in Fig. 1g. When the STM image is compared with the atomic model in Fig. 1e, it turns out that the individual Fe atoms are not resolved. Instead, six equally bright spots appear on the branches of the David star, which agrees with the STM image simulated using the DFT calculations (Fig. 1k).

Figure 1h shows the height profile taken along the yellow line in Fig. 1f. The step height between the Fe_{3}Sn layers is identified as ~4.4 Å, which is consistent with the literature^{21}. The Sn_{2} surface is not positioned right in the middle of the two Fe_{3}Sn layers because the inversion symmetry is broken at the Sn_{2} surface. We measured the differential conductance (*dI/dV*) spectra on the Fe_{3}Sn and Sn_{2} surfaces, respectively (Fig. 1i). In the *dI/dV* spectrum of Fe_{3}Sn surface, spectral peaks are found at the bias voltages of −50 mV and −260 mV. No notable features are observed in the *dI/dV* spectrum of Sn_{2} surface.

To understand the origin of peaks in the *dI/dV* spectrum of Fe_{3}Sn surface, we performed DFT calculations on FeSn slabs terminated with a Fe_{3}Sn surface (see Methods). Figure 1j shows the calculated band structure. In the band structure, the Dirac bands of the *d*_{xy} and *d*_{x2 − y2} orbitals cross the Fermi energy and continue to *E*_{F} − 0.6 eV. Interestingly, the Dirac bands are gapped near *E*_{F} − 0.26 eV because the *d*_{xy} and *d*_{x2 − y2} orbitals hybridize with the *d*_{xz} and *d*_{yz} orbitals due to the broken inversion symmetry at the surface^{21}. Therefore, we attribute the peak at *V*_{bias} = −260 mV to the less-dispersive surface bands consisting of the hybridized orbitals. The peak at *V*_{bias} = −50 mV arises from the van Hove singularity (VHS) of a dispersive band of FeSn at M-point, as indicated by the yellow arrow in Fig. 1j.

In SPSTM, the tunneling current is sensitive to the spin states of the sample with a spin-polarized STM tip^{25}. For this SPSTM operation, we intentionally indented the STM tip into the Fe_{3}Sn surface (see Methods). Since FeSn is a fragile material consisting of layers, it is possible that the tip picks up small crystalline FeSn fragments by indentation. Once the tip is terminated with Fe_{3}Sn layer, the tip is inherently spin-polarized because the Fe_{3}Sn layer is ferromagnetic (Fig. 2a). By this method, we could easily obtain a spin-polarized STM tip.

To confirm the spin-polarization of the STM tip, we measured the *dI/dV* spectra of two neighboring Fe_{3}Sn terraces. As the Fe_{3}Sn layers are antiferromagnetically coupled, the spin orientations of the two terraces are opposite. In Fig. 2b, the red curve represents the case when the tip spin is parallel to the sample spin. The blue curve is for the case when the spins are antiparallel. Note that we do not determine the absolute spin direction of tip or sample, but we measure the spin-polarized signals depending on the relative spin alignment between tip and sample. The maximum spin-polarization is found at *V*_{bias} = −50 mV, most likely due to the spin-polarization of the VHS with high density of states. We then measure a topographic image across several Fe_{3}Sn terraces at this bias voltage (Fig. 2c). Figure 2d presents a *dI/dV* map which is simultaneously obtained with Fig. 2c. Interestingly, there exists a contrast in the *dI/dV* map, implying that the spins of the terraces are oriented differently.

To understand the spin contrast in the *dI/dV* map, we have measured the height profile across the terraces along the dashed line in Fig. 2c, as shown in Fig. 2e. The vertical bars represent the single-step heights of Fe_{3}Sn, which is identified as ~4.4 Å in Fig. 1h. The height of the terraces is determined from the bottom terrace in units of single-step height. It is numbered on the terraces in Fig. 2e. We confirm that the spin-polarization of the terrace does not affect the count of the terrace height (see Supplementary Figs. 1–4). Because the Fe_{3}Sn layers are antiferromagnetically coupled, the even-numbered terraces hold the same spin as the bottom terrace. In contrast, the odd-numbered terraces have the opposite spins. According to this interpretation, the relative spin orientations of the terraces are assigned by the blue and red arrows in Fig. 2e. Remarkably, the spin configuration perfectly agrees with the spin contrast measured in Fig. 2d. The spin contrast in the *dI/dV* map was not observed when a clean tip was used, confirming that the contrast is due to the spin-polarization of the tip (see Supplementary Fig. 5).

