Probing the Dzyaloshinskii-Moriya interaction via the propagation of spin waves in ferromagnetic thin films
PProbing the Dzyaloshinskii-Moriya interaction via the propagation of spin waves inferromagnetic thin films
Zhenyu Wang, Beining Zhang, Yunshan Cao, and Peng Yan ∗ School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Film and Integrated Devices,University of Electronic Science and Technology of China, Chengdu 610054, China
The Dzyaloshinskii-Moriya interaction (DMI) has attracted considerable recent attention owingto the intriguing physics behind and the fundamental role it played in stabilizing magnetic solitons,such as magnetic skyrmions and chiral domain walls. A number of experimental efforts have beendevoted to probe the DMI, among which the most popular method is the Brillouin light scatteringspectroscopy (BLS) to measure the frequency difference of spin waves with opposite wave vectors ± k perpendicular to the in-plane magnetization m . Such a technique, however, is not applicablefor the cases of k (cid:107) m , since the spin-wave reciprocity is recovered then. For a narrow magneticstrip, it is also difficult to measure the DMI strength using BLS because of the spatial resolutionlimit of lights. To fill these gaps, we propose to probe the DMI via the propagation of spin waves inferromagnetic films. We show that the DMI can cause the non-collinearity of the group velocitiesof spin waves with ± k (cid:107) m . In heterogeneous magnetic thin films with different DMIs, negativerefractions of spin waves emerge at the interface under proper conditions. These findings enable usto quantify the DMI strength by measuring the angle between the two spin-wave beams with ± k (cid:107) m in homogeneous film and by measuring the incident and negative refraction angles in heterogeneousfilms. For a narrow magnetic strip, we propose a nonlocal scheme to determine the DMI strengthvia nonlinear three-magnon processes. We implement theoretical calculations and micromagneticsimulations to verify our ideas. The results presented here are helpful for future measurement ofthe DMI and for designing novel spin-wave spintronic devices. I. INTRODUCTION
The Dzyaloshinskii-Moriya interaction (DMI) is the an-tisymmetry component of exchange couplings, which wasinitially proposed to explain the weak ferromagnetism ofantiferromagnets [1, 2]. This interaction originates fromthe spin-orbit coupling in magnetic materials with bro-ken inversion symmetry, either in bulk or at the interface.Recently, the DMI has drawn extensive research interestdue to two main reasons: (i) its fundamental role in sta-bilizing topological magnetic solitons, such as skyrmions[3–7] and chiral domain walls [8–12], which are promis-ing candidates for future spintronic applications; (ii) theintriguing physics associated with the nonreciprocal prop-agation of spin waves (magnons) [13–16], the elementaryexcitations in ordered magnets. The determination of theDMI is thus an important issue.Several experimental schemes have been proposed tomeasure the DMI strength. For example, it can be quan-tified by imaging the profile of chiral domain walls [8–10] or by analyzing their dynamical behaviors [11, 17–20]when the driving electric currents and/or magnetic fieldsare applied. When the DMI is not strong enough to sta-bilize the inhomogeneous magnetic texture (such as thedomain wall), the spin-wave excitation carries the uniqueinformation of the DMI. Recent experiments have demon-strated that the DMI constant can be determined by mea-suring the frequency difference [∆ ω = ω ( k ) − ω ( − k )] ofspin waves with opposite wave vectors ( ± k ) perpendic-ular to the magnetization ( m ) using the Brillouin lightscattering spectroscopy (BLS) [20–25], the spin-polarized ∗ Corresponding author: [email protected] electron energy loss spectroscopy [26], and the propagat-ing spin wave spectroscopy [27]. For the case of k (cid:107) m ,the frequency difference of spin waves with ± k vanishesand these schemes are unfeasible. We note that spin-waveexcitations in these experiments are in the long wave-length regime, where the anisotropic dipolar interactioncannot be ignored. The nonreciprocal nature of dipo-lar interactions may blur the quantification of the DMI.Moreover, for a magnetic strip with the width well below100 nm, it is difficult to utilize the BLS to measure theDMI owing to the diffraction limit of lights. It is, there-fore, necessary to develop new methods to measure theDMI for the situations mentioned above.In this work, we propose to probe the DMI strength inferromagnetic films via the propagation of spin waves. Tothis end, we systematically investigate the effects of theDMI on the propagation, the scattering, and the interac-tion of spin waves in different magnetic structures. Wefirst consider a homogeneous ferromagnetic film, and findthat the DMI induces a non-collinearity between the wavevector and group velocity ( v g = d ω/ d k ) of spin waveswhen k is not perpendicular to m . Spin-wave cantinginduced by the DMI has been reported in ferromagneticnanowires [28] with v g (cid:107) m . Here, we predict anothernon-collinearity between two spin-wave beams with op-posite wave vectors ± k (cid:107) m . The angle between thetwo spin-wave beams is derived analytically. Since thisnon-collinearity comes from the DMI but not the dipo-lar interaction, we can exclusively determine the DMIstrength by measuring the angle between the two beams.Inspired by recent advances of spatially modulated DMIin heterogeneous ferromagnetic films [29–31], we then in-vestigate the spin-wave scattering at the interface sepa-rating two co-planar ferromagnets with different DMIs.We focus on the exchange spin-wave region, where the a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t nonlocal dipolar effect can be approximated by local de-magnetizing fields. We obtain the generalized Snell’s law,and show the emergence of both the negative refractionand the total reflection under proper spin-wave incidentangles. These peculiar phenomena and the generalizedSnell’s formula can be used to quantify the DMI strengthby simply measuring the incident and refracted angles ofspin-wave beams, which