Graded Index Confined Spin Waves in an Intermediate Domain Wall
GGraded Index Confined Spin Waves in an Intermediate Domain Wall
D. Osuna Ruiz * , ∗ A. P. Hibbins , and F. Y. Ogrin Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom. (Dated: September 14, 2020)We propose a mathematical model for describing propagating confined modes in domain wallsof intermediate angle between domains. The proposed model is derived from the linearised Blochequations of motion and after reasonable assumptions, in the scenario of a thick enough magneticpatch, are accounted. The model shows that there is a clear dependence of the local wavenumberof the confined spin wave on the local angle of the wall and excitation frequency used, which leadsto the definition of a local index of refraction in the wall as a function of such angle and frequency.Therefore, the model applies to 1-D propagating modes, although it also has physical implicationsfor 2-D scenarios where a domain wall merges with a saturated magnetic region. Micromagneticsimulations are in good agreement with the predictions of the model and also give insight on theeffects of curved finite structures may have on the propagating characteristics of spin waves indomain walls.
I. INTRODUCTION
Due to their low loss and shorter wavelength comparedto electromagnetic waves in free space, spin waves area promising candidate for information carrier in micronand sub-micron scale magnonic circuits [1–3]. Inhomo-geneities such as vortex core have been widely studiedas spin waves emitters [4–6]. Once the spin wave is ex-cited, an adequate control on its propagation is key forthe developing of circuits that conducts the spin wavethrough channels. Local excitation of spin waves andits spatial confinement has been widely studied in termsof local ferromagnetic resonances due to inhomogeneities[7–9], confinement along the edges [10, 11], along domainwalls and by domain wall natural fluctuations modes orso called Winter magnons [12]. Domain walls do actuallyact as natural channels for spin waves due to the energywell. More importantly, Winter magnons are very usefulfor efficiently channeling in a wide range of frequenciessince they are gapless modes [13].Graded index media for wave propagation have beenwidely studied specially in electromagnetics, in the fieldknown as transformation optics [14–16]. The develop-ment of structures ranging from nanometre size to cen-timetres, have proved interesting properties not shown innature, leading to the application of novel graded-index(GRIN) lenses in an extremely wide range of frequencies,from microwaves to visible light.In the field of magnonics, graded index magnetic me-dia serve to similar purposes, taking advantage of thehigh anisotropic behaviour of spin waves [7, 17]. Forexample, tailoring the spin wave propagation in mag-netic domains, allow the development of lenses for spinwaves[18–20]. Regarding uni-directional propagation indomain walls, previous studies have dealt mainly withredirection and steering the spin wave path [21, 22], in-ducing phase changes [23, 24] or non-reciprocal paths bymeans of non-linear effects [25]. ∗ [email protected] However, anisotropic magnetic media can not onlymodify direction, intensity, or temporal frequency of thespin wave. Spatial frequency modulation or spatial chirp-ing is a technique widely used in telecommunication en-gineering and photonics, for example, to use in fibre-Bragg gratings or other chirped mirrors as filters, wherewavenumber or equivalently, wavelength ( λ = 2 π/k ),spatially changes. Analysis of spatially chirped sig-nals can be extended even to the processing of imageswhere periodic features are visualised in perspective. Thegraded-index technique for EM waves could also haveits equivalent for spin waves, and therefore, mathemati-cal tools are required to control this feature. Using thenon-uniform demagnetising field in a saturated YIG non-ellipsoidal rod, the pioneer work from Schlmann [26] forBackward Volume Spin waves and from Stancil [27] forSurface Spin waves and further experimental results us-ing spatially varying external fields [28] confirmed therealization of this technique for spin waves. A changeof wavelength has also been observed by using taperedsaturated magnonic waveguides [29]. In this article, wedemonstrate a spatially dependent wavenumber (spatialdispersion) of confined modes in domain walls (and there-fore, in non-saturated films), providing with an equa-tion that allows to model their local wavenumber andpropagation properties. Micromagnetic simulations on aspecifically designed shape of patch, that retains in rema-nence a single intermediate domain wall of variable anglebetween two vortex cores, show good agreement with ourproposed model. II. MATERIALS AND METHODS
To obtain insight into the dynamics, we performeda set of micromagnetic simulations using Mumax3 [30].We simulated a ‘rounded bow-tie’ shaped patch (see Fig.3(a)) of 6000 nm length, 80 nm thickness ( t ) and 2000 nmdiameter ( d ) of the circumscribed circles at the ends, withthe typical material parameters of permalloy at roomtemperature with saturation magnetization M s = 7 . × a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Am − , exchange constant A ex = 1 . × − Jm − and Gilbert damping constant α = 0 .