Now, we take a closer look into the Fe_{3}Sn surfaces using the Fe_{3}Sn-terminated STM tip. Surprisingly, the STM image obtained with this functionalized tip is very different from that measured using a clean tip. Figure 3a shows a typical Fe_{3}Sn surface imaged by the Fe_{3}Sn-terminated tip. The tunneling conditions are same as those for Fig. 1b (*V*_{bias} = −50 mV and *I*_{t} = 100 pA). Strikingly, three spots are brighter than the other three spots on the David star, which breaks the apparent C_{6} rotational symmetry around the David star in Fig. 1g. The bright spots constitute a triangle, and the triangular orientation represents the flavor of the broken symmetry. Figure 3b displays the height profile recorded along the blue line in Fig. 3a. It clearly shows that the brighter spots are 1.7 pm taller than the darker spots.

One can attribute the broken symmetry to the possibility that the Sn_{2} layer affects the symmetry of the Fe_{3}Sn layer. In fact, the underlying layers break the symmetry of KLs for several 3d-transition kagome metals, such as Fe_{3}Sn_{2} and Co_{3}Sn_{2}S_{2}^{1,23,26}. In these materials, three spots of the David star are brighter than the other three spots, similar to our observation. However, in FeSn, the Sn atoms in Sn_{2} layer are equally positioned under the branches of the David star (Fig. 1c), preserving the symmetry of the KL. The *dI/dV* spectra measured using a clean tip show no spectral difference between those spots (see Supplementary Fig. 6). Therefore, this possibility can be safely discarded. Another possibility is that the magnetization of the Fe_{3}Sn layer breaks the symmetry of the KL. This scenario is plausible because the Fe_{3}Sn layer is in-plane magnetized, and so the spin texture can break the in-plane rotational symmetry. To verify this possibility, we examined the Fe_{3}Sn terraces of different spins (Fig. 3c). If the symmetry breaking is due to the magnetization, the triangular orientation will be reversed depending on the spin of the Fe_{3}Sn terraces. However, Fig. 3c reveals that the triangular orientation remains the same regardless of the spin direction, which concludes that the symmetry breaking is not caused by the magnetization. Thus, this scenario should also be excluded. Since the symmetry breaking is only observed by the functionalized STM tip (see Supplementary Fig. 7), we suggest that this phenomenon is associated with the pseudospin physics of KLs.

In graphene, the pseudospin denotes a sublattice degree-of-freedom. Here, up- and down-pseudospins are imposed on the sublattices A and B, respectively, as eigenstates of \({\hat{S}}_{z}\)^{27,28}. In momentum space, a linear combination of the up- and down-pseudospins leads to a chiral pseudospin texture around a Dirac cone. Compared to graphene, it is not straightforward to construct the sublattice bases corresponding to up- and down-pseudospins in a KL. Since there are three atoms in the unit cell of KL, the pseudospin structure can be only revealed by the superposition of the three atomic bases. Equation 1 shows the tight-binding Hamiltonian of a KL in the basis space of \(\bar{\varphi }={[{\varphi }_{A}{\varphi }_{B}{\varphi }_{C}]}^{T}\), where *φ*_{A}, *φ*_{B}, and *φ*_{C} are three atomic bases of the KL. **q** denotes the momentum state measured at the K-point of the Brillouin-zone. We perform a basis transformation using a unitary matrix (*U*) to obtain a Hamiltonian representing Dirac bands and a flat band (see Methods). We find that the basis space \(\bar{\psi }={U}^{{\dagger} }\bar{\varphi }={[{\psi }_{+}{\psi }_{-}{\psi }_{F}]}^{T}\) transforms the Hamiltonian in Eq. 1 into a Hamiltonian that contains a 2 × 2 Dirac Hamiltonian (left-upper block) and a 1 × 1 flat band Hamiltonian (right-lower block) (Eq. 2). In the Dirac Hamiltonian, *ψ*_{+} and *ψ*_{−} are analogous to the sublattices A and B in graphene, corresponding to the up- and down-pseudospins, respectively. *ψ*_{F} represents the flat band of the KL.

