Spin-Wave Modes in Transition from a Thin Film to a Full Magnonic Crystal M. Langer,1,2 R. A. Gallardo,3 T. Schneider,1,4 S. Stienen,1 A. Rolda´n-Molina,5 Y. Yuan,1 K. Lenz,1 J. Lindner,1 P. Landeros,3 and J. Fassbender1,2 1Helmholtz-Zentrum Dresden – Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr. 400, 01328 Dresden, Germany 2Institute for Physics of Solids, Technische Universita¨t Dresden, Zellescher Weg 16, 01069 Dresden, Germany 3Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, 7 Avenida Espan˜a 1680, 2390123 Valpara´ıso, Chile 1 4Department of Physics, Technische Universita¨t Chemnitz, 0 Reichenhainer Str. 70, 09126 Chemnitz, Germany 2 5Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Avenida Brasil 2950, 2390123 Valpara´ıso, Chile b (Dated: February 23, 2017) e F Spin-wave modes are studied under the gradual transition from a flat thin film to a ‘full’ (one- dimensional) magnonic crystal. For this purpose, the surface of a pre-patterned 36.8 nm thin 2 permalloy film was sequentially ion milled resulting in magnonic hybrid structures, referred to as 2 surface-modulated magnonic crystals, with increasing modulation depth. After each etching step, ferromagnetic resonance measurements were performed yielding the spin-wave resonance modes in ] backward-volume and Damon-Eshbach geometry. The spin-wave spectra of these hybrid systems l l reveal an even larger variety of spin-wave states compared to the ‘full’ magnonic crystal. The a h measurements are corroborated by quasi-analytical theory and micromagnetic simulations in order - to study the changing spin-wave mode character employing spin-wave mode profiles. In backward- s volume geometry, a gradual transition from the uniform mode in the film limit to a fundamental e mode in thethin part of the magnonic crystal was observed. Equivalently,thefirst and thesecond m filmmodesaretransform intoacenterandanedgemodeofthethickpartofthemagnonic crystal. . Simple transition rules from the nth film mode to the mth mode in the ‘full’ magnonic crystal are t a formulated unraveling the complex mode structure particularly in the backward-volume geometry. m Ananalogous analysis was performed in the Damon-Eshbach geometry. - d n I. INTRODUCTION to assurface-modulatedmagnoniccrystal(SMMC). The o transitional SMMCs are fabricated by sequential ion c milling of a lithographically surface-patternedpermalloy [ Periodically patterned magnetic materials with pe- riodicities ranging from micrometers down to several (Ni80Fe20) thin film offering the benefit of studying the 2 transition gradually on the same sample. Subsequent tens of nanometers are referred to as magnonic crys- v to each ion-milling step, broadband ferromagnetic reso- 5 tals (MCs).1–8 In the last decade, this group of meta- nance (FMR) measurementsareperformedandcorrobo- 7 materials, such as bi-component systems,9–14 free stand- ratedbymicromagneticsimulationsandquasi-analytical 6 ing structures,15–20 and continuous films with peri- theory. 5 odic structures on top (surface-modulated magnonic 0 crystals),21–26 experienced a growing scientific interest. In an ongoing miniaturization process, future devices . will require small SW wavelengths below 100 nm in an 2 As spin waves (SWs) offer unique properties such as exchange-dominated regime. For this reason, a base pe- 0 charge-less propagation and high group velocities, there 7 are multiple applications conceivable since industry is in riodicity of a0 = 300 nm, and an individual nominal 1 wire width of approximately w = 140 nm was chosen, need for higher efficiencies as well as high performances : i.e., characteristic SW mode wavelengths in this system v in information technology including the transport and are ranging from 60 to 300 nm. Even though, there i processing of data.8,27–36 X are other advanced methods to investigate such transi- r Thepossibilitytomanipulatethebandgaps9,37–43 and tion, as e.g. thermal landscape modulation50 or periodic a to tailor the SW properties paves the way for many Oersted-fields of current-carrying meander structures,51 magnonic applications.44 such as magnonic filters,45 log- theseapproachesaresofarnon-applicabletoperiodicities ical circuits29,30,46,47 and magnonic grating couplers.11 of 500 nm and below. Moreover, fast developments in spintronics and spin- Uptonowthetwolimits—thethinfilmlimit21,22,24,26 caloritronics48 hold out the perspective of many novel and the ‘full’ MC37,39,40,52–57 — are already intensively promisinghybrid-topics8 