008 for a weightedaverage of iron and nickel. The grid was discretized inthe x, y, z space into 1536 × ×
16 cells. The cellsize along x and y was 3.9 nm, while the cell size along z was fixed to 4 nm. The cell size along three dimen-sions is always kept smaller than the exchange length ofpermalloy (5.3 nm). The number of cells was chosen tobe powers of 2 for sake of computational efficiency. Wealso set a ‘smooth edges’ condition with value 8. A keypoint in micromagnetic simulations is to achieve a stableequilibrium magnetization state. We first set a doublevortex state with polarity and ‘vorticity’ numbers of (1, −
1) and (1,+1) and then executed the simulation with ahigh damping ( α = 1) ) to relax the magnetization un-til the maximum torque (‘maxtorque’ parameter in Mu-max3), which describes the maximum torque/ γ over allcells, where γ is the gyromagnetic ratio of the material),reached 10 − T indicating convergence and the achieve-ment of a magnetization equilibrium state. The typicaltime to achieve the equilibrium state was 100 ns. Notethat this value has no direct physical meaning due to theartificial high damping. Once the ground state was ob-tained, damping was set back to original ( α = 0.008),relaxation process was repeated, the spin configurationwas recorded as the ground state of the sample and thenused for the simulations with the dynamic activation. Foranalyzing time evolution of the magnetic signal, we ap-ply a continuous wave excitation with a magnetic field B at a specific frequency f in the first vortex core regiononly, B ( t ) = B sin(2 πf t )) , (1)and each mode is excited with a relatively small oscil-lating field, B = 0 . T s =25ps was used, recording up to 300 simulated snap-shotsin space and time, only after the steady state is reached.With these parameters, the time window of observationof the spin waves propagating in the domain wall coversup to 7.5 ns.To numerically validate the change in wavelength pre-dicted by the model, we run micromagnetic simulationswith an excitation frequency of 1.5 GHz and 3 GHz soWinters magnons [13] can be efficiently launched fromthe core region and travel along the domain wall [25].Numerical results for the spatially dependent wavelength( λ (x)) are obtained from the time-averaged spectrogramsof the channelled spin wave profiles using a Hanning win-dow of 128 FFT points and 100 overlapping points. Sincethe spatial wavelength is constantly changing along thedomain wall length, a fixed width of the window will in-troduce a trade-off between spatial frequency resolutionand position accuracy in the x-direction. A wider spatialwindow yields less position accuracy to the wavenumberand a very narrow window cannot properly resolve thespatial frequency at large wavelengths, leading to spatialfrequency leakage per frequency bin. The chosen param- eters of window width and overlapping points are ob-tained after an optimisation process considering severaldifferent outcomes. Another solution could be using awidth-variable window as described in [31]. III. DESCRIPTION OF THE MODEL
As a first step to a reliable model and similarly to whatis done in Ref.[23], we search for an expression of a spa-tially dependent wavelength for a spin wave travellingin a domain wall, based on a Wentzel-Kramers-Brillouin(WKB) approximation [32]. Assuming a Neel-type do-main wall along the x-direction, this implies that dy-namic magnetization components at its centre can be ex-pressed as: m x = m e iωt , m y = M s , m z = m e iωt e ik ( x ) x .As it will be shown later in this section, this description ofstatic and dynamic magnetization is still approximatelycorrect even for a Bloch wall of an arbitrary angle andsuitable, at least, for the in-plane component of magne-tization. If we assume an internal magnetic field per-pendicular to the wall and only related to dipolar andexchange interactions, H i = ( H d + H ex ) y , where the de-magnetising field is H d = H d ( x ) and the exchange field is H ex = 2 A ex ∇ m ( x ) / ( µ M s ). As in Ref. [23], in order tofind an expression for a spatial dependent wavenumber k ( x ) accounting for the demagnetising field magnitude H d ( x ) = | H d ( x ) | , we