To plot |*ψ*_{+}|^{2} and |*ψ*_{−}|^{2} in real space, we assume that the wavefunction localized on the kagome atoms is isotropic, referring to s-orbitals (see Methods). The essence of the pseudospin is captured within this assumption, revealing the pseudospin’s geometric origin. Figure 3d, e present the distribution of |*ψ*_{+} (x, y)|^{2} and |*ψ*_{−} (x, y)|^{2} in real space, respectively. Obviously, the lattice sites for the up- and down-pseudospins are spatially separated in the KL; the up-pseudospin is localized within the upward-triangles of the KL, and the down-pseudospin is in downward-triangles.

Since the pseudospins are spatially polarized on the Fe_{3}Sn layer, the STM tip can be also pseudospin-polarized if it is terminated with the Fe_{3}Sn layer, realizing pSPSTM (Fig. 3f). The pseudospins are real angular momenta^{27}, which are represented by the eigenstates of \({\hat{S}}_{z}\) in |*ψ*_{+} (x, y)|^{2} and |*ψ*_{−} (x, y)|^{2} ^{19,28,29,30}. Pseudospin-excitations of such sublattices are recently demonstrated in artificial photonic graphene and Lieb lattices^{31,32}. Further, it has been shown that the pseudospins of the sublattices are aligned along the pseudo-magnetic field in straining graphene^{29,33,34,35,36}. For pSPSTM, when the pseudospins are parallel between the tip and the sample, the wavefunction overlap is maximized, with large tunneling current flows. When the pseudospins are antiparallel, the wavefunction overlap should be zero, suppressing the electron tunneling. However, the pseudospin-polarization does not reach 100 % in FeSn due to the bands irrelevant to pseudospin (Fig. 1j). The maximum pseudospin-polarization is obtained about 38 % in the experiment. The pseudospin-polarization depending on the bias voltage is provided in Supplementary Information (Supplementary Fig. 8).

It is also worth considering how the pseudospin-dependent tunneling takes place in momentum space. Similar to graphene, the chiral pseudospin textures are imposed on the Dirac cones of the KL at K- and K’-points. The K- and K’-points are distinguished by the valley index characterized by pseudospin chirality. Figure 3g depicts the electron tunneling depending on the pseudospin configuration in momentum space. In the parallel pseudospin configuration (left panels in Fig. 3g), the tip KL coincides with the sample KL when viewed from the top. Thus, the Dirac cones of the tip and sample KLs have the same valley indices at the Brillouin-zone-corners, enabling tunneling between the tip and the sample. By contrast, in the antiparallel configuration (right panels in Fig. 3g), the up-pseudospin site of the tip KL is placed on top of the down-pseudospin site of the sample KL. Two KLs are then inversion symmetric. Therefore, the Dirac cones of the tip and sample KLs have opposite valley indices at the Brillouin-zone-corners, suppressing the electron tunneling. This valley-selective tunneling is responsible for the broken symmetry in Fig. 3a, and its origin is rooted in the pseudospin-dependent tunneling.

Equation (2) shows that the Hamiltonian for Dirac bands is block-diagonalized, and the pseudospin is only associated with the Dirac bands. Therefore, tunneling into parabolic bands will be insensitive to the pseudospin-polarization of the tip. To confirm this, we measured the *dI/dV* maps at the bias voltages of −50 mV and −260 mV. Figure 4a shows the simultaneously measured topography and *dI/dV* map at *V*_{bias} = −50 mV, where the Dirac band is located. The symmetry of the David star is broken due to the pseudospin-dependent tunneling. Figure 4b shows the topography and *dI/dV* map measured at *V*_{bias} = −260 mV, where the Dirac band is gapped. Remarkably, the *dI/dV* map in the flat band exhibits the full KL without symmetry breaking. To understand our observation, we calculated STM images of KL based on the tight-binding model (see Methods). When a pseudospin-polarized tip is used in the calculation, strong pseudospin-polarization is observed in the simulated image, agreeing very well with the experiment. This further advocates our claim that pseudospin is involved in the electron tunneling between KLs.

In summary, we have used STM to investigate spin and pseudospin textures of kagome metal FeSn. We have found that the STM tip becomes spin-polarized when it is terminated with Fe_{3}Sn layers, realizing SPSTM. We have confirmed that the Fe_{3}Sn layers are ferromagnetic while the interlayer coupling is antiferromagnetic. Furthermore, we have shown that the Fe_{3}Sn-terminated tip could be also pseudospin-polarized, which is supported by the DFT and tight-binding calculations. Our experiments suggest possible lattice symmetry-preserving tunneling in Dirac materials. If the angle between the tip and sample KLs can be controlled, one could study the pseudospin-dependent tunneling in a more systematic way. The in-situ rotation of the sample would be one of such ways, but requires further instrumental development in STM.

## Methods

### Sample growth

Single crystals of FeSn were synthesized by the Sn-flux method with a molar ratio of Fe:Sn = 1:49. Fe granule (Alfa Aesar 99.95%) and Sn shot (Alfa Aesar 99.99+ %) were placed in an alumina crucible. The crucible was then sealed in a quartz tube under a partial Argon atmosphere. The quartz tube was placed in a furnace and kept at 900 °C for 2 days to obtain a homogeneous metallic solution, and then was cooled from 800 °C to 650 °C at a rate of 2.5 °C per hour. Single crystals of millimeter-size were obtained by removing Sn flux using a centrifuge at 650 °C.

### STM measurements

STM experiments have been performed using a cryogen-free low temperature STM (Panscan Freedom, RHK) working at the temperature of 15 K. The FeSn crystal which had been pre-cooled at 80 K was cleaved in the ultra-high vacuum (UHV) chamber and immediately plugged into the STM head for the measurement. To acquire *dI/dV* spectra and *dI/dV* maps, we used a standard lock-in technique with a modulation frequency of *f* = 718 Hz. In the tip indentation, we indented the tip 5 nm deep into the FeSn with a speed of 1 Å per second, stayed for 10 s, and then retraced the tip back with the same speed. The probability that the indented tip was spin-polarized was ~60 % and the probability that the spin-polarized tip was pseudospin-polarized was ~50 %. The quality of the Fe_{3}Sn-terminated tip was checked on Cu(111) after the measurements (see Supplementary Fig. 9).

### DFT calculations

Calculations for ground states of the layered FeSn were computed by using the all-electron linearized augmented planewave (APW) method implemented in ELK code (http://elk.sourceforge.net). The maximum length for the momentums multiplying averaged muffin-tin radius is fixed to be 6.8. To stabilize variational self-consistent calculations for the slab structure, we adopt large cutoff for reciprocal lattice to be 20 a.u. (atomic unit) with the Broyden mixing^{37}. To simulate kagome (Fe_{3}Sn)-terminated surface states, we considered the four slabs FeSn as a model system where a single slab of FeSn is composed of Fe_{3}Sn layer and Sn_{2} layer as shown in Fig. 5. The two kinds of surfaces can be found to have Fe_{3}Sn-terminated surface and Sn-terminated surface. By considering the experimental results, we mainly focus on the Fe_{3}Sn-terminated surface which is denoted by the top surface Fe_{3}Sn of the 1st slab as given in Fig. 5. The lattice parameters of in-plane the bulk hexagonal P6/mmm (191) is 5.3 Å and that of out-of-plane is 4.48 Å^{38}. We performed the spin-polarized PW-CA local-density approximation with non-collinear spin basis aligned to the in-planar direction^{39}. We confirmed the convergence of the electronic structures with a sampling of the first Brillouin zone of 15 × 15 × 1.

### STM image simulation

STM image simulation was performed based on the Tersoff-Hamann model. Since the tip wavefunction is ignored here, we assume the spectral density of the Fe_{3}Sn is directly proportional to the measured topography intensity, i.e., \(I\left(x,y,{h;V}=-50{{{\rm{meV}}}}\right)\propto {\sum }_{n}{\int }_{{FBZ}}d{{{\boldsymbol{q}}}}{\int }_{{E}_{F}-V}^{{E}_{F}}{dE}{\left|{\psi }_{n{{{\boldsymbol{q}}}}}\left(x,y,h\right)\right|}^{2}\delta ({E}_{n{{{\boldsymbol{q}}}}}-E)\) with smeared delta-function with standard deviation of 20 meV. The *ψ*_{nq} (** r**) is the real-space Kohn-Sham wave function of FeSn found in DFT calculation.

*h*is the height from the topmost surface of FeSn to the measured point of spectral density, i.e.,

*h*= 2 Å.

### Tight-binding models

We have generated 3 × 3 tight-binding Hamiltonian with three atomic bases belonging to the three Fe atoms in the kagome lattice (Fig. 6),

From the tight-binding geometry shown, each hopping element can be defined to \({H}_{{AC}}\left({{{\boldsymbol{k}}}}\right)={t}_{0}\left({e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{1}}}}}}+{e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{4}}}}}}\right)\), \({H}_{{AB}}\left({{{\boldsymbol{k}}}}\right)={t}_{0}\left({e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{2}}}}}}+{e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{3}}}}}}\right)\), and \({H}_{{BC}}\left({{{\boldsymbol{k}}}}\right)={t}_{0}\left({e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{5}}}}}}+{e}^{-i{{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{\delta }}}}}_{{{{\boldsymbol{6}}}}}}\right)\). The *t*_{0} indicates the nearest-neighbor hopping between atomic bases. The Hamiltonian (3) can be written in the matrix representation as,

To understand electronic structures around the Dirac cone vertex, we defined momentum ** q** to be expanded from the Dirac cone vertex, i.e.,

**=**

*q***+**

*k***or \({{{\boldsymbol{q}}}}={{{\boldsymbol{k}}}}+{{{\boldsymbol{K}}}}^{{\prime}}\) and |**

*K***| ≪ 1. For the brevity, we set**

*q**t*

_{0}= 1/2 and

*L*

_{0}= 1 in the calculation. The expanded Hamiltonian is given,

From the Hamiltonian (5) in the atomic basis space, one can directly obtain the diagonalized Hamiltonian in the eigenstate basis space by rotating the *H*_{TB} (** k**), that is, \({U}_{E}^{+}\left({{{\boldsymbol{q}}}}\right){H}_{{TB}}\left({{{\boldsymbol{q}}}}\right){U}_{E}\left({{{\boldsymbol{q}}}}\right)={E}_{{{{\boldsymbol{q}}}}}\). To determine the pseudospin components from the atomic basis space, we additionally defined modified-O(3) rotation matrix to rotate the

*H*

_{TB}(

**) to the pseudospin basis space through the**

*q**U*

_{E}(

**). Here we assume the rotation matrix as,**

*q*where the *θ*_{q} denotes the polar angle between the momentum ** q** and the

*k*_{x}-direction and thus the

*q*

_{x}and

*q*

_{y}become

*qcosθ*

_{q}and

*qsinθ*

_{q}. Finally, from the successive rotation by using the

*U*

_{E}and

*U*

_{U}, one can reach the Hamiltonian in the pseudospin basis space,

Here the *E*_{0} and *E*_{flat} the energy level of the up- and down-pseudospin states and flat band of the kagome lattice, respectively. It is notable that the upper left 2 × 2 block of the rotated matrix (7) clearly corresponds to the general formulation of the Dirac Hamiltonian and, therefore, the first and second column vectors of the *U*_{E} (** q**)

*U*

_{U}(

**)( =**

*q**U*(

**)) directly delivers coefficients to connect between the atomic basis space and the pseudospin basis space. Eventually, each pseudospin basis |**

*q**ψ*

^{+}〉 and |

*ψ*

^{−}〉 can be obtained as linear combinations of the atomic basis |

*ϕ*

_{τ}〉, that is, \(|{\psi }^{+}\rangle =1/\sqrt{2}({\sum }_{\tau }{u}_{\tau ,1}({{{\boldsymbol{q}}}})|{\phi }_{\tau }\rangle -{e}^{-i{\theta }_{{{{\boldsymbol{q}}}}}}{\sum }_{\tau }{u}_{\tau ,2}({{{\boldsymbol{q}}}})|{\phi }_{\tau }\rangle )\) and \(|{\psi }^{-}\rangle =1/\sqrt{2}({e}^{i{\theta }_{{{{\boldsymbol{q}}}}}}{\sum }_{\tau }{u}_{\tau ,1}({{{\boldsymbol{q}}}})|{\phi }_{\tau }\rangle +{\sum }_{\tau }{u}_{\tau ,2}({{{\boldsymbol{q}}}})|{\phi }_{\tau }\rangle)\), which corresponds to the up- and down-pseudospin basis states, respectively. In the above relation,

*u*

_{n,m}(

**) is the matrix element of (**

*q**n*,

*m*) of

*U*

_{E}(

**).**

*q*### Real space representation

The real space representation of the pseudospin basis |*ψ*^{+}〉 and |*ψ*^{−}〉 can be directly computed by the wavefunction representations of the pseudospin eigenvectors, 〈** r**|

*ψ*

^{+}〉 and 〈

**|**

*r**ψ*

^{−}〉, respectively. In case of up-pseudospin basis, one can write the wavefunction as,

where the wavefunction of the atomic basis |*ϕ*_{τ}〉 becomes infinite combination of identical atoms of *ϕ*(** r**), so that \(\left\langle {{{\boldsymbol{r}}}}|{\phi }_{\tau }\right\rangle ={\sum }_{l}{\sum }_{\tau }e\left(i{{{\boldsymbol{q}}}}\cdot\left({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{l}}}}}+{{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{\tau }}}}}\right)\right)\phi \left({{{\boldsymbol{r}}}}{{{\boldsymbol{-}}}}{{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{l}}}}}{{{\boldsymbol{-}}}}{{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{\tau }}}}}\right)\). The

*R*_{l}and

*R*_{τ}are the position of

*l*-th unit cell and

*τ*-th atomic position inside a single unit cell, respectively. Down-pseudospin basis could be obtained in a similar manner. The real-space atomic orbital

*ϕ*(

**) is assumed to be 1s-like state. Finally, |\(\langle {{{\boldsymbol{r}}}}{\left|{\psi }^{+}\right\rangle |}^{{{{\boldsymbol{2}}}}}\) and \({{{{\rm{|}}}}\langle {{{\boldsymbol{r}}}}\left|{\psi }^{-}\right\rangle |}^{2}\) represent the real-space electronic density of pseudospin up and down, respectively.**

*r*### Pseudospin-polarization simulation

To simulate the pseudospin-polarization in the *dI/dV* maps, we have employed 6 × 6 tight-binding Hamiltonian that describes bilayer kagome lattice, which reads,

where the index *i* represents the top layer (*i* = 1) and the bottom layer (*i* = −1) of the bilayer kagome lattice. The top and bottom layers correspond to the tip and the sample, respectively, and they do not interact. *V* is a small bias voltage introduced for the computational purpose of lifting the degeneracy of the Dirac cones.

By diagonalizing the Hamiltonian (9), we obtain the eigenstates of the Dirac bands; \({\psi }_{n{{{\boldsymbol{k}}}}}^{\{i\}}({{{\boldsymbol{r}}}})={\sum }_{l,{\tau }_{j}}{u}_{n{{{\boldsymbol{,}}}}{{{\boldsymbol{k}}}}}({\tau }_{j}){e}^{i{{{\boldsymbol{k}}}}\cdot ({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{l}}}}}{{{\boldsymbol{+}}}}{{{{\boldsymbol{R}}}}}_{{\tau }_{j}})}\phi ({{{\boldsymbol{r}}}}{{{\boldsymbol{-}}}}{{{{\boldsymbol{R}}}}}_{l}{{{\boldsymbol{-}}}}{{{{\boldsymbol{R}}}}}_{{\tau }_{j}})\), where the superscript {*i*} denotes the tip layer (*i* = 1) and the sample layer (*i* = 2). *R*_{l} and \({{{{\boldsymbol{R}}}}}_{{\tau }_{j}}\) represent the *l*-th unit cell and *τ*_{j}-th atomic position in the unit cell, respectively. \({\tau }_{1}\in \) {sample; *A*_{1}, *B*_{1}, and *C*_{1}} and \({\tau }_{-1}\in \) {tip; *A*_{−1}, *B*_{−1}, and *C*_{−1}}. *u*_{n,k} (*τ*_{j}) is the diagonalization coefficient. The subscript *n* is the band index. Therefore, \({\psi }_{n=1,{{{\boldsymbol{k}}}}}^{\left\{1\right\}}\) represents the bottom Dirac band (*n* = 1) of the tip kagome lattice composed of three tight-binding basis orbitals at \({{{{\boldsymbol{R}}}}}_{{\tau }_{1}}{\psi }_{n=2,{{{\boldsymbol{k}}}}}^{\left\{-1\right\}}\) indicates the bottom Dirac band (*n* = 2) of the sample kagome lattice made of the three orbitals at \({{{{\boldsymbol{R}}}}}_{{\tau }_{-1}}\). The band indices *n* = 3 and *n* = 4 represent the top Dirac bands of the tip and sample, respectively. Here, we only consider tunneling between the bottom Dirac bands (*n* = 1 and *n* = 2).

The wavefunction of the bottom Dirac band of the top layer is,

Since STM operation demands local tunneling, we insert a local tunneling parameter \({e}^{-\left({{{{\boldsymbol{R}}}}}_{l}-{{{{\boldsymbol{R}}}}}_{{l}^{{\prime} }}\right)/{L}_{0}}\) into Eq. 10, defining the tip wavefunction \(({\psi }_{{Dirac}}^{{tip}}\left({{{\boldsymbol{r}}}}\right))\). This term says that only three Fe atoms at a given unit cell \({{{{\boldsymbol{R}}}}}_{{l}^{{\prime} }}\) mainly contribute to constructing the STM image.

We also define the non-Dirac tip wavefunction (\({\psi }_{{non}-{Dirac}}^{{tip}}\left({{{\boldsymbol{r}}}}\right)\)) by dropping the coefficient \({u}_{n{{{\boldsymbol{,}}}}{{{\boldsymbol{k}}}}}({\tau }_{j})\) in Eq. 11. It describes three Fe atoms at the tip apex without a Dirac band.

As we have already found, the Dirac sample wavefunction (\({\psi }_{{Dirac}}^{{sample}}\left({{{\boldsymbol{r}}}}{{{\boldsymbol{,}}}}\Delta {{{\boldsymbol{r}}}}\right)\)) is,

where Δ** r** denotes the in-plane displacement between the \({{{{\boldsymbol{R}}}}}_{{\tau }_{1}}\)and \({{{{\boldsymbol{R}}}}}_{{\tau }_{-1}}\), which corresponds to the lateral displacement of the tip in the experiment. We then construct the STM tunneling matrix by plane integration at the middle of the tip-sample gap;

^{40}

We finally obtain the calculated *dI/dV* map by integrating Eq. 14 over the momentum space,

In the calculation, we set the tip-to-sample distance of *z*_{h} = 2*L*_{0}, and assume the s-wave orbital as a tight-binding basis, \(\phi ({{{\boldsymbol{r}}}}{{{\boldsymbol{-}}}}{{{{\boldsymbol{r}}}}}_{{{{\boldsymbol{0}}}}}){{{\boldsymbol{=}}}}{(\pi {{a}_{0}}^{3})}^{{{{\boldsymbol{-}}}}0.5}{e}^{-\frac{\left|{{{\boldsymbol{r}}}}{{{\boldsymbol{-}}}}{{{{\boldsymbol{r}}}}}_{{{{\boldsymbol{0}}}}}\right|}{{a}_{0}}}\), with a radius parameter of *a*_{0} = 0.3*L*_{0}.

The calculated *dI/dV* maps are shown in Fig. 7. If the coefficient of eigenvectors \({u}_{n{{{\boldsymbol{,}}}}{{{\boldsymbol{k}}}}}({\tau }_{j})\) is not taken into account for the tip, the tunneling current results from a trivial overlap between the tip wavefunction and the sample wavefunction. In this case, the pseudospin-polarization is not observed (Fig. 7a). However, considering the eigenvector coefficients to compute the tunneling matrix elements, the pseudospin alignment between the tip and the sample is naturally defined and directly affects the tunneling matrix. As a result, the *dI/dV* map shows a strong pseudospin-polarization (Fig. 7b).

## Data availability

The data used in this paper are available from K.P. or J.L. or J.S. on reasonable request.

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## Acknowledgements

This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (No. 2020R1A2C2102838), the CoE program (19-CoE-NT-01) at DGIST and the DGIST institution specific program (21-BRP-07). The authors thank the Center for Core Research Facilities (CCRF) at DGIST for providing technical support for the material analyses.

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J.S. conceived and designed the experiment. S.-H.L., M.-S.K., H.J., and J.S. performed the STM experiment. B.C. and J.P. synthesized the FeSn single crystals under the supervision of K.P. K.P., and M.J. carried out the STEM measurement. Y.K. conducted the theoretical calculations under the supervision of J.L. All the authors discussed the results and contributed to the writing of the manuscript.

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Lee, SH., Kim, Y., Cho, B. *et al.* Spin-polarized and possible pseudospin-polarized scanning tunneling microscopy in kagome metal FeSn.
*Commun Phys* **5**, 235 (2022). https://doi.org/10.1038/s42005-022-01012-z

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DOI: https://doi.org/10.1038/s42005-022-01012-z

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