Field‐Free Spin‐Orbit Torque Driven Perpendicular Magnetization Switching of Ferrimagnetic Layer Based on Noncollinear Antiferromagnetic Spin Source

The utilization of novel noncollinear antiferromagnetic materials holds great promise for the development of energy‐efficient spintronic devices. However, only a few studies have reported on the all‐electrical control of perpendicular magnetization switching using noncollinear antiferromagnets as the spin source, and the underlying mechanism behind the unconventional spin‐orbit torque (SOT) is still a topic of debate. In this work, deterministic perpendicular magnetization switching in Mn3Sn/CoTb bilayers is successfully achieved. Compared to the control samples with heavy metal as the spin source, the critical switching current density is over one order of magnitude reduced, indicating an enhanced efficiency of the out‐of‐plane charge‐to‐spin conversion in the textured Mn3Sn films. The influence of film thickness and growth temperature on the efficiency of different spin polarizations suggests potential roles of crystal quality and spin texture in spin diffusion with different spin polarization directions. These findings provide valuable insights into the crystal structure, spin‐orbit torque effects, and charge‐to‐spin conversion in Mn3Sn films, highlighting the importance of understanding interface and bulk contributions in antiferromagnetic spin transport phenomena.


Introduction
The spin-orbit torque (SOT) effect is a critical research frontier in the field of spintronics, arising from the spinorbit coupling of materials that act as the spin source. [1,2]This effect induces a spin current or accumulates spin at the interface of adjacent magnetic layers through a charge current.[13] However, in heavy metalbased systems, the conventional spin Hall effect demands that the charge current, spin current, and spin polarization direction be mutually orthogonal, mandating an in-plane auxiliary magnetic field to break the symmetry for driving the perpendicular magnetization switching. [14,15]As a result, power consumption and circuit complexity increase, rendering the approach impractical and emphasizing the need to explore novel physical mechanisms and material systems.[18][19] Noncollinear antiferromagnets based on hexagonal Mn 3 X (X = Ga, Ge, and Sn) are particularly important in antiferromagnetic spintronics due to their nontrivial magnetic properties.Among these compounds, Mn 3 Sn is a prime example of a noncollinear antiferromagnet.In Mn 3 Sn, the Mn atoms form a triangular sublattice in the c-plane (0001) known as the Kagome lattice, and their net magnetic moments are disclosed from a noncollinear 120°ordered chiral spin texture. [20,21]his unique spin texture leads to novel physical properties, including magnetic spin Hall effect (MSHE), [22][23][24] controllable spin polarization direction in spin current, [25,26] enhanced charge-spin conversion efficiency, [27,28] and the damping-like torque along the out-of-plane direction that can efficiently drive perpendicular magnetization switching. [29,30]ecent studies have investigated noncollinear antiferromagnetic Mn 3 Sn as either a magnetization switching layer [31][32][33][34][35][36] or a spin source layer [29,30,37] in various heterostructures.However, research on the SOT effect in Mn 3 Sn/ferrimagnet heterostructures is still limited, and unclear physical mechanisms remain, such as 1) the accurate determination of the spin diffusion length in noncollinear antiferromagnetic Mn 3 Sn material, 2) the role of self-generated spin current in the electrical manipulation of noncollinear antiferromagnetic state, particularly in polycrystalline samples, [34,38] 3) the thermal contribution to electrical manipulation of noncollinear antiferromagnetic state, particularly in thick samples, [35,36] and 4) the origin of out-of-plane polarized spin current either from chiral spin texture or spin swapping at the interface. [39,40]All of the above controversies highlight the importance of understanding the material properties of Mn 3 Sn as a precondition for any analysis of the SOT effect due to its unique spin texture.Moreover, regarding potential applications, it is noteworthy that only a few studies have reported on the all-electrical control of perpendicular magnetization switching utilizing Mn 3 Sn as the spin source. [29,30]n this work, we investigate the impact of crystal quality and film thickness on the efficiency of SOT in highly textured Mn 3 Sn films.We demonstrate the successful deterministic switching of magnetization in a perpendicular CoTb alloy layer by utilizing the textured Mn 3 Sn film as the spin source.The critical switching current density is more than one order of magnitude smaller than that of the control sample Pt/CoTb, attributed to the enhanced out-of-plane spin polarization stem from the chiral spin texture inherent to the bulk Mn 3 Sn.These results highlight the potential of noncollinear antiferromagnetic materials in developing new spintronic devices and effective control of SOT.

