Room temperature observation of the anomalous in-plane Hall effect in a Weyl ferromagnet

Abstract

Topologically nontrivial electronic states can lead to novel anomalous Hall effects, with room temperature manifestations promising for applications in magnetic sensing, spintronics, and energy harvesting. The anomalous in-plane Hall effect is expected in topological magnetic materials under an in-plane magnetic field, but its detection has been challenging because of strict symmetry requirements. Here, we combine molecular beam epitaxy of the kagome metal Fe₃Sn, electric Hall effect measurements, and theoretical calculations to propose and demonstrate that the kagome lattice motif combined with spin-orbit coupling and canted ferromagnetism induces the anomalous in-plane Hall effect at room temperature via topological Weyl points. Additionally, we synthesize a topological thin-film heterostructure with Fe₃Sn and ferromagnetic CoFeB, showing enhanced anomalous in-plane Hall effect amplitude due to CoFeB’s magnetic stray field. This work establishes a design framework for topological magnets and heterostructures aimed at discovering and controlling anomalous Hall effects for technological applications.

Introduction

Topologically nontrivial electronic states can give rise to novel electric transport phenomena, such as anomalous Hall effects (AHE)^1,2,3,4,5. These phenomena provide deep insight into the fundamental properties of topological quantum materials and have garnered increasing attention for their applications in magnetic sensing, spintronics, and energy harvesting technology^6,7,8. The conventional AHE requires the breaking of the time-reversal symmetry ${{\mathcal{T}}}$ resulting in a finite Berry curvature of the occupied electronic states. Therefore, AHE can be experimentally detected in magnetic materials and heterostructures, where ${{\mathcal{T}}}$ is broken by long-range magnetic order^1,9,10. As long as the energy scale of this ${{\mathcal{T}}}$-breaking mechanism, such as the magnetic exchange interaction J, is sufficiently large, Berry curvature-induced AHEs can be detected even at and above room temperature^{5,11,12,13,14,15,16}. This favors their potential integration in technological applications and motivates the search for new classes of topological magnetic materials and heterostructures, which can be manufactured using scalable synthesis methods, to discover and control Hall effects at room temperature.

Novel Hall effects can arise by the additional breaking of crystalline symmetries, such as inversion and rotation symmetries. Governed by distinct symmetry restrictions, these Hall effects can be qualitatively distinguished from the conventional AHE^9,10,17, and their unique experimental characteristics are attracting an increasing amount of interest. For example, breaking of inversion symmetry permits the observation of the second-order nonlinear anomalous Hall effect that appears as a Hall voltage at the second harmonic of a longitudinal a.c. current drive^2,3,4,9. It has also been predicted that breaking of out-of-plane rotation symmetries^17,18 can give rise to an anomalous in-plane Hall effect (IPHE). The IPHE describes the appearance of an electric Hall voltage when an external magnetic field is applied within the sample plane. Because of this rigorous restriction with respect to rotation symmetries, the observation of the anomalous IPHE has proven challenging. To date, IPHEs are mostly field-induced, where the breaking of the relevant crystalline symmetries is realized by an external magnetic field^19,20,21; moreover, a temperature-independent anomalous IPHE, resulting from topological electronic states, has not yet been reported²².

In this study, we overcome this challenge by combining electric Hall measurements of epitaxially grown thin films of the Weyl ferromagnet Fe₃Sn with systematic theoretical calculations. We propose and experimentally demonstrate that the interplay of the kagome lattice motif with spin-orbit coupling and canted in-plane ferromagnetism with large exchange interactions J ≈ 40 meV²³ breaks the relevant crystalline symmetries and gives rise to the anomalous IPHE at room temperature.

Results

Prediction of the anomalous in-plane Hall effect in the Weyl ferromagnet Fe₃Sn

The kagome metal Fe₃Sn (a = b = 5.487 Å and c = 4.3 Å, space group P6₃/mmc) belongs to the class of binary kagome metals X_pY_q, which attracts a lot of interest due to its topological and flat band electronic states (X = Fe, Co, Ni, Y = Sn, In, Ge, with chemical stoichiometry p,  q)^{5,11,24,25,26,27,28,29,30}. Common among this material class are strong nearest-neighbor magnetic exchange interactions between transition metal atoms arranged on the kagome lattice, as seen in Fig. 1a that can result in a long-range magnetic order much above room temperature^11,23,31.

Fig. 1: Prediction of the anomalous Hall in-plane Hall effect in the Weyl ferromagnet Fe₃Sn.

a Left panel: shown is the top view of the lattice structure of Fe₃Sn within the two-dimensional kagome plane (xy-plane). The hexagonal unit cell (black solid line) and definition of Cartesian coordinates with respect to the crystallographic axes are indicated. Right panel: shown is an isometric view of the bilayer stacking of the kagome layer along the z-direction. b, c shown are the schematic distributions of the total Berry curvature within the xz-plane at k_y = 0 for an ideal in-plane ferromagnet with magnetization vector M = (M_x,  0,  0) and an in-plane ferromagnet with out-of-plane canting M = (M_x,  0,  M_z), respectively. The mirror plane ${{{\mathcal{M}}}}_{{{\rm{x}}}}$ is indicated. In case without canting, ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ is preserved and the total Berry curvature integrates to zero. A finite spin canting M_z breaks ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ resulting in a finite total Berry curvature. d, e shown are the energy-integrated calculated Berry curvature obtained from a Wannierized ab initio band structure of Fe₃Sn within the first hexagonal Brillouin zone (black solid line) at k_z = 0 without and with out-of-plane spin canting, respectively. f Shown is the calculated anomalous in-plane Hall conductivity σ_IPHE(ϕ_B). The azimuthal angle ϕ_B is defined as the angle between the direction of an in-plane magnetic field B_∥ and the in-plane component of the magnetization M_x, as schematically indicated. Without canting, M = (M_x,  0,  0), the symmetry ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ is preserved, resulting in σ_IPHE = 0 (blue solid line). Breaking of this symmetry by a finite out-of-plane canting M_z ≠ 0 and M = (M_x,  0,  M_z) permits a finite σ_IPHE(ϕ_B) that is modulated by B_∥ (solid red dots). Shown is the result for M_z/M_x = 0.1. Details of the model calculations are presented in the “Methods” section.

In Fe₃Sn, the Fe ions form a kagome lattice in the crystallographic ab plane, and a Sn ion occupies the honeycomb center, as seen in Fig. 1a. The primitive unit cell of Fe₃Sn is composed of kagome bilayers stacked along the crystallographic [0001]-direction. For simplicity, we introduce the Cartesian coordinates x, y, and z corresponding to the crystalline directions $[2\bar{1}\bar{1}0]$, $[01\bar{1}0]$, and [0001], respectively. An easy-plane magneto-crystalline anisotropy imparts in-plane ferromagnetism with the magnetization M pointing along the x direction, where large ferromagnetic exchange interactions J ≈ 40 meV²³ favor a Curie temperature T_C ≈ 705 K above room temperature^{31,32,33,34,35}.