can be readily realized by directimagings [32].Recently, we developed a three-magnon interactionmethod to detect spin waves localized in the magneticdomain wall nanochannels [33]. The approach can be par-allelly applied for probing the DMI in narrow magneticstrips. In general, the three-magnon process is triggeredby the weak nonlocal magnetic dipole-dipole interactionin uniform ferromagnets [34]. It can also occur in mag-netic textures such as skyrmions [35] and domain walls[33] without the dipolar interaction. Here, we consideranother three-magnon effect induced by the DMI in uni-form ferromagnets. The idea is analytically formulatedwith micromagnetic simulations performed to verify thetheoretical predictions. All micromagnetic simulationsin this work are performed using the OOMMF package[36, 37].The structure of this paper is organized as follows. InSec. II, we derive the dispersion relation of spin waves in achiral magnetic film. The group-velocity non-collinearityof two propagating spin-wave beams with anti-parallelwave vectors is presented. We also investigate the spin-wave scattering across the interface of two ferromagnetswith different DMIs. In Sec. III, the three-magnon pro-cesses arising in a narrow magnetic strip with the DMIare studied. We demonstrate that this nonlinear effectcan be utilized to accurately quantify the DMI constantin the strip. Conclusions are drawn in Sec. IV. II. THE LINEAR DYNAMICS OF SPIN WAVESIN HOMOGENEOUS AND HETEROGENEOUSCHIRAL MAGNETIC FILMS
We first consider the spin-wave propagation in a mag-netic thin film with the interfacial DMI of the followingform [38], H DM = 2 Dµ M s [ ∇ m z − ( ∇ · m )ˆ z ]= 2 Dµ M s ( ∂m z ∂x , ∂m z ∂y , − ∂m x ∂x − ∂m y ∂y ) , (1)where D is the DMI constant, M s is the saturation mag-netization, and m = ( m x , m y , m z ) is the unit magnetiza-tion vector. The magnetization dynamics is described bythe Landau-Lifshitz-Gilbert (LLG) equation, ∂ m ∂t = − γµ m × H eff + α m × ∂ m ∂t , (2)with the gyromagnetic ratio γ = 1 . × rad s − T − ,the vacuum permeability µ , and the Gilbert damping (a) v v k k m ˆsin( ) h t z ω z m δ − θ (nm) x ( n m ) y (mJ/m ) D
2 (GHz) ω π () θ (c) x y z x L y L z L m H (d) -0.3 0 0.3-0.300.3 x − k x + k y + k y − k +gy v gy − v g − v g + v (nm ) x k − ( n m ) y k − m (b) + k − k FIG. 1. (a) Schematic illustration of an ultrathin film withwidth L x , length L y , and thickness L z . A magnetic field H is applied along +ˆ y , which makes the magnetization m lyingin the film plane ( m (cid:107) +ˆ y ). (b) The isofrequency curve calcu-lated based on the dispersion relation Eq. (4). k + x and k − x arethe wave vectors of spin waves propagating along the direc-tion perpendicular to the magnetization. v +g and v − g are thegroup velocities of spin waves with k ± y (cid:107) m . k + and k − arethe wave vectors of spin waves propagating along the mag-netization ( v ± gy (cid:107) m ). (c) Non-collinear propagation of twospin-wave beams with opposite wave vectors ( k = − k ). Thespin-wave beams are excited by a sinusoidal monochromaticmicrowave source with µ h = 0 . ω/ π = 80 GHz ina rectangular region (black bar). (d) The angle between thetwo spin-wave beams as a function of the excitation frequencyfor various DMI constants D . The dots and curves correspondto the numerical simulation results and the analytical formulaEq. (6), respectively. constant α . The effective field H eff comprises of the ex-change field, the DM field, the demagnetization field, andthe external field. Although the DMI facilitates the in-homogeneous magnetic texture, it is possible to stabilizea single-domain structure as the ground state, when theexternal field is sufficiently strong. Given that the DMIhas no effect on spin waves when m is perpendicular tothe film plane [15, 39], an external field H ext = H ˆ y is applied to make m in the film plane ( m = +ˆ y ), asshown in Fig. 1(a). For simplicity, the dipolar interac-tion is approximated by the static demagnetizing field foran extend film H d = − M s m z ˆ z . Neglecting the dampingterm ( α = 0), the spin wave dispersion relation can beobtained by solving the linearized LLG equation [39, 40], ω ( k ) = (cid:112) ( A ∗ k + ω H )( A ∗ k + ω H + ω m ) − D ∗ k x , (3)where A ∗ = 2 γA/M s with the exchange constant A , ω H = γµ H , ω m = γµ M s , D ∗ = 2 γD/M s , and k = ( k x , k y ) the wave vector of spin wave. Here we fo-cus on the exchange spin waves with high frequencies andsimplify Eq. (3) to ω ( k ) = A ∗ k − D ∗ k x + ω H + ω m , (4)and obtain the group velocity v g = ∂ω∂ k = (2 A ∗ k x − D ∗ )ˆ x + 2 A ∗ k y ˆ y. (5)The influence of the DMI on spin-wave propagations ina chiral ferromagnetic film can be analyzed by the isofre-quency curve, as shown in Fig. 1(b). In the presence ofthe DMI, the isofrequency circle shifts away from the ori-gin in the k space. For k ⊥ m , the magnitudes of wavevectors with opposite directions are different, which in-dicates an asymmetry of the spin-wave wavelength withrespect to the propagation direction [40]. When the prop-agation direction ( v +gy and v − gy ) of spin wave is along themagnetization, the wave vectors ( k + and k − ) of spinwaves become oblique with respect to the propagationdirection, which is responsible for the spin-wave cantingnumerically observed in Ref. [28]. In the case of k (cid:107) m ( k x = 0), the group velocity in Eq. (5) can be writtenas v g = − D ∗ ˆ x + 2 A ∗ k y ˆ y . In the presence of the DMI( D (cid:54) = 0), the group velocities ( v +g and v − g ) of spin waveswith opposite wave vectors ( k +y and k − y ) are not collinear,as shown in Fig. 1(b). The angle between the two groupvelocities can be obtained: θ = arccos v +g · v − g | v +g || v − g | = arccos (cid:104) ( D ∗ ) − A ∗ ( ω − ω H − ω m / D ∗ ) + 4 A ∗ ( ω − ω H − ω m / (cid:105) . (6)Based on Eq. (6), the DMI strength can be evaluated bymeasuring the angle of the two spin-wave beams.We confirm the above results using micromagnetic sim-ulations. We consider a ferromagnetic thin film withlength 2000 nm, width 2000 nm, and thickness 1 nm,which lies in the x − y plane. Magnetic parameters ofPermalloy were used in simulations: M s = 8 × A / m, A = 13 pJ / m, and α = 0 .