Structural Characterization
Figure 1a,b illustrates the crystal structure of the Mn 3 Sn compound (space group P6 3 /mmc).The Mn sublattice forms a Kagome plane, and the spin of Mn atoms exhibits a coplanar 120°chiral structure due to the geometrical frustration and the Dzyaloshinskii-Moriya interaction within two Kagome (0001) planes.In this study, Mn 3 Sn films were deposited at 600 °C on single crystal MgO (111) substrates using ultrahigh vacuum magnetron sputtering (details in the Supporting Information).The crystal structure of the Mn 3 Sn film was characterized using highresolution X-ray diffraction (XRD).As shown in Figure 1c, the out-of-plane XRD patterns display two sets of (000l) peaks corresponding to Mn 3 Sn and additional peaks originating from the MgO substrate.This observation confirms the successful growth of a c-plane-oriented Mn 3 Sn film.To determine the specific epitaxial relationship, we conducted pole figure measurements using Φ and Ψ scans with a fixed 2 angle at 39.71°for the (0002) reflection of the Mn 3 Sn film, as presented in Figure 1d.The pole figure exhibits periodic reflections at 60°intervals for Mn 3 Sn and 120°intervals for MgO, indicating the production of nontwinned, highly textured films.The peak positions further reveal that the textured film has an epitaxial relationship of MgO  (111)[002] || Mn 3 Sn (0001) [20 21], consistent with the previous results. [30]Furthermore, Figure 1e shows a cross-sectional highresolution transmission electron microscopy (HRTEM) image taken with the primary electron beam aligned parallel to the [001] direction of the MgO substrate.The image demonstrates the epitaxial growth of the Mn 3 Sn thin film with a well-defined interface to the MgO substrate.The alignment of the Mn 3 Sn unit cell along the [2 11 0] direction matches well with the MgO unit cell along the [1 10] direction, consistent with the XRD results.In Figure 1f, the energy-dispersive X-ray spectroscopy (EDS) mapping reveals the elemental composition layer by layer, represented by distinct colors.This result confirms the presence of well-defined interfaces and minimal diffusion between each layer within the entire sample stack.