Ferromagnetism combined with spin-orbit coupling profoundly influences the electronic structure and topology of kagome metals^5,11,26. In Fe₃Sn, it has been shown that this combination imparts a set of Weyl points when the magnetization vector points along the x direction³². An additional magnetization component along the z direction, such as resulting from a canting of the magnetic moments of Fe, further lifts characteristic degeneracies of the electronic band structure of the kagome lattice, such as occurring at the crystallographic Γ and K points of the hexagonal Brillouin zone. This results in a set of Weyl points near the Fermi energy³². Using symmetry analyses and ab-initio model calculations, we show that these Weyl points give rise to an anomalous in-plane Hall conductivity σ_IPHE in spin-canted Fe₃Sn.

Given that the magnetization vector is oriented along the x-direction, the magnetic space group (MSG) of Fe₃Sn is $Cm{c}^{{\prime} }{m}^{{\prime} }$, which consists of three mirror (glide-mirror) symmetries ${{{\mathcal{M}}}}_{{{\rm{x}}}}$, ${{{\mathcal{M}}}}_{{{\rm{y}}}}t{{\mathcal{T}}}$, and ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$, where t is a fractional translation. Note that ${{\mathcal{M}}}=I\cdot {C}_{{{\rm{2}}}}$, where I and C₂ correspond to the inversion and two-fold rotation symmetry, respectively. Since the Berry curvature is invariant under I, ${{\mathcal{M}}}$ and C₂ are equivalent in symmetry analysis. Determined by all the symmetries in the MSG, the total Berry curvature (as seen in Fig. 1b, d) and thus the anomalous Hall conductivity σ_xy within the sample xy-plane vanishes; only an out-of-plane component σ_yz is permitted (see Section I of the Supplementary Materials for more details). On the other hand, an out-of-plane magnetic field or out-of-plane canting of the magnetization along the z-direction breaks ${{{\mathcal{M}}}}_{{{\rm{x}}}}$ and ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ and results in a finite total Berry curvature Ω_z (as seen in Fig. 1c, e).

This understanding is also consistent with our results of ab-initio calculations of the electronic band structure of Fe₃Sn (see the “Methods” section for calculation details). While the total Berry curvature vanishes in the absence of canting (as seen in Fig. 1d), the presence of canting M = (M_x,  0,  M_z) results in a finite Berry curvature that is induced by Weyl points near the Fermi energy, as seen in Fig. 1e. The detailed analysis of the electronic structure, Weyl point chirality, and resulting Berry curvature is shown in Section II of the Supplementary Materials.

An externally applied in-plane magnetic field ${{{\bf{B}}}}_{\parallel }={{\rm{B}}}\left(\cos {\phi }_{{{\rm{B}}}},\,\sin {\phi }_{{{\rm{B}}}},\,0\right)$ can further modulate the associated anomalous in-plane Hall conductivity σ_IPHE through the exchange coupling between B_∥ and the spin of the itinerant electrons. Here, ϕ_B is defined as the azimuthal angle between M_x and B_∥ within the xy-plane, as illustrated in Fig. 1f. We have analyzed this effect using a Wannier tight-binding model of the electronic band structure of Fe₃Sn (see the “Methods” section for details). The resulting σ_IPHE(ϕ_B) is shown in Fig. 1f. Our calculations establish that B_∥ modulates σ_IPHE with 2π periodicity in the presence of canting. The $\cos ({\phi }_{{{\rm{B}}}})$ dependence arises because the preserved ${{{\mathcal{M}}}}_{y}t{{\mathcal{T}}}$ symmetry requires σ_IPHE to be an even function of the magnetic field angle ϕ_B (see details in Section III of the Supplementary Materials). Therefore, our symmetry analysis and model calculations propose that Fe₃Sn should allow the observation of the anomalous IPHE when finite spin canting in the z direction is present. Note that the anomalous IPHE is odd under magnetic field reversal and exhibits an antisymmetric conductivity tensor. It should thus be invariant under a rotation of the current direction within the xy-plane.

Classification of Hall effects

This distinct dependence on the relative angle between B_∥ and M_x (seen in Fig. 1f), as well as the magnetic-field-reversal symmetry of σ_IPHE enables us to separate the anomalous IPHE from possible other contributions to the measured transverse voltage V_xy that can occur in the presence of an in-plane magnetic field B_∥ and canted ferromagnetism with M = (M_x,  0,  M_z), namely the symmetric transverse resistivity (STR), the planar Hall effect (PHE), the AHE, and the ordinary Hall effect (OHE). To this end, it is instructive to briefly introduce these effects and discuss their properties. The detailed characteristics of these phenomena are summarized in Table 1. The STR is an anisotropy effect that originates from the breaking of the C_3z-rotation symmetry by the in-plane magnetization component M_x. Therefore, it only depends on the azimuthal angle ϕ_E between the longitudinal electric field E, which lies in the xy-plane of the sample and is always parallel to the longitudinal current direction, and M_x³⁶. The associated conductivity is symmetric under magnetic field reversal and follows the relation ${\sigma }_{{{\rm{STR}}}}\propto \sin (2{\phi }_{{{\rm{E}}}})$ (also see Section IV of the Supplementary Materials for a detailed derivation of the STR effect). The PHE arises when both B_∥ and E lie within the sample plane. It results from a field-induced anisotropic magnetoresistance and its conductivity ${\sigma }_{{{\rm{PHE}}}}\propto \sin (2{\phi }_{B}-2{\phi }_{{{\rm{E}}}})$ exhibits a sinusoidal dependence on the relative angle between B_∥ and E³⁶. Critically, PHE displays a symmetric conductivity tensor, classifying it as an anisotropic magnetoresistance effect rather than a true Hall signal. The AHE results from a finite out-of-plane magnetization M_z. Hence, the anomalous Hall conductivity is independent of ϕ_B and ϕ_E and the AHE is isotropic within the xy-plane¹. Moreover, its conductivity tensor is odd under magnetization reversal, σ_xy(M_z) = −σ_xy(−M_z).

Table 1 Classification of different types of Hall effects

Finally, a slight misalignment of the sample xy-plane with the direction of B_∥ results in a finite out-of-plane component B_z of the magnetic field, which can give rise to the OHE. The OHE arises from the B_z-induced Lorentz force acting on the moving charge carriers, and its conductivity tensor is odd under magnetic field reversal. There are two possible misalignment configurations, which are described in detail in Section V of the Supplementary Materials: When the surface normal of the sample xy-plane is parallel to the rotation axis of B_∥, the associated Hall conductivity σ_H is independent of both ϕ_B and ϕ_E due to a fixed B_z. When additionally the surface normal of the sample xy-plane is misaligned with the rotation axis of B_∥, the associated Hall conductivity exhibits a sinusoidal dependence as ${\sigma }_{{{\rm{H}}}}\propto \sin ({\phi }_{{{\rm{B}}}})$, since the in-plane rotation modulates the relative orientation between B_z and the surface normal. Overall, this classification shows that different contributions to the transverse voltage V_xy can be distinguished by their distinct angle dependence and by using suitable symmetrization operations with respect to the externally applied magnetic field.

