Hippopede curves for modelling radial spin waves in an azimuthal graded magnonic landscape
HHippopede curves for modelling radial spin waves in an azimuthal graded magnoniclandscape
D. Osuna Ruiz * , ∗ A. P. Hibbins , and F. Y. Ogrin Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom. (Dated: August 28, 2020)We propose a mathematical model for describing radially propagating spin waves emitted fromthe core region in a magnetic patch with n vertices in a magnetic vortex state. The azimuthalanisotropic propagation of surface spin waves (SSW) into the domain, and confined spin waves (orWinter’s Magnons, WM) in domain walls increases the complexity of the magnonic landscape. Inorder to understand the spin wave propagation in these systems, we first use an approach based ongeometrical curves called ‘hippopedes’, however it provides no insight into the underlying physics.Analytical models rely on generalized expressions from the dispersion relation of SSW with an arbi-trary angle between magnetization M and wavenumber k . The derived algebraic expression for theazimuthal dispersion is found to be equivalent to that of the ‘hippopede’ curves. The fitting curvesfrom the model yield a spin wave wavelength for any given azimuthal direction, number of patchvertices and excitation frequency, showing a connection with fundamental physics of exchange dom-inated surface spin waves. Analytical results show good agreement with micromagnetic simulationsand can be easily extrapolated to any n-corner patch geometry. I. INTRODUCTION
Due to their low loss and shorter wavelength comparedto electromagnetic waves in free space, spin waves are apromising candidate for comunicating information in mi-cron and sub-micron scale magnonic circuits [1–3]. Spinwave spectra of magnetic circular nanodots have beenstudied intensively [4–7]. When an in-plane magneticfield excitation is applied to a vortex spin configuration,the lowest energy mode that can be excited is the gyra-tion of the vortex core, which depends upon the aspectratio of the disc [8]. At higher frequencies, higher ordergyrotropic modes [9, 10] and a complete set of modes re-lated to azimuthal and radial spin waves appear [4]. Thelatter type of spin waves are related to Damon-Eshbachmodes where k is perpendicular to M in a vortex coreconfiguration [11], and their spectra is strongly depen-dent on thickness and more generally, on the physicalgeometry of the patch. Moreover, magnetization inho-mogeneities such as vortex cores have attracted atten-tion as spin waves emitters [12, 13]. It is known that,due to the confinement or the natural magnetic stateof the sample, inhomogeneities of the internal magneticfield can be sources of spin waves due to a graded in-dex in the magnonic landscape [14, 15]. Spiralling spinwaves found in vortex configurations have been explainedas hybridization of stationary azimuthal spin waves andhigher order gyrotropic modes, therefore showing no ra-dial propagation [12, 16, 17].Spin waves with spiral or circular wavefronts have beenreported through micromagnetic simulations and experi-ments in simple elements such as circular discs or squarepatches [12, 13, 18]. In Ref. [12], the authors proposean analytical expression for the dispersion relation of ra-dially propagating exchange-dominated spin waves from ∗ [email protected] the core region, which are explained as laterally emittedspin waves from a first order gyrotropic mode of the vor-tex core. Therefore, they manifest a ’Surface Spin Wave-like’ (’SSW’) propagating behavior (since M is perpen-dicular to k ).In the past, a vast work on analytical modelling hasdealt with magnetically non-saturated structures pre-senting domain walls. Some examples are spin waveemission from Bloch domain walls [14], reflection andtransmission across domain walls [19, 20] or magneticconfiguration in a transition between domain walls types[21]. These models help to provide insight into the dy-namics and a tool for modelling spin wave phenomena inconfined structures that show a more complex magnoniclandscape than saturated dots. To the best of our knowl-edge, we have not found in the literature a generalizedmathematical model dedicated to the particular physicsof the spiral or circular spin waves emitted from a pointsource in confined structures of more complex geometriesthan a circular disc, which are expected to considerablyreshape the radial wavefront [18].Following on these studies, in this work we report ona model for the observed wavefront of spin waves emit-ted from an almost point source (e.g., a vortex core) inany n-vertex patch, which implies the existence of do-main walls, azimuthally distributed across the geometry.The final expression is derived from a generalisation ofthe dispersion relation of surface waves for an arbitraryangle between magnetisation M and wavenumber k . Forn >
