Observation of Omnidirectional Exchange Bias at All‐Antiferromagnetic Polycrystalline Heterointerface

Due to promising functionalities that may dramatically enhance spintronics performance, antiferromagnets are the subject of intensive research for developing the next‐generation active elements to replace ferromagnets. In particular, the recent experimental demonstration of tunneling magnetoresistance and electrical switching using chiral antiferromagnets has sparked expectations for the practical integration of antiferromagnetic materials into device architectures. To further develop the technology to manipulate the magnetic anisotropies in all‐antiferromagnetic devices, it is essential to realize exchange bias through the interface between antiferromagnetic multilayers. Here, the first observation on the omnidirectional exchange bias at an all‐antiferromagnetic polycrystalline heterointerface is reported. This experiment demonstrates that the interfacial energy causing the exchange bias between the chiral‐antiferromagnet Mn3Sn/collinear‐antiferromagnet MnN layers is comparable to those found at the conventional ferromagnet/antiferromagnet interface at room temperature. In sharp contrast with previous reports using ferromagnets, the magnetic field control of the unidirectional anisotropy is found to be omnidirectional due to the absence of the shape anisotropy in the antiferromagnetic multilayer. The realization of the omnidirectional exchange bias at the interface between polycrystalline antiferromagnets on amorphous templates, highly compatible with existing Si‐based devices, paves the way for developing ultra‐low power and ultra‐high speed memory devices based on antiferromagnets.


Introduction
The study of exchange coupling through an interface of magnetic nanostructures has propelled the advance of spintronics and its related technologies.One of the typical examples is the exchange bias (EB) that arises at the interface between magnetic particles or thin films.4] In spintronic devices, active elements have been restricted to FMs because they exhibit large responses linearly proportional to their magnetization.[11] The zero or negligibly small magnetization of AFMs generally makes it very challenging to control and detect their magnetic textures, having hindered the application of AFMs as functional elements in spintronic devices.16][17][18][19][20][21][22][23] Among such functional AFMs, Mn 3 Sn is one of the beststudied AFMs possessing magnetic and spintronic functionalities similar to FMs. [14][15][16]24] Despite its vanishingly small magnetization, Mn 3 Sn shows various large responses, including the anomalous Hall effect (AHE), [14][15][16]24,25] the anomalous Nernst effect (ANE), [26,27] and the magneto-optical Kerr effect [28] owing to its unique magnetic structure combined with the topological Weyl semimetallic state.[29] Mn 3 Sn has the hexagonal D0 19 structure with the ABAB stacking sequence of the kagome planes along the [0001] direction (Figure 1a, left).The three-sublattice Mn moments lie in the kagome plane and form the anti-chiral antiferromagnetic (AF) order characterized by the ferroic order-ing of cluster magnetic octupoles (Figure 1a, right) below the Néel temperature T N ≈ 420 K. [15,30,31] The Mn moments slightly cant and produce a spontaneous uncompensated FM moment of ≈0.003 μ B /Mn within the (0001) plane, allowing the magnetic field control of the AF ordering.[15] In Mn 3 Sn, the octupole ordering is the order parameter breaking the time-reversal symmetry macroscopically, producing the large transverse responses described above.[31] It is thus the polarized direction of the magnetic octupole, not the canting moment, that determines the distribution of the Berry curvature in momentum space, acting as an effective magnetic field and giving rise to the gigantic AHE and ANE.[15,26,27,29,31] The recent success in microfabrication and high-quality film fabrication of Mn 3 Sn has triggered extensive research in the nanoscale regime, leading to significant discoveries, including the observation of the novel magnetic spin Hall effect [32][33][34] and the electrical control of the chiral AF order by the spin-orbit torque in the Mn 3 Sn/heavy metal heterostructures.[35][36][37] Among them, the notable advance is the observation of the tunnel magnetoresistance (TMR) effect in Mn 3 Sn/MgO/Mn 3 Sn multilayered devices, [38] the first case based on purely antiferromagnetic electrodes. Following th historical development of ferromagnetic devices, [39,40] the next step toward memory applications requires the demonstration of the interlayer coupling and EB effect capable of controlling the local magnetic state in multilayers.In fact, the EB effect has been reported at the epitaxial heterointerface between similar AFM Mn 3 Pt and the pinning AFM MnPt, contributing to the observation of the TMR effect.[41] On the other hand, for the industrial application and its compatibility with Si-based electronics, the fabrication of the polycrystalline AFM-based heterostructure on amorphous substrates will be the mainstream of research and development.
Here we report the first observation of the exchange bias at the interface between polycrystalline AFMs fabricated on a Si/amorphous-SiO 2 substrate.[44][45] The observed properties, such as the coupling energy and temperature dependence, are similar to those in conventional FM/AFM bilayers.However, contrary to the previous studies on the EB effects, our observation confirms the omnidirectional control of the EB field and indicates that the chiral AF order without a demagnetizing field is responsible for the effect. [46]These findings pave the way for developing antiferromagnetic spintronic devices.