in the future, where the unique studied. This work is focused on the properties of tran- properties of MCs are combined with functional entities, sitionalsystemsofferingevenricherspectraofSWstates such as recently shown for spin-torque oscillators.49 with adjustable SW amplitudes. However, the huge This work focuses on the understanding of the transi- variety of modes in SMMCs, the strong coupling be- tion of SW modes from the film limit to the limit of a tween them, and the distortion of the mode profiles due one-dimensionalMC.Alltransitionalstatesconsistingof to the inhomogeneous internal demagnetizing fields hin- periodic array of wires on top of a thin film are referred der a straightforwardinterpretationof the mode charac- 2 excitation method. The etching depths ofeachSMMC was determinedby fitting of the f(H ) dependence of SW modes standing 0 vertically in the film, also referred to as perpendicular standing spin-wave (PSSW) modes.59,60 For largermod- ulation heights ∆d > 10 nm, vibrating sample magne- tometry measurements were employed to determine the magnetic volume of anetched referencefilm before (M ) 1 and after (M ) the ion milling step. This approach es- 2 timates the modulation height ∆d = d×(1−M /M ) 2 1 withtheratioofthemeasuredmagneticvolumeM /M . 2 1 The following discussion of SW modes in SMMCs is subdivided by the in-plane orientation of the external field with respect to the modulation axis of the SMMC, i.e. backward-volumeand Damon-Eshbach geometry. FIG.1. (Coloronline)(a)Sketchofthesequentialfabrication ofseveralSMMCsbyion-millingofasinglesample. (b)TEM imageofasamplemilled to∆d=10nmsurfacemodulation. (c-d) SEM images of the top view of the same sample before III. BACKWARD-VOLUME GEOMETRY theremoval of theresist mask. SWs in an SMMC with field orientation parallel to the modulation axis, i.e. kkM (referred to as backward- ter. This is where the approach to follow the modes of volumegeometry),areparticularlyinterestingduetothe SMMCs with an increasing modulation height becomes highly inhomogeneous demagnetizing fields H , which d particularly favorable in order to disentangle these ef- are largest compared to all other in-plane field orien- fects. tations. Before different transitional SMMCs are ad- The manuscript is organized as follows. Detail about dressed,both,theplanarfilmlimitaswellasthelimitof the sample fabrication and the measurement technique a ‘full’ MC of separate wires shall be addressed. are specified in Sec. II followed by Sec. III and Sec. IV In order to understandthe complex mode structure of containing the results for the backward-volume and the theSMMCs,micromagneticsimulationswerecarriedout Damon-Eshbach geometry, respectively. At last, a brief usingtheMuMax3-code.61 Forallsimulations,themate- summary is provided in Sec. V. rialparameterspresentedinSec.IIwerere-employedand several SMMCs were micromagnetically reconstructed usingafixedfilmthicknessofd=36nmandsixmodula- II. EXPERIMENT tion heights of ∆d = 2,4,9,13.5,18 and 36 nm with the latterrepresentingthe‘full’MC.Fortherealizationofan The experiments are based on a single polycrystalline extendedMC,periodicboundaryconditionswereapplied d = 36.8 nm thin permalloy (Ni Fe ) film deposited to the in-plane axes. To obtain the spin-wave spectra of 80 20 by electron beam deposition on surface-oxidized Si(001) each SMMC, a pulsed excitation62 was realized and the substrate. The surface of the film was lithographically integraldynamic magnetizationwas plottedas the FMR stripe patterned using ma-N 2401 negative resist. In or- response. der to stepwise remove the magnetic material between the resist-covered stripes, sequential Ar-ion milling was employed. The procedure is schematically depicted in A. The Limits: Thin Film and Full Magnonic Fig.1(a). Figures1(b)–(d)showthecorrespondingcross- Crystal section and top view of a patterned permalloy film after the milling of ∆d = 10 nm. Altogether, the film was milled five times until an array of separate wires, i.e. a Thin Film Limit. The film limit has previously ‘full’ MC, was achieved. been studied in experiment21,26,63 and theory using two- After each ion-milling step, the frequency-field- magnon scattering perturbation theory.22,24 It is coher- dependence f(H ) was measured using a broadband fer- ently found that for SMMCs with small modulation 0 romagnetic resonance (FMR) setup, as described in ref- height, the demagnetizing field H has only substantial d erence [26]. An FMR