Spin Torque Ferromagnetic Resonance Measurement
Next, we utilize the spin torque ferromagnetic resonance (ST-FMR) technique to investigate the current-induced SOT effect. [41,42]The samples comprise a deposited Mn 3 Sn (5 nm)/Py (5 nm) bilayer, which is subsequently patterned into a microstrip with microwave-compatible contacts to improve impedance matching.During the ST-FMR measurement, a microwave current (I rf ) sourced by a signal generator is injected along the microstrip, resulting in a torque effect that drives the magnetization precession.The anisotropy magnetoresistance (AMR) of the oscillating magnetic moment induces resistance oscillations at the same frequency as I rf , leading to the generation of a DC voltage V mix through the rectification effect.For angle-dependent measurements, we initially applied an in-plane magnetic field (H ext ) to fully saturate the magnetization of the Py layer in an arbitrary direction.The angle between I rf and H ext is defined as ϕ H .The resonance in V mix is obtained by sweeping the magnetic field across the Py resonance condition.The value of V mix is detected by a lock-in amplifier and fitted using the following equation.
where ΔH and H res represent the linewidth and resonant field, respectively.On the right side of Equation ( 1), the first term describes a symmetric component with an amplitude V s , attributed to the I rf -induced in-plane torques ( // ) as sketched in Figure 2a.
The second term describes an antisymmetric component with an amplitude V a resulting from the I rf -induced out-of-plane torques ( ⊥ ).As shown in Figure 2b, representative ST-FMR spectra for the Mn 3 Sn/Py bilayer are measured at 7 GHz with ϕ H = 45°and 225°, respectively.The final signal, labeled as V mix − V offset , results from the subtraction of the background.The resonance peaks are the sum of symmetric (V s ) and antisymmetric (V a ) Lorentzian line shapes, which can be well-fitted using Equation (1).Compared with the conventional heavy metal/ferromagnet bilayer where V mix always retains the same amplitude but changes the sign when rotating ϕ H by 180°(given by V mix (ϕ H ) = −V mix (ϕ H + 180°), it is observed that V mix of the Mn 3 Sn/Py bilayer severely deviates from the symmetry relation under positive (45°) and negative (225°) magnetic fields, indicating the presence of unconventional SOT generated by the out-of-plane polarized ( z ) spin current. [43,44]More recently, the generation of  z has been demonstrated in noncollinear antiferromagnets due to their chi-ral spin texture, which creates low magnetic symmetry while maintaining high crystalline symmetry. [25,43]To analyze the spin current with different spin polarizations that existed in our Mn 3 Sn films, angle-dependent V mix measurements were systematically conducted for the Mn 3 Sn/Py bilayer with ϕ H ranging from 0°to 360°at 10°intervals.Considering all cases of spin torques induced by the y-( y ), x-( x ), and z-( z ) component spin currents, the ST-FMR signals of the Mn 3 Sn/Py bilayer can be described using the following equations.
where S Y DL , S X DL , and A Z DL are coefficients for the damping-like torque generated by the y-( y ), x-( x ), and z-( z ) component spin current, respectively.Similarly, A Y FL , A X FL , and S Z FL correspond to the field-like spin torque components.It should be noted that the coefficient A Y FL includes the contribution of the field-like torque from both the  y spin current and the Oersted field h Oe ; However, in this context, we assume the dominance of h Oe in determining A Y FL . [37,43]As shown in Figure 2c, the angle dependence of the separated V a and V s amplitudes are plotted, which can be well fitted using Equations ( 2) and (3) to obtain the corresponding coefficients, respectively.It is also important to point out that throughout the measurement, the direction of the magnetic octupole moment in Mn  polarizations  i (i = x, y, z) is estimated by the following equations.
where e, ℏ, t source , t FM , and M s represent the elementary charge, the reduced Planck constant, the thickness of the spin source layer (Mn 3 Sn) and the ferromagnetic (Py) layer, and the saturation magnetization of the ferromagnetic layer, respectively.The effective magnetization M eff can be obtained by fitting the frequencydependent resonance field to the Kittel formula (see Figure S3, Supporting Information).Figure 2d shows the calculated  i as a function of the thickness of Mn 3 Sn in which  y and  z are clearly observable, while  x is negligible in comparison.
We now discuss the different behavior between  y and  z .We note that the value of  y increases with increasing t Mn 3 Sn , similar to the behavior observed in the Mn 3 Pd [45] and RuO 2 [46] systems, which can be described by the spin drift-diffusion model.In com-parison,  z decreases with increasing t Mn 3 Sn , indicating a potential influence from the interface effect.However, the suppression of  z may also be related to the distorted spin texture, particularly as thicker films can experience heightened strain during growth.This hypothesis is reasonable since  y ( y ) and  z ( z ) arise from different mechanisms: Bulk spin-orbit coupling and chiral spin texture, respectively, indicating the possibility of distinct spin diffusion lengths.In the subsequent discussion related to Figures 3  and 4, we will provide evidence supporting the significant role of crystal quality in enhancing  z , while excluding the contribution from interfaces.

Spin-Orbit Torque Driven Magnetization Switching
With the confirmed unconventional spin torque  z , we first demonstrate the deterministic perpendicular magnetization switching as sketched in Figure 3a.The spin source layer of the device consists of a 5 nm thick Mn 3 Sn (0001) film, responsible for generating the spin current.The upper layer comprises a ferrimagnetic CoTb layer with perpendicular magnetic anisotropy, serving as the switching layer.The injected current direction aligns the same as the depiction in Figure 2a. Figure 3b shows the Hall loop which confirms the good perpendicular magnetic anisotropy of the CoTb layer.To achieve controllable SOT for driving perpendicular magnetization switching, a series of experiments were conducted to measure the Hall resistance of the CoTb layer under different magnetic fields (H x ).
As shown in Figure 3c, the measured Hall loop exhibits hysteresis behavior and undergoes a sign change even in the absence of an applied magnetic field, providing evidence for fieldfree magnetization switching of the ferrimagnetic CoTb layer.This phenomenon arises due to the generation of an out-of-plane damping-like torque by the z-polarized spin current, effectively breaking the spatial symmetry of perpendicular magnetization.A similar result has been reported in the Mn 3 Sn/[Co/Ni] 3 multi-layer, and the sign of the z-polarized spin current can be modulated by the antiferromagnetic order under varying in-plane magnetic fields. [30]To confirm that the magnetization switching in CoTb is caused by  z rather than thermal effects, the sample is initially saturated using an in-plane field to set the antiferromagnetic order.Subsequently, the magnetic field is removed, and the Hall resistance is measured without the presence of any external field.The results are represented by the green line in Figure 3c.Similarly, the antiferromagnetic order is set to the opposite direction using an opposite in-plane field, followed by the measurement of the Hall resistance without an external field.The corresponding results are shown by the orange line in Figure 3c, clearly demonstrating opposite polarities between the two configurations.Combining these findings with those from ST-FMR, it can be concluded that  z is the primary factor responsible for field-free perpendicular magnetization switching in CoTb.
Figure 3d,e shows the switching behavior driven by SOT under different external fields.Notably, the polarity of the switching reverses upon reversing the magnetic field in the opposite direction, consistent with the typical SOT switching behavior observed in perpendicular magnetized ferromagnets. [5]The critical switching current density for the Mn 3 Sn/CoTb bilayer is found to be 1.3 × 10 6 A cm −2 , which is at least one order of magnitude smaller than that of the control sample Pt/CoTb (see Figure S6, Supporting Information), and even lower than the reported value in Mn 3 Sn/[Co/Ni] 3 multilayer. [30]It is well known that due to the ferrimagnetic nature of the CoTb layer, the critical switching current density can be further reduced compared to traditional heavy metal/ferromagnet structures. [47]While with the comparison of the Mn 3 Sn/CoTb bilayer (Figure 3d) and Pt/CoTb bilayer (Figure S6b, Supporting Information), it can be inferred that the achievement of deterministic switching and a significant reduction in critical switching current density are primarily attributed to the enhanced  z of the well-textured Mn 3 Sn films.
As highlighted in the introduction, the origin of out-of-plane polarized spin current in Mn 3 Sn is still under debate.Pioneering studies have reported the presence of a predominant odd MSHE generating an out-of-plane spin polarization, accompanied by a minor contribution from the conventional even SHE in single crystals of Mn 3 Sn. [24,26]However, recent experimental findings challenge this interpretation by suggesting that the outof-plane spin polarization is not associated with the chiral spin texture but rather arises due to spin scattering at the interface between the noncollinear antiferromagnetic thin film and the ferromagnetic layer. [40,48]We speculate that the inconsistent results observed could potentially be attributed to variations in sample quality.In comparison, we have identified three common features among the results that indicate  z (or  z ) originating from the interface rather than the bulk: 1) The  z spin current primarily corresponds to the field-like