2, the patch adopts the form of a regular polygon,for n = 2 and n = 1, the model considers two or one sin-gle vertices. Finally, n = 0 implies a circular disc. Theobtained curves from the model agree well with numer-ical results. In section II we describe our models andthe numerical method used for their validation. In sec-tion III, we provide with a comparative of both meth-ods and their validation through numerical simulationsas well as discussion of results. This model can help on a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug the description of complex spin wave wavefronts in non-saturated elements, which avoids running numerical sim-ulations for particular shapes. Due to the attention thatthe experimentally observed short-wavelength radial spinwaves have recently drawn, we believe this model is of in-terest to researchers, experimentalists mainly, working inthe field of spin waves emitted from a point source. II. NUMERICAL METHODS ANDCALCULATIONS
In order to obtain an analytical solution for the spinwave wavefront in a non-saturated patch, we must makesome initial approximations. This is due to the remark-able complexity of the magnetic configuration within thepatch along the azimuthal direction, specially near mag-netically inhomogeneous regions, which are determinedby shape anisotropy and dipolar and exchange interac-tions solely. Thus, in our first approach we mathemati-cally infer a general equation from numerical results andin our second approach, we extrapolate analytical resultsfrom a simpler case scenario to ours. Despite of the ap-parent crudity of these extrapolations, both models showgood agreement and therefore considered reliable for, atleast, descriptive purposes. However, our second ap-proach is more fundamental and still, even after a crudegeneralization, it also shows very good agreement withmicromagnetic results.To obtain more insight into the dynamics and con-firm the performance of our model, we performed a setof micromagnetic simulations using Mumax3 [22]. Wesimulated a circular microdisc with the typical materialparameters of permalloy at room temperature with satu-ration magnetization M s = 8 × Am − , exchange con-stant A ex = 1 . × − Jm − , Curie temperature froma weighted average of iron and nickel T C = 270 K andGilbert damping constant α = 0 . d of 900 nm and thickness t of 80 nm were simulated. The grid was discretized inthe x, y, z space into 512 × ×
16 cells. The cell sizealong x and y was 3.9 nm, while the cell size along z wasfixed to 4 nm. The cell size along three dimensions is al-ways kept smaller than the exchange length of permalloy(5.3 nm). The number of cells was chosen to be powersof 2 for sake of computational efficiency. We also set a‘smooth edges’ condition with value 8 [22]. A key pointin micromagnetic simulations is to achieve a stable equi-librium magnetization state. We first set a vortex statewith polarity and chirality numbers of (1, −
1) and thenexecuted the simulation with a high damping ( α = 1) )to relax the magnetization until the maximum torque/ γ (‘maxtorque’ parameter in Mumax3) reached 10 − T in-dicating convergence and the achievement of a magneti-zation equilibrium state. The typical time to achieve theequilibrium state was 100 ns. Once the ground state wasobtained, damping was set back to α = 0.008 and the relaxation process repeated. The microdisc spin config-uration was recorded as the ground state of the sampleand then used for the simulations with the dynamic ac-tivation.For analyzing time evolution of the magnetic signal,we apply a continuous wave excitation at the core regionwith a magnetic field B at a specific frequency f , B ( t ) = A sin(2 πf t )) , (1)where f is the microwave excitation frequency andpulse amplitude A = 0.3 mT. This is small enough toremain in the linear excitation regime and avoid anychanges to the equilibrium state. A sampling period of T s =25 ps was used, recording up to 200 simulated sam-ples in space and time, only after the steady state isreached.In the next sections we describe the proposed mod-els and their derivations. Finally, validations for each ofthem through numerical simulations are shown. A. First approach
