Abstract
Noncoplanar spin textures with finite scalar spin chirality can be artificially induced at surfaces and interfaces through the interfacial DzyaloshinskiiMoriya interaction. However, stabilizing a proper magnetic skyrmion crystal via this route remains elusive. Here, using an epitaxial bilayer of platinum and geometrically frustrated kagomelattice ferromagnet Fe_{3}Sn, we show the possible formation of a twodimensional skyrmion crystal under wellregulated Fe_{3}Sn thickness conditions. Magnetization measurements reveal that the magnetic anisotropy is systematically varied from an inherent inplane type to a perpendicular type with the thickness reduction. Below approximately 0.5 nm, we clearly detect a topological Hall effect that provides evidence for finite scalar spin chirality. Our topological Hall effect analysis, combined with theoretical simulations, not only establishes its interfacial DzyaloshinskiiMoriya interaction origin, but also indicates the emergence of a stable skyrmion crystal phase, demonstrating the potential of kagomelattice ferromagnets in spin chirality engineering using thinfilm nanostructures.
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Introduction
Scalar spin chirality, \({{{{{{\bf{S}}}}}}}_{i}\cdot ({{{{{{\bf{S}}}}}}}_{j}\times {{{{{{\bf{S}}}}}}}_{k})\), is a quantity that corresponds to a solid angle subtended by three spins S_{i}, S_{j}, and S_{k} on neighboring triangular sites^{1,2,3}. Electrons hopping through noncoplanar spin textures with finite scalar spin chirality acquire Berry phase equivalent to a fictitious magnetic field, leading to topological Hall effect (THE)^{4,5} that is distinct from other Hall effects. Among various noncoplanar spin textures, a swirling spin texture called the magnetic skyrmion^{4,5,6} is particularly interesting because it can not only behave like an individual particlelike object but also can form an ordered skyrmion crystal (SkX) under the delicate competition of magnetic interactions and external perturbations. Understanding the stability and controllability of SkX phase is currently at the frontier of condensed matter research.
The established approach for the formation of SkX phase is to use noncentrosymmetric bulk crystals^{4,5} in which finite contributions of spinorbit coupling (SOC) induce the DzyaloshinskiiMoriya interaction (DMI)^{7,8}, expressed as \({H}_{{{{{{\rm{DMI}}}}}}}={{{{{\bf{D}}}}}}\cdot ({{{{{{\bf{S}}}}}}}_{i}\times {{{{{{\bf{S}}}}}}}_{j})\) with D being the DMI vector. This interaction favors a relative twist between otherwise parallel or antiparallel spins, giving rise to complex magnetic states such as canted ferromagnetism^{7,8}, noncoplanar and helical/spiral spin states^{4,5}. In thinfilm heterostructures, the role of SOC in bulk crystals can be activated by fabricating an asymmetric interface with individual magnetic and SOC layers, which is known as the interfacial DMI^{9,10}. In various interface systems including metals^{11,12,13,14,15,16}, perovskitetype oxides^{17,18}, and topological insulators^{18,19}, the formation of isolated skyrmions and the detection of THE have been reported. However, the twodimensional SkX has rarely been achieved by the interfacial DMI mechanism^{20}. Given the crucial role of crystal symmetry in the bulk approaches, a choice of specific lattice systems that enable the tuning of complex magnetic interactions is essential.
With this perspective, we focused on kagomelattice magnets with a triangularbased lattice, which is suitable for inducing noncoplanar magnetic states required for the SkX phase. We selected a ferromagnetic intermetallic compound Fe_{3}Sn with the D0_{19} structure. Bulk Fe_{3}Sn is ferromagnetic below Curie temperature T_{c} of 743 K with inplane magnetic anisotropy in the Fe_{3}Sn kagome plane (ab plane)^{21}. Very recently, it has been reported that Fe_{3}Sn can be epitaxially grown as a thin film on Pt(111) (ref. ^{22}). The combination of Fe_{3}Sn and Pt offers the following advantages for spin chirality engineering via the interfacial DMI: (i) Fe_{3}Sn consists of ideal twodimensional kagome planes (Fig. 1a), (ii) the epitaxial interface across Pt with strong SOC effectively activates the interfacial DMI, (iii) large magnetization derived from Fe enables the characterization of magnetic properties in the ultrathin films.