define a compact expression formagnetisation φ = m x + im z , assuming φ ∼ φ e ik ( x ) x andneglecting the imaginary terms in order to obtain real-valued solutions, we reduce the linearised Bloch equa-tions of motion to the following first-order, non-lineardifferential equation, ω | γ | = | H d ( x ) | + 2 A ex µ M s (cid:32) x (cid:18) dkdx (cid:19) + 2 x dkdx + k (cid:33) , (2)where ω is the excitation frequency and γ the gyro-magnetic ratio. As a first approach to solve this dif-ferential equation, we can naively assume that variationsof spin wave wavelength along the domain wall will besmooth, since no sudden changes in the demagnetizingfield along the longitudinal direction are expected in astraight domain wall configuration. This implies thatthe first derivatives with respect to x can be neglected.Similarly to Schlmanns work [26], clearing k ( x ) in Eq.(2)gives an approximate description for a spatially depen-dent wavenumber, as a function of the module of thedemagnetizing field in the wall, k ( x ) = (cid:114) k − µ M s A ex | H d ( x ) | , (3)where k = (cid:113) µ M s A ex γ ω is the wavenumber for a confinedspin wave of frequency ω when the transversal in-planedemagnetizing field is zero, or in other words, in a 180degrees Bloch domain wall [25].Finding a general demagnetizing field expression in anon-saturated ferromagnet of a non-ellipsoidal, arbitraryshape can be a very tough task. To address this, wefirst provide a magnetostatic first approach for such ascenario based on how in-plane magnetization transver-sally changes through the wall, this is, from one domaininto the other. In other words, we aim to establish a linkbetween the Eq. (3) and the angle ( α ) between domains. FIG. 1. Schematic of a domain wall in terms of the domainmagnetisation (red arrows) and their angle ( α ) with respectto the domain wall of initial width ∆(0) = ∆ in a sample ofthickness t . The domain wall borders (green areas) and thedomain wall centre (black area) are shown. The absolute andrelative coordinate systems, chosen for the calculations, arealso shown. Let us assume the scenario from Fig. 1 for the domainwall of initially constant width ∆( x ) = ∆ and angle α that magnetisation m , makes with the straight domainwall. Solving the static Landau-Lifshitz-Gilbert equationyields the Walkers profiles across the domain wall [33].Below a critical angle α c , a mixed Bloch-Neel behaviourof the wall is obtained, where the Neel component is al-ways dominant if t ∼ ∆ [34]. A sin( α )-dependent co-efficient is naively added to the magnetisation in-planecomponents transverse to the wall ( m y ) to consistentlymodel the angle dependence of a mixed wall. Followingsimilar calculations to those shown in Ref. [35], chap-ter 3.9 (a more detailed mathematical description of thederivation of the following equations can be found in Sup-plemental Material (1)), and considering that M s = | m | must be satisfied for every y-position across the domainwall, the three components of magnetisation in the wallregion can be regarded as, m x = M s (cid:115) − sin ( α )sech (cid:18) y∆ (cid:19) − cos ( α ) ξ ( α, y) (1 − sin( α )) (4) m y = M s sin( α )sech (cid:18) y∆ (cid:19) (5) m z = M s cos( α )1 − sin( α ) ξ ( α, y) (6) where ξ ( α, y) = 1 + sin( α )cosh (cid:16) cos( α ) y∆ (cid:17) sin( α ) + cosh (cid:16) cos( α ) y∆ (cid:17) − sin( α ) (7)Following a magnetostatic approach and assuming that dm x dx ≈ dm z dz ≈ ρ m ( r (cid:48) ) = ρ m (y) = ∇ M (y) = dm y dy ) can be found, ρ m ( r (cid:48) ) = − M s sin( α )tanh (cid:16) y∆ (cid:17) sech (cid:16) y∆ (cid:17) ∆ (8)The antiderivative of the latter expression effectively re-trieves m y (Eq. (5)), which implies a ‘slow’ variation ofthe demagnetising field at the central region of the do-main wall (y → α = α ( x ), ∆ =∆ ( x ) and a cross-tie wall were formed, the term dm x dx is not necessarily zero, and therefore, it should countinto the expression of the bulk magnetic density charge.However, for the sake of simplicity, in this case we con-sider slow variations of the domain angle and domainwalls width along the domain walls length ( dαdx ≈ d ∆ dx ≈
0) and no formation of a cross-tie profile. Thisimplies that dm x dx ≈