pre-characterization of the thin field strength at the relatively small edges on the film film properties was carried out prior to the sequential surface and is negligible in the volume of the structure. ion milling yielding the magnetic polarization µ M = Thus, H acts as a small periodic perturbation intro- 0 S d 0.9236T,theg-factorg =2.11andtheexchangestiffness ducing a base periodicity a crucial for the presence of 0 D = 23.6 Tnm2.26,58 It shall be noted that the investi- standing SWmodes. While the dispersionremainsunaf- gations concentrate on symmetric modes only, since the fected, an introduction of periodic perturbations in the dynamicmeasurementsarecarriedoutusingasymmetric realspaceleadstoadiscretizationofstandingSWmodes 3 measurement simulation GHz)200 (a) (b42)5 450 475 11117896f (GHz) m = 7 Frequency (11505 smµina0gHjol0er (mwmoiTred)e m =c e5nteremdgoedme(omde=(1m) = 3) 0 minor mode 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 Field (T) Field (T) FIG. 3. (Color online) (a) Measurement and (b) simulation of the frequency-field dependence of an array of wires (one- dimensional MC) with the orange lines indicating the simu- lated resonance branches of a single wire of same properties. Theedge modeisclearly suppressedinthemeasurement due to beveled edges and theroughness of the measured sample. FIG. 2. (Color online) Measurement and simulation of the frequency-fielddependenceof (a),(c) aflat film and (b),(d)a surface-modulated magnonic crystal with tiny modulations (∆d = 2 nm). Orange lines represent the spin waves’ and thus, with the transition from film physics to the frequency-fielddependencescalculated from Eq.(1). physics of MCs. in the k-space according to k = 2πn/a with k being Full MC. The opposite limit is the periodic array 0 the in-plane wave vector and n = 1,2,3,... . As men- of wires, also known as ‘full’ or one-dimensional MC,5,10 tioned above, the SW dispersion follows the well known with the f(H ) dependence shown in Fig. 3. Figure 3(a) 0 relation:64 is the measurement and 3(b) the simulation with orange linesindicatingthesimulatedresonancemodesofasingle 2 ω wire. = µ H +Dk2 0 0 (cid:18)γ(cid:19) (cid:0) 1(cid:1)−e−kd (1) Ifthestrayfieldcontributionofneighboringwires(ap- × µ H +µ M +Dk2 proximately 30 mT) is considered to shift the modes 0 0 0 S (cid:20) (cid:18) kd (cid:19) (cid:21) to lower resonance fields, both the MC (colorplot in Fig. 3(b)) and the single wire (orange lines in Fig. 3(b)) with f =ω/(2π) the FMR frequency. reveal very similar results for the given dimensions. Equation (1) yields a parabolic shape of the disper- Due to the strong locally varying internal demagnetiz- sion for dipole-exchange spin waves in backward-volume ing fields, different fundamental modes can be excited, geometry. The characteristic shape implies an energy namely the wire and the edge mode. Above, there degeneracy,whichtogetherwiththe scatteringcondition are also higher order modes standing within the bound- k = 2πn/a0 enables the two-magnon scattering channel aries of the structure. Note that a symmetric excitation fromtheuniformmodetothek 6=0standingSWmodes. onlyallowsforthemeasurementofsymmetricSWeigen- In Figs. 2(a),(c), the measured and the simulated modes and due to symmetry arguments, the peak num- frequency-field dependences f(H0) are depicted for a ber m of SW modes in the ‘full’ MC limit must be odd flat film and in Figs. 2(b),(d) for an SMMC with 2 nm (m=1,3,5,...). modulation height. Fig. 2(d) additionally provides the f(H0)dependenceofthen=1...4modescalculatedfrom The mode profiles corresponding to each resonance Eq. (1) and plotted as orange solid lines. Clearly, stand- branch in Fig. 3(b) are shown in Fig. 4 at an external ing SW modes inanSMMC with tiny modulationfollow field of 300 mT. The two fundamental branches can be the f(H0) dependence of propagating modes in a flat identified as the edge mode (m = 3) in Fig. 4(a) and film of the same properties quite well. However, devia- the wire mode (m = 1, also termed ‘wire center mode’) tionsfromthefilmlimitcanbeobservedinthevicinityof in Fig. 4(b), where the former appears to be completely crossingpoints inthe f(H0)dependence, wheredifferent suppressed in the measurement due to the beveled edge standing SW modes couple to each other. shapeandedgeroughnessofthesample. Figures4(c),(d) In analogy to coupled oscillators in mechanics, both reveal the mode profiles of higher modes (m = 5,7) modes split up into an acoustical and an optical branch standingwithintheboundariesofthewire. Thesemodes with in-phase and anti-phase oscillation, respectively. onlycarrysignificantintensitieswhilecrossingoneofthe The evolving gap between acoustical and optical mode fundamentalbranchestowhichtheycouple(seetheinset is connected with a band gap formation in the k-space in Fig. 3(b)). 