Experimental detection of spin-canted in-plane ferromagnetism in Fe₃Sn thin films

We developed the molecular beam epitaxy on Fe₃Sn thin films (typical thicknesses 30–60 nm) on the (0001) surface of sapphire substrates using a Pt(111) buffer layer of 5 nm thickness to facilitate layer-by-layer growth of continuous films (see the “Methods” section and Section VI of the Supplementary Materials for results of the chemical and structural characterization). Transmission electron microscopy measurements reveal a defined Pt(111)-Fe₃Sn interface and confirm the characteristic bilayer stacking of the Fe₃Sn kagome layers along the z direction, as seen in Fig. 2a. Measurements of the temperature (T) dependent longitudinal resistivity shown in Fig. 2b demonstrate the metallic character of the Fe₃Sn thin films. The residual ρ_xx ≈ 72 × 10⁻⁶ Ω cm at T = 2 K agrees well with previously reported values for single crystals^31,32,35, indicating a high crystalline quality and a low defect density of our samples. Note that the Pt(111) buffer layer is metallic and could shunt the electric conduction through the Fe₃Sn layer in current-biased measurements. Our analysis of this shunt effect, discussed in Section VII of the Supplementary Materials, shows that electric conduction through the Pt(111) buffer layer enhances the transverse voltage ${V}_{{{\rm{xy}}},{{{\rm{Fe}}}}_{3}{{\rm{Sn}}}}=\zeta \,{V}_{{{\rm{xy}}}}$ across the Fe₃Sn layer over the measured transverse voltage V_xy by a shunt factor ζ ≤ 1.19 that is nearly temperature independent. All transverse resistivities and voltages across the Fe₃Sn layer shown in the main text and Supplementary Materials are corrected for this shunt effect.

Fig. 2: Detection of spin-canted in-plane ferromagnetism in Fe₃Sn thin films.

a Shown is a transmission electron microscopy (TEM) cross-sectional image of the Fe₃Sn(001)/Pt(111)/sapphire(0001) thin film within the $[11\bar{2}0]-[0001]$-plane, where white and black color correspond to high and low TEM signal intensity. The different layers are annotated. The inset shows a magnification of the atomic layer structure of the Fe₃Sn layers and highlights their bilayer stacking along the z-direction. Fe atoms are schematically indicated by red spheres. b Shown is the temperature (T) dependence of the longitudinal resistivity ρ_xx (solid red line) of Fe₃Sn (60 nm)/Pt (5 nm)/sapphire. c Shown is the magnetization M of Fe₃Sn (30 nm)/Pt (5 nm)/sapphire for magnetic fields B applied within the xy-plane (red solid line) and along the z-direction (blue solid line) at T = 300 K. The inset shows the raw data of the measured magnetic moment m(B) for B∥xy before the subtraction of the diamagnetic background contributed by the sapphire substrate (see “Methods” section). d Optical micrograph of the circular device structure fabricated from the Fe₃Sn/Pt(111)/sapphire(0001) thin films. The numerical contact labels and Cartesian x- and y-axes are indicated. The in-plane component of the sample magnetization M_x is defined to be parallel to the  + x-direction. The azimuthal (within xy-plane) angles ϕ_E and ϕ_B are defined as the angles between the direction of the electric field E and the x-axis and the direction of the in-plane magnetic field B_∥ and the x-axis, respectively. Here, E is always parallel to the direction of the electric bias current I applied between different contact pairs. e Shown is the Hall resistivity ρ_H at ϕ_E = 0 measured as a function of a magnetic field B_z applied along the z−direction at T = 300 K. f Shown is a magnification of ρ_H in (e) at small magnetic field amplitudes for upward and downward sweeps of Bz.

The results of our magnetization measurements (see “Methods” section) at T = 300 K as a function of an externally applied magnetic field B are shown in Fig. 2c. In agreement with previous results on sputtered thin films and bulk crystals^31,32,33, we detect an anisotropic response to an externally applied magnetic field B; the saturation field is smaller when B is applied within the xy-plane than when B is applied along the z-axis. This result is consistent with the presence of in-plane ferromagnetism along the x-axis. The detected magnetic moment of  ≈ 2.3 μ_B/Fe atom matches well with previous reports^31,32,33 and is dominated by the spin moment of the Fe atoms³¹. A closer examination of the magnetization curve at small out-of-plane fields further reveals the presence of a narrow hysteresis loop with remnant out-of-plane magnetization M_z ≈ 0.11 μ_B/Fe atom at B = 0. This observation indicates the presence of a finite out-of-plane canting of the magnetic easy plane of Fe in our thin films. Note that the absence of an in-plane hysteresis, as seen in Fig. 2c could result from the combination of small energy barriers for magnetization reversal within the ab-plane³¹ and finite size effects. These factors could enable nearly reversible magnetization processes through coherent spin rotation³⁷ or free domain wall motion at small field sweep rates³⁸.

Fe₃Sn films were shaped into circular Hall bar devices with 12 terminals, as seen in Fig. 2d (see “Methods” section for details of the device fabrication). As discussed above, this device geometry enables us to identify and extract the different contributions to measured transverse voltage V_xy, when an externally applied magnetic field ${{{\bf{B}}}}_{\parallel }={B}_{\parallel }(\cos {\phi }_{{{\rm{B}}}},\,\sin {\phi }_{{{\rm{B}}}},\,0)$ is rotated within the xy-plane of the sample and when a longitudinal electric field E, which is parallel to the direction of the applied bias current I, is applied between different terminal pairs. In our experiment, we define the Cartesian coordinate system, shown in Fig. 2d, such that the in-plane magnetization component M_x points along the  + x-direction. Moreover, ϕ_B = ϕ_E = 0 when both B_∥ and E are aligned in parallel to the  + x-direction. For example, ϕ_E = 0 and ϕ_E = π/2 correspond to I applied from contacts 1 to 7 and 4 to 10, respectively. The azimuthal angle ϕ_B can be independently controlled by rotating the circular Hall bar sample within a static magnetic field applied within the xy-plane of the sample (also see “Methods” section for measurement details).

We first characterize the conventional Hall response of a Fe₃Sn thin film (thickness d₁ = 60 nm) at room temperature when the magnetic field B_z is applied in the out-of-plane direction along the z-direction. The Hall resistivity ρ_H(B_z,  ϕ_E = 0) is shown in Fig. 2e and corresponds to the magnetic field-antisymmetric contribution to the transverse resistivity tensor ρ_xy. Specifically, it can be obtained by using the symmetrization operation ρ_H(B_z) = (ρ_xy(B_z) − ρ_xy(−B_z))/2. The magnetic-field symmetric contribution to ρ_xy that depends on ϕ_E is discussed in Section IV of the Supplementary Materials. As can be seen, ρ_H(B_z) ∝ B_z in small fields and saturates at ρ_H(B_z) ≈ 6 × 10⁻⁶ Ω cm at B_z > 1  T, consistent with the results of magnetization measurements (c.f. Fig. 1c). These observations agree well with previous results^31,32,33. This saturation effect was reported to be associated with the magnetic field-induced reorientation of Fe magnetic moments from the x-axis to the z-axis³¹. Examining ρ_xy over a smaller field range for upward and downward sweeps of B_z reveals the presence of a hysteresis loop, as seen in Fig. 2f. The observation of hysteresis in both the electric Hall and magnetization measurements experimentally indicates an out-of-plane canting of the Fe magnetic moments at zero applied magnetic field. As suggested by our symmetry analysis and model calculation, the presence of this canting should permit the observation of the anomalous IPHE by breaking the ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ symmetry.