01. In the simulations, theGilbert damping constant close to the film edges is setto linearly increase to 1.0 to avoid the spin-wave reflec-tion by the boundaries [41]. We apply an external field µ H = 1 T along +ˆ y that is sufficiently strong to satu-rate the magnetization in the film plane. Then, we stim-ulate the propagations of two spin-wave beams with op-posite wave vectors parallel and antiparallel to the mag-netization ( ± k (cid:107) m ). To this end, we apply a sinusoidalmonochromatic microwave source H ext = h sin( ωt )ˆ z ina narrow rectangular region (300 ×
10 nm ) [black barshown in Fig. 1(c)], where the field amplitude h has aGaussian profile in the transverse direction (ˆ x ) [42]. Fig-ure 1(c) shows two spin waves with ω/ π = 80 GHz and ± k (cid:107) m in the presence of DMI ( D = 3 . / m ). It is clear to see that the group velocities v and v of the twospin-wave beams are non-collinear. Both beams propa-gate towards the left, which is fully in line with formula v g = − D ∗ ˆ x + 2 A ∗ k y ˆ y . The angle θ = 120 . ◦ obtainedfrom simulation is also consistent with the theoretical pre-diction Eq. (6) with a deviation less than 0 . ω/ π (cid:54)
50 GHz) where the wave vectors ofspin waves have a slight deviation from the presumed ˆ y direction (not shown). For D = 0, the angle between thetwo spin-wave beams is always equal to 180 ◦ . This indi-cates that the non-collinearity of the two spin-wave beamsobserved above is exclusively induced by the DMI ratherthan the dipolar interaction. Thus, the DMI strength canbe determined by measuring the angle between the twospin-wave beams. This method requires the spin waveimaging with the spatial resolution in the range of about200 nm (the width of the spin-wave beam), which can beachieved by X-ray magnetic circular dichroism (XMCD)[43] or the near-field BLS [44].Next, we proceed to investigate the scattering of ex-change spin waves across the interface of two ferromag-nets with different DMIs. One effect of the spatially mod-ulated DMI is the equilibrium spin canting at the DMIinterface [29]. Since spin canting only occurs at a nar-row range around the DMI interface, its effect on thespin-wave propagation is rather weak. Hence, we viewthe magnetization of the heterogeneous film as uniformalong the +ˆ y direction. Based on Eq. (4), the isofre-quency curves of spin waves propagation in no-DMI andDMI regions are plotted in k space, as shown in Fig. 2(a).In the absence of the DMI ( D = 0), the spin-wave isofre-quency curve at a given frequency ω is a circle centered atthe origin with the radius k = (cid:112) ( ω − ω H − ω m / /A ∗ .With the DMI ( D (cid:54) = 0), the isofrequency circle is shiftedby ∆ = D ∗ / A ∗ along + k x axis and its radius increasesto k Dr = (cid:112) ( k ) + ∆ . According to the conservationof momentum parallel with the interface, we obtain thegeneralized Snell’s law k sin θ i = k Dr sin θ t + ∆ , (7)where θ i and θ t are the incident angle and refraction angleof spin-wave beams with respect to the interface normal,as shown in Fig. 2(a). Similar results have been pre-sented for spin waves propagation in different magneticsystems [45, 46]. For instance, the generalized Snell’s lawdescribes the spin wave refracted at the domain wall in achiral magnetic film in Ref. [45].The spin-wave scattering at the DMI interface can bedivided into three cases: total reflection, negative re-fraction and normal refraction. The critical angles fortotal reflection ( θ Tc ) and negative refraction ( θ Nc ) arethe incident angles corresponding to the refracted angles θ t = − ◦ and θ t = 0 ◦ , respectively. Using Eq. (7), weobtain θ Tc = arcsin[(∆ − k Dr ) /k ] and θ Nc = arcsin[∆ /k ].Below we examine the scattering of spin waves at the i θ i r θ θ= = − ( n m ) y z m δ r θ t θ i θ = t θ = − i θ t θ i θ = t θ = − (nm) x tg v ig v rg v i θ r θ t θ t θ = i r θ θ= = (nm) x ( n m ) y ig v rg v (b) i θ t θ i θ = − t θ = − ( n m ) y (c) (d)(e) (f)(a) x k x k y k i θ t θ ig v tg v ∆ Dr k k mm x y z D = D ≠ FIG. 2. (a) Schematic plot of the generalized Snell’s law forspin-wave scattering at the DMI interface. The magnetizationof ultrathin film is saturated along +ˆ y direction. The green(lower) and magenta (upper) circles correspond to the isofre-quency curves in momentum space for spin-wave propagationin no-DMI and DMI regions, respectively. (b) ∼ (f) Refractionand reflection of spin waves scattering at the DMI interfacewith different incident angles. (b) θ i = − ◦ , (c) θ i = − ◦ ,(d) θ i = 0 ◦ , (e) θ i = 18 ◦ , and (f) θ i = 60 ◦ . Arrows label thegroup velocities of spin waves. DMI interface ( D = 3 . / m ) for different incidentangles. For spin waves with ω/ π = 80 GHz, the crit-ical angles for total reflection and negative refractionare θ Tc = − . ◦ and θ Nc = 34 . ◦ . For incident angles − ◦ < θ i < − . ◦ , there is no real solution for therefracted angle θ t and total reflection takes place. Thissituation is plotted in Fig. 2(b), showing that the incidentspin wave with θ i = − ◦ cannot transmit into the DMIregion (upper gray region) and is completely reflected.For − . ◦ < θ i < ◦ , the incident and refracted angleshave the same sign corresponding to normal refraction, asshown in Fig. 2(c) with θ i = − ◦ . In the case of verticalincidence, the refracted angle is − . ◦ rather than 0 ◦ ,which is different from