0. Also, a thick enough patch is con-sidered, so at the central region of the patch, far enoughfrom the surface pinning effects, variations of magnetiza-tion across the thickness can also be neglected ( dm z dz ≈ σ m = m y , − m y , = 0). Choosing the right integra-tion volume under reasonable assumptions, for examplea rectangular prism of chosen dimensions equivalent tothe domain wall width (∆ ) in the x- and y-directionsand to the thickness ( t ) in the z-direction, an expressionof the demagnetising field can be found. This expres-sion, derived from the magnetostatic potential obtainedin turn from the defined magnetic density charge, Eq. (8)(see Supplemental Material (2)), is, H d (y) = − tM s sin( α )tanh (cid:16) y∆ (cid:17) sech (cid:16) y∆ (cid:17) π y (9)where t is the thickness of the magnetic patch and ∆ the‘constant’ domain wall width. A more detailed deriva-tion of Eq.(9) is shown in the Supplemental Material (2).Note that Eq. (9) is only valid for values of y in thewall region ( − ∆ < y < ∆ ). Also, Eq. (9) yields a nu-merical indeterminate at the centre of the wall althoughthis is not a physically realisable solution. At the mid-width (taking the limit y →
0) of the domain wall, wherethe confinement of the mode is strongest (in accordancewith the maximum of the ‘sech’ function), and assuming t ∼ ∆ in a thick enough sample, Eq. (9) finally leads to H d (y) = − M s sin( α )4 π (10)This equation allows to express the demagnetizing fieldperpendicular to the wall, in terms of the arbitrary angleof magnetization between the magnetic domains and thewall. Moreover, it is also consistent with the assumptionof a dominant Neel component in the domain wall profile.After obtaining the compact expression from Eq.(10),it is worth noting that at an angle of α = π/ H d ( x ), an additional demagnetising field should be de-rived from the respective magnetic potential, again de-rived from the magnetic density charges defined by dm x dx .These considerations along with a variable domain wallwidth ∆( x ) add more complexity to the model. However,as described before, the following reasonable assumptionscan be made for sake of simplicity in our approximatemodel: (1) Since the maximum intensity of the confinedmode will be at the centre of the domain wall (y → y∆ is very small regardless of the domain wallwidth while ∆ >
0, or equivalently, for α < π/
2. Underthis condition is where the proposed model is intended tobe used. Due to the equivalence between y∆ and ‘y’ wheny →
0, replacing the first term by just ‘y’ in the equationavoids the dependence on the wall width ( y∆ := y)); and(2), the main assumption for these results is that the de-magnetising field is orientated, almost fully in-plane andperpendicular to the domain wall. This assumption isnot far from reality, since it has been proved that, inintermediate domain walls (between 180-Bloch and Neelwall), the Neel component is dominant. This behaviouris observed in arbitrary -Bloch walls where α < t ∼ ∆ [34]. Eq.(10) is consistent with theseresults, since it also reduces in magnitude as α reduces.After all these considerations, we need to stress, onceagain, that finding an exact solution for a local de-magnetising field is very often, a very complicated taskfor non-ellipsoidal shapes, even in saturation or quasi-saturation [38, 39]. When not found numerically throughmicromagnetic simulations, approximate analytical ap-proaches are usually taken under reasonable assumptions [40]. For all this and realizing that Eq. (10) is a phys-ically consistent model for the defined scenario, we con-sider it from now on, as a valid first approximation forthe transverse demagnetising field along a domain wallof variable domain angle. Therefore, this expression canbe combined with Eq. (3) to give a new one where thedependence is now with the variable angle between mag-netic domains, k ( x ) = (cid:115) k − µ M πA ex sin( α ( x )) , (11)This equation relies on an initial k ( k x ) which is foundto be the wavenumber of a Winters magnon for a fre-quency ω through a 180 domain wall (i.e., when α = 0).Regarding this, through the dispersion relation of Win-ters magnons, a