4 FIG. 4. (Color online) Mode profiles in an array of wires in backward-volumegeometry(asshowninFig.3(b))at300mT. B. Transitional SMMCs The sequential ion-milling was performed with very small etchings steps in the beginning (∆d −→ 0) and large etching steps in the end (∆d −→ d). This is due tothe demagnetizingfieldincreasingsubstantiallyinthe range of ∆d=0−−10 nm (see inset Fig. 5) and gradu- allysaturatingabovethatrange. ForSMMCswithlarger modulation heights, the changes become less significant with ∆d and thus, the alteration of the mode character is uncritical. Figure5showsthef(H )plotsofseveralSMMCswith 0 a modulation height ∆d in the range of 2.0–13.5 nm. In Figs.5(a)–(d)and5(e)–(h),theFMRmeasurementsand the simulations are illustrated, respectively. In the in- set,thelevelofthemodulationisprovidedtogetherwith thex-componentofthesimulatedinternaldemagnetizing fieldH ascolorplot. Insidethethickpart(alsotermed d,x FIG. 5. (Color online) (a)–(d) FMR measurements of MCs ‘wire’) of the MC, the demagnetizing field acts against with different modulation heights as sketched in the insets. (negative sign) the local magnetization direction as usu- (e)–(h) The dynamic response with the internal demagnetiz- ally. Interestingly, inside the thin part (also ‘trench’), it ing field in the inset both calculated using pulsed micromag- acts with the magnetization and is, thus, a magnetizing netic simulations. Colored dots represent the field-frequency field. The increasing contrast between the magnetizing- position of themode profiles illustrated in Fig. 6. and demagnetizing fields with the modulation height is easily inferred from the insets in Figs. 5(e)–(h). This ef- fectisimportantfortheunderstandingoftheevolutionof mode from the film to the ‘full’ MC shall be addressed. SWmodesinSMMCswithincreasingmodulationheight ∆d. Quasi-Uniform Mode. A comparisonbetween the It is again distinguished between fundamental modes uniform mode in the film limit (blue dotted line in and higher orderSW modes. The former canbe directly Fig.2(d))andthen=0modeinSMMCs(orangedashed excited uniformly whereas the latter gain intensity from line in Fig. 5(e)) reveals the transition of the uniform couplingtoafundamentalbranch. InFigs.5(e)–(h),fun- mode to a localized quasi-uniform excitation of the thin damentalmodesaremarkedbydashedcoloredlines. The partoftheSMMC(heretermedtrench mode). Thetran- color of each mode is chosen corresponding to the H sitionofthemodecharactercanbeunderstoodbestwith d,x fieldwhereorangecolorrepresentsmagnetizing fieldsand the help ofFig. 6 where the ∆d-dependent mode profiles blue color demagnetizing fields. atthemarkedresonancepositionsinFig.5aredisplayed. Priorto a detailed discussion,a splitting into two fun- Figure 6(a) shows the mode profiles of the trench mode damental modes, a SW excitation in the trench (trench (n = 0) within one period a0. Moreover, a colorplot of mode) and an excitation of the wire (wire mode), can the ∆d-dependent mode amplitude within the trenchre- already be concluded. Now, due to the increasing con- gion can be found in Fig. 6(f). trast between the (de-)magnetizing fields in the trench A clear maximum in the center of the trench is ob- and in the wire, both modes undergo a vast separation served, which becomes more confined with increasing leadingtotheevolutionofalargegapbetweenthem(see modulation height. Moreover, the amplitude in the wire Figs. 5(g) and 5(h)). Next, the transition of each SW region vanishes as ∆d increases. These observations are 5 evidencesforatransitionfromtheuniformmodetowards a quasi-uniform (k = 0) excitation of the trench center, which vanishes if the trench is completely removed. Theconfinementofthecenterpeakofthetrench mode is explicitly explained in Ref. [58] and increasing with ∆d. The effect is connected with the diverging inter- nal H fields. For spins in neighboring regions to the d trenchcenter,the higher internalfields shift the solution of the SW dispersion in Eq. (1) to k 6= 0. And due to an increasing internal field contrast, the local k-vector becomes larger with ∆d leading to a narrowing of the center peak. 1st SW Mode. The bright blue dashed lines in