its optical analog, as illustratedin Fig. 2(d). For 0 ◦ < θ i < . ◦ , the refracted angle isnegative. As an example, we set θ i = 18 ◦ in Fig. 2(e), and find that the refracted angle is θ t = − . ◦ . Boththe incident and refracted spin-wave beams are on thesame side of the interface normal, which indicates the oc-currence of negative refraction. For the incident anglesat 34 . ◦ < θ i < ◦ , the refracted angles become positiveand have the same sign with θ i , recovering the normalrefraction again, as shown in Fig. 2(f) with θ i = 60 ◦ .As we have shown, both the total reflection and thenegative refraction can happen at the DMI interface fora certain range of the incident angles. Utilizing total re-flection at the DMI interface, a spin-wave fiber or guidecan be designed, analogous to the cases studied in Refs.[45] and [46]. It is worth noting that total reflection is nota unique feature at the DMI interface. In non-chiral ferro-magnetic heterostructures with different material param-eters such as the exchange constant, saturation magneti-zation, and thickness, it can happen as well [47–50]. Incontrast, the negative refraction is exclusively due to theexistence of a DMI step for the exchange spin wave andwould disappear without the DMI. By measuring the in-cident and negative-refracted angles, one can determinethe DMI constant based on the generalized Snell’s law[Eq. (7)].One issue of our methods is how to excite the planespin waves with the frequencies in the range of a few tensGHz in experiments. Very recently, it has been demon-strated that the excitation of the exchange spin waveswith short wavelength (50 nm or shorter) and frequencyup to 30 GHz can be realized by using the magnetizationprecession in periodic ferromagnetic nanowires to drivespin waves in a neighboring magnetic film [51]. III. THREE-MAGNON INTERACTIONSARISING IN A MAGNETIC STRIP
In the above discussions, we focused on how to mea-sure the DMI in ferromagnetic thin films of large scales.Due to the big laser spot size subjected to the diffrac-tion limit of lights in the wavevector-resolved BLS, it isdifficult to detect spin waves propagating in a rather nar-row magnetic strip or nanowire. To address this prob-lem, we propose a novel method to measure the DMI pa-rameter in magnetic strip by analyzing the spectrum ofspin waves outside the strip, which involves the nonlinearthree-magnon processes.Firstly, we show that the three-magnon processes in-deed can be induced by the DMI in uniform ferromagnets,even without the magnetic dipolar interaction. We startfrom the interfacial DMI Hamiltonian H DM = DM s (cid:90) d r [ M z ( ∇ · M ) − ( M · ∇ ) M z ] , (8)where the static magnetization M = ( M x , M y , M z ) liesin-plane and deviates from ˆ x direction with an arbitraryangle ϕ . We consider small oscillation of the magne-tization over the ground state and represent M in theform M = M + s ( r , t ), where M is the backgroundmagnetization, and s corresponds to the small oscilla-tions. We construct a new coordinate system for mag- w D ≠ D = m x y n m x m y m z m (nm ) x k − ( GH z ) ω π (a) (b) (c) b1 k b2 k FIG. 3. (a) The heterogeneous ultrathin film with a DMI strip(gray region) in the center. The width of the DMI strip is w =50 nm. The sinc-function field is applied in the regions of blackbar. (b) Three components of the equilibrium magnetization m along longitudinal direction at x = 500 nm. (c) FFTspectrum along the DMI strip center y = 900 nm in (a). In(c), the black curve corresponds to the analytical formula Eq.(3), while the yellow one represents the modified formula Eq.(13) with the fitting parameter δ ≈ . ◦ . The wave vectors ofspin waves with 35 GHz are k b1 = 0 .
039 nm − and k b2 = 0 . − , which correspond to the backward and forward spinwaves, respectively. netization ( e , e , e ) by rotating the coordinate system( x, y, z ) around ˆ z over a angle ϕ , making e and M par-allel. Expressing the magnetization in the rotated coor-dinate ( e , e , e ) yields M x = M cos ϕ − M sin ϕ,M y = M sin ϕ + M cos ϕ,M z = M , (9)where M = M + s , M = s , and M = s . ByHolstein-Primakoff transformation, we can express themagnetization in terms of boson operators ( a and a + ), M = 2 µ B ( S − a + a ) ,M = µ B √ S ( a + a + ) − µ B √ S ( a + aa + a + a + a ) ,M = − iµ B √ S ( a − a + ) + iµ B √ S ( a + aa − a + a + a ) , (10)where S = M s / (2 µ B ) is the spin of an atom with theBohr magneton µ B . Substituting (10) and (9) into (8)and keeping the third-order terms of boson operators, wehave H (3)DM = iDS − / √ (cid:90) d r (cid:26) cos ϕ (cid:20) a − a + ) ∂∂x ( a + a ) − a + a ) ∂∂x ( a − a + ) − ∂∂x ( a + aa − a + a + a ) (cid:21) + sin ϕ (cid:20) a − a + ) ∂∂y ( a + a ) − a + a ) ∂∂y ( a − a + ) − ∂∂y ( a + aa − a + a + a ) (cid:21) (cid:27) , (11)which is the three-magnon interaction Hamiltonian.Next, we numerically examine the three-magnon pro-cesses arising in the DMI strip. To this end, we constructa heterogeneous ferromagnetic thin film with length 1800nm, width 1000 nm, and thickness 1 nm, in which a DMIstrip ( D = 3 . / m ) with the width w = 50 nm lo-cates in the center while the rest parts have no DMI, asshown in Fig. 3(a). To ascertain the spin-wave spec-trum in the DMI strip, we apply a sinc-function field h ( t ) = h sin[ ω H ( t − t )] / [ ω H ( t − t )]ˆ z for 10 ns with µ h = 0 .