Observation of the in-plane Hall effect at room temperature

In the next step, we characterize the Hall response of Fe₃Sn at T = 300 K when the magnetic field is applied within the sample plane. To this end, we record the transverse resistivity ρ_xy(ϕ_B,  ϕ_E,  B_∥) as a function of ϕ_E and ϕ_B at T = 300 K and B_∥ = ±20 mT. Using symmetrization operations ${\rho }_{{{\rm{IPHE}}}}({\phi }_{{{\rm{B}}}},{\phi }_{{{\rm{E}}}})=\left({\rho }_{{{\rm{xy}}}}\right.({\phi }_{{{\rm{B}}}},\,{\phi }_{{{\rm{E}}}},\,{B}_{\parallel })-{\rho }_{{{\rm{xy}}}}({\phi }_{{{\rm{B}}}},\,{\phi }_{{{\rm{E}}}},\,-{B}_{\parallel })/2$ and ${\rho }_{{{\rm{sym}}}}({\phi }_{{{\rm{B}}}},\,{\phi }_{{{\rm{E}}}})=\left({\rho }_{{{\rm{xy}}}}\right.({\phi }_{{{\rm{B}}}},\,{\phi }_{{{\rm{E}}}},\,{B}_{\parallel })+{\rho }_{{{\rm{xy}}}}({\phi }_{{{\rm{B}}}},\,{\phi }_{{{\rm{E}}}},\,-{B}_{\parallel })/2$, we obtain the magnetic field-antisymmetric and symmetric contributions to ρ_xy(ϕ_B,  ϕ_E,  B_∥), respectively. Representative raw data of ρ_xy are presented in Section VIII of the Supplementary Materials.

We first consider the antisymmetric part ρ_IPHE, which is a true Hall effect, because it is odd under magnetic field reversal. In Fig. 3a, we plot the angular dependence of ρ_IPHE(ϕ_B) for device D1 recorded at ϕ_E = 0, and ϕ_E = π. The complete 4π angle dependence of ρ_IPHE within the (ϕ_B,  ϕ_E)-plane is shown in Fig. 3b. We detect a characteristic 2π-periodicity of ρ_IPHE(ϕ_B,  ϕ_E) with respect to ϕ_B. Moreover, as clearly seen in Fig. 3a, ρ_IPHE(ϕ_B,  ϕ_E) is independent of ϕ_E. We can accurately describe the ϕ_B-dependence of ρ_IPHE using ${\rho }_{{{\rm{IPHE}}}}({\phi }_{{{\rm{B}}}})={\rho }_{{{\rm{0}}}}+{\rho }_{{{\rm{IPHE}}}}^{0}\cos ({\phi }_{{{\rm{B}}}}+\delta )$ in a least square fit; the fit parameters are listed in Section VIII of the Supplementary Materials. The phase factor δ accounts for small angular offsets, which possibly arise from small misalignment of ϕ_B and ϕ_E with M_x in our measurements and give rise to the curved contour of ρ_IPHE along the ϕ_E-axis, as seen in Fig. 3b. Also note that these fits reveal the absence of a sizable vertical offset ρ₀, as shown in the inset of Fig. 3a (ρ₀/max(ρ_IPHE) < 1 %). Hence, our experimental observations indicate that ρ_IPHE only depends on the relative angle ϕ_B between the in-plane magnetic field B_∥ and the in-plane magnetization M_x and exhibits a characteristic $\cos ({\phi }_{{{\rm{B}}}})$ dependence. These findings are in excellent agreement with our theoretical predictions shown in Fig. 1f (also see Sections I–III of the Supplementary Materials) and suggest the experimental detection of anomalous IPHE in our measurements.

Fig. 3: Experimental observation of the anomalous in-plane Hall effect in Fe₃Sn at room temperature.

a Shown is the in-plane Hall resistivity ρ_IPHE as a function of the angle ϕ_B recorded on device D1 at different, indicated values of the angle ϕ_E at a temperature T = 300 K and an in-plane magnetic field B_∥ = 20 mT. Shown are the data (symbols) and fit to data (solid lines). The inset shows the vertical offset ρ₀ and corresponding error bars obtained from a fit to the data (see main text). b Shown is the full angle dependence of ρ_IPHE within the (ϕ_B,  ϕ_E)-plane recorded on device D1 at a temperature T = 300 K and an in-plane magnetic field B_∥ = 20 mT. c Shown is the full angle dependence of the symmetric contribution ρ_sym to the transverse resistivity within the (ϕ_B − ϕ_E,  ϕ_E)-plane recorded on device D1 at a temperature T = 300 K and an in-plane magnetic field B_∥ = 20 mT. d Shown is the symmetric transverse resistivity ${\rho }_{{{\rm{STR}}}}$ recorded as a function of the angle ϕ_E on device D1 at ϕ_B = π/2, T = 300 K, and B_∥ = 20 mT. e Shown is the resistivity ρ_PHE of the planar Hall effect (PHE) plotted as a function of ϕ_B − ϕ_E on device D1 at ϕ_E = 0 and ϕ_E = π at T = 300 K, and B_∥ = 20 mT. Shown are the data (symbols) and fit to data (solid lines) with fitted amplitudes ρ_PHE(ϕ_E = 0) = (33.6 ± 1.7) nΩ cm and ρ_PHE(ϕ_E = π) = (34.6 ± 0.9) nΩ cm. The two-dimensional resistivity maps in b and c result from interpolating ϕ_B-dependent resistivity curves recorded at 12 different values of ϕ_E. The hatched areas in b and c indicate missing data points (also seen in d and e), which could not be reached owing to the travel limit of the rotator probe (see “Methods” section).