Characterization of the Film
Figure 1c shows the X-ray diffraction (XRD) pattern of the Mn 3 Sn(30)/MnN (30) bilayer on the Si/SiO 2 (500) substrate.All peaks are identified as signals from Mn 3 Sn, MnN, or the substrate, and no peaks due to other phases such as Mn 3 Sn 2 or Mn 3 N 2 are observed.The spectra reveal the polycrystalline character of the Mn 3 Sn and MnN layers.To verify the magnetic and transport properties of Mn 3 Sn, the magnetic field dependence of the Hall resistivity and the anomalous Nernst voltage of the Mn 3.06 Sn 0.94 (30)/MnN (30) film are measured at T meas = 300 and 320 K, respectively (Figure 1d).The Hall resistivity is measured with a perpendicular magnetic field B meas, ⊥ and in-plane read current I, and the Nernst voltage is measured with an in-plane magnetic field B meas, ║ and perpendicular heat current Q of 1 W (Figure 1e, Experimental Section).Both the AHE and ANE measurements show clear hysteric behaviors with relatively large coercivity (B c = 1.0-1.5 T) around room temperature.Figure 1f shows the magnetization of the Mn 3.06 Sn 0.94 (30) monolayer as a function of the perpendicular B meas, ⊥ at T meas = 300 K (black), and its spontaneous component estimated by subtracting the B linear component, mainly from the diamagnetism of the Si substrate (blue).Spontaneous magnetization at B meas, ⊥ = 0 is found to be ≈4 emu cm −3 (≈9 mμ B /Mn), comparable to the reported values for bulk samples and polycrystalline films. [15,25]The temperature dependence of the signals is shown in Figure S1 (Supporting Information).