01 T, ω/ π = 100 GHz, and t = 1 ns, over theblack bar with volume 10 × × shown in Fig.3(a). In Fig. 3(c), the dispersion relation is obtained byperforming the FFT of the spatiotemporal oscillation ofthe z -component magnetization ( δm z ) over the latticesalong the DMI strip center ( y = 900 nm) in Fig. 3(a).One can immediately see that the theoretical result (3)[black curve shown in Fig. 3(c)] does not quite agreewith the micromagnetic simulations. The reason for thisdeviation is attributed to the spin canting at the DMIinterface [29], which is not a negligible effect any morebecause the tilting range is comparable with the stripwidth, as shown in Fig. 3(b). The largest tilting angleis obtained at the interface and is equal to 13 . ◦ . Theexact solution of the spin-wave spectrum on top of this strongly inhomogeneous magnetization texture is unlikelyto obtain. However, to describe spin wave over this tiltedground state accurately enough, we assume that the DMIstrip still has the uniform magnetization but deviatingfrom the film plane with an angle δ . The stabilization ofthis uniform tilting magnetization state requires an addi-tional effective field H (cid:48) along ˆ z , which may originate fromthe DMI step [30] or the pinning of magnetizations closeto the interface. According to the equilibrium condition m × H eff = 0, we can get H (cid:48) = ( H tan δ + M s sin δ )ˆ z .Then, the effective field is given by H eff = 2 Aµ M s ∇ m + H DM + H ˆ y − M s m z ˆ z + H (cid:48) . (12)Substituting (12) into the LLG equation (2), the spin-wave dispersion relation in DMI strip can be calculated, ω = (cid:112) ( A ∗ k + ω H / cos δ )( A ∗ k + ω H / cos δ + ω m cos δ ) − D ∗ k x cos δ. (13)The simulated dispersion relation is well fitted by theabove formula (13) with δ ≈ . ◦ [see the yellow curvein Fig. 3(c)]. However, we notice that the fitting pa-rameter δ is twice as large as the actual tilting angle mmm D ≠ D = D = b k b k i k i k k = k k k confluence splitting FIG. 4. Schematic picture of nonlinear three-magnon pro-cesses in the DMI strip. In dashed red square, it shows thethree-magnon confluence of k i and k b into k . In dashedblue square, we plot the stimulated three-magnon splittingof k i into two modes k = k b and k , assisted by a localizedmagnon k b (gray arrow). at the interface. This obvious disagreement is due tothat approximating a strongly inhomogeneous magneti-zation texture by a globally tilted magnetization stateis too simplified. However, this approximation gives agood description of the dispersion relation of spin waveslocalized in the DMI strip. Interestingly, we find thatthe presence of the DMI reduces the spin-wave band gapfrom (cid:112) ω H ( ω H + ω m ) / π ≈ . D = 3 . / m , as shown in Fig. 3(c).In other words, spin waves with frequencies in the range(31.4, 39.7) GHz will be localized in the DMI strip. Thismotivates us to consider the three-magnon processes inthe strip channel, while the present authors have consid-ered a similar issue but in magnetic domain wall channelsin Ref. [33].In our strategy, we input a propagating spin wave( ω i , k i ) in the lower part of the heterogeneous films, tointeract with the localized spin wave ( ω b , k b ) boundedin the strip. In general, two kinds of three-magnon pro-cesses, i.e., confluence and splitting, can occur, as illus-trated in Fig. 4. We first consider the three-magnonconfluence. In this process, both the energy and the mo-mentum parallel with the strip are conserved. Thus wehave ω k = ω i + ω b , ( k − k i − k b ) · ˆ x = 0 . (14)For a normal incident, i.e., k i = k i ˆ y , we obtain the wavevector of the the three-magnon confluence k = k b ˆ x + k y ˆ y, (15)where k y ≈ (cid:112) k + C with a positive constant C =[ ω H / cos δ +( ω m cos δ ) / − D ∗ k b cos δ ] /A ∗ . For the three-magnon splitting process, the energy-momentum conser-vation gives rise to ω + ω = ω i , ( k + k − k i ) · ˆ x = 0 . (16)In analogy to the case in Ref. [33], the presence of spinwaves localized in the DMI strip can trigger a stimulatedsplitting, implying k = k b in (16). We are also inter-ested in the normal-incident case, so the wave vector ofthe three-magnon splitting can be determined as k = − k b ˆ x + k y ˆ y, (17)
700 nm xy H
35 GHz80 GHz0.01 − z m δ -4 -3 -2 -1
80 115 FF T a m p lit ud e ( a r b . un it s )
2 (GHz) ω π (nm ) x k − ( n m ) y k −
80 GHz (0,0.215)
45 GHz ( 0.045,0.063) −
115 GHz (0.045,0.296) (nm ) x k − ( n m ) y k − ( n m ) y k − (nm ) x k − (a) (b)(c)(e)(d) FIG. 5. (a) Micromagnetic simulations of three-magnon pro-cesses. The incident and localized spin waves are excited atthe lower part of the magnetic film (horizontal black bar) andthe right side of the DMI strip (vertical black bar), respec-tively. (b) FFT spectrum at a single lattice [black dot in(a)]. (c) ∼ (e) Spatial FFT spectra analyses for three peaks at(c) 80 GHz, (d) 45 GHz and (e) 115 GHz, observed in (b)where the incident spin-wave frequency is ω i / π = 80 GHzand the localized spin-wave frequency is ω b / π = 35 GHz.The FFT analysis is implemented over the region inside thedashed black square with the side length 700 nm in (a). where k y ≈ (cid:112) k − k − C . This indicates that thethree-magnon splitting processes can only happen when k i ≥ (cid:112) k + C , which requires the frequency of the in-coming spin waves higher than a critical value ω i , c = A ∗ (2 k + C ) + ω H + ω m / . (18)Micromagnetic simulations are performed to verify thethree-magnon processes arising in the DMI strip. Weapply two sinusoidal monochromatic microwave fields si-multaneously to excite the propagating spin waves ( ω i , k i )in the lower part of the magnetic film and the localizedspin waves ( ω b , k b ) in the DMI strip, respectively [see Fig.5(a)]. Here, we consider ω i / π = 80 GHz and ω b / π = 35GHz and focus on the normal incident case, i.e., k i (cid:107) ˆ y .Because of the conservation of both the energy and themomentum along the DMI strip (ˆ x ), the transmitted spinwaves carry the information ( ω i , b , k i , b ) from the incidentand localized spin waves. This result is confirmed by thetemporal FFT spectrum at a single cell [the black dot inFig. 5(a)], which shows three peaks at 45 GHz, 80 GHz,and 115 GHz, as plotted in Fig. 5(b). The main peak of80 GHz is from the incident spin wave excited at the lower − z m δ xy