Before we discuss this key result of our work in a broader context, we also consider the full 4π angle dependence of the magnetic-field symmetric resistivity ρ_sym(ϕ_B,  ϕ_E). To this end, we plot its angular dependence in the (ϕ_B − ϕ_E,  ϕ_E)-plane, as shown in Fig. 3c. The circular Hall bar geometry allows us to decompose ρ_sym(ϕ_B,  ϕ_E) into two separate contributions, ${\rho }_{{{\rm{STR}}}}({\phi }_{{{\rm{E}}}})$ and ρ_PHE(ϕ_B − ϕ_E) through their separate dependencies on ϕ_E and ϕ_B − ϕ_E, respectively, as seen in Fig. 3c. Here, ${\rho }_{{{\rm{STR}}}}({\phi }_{{{\rm{E}}}})$, shown in Fig. 3d, only depends on ϕ_E and exhibits a π-periodicity with respect to ϕ_E. These observations indicate the presence of the symmetric transverse resistivity effect in our measurements (see Section IV of Supplementary Materials for a detailed analysis of STR), which describes the projection of the anisotropic magnetoresistance effect into transverse resistivity³⁶. On the other hand, ρ_PHE exhibits a sinusoidal dependence on (ϕ_B − ϕ_E) with π-periodicity. It can be accurately fitted using ${\rho }_{{{\rm{PHE}}}}({\phi }_{{{\rm{B}}}}-{\phi }_{{{\rm{E}}}})={\rho }_{{{\rm{0}}}}^{{{\rm{PHE}}}}\sin (2{\phi }_{{{\rm{B}}}}-2{\phi }_{{{\rm{E}}}})$, as seen in Fig. 3e, indicating the underlying AMR origin of PHE. More broadly, the detection and characteristics of the STR and PHE in our measurements are consistent with theoretical expectations for a ferromagnetic metal with an in-plane magnetization component, as discussed above. Finally, the regular periodicity of ρ_IPHE(ϕ_B,  ϕ_E) and ρ_sym(ϕ_B − ϕ_E,  ϕ_E) suggests that the effect of magnetic domains, if present, on the in-plane Hall response is negligible.

Discussion

Origin of the in-plane Hall effect

The observation of the anomalous IPHE in spin-canted Fe₃Sn in our measurements agrees with our theoretical analyses, which predict a finite in-plane Hall response induced by topological Weyl points in the kagome band structure when the relevant magnetic symmetry (${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$) is broken. Performing measurements as a function of the magnetic field amplitude, we can detect a 2π-periodic IPHE in fields as small as B_∥ = 5 mT, as shown in Fig. 4a. Moreover, we find that the IPHE amplitude remains nearly field independent at B_∥ < 100 mT. This observation is consistent with our theoretical proposal, shown in Fig. 1f, that B_∥ only modulates ρ_IPHE(ϕ_B) through Zeeman coupling as a function of ϕ_B, while ${{{\mathcal{M}}}}_{{{\rm{z}}}}t{{\mathcal{T}}}$ is already broken by the canting of the magnetic moments rather than by a strong in-plane magnetic field. Indeed, we also detect a finite transverse resistivity at zero magnetic field, such as ρ_xy ≈ 140 nΩcm at ϕ_B = π where ρ_IPHE has the maximum amplitude. Note that the quantitative assessment of the IPHE contribution to this value is challenging. An unavoidable, yet small, misalignment of our Hall bar electrodes with the direction of the magnetization M results in a finite contribution of the symmetric transverse resistivity to ρ_xy. Therefore, a symmetrization with respect to the magnetic field at B ≠ 0 is needed to extract the contributions of ρ_IPHE(ϕ_B) (antisymmetric in B), as well as ${\rho }_{{{\rm{STR}}}}({\phi }_{{{\rm{E}}}})$ and ρ_PHE(ϕ_B, ϕ_E) (both symmetric in B) from ρ_xy.

Fig. 4: Origin of the anomalous in-plane Hall effect in Fe₃Sn.

a Shown is the in-plane Hall resistivity ρ_IPHE of device D1 plotted as a function of ϕ_B recorded at ϕ_E = 0 and a temperature T = 300 K as well as at different amplitudes of the in-plane magnetic field B_∥ (see legend). The inset shows the amplitude of ρ_IPHE at ϕ_B = 0 plotted as a function of B_∥ as extracted from the data in the main panel. b Shown is the temperature (T) dependence of the anomalous in-plane Hall conductivity σ_IPHE of device D1 recorded at B_∥ = 50 mT, ϕ_B = 0, and ϕ_E = π. The inset displays the temperature dependence of the corresponding longitudinal conductivity σ_xx(T). c Shown is the dependence of σ_IPHE on σ_xx of the data shown in (b). The different temperature ranges and conductivity regimes are indicated. d Shown is the ϕ_B dependence of the magnetic field-antisymmetric contribution V_IPHE(ϕ_B) to the measured transverse voltage V_xy(ϕ_B) recorded on devices D1 (d₁ = 60 nm) and D2 (d₂ = 30 nm) at T = 300 K, B_∥ = 20 mT, and ϕ_E = 0. Data are shown as symbols, and fits to the data are shown as solid lines. The fitted voltage amplitudes are V_IPHE(60 nm) = (3.4 ± 0.1) μV and V_IPHE(30 nm) = (8.7 ± 0.2) μV.

In the presence of a small misalignment of the sample plane with the direction of B_∥, the OHE Hall effect can generally contribute to ρ_IPHE, because both the IPHE and the OHE are odd under magnetic field reversal, as discussed above (also see Table 1 for a detailed comparison). Given an alignment accuracy of our sample plane with respect to the magnetic field axis of equal or better than 2°, an in-plane magnetic field amplitude B_∥ = 20 mT could cause an out-of-plane component of the magnetic field B_z < 0.7 mT. However, we can safely exclude measurable contributions of the OHE to the experimentally detected ρ_IPHE(ϕ_B) for two key reasons. First, temperature (T)-dependent measurements show that the anomalous in-plane Hall conductivity σ_IPHE(T) is temperature independent over a wide temperature range from 100 K to 300 K, as seen in Fig. 4b. This observation is inconsistent with a temperature dependence reported on the Hall coefficient of Fe₃Sn within the same temperature range³⁵. Second, the small Hall coefficient of Fe₃Sn at 300 K, R₀ ≈ −1 × 10⁻⁴ cm³C^{−1 35} would result in a vanishing voltage signal V_H < 10 nV of the OHE in our measurements when B_z < 0.7 mT; by comparison, the voltage signal of the anomalous IPHE is on the order of microvolts, as seen in Fig. 4d.

Moreover, the out-of-plane magnetization M_z of the canted moments breaks the mirror symmetry ${{{\mathcal{M}}}}_{{{\rm{x}}}}$ and permits a finite out-of-plane Berry curvature Ω_z, which is isotropic within the xy-plane. Hence, M_z can give rise to the conventional AHE with resistivity ρ_AHE that is independent of ϕ_E and ϕ_B (also see Table 1 for a detailed overview of the AHE properties). If the out-of-plane component of the magnetic field B_z is smaller than the coercive field of  ≈50 mT, as seen in Fig. 2f, it cannot switch the out-of-plane magnetization and ρ_AHE(B) = ρ_AHE(−B). In our measurements B_z < 0.7 mT. Therefore, the conventional AHE would thus appear as a ϕ_B and ϕ_E-independent contribution to the magnetic field-symmetric part ρ_sym(ϕ_B,  ϕ_E). Note that it is difficult to quantitatively determine and separate the contribution of ρ_AHE to ρ_sym due to the large amplitude of ${\rho }_{{{\rm{STR}}}}$ and the scarce point density along the ϕ_E-axis limited by the number of Hall bar terminals.