For this study, we mathematically infer a fitting modelfrom numerical results on the first obtained shapes whenn = 0, 1, 2, 3, 4... and so on. We then generalize it to anyn-vertex patch. For the case of n = 1 we take an internalangle assumed to be π/ n = 2, we also make a similar assumption and the patchresembles to a ‘double teardrop’ shape. For larger valuesof n , the internal angles of the vertices are the internalangles of the regular polygons, defined as, 2 π ( n − /n .Of course, for n = 0 we have a circle. The vertices aredistributed around the shape, separated by 2 π/n radiansand the resulting domain walls spaced by π/n .Regarding magnetic configuration in equilibrium af-ter a relaxation process, and assuming a centred vor-tex core, ( n −
1) triangular domains and n domain wallswill form in the patch. In contrast to the circular dot,the azimuthal distributed domain walls will distort thewavefront of the propagating spin wave from the coreregion, introducing an azimuthal dependence (or equiv-alently, n − dependence) to the wavelength of the radialwave ( λ ( θ )). Also, two known values for the spin wavewavelength can be analytically deduced for any n-cornerpatch: the characteristic wavelength of an exchange dom-inated surface spin wave ( λ SSW ), when k is perpendicularto M (this is, when the spin wave propagates into the do-main), and the characteristic wavelength of the confinedspin wave along the domain wall ( λ WM ), also known asWinter’s magnon [23]. Fig. 1 shows the characteristicdispersion relations for the laterally emitted spin wavefrom the vortex core from Ref. [12] (blue curve) and theexchange-dominated Winter’s magnon in an ideal 180 de-grees Bloch wall [24] (orange curve). It is worth notingthat, in an n-vertex patch, the formed domain walls willbe of the angle of the vertex. For example, in a square (n= 4), this is an angle of 90 degrees (see the top-right insetin Fig. 1). Micromagnetic simulations (not shown here)show that the expected wavenumber is reduced with re-spect to the 180 degrees Bloch wall, due to the shapeanisotropy of the sharp corner. This latter study is notin the scope of this article (although it will be addressedin a future work) and the effects of an intermediate do-main wall are not included in our model. For practicalpurposes, we obtain k WM from the dispersion relation ofan ideal 180 degrees Bloch wall (see orange curve in Fig.1). For excitation frequencies at which both modes co-exist, the wavenumbers (or wavelengths) that fall in thegrey area, delimited by k SSW and k WM , can relate to theazimuthal-dependent wavelength of the radial spin wavein the patch. FIG. 1. Dispersion relations for the laterally emitted spinwave from a first higher order dynamical core, see Ref.[12],and a Winter’s magnon (WM), for the material parametersindicated in section II, as limiting cases. Dashed horizontalline indicates an excitation frequency of 8.8 GHz. Grey areahighlights the expected magnitudes for intermediate wavevec-tors (see bottom-right inset) between the two limiting cases,when being simultaneously excited. (Insets) Colour arrowsshow the wavevectors for the laterally emitted spin wave (ina SSW configuration) and the Winter’s magnon, which are as-sumed k WM ≈ k SSW for the excitation frequency of 8.8 GHz(length of the vectors represent their magnitudes). Black ar-rows show the orientation of magnetisation in the domains.Bottom inset shows a simplified schematic of the angular dis-tribution of wavevector at the top-right corner of a squarepatch (n = 4). Grey arrow is an intermediate case for thespin wave wavevector between the limiting cases, from whatan angular dependence can be inferred.