Here, we report a comprehensive study that strongly suggests the formation of a twodimensional SkX phase in Fe_{3}Sn/Pt epitaxial bilayers (Fig. 1a). By tuning the magnetic anisotropy and the interfacial DMI via the reduction of the Fe_{3}Sn thickness, t, we attempt to achieve the noncoplanar spin state with finite scalar spin chirality (Fig. 1b, c). The evolution of magnetic anisotropy and the occurrence of THE revealed by the systematic tdependent measurements are consistent with the formation of SkX in a theoretical model based on a ferromagnetic kagome lattice.
Results and discussion
Fabrication of kagomelattice Fe_{3}Sn/Pt bilayers
We fabricated tnmthick Fe_{3}Sn/10nmthick Pt bilayers on Al_{2}O_{3}(0001) substrates by radiofrequency magnetron sputtering (see Methods). Figure 1d displays a crosssectional transmission electron microcopy (TEM) image of a bilayer sample with t = 8 nm. The selectedarea electron diffraction, shown in Fig. 1e, together with the macroscopic Xray diffraction (Supplementary Fig. 1), identifies the crystallization of the hightemperature Fe_{3}Sn phase (JCPDS PDF 010745857) with an epitaxial orientation relationship of Fe_{3}Sn\([01\bar{1}0]\)(0001)/Pt\([11\bar{2}]\)(111)/Al_{2}O_{3}\([11\bar{2}0](0001)\) (see Supplementary Fig. 2 for the analysis of fast Fourier transformation diffraction patterns). The wellcrystallized and homogeneous growth of Fe_{3}Sn was also observed for t = 1.6 nm (Supplementary Fig. 3). Because it is difficult to apply the same analysis to much thinner films (t < 1 nm) shown below, we hereafter adopt the nominal t value based on the sputtering rate assuming uniform film deposition. As shown in Fig. 1f, M versus magnetic field μ_{0}H curves (where μ_{0} is vacuum permeability and H is magnetic field strength) measured for t = 0.80 nm (~2 unit cells, the bulk caxis length = 0.436 nm) in an outofplane H show hardaxis behavior without remanent M (see Supplementary Fig. 4 for the data of other t values). The saturation magnetization M_{sat} of 1.2 × 10^{6} J m^{−3} T^{−1} agrees with the bulk value reported in the literature^{21,23}. The t dependence of M_{sat} at temperature T = 400 K (Fig. 1g) shows that the M_{sat} decreases below t = 0.80 nm. However, the bulklike M_{sat} value at low T (see Supplementary Fig. 5 for the T dependence of M_{sat}) persists down to t = 0.32 nm, indicating that Fe spins order ferromagnetically even in these ultrathin samples. The M_{sat} significantly decreases at t = 0.24 nm. The clear qualitative change in the magnetic behavior occurs between t = 0.32 nm and 0.24 nm. Combined with the TEM results (Fig. 1d and Supplementary Fig. 3), this implies that the nominal t value corresponds to the actual film thickness even in these ultrathin films. The estimated T_{c} is significantly higher than 400 K for t ≥ 0.48 nm and becomes comparable to 400 K for 0.32 nm ≤ t ≤ 0.40 nm. A more detailed analysis of the T dependence of magnetization across the ferromagnetic transition will clarify the accurate T_{C} and possible role of disorder on the magnetic behavior^{24}.