Berry curvature contributions to the Hall conductivity are generally independent of the charge carrier scattering time. They can thus be distinguished from other contributions, such as skew scattering and side jumps^1,39,40,41, which depend on the scattering time. Accordingly, Berry curvature contributions to the Hall effect can be identified by analyzing the temperature dependence of the anomalous Hall conductivity. To examine this relation, we characterize the IPHE as a function of the temperature. The resulting anomalous in-plane Hall conductivity σ_IPHE(T,  ϕ_B = 0,  ϕ_E = π ), as well as the longitudinal conductivity σ_xx(T) are shown in Fig. 4b. As can be seen, the amplitude of σ_IPHE is temperature independent at T ≥ 100 K but increases at T < 100 K. These characteristics contrast with those of σ_xx(T), which depends on T throughout the examined temperature range. The experimental observation of a temperature-independent contribution $| {\sigma }_{{{\rm{IPHE}}}}^{0}| \approx 0.68\,{{\rm{S}}}\,{{{\rm{cm}}}}^{-1}$ to the IPHE suggests that the Hall response at T ≥ 100 K is dominated by an intrinsic Berry curvature contribution¹, consistent with our expectations from symmetry analyses and model calculations. Moreover, the experimentally detected $| {\sigma }_{{{\rm{IPHE}}}}^{0}| \approx 0.7\,{{\rm{S}}}\,{{{\rm{cm}}}}^{-1}$ is also in relatively good quantitative agreement with the calculated in-plane Hall conductivity $| {\sigma }_{{{\rm{IPHE}}}}^{{{\rm{calc}}}}| \approx 1.5\,{{\rm{S}}}\,{{{\rm{cm}}}}^{-1}$, as shown in Fig. 1f. At lower temperatures, the scaling σ_IPHE(T) ∝ σ_xx, as seen in Fig. 4c, indicates that the Hall conductivity is dominated by skew-scattering contributions^1,39,40.

Breaking inversion symmetry at the Fe₃Sn/Pt(111) interface could induce a canting of the Fe magnetic moments near the interface via the Dzyaloshinskii-Moriya (DM) interaction⁴². To test whether the experimentally observed canting in our samples originates from this effect, we examined in the in-plane Hall response of thin films with a smaller thickness (d₂ = 30 nm). If the IPHE arises from interfacial symmetry-breaking effects, the corresponding antisymmetric part V_IPHE(ϕ_B) of the Hall voltage V_xy(ϕ_B) is independent of the film thickness. The V_IPHE(ϕ_B,  ϕ_E = 0) of devices D1 (d₁ = 60 nm) and D2 (d₂ = 30 nm) are shown in Fig. 4d. Applying a sine function fit as before, we find V_IPHE(30 nm)/V_IPHE(60 nm) ≈ 2.5, i.e., V_IPHE scales approximately with the thickness of the film. Since ρ_xy ∝ V_xyd (d is the thickness of the film), it follows that ρ_IPHE(ϕ_B,  ϕ_E) and ρ_sym(ϕ_B,  ϕ_E) of devices D1 and D2 are in qualitative and approximate quantitative agreement (ρ_IPHE(ϕ_B,  ϕ_E) and ρ_sym(ϕ_E,  ϕ_B − ϕ_E) of device D2 are shown in Section IX of the Supplementary Materials).

Our results suggest that the IPHE is independent of the film thickness, and thus arises from spin canting within the bulk of the film. Since our films are comparably thick (d ≤ 60 nm), it is difficult to reconcile the fact that epitaxial strain at the Pt(111)-Fe₃Sn interface gives rise to the detected canting effect. Although it remains an interesting question to what extent this canting is generally present in Fe₃Sn^31,35, it has been reported that the magnetization of the iso-structural ferromagnet Fe₃Ge undergoes a reorientation from an out-of-plane to an in-plane magnetization via a second-order transition below 380 K^43,44. It is conceivable that such a transition is present in Fe₃Sn, leading to finite out-of-plane canting at finite temperatures. Thus, it would be desirable to conduct more temperature-dependent neutron scattering measurements on bulk crystals to obtain more detailed information on the magnetic structure of Fe₃Sn.

Tuning the in-plane Hall effect in a topological heterostructure

The amplitude of a Hall effect induced by Berry curvature generally depends on the magnitude of the underlying mechanism of symmetry breaking¹. Tuning the spin-canting in Fe₃Sn through external control parameters would thus allow the control of the anomalous IPHE. We will show that the magnetic stray field of ferromagnetic CoFeB layers augments the IPHE amplitude via the coupling to the magnetic moments of Fe. To this end, we realize a topological heterostructure structure comprising of TiO₂ (3 nm)/CoFeB (20 nm)/Al₂O₃ (20 nm)/Fe₃Sn (30 nm)/Pt(111) (5 nm) layers deposited on sapphire(0001), as schematically shown in Fig. 5a (see “Methods” section for fabrication details). The electrically isolating Al₂O₃ spacer prevents electric shunting of Fe₃Sn by the metallic CoFeB. The successful fabrication of the topological heterostructure is confirmed by scanning transmission electron microscopy and energy-dispersive X-ray spectroscopy measurements shown in Fig. 5b–d (see “Methods” section for measurement details). In addition, we performed a detailed characterization of the crystallographic and magnetic properties of the CoFeB films, as shown in Section X of the Supplementary Materials. These measurements indicate that CoFeB exhibits an amorphous crystal structure with a tilted easy-plane magnetic anisotropy. Hence, the CoFeB films of device D2hs are expected to have a magnetic stray field with a component in the z direction.

Fig. 5: Tuning the anomalous IPHE in a topological heterostructure.

a Shown is a schematic overview of the layer structure and chemical composition of the topological heterostructure device D2hs. b Shown is a corresponding cross-sectional scanning transmission electron microscopy (STEM) image of a 100 nm × 100 nm area D2hs. The scale bar corresponds to a length of 25 nm. The individual layers of the multilayer structure can be distinguished by their different black-white contrast. c, d display the elemental distribution of iron and cobalt, respectively in the same field of view as b recorded using energy-dispersive X-ray spectroscopy (EDS) with the STEM. A more detailed characterization of the topological heterostructure is presented in Section X of the Supplementary Materials. e Shown is the ϕ_B-dependence of the anomalous in-plane Hall resistivity ρ_IPHE (symbols) and fit to the data (solid lines) recorded on device D2 and the heterostructure device D2hs at T = 300 K, B_∥ = 20 mT, and ϕ_E = 0. The fitted amplitudes are ${\rho }_{{{\rm{IPHE}}}}^{0}(\,{\mbox{D2}})=(31.2\pm 0.7)\,\,{\mbox{n}}\,\Omega \,\,{\mbox{cm}}\,$ and ${\rho }_{{{\rm{IPHE}}}}^{0}(\,{\mbox{D2hs}})=(43.3\pm 0.7)\,\,{\mbox{n}}\,\Omega \,\,{\mbox{cm}}\,$. The heterostructure and the effect of the magnetization M of CoFeB on the magnetization of Fe₃Sn via its magnetic stray field are schematically indicated.