torque rather than the dampinglike torque (i.e., S Z FL ≫ A Z DL ); 2) V mix exhibits changes in amplitude while maintaining the same sign when rotating ϕ H by 180°i n angle-dependent ST-FMR measurements (i.e., the contribution of  z to V mix is not strong enough to break the symmetry relation); 3) None of the results demonstrate deterministic perpendicular magnetization switching using a noncollinear antiferromagnet as the spin source.In contrast, our key findings shown in Figures 2 and 3 significantly differ from the aforementioned observations: A Z DL ≫ S Z FL , V mix exhibits changes in both amplitude and sign, and most importantly, field-free perpendicular magnetization switching has been successfully demonstrated in the Mn 3 Sn/CoTb bilayer.
It is important to point out that the enhanced  z is expected to be strongly correlated with the highly ordered spin texture, as evidenced by the good crystallization of the samples.This relationship is further supported by the findings discussed in Figure 2d, which indicate that films with a thickness greater than 5 nm may encounter heightened strain during growth.Consequently, this strain can give rise to crystal defects, leading to the distortion of the chiral spin texture and the suppression of  z diffusion.The growth temperature may also influence the quality of crystallization, thereby impacting the spin texture.In order to demonstrate the significant role of growth temperature in enhancing  z , a control experiment was performed using a polycrystalline Mn 3 Sn film.As shown in Figure S1a,b, Supporting Information, with the growth temperature decreasing from 600 to 400 °C, the featured (000l) peaks corresponding to Mn 3 Sn almost vanish, and no periodic reflections are found in the Φ and Ψ scans.Furthermore, the HRTEM image confirms the polycrystalline nature of the Mn 3 Sn film.Additionally, evident interdiffusion between Mn 3 Sn and CoTb is observed in the EDS mapping, all the above results indicate bad crystal and interface quality due to the insufficient growth temperature.We further utilize the polycrystal Mn 3 Sn film as a spin source to drive perpendicular magnetization switching in the CoTb layer.The Hall loop shown in Figure S2a, Supporting Information confirms the good perpendicular magnetic anisotropy of the CoTb layer.Compared with the result in Figure 3b, the coercivity of CoTb on top of the polycrystal Mn 3 Sn is larger than that on the textured Mn 3 Sn, potentially attributed to the rougher surface of the polycrystalline Mn 3 Sn.The critical switching current density of the polycrystal Mn 3 (Figure S2b, Supporting Information) is approximately one order of magnitude higher than that of the textured Mn 3 Sn (Figure 3d), and nearly comparable to that of the polycrystal Pt layer (Figure S6d, Supporting Information).Most notably, the absence of deterministic switching using the polycrystalline Mn 3 Sn as the spin source suggests that an appropriate growth temperature is crucial for inducing a stable chiral spin texture in Mn 3 Sn, leading to the enhancement of  z for driving perpendicular magnetization switching.
The interfacial Dzyaloshinskii-Moriya interaction (iDMI) plays a crucial role in stabilizing chiral spin structures, such as the Neel-type domain wall (DW), which possesses a spin texture that enables highly efficient SOT-driven DW motion. [49]This interaction is instrumental in SOT-induced magnetization switching, making it a pivotal element in the design of SOT-based spintronic devices. [50]More recently, it has been reported that the broken inversion symmetry at the interface of IrMn 3 /Py, resulting from the iDMI between Mn atoms and Ni (Fe) atoms, induces spin scattering that generates  z . [48]To verify the existence of iDMI, we fabricated Mn 3 Sn/Py bilayer for Brillouin light scattering (BLS) measurements.Typical BLS spectra are depicted in Figure 4a,b.Notably, the frequency separation between the Stokes peak and the anti-Stokes peak remains nearly constant.To further investigate, we performed a series of measurements by varying the angle of incidence to modify the wave vector k x .Figure 4c illustrates the functional relationship between the iDMI-induced frequency shift Δf and k x .Significantly, Δf does not increase with k x but rather decreases, indicating the absence of iDMI in our Mn 3 Sn/Py sample.Hence, the observed enhancement of  z stem from the chiral spin texture inherent to Mn 3 Sn, rather than the Mn 3 Sn/Py interface.

Conclusion
In summary, we successfully fabricated highly textured Mn 3 Sn thin films and investigated the efficiency of charge-to-spin conversion ( i ) with different spin polarizations.Our findings reveal a significant contribution of the out-of-plane polarized ( z ) spin current to the SOT effect.The efficiency of different spin polarizations is influenced by the thickness and growth temperature of the Mn 3 Sn film, suggesting that crystal quality and strain play a potential role in spin diffusion with different spin polarization directions.Moreover, we demonstrated deterministic perpendicular magnetization switching in Mn 3 Sn/CoTb bilayers, with a significantly reduced critical switching current density compared to control samples.This reduction indicates an enhanced  z in the textured Mn 3 Sn films.Our study provides valuable insights into the crystal structure, SOT effects, and chargeto-spin conversion efficiency in Mn 3 Sn films.The presence of unconventional SOT and the potential influence of crystal quality on spin diffusion lengths highlight the importance of understanding the interface and bulk contributions in spin transport phenomena.These findings contribute to the advancement of spintronic devices utilizing noncollinear antiferromagnetic materials.

Experimental Section
Sample Preparation: Samples of Mn 3 Sn, Mn 3 Sn/CoTb, and Mn 3 Sn/Py films were deposited onto MgO (111) substrates by DC/RF magnetron sputtering under a base pressure of 1 × 10 −7 Torr.The growth temperature for Mn 3 Sn was selected as 600 °C to achieve a well-textured structure or 400 °C for a polycrystal structure.Subsequently, after cooling down to room temperature, CoTb/SiN or Py/SiN multilayers were deposited onto the Mn 3 Sn film.A 2 nm thick SiN was used as a capping layer.
Sample Characterization: The magnetic properties of the deposited films were characterized using a vibrating sample magnetometer (VSM) with a resolution of 10 −7 emu.The thickness and crystal structure analysis were performed using X-ray reflectivity (XRR) and high-resolution X-ray diffraction (XRD) techniques.A Bruker D8 Discover diffractometer with Cu K radiation ( = 0.15419 nm) was employed for this purpose.To enable cross-sectional high-resolution transmission electron microscopy (HRTEM) characterization, the sample was prepared using the focused ion beam technique (FIB).
Spin Torque Ferromagnetic Resonance (ST-FMR) Measurements: The ST-FMR signals (V mix ) were detected by a Stanford Research SR830 lock-in amplifier.For the angle-dependent ST-FMR measurements, a microwave current with a frequency of 7 GHz and power of 18 dBm was applied.
Current-Induced Magnetization Switching Measurements: The currentinduced magnetization measurements were performed using a Keithley 6221 current source and a 2182 nanovoltmeter.In each data point of the R Hall -I loop, a 200 μs pulse was applied as the write current for the Hall bar device.Subsequently, a small pulse current of 0.1 mA, lasting for 2 ms, was applied to measure the R Hall .The amplitude of the write pulse current was adjusted to obtain a complete R Hall -I loop.
Brillouin Light Scattering (BLS) Measurements: A polarized monochromatic laser beam with a wavelength of  = 532 nm was focused onto the surface of the sample at an incidence angle of .The p-polarized backscattered light was collected and directed to a Sandercock-type (3 + 3)-pass tandem Fabry-Perot interferometer.The observed Stokes and anti-Stokes peaks in the BLS spectra correspond to the creation and annihilation of magnons, respectively.The in-plane wave vector of the magnons, denoted as k x , represents the projection of the magnon wave vector in the x-direction.

Figure 1 .
Figure 1.Structural characterization of well-textured MgO//Mn 3 Sn and MgO//Mn 3 Sn/CoTb thin films at a growth temperature of 600 °C for Mn 3 Sn.a) Schematic representation of the crystal structure within a unit cell of Mn 3 Sn.b) Schematic illustration of the spin texture within a Kagome plane of Mn 3 Sn.c) Comparison of out-of-plane XRD results between the MgO//Mn 3 Sn sample and bare MgO substrate.d) Pole figure measurements using Φ and Ψ scans at a fixed 2 angle at 39.71°for the (0002) reflection of the Mn 3 Sn film.e) Cross-sectional HRTEM image taken with the primary electron beam aligned parallel to the [001] direction of the MgO substrate.f) Layer-by-layer EDS mapping of the entire sample stack.

3
Sn remains unchanged, which is confirmed by the symmetry relation of S Y DL ( H ) = −S Y DL ( H + 180 • ) in Figure 2c.To quantitatively analyze the charge-to-spin conversion efficiency of Mn 3 Sn, the SOT efficiency  i with different spin

Figure 2 .
Figure2.ST-FMR analysis of current-induced torques in the Mn 3 Sn/Py bilayer with 5 nm thick textured Mn 3 Sn grown at 600 °C.a) Depiction of the schematic device configuration and the induced torques ( // and  ⊥ ) by the injected microwave current (I rf ).b) Representative ST-FMR spectra measured at 7 GHz with ϕ H = 45°and 225°, respectively.Here, ϕ H is defined as the angle between I rf and H ext .c) Fit of angle-dependent symmetric (V s ) and antisymmetric (V a ) components extracted from the ST-FMR spectra.The gray line shows the contribution of the conventional damping-like torque induced by  y spin current.d) Calculated SOT efficiency  i with different spin polarizations  i (i = x, y, z) as a function of the Mn 3 Sn thickness.

Figure 3 .
Figure 3. Demonstration of the deterministic perpendicular magnetization using textured Mn 3 Sn as the spin source.a) Schematic illustration of the out-of-plane spin polarization  z due to the chiral spin texture in Mn 3 Sn.b) Normalized Hall resistance contributed from the CoTb layer as a function of the applied magnetic field.c) Field-free magnetization switching of the CoTb layer with different initial magnetization.d) and e) Switching behavior driven by SOT under different external fields.

Figure 4 .
Figure 4. Characterization of the interfacial Dzyaloshinskii-Moriya interaction (iDMI) by Brillouin light scattering (BLS) measurements.a,b) BLS spectra measured with an in-plane wavelength of  = 532 nm under oppositely oriented external fields H. Solid lines represent the Lorentzian fit.c) Relationship between the iDMI-induced frequency shift Δf and the wave factor k x .