Thicknessdependent magnetic anisotropy in Fe_{3}Sn/Pt bilayers
By comparing M versus μ_{0}H curves in outofplane and inplane configurations, we detected the expected variation of magnetic anisotropy, as displayed in Fig. 2a–c. For t = 0.64 nm (Fig. 2a), the M saturates easily under the application of inplane H (blue curve), showing the inplane magnetic anisotropy consistent with the bulk behavior^{21}. In the sample with t = 0.48 nm (Fig. 2b), however, the M saturates at almost comparable μ_{0}H values in the two measurement configurations. The even thinner condition of t = 0.24 nm appears to favor perpendicular magnetic anisotropy (Fig. 2c). From the \(M/{M}_{{{{{{\rm{sat}}}}}}}\) versus μ_{0}H curves, we calculated the magnetic anisotropy energy as the effective anisotropy field μ_{0}H_{A,eff} in the unit of T (μ_{0}H_{A,eff} > 0 for perpendicular magnetic anisotropy and < 0 for inplane magnetic anisotropy). As plotted in Fig. 2d, μ_{0}H_{A,eff} gradually varies from negative to positive with decreasing t, changing its sign at approximately 0.5 nm (~1 unit cell). This result captures that the inherent inplane magnetic anisotropy turns to perpendicular magnetic anisotropy at a crossover thickness of t_{MA} ~0.5 nm. To gain insight into spin textures in these ultrathin Fe_{3}Sn bilayer samples, we performed a numerical simulation for a single kagome plane that took into account the ferromagnetic Heisenberg interaction J, magnetic anisotropy A (inplane and perpendicular magnetic anisotropy for A < 0 and > 0, respectively), and DMI (see Methods). As schematically shown in Fig. 2e, this model considers spin interactions between the neighboring spins on the kagome plane in an outofplane magnetic field. Bulk and interfacial DMIs contribute to the outofplane and inplane components of D (D_{ǁ} and D_{⊥}), respectively, which modifies the local spin interactions to produce a finite spin solid angle. Figure 2f displays the total spin solid angle per the unit cell of Fe_{3}Sn, \({\varOmega }_{{{{{{\rm{tot}}}}}}}\), as a function of A and D_{⊥} (J = D_{ǁ} = 1 and outofplane magnetic field gμ_{B}H = 0.05, where g is the gfactor and μ_{B} is the Bohr magnetron). In the upperleft region of large negative A and small D_{⊥}, the spins basically lie in the kagome plane with slightly tilting to the outofplane H direction (see Supplementary Fig. 6 for the simulated magnetic moments). This corresponds to a ferromagnetically ordered phase with inplane magnetic anisotropy in the kagome plane as in the bulk^{21}. Increasing A to the positive side stabilizes a different ordered phase where spins fully point to the outofplane H direction. Although these ordered phases do not generate \({\varOmega }_{{{{{{\rm{tot}}}}}}}\), we found that a moderate contribution of D_{⊥} stabilizes a twodimensional SkX phase with a finite \({\varOmega }_{{{{{{\rm{tot}}}}}}}\) (inset, also see Supplementary Fig. 7). By a further increase of D_{⊥}, the spiral phase without a finite \({\varOmega }_{{{{{{\rm{tot}}}}}}}\) is induced. This magnetic phase diagram highlights the critical importance of the interfacedriven modification of DMI for the stabilization of SkX phase in the ferromagnetic kagome lattice. In fact, the clear switching of magnetic anisotropy shown in Fig. 2d corroborates that our tcontrolled Fe_{3}Sn/Pt bilayers are an excellent arena for experimentally verifying scalar spin chirality via the THE analysis.