the Figs. 5(e)–(h) indicate the wire mode. Originating from the 1st standing SW mode in the film limit (see n = 1 mode in inset of Fig. 3(d)), this mode evolves from a sinusoidal harmonic standing SW mode to one central peak localized inside the wire region. The mode profiles in Figs. 6(b) and 6(g) demonstrate this transition. The reason for the modes’ confinement in the wire re- gion lies in the energy of the mode in the film limit. Be- ing below the uniform mode in the f(H ) dependence 0 (Fig. 2(b)), the energy of the 1st SW mode is insuffi- cientforanexcitationofthecompleteSMMC.Thus,the mode localizes in regions, where the energy is internally reduced due to the presence of a demagnetizing field. In regions with magnetizing fields (e.g., the trench), this mode becomes strongly damped. It shall be mentioned that the wire mode is of a dif- ferent nature than the trench mode with respect to the local k-vector. Even though both modes exhibit a sim- ilar shape within the respective region, the wire mode is localized at a local maximum of the internal H field, d whereas the trench mode is excited at a local minimum. As mentioned above, for small wavelength excitations, the k-vector locally rises with higher internal fields and in-turn, dropswithlowerinternalfields due toEq.(1).58 So, if the center of the wire was excited quasi-uniformly (k = 0), like in case of the trench mode, there would be no possible solution solving Eq. (1) for the neighboring spins, which are exposed to a reduced H field. This d wouldimmediatelydampoutthismode. Hence,thefun- damental wire mode must exhibit a k 6= 0 excitation in thecenter,andhence,shouldbeinterpretedasanm=1 standing spin-wave mode within the wire region. Eventually, mode 1 is following the transition from an n=1 film mode to an m=1 mode in the ‘full’ MC. FIG. 6. (Color online) (a)–(e) Mode profiles at the marked resonance positions in Fig. 5 revealing thetransitional states 2nd SW Mode. With moderate modulation heights between the film limit and the limit of an array of wires. of∆d≈10nm, the 2nd SW modeinthe filmlimitstarts (g)–(j) The modulation-height dependent normalized mode togainintensity,asillustratedintheFigs.5(g)and5(h), profiles inside the wire region being an important proof for andevolvestowardsanedge modeinsidethewireregion. thetransitionfromthenth filmmodetothemth modeinthe This transition is again reflected by the mode profiles in ‘full’ MC limit. the Figs. 6(c) and 6(h). In general, two transformations are observed. (i) The localizationof 2n−1=3 nodes inside the wire and sub- alent to the explanation for the localization of the wire sequently,(ii)thesuppressionofthecentralpeakandthe mode. In the film limit, the 2nd SW mode lies even be- localizationatthe wire edge. The reasonfor (i) is equiv- lowthe 1st mode(atfieldsabove100mT)withthe same 6 consequence, the mode cannot extend over the ‘full’ MC sameperiodicitya =300nm,thicknessd=36.8nmand 0 and localizes with m = 3 peaks within the wire region materialparameters was analyzed. This MC exhibited a instead. The reason for (ii) is due to the formation of a significantly larger nominal wire width of w = 166 nm distinct spin-wave well,65 i.e. a local minimum of H (x), resulting in similar mode characteristics with a different d coinciding with the outermost peaks of this mode close pinning condition at the wire edges. to the wire edge. Since the energy of the mode is lowest Itiscoherentlyobservedthatthenthfilmmodeevolves below the trench and the wire mode, the excited reso- intothemthmodeinthe‘full’MCwithm=2n−1. Only nant region must be of lower energy, which is only the the n=0 mode maintains a local quasi-uniform (n=0) case at the wire edge. Thus, the mode profiles of the character and does not follow this general rule. This second mode in Fig. 6(c) are interpreted as a transition observation is believed to reflect the momentum quan- from the n=2 mode to an m=3 mode with increasing tization of standing SW modes throughout this process mode localizationatthe edgesanddecreasingamplitude allowing only for even numbers of 2n nodes and peaks (mode damping) at the wire center. within one period a . 0 Higher Order Modes. Higher order (n > 2) spin- wave modes are also present throughout the transition IV. DAMON-ESHBACH GEOMETRY fromafilmtoa‘full’MC.Andasmentionedbefore,these modes only carryintensity when crossingone of the fun- In the Damon-Eshbach orientation (k⊥M) when the damentalmodes(seeFig.5)andcouplingtothem. Thus, fieldisappliedparalleltothesurfaceedges,standingSW thesemodesexhibitquitedifferentmodeprofilesdepend- modesarealsopresent. Forthesakeofcompleteness,the ing on whether they couple to the trench or to the wire measurement results for this orientation are provided as mode. In the former case, the modes reveal major am- well. However, the micromagnetic part of the study will plitude inside the thin part of the MC and in the latter reveal that major changes of the mode character occur theamplitudesconcentrateinsidethethickpart(seepro- at a very high level of surface modulation where the ex- files of mode 3 and 4 in Fig. 6(d),(e)). However, for the perimental data remains incomplete. transition to the ‘full’ MC, the ∆d-dependent amplitude inside the wire is crucial revealing a steady evolution of the mode characterto the ‘full’ MC. This steady process A. The Limits: Thin Film and Full Magnonic is depicted in Fig. 6(i),(j). Crystal Generally, it is observed, that film modes localize m=2n−1peakswithinthewireregionduringthistran- Thin Film Limit. In contrast to the backward- sition process. Together with a single ‘silent’ (damped) volume orientation, there is no energy degeneracy in the minimuminthecenterofthetrench,thetotalnumberof Damon-Eshbach orientation. For this geometry, the dis- 2n peaks within one period a is conserved throughout 0 persion reads24,64 this process. In some cases, like for mode 3 and 4, the mode profiles in the ‘full’ MC (see Figs. 4(c),(d)) rather ω 2 1−e−kd appearasm=3andm=5 modesinsteadofm=5 and = µ0H0+µ0MS 1− +Dk2 (cid:18)γ(cid:19) (cid:20) (cid:18) kd (cid:19) (cid:21) m = 7. In both cases, the last peaks at the edges are (2) incomplete raising the question whether they contribute × µ H +µ M 1−e−kd +Dk2 . 0 0 0 S or not to the peak number m in the limit of the ‘full’ (cid:20) (cid:18) kd (cid:19) (cid:21) MC.Toanswerthisquestion,itisusefultoidentifytheir Hence, there are no scattering channels presentto trans- role in the transition to the ‘full’ MC. Take for example fer intensity from the uniform mode to a k 6=0 standing the 3rd mode at ∆d=18 nm (see Fig. 6(d)). Only when SW mode. This is why Damon-Eshbach modes are of both edge peaks are fully taken into account, the total no (or extremely small) intensities for small modulation number of nodes within one period remains 6, making heights ∆d in contrast to standing spin-wave modes in n a conserved quantity. At this stage of the transition, backward-volumeorientation. Aprominentlevelofmod- the modes’ profile inside the wire region is already set ulationinthemagnitudeofd/4wasfoundtobenecessary and since both peaks amplitude is clearly damped out- for the detection of at least a small intensity of the 1st side the wire, they should fully contribute to the mode mode. Thus, the film limit in Damon-Eshbachgeometry number m=5 inside the wire. is governed by the unaltered uniform mode. Certainly, modes of a high peak number m tend to cut off their edge peaks, since in that way, the high cost Full MC. Inthis limit,the mainmode(heretermed of exchange energy Dk2 ∝ (m/w)2 is minimized when wire mode) can be calculated using the demagnetizing condensing 2n−1 peaks within the narrow wire region. factor of the wires57,66 Thus,theindividualpinningofthemth modeatthewire 2 ω edges is likely depending on the width w of the wire. =[µ H +N µ M ] 0 0 x 0 S ThisissupportedbythemodeprofilesshowninRef.[58] (cid:18)γ(cid:19) (3) whereanSMMCofadifferentwirewidthw,butwiththe ×[µ H −N µ M +µ M ] 0 0 x 0 S 0 S 7 300 0 uniform e 1 DE1b d o m T)200 e r m wi ( d 1 DE1a e el d o Fi100 m Measurement 2 DE2 3 DE3 ge (f = 17 GHz) d e uniform mode linear fit 1st DE mode 1. DE mode [Eq. (2)] 0 0.2 0.4 0.6 0.8 1.0 Dd / d FIG. 8. (Color online) Micromagnetic Simulation of the SW resonances at f = 17 GHz depending on the modulation FIG. 7. (Color online) Measurement of the frequency-field height∆dinDamon-Eshbachorientation. Symbolsrepresent dependencesofthe1st Damon-Eshbachmode(opensymbols) the results from the measurement. Note the fading uniform forthemodulationheights∆d=8.7nm,13.2nmand36.8nm modeandtheemergingofthewiremodeinthe‘full’magnonic with fits (dashed and solid lines). Crossed symbols represent crystal from the 1st and the2nd spin-wave mode. the results of micromagnetic simulations. The standard ma- terial parameters were used as fitting parameters yielding an effectivethicknessd=27.6nmand23.0nmfor∆d=8.7nm, and 13.2 nm. In case of the ‘full’ magnonic crystal, the fit frequencyoftheDamon-Eshbachmodeisreduceddueto yields a demagnetizing factor of Nx =0.115. thereduceddipolarenergy. Theexpectedincreaseofthe mode’s resonance field according to Eq. (2) is depicted in Fig. 8 as a blue dotted line. At a modulation height With N being the demagnetizing factor along the wire of about half the film thickness, applying Eq. (2) leads x in-plane short (x-) axis and with the shape-anisotropy to major deviations from the measured (blue asterisks N µ M . ThesemodescanbeexpressedbyEq.(2)with in Fig. 8) and simulated (colorplot) resonance positions. x 0 S the wavevector being quantized with k = mπ/w, m = This is due to the fact that the thin-film approximation 1,3,5... due to a defined number of nodes fitting inside becomesincreasinglyinappropriatetoreflectthegeomet- thewirewidthw. Notethatthepinningatthewireedges rical reality. canstronglyinfluencethef(H )dependenceofthemode. However, micromagnetic simulations (colorplot in 0 Therefore, a modified mode number m → m+λ can be Fig. 8) indicate that the 1st Damon-Eshbach mode (to- defined with λ being the pinning parameter leading to a gether with the 2nd mode) evolves into a fundamental higher accuracy of the resonance positions.57 (wire) mode in the full magnonic crystal. At the same In case of the measurements carried out in this work, time, the quasi-uniform mode (orange lines and symbols only the wire mode was detected (black open pentagons in Fig. 8) gradually disappears. in Fig. 7). However, micromagnetic simulations (black In the following the mode profiles of these two main crossedpentagons and squares in Fig. 7) revealthe exis- modesshallbediscussedtogetherwithsomehigherorder tence of higher (m>1) modes with small intensity. The Damon-Eshbachmodes. simulatedresonancepositionofthesecondhighestinten- sitymodewithm=3(termed‘edge mode’)isalsoshown Quasi-UniformMode. InFig.9the∆d-dependent (crossed squares) in Fig. 7 where the crossed pentagons mode profiles of the quasi-uniform mode and the first represent the n=1 (wire) mode. threeDamon-Eshbach-typestandingSWmodesareplot- ted. On the left side of Figs. 9(a)–(e), the space- dependent spin-wave amplitude is plotted, whereas the colorplots on the right side (Figs. 9(f)–(j)) reveal the ef- B. Transitional SMMCs fective amplitudes, i.e. the same profiles scaled with the localfilm thickness. In caseofthe n=0 (quasi-uniform) In the experimental study, the first measurable inten- mode, the dynamic activity appears to concentrate in- sity of an n = 1 Damon-Eshbach mode was detected side the thin part, which is analogous to the character for ∆d = 8.7 nm (circles in Fig. 7). At a modulation of the trench mode in the backward-volume geometry. height of ∆d = 13.2 nm, the mode gains intensity and However, when scaled with the local film thickness, the is slightly shifted to lower frequencies and higher field response of both, thick and thin part remains equally values. In general, this is what one would expect from high in the center of both regions. This observation Eq. (2), since with lower effective film thickness d the is a major difference compared to the behavior of the 8 n = 0 mode in backward-volume direction. Here, in the Damon-Eshbach orientation, the effective behavior remains quasi-uniform in the full structure. And as the modulationbecomesdominant,theeffectivedynamicac- tivity is reduced close to the edges of the wire, where dynamic demagnetizing fields become relevant altering the SW dispersion. This leads to the picture illustrated in Fig. 9(f), where a constant effective SW amplitude is visible together with channels of reduced dynamic activ- ity at the boundaries between both regions. 1stSWMode. ThecolorplotinFig.8revealsasplit- tingofthe1stSWmodeintotwomodes. Ahigh-intensity modechanginggraduallyintoawire modeand,closerto the dotted line corresponding to Eq. (2), another low- intensity mode. The latter fades out for high ∆d. The profiles of both modes are illustrated in Figs. 9(b),(c) and 9(g),(h). Starting from the film limit, the character of both modes is dominated by the harmonic character of an n = 1 film mode. Only for pronounced modula- tions (∆d≥27 nm), an asymmetry within the thin part occurs in case of the low-intensity mode (Figs. 9(c),(h)), while the high-intensity mode (Figs. 9(b),(g)) remains fully symmetric throughout the whole transition. This mode keeps one peak within the thick part throughout the transition whereas the number of peaks within the thin region increases with ∆d. Once the thin part of the MC is fully removed, only the peak within the thick part remains as the wire mode of the full MC. It shall be noted, that the origin of the low-intensity mode re- mains unclear but might be due to the breaking of the symmetry by the dynamic demagnetizing fields. 2nd SW Mode. Due to the coupling to PSSW-type modes, the mode profiles of the 2nd and 3rd Damon- Eshbach modes are significantly asymmetrically dis- torted. Eventhoughthe distortioncausescomplextime- dependent SW amplitude oscillations, the main mode character is conserved. In Figs. 9(d),(i) the 2n peaks ofthe 2nd mode areclearlyidentifiedaswellasthe grad- ual transition of the mode profile within the thick part towards the wire mode. In fact, the two mode profiles of the n = 1 and the n = 2 mode become less different as very high modulations are reached. For example, at ∆d=35 nm, both modes consist of a clear central peak within the thick part and another small-wavelength ex- citation(consistingofapproximately3peaks)withinthe FIG.9. (Coloronline)(a)–(e)Simulatedspin-wavemodepro- filesinDamon-Eshbachgeometryinthetransitionfromafilm thin part. The only difference is that the phase of the toafullMC.(f)–(j)The∆d-dependentcolorplotsofthemode small-wavelength excitation in the thin part is shifted profiles weighted by thelocal film thickness. around π. From this observation, it can be concluded that closeto the full MC the mode characterofboth the n = 1 mode and the n = 2 mode is dominated by a wire mode character inside the thick part with an opti- part,thismodewasobservedtoevolveintoanedge mode cal (n = 2 mode) and acoustical (n = 1 mode) lateral inthefullMClimit(seeFigs.9(e),(j)). Thetypicalmode coupling low-wavelengthexcitation inside the thin part. profilecharacteristicsofthismodearealreadyestablished at ∆d = 31.5 nm inside the thick part with pronounced 3rd SW Mode. The general mode character of the peaksatbothedgesandasmalleranti-phasepeakinthe n = 3 mode with its 2n = 6 peaks in total is also pre- center. According to the above findings for the n = 1 served during the transition. However inside the thick and n = 2 modes, one would expect the presence of an 9 analogous second mode close to the full MC limit with FONDECYT 1161403, CONICYT PCCI (grant no. ananti-phasedynamicactivityinsidethethin part. Such 140051), the project 111559 DGIP USM, DAAD PPP mode exists in the spin-wavespectrapresentedin Fig. 8, ALECHILE(grantno.57136331)andfromtheDeutsche but reveals a much smaller (negligible) resonance ampli- Forschungsgemeinschaft(grant no. LE2443/5-1). tude. The different behavior is due to the symmetry of themodeprofileinsidethethickpart. Thecharacteristic pronounced peaks at the edges of the thick part hinder thelateralcouplingofananti-phaseexcitationinsidethe thin part due to the high cost of exchange energy. In order to summarize the transition of spin-wave modes in Damon-Eshbach geometry, it is noted that the quasi-uniform mode keeps its character throughout the transitionbutwithavanishingamplitude inthe fullMC limit where it, finally, disappears. The n = 1 and n = 2 modesarefoundtodevelopintolaterallycouplingacous- tical and optical modes of the wire mode with a low- wavelength excitation inside the thin part. Moreover, the egde mode of the full MC gradually evolvesfromthe n=3 Damon-Eshbachmode. V. CONCLUSION The gradualevolutionofspin-wavemodes inthe tran- sition from a thin film to a full magnonic crystal has been investigated in the backward-volume and Damon- Eshbach geometry. Experimentally obtained spin-wave spectra were compared to analytical theory and micro- magneticsimulationsinordertoidentifythemodulation- height-dependent mode character. In the backward- volumeorientation,anevenricherspin-wavespectrumis reported compared to the full magnonic crystal. The 1st and the 2nd SW modes are found to evolve into the wire andthe edge mode in the full-magnonic-crystallimit, re- spectively. Forthesemodesaswellasformodesofhigher modenumbern,generaltransitionrulestowardsthemth mode in the full magnonic crystal are formulated and supported by the mode profiles. In the Damon-Eshbach orientation, major changes of the mode character are found at very high levels of sur- face modulation with ∆d → d. Here, the 1st and the 2nd spin-wave mode are observed to merge into the wire mode whereas the 3rd SW mode evolves into the edge mode. VI. ACKNOWLEDGMENT We thank B. Scheumann for the film deposition, A. BanholzerandC.Fowleyforthetechnicalsupportinthe patterning process and P. C. Grