Spin excitations in metallic kagome lattice FeSn and CoSn

In two-dimensional (2D) metallic kagome lattice materials, destructive interference of electronic hopping pathways around the kagome bracket can produce nearly localized electrons, and thus electronic bands that are flat in momentum space. When ferromagnetic order breaks the degeneracy of the electronic bands and splits them into the spin-up majority and spin-down minority electronic bands, quasiparticle excitations between the spin-up and spin-down flat bands should form a narrow localized spin-excitation Stoner continuum coexisting with well-defined spin waves in the long wavelengths. Here we report inelastic neutron scattering studies of spin excitations in 2D metallic Kagome lattice antiferromagnetic FeSn and paramagnetic CoSn, where angle resolved photoemission spectroscopy experiments found spin-polarized and nonpolarized flat bands, respectively, below the Fermi level. Although our initial measurements on FeSn indeed reveal well-defined spin waves extending well above 140 meV coexisting with a flat excitation at 170 meV, subsequent experiments on CoSn indicate that the flat mode actually arises mostly from hydrocarbon scattering of the CYTOP-M commonly used to glue the samples to aluminum holder. Therefore, our results established the evolution of spin excitations in FeSn and CoSn, and identified an anomalous flat mode that has been overlooked by the neutron scattering community for the past 20 years.

In general, spin-flip excitations in a magnet can be interpreted in terms of either a quantum spin models [1,2] with local moments on each atomic site [ Fig. 1(a)], or a Stoner [3][4][5][6]13] itinerant electron model. In insulating ferromagnets such as EuO, magnetic excitations can be fully described by a Heisenberg Hamiltonian [31] with spins on Eu lattice sites. In ferromagnetic metals, magnetic order breaks the degeneracy of the electronic bands, splitting spin-up majority and spin-down minority electrons [ Fig. 1(b)] [6] . For 3D metallic FM Fe and Ni, the low-energy spin waves are strongly damped when they enter a broad Stoner continuum of band-electron spin-flips that extends over several eV in energy [ Fig. 1(c)] [8][9][10][11][12] .
For a paramagnetic metal, there is no splitting of the degenerate electronic bands, and one would not expect to observe a Stoner continuum [13] . In strongly correlated materials like copper and iron-based superconductors, the subtle balance between electron kinetic energy and short-range interactions can lead to debates concerning whether magnetism has a localized or itinerant origin [32,33] .
In some 2D crystals, electrons can be confined in real space to form flat bands, for example through geometric lattice frustration [19][20][21] . The flat bands of magic-angle twisted bilayer graphene [23] provide one example of this route toward strong electronic correlation [24] . The kagome lattice depicted in Fig. 1 [20] , provides a second. Recently, a spin-polarized flat electronic band has been identified in the AF kagome metal FeSn at an energy E = 230 ± 50 meV below the Fermi level by angle-resolved photoemission spectroscopy (ARPES) experiments [25] . FeSn is a A-type AF with antiferromagnetically coupled FM planes [34] , which we will view as 2D ferromagnets. Neutrons should in principle detect the electron-hole-pair Stoner excitations from the majority-spin flat band below the Fermi level to minority-spin bands near or above the Fermi level [ Fig. 1(b)] [6,13] . Since neutron scattering measures electron-hole-pair excitations, having a flat spin-up electronic band below the Fermi level is a necessary, but not a sufficient condition to observe a flat Stoner continuum band. Instead, such a dispersionless narrow energy spin excitation band also requires a flat spin-down electronic band above (or near) the Fermi level [13] . Unfortunately, ARPES measurements cannot provide any information concerning such an electronic band above the Fermi level, although density functional theory (DFT) calculations suggest its presence [25] . For comparison, although ARPES measurements have also identified flat band at an energy E = 270 ± 50 meV below the Fermi level in CoSn [26,27] , one would not expect to observe a flat Stoner continuum band due to degenerate electronic bands and paramagnetic nature of the system [35] .
In this paper, we report inelastic neutron scattering (INS) studies of spin excitations in 2D metallic Kagome lattice antiferromagnetic FeSn [34] and paramagnetic CoSn [35] . For FeSn, our initial measurements reveal well-defined spin waves extending well above 140 meV and a narrow 24 meV wide band of excitations that cannot be described by a simple spin-wave model. While these data suggest the presence of electron-hole-pair Stoner excitations from the majority-spin flat band below the Fermi level to minority-spin bands near or above the Fermi level in FeSn, subsequent experiments on paramagnetic CoSn also have the same flat mode coexisting with expected paramagnetic spin excitations. Through careful analysis of INS spectra under different conditions, we conclude that the observed flat mode actually arises mostly from hydrocarbon scattering of the CYTOP-M commonly used to glue the samples to aluminum holder [36] . Therefore, our results established the evolution of spin excitations in FeSn and CoSn, and identified an anomalous flat mode that has been overlooked by the neutron scattering community for the past 20 years.

We have carried out INS experiments to study spin waves and search for anomalous
Stoner excitations in AF kagome metallic FeSn [34] and paramagnetic CoSn [35] . The structure of FeSn consists of 2D kagome nets of Fe separated by layers of Sn, and exhibits AF order below T N ≈ 365 K with in-plane FM moments in each layer stacked antiferromagnetically along the c-axis [ Fig. 1(d)] [34] . Since each unit cell contains three Fe atoms [ Fig. 1(e)], we expect one acoustic and two optical spin-wave branches in a local moment Heisenberg Hamiltonian [34,37,38] . Figures 1(f,g) show the reciprocal spaces corresponding to the crystal structures of FeSn depicted in Figs. 1(d,e), respectively. CoSn has the same crystal structure as that of FeSn but is paramagnetic at all temperatures [35] . ferromagnetic kagome planes. We also observe an easy-axis anisotropy gap ∆ a ≈ 2 meV due to single-ion magnetic anisotropy [ Fig. 2(h)] [34] .
To understand these observations, we start with a local moment Heisenberg Hamiltonian (x ) direction [39,40] , we define A (< 0) to be the single-ion anisotropy. The experimental in-plane FM exchange couplings obtained from this fit are smaller than theoretical predictions, while the the c-axis exchange coupling is larger by a factor of two [34] . To determine whether or not the magnetic excitations of FeSn can be understood within a Heisenberg Hamiltonian with S = 1 [34] , we consider the energy dependence of the local dy- , obtained by integrating the imaginary part of the generalized dynamic spin susceptibility χ ′′ (Q, E) over the first Brillouin zone [the green shaded region in Fig. 1(g)] at different energies [41] using where S(Q, E) is the measured magnetic scattering in absolute units, E is the neutron energy transfer, and k B is the Boltzmann's constant. Since the static ordered moment per Fe is M ≈ 1.85 µ B at 100 K [34] , the total magnetic moment M 0 of FeSn satisfies per Fe, where the local fluctuating moment m 2 ≈ 2 µ 2 B is obtained by integrating χ ′′ (E) at energies below 150 meV. In the local moment Heisenberg Hamiltonian with S=1, the total moment sum rule implies that M 2 0 = g 2 S(S + 1), where g ≈ 2 is the Lande g-factor, requiring a fluctuating moment contribution of g 2 S = 4 µ 2 B per Fe, which is a factor of 2 larger than the measured m 2 ≈ 2 µ 2 B per Fe. The solid and dashed lines in Fig. 3(h) are the calculated χ ′′ (E) in absolute units assuming S = 1, and 0.5, respectively. These unusual fluctuation properties suggest that electronic itineracy contributes plays a role in both the observed static ordered moment of FeSn [34] . We remark that flat spin wave flat bands can occur in kagome lattice ferromagnets with Dzyaloshinskii-Moriya (DM) interactions [37] , but these have an entirely different origin [38] .
If the flat band observed in Figs suggesting that the mode may not have a magnetic origin. Since our FeSn and CoSn samples are glued on the aluminum plates by CYTOP-M which is an amorphous fluoropolymer but contains one hydrogen to facilitate bonding to metal surface [36,42] , the hydrogenated amorphous carbon films formed between sample and aluminum plates should have C-H bending and stretching vibrational modes occurring around 150-180 meV and 350-380 meV, respectively [43,44] .  Fig. 5(c) confirms that the scattering at 170 meV arises from the C-H bending mode [43,44] . To further test if pure FeSn without CYTOP-M can also be contaminated by hydrocarbons, we prepared fresh single crystals of FeSn and carried out measurements at 5 K using unaligned single crystals on SEQUOIA. We find weak and broad excitations at 170 meV and shifting the incident beam neutron away from the sample using a motorized mask, the hydrocarbon contamination is still present with the similar intensity ratio between 170 meV and 360 meV modes (Table 1). Our careful infrared absorption spectrum analysis on the thermal shielding suggests the presence of hydrocarbon contamination, probably a silicone oil accidentally contaminating the vacuum system at SEQUOIA. Therefore, we conclude that the observed scattering at 170 meV in FeSn arises mostly from solid CYTOP-M with small additional contamination from hydrocarbons on thermal shielding.