The proposed model is based on the mathematical ex-pressions of a family of curves known as ‘hippopedes’.Since the polar representation of these curves allow asmooth azimuthal transition from a certain wavelength(maximal) to another finite value (minimal), these curvescan be used here as a generalization of the problem sce-
FIG. 2. (a) Schematic of various magnetic patches dependingof the number of vertices n with magnetization around thecore (solid black line). Domain walls bisect the sharp corners.(b) Polar representation of Eq. (2) showing the azimuthalvariation of wavelength within the shape, with the appropri-ate rotation of the patch ( φ ) so the equation describes cor-rectly the shape of the patch. Minimum and maximum radialamplitudes are assumed λ SSW = 2 λ WM (length of the vec-tors represent their magnitudes). In terms of the hippopedesparameters a and b , this means a ratio of b/a = 0 . nario. By applying these expressions to the particularscenario, i.e. a magnetic patch with regularly distributedvertices and domain walls, a simple expression for thespin wave wavelength can be obtained. From a coniccanonical equation with geometrical parameters a and b ,see Supplemental Material (1) for a more detailed de-scription, then the generic equation of the resulting hip-popede in polar coordinates is, f ( θ ) = 2 √ b (cid:114) a − b · sin ( θ n φ ) , (2)where the phase parameter φ sets an initial rotation an-gle for the patch. For n = 2 and b = 2 a , Eq.(2) leads to aspecial case known as Bernoulli’s Lemniscate. In fact, awhole family of lemniscates can be obtained from the hip-popedes if b > a . More information on the ‘Hippopedecurves’ can be found in the Supplemental Material (1).The ‘hippopedes’ when b < a , known as Booth’s ovals, al-low a transition from a finite wavelength maximum value( λ SSW ) to another non-zero minimum value ( λ WM ), seeFig. 2(b). The particular values of the geometrical pa-rameters a and b can be found from the ‘hippopede’ gen-eral equation particularized to the wavelength limitingconditions of a maximum λ SSW at every 2 π/n angle anda minimum λ WM at every 2 π/n + π/n angle. An initialrotation angle of φ = 0 is assumed. The ratio b/a isfound to be equal to 1 − λ /λ . In the range of fre-quencies under study, λ SSW > λ WM is satisfied, so thisimplies we can model our system with hippopede curveswhere b < a . The initial phase rotation φ in Fig. 2(b)(and hereafter) is chosen so it properly coincides with thenumerically modelled patch. Therefore, a more completeexpression for our model is, λ ( θ, n ) = λ SSW (cid:115) − (1 − λ λ )sin ( θ n φ ) . (3)Fig. 2(b) shows a collection of curves from Eq.(3) fordifferent number of vertices n . Following from the mag-netic configuration of the patch in the vortex state andassuming for example λ SSW = 2 λ WM , which yields a ra-tio b/a = 0 .
75 ( b < a ), results qualitatively show anazimuthal changing wavelength around the centre of theshape according to a hippopede curve. In Section II.B,we derive a more generalized expression that yields a con-nection with the fundamental physics.
B. Second approach
In this case, we start from the analytical expression ofthe dispersion relation of surface spin waves given an ar-bitrary angle α between M and k . Damon and Eshbach[11] found a generalised expression for the dispersion re-lation of surface spin waves in a semi-infinite stripe forarbitrary angles α and β , being α the angle between theferromagnetic planar body surface and effective field H i ,and β the angle between the orthogonal direction of thateffective field and wavenumber k . Extended to the ex-change regime, it can be expressed as, ω = γH i α cos β + γB i α cos β + ω M λ ex k . (4)In Ref. [11], the original expression is derived for mag-netostatic spin waves in a semi-infinite stripe. It shouldbe noted that our problem scenario, although being a fi-nite sample, can still be regarded equivalent due to the short wavelength of the spin wave modes under study[11]. In Ref. [11], the expression shows a continuousvariation of the spin wave wavelength as angle increases,towards the limiting scenario of a Backward Volume spinwave (BVSW) configuration. It is worth to note that,in our problem scenario, that limiting case would not bethe BVSW dispersion relation but the Winter’s magnon’s(see Fig. 1). Also, in non-saturated samples, Eq. (4)can be reduced by specifying only in-plane magnetiza-tion ( α = 0), assuming the internal field in the magneticdomain is H i = − M S (therefore, H i = M S and B i = 0)in absence of an external biasing field. For β = 0, k isperpendicular to M ( k ⊥ M , and therefore λ = λ SSW (seeinset in Fig. 1). Hence, for a specific excitation frequency ω = ω we can rewrite Eq. (4) in terms of a variablewavelength ( λ = 2 π/k ) in the azimuthal direction β (asdefined in Fig. 1 and in Ref. [11]) as (a step-by-stepderivation is shown in Supplemental Material (2)), λ ( β ) = λ SSW (cid:115) (1 − p )cos β cos β − p , (5)where β = θ , being θ the azimuthal direction as definedin Eq. (2) and Eq. (3) that coincides with the angle β , setting the reference for an azimuthal dependence of k at θ = 0, and p = ω M / ω , where ω M = γM s . Weobtain a classical surface spin wave dispersion behaviorfrom Eq. (5) when θ = 0. In Eq. (2), the initial arbitraryrotation phase of the patch can be conveniently chosenas φ = π/ θ = 0, as reference point. Thisimplies we can substitute sin( θ + φ ) → cos( θ ) in Eq. (2),which keeps the reference λ (0) = λ SSW consistent withEq. (5). Eq. (5) implies a decreasing wavelength as theangle θ (or equivalently, β ) increases. For a flux closuremagnetisation in the patch, the reference angle coincideswith the direction of propagation into a first magneticdomain.In the model, parameter p = ω M / ω yields a connec-tion between the observed radial wavefront in simulationsand the magnetic properties of the material but sets anupper frequency bound for the model, which is not phys-ically meaningful. The model from Eq. (5) yields imag-inary values for cos( θ ) < p ≤ ω < ω M / θ from 0, although it does not apply when θ → π/
2, where backward volume spin wave propagatesinstead of surface spin waves according to Ref.[11].Also, this preliminary model assumes in-plane mag-netisation for all azimuthal directions, so it still does nottake into account effects of magnetic inhomogeneities,i.e., the domain walls. Fig. 3 shows a collection of curvesfrom Eq. (5) illustrating a periodic effect when an n num-ber of corners is included ( θ → nθ/ − π/n < θ < π/n and n > FIG. 3. Collection of curves from Eq.(5) for n = 0 , , n = 3 and higher orders ( n >
4) can be easilyinferred. Physically non-realizable zeroes are placed at every π/n angle.
As stated above, imaginary values and zeroes of Eq.(5) due to the frequency dependence and geometry, arenot physically meaningful in our scenario. This is due tothe imposed lower frequency gap for the FMR in the discand the presence of domain walls, respectively. To in-clude these phenomena and avoid the zeroes in the model,Eq. (5) has to be generalised with the undefined parame-ter σ and re-scaling factor (cid:15) so the expression is extendedas, λ ( θ, n ) = (cid:15) · λ SSW (cid:115) σ + (1 − p )cos( n θ )cos( n θ ) − p . (6)In an n-corner patch, the spin wave shows a wave-length of λ SSW at every 2 π/n angle and of λ WM at ev-ery π/n angle. These limiting conditions allow to findthe values of parameters (cid:15) and σ . The values for (cid:15) and σ are found to be: (cid:15) = ( (cid:112) λ − λ ) /λ SSW and σ = λ / ( λ − λ ), which simplified for λ WM = 0,lead to the equation from Ref. [11]. Due to the az-imuthal periodicity every π/n (in contrast to the sce-nario described in Ref. [11]), cosine terms in Eq. (6)must only take positive values. Also, since taking theirabsolute value would yield a non differentiable functionat the angle where the domain wall is encountered, thecosine terms are replaced by their squared values. As-suming λ WM (cid:54) = 0 and after algebraic transformations,the modified equation is, λ ( θ, n ) = λ WM (cid:115) − (1 − λ λ ) (1 − p )cos ( n θ ) − p cos ( n θ , (7) where p = ω M / ω . Our key result is obtained if p >> − p ) / (cos ( nθ/ − p ) ≈
1. Then, theresultant equation is indeed the Hippopede curve equa-tion (assumed φ = π/ n , so the directions for λ WM and λ SSW are exchanged) described in Section II.A, whichexplains the good fitting to these curves at low frequen-cies ( ω << ω M / FIG. 4. Ratio (black solid curve) between wavelengths fromthe dispersion relation from [11] ( λ DE ) and the extendedmodel linked to the Hippopedes ( λ Hp ) for p ≈ .
75 showinggood agreement, and practically no deviation from unity atany angle apart from the domain wall region for p ≈ . >> n = 0(blue), n = 1 (red), n = 2 (yellow) and n = 4 (purple). Inset(right): Comparison of the normalised results from Eq. (7)(solid purple curve) and Eq. (5) (dashed purple curve) for n = 4 and p ≈ . Fig. 4 (left inset) shows a collection of curves ob-tained from Eq. (7) normalized to λ SSW , assuming λ SSW = 2 λ WM (see Fig. 1). In the right inset, a par-ticular case for n = 4 is shown in comparison with therespective solution from Eq. (5) and the relative er-ror between both equations for all angles between 0 and π/n , normalised to the number of vertices n for a value p ≈ .