Verification of interfacial DMI
We characterized three sample structures schematically shown in Fig. 3a–c: 0.80nmthick Fe_{3}Sn/Pt (thickbilayer, t > t_{MA}), 0.40nmthick Fe_{3}Sn/Pt (thinbilayer, t < t_{MA}) and Pt/0.40nmthick Fe_{3}Sn/Pt (thintrilayer, t < t_{MA}; see Supplementary Fig. 8 for the M data). In the thickbilayer and thinbilayer structures, the interfacial DMI is activated by the bottom Pt layer, whereas in the thintrilayer structure, the effective interfacial DMI would be canceled by opposite contributions from the top Pt/Fe_{3}Sn and bottom Fe_{3}Sn/Pt interfaces. Although there is no direct correspondence between the actual samples and simulation conditions, the thickbilayer, thinbilayer, and thintrilayer structures can be compared with the simulation conditions of negative A and small D_{⊥}, small positive A and large D_{⊥}, and small positive A and small D_{⊥}, respectively. Note here that a large part of conduction in these heterostructures is governed by the highly conducting Pt layer (See Supplementary Figs. 9 and 10 for the tdependence of sheet resistance and the magnetoresistance data, respectively). Figure 3d shows Hall resistance R_{yx} versus μ_{0}H curves of the thickbilayer structure at T = 400 K. In addition to the linear ordinary Hall effect of Pt (Supplementary Fig. 11k), a nonlinear response in the R_{yx} is clearly discernable. A fit using an empirical relation^{15,16,17,18,19}, \({R}_{yx}={R}_{0}{{\hbox{'}}}{\mu }_{0}H+{R}_{{{{{{\rm{A}}}}}}}{{\hbox{'}}}M+{R}_{yx}^{{{{{{\rm{T}}}}}}}\), where \({R}_{0}^{{\prime} }\) and \({R}_{{{\rm{A}}}}^{{\prime} }\) are the ordinary and anomalous Hall coefficients in the unit of Ω T^{−1}, and \({R}_{{{{{{\rm{A}}}}}}}^{{\prime} }M\,(={R}_{yx}^{{{{{{\rm{A}}}}}}})\) and \({R}_{yx}^{{{{{{\rm{T}}}}}}}\) are the anomalous and topological Hall resistances, reveals the dominant contribution of \({R}_{yx}^{{{{{{\rm{A}}}}}}}\) to the R_{yx} (Fig. 3e inset), which is attributed to the anomalous Hall effect (AHE) of Fe_{3}Sn. The negligibly small residual \({R}_{yx}^{{{{{{\rm{T}}}}}}}\) (Fig. 3e) indicates the absence of THE in the thickbilayer structure. This is consistent with that the bulklike inplane magnetic anisotropy favors the coplanar spin state without scalar spin chirality (Fig. 1b). Contrastingly, in the thinbilayer structure, the extracted \({R}_{yx}^{{{{{{\rm{T}}}}}}}\) (Fig. 3g) at T = 400 K ~T_{c} (Supplementary Fig. 5) overwhelms both ordinary and anomalous Hall resistances (Fig. 3f and the inset of Fig. 3g), indicating the presence of finite scalar spin chirality that contributes to THE. The addition of top Pt layer in the thintrilayer structure, as intended, completely diminishes the THE (Fig. 3h, i). On the other hand, the comparable AHE for the thintrilayer and thinbilayer structures at T = 300 K (the insets of Fig. 3f, h; also see Supplementary Fig. 11 for the data at various T) indicates their macroscopically similar ferromagnetic states. In recent studies on SrRuO_{3} ultrathin films and SrRuO_{3}based perovskite multilayers and superlattices^{24,25,26,27,28}, the impact of inhomogeneity on the occurrence of THElike R_{yx} anomalies has been argued. When AHE changes its sign depending on thickness as in the present system (Supplementary Fig. 11), local thickness fluctuation could give rise to humplike R_{yx} behavior via the superposition of AHE components with different signs. To understand the origin of the observed \({R}_{yx}^{{{{{{\rm{T}}}}}}}\), we evaluated asymmetric Ta/Fe_{3}Sn/Pt and W/Fe_{3}Sn/Pt trilayer structures (Supplementary Fig. 12 and Supplementary Note 1). The results were consistently explained by considering different magnitudes of the interfacial DMI contributions from the top and bottom interfaces. In conjunction with the insignificant thickness fluctuation suggested by the TEM and magnetization measurements, this strongly supports that the interfacial DMI plays a more decisive role in the occurrence of \({R}_{yx}^{{{{{{\rm{T}}}}}}}\) than inhomogeneity. Furthermore, the superposition of R_{yx} versus μ_{0}H curves for different t values at T = 400 K (Supplementary Fig. 11) cannot reproduce the sharp R_{yx} peaks detected in the thinbilayer structure (Fig. 4f). These observations are fully consistent with the generation of finite scalar spin chirality owing to the local modification of spin interactions by the interfacial DMI.