spatial index of refraction can be definedas n ( x, ω ) = ( k ( x )) k , n ( x ) = (cid:115) − | γ | ω | H d ( x ) | = (cid:114) − ω M πω sin( α ( x )) , (12)where ω M = γM s and ω is the frequency of a continu-ous wave excitation. This equation predicts the change inwavenumber (or wavelength) from a given k along thedomain wall. In other words, a different initial k willgive different values of local wavenumbers, but alwaysvarying in accordance to this model. Therefore, this ap-plies as well to the index of refraction if we assume anindex of unity for an arbitrary initial k . Fig. 2(a) showsa contour plot of the real values of Eq. (12) as a functionof the magnitude of H d and f = ω / π . It clearly shows,for the right combination for excitation frequency and ageneric demagnetising field magnitude, the different val-ues for the real part of the index of refraction (dark bluearea shows an entirely imaginary index). Most impor-tantly, it shows how at lower frequencies, the change inwavenumber is more sensitive to the transversal demag-netising field than at higher frequencies. Fig. 2(b) showsthe same behaviour even when Eq. (10) is introducedinto Eq. (12), which reflects now the dependence on theangle α .It can be inferred from Eq.(12) that, in absence of abiasing field, there is an upper limit for the frequencyat which an index of refraction of zero can be obtained.This maximum is reached when the demagnetising field ismaximal. In principle, in terms of the magnetic momentsorientation, this is when α = π/
2. However, notice that,in this situation, the definition of domain wall is mean-ingless. The scenario would imply the vanishing of thedomain wall into a saturated region in the y-direction(i.e., a magnetic domain). In that region, Eq.(12) canbe modified by replacing the formerly dominant demag-netising field from the (also former) domain wall by anexpression of an effective or internal magnetic field H i .Interestingly, Eq.(12) implies that an index of refraction FIG. 2. Contour plots showing the real part of (a) Eq. (12) as a function of the magnitude of the demagnetising field and (b) Eq.(12) as a function of the angle α for M S = 720 kAm − , a gyromagnetic ratio = 2 . × Hz(Am) − and A ex = 1 . × − Jm − .Red dashed line with slope γ shows the ‘n = 0’ condition in (a). of zero would be obtained at ω = | γ | H i , which is actuallythe ferromagnetic resonance main mode (i.e., all preces-sions in phase) of a saturated film for a given internalfield H i . This result agrees with the wave perspectiveof a uniform FMR precession, which lays a wavenumberof k = 0 (infinite wavelength and n = 0). Additionalsimulations (see Supplemental Material (3)), show thatat this combination of frequency and effective field, theFMR main mode is not necessarily excited in the satu-rated region but instead, a spin wave mode with at least k x = 0 (but not necessarily k y = 0) can be excited (i.e.,a Backward Volume Spin Wave mode). This result stillagrees with the perspective of a ‘confined mode’ in thex-direction , since the model becomes a loose approxima-tion to a scenario where domains or saturated regions arepresent. In other words, it does not account for modesthat may show non-zero wavenumber in a different direc-tion of propagation, such as the y-direction, orthogonalto the original domain wall, although it effectively re-trieves the expected zero wavenumber in the x-direction.More interestingly, the model predicts the existence of,conceptually speaking, evanescent spin waves below that‘pseudo-FMR’ frequency condition. IV. NUMERICAL VALIDATION