700 nm
35 GHz80 GHz H -4 -3 -2 -1
70 11580 FF T a m p lit ud e ( a r b . un it s ) (nm ) x k − (nm ) x k − ( n m ) y k − ( n m ) y k −
115 GHz (0.171,0.233)
70 GHz (a)(b) (c)(d)
2 (GHz) ω π
FIG. 6. (a) Micromagnetic simulations of three-magnon pro-cesses with a different localized spin-wave mode. The incidentand localized spin waves are excited at the lower part of themagnetic film (horizontal black bar) and the left side of theDMI strip (vertical black bar), respectively. (b) FFT spec-trum at a single lattice cell [black dot in (a)]. Spatial FFTspectra analyses for the frequency ω/ π = 70 GHz (c) and 115GHz (d) are implemented over the region inside the dashedblack square with the side length 700 nm in (a). part of the magnetic film. Two relatively weaker peaksat 45 GHz and 115 GHz are due to the three-magnonsplitting and confluence processes, which satisfy the en-ergy conservation ω k = ω i ∓ ω b , respectively. The wavevectors of spin waves for three frequency peaks can beobtained by the spatial FFT spectrum analysis over theregion inside the dashed black square in Fig. 5(a). FFTresults are shown in Figs. 5(c) − (e). The magnon wavevector at 80 GHz is k i = 0 . y in the unit of nm − ,which agrees with the dispersion relation Eq. (3). Whilethe magnon wave vectors at 45 GHz and 115 GHz are k = − . x + 0 . y and 0 . x + 0 . y in the units ofnm − , respectively. These values excellently agree withthe wave vector formulas of the three-magnon splittingand confluence Eq. (17) and (15), respectively. Accord-ing to the conservation of momentum parallel with theDMI strip, the x -components of the wave vectors k ofthe transmitted spin waves for three-magnon confluenceand stimulated splitting are k b and − k b , respectively.Therefore, we can determine the wave vector of the local-ized spin wave in the DMI strip, k b = 0 . x in unit ofnm − , which is consistent with the direct FFT analysisin the strip.Now, considering the inverse problem by assuming thatboth the DMI constant D and the canting angle δ aretwo unknown parameters in Eq. (13), only one group of( ω b , k b ) is insufficient to determine them. We need an- other set of ( ω b , k b ) to completely quantify the DMI. Tothis end, we apply the same sinusoidal microwave fieldon the left side of the DMI strip, as shown in Fig. 6(a).A different microwave field on the same side also servesthe same purpose (not shown). Although they have thesame frequency, the localized spin waves excited at twosides of the DMI strip carry different wave vectors dueto their non-reciprocal nature, as shown in Fig. 5(a) andFig. 6(a). Temporal FFT spectrum analysis at a singlecell [the black dot in Fig. 6(a)] shows two peaks at 80 and115 GHz in Fig. 6(b), which are from the incident spinwave and the three-magnon confluence event discussedabove. As compared with the FFT spectrum in Fig. 5(b),the frequency peak at 45 GHz disappears, but there is avery weak new peak at 70 GHz. The reason for the dis-appearance of 45 GHz peak is that the frequency of theincident spin wave is not high enough for generating thestimulated three-magnon splitting process. According tothe criterion Eq. (18), the lowest incident frequency togenerating the splitting process is ω i , c / π ≈ . ω b / π = 115 −
80 = 35 GHz. The conservation of mo-mentum parallel with the DMI strip indicates that thewave vector of the localized spin wave is k b = 0 . x inunit of nm − , as shown in Fig. 6(d). Further, the wavevector 0 .
171 nm − corresponds to a spin-wave wavelength36.7 nm, which can be measured by an antenna as demon-strated in Ref. [51]. Substituting the two sets ( ω b , k b )into the dispersion relation (13), and solving the followingcoupled equations ω b = (cid:114) ( A ∗ k + ω H cos δ )( A ∗ k + ω H cos δ + ω m cos δ ) − D ∗ k b1 cos δ,ω b = (cid:114) ( A ∗ k + ω H cos δ )( A ∗ k + ω H cos δ + ω m cos δ ) − D ∗ k b2 cos δ, (19)we obtain the DMI constant D = 3 . / m and δ = 32 . ◦ , which is consistent with the input parame-ter D = 3 . / m and the fitting δ = 25 . ◦ obtainedearlier. We also perform micromagnetic simulations withsmaller DMI constants (1 . ∼ . / m ), which arethe typical values measured in experiments [24, 52]. TheDMI constants obtained by solving Eq. (19) are excel-lently consistent with the simulation parameters. Theseresults suggest that the local DMI of a narrow magneticstrip can be accurately probed by non-locally detectingthe spectra of both the incident and the transmitted spinwaves involving in the nonlinear three-magnon processes. IV. CONCLUSION
To conclude, we systematically investigate the prop-agation, scattering, and interaction of spin waves invarious ferromagnetic mediums and structures. In ho-mogeneous ferromagnetic thin films, we predict a non-collinearity of two spin-wave beams with ± k (cid:107) m , whichsolely comes from the DMI rather that the dipolar inter-action. By measuring the angle between the two beams,one can determine the DMI parameter. We also con-sider a magnetic interface in the heterogeneous ultrathinfilms with different DMIs, and obtained the spin-waveSnell’s law confirmed by micromagnetic simulations. To-tal reflection and negative refraction are observed at theDMI interface for certain incident angles. The total re-flection induced by the DMI can be used to design spin-wave fiber with unidirectional transmission functionality.Negative refraction found here is exclusively induced bythe DMI. These effects would provide an alternative ap- proach to BLS for probing the DMI strength. Moreover,we propose a nonlocal scheme to measure the DMI pa-rameter in a narrow ferromagnetic strip or nanowire bythree-magnon processes, which is not accessible for thewavevector-resolved BLS due to the detection limit. Ourresults would be helpful to extend the present method forprobing the DMI in experiments and for designing novelmagnonic devices in the future. V. ACKNOWLEDGMENT
We thank X.S. Wang, C. Wang, and Z.-X. Li for helpfuldiscussions. This work is supported by the National Nat-ural Science Foundation of China (Grants No. 11604041and 11704060), the National Key Research DevelopmentProgram under Contract No. 2016YFA0300801, and theNational Thousand-Young-Talent Program of China. [1] I. Dzyaloshinsky, A thermodynamic theory of “weak”ferromagnetism of antiferromagnetics, J. Phys. Chem.Solids , 241 (1958).[2] T. Moriya, Anisotropic Superexchange Interaction andWeak Ferromagnetism, Phys. Rev. , 91 (1960).[3] S. M¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.Rosch, A. Neubauer, R. Georgii, and P. B¨oni, SkyrmionLattice in a Chiral Magnet, Science , 915 (2009).[4] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,Y. Matsui, N. Nagaosa, and Y. Tokura, Real-space ob-servation of a two-dimensional skyrmion crystal, Nature(London) , 901 (2010).[5] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A.Fert, Nucleation, stability and current-induced motionof isolated magnetic skyrmions in nanostructures, Nat.Nanotechnol. , 839 (2013).[6] S. Woo, K. Litzius, B. Kr¨uger, M. Y. Im, L. Caretta,K. Richter, M. Mann, A. Krone, R. M. Reeve, M.Weigand, P. Agrawal, I. Lemesh, M. A. Mawass, P. Fis-cher, M. Kl¨aui, and G. S. D. Beach, Observation of room-temperature magnetic skyrmions and their current-drivendynamics in ultrathin metallic ferromagnets, Nat. Mater. , 501 (2016).[7] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de SouzaChaves, A. Locatelli, T. O. Mentes, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussign´e, A.Stashkevich, S. M. Ch´erif, L. Aballe, M. Foerster, M.Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Room-temperature chiral magnetic skyrmions in ultrathin mag-netic nanostructures, Nat. Nanotechnol. , 449 (2016).[8] M. Heide, G. Bihlmayer, and S. Bl¨ugel, Dzyaloshinskii-Moriya interaction accounting for the orientation of mag-netic domains in ultrathin films: Fe/W(110), Phys. Rev.B , 140403 (2008).[9] G. Chen, J. Zhu, A. Quesada, J. Li, A. T. N’Diaye, Y.Huo, T. P. Ma, Y. Chen, H. Y. Kwon, C. Won, Z. Q.Qiu, A. K. Schmid, and Y. Z. Wu, Novel Chiral MagneticDomain Wall Structure in Fe/Ni/Cu(001) Films, Phys.Rev. Lett. , 177204 (2013). [10] G. Chen, T. Ma, A. T. N’Diaye, H. Kwon, C. Won, Y.Wu, and A. K. Schmid, Tailoring the chirality of magneticdomain walls by interface engineering, Nat. Commun. ,2671 (2013).[11] M. J. Benitez, A. Hrabec, A. P. Mihai, T. A. Moore, G.Burnell, D. McGrouther, C. H. Marrows, and S. McVi-tie, Magnetic microscopy and topological stability of ho-mochiral N´eel domain walls in a Pt/Co/AlO x trilayer,Nat. Commun. , 8957 (2015).[12] J. P. Tetienne, T. Hingant, L. J. Martinez, S. Rohart, A.Thiaville, L. H. Diez, K. Garcia, J. P. Adam, J. V. Kim,J. F. Roch, I. M. Miron, G. Gaudin, L. Vila, B. Ocker, D.Ravelosona, and V. Jacques, The nature of domain wallsin ultrathin ferromagnets revealed by scanning nanomag-netometry, Nat. Commun. , 6733 (2015).[13] J. Lan, W. C. Yu, R. Q. Wu, and J. Xiao, Spin-WaveDiode, Phys. Rev. X , 041049 (2015).[14] X. Xing and Y. Zhou, Fiber optics for spin waves, NPGAsia Mater. , e246 (2016).[15] F. Garcia-Sanchez, P. Borys, A. Vansteenkiste, J.-V.Kim, and R. L. Stamps, Nonreciprocal spin-wave chan-neling along textures driven by the Dzyaloshinskii-Moriyainteraction, Phys. Rev. B , 224408 (2014).[16] T. Br¨acher, O. Boulle, G. Gaudin, and P. Pirro, Creationof unidirectional spin-wave emitters by utilizing interfa-cial Dzyaloshinskii-Moriya interaction, Phys. Rev. B ,064429 (2017).[17] R. Hiramatsu, K. J. Kim, Y. Nakatani, T. Moriyama,and T. Ono, Proposal for quantifying the Dzyaloshinsky-Moriya interaction by domain walls annihilation measure-ment, Jpn. J. Appl. Phys. , 108001 (2014).[18] D. H. Kim, S. C. Yoo, D. Y. Kim, B. C. Min, and S.B. Choe, Wide-Range Probing of Dzyaloshinskii-MoriyaInteraction, Sci. Rep. , 45498 (2017).[19] A. L. Balk, K. W. Kim, D. T. Pierce, M. D. Stiles,J. Unguris, and S. M. Stavis, Simultaneous controlof the Dzyaloshinskii-Moriya interaction and magneticanisotropy in nanomagnetic trilayers, Phys. Rev. Lett. , 077205 (2017). [20] R. Soucaille, M. Belmeguenai, J. Torrejon, J. V. Kim, T.Devolder, Y. Roussign´e, S. M. Ch´erif, A. A. Stashkevich,M. Hayashi, and J. P. Adam, Probing the Dzyaloshinskii-Moriya interaction in CoFeB ultrathin films using domainwall creep and Brillouin light spectroscopy, Phys. Rev. B , 104431 (2016).[21] M. Belmeguenai, J.-P. Adam, Y. Roussign´e, S. Eimer, T.Devolder, J.-V. Kim, S. M. Cherif, A. Stashkevich, and A.Thiaville, Interfacial Dzyaloshinskii-Moriya interaction inperpendicularly magnetized Pt/Co/AlO x ultrathin filmsmeasured by Brillouin light spectroscopy, Phys. Rev. B , 180405 (2015).[22] J. Cho, N. H. Kim, S. Lee, J. S. Kim, R. Lavrijsen,A. Solignac, Y. Yin, D. S. Han, N. J. van Hoof, H. J.Swagten, B. Koopmans, and C. Y. You, Thickness depen-dence of the interfacial Dzyaloshinskii-Moriya interactionin inversion symmetry broken systems, Nat. Commun. ,7635 (2015).