Analysis of THE and possible formation of SkX phase
Having verified the interfacial DMI origin, we demonstrate the tcontrolled variation of scalar spin chirality via the detection of THE. As shown in Fig. 4a–c, a slight reduction of t effectively lowers the T range where THE appears (see Supplementary Fig. 13 for the corresponding \({R}_{yx}^{{{{{{\rm{T}}}}}}}\,\)data). Upon comparing the R_{yx} data including those of other t values (Supplementary Fig. 11), we noticed the sign reversal of R_{yx} at approximately T = 370 K for t = 0.40 nm (Fig. 4b) and at T = 340 K for t = 0.32 nm (Fig. 4c). Figure 4d shows a contour plot of \(\varDelta {R}_{yx}(\pm 3\,{{{{{\rm{T}}}}}})={R}_{yx}(+3\,{{{{{\rm{T}}}}}}){R}_{yx}(3\,{{{{{\rm{T}}}}}})\) on t–T plane; its sign reversal occurs in close proximity to the THE region (surrounded by dashed lines). According to recent band structure calculation^{29}, Fe_{3}Sn is classified as a magnetic Weyl semimetal with Weyl nodes^{30} near the Fermi level. The Tinduced shift of the Fermi level and the resulting change in the intrinsic AHE contribution may play a role in the Tinduced sign reversal of AHE.
More importantly, as shown in Fig. 4e, the magnitude of \({R}_{yx}^{{{{{{\rm{T}}}}}}}\) increases with increasing T (up to our measurement limit of 400 K), concomitantly with the increase of the peak magnetic field at which R_{yx} shows local maxima/minima due to THE, μ_{0}H_{peak} (black circles). Using the standard linearresponse theory for a Kondolattice model, we performed the analysis of finitetemperature Hall conductivity \({\sigma }_{{{{{{\rm{H}}}}}}}\) of the intrinsic Berry phase mechanism (Fig. 4f, also see the \({\varOmega }_{{{{{{\rm{tot}}}}}}}\) result for Supplementary Fig. 14). Overall, the simulation results reproduce the observed experimental trend, indicating the contribution of thermal fluctuation^{18,31} to the intrinsic (topological) Hall effect. These excellent agreements between the experiments and simulation suggest the formation of SkX phase in the THE region. To support this, we attempted to extract realspace features from the THE data using the relation^{5,16,17}: \({\rho }_{yx}^{{{{{{\rm{T}}}}}}}=P{R}_{0}{n}_{{{{{{\rm{sk}}}}}}}{\phi }_{0}\), where \({\rho }_{yx}^{{{{{{\rm{T}}}}}}}\) is the topological Hall resistivity, P is the spin polarization, R_{0} is the ordinary Hall coefficient, n_{sk} is the skyrmion density, and \({\phi }_{0}\) is one magnetic flux quantum (= h/e with h being the Planck constant and e being the elementary charge). The \({n}_{{{{{{\rm{sk}}}}}}}^{0.5}\) corresponds to the average separation of skyrmions. Assuming a parallel circuit consisting of two conducting layers, we calculated the resistivity and Hall resistivity (Supplementary Fig. 15) of the Fe_{3}Sn layer from the 0.40nmthick bilayer data and reference Pt monolayer data at T = 350 K, yielding \({\rho }_{yx}^{{{{{{\rm{T}}}}}}}\) = 37.4 nΩ cm and R_{0} = 8.48 × 10^{−5} cm^{3} C^{−1}. These values give \({n}_{{{{{{\rm{sk}}}}}}}^{0.5}\) = 9.68 nm and 30.6 nm for P = 0.1 and 1, respectively, which are reasonable as compared with the size of skyrmions reported for other bilayer systems (Supplementary Table 1). We therefore think that densely arranged skyrmions like SkX exist in the Fe_{3}Sn/Pt bilayer samples. Direct observation of the spin textures using microscopy will be an interesting future study.