A mathematical model is only as good as the assump-tions. In order to validate our model for the real valuesobtained from Eq.(12), micromagnetic simulations on themagnetic structure shown in Fig. 3 are performed and recorded for comparison with the analytical model. Sinceit is inevitable to start from an angle of approximately π/ α and therefore of n, seeFig.4. Therefore, for the shape of this patch, the demag-netizing field transversal to the wall is not constant alongits length: The angle α at both sides of the wall changesfrom π/ π/ k is that ofa Winters magnon along a 180 Bloch wall. For a spinwave propagating from one of the core regions, reductionin α (or equivalently, reduction in the transverse demag-netizing field, see Fig. 3(c)) implies an increase in thelocal wavenumber. From numerical results on this par-ticular shape, small and smooth variations of the angleare considered far from the core regions, so the assump-tion dαdx ≈ d ∆ dx ≈ dmdy ≈ dm y dy . Close to the core regions, the domain wallangle is large enough to assume the domain wall Neelcomponent to be dominant [37] and therefore, dmdy ≈ dm y dy is also satisfied at the centre of the domain wall. In otherwords, the assumptions held in the previous section arestill valid as well as the derived model for the demag-netising field transverse to the wall.Fig. 4(a) shows the spatial position profile of parallelto the domain wall at 50 nm from the centre of the wall,calculated as tan − ( m y /m x ), where the components of FIG. 3. (a) Schematic of the proposed structure of 2000 nm6000 nm 80 nm. A Bloch domain wall is induced and leftto relax before running dynamic excitations.(b) Normalisedout-of-plane component of magnetisation is shown, demon-strating the formation of a Bloch domain wall in the mid-dle of the structure. (c) Normalised in-plane y-componentof the demagnetising field, showing a reduction in magnitudein the centre of the structure. Inset in (c) shows the mag-nitude of the in-plane component of the demagnetising field,perpendicular to the wall, at the center of the wall obtainedfrom micromagnetic simulations (blue curve) and a sinusoidaldependence with x-position between the two vortex core po-sitions (x = 2000 nm and x = − − ) is obtained from thespatially-dependent angle between magnetic moments in theshape (Fig. 6.5(a)). magnetisation are extracted from numerical simulationsand magnetisation is assumed in-plane. The ripples closeto the central region come from the numerically obtainedequilibrium state of magnetic moments. More specifi-cally, from those in the region where the curved contouris closer to the straight domain wall, which makes mag-netization to not smoothly follow the contour neither thedomain wall in a straight line but rather, in an apparentzig-zagging path as a middle-ground solution. This al-ters the calculation of the angle α ( α = tan − ( m y /m x )),introducing the rippling. Ideally, this should not be hap-pening, and magnetisation should be laying completelyin-plane and smoothly following the shape contour even FIG. 4. (a) Profile of angle α for the structure, calculatedas tan − ( m y /m x ) along a line, parallel to the wall, at 50 nmfrom the domain wall centre. Red dotted line is a linear fitto the values from the vortex core region to the centre of thestructure. (b) Local index of refraction from Eq. (12) withthe wavelength of a Winters magnon in a 180 Bloch wall asreference for 3 GHz and 1.5 GHz. close to the domain wall. Still, the shape of the structureallows to approximate the dependence of angle with xby a fitting linear dependence α ( x ) − . x + 87, see reddotted line in Fig. 4(a), making it closer to the ideal case.Including this linear fitting into the proposed model forthe demagnetising field (Eq. 10) shows also good qualita-tive agreement with the values obtained from simulations(see inset in Fig. 3(c)).Quantitatively, result from simulation fits better to asinusoidal function with a maximum value of 10 Am − .While in the same order of magnitude, this value is aboutthree times the maximum of the model from Eq. (10)for α = π/ M s sin( π/ / π ≈ . × Am − , (for asaturation magnetisation of 7 . × Am − ). This canbe explained by the neglection of other components ofthe magnetostatic potential in the model (which leadsto a ‘weaker’ demagnetising field than in reality, due tofewer contributions), that should be accounted for a moreaccurate description.Fig. 4(b) shows the analytical index of refraction fromEq. (12) with the obtained values of as inputs for theexcitation frequencies of 1.5 GHz and 3 GHz. This agreeswith analytical results from Fig. 2(b), at 3 GHz excita-tion frequency, which show that the wavevector can bereduced as much as approximately 0.7 times the refer-ence wavevector k (i.e., n ≈ .