[23] K. Di, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok,J. Yu, J. Yoon, X. Qiu, and H. Yang, Direct observationof the Dzyaloshinskii-Moriya interaction in a Pt/Co/Nifilm, Phys. Rev. Lett. , 047201 (2015).[24] H. T. Nembach, J. M. Shaw, M. Weiler, E. Ju´e, and T. J.Silva, Linear relation between Heisenberg exchange andinterfacial Dzyaloshinskii–Moriya interaction in metalfilms, Nat. Phys. , 825 (2015).[25] A. Hrabec, M. Belmeguenai, A. Stashkevich, S. M. Ch´erif,S. Rohart, Y. Roussign´e, and A. Thiaville, Making theDzyaloshinskii-Moriya interaction visible, Appl. Phys.Lett. , 242402 (2017).[26] K. Zakeri, Y. Zhang, J. Prokop, T. H. Chuang, N. Sakr,W. X. Tang, and J. Kirschner, Asymmetric spin-wave dis-persion on Fe(110): direct evidence of the Dzyaloshinskii-Moriya interaction, Phys. Rev. Lett. , 137203 (2010).[27] J. M. Lee, C. Jang, B. C. Min, S. W. Lee, K. J. Lee,and J. Chang, All-Electrical Measurement of InterfacialDzyaloshinskii-Moriya Interaction Using Collective Spin-Wave Dynamics, Nano Lett. , 62 (2016).[28] J. Guo, X. Zeng, and M. Yan, Spin-wave canting in-duced by the Dzyaloshinskii-Moriya interaction in ferro-magnetic nanowires, Phys. Rev. B , 014404 (2017).[29] J. Mulkers, B. Van Waeyenberge, and M. V. Miloˇsevi´c,Effects of spatially engineered Dzyaloshinskii-Moriya in-teraction in ferromagnetic films, Phys. Rev. B , 144401(2017).[30] S.-J. Lee, J.-H. Moon, H.-W. Lee, and K.-J. Lee, Spin-wave propagation in the presence of inhomogeneousDzyaloshinskii-Moriya interactions, Phys. Rev. B ,184433 (2017).[31] I.-S. Hong, S.-W. Lee, and K.-J. Lee, Magnetic domainwall motion across a step of Dzyaloshinskii-Moriya inter-action, Curr. Appl. Phys. , 1576 (2017).[32] J. Stigloher, M. Decker, H. S. Korner, K. Tanabe, T.Moriyama, T. Taniguchi, H. Hata, M. Madami, G. Gub-biotti, K. Kobayashi, T. Ono, and C. H. Back, Snell’s Lawfor Spin Waves, Phys. Rev. Lett. , 037204 (2016).[33] B. Zhang, Z. Wang, Y. Cao, P. Yan, and X. R. Wang,Eavesdropping on spin waves inside the domain-wallnanochannel via three-magnon processes, Phys. Rev. B , 094421 (2018).[34] R. Costa Filho, M. Cottam, and G. Farias, Microscopictheory of dipole-exchange spin waves in ferromagneticfilms: Linear and nonlinear processes, Phys. Rev. B ,6545 (2000).[35] D. N. Aristov and P. G. Matveeva, Stability of a skyrmionand interaction of magnons, Phys. Rev. B , 214425 (2016).[36] M. J. Donahue and D. G. Porter, OOMMF User’s Guide,Version 1.0, Interagency Report NISTIR 6376, (1999).[37] S. Rohart and A. Thiaville, Skyrmion confinementin ultrathin film nanostructures in the presence ofDzyaloshinskii-Moriya interaction, Phys. Rev. B ,184422 (2013).[38] A. N. Bogdanov and U. K. Rossler, Chiral symmetrybreaking in magnetic thin films and multilayers, Phys.Rev. Lett. , 037203 (2001).[39] D. Cort´es-Ortu˜no and P. Landeros, Influence of theDzyaloshinskii-Moriya interaction on the spin-wave spec-tra of thin films, J. Phys. Condens. Matter , 156001(2013).[40] J.-H. Moon, S.-M. Seo, K.-J. Lee, K.-W. Kim, J. Ryu, H.-W. Lee, R. D. McMichael, and M. D. Stiles, Spin-wavepropagation in the presence of interfacial Dzyaloshinskii-Moriya interaction, Phys. Rev. B , 184404 (2013).[41] D. V. Berkova and N. L. Gorn, Micromagnetic simula-tions of the magnetization precession induced by a spin-polarized current in a point-contact geometry (Invited),J. Appl. Phys. , 08Q701 (2006).[42] P. Gruszecki, Y. S. Dadoenkova, N. N. Dadoenkova, I.L. Lyubchanskii, J. Romero-Vivas, K. Y. Guslienko, andM. Krawczyk, Influence of magnetic surface anisotropyon spin wave reflection from the edge of ferromagneticfilm, Phys. Rev. B , 054427 (2015).[43] S. Bonetti, R. Kukreja, Z. Chen, F. Maci´a, J.M.Hern´andez, A. Eklund, D. Backes, J. Frisch, J. Katine,G. Malm, S. Urazhdin, A.D. Kent, J. St¨ohr, H. Ohldag,and H.A. D¨urr, Direct observation and imaging of a spin-wave soliton with p-like symmetry, Nat. Commun. ,8889 (2015).[44] J. Jersch, V. E. Demidov, H. Fuchs, K. Rott, P. Krzys-teczko, J. M¨unchenberger, G. Reiss, and S. O. Demokri-tov, Mapping of localized spin-wave excitations by near-field Brillouin light scattering, Appl. Phys. Lett. ,152502 (2010).[45] W. Yu, J. Lan, R. Wu, and J. Xiao, Magnetic Snell’s lawand spin-wave fiber with Dzyaloshinskii-Moriya interac-tion, Phys. Rev. B , 140410 (2016).[46] J. Mulkers, B. Van Waeyenberge, and M. V. Miloˇsevi´c,Tunable Snell’s law for spin waves in heterochiral mag-netic films, Phys. Rev. B , 104422 (2018).[47] D.-E. Jeong, D.-S. Han, and S.-K. Kim, Refractive In-dex And Snell’s Law for Dipole-Exchange Spin Waves InRestricted Geometry, Spin , 27 (2011).[48] H. Xi, X. Wang, Y. Zheng, and P. J. Ryan, Spinwavepropagation and coupling in magnonic waveguides, J.Appl. Phys. , 063921 (2008).[49] Yu. I. Gorobets and S. A. Reshetnyak, Reflection andrefraction of spin waves in uniaxial magnets in thegeometrical-optics approximation, Tech. Phys. , 188(1998).[50] A. V. Vashkovskii, A. V. Stal’makhov, and D. G.Shakhnazaryan, Formation, reflection, and refraction ofmagnetostatic wave beams, Sov. Phys. J. , 908 (1988).[51] Chuanpu Liu, Jilei Chen, Tao Liu, Florian Heimbach,Haiming Yu, Yang Xiao, Junfeng Hu, Mengchao Liu,Houchen Chang, Tobias Stueckler, Sa Tu, YouguangZhang, Yan Zhang, Peng Gao, Zhimin Liao, DapengYu, Ke Xia, Na Lei, Weisheng Zhao, and MingzhongWu, Long-distance propagation of short-wavelength spinwaves, Nat. Commun. , 738 (2018).[52] A. A. Stashkevich, M. Belmeguenai, Y. Roussign´e, S. M.Cherif, M. Kostylev, M. Gabor, D. Lacour, C. Tiusan, and M. Hehn, Experimental study of spin-wave disper-sion in Py/Pt film structures in the presence of an inter- face Dzyaloshinskii-Moriya interaction, Phys. Rev. B91