To maintain direct comparability of the measured IPHE amplitudes, we deposited the (TiO₂/CoFeB/Al₂O₃)-layer structure directly on the circular Hall bar device D2 using a two-step lithography process. We then recorded the electric in-plane Hall response of device D2hs (hs - heterostructure). The resulting ρ_IPHE(ϕ_B) of the devices D2hs and D2 at ϕ_E = 0 are shown in Fig. 5e. We find that the amplitude of the IPHE is increased by approximately 38% after the deposition of the CoFeB film whereas the angular characteristics of the IPHE remain unchanged, as can be seen from the full angle dependence of ρ_IPHE and ρ_sym shown in Section X.D of the Supplementary Materials. On the other hand, the symmetric part ρ_sym(ϕ_B − ϕ_E,  ϕ_E) of the Hall resistivity of device D2hs and D2 are in qualitative and quantitative agreement. At a basic level and disregarding effects from the microscopic magnetic structure of CoFeB, these observations suggest that the CoFeB layer leaves the in-plane magnetic structure of Fe₃Sn invariant, but enhances the out-of-plane canting effect through the coupling of the magnetic moments of Fe to its magnetic stray field. Note that such enhanced canting could increase the amplitude of the conventional AHE. However, we do not detect a measurable ϕ_B- and ϕ_E-independent background of ρ_sym(ϕ_B − ϕ_E,  ϕ_E). We anticipate that further optimization of the heterostructure layers and the use of other kagome materials^11,45 could enhance the IPHE amplitude even further beyond what was detected in this proof-of-concept experiment.

To conclude, we report the experimental observation of the anomalous IPHE in epitaxially grown thin films of the Weyl ferromagnet Fe₃Sn at room temperature. The combination of the kagome lattice motif with spin-orbit coupling and ferromagnetism imparts a topological electronic structure³² and canting of the magnetic moments of Fe breaks the relevant crystalline symmetry for the anomalous IPHE to appear. Meanwhile, strong ferromagnetic exchange interactions J ≈ 40 meV between the transition metal atoms arranged on the kagome lattice²³ enable the room temperature observation of this phenomenon. The temperature-independent anomalous in-plane Hall conductivity detected over a wide temperature range (100–300 K) provides strong evidence for the topological origin of the anomalous IPHE, consistent with the presence of topological Weyl points near the Fermi energy. Our work also presents a technical advance. We demonstrate that the combination of circular Hall bar structures with magnetic field-angle-dependent measurements and suitable symmetrization operations can be used to distinguish different contributions to the transverse voltage, such as IPHE, AHE, OHE, and anisotropy effects (PHE and STR). More broadly, establishing the thin-film heteroepitaxy of topological kagome magnets and heterostructures, our work establishes a design paradigm to examine new electric Hall effects^2,3,4,5,9,10 that result from crystalline symmetry breaking toward their use in technological applications, such as magnetic sensing, spintronics, and energy harvesting.

Methods

Molecular beam epitaxy of Fe₃Sn films

Epitaxial Fe₃Sn films were grown on top of a 5 nm thick Pt(111) buffer layer on [0001]-oriented 5 mm × 5 mm × 0.5 mm sized sapphire substrates (from CrysTec GmbH) using a home-build molecular beam epitaxy (MBE) system (effusion cells from MBE Komponenten GmbH). The as-received sapphire substrates were chemically cleaned using sonication in acetone and isopropyl alcohol (IPA) for 5 min each. Atomically flat surfaces with single-step height terraces were obtained by thermal annealing in air at 1300 K for 1 h using a tube furnace. Prior to the thin-film deposition, the substrates were degassed inside the MBE chamber at 900 K at a base pressure p ≤ 5 × 10⁻¹⁰ mbar. Adopting the recipe of Cheng et al.⁴⁶, high purity Pt(99.999%) was evaporated using an electron beam evaporator (EFM4 from Focus GmbH) at an approximate rate of 0.2 Å/min to obtain a 5 nm thick Pt layer oriented along the [111]-direction. The initial 6 Å of Pt were deposited at a substrate temperature of 740 K. The remaining 4.4 Å were deposited while the substrate temperature was reduced from 440 K to 380 K, followed by annealing at 600 K for 10 min. Onward, high-purity Fe(99.999%) and Sn(99.999%) were co-evaporated using effusion cells at an approximate growth rate of 1.2 Å/min to obtain Fe₃Sn films. The structural and chemical characterization of the fabricated thin films is described in Section V of the Supplementary Materials.

Device fabrication and deposition of a topological heterostructure

The as-grown Fe₃Sn thin films were shaped into 12-terminal circular Hall bar devices using Ar⁺ ion milling and optical UV lithography. The electrical contacts to the Hall bars were fabricated by evaporating 5 nm/100 nm titanium/gold electrodes. The heterostructure device D2hs was prepared directly on top of the Fe₃Sn circular hall bars of device D2 using a UV lithography mask. An e-beam evaporator (Innovative Vacuum System) was used to deposit 20 nm of amorphous Al₂O₃. In the next step, a 20 nm thick amorphous CoFeB layer was deposited using DC magnetron-sputtering (AJA International Ltd.) at a base pressure of 1 × 10⁻⁸ torr with a 3 mtorr partial pressure of argon and a d.c. power of 60 W. Following CoFeB deposition, a 3 nm thick titanium (Ti) capping layer was sputter-deposited to protect the heterostructure from oxidation, where the Ti layer naturally oxidizes into TiO₂ via exposure to an ambient atmosphere. The material characterization of the CoFeB films is described in Section IX of the Supplementary Materials.

Electric transport measurements

All electric transport measurements were carried out in a Quantum Design PPMS 6000 system using a four-probe contact geometry. The circular Hall bar devices were wire bound to the chip holder using 25 μm thick aluminum wire. An a.c. bias current of $I={I}_{{{\rm{0}}}}\sin (\omega t)$ with a peak amplitude of I₀ = 1.1 mA at a frequency of f = ω/2π = 19.375 Hz was applied by using an a.c. current source (6221A, Keithley). Two lock-in amplifiers (SRS 830, Stanford Research Systems), which are phase-matched to the bias current output, were used to record the longitudinal V_xx and transverse V_xy voltages as a function of the azimuthal angles ϕ_E and ϕ_B. As illustrated in Fig. 2d, ϕ_E and ϕ_B are defined as the angles between the direction of the electric field E and the x-axis and the direction of the in-plane magnetic field B_∥ and the x-axis, respectively. Within this definition, ϕ_E can be varied by applying the bias current along different contact pairs of the circular Hall bar, and ϕ_B can be independently tuned from ϕ_B = 0° to ϕ_B = 350° by rotating the sample within an externally applied uniaxial field using a commercial rotator probe by Quantum Design. The travel limitation of the rotator probe to ϕ_B ≤ 350° prevents a complete 2π rotation of ϕ_B and causes the missing data points seen in Fig. 3. The longitudinal and transverse resistivities were obtained by using the relations ρ_xx = (d_1,2L_y/L_x)(V_xx/I_x) and ρ_xy = (d_1,2L_y/L_x)(V_xy/I_x) with the device dimension, L_x = L_y = 200 μm and thicknesses d₁ = 60 nm and d₂ = 30 nm, respectively.