7) when the spin waveapproaches the core regions (at 900 nm from the vor-tex core) and α is approximately π/ . ≈ . π radians(i.e., 37, see Fig. 4(a)).Fig. 5(a) shows the wave profile of the propagatingmode and the respective wavelengths found in the re-spective regions I ( λ I ) and II ( λ II ) (insets). A signifi-cant attenuation of the spin wave is observed in regionII (the colour intensity in the insets is manually adjustedfor ease of comparison), we believe this is originated bythe ‘zig-zagging’ path of magnetisation in the domainswhich induces a pronounced gradient in the local demag-netising field halfway between the two regions (see bluecurve in the inset of Fig. 3(c)), modifying in turn thespatially local ferromagnetic resonance in the wall andthus presenting low transmission between regions in thewall [41, 42], as a 1-D analogy of the 2-D scenario ex-plored in Ref.[42]. Fig. 5(b) shows the index of refrac-tion profiles for 1.5 GHz and 3 GHz (solid lines) obtainedfrom Eq.(12), with the range of angles found from sim-ulations as input, and numerical results (dots) for thespatial frequency in the x-direction, normalised to thevalue at 3000 nm (or in other words, the simulated spa-tial index of refraction). Numerical results confirm whatthe equation predicted: The change in wavenumber ismore pronounced, and more sensitive to the transversedemagnetising field (or equivalently, to the domain angle α ) when the frequency is smaller. Therefore, the cov-ered graded-index profile is wider at lower frequencies.Also, as expected from the model, the values of the ob-tained index of refraction from simulations agrees withthe predicted values from Eq. (12) with the specifiedangle α = 37 at the corresponding position (x = 1900nm) in the shape ( n ≈ . n ≈ . FIG. 5. (a) Simulated wave profiles for an excitation fre-quency of 3 GHz and 1.5 GHz, with the chosen regions high-lighted in dashed lines (they are not representing the Hanningwindow): Region I centred around x=1900 nm and region IIcentred around x=2800 nm, showing the spin wave profile inone half of the structure. Note that, for ease of comparison,the colour scale in the insets for region II is not to scale (itis reduced) to that of region I. The local wavelengths are in-dicated in the insets, along a length of 400 nm. (b) Resultsfrom the analytical model from Eq.(12) for 1.5 GHz and 3GHz (solid lines) and the ratios λ II /λ (x) from micromagneticsimulations (dots) for each frequency and per Hanning win-dow. different demagnetising fields or shape contour effects.The model helps to predict how these effects modify thewavelength of a confined mode in a domain wall. V. SUMMARY
The main result of this work is the proposed mathe-matical model for an effective spatial frequency depen-dence of confined spin waves in an ‘intermediate angle’domain walls as a function of that angle. The equation ofthe model is derived from the fundamental Bloch equa-tions of motion, an approximated (under reasonable as-sumptions, similar to those in Ref. [40] expression of thedemagnetising field transversal to the wall, that is ap-plied to a variety of magnonic scenarios. The model canbe applied to straight domain walls of variable domainangles, which can be useful as a first approximation tothe study of more complex scenarios such as spin wavesin confined structures of arbitrary shapes showing mag-netic domains. Reciprocally, the connection between theshape of the magnetic ‘patch’ and the shape-induced de-magnetising field transverse to the wall, allows us to de-sign its shape for a particularly desired channelled spinwave profile.The model also leads to conditions that have physicalmeaning such as the FMR main mode frequency or Back-ward Volume Spin Waves, and therefore further extend- ing the applicability of the model to not localised modesin domain walls. In summary, an equation for a spatial-dependent wavenumber for spin waves is proposed, per-forming as a good model for their propagating behaviourin domain walls. This result may help in the developmentof more complex models for spin wave propagation innon-saturated nanostructures, channelled along domainwalls or propagating into magnetic domains.
VI. ACKNOWLEDGEMENTS
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