Conclusion
The epitaxial interface of kagomelattice ferromagnet Fe_{3}Sn and Pt enables the rational control of magnetic and electrical properties based on the interfacial DMI. Considering the rich variety of kagomelattice magnets such as Fe_{3}Sn_{2}, FeSn, and Co_{3}Sn_{2}S_{2} that have been discovered from the aspect of topological physics, the development of heterointerfaces and superlattices is worthy of investigation. These fascinating features of kagomelattice magnets will offer tremendous opportunities for exploring new functionalities of SkXbased phenomena.
Methods
Thinfilm growth
The films were fabricated on Al_{2}O_{3}(0001) substrates by radiofrequency magnetron sputtering at an Ar gas pressure of 0.5 Pa. The Pt, Fe_{3}Sn, and SiO_{x} layers were in situ deposited at 600, 400, and 100 °C with Pt, FeSn [ref. ^{32}], and SiO_{x} targets, respectively. The 2nmthick Pt top layer of the thintrilayer structure (Fig. 3c) was deposited at 100 °C before the SiO_{x} capping. The crystal structure of the films was characterized by TEM and Xray diffraction using Cu K_{α} radiation. The t values were calculated based on the sputtering rate that was calibrated with the crosssectional TEM image shown in Fig. 1d.
Measurements
The magnetization was measured with a superconducting quantum interference device magnetometer (MPMS3, Quantum Design) upon decreasing μ_{0}H from 7 T to −7 T and increasing μ_{0}H from −7 T to 7 T. By subtracting a diamagnetic contribution from Al_{2}O_{3} substrate, which was estimated from the highfield data at μ_{0}H = 4–7 T, the M was calculated. By antisymmetrizing the decreasingfield and increasingfield M data, the two antisymmetrized M curves shown in Figs. 1f and 2a–c and Supplementary Figs. 4 and 8 were obtained. The electrical properties were measured with a physical property measurement system (PPMS, Quantum Design). The films were patterned into a Hallbar shape by mechanical scratch, and electrical contacts were made with indium solder. For the analysis of THE, the decreasingfield and increasingfield R_{yx} data were antisymmetrized against μ_{0}H to eliminate spurious contributions arising from thermoelectric effect and misalignment of the Hall voltage probes.
Simulation
We considered a classical Heisenberg model on the kagome lattice to understand the magnetism of Fe_{3}Sn thin films. The Hamiltonian reads
where \({{{{{{\bf{S}}}}}}}_{i}\) represents a classical spin with fixed length \({{{{{{\bf{S}}}}}}}_{i}=1\) on ith site. The first sum of \(\langle i,j\rangle\) runs over all the nearest neighbor sites; J and \({{{{{{\bf{D}}}}}}}_{ij}\) represent the ferromagnetic Heisenberg coupling and the DMI, respectively. The second sum represents the single ion anisotropy A and the Zeeman coupling to the external magnetic field H perpendicular to the kagome plane. We set the direction of \({{{{{{\bf{D}}}}}}}_{ij}\) as shown in Fig. 2e from the symmetry point of view^{8}: The outofplane component D_{ǁ} is inherent to the inversion symmetry breaking on the bond centers on the kagome lattice (bulk DMI), while inplane component D_{⊥} arises from the breaking of mirror symmetry due to the attached Pt layer (interfacial DMI). Note that the latter satisfies C_{3} and C_{6} rotational symmetries around the center of the triangular and hexagonal plaquettes of the kagome lattice, respectively. We adopted \(J=\,{D}_{}\); similar conditions have been used in previous studies^{33,34}. The magnetic field H was set to be weak so that the magnetization is not forced to be parallel to the applied H direction. In the finitetemperature analysis shown in Fig. 4f, we set \({D}_{\perp }=0.25\, < \, J={D}_{}\). Because D_{⊥} is a contribution that is effective only at the interface, this assumption is qualitatively valid. Positive and negative D_{⊥} gave the identical simulation results. As for the singleion anisotropy A, we set \(A=0.1\) by considering the saturation field ~0.5 T and T_{C} ≥ 400 K (i.e., A << J) in the experimental magnetization data.
To obtain the groundstate spin configuration of (1), we used a combined method of simulated annealing and local optimization. The annealing was performed from the temperature 1 to 0.001 with 100 steps in the logarithmic scale. In each step, we spent 1000 Monte Carlo (MC) sweeps. After the annealing, we optimized the spin directions one by one so as to minimize the local energy with fixed surrounding spins. We repeated 20000 sweeps of this optimization process. For analyzing the finitetemperature properties, we performed MC simulations with 100000 MC sweeps after 100000 thermalization at each temperature. In all the calculations, we considered \(N=3{L}^{2}\) spins with \(L=48\) under the periodic boundary condition.
To detect the noncoplanar spin structure in the SkX, we computed the solid angle defined as
with the solid angle^{35}
which is defined in the range of \((2{{{{{\rm{\pi }}}}}},2{{{{{\rm{\pi }}}}}})\). The sums of \((i,j,k)\) and \((i,j,k,l,m,n)\) run over all the triangular and hexagonal plaquettes of the kagome lattice, respectively, in which i, j, … were assigned in the counterclockwise order when viewed from the z direction. We also computed the magnetizations (Supplementary Fig. 6):
We calculated the Hall conductivity σ_{H} for the spin configurations obtained by the MC simulations. For this purpose, we introduced the Kondo lattice model;
with the given spin configurations S_{i}, and computed σ_{H} by using the standard Kubo formula. The first term represents the kinetic energy of itinerant electrons with the nearest neighbor hopping \({t}_{{{{{{\rm{K}}}}}}}\), and the second term represents the Hund’s coupling \({J}_{{{{{{\rm{H}}}}}}}\) between itinerant electron spins defined by the Pauli matrices σ and localized spins S_{i}. In the calculations, we took \({J}_{{{{{{\rm{H}}}}}}}=4{t}_{{{{{{\rm{K}}}}}}}\) and fixed the electron density at \({n}_{{{{{{\rm{el}}}}}}}=\mathop{\sum}\limits_{i,s}\langle{c}_{is}^{{{\dagger}} }{c}_{is}\rangle/N=0.1\). This low \({n}_{{{{{{\rm{el}}}}}}}\) is regarded as a dilute limit^{36,37}, which is appropriate to capture the generic feature of the single kagome plane. For simplicity, we set the temperature of the itinerant electrons sufficiently low as \({T}_{{{{{{\rm{KLM}}}}}}}={t}_{{{{{{\rm{K}}}}}}}/40\) to focus on the fluctuations of spins.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank S. Ito for the TEM analysis and N. Kanazawa for helpful discussions. This work was supported by JST CREST (JPMJCR18T2), JSPS KAKENHI GrantinAid for Scientific Research (S) (JP18H05246) and (B) (20H01830), SEI Group CSR Foundation, and Foundation for Interaction in Science and Technology.
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Contributions
K.F. grew the films and characterized their structural, magnetic, and electrical properties. T.S. and K.T. contributed to the magnetization measurements and analysis. Y.K. and Y.M. performed the Monte Carlo simulation and the calculation using the linearresponse theory. K.N. contributed to theoretical interpretations of the experimental results. K.F., Y.K., Y.M. and A.T. wrote the manuscript with input from other authors. A.T. supervised the project.
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Fujiwara, K., Kato, Y., Seki, T. et al. Tuning scalar spin chirality in ultrathin films of the kagomelattice ferromagnet Fe_{3}Sn. Commun Mater 2, 113 (2021). https://doi.org/10.1038/s4324602100218y
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DOI: https://doi.org/10.1038/s4324602100218y
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