DISCUSSION
To account for electronic itineracy, we calculate the electronic structure of FeSn in the paramagnetic and AF ordered states using a combination of DFT and dynamical mean field theory (DFT+DMFT) [45] . In the paramagnetic state, the mass enhancements of the Fe to the values in iron arsenide superconductors, we conclude that FeSn is a Hund's metal [46] with intermediate strength correlations. Figure 4 Although CYTOP-M has been used by neutron scattering community as a glue to mount small samples for over 20 years, its characteristics at high energies have not been reported [36] . This is mostly because the difficulty in carrying INS at energies above 100 meV at traditional reactor sources. The development of neutron time-of-flight measurements at spallation sources allows measurements at energies well above 200 meV, and the flat mode was missed in previous work [41] because of its weak intensity and its weak Q dependence. Our identification of the flat C-H bending and stretching vibrational modes should help future neutron scatterers to separate these scattering from genuine magnetic signal.

Sample synthesis, structural and composition characterization.
Single crystals of FeSn and CoSn were grown by the self-flux method. The high-purity Fe (Co) and Sn were put into corundum crucibles and sealed into quartz tubes with a ratio of Fe (Co) : Sn = 2 : 98. The tube was heated to 1273 K and held there for 12 h, then cooled to 823 K (873 K) at a rate of 3 (2) K/h. The flux was removed by centrifugation, and shiny crystals with typical size about 2×2×5 mm 3 can be obtained. The single crystal X-ray diffraction (XRD) pattern was performed using a Bruker D8 X-ray diffractometer with Cu K α radiation (λ = 0.15418 nm) at room temperature (Fig. S1).
The elemental analysis was performed using energy-dispersive X-ray (EDX) spectroscopy analysis in a FEI Nano 450 scanning electron microscope (SEM). In order to determine composition of FeSn accurately, we carefully polished FeSn surface using sandpaper and carried out EDX measurements on five FeSn crystals (Fig. S2). The average stoichiometry of each crystal was determined by examination of multiple points (5 positions). As shown in Table S1, the atomic ratio of Fe:Sn is close to 1:1.
To further determine the crystalline quality and stoichiometry of the samples used in neutron scattering experiments, we took X-ray single-crystal diffraction experiments on two pieces of these samples at the Rigaku XtaLAB PRO diffractometer housed at Spallation Neutron Source at Oak Ridge National Laboratory (ORNL). The measured crystals were carefully suspended in Paratone oil and mounted on a plastic loop attached to a copper pin/goniometer (Fig. S3). The single-crystal X-ray diffraction data were collected with molybdenum K radiation (λ = 0.71073Å). More than 2800 diffraction Bragg peaks were collected and refined using Rietveld analysis (Table S2). We find no evidence of superlattice peaks indicating possible Fe vacancy order (Fig. S3). The refinement results indicate less than 1.5% possible Fe vacancy (Fig. S4), suggesting that the single crystals are essentially fully stoichiometric.
To determine whether the AF phase transition in our sample is consistent with earlier work [34] , we carried out temperature and field dependence of the magnetization measurement. work [34] .
INS measurements on FeSn were carried out using the MAPS time-of-flight chopper spectrometer at the ISIS Spallation Neutron Source, the Rutherford Appleton Laboratory, UK [47] . INS measurements on CoSn and FeSn are also performed using the SEQUOIA spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratory [48] . Fifty pieces of single crystals of FeSn with the total mass of 0.97 g were co-aligned on one single piece of aluminum plate and mounted inside a He displex. Figure S6 shows that the mosaic of aligned single crystals is about 6 degrees. The crystal structure of FeSn is hexagonal with space group P 6 /mmm with lattice parameters a = b = 5.529Å, and c = 4.4481Å [34] .
The lattice parameters of CoSn are a = b = 5.528Å, and c = 4.26Å [35] . We define the momentum transfer Q in 3D reciprocal space inÅ temperature is set at 5 K. The neutron scattering data is normalized to absolute units using a vanadium standard, which has an accuracy of approximately 30% [41] .
Heisenberg model fitting to spin waves of FeSn.
We use the Heisenberg model and least-square method to fit spin waves of FeSn (Figs. S7-S9). The software package used was SpinW and Horace [49] . The Heisenberg Hamiltonian as discussed in the main text. Note in our Heisenberg Hamiltonian fit to spin wave data, we only used dispersion relations from experiments, and assumed S = 1, which is close to the 1.86 µ B per Fe ordered moment [34] . The overall intensity from SpinW fit, when considered in absolute units, is considerably higher than the experiment [ Fig. 2(h)]. This suggests that Heisenberg Hamiltonian overestimates the spin wave intensity contribution from the ordered moment. We first determine the interlayer coupling J c . Using linear spin wave theory, we find that the spin-wave band top along the c Note that the error bars of these parameters are estimated as follows: First, calculate the least square error using best-fit parameter J 0 and denote it as R 0 ; then determine the parameter J ′ 0 when the square error values give 2R 0 , and the error bar is given by ∆J = |J ′ 0 − J 0 |. This error range gives 68% confidence interval, meaning that if the residual of the fit has a Gaussian distribution, the calculation generated by range [J 0 − ∆J, J 0 + ∆J] can cover 68% (i.e. 1σ in Gaussian distribution) of the data points.

DFT+DMFT calculations.
The electronic structures and spin dynamics of FeSn in the paramagnetic and magnetically ordered states are computed using DFT+DMFT method [45] . The density functional theory part is based on the full-potential linear augmented plane wave method implemented in Wien2K [50] . The Perdew-Burke-Ernzerhof generalized gradient approximation is used for the exchange correlation functional [51] . DFT+DMFT was implemented on top of Wien2K and was described in detail before [52] . In the DFT+DMFT calculations, the electronic charge was computed self-consistently on DFT+DMFT density matrix. The quantum impurity problem was solved by the continuous time quantum Monte Carlo (CTQMC) method [53,54] with a Hubbard U = 4.0 eV and Hund's rule coupling J = 0.7 eV in both the paramagnetic state and the magnetically ordered state. Bethe-Salpeter equation is used to compute the dynamic spin susceptibility where the bare susceptibility is computed using the converged DFT+DMFT Green's function while the two-particle vertex is directly sampled using CTQMC method after achieving full self-consistency of DFT+DMFT density matrix [55] .
For the magnetically ordered state, the averaged Green's function of the spin up and spin down channels is used to compute the bare susceptibility. In the paramagnetic state, an electronic flat band of dominating d xz and d yz orbital characters locates a few meV above the Fermi level (Fig.4b).
In the magnetic state, the spin exchange interaction leads to about  Figure S10 shows orbital-resolved band structures of FeSn in the paramagnetic, spin up, and spin down magnetically ordered state. We also note that the possible ∼1.2% iron deficiency in FeSn obtained from X-ray refinement (Table S2) is not expected to modify the band structure.
The infrared absorption measurements.
To determine if the thermal heat shielding of SEQUOIA acquired an organic coating, we cut a small piece of the shielding right after the experiment and carried out the infrared absorption spectrum measurement on that piece. The spectrum in Fig. S11 shows The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
The codes used for the DFT+DMFT calculations in this study are available from the corresponding authors upon reasonable request.

ACKNOWLEDGEMENT
First and foremost, we wish to express our sincere appreciation to the anomalous referees who reviewed this paper, particularly referee 2. In the original draft of the paper, we only have data for FeSn. It is the comment of referee 2 that inspired us to carry out measure- In the present study, the color bars represent the vanadium standard normalized absolute magnetic excitation intensity in the units of mbarn meV −1 per formula unit, unless otherwise specified. The calculated spin wave intensity in (c,e,g) is in absolute units assuming S = 1 in the SpinW+Horace program [49] . The error bars in (h) represent statistical errors of 1 standard deviation.