75 (black solid curve), obtained from assuming ω / π = 8 . ω M / π ≈
27 GHz. For a large enough value of p ,assumed to be about six times larger ( p ≈ . p >>
1, results show indeed a minimal differ-ence with the hippopede curves (red solid curve), almostnegligible away from the domain wall regions. The pro-posed model, as an extension of Eq. (5), avoids the az-imuthal zeroes and shows a minimal difference with thevalues from the exchange dominated ‘surface spin wave’dispersion. Therefore, Eq. (7) is a suitable model, de-rived as a generalization from Eq. (5). In Section III, weprovide numerical evidence of its reliability.
III. NUMERICAL RESULTS AND DISCUSSION
In this section we compare the models previously de-scribed to numerical micromagnetic simulations in orderto validate them. Radial spin waves with a spiral pro-file from the core region can also propagate, when theexcitation signal is applied in-plane of the patch [13].This effect can be added to the model in terms of anormalised azimuthal factor that creates a counterclock-wise spiralling effect as observed in the simulated wave-fronts ( λ ( θ ) → θ π λ ( θ )). In micromagnetic simulations,a continuous out-of-plane wave excitation of frequency ω / π = 8 . λ SSW = 135nm and λ WM = 89 nm. The condition λ SSW > λ WM issatisfied by applying an oscillating magnetic field in theGHz range. We need to address that the main objectiveis to test the relative change between these two wave-lengths, regardless of their absolute values. Fig. 5(a)shows snapshots of the dynamic out-of-plane magneti-sation from micromagnetic simulations of two differentshapes, a ’double’ teardrop shape (two vertices) and asquare (four vertices). Their respective k-space mapsfrom each image are shown on the right, where the whitearrows indicate the propagation of the main modes, sur-face spin-waves and Winter’s magnons. The images areinterpolated for clarity. Before performing a spatial FFT,a Hamming window of 256 points is applied to the dataset to avoid image artifacts due to reflections at the edgesand spurious high frequency values.Values of wavelength are extracted from the simulatedk-space images at angles from 0 rad to π/ π/ π/
16. The error bars are found after interpolation, yield-ing an error in wavelength of approximately 8 nm. Thechange in wavelength given by the model shows verygood agreement with numerical results and follow thepredicted trend. The analytical results can easily be ex-trapolated to any n-corner shape. Taking this into ac-count, we can confidently say that the proposed modeldescribes the spin wave wavefront of an emitted spin wavefrom the core in any n-corner shape with accuracy.Finally, in any n-corners shape presenting angular peri-odicity, the azimuthal transition from a surface spin waveof wavelength λ into that of a confined mode along a do-main wall λ < λ is of the form of an Hippopede curveor, more generalised, FIG. 5. (a) (Left) Snapshots of numerical results for a ‘dou-ble’ teardrop shape ( n = 2) and a square ( n = 4). Red arrowsindicate the direction of propagation of the two main modes(surface spin wave and Winter’s magnon). (Right) k-space ofthe snapshots where the wavenumber profile (inverse of thewavelength profile) is shown. Results are interpolated to 5 ex-tra points between data points for clarity. White arrows showthe direction of propagation of the main modes. (b) Compar-ison of the maximum values of λ = 2 π/k found in (a) for thelq double’ teardrop shape (blue) and the square (orange) withthe results from Eq. (6) where λ SSW = 135 nm and λ WM = 89nm found from numerical results at ω / π = 8 . p ≈ .
75. Error bars are found after the interpolation pro-cess, which introduces a measured error of approximately 8nm. λ ( λ , λ , θ, n ) = λ f ( λ , λ , θ, n ) (cid:114) − g ( λ , λ )cos ( θ n , (8)where functions f and g are generic functions of the in-dicated magnitudes.As a suggested improvement to the model, a radialdependence could be included into Eq. (7).The radialdependence should consider that, as n increases, the do-main walls will merge closer to the core region, whichimplies no azimuthal gradient and the wavefront profilewill be that of a disk (equivalent to n = 0) under a criticaleffective radius.The Hippopede curves are obtained from Eq. (7) underthe condition: p >>
1. It is worth noting that the con-dition √ λ WM > λ SSW > λ WM , which implies an ‘hip-popede wavefront’ with geometrical parameters b < a , isnot necessary satisfied for all values of λ ( θ ) at every ω ,as explained before. Although the extended model avoidsan upper frequency bound, there is a lower frequencybound from which only spin waves will be radiated intothe domains from the core due to the non-zero internalfield there. In contrast, a gapless mode can propagatein the domain walls [24]. As ω increases, at the limitwhen λ SSW ≈ λ WM , the wavefront tends to a circularprofile, as explained elsewhere [18] and as Eq. (7) con-sistently predicts. This frequency dependence is alreadyimplicit in the model in parameter p = ω M / ω . At highfrequencies where ω ≈ ω M / p ≈ p >> λ SSW ≈ λ WM , Eq. (7) ef-fectively yields a wavelength of λ ( θ, n ) = λ SSW , this is, acircular wavefront, which is confirmed by micromagneticsimulations and elsewhere.At even higher frequencies ( ω >> ω M ), p << − p ) / (cos ( nθ/ − p ) ≈ / cos ( nθ/ λ ( θ, n ) = λ SSW , regardless of λ SSW and λ WM values. This implies that Eq. (7) is still a validmodel even for p ≈ p <<
1, or in other words, itdoes not show an upper frequency bound for radial waves.This is consistent with the physical scenario to describeand previous work on radial propagating spin waves.Previous work on exchange-dominated radial spinwaves have predominantly dealt with direct observationor experimental detection, as referred to in section I. Webelieve that our results may help in obtaining furtherinformation of these spin waves such as an expressionfor a spatial-dependent wavelength, potential detectionof magnetic inhomogeneities in magnetic films of variousgeometries or characterization of the material properties,when used, for example, as a fitting tool. We would alsolike to highlight the applicability of these mathemati-cal curves itself. We believe that, in addition to theirknown applications in mechanical linkages (see Supple-mental Material (1)), the work proposed here is anotherinteresting use of these curves for modeling in physicsand in particular, a novelty in Magnetism.
IV. SUMMARY
We have used geometrical expressions to successfullymodel propagating spin waves from the vortex core regionin n-corners elements in a magnetic flux closure config-uration where domain walls are present. The proposedmodels are validated and all show very good agreementwith numerical results. The equations can be generalisedto any n-corner shape, including non regular shapes suchas a teardrop shape (one corner) and a double teardropshape (two corners). A first model is based on a spe-cial case of the ‘hippopede’ curves, known as Booth’sovals, since they allow smooth transitions between twoknown wavelength values. A more exact model is ob-tained straight from generalising the fundamental equa-tion of surface spin wave dispersion, which can be re-trieved by setting λ WM = 0. This more compact modeldescribes the spin wave wavefront accurately at positionsfar from the core and specially, close to the inhomoge-neous areas (i.e., domain walls). Interestingly, throughalgebraic transformations, the final equation of the model(where λ WM (cid:54) = 0 and p >> λ WM and λ SSW ) can be retrieved from the geometrical parametersof the plotted Hippopede curve ( a and b ). Given the fre-quency of the oscillating field ω , parameter p and there-fore ω M and M S can also be retrieved.The model from Eq.(7) also takes into account thefrequency dependence of the oscillating field. At lowerfrequencies, the model yields hippopede curves for thespin wave wavefront profile. At higher frequencies, it ef-fectively leads to circular wavefronts, as expected fromnumerical results.The models can also be applied on spiral wavefronts,although they are originally defined for in-phase wave-fronts, which makes them suitable for also describingcircular/non-spiralling wavefronts. For modelling spiralwavefronts, the equation must be modified accordinglyby simply introducing a normalised spiralling effect fac-tor. We hope these results help to better understand thepropagating features of spin waves in confined structures,more specially those emitted from quasi-punctual sourcesand how to control their dynamical properties. V. ACKNOWLEDGEMENTS
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