Magnetization measurements

Measurements of sample magnetization were performed using a vibrating sample magnetometer (MPMS3 from Quantum Design, resolution  <10⁻⁸ emu). The samples were attached to standard quartz or brass holders using GE varnish. All measurements were performed at a temperature of 300 K. The magnetic moment per Fe atom was calculated considering the film thickness, as determined from X-ray reflection measurements, and the effective sample area ${A}^{{\prime} }$. ${A}^{{\prime} }$ is smaller by 4 mm² than the actual surface area A = 5 × 5 mm² of the sapphire substrate due to the four corner clamps used to mount the substrate to the sample holder of the molecular beam epitaxy system. A linear diamagnetic background, as seen in the inset of Fig. 2c, caused by the sapphire substrate, was subtracted from the magnetization data. Note that this diamagnetic contribution (magnetic moment m ≈ 1.5 × 10⁻⁴ emu at B = 1 T) dominates over the paramagnetic contribution of the Pt(111) buffer layer, which is estimated to be on the order of 10⁻⁸ emu at B = 1 T.

Scanning transmission electron microscopy measurements

Lamellas suitable for carrying out scanning transmission electron microscopy (STEM) measurements were prepared using a focused ion beam system (FEI Helios G4 UX). To protect the heterostructure from potential damage caused by the Ga⁺ ion beam, an amorphous platinum (Pt) layer was deposited on the surface of the sample. Following the ion milling process, the as-prepared lamella was transferred to a copper grid. Cross-sectional STEM measurements were performed using a commercial JEOL JEM-ARM200F system. Using an acceleration voltage of 60 kV and a probe corrector, an image resolution below 0.1 nm was achieved. Energy-dispersive X-ray spectroscopy measurements were performed to map out the elemental distribution in an area measuring 100 μm × 100 μm.

Density functional theory calculations

Density functional theory (DFT) calculations of the electronic structure of Fe₃Sn were performed using the density functional theory framework as implemented in the Vienna ab initio simulation package^47,48. The projector-augmented wave potential was adopted with the plane-wave energy cutoff set to 600 eV (convergence criteria 10⁻⁶ eV). The exchange-correlation functional of the Perdew-Burke-Ernzerhof type was used⁴⁹ with a 11 × 11 × 13 gamma-centered Monkhorst-Pack mesh. We constructed the Hamiltonian of a Wannier tight-binding model using the WANNIER90 interface⁵⁰, including the Fe d-and Sn p-orbitals. We then calculated the anomalous Hall conductivity from this Wannier model using a 131 × 131 × 141 k-mesh and the WannierBerri code⁵¹. The surface states are calculated with the surface Green’s function for the semi-infinite system⁵².

Anomalous in-plane Hall effect calculated using the Wannier tight-binding model

To analyze the characteristics of in-plane Hall effect (IPHE) under an in-plane magnetic field B_∥, we computed the anomalous in-plane Hall conductivity σ_IPHE using the Wannier tight-binding model projected by DFT calculations. The effective Hamiltonian is given by:

$$H={H}_{0}+g{{{\bf{B}}}}_{\parallel }\cdot {{\boldsymbol{\sigma }}},$$

(1)

where H₀ refers to the Wannier Hamiltonian of Fe₃Sn with or without canting, the second term denotes the Zeeman coupling with ${{{\bf{B}}}}_{\parallel }={{\rm{B}}}\left(\cos {\phi }_{{{\rm{B}}}},\,\sin {\phi }_{{{\rm{B}}}},\,0\right)$, with ϕ_B as the angle between the in-plane magnetization of Fe₃Sn and B_∥, g as the effective coupling factor, and σ as the Pauli matrices for spin. We calculate σ_xy ≡ σ_IPHE using the effective model through the Kubo formula^1,53:

$${\sigma }_{{{\rm{xy}}}}=\frac{e}{\hslash }\sum _{{{\rm{n}}}}{\int}_{{{\rm{BZ}}}}\frac{{{{\rm{d}}}}^{3}{{\bf{k}}}}{{(2\pi )}^{3}}{f}_{{{\rm{n}}}}({{\bf{k}}}){\Omega }_{{{\rm{n}}},{{\rm{xy}}}}({{\bf{k}}}),$$

(2)

$${\Omega }_{{{\rm{n}}},{{\rm{xy}}}}({{\bf{k}}})=2i{\hslash }^{2}\sum _{{{\rm{m\ne n}}}}\frac{\left\langle {\psi }_{n{{\bf{k}}}}\right\vert {\hat{v}}_{{{\rm{x}}}}\left\vert {\psi }_{{{\rm{m}}}{{\bf{k}}}}\right\rangle \left\langle {\psi }_{m{{\bf{k}}}}\right\vert {\hat{v}}_{{{\rm{y}}}}\left\vert {\psi }_{{{\rm{n}}}{{\bf{k}}}}\right\rangle }{{({E}_{{{\rm{n}}}{{\bf{k}}}}-{E}_{{{\rm{m}}}{{\bf{k}}}})}^{2}},$$

(3)

where ${f}_{{{\rm{n}}}}\left({{\bf{k}}}\right)$ denotes the Fermi-Dirac distribution, ψ_nk and E_nk refer to the Bloch wave functions and eigenvalues of band n, respectively, and $\hat{v}$ is the velocity operator. The calculated σ_IPHE using gB_∥ = 0.001 eV is shown in Fig. 1f. As expected from our symmetry analysis, σ_IPHE vanishes without canting, while it exhibits a finite amplitude with 2π periodicity with respect to ϕ_B when canting is included. Specifically, in the absence of canting, the magnetic space group symmetry ${{{\mathcal{M}}}}_{z}t{{\mathcal{T}}}$ is preserved under an in-plane magnetic field, resulting in opposite values of Ω_n,xy(k) at symmetry-related reciprocal points: ${\Omega }_{{{\rm{n}}},{{\rm{xy}}}}\left({{\bf{k}}}\right)=-{\Omega }_{{{\rm{n}}},{{\rm{xy}}}}\left({{{\mathcal{M}}}}_{z}t{{\mathcal{T}}}\cdot {{\bf{k}}}\right)$. Thus, σ_IPHE vanishes without canting, as it involves integrating Ω_xy over the reciprocal space for all symmetry-determined relations of the Berry curvature (see Section I of Supplementary Materials). Meanwhile, when canting is present, ${{{\mathcal{M}}}}_{z}t{{\mathcal{T}}}$ is broken, allowing for a non-zero σ_IPHE based on symmetry considerations.