Perpendicular Exchange Bias
Let us first show our main results on the perpendicular exchange bias in our polycrystalline all-antiferromagnetic multilayer films.In this paper, T FC and B FC denote the highest temperature in the field cooling (FC) process and the cooling field, respectively.Figure 2a shows the field dependence of the Hall resistivity of Mn 3.06 Sn 0.94 (30)/MnN(30) at T meas = 300 K after the zero-field cooling (ZFC) and the FC processes with the perpendicular magnetic field.The Hall loops between ±9 T are shown in Figure S2a (Supporting Information).Notably, the exchange bias is found through a horizontal shift of the Hall resistivity in the negative(positive) direction from that for the ZFC process with a positive(negative) cooling field.This EB is stabilized in the FC process from T FC = 400 K with the perpendicular cooling field B FC, ⊥ = ±9 T. The bias field B ex, ⊥ is estimated to be ≈0.05T as B ex, ⊥ = (B 1 + B 2 )/2 using the two magnetic fields at which the Hall voltage becomes zero (B 1 and B 2 in Figure 2a).[4] Here, M Mn 3 Sn denotes the magnetization and t Mn 3 Sn the thickness of the Mn 3 Sn layer, respectively.Note that the interfacial roughness between Mn 3 Sn and MnN is relatively large (Figure S3, Supporting Information), leaving room for further enhancement in the interfacial energy.Although t Mn 3 Sn of 30 nm is relatively thick compared with the conventional case using FMs (typically a few nanometers), the relation for the interfacial energy is expected to be still valid due to its long exchange length (Table S1, Supporting Information).These results demonstrate that the EB effect can be obtained in the chiral-AFM/collinear-AFM multilayer in which the polycrystalline form of the chiral AFM is the functional layer.
As discussed above, Mn 3 Sn has the anti-chiral AF order that macroscopically breaks the time-reversal symmetry, resulting in large transverse responses beyond the empirical law of the magnetization scaling confirmed in the conventional FMs. [15,47]herefore, the magnetic octupole can be controlled with a negligibly small demagnetizing field.In other words, the anti-chiral AF order can be aligned with the direction of an external magnetic field without the influence of the shape anisotropy. [46]In sharp contrast to the ferromagnetic case, the absence of the shape anisotropy in Mn 3 Sn enables the perpendicular EB without taking advantage of the interfacial anisotropy.
The Mn 3 Sn composition dependence of B ex, ⊥ is evaluated and found to be almost constant (Table S2, Supporting Information) in the composition range of Mn 3.02 Sn 0.98 -Mn 3.14 Sn 0.86 , where the D0 19 structure is stable at room temperature. [15,48]In the following, we employ Mn 3.14 Sn 0.86 as a representative composition and use "Mn 3 Sn" to refer to it for clarity.
In contrast with the EB seen in FM/AFM multilayers, a distinct feature is seen in the cooling field B FC, ⊥ dependence of the bias field (Figure 2b).We find that B ex, ⊥ systematically decreases with increasing B FC, ⊥ after saturating at B FC, ⊥ ≈ 1.5 T and the sample cooled without a field after 9 T was applied at 400 K shows the maximum bias field.The saturation field of 1.5 T corresponds to the field sufficient to align magnetic domains of the chiral AF order at the T FC of 400 K (Figure S2b, Supporting Information). [49,50]Further increase in B FC, ⊥ should enhance the magnetization with the canting of the Mn moments composing the magnetic octupole order. [15,16,25,46]In contrast, the clear saturation of B ex, ⊥ indicates that the EB is controlled by the chiral AF order characterized by the magnetic octupole, not by the canting moment in our chiral-AFM/collinear-AFM multilayers; the size and direction of the uncompensated magnetization is not an important factor for the effect, which will be further justified in the discussion on the angular dependence.Note that all the data are obtained using closed loops starting at 9 T. The initial curves from B FC to this starting field of 9 T are shown in Figure S2c (Supporting Information).Cross-sectional images of the crystal structures are shown in Figure S4 (Supporting Information).
On cooling, the observed EB becomes more significant.Figure 2c shows the field dependence of the Hall resistivity at various measurement temperatures (T meas = 100-300 K), obtained after the FC process.From this measurement, the temperature dependence of B ex, ⊥ is derived (Figure 2d).B ex, ⊥ increases with decreasing T meas and reaches a remarkably large value of 0.41 T at 100 K.[6][7][51][52][53][54] The training effect, which is one of the unique features of the EB effect, is also observed (Figure S5, Supporting Information).
The observed EB must come from the interfacial coupling between Mn 3 Sn and MnN.To control the coupling, a nonmagnetic metal Ta is inserted between the Mn 3 Sn and MnN layers.This Ta insertion layer changes the coupling as a function of the Ta thickness t Ta .The t Ta dependence of the bias field after the FC process is shown in Figure 3a.In fact, the Ta layer induces an abrupt suppression of B ex, ⊥ , revealing that the EB effect is induced by the coupling through the interface between the Mn 3 Sn and MnN lay-ers.[57] Among them, the coupling with  ≤ 1 nm is thought to be a short-range coupling, a direct coupling via exchange interaction between atoms in the magnetic layers through pinholes in the nonmagnetic interlayer. [55,56]he abrupt decay of B ex, ⊥ in our system demonstrates that the interlayer coupling is induced by this short-range coupling.The presence of a small bias field of ≈5 mT in the samples with a thick Ta layer would be attributed to pinholes in the Ta layer.The absence of a bias effect in the Mn 3 Sn monolayer supports this conclusion (Figure S2d, Supporting Information).
As widely observed in FM/AFM systems, [58,59] the interfacial coupling causes an enhancement of the coercivity.Figure 3b shows the t Ta dependence of the coercivity B c, ⊥ , which slightly decreases as t Ta increases.Another measure known to gauge the uniaxial magnetic anisotropy is d n, yx /dB(B = B ex, ⊥ ) (Figure 3c), which is defined as the average of the slope of the ascending and descending curves for normalized Hall resistivity  n, yx at  n, yx = 0 μΩ cm. [43]In our case, d n, yx /dB(B = B ex, ⊥ ) varies depending on t Ta .The t Ta dependence of the coercivity and slope of the Hall loop indicates that the interlayer coupling also affects the effective uniaxial magnetic anisotropy, as well as the unidirectional anisotropy inducing the bias effect.0][61][62]

In-Plane Exchange Bias
To further demonstrate the effect of the absence of shape anisotropy, we first evaluate the in-plane EB at the Mn 3 Sn/MnN interface.Because the AHE is absent for this measurement geometry with an in-plane magnetic field, we employ the ANE [26,27,63,64] with a perpendicular heat current and an in-plane magnetic field.Figure 4a shows the in-plane field dependence of the ANE in Mn 3 Sn(30)/MnN(30) at 300 and 220 K before and after the FC process from T FC = 400 K with the in-plane B FC, ║ of 9 T. We find shifts of the switching fields in the negative field direction.In addition, B ex, ║ decreases on heating, as shown in Figure 4b.The magnitude of B ex, ║ under the in-plane field is slightly smaller than that under the perpendicular magnetic field.7]

Omnidirectional Control of the Unidirectional Anisotropy
In addition to the above case of the in-plane EB, we demonstrate the omnidirectional control of the magnetic anisotropies in the all-antiferromagnetic bilayer by changing the cooling field direction.Before going to the detailed results on the field angle dependence, let us first examine the results obtained under the in-plane B FC, || , which is perpendicular to B meas, ⊥ used for the Hall effect measurements.Figure 5a shows the perpendicular field dependence of the Hall resistivity at 300 K after the ZFC (black) and in-plane FC ( = 90°, red) processes.Here,  is defined as the angle between B FC and the film normal.In this configuration, no bias is observed in the field-cooled sample, as expected for the symmetry of the measurement setup.
On the other hand, the shape of the Hall loop is slightly modulated from the ZFC case under the influence of the in-plane FC.
Next, we measure the Hall effect while changing the direction of B FC from perpendicular ( = 0°) to in-plane ( = 90°).Figure 5b,c, respectively, show the B FC angle dependence of the perpendicular B ex, ⊥ and B c, ⊥ obtained in the Hall resistivity measurements made in the perpendicular B meas, ⊥ .We find a gradual decrease in both B ex, ⊥ and B c, ⊥ with increasing .Their behavior demonstrates the omnidirectional character of the unidirectional and uniaxial anisotropies; the axis related to the anisotropies is solely determined by the direction of B FC .
[70][71][72] The small deviation from cos shown in Figure 5b further confirms that the AF order or magnetic octupole restricted inside the kagome plane is essential for the EB effect.Namely, the direction of the bias field should be determined by the projection of B FC onto the kagome plane denoted by B eff instead of the magnetization, which is expected to be rather parallel to B FC .For simplicity, here we consider three representative kagome plane configurations: i) the kagome plane is parallel to the film normal and B FC (Figure 5d(i)); ii) the kagome plane is rotated around the film normal by 90°(Figure 5d(ii); iii) the kagome plane is perpendicular to the film normal (Figure 5d(iii)).Configuration (i) is omnidirectional in the plane and can be treated as a soft magnet without a demagnetizing field because B FC of 9 T is large enough to omnidirectionally align the magnetic order in the kagome plane.The induced anisotropy denoted by B ex can also be aligned with the direction of B FC , and its magnitude should reach the maximum value denoted by B ex, max .Therefore, B ex, ⊥ follows the cos dependence.On the other hand, in configuration (ii), the induced anisotropy should be perpendicular to the film plane for any .This case can be seen as a very hard magnet with a perpendicular magnetic anisotropy.Considering the result in Figure 2b, the size of B ex remains almost the same as B ex, max for any  with a large B FC .In the last case (configuration (iii)), no AHE should be generated, and this configuration is not directly responsible for the bias effect measured in the present case.In Figure 5b, B ex, ⊥ expected for the above three configurations is plotted.We note that B ex, ⊥ obtained experimentally in our sample falls between those for configuration (i) and (ii).⃗ n represents the normal vector of the kagome plane, and it is parallel to the y-axis, z-axis, and x-axis in configuration (i-iii), respectively.e) Schematics of the effective field B eff and bias field B ex in a kagome plane which is rotated by the angle ϕ around the z-axis from the original configuration (i), that is, the x-z plane. is the angle between B eff and the z-axis.
To further discuss our results quantitatively, we simulate the expected B ex, ⊥ using the following model.Here, among randomly oriented grains in the polycrystalline sample, we consider the grains that host the kagome planes that are perpendicularly oriented to the substrate and rotated by angle ϕ from the x-z plane as shown in Figure 5e because these grains provide the largest contributions to the AHE with the largest size of bias field in the measurement and thus determine the observed B ex, ⊥ .The direction of the effective field B eff (, ϕ) (described by the angle ) and the observable bias field in the perpendicular direction B ex, ⊥ (, ϕ) can be calculated as a function of  and ϕ using the discussion made for three distinct configurations (Details of the calculation are shown in Supporting Information.); The perpendicular bias field for B FC with the angle  can be calculated by integrating B ex, ⊥ (, ϕ) from ϕ = 0°to ϕ = 180°, and the result is plotted in Figure 5b.
This model reproduces the small deviation from the cos dependence for the omnidirectional case of configuration (i) and indicates that the bias field direction is mainly determined by the cooling field direction.This is in sharp contrast with the ferromagnetic case where the direction of the effective field, thus EB cannot be determined simply by the cooling field direction unless the effects of the cooling field well exceed the energy scale of the shape anisotropy.This concept would be more clearly confirmed in epitaxial Mn 3 Sn films.For example, the omnidirectional exchange bias effects could be designed by fabricating a specific kagome plane configuration among those discussed in Figure 5d.In the present study, the anisotropy of the EB effect contributed by MnN should be almost averaged out to vanish due to the random orientation of the MnN layer, helping the observation of the omnidirectional properties.If MnN is an epitaxial film, an angular dependence of the EB effect might be observed, following the magnetic anisotropy of MnN.Although there have been a number of studies on the angular dependence of the EB effect, [71][72][73][74][75][76][77][78][79][80] our observation of the omnidirectional EB is the first case for the EB effect in a magnetic heterointerface and is peculiar to allantiferromagnetic heterointerface.

Conclusion
In conclusion, we report our first observation of the omnidirectional exchange bias effect in an all-antiferromagnetic polycrystalline multilayer.We demonstrate that i) the obtained interfacial energy is comparable to those observed in conventional FM/AFM bilayers, ii) the bias effect is controlled by the chiral antiferromagnetic order characterized by the magnetic octupole and thus by the effective magnetic field projected onto the kagome plane, rather than the canting moment, and iii) in sharp contrast with previous reports using ferromagnets, the omnidirectional character of the exchange bias effect with the negligible demagnetizing field is observed owing to the absence of the magnetization in antiferromagnets.Our demonstration the spin texture control in the polycrystalline chiral antiferromagnet via the exchange bias lays the foundation for developing various antiferromagnetic spintronic devices, including MRAM using the antiferromagnetic tunnel junctions, [38,41] essential for ultrafast and ultra-power-efficient computing.

Experimental Section
Sample Preparation: All the depositions were conducted in a sputtering chamber with a base pressure of <5 × 10 −7 Pa.Mn 3 Sn(30 nm) films were deposited on thermally oxidized Si (Si/SiO 2 (500 nm)) substrates by DC magnetron sputtering from the Mn 2.9 Sn target at room temperature.The Ar pressure and sputtering power were 1.0-1.2Pa and 60 W, respectively.The composition of Mn 3 Sn was calibrated by the scanning electron microscope-energy dispersive X-ray spectroscopy and X-ray fluorescence (XRF) measurements, and it was between Mn 3.02 Sn 0.98 -Mn 3.14 Sn 0.86 .The films were subsequently annealed at 500 °C for 30 min.After the films were cooled to room temperature, Ta (t Ta nm) interlayer and/or MnN (30 nm) were deposited.MnN was deposited by RF reactive sputtering from the Mn target using the gas pressure of 0.5 Pa and sputtering power of 50 W.During the deposition of MnN, Ar and N 2 gas flow were 6.6 and 10 sccm, respectively.The composition of the MnN monolayer was estimated by XRF measurement and it was Mn:N = 51:49.The growth rates of each layer were determined by X-ray reflectivity measurement.The structural characterization was performed using XRD measurement equipped with a Cu K  1 ( = 1.54059Å) source.
Magnetization and Transport Measurements: The magnetization was measured using a commercial SQUID magnetometer (MPMS, Quantum Design).The Hall resistivity and angular dependence were measured by a standard four-probe method in a commercial physical property measurement system (PPMS, Quantum Design) equipped with the horizontal rotator option.The Nernst signal was measured with a sample stage equipped with a resistance heater and Cu block that worked as a heat sink (Figure S6, Supporting information).The temperature difference between the sample and the PPMS sample puck due to the heat from the heater was estimated by the longitudinal resistivity of the sample.

Figure 1 .
Figure 1.a) Crystal and magnetic structures of the chiral antiferromagnet Mn 3 Sn.The blue and yellow spheres and arrows represent the Mn atoms and their spins, respectively, and the gray and black spheres represent the Sn atoms.The orange arrow in the right figure represents the direction of the magnetic octupole.b) Schematic of the interfacial coupling between the polycrystalline chiral antiferromagnet and collinear antiferromagnet on an amorphous substrate.The green and purple spheres and arrows represent the Mn atoms and their spins in the MnN layer.c) XRD spectra for the Mn 3 Sn(30)/MnN(30) film on a Si/SiO 2 substrate.The calculated spectra for Mn 3 Sn and MnN are shown on the bottom.The numbers in the brackets right after the materials indicate the thickness in nm.d) Field dependence of the Hall resistivity (blue) and the anomalous Nernst voltage (red) of Mn 3 Sn/MnN at 300 and 320 K, respectively.A heat current of 1 W is applied in the Nernst measurement.The arrows indicate the direction of the field sweep.e) Schematics of the measurements.A perpendicular magnetic field is applied in the Hall measurement (top), while an in-plane magnetic field and perpendicular heat current are applied in the anomalous Nernst measurement (bottom).f) Field dependence of the magnetization of the Mn 3 Sn(30) monolayer under a perpendicular magnetic field at 300 K before (black) and after (blue) subtracting the linear component mainly from the diamagnetism of the Si substrate.The arrows indicate the direction of the field sweep.

Figure 2 .
Figure 2. a) Field dependence of the Hall resistivity of the Mn 3 Sn(30)/MnN(30) bilayer at 300 K after zero-field cooling (black) and the field cooling with B FC, ⊥ = +9 T (red) and B FC, ⊥ = −9 T (green) from T FC = 400 K.The arrows indicate the direction of the field sweep.The dotted lines show the positions of the magnetic fields at which the Hall resistivity becomes zero and the bias field of ≈0.05 T for the +FC loop.The inset provides a magnified figure of a part of the Hall loops around which the Hall signal becomes 0. b) Cooling field dependence of the bias field at 300 K after the field cooling from T FC = 400 K.The red point shows the value for the sample cooled from 400 K without a magnetic field after 9 T was applied at 400 K. c) Field dependence of the Hall resistivity of the Mn 3 Sn/MnN bilayer at various temperatures from 100 to 300 K after the field cooling from T FC = 400 K with B FC, ⊥ = 9 T. The arrows indicate the direction of the field sweep.d) Temperature dependence of the bias field after the field cooling from T FC = 400 K with B FC, ⊥ = 9 T.

Figure 3 .
Figure 3. a-c) Ta thickness dependence of the bias field (a), coercivity (b), and the average of the slope of the normalized Hall loop at  n, yx = 0 μΩ cm (c) of the Mn 3 Sn /Ta(t Ta )/MnN film at 300 K. T FC = 400 K and B FC, ⊥ = 9 T are used.

Figure 4 .
Figure 4. a) Field dependence of the anomalous Nernst voltage of the Mn 3 Sn/MnN bilayer at 300 K after zero-field cooling (black) and the field cooling from T FC = 400 K with the in-plane field of B FC, || = 9 T (red), and at 220 K after the field cooling (orange).The heater power of 1 W is used.The arrows indicate the direction of the field sweep.b) Temperature dependence of the in-plane bias field after the field cooling from T FC = 400 K with B FC, || = 9 T. The heater power Q = 1 W is used.

Figure 5 .
Figure 5. a) Field dependence of the anomalous Hall resistivity of Mn 3 Sn/MnN at 300 K after zero-field cooling (black) and the field cooling from T FC = 400 K with the in-plane field of B FC, || = 9 T (red).The arrows indicate the direction of the field sweep.b) Cooling field angle dependence of the bias field at 300 K measured with a perpendicular magnetic field after the field cooling from T FC = 400 K with a cooling field of B FC = 9 T tilted from the film normal by .The angle dependence of B ex, ⊥ expected for three configurations (i-iii) shown in Figure 5d, and the result of the calculation for the random kagome plane configuration in Figure 5e are also plotted.c) Cooling field angle dependence of the coercivity at 300 K after the field cooling from T FC = 400 K with a cooling field of B FC = 9 T. d) Schematics of the induced anisotropy in the measurement in Figure 5b.The cooling field B FC lies in the x-z plane as shown in the top figure.The film plane is in the x-y plane and it is shown in each kagome plane configuration.The orange arrows in each kagome plane represent the direction and size of the effective magnetic field B eff , which is the projection of the cooling field B FC on the kagome plane.The green arrows represent the direction and size of the induced anisotropy or bias field B ex in Mn 3 Sn.In configuration (i-iii), three representative kagome plane orientations are shown.⃗n represents the normal vector of the kagome plane, and it is parallel to the y-axis, z-axis, and x-axis in configuration (i-iii), respectively.e) Schematics of the effective field B eff and bias field B ex in a kagome plane which is rotated by the angle ϕ around the z-axis from the original configuration (i), that is, the x-z plane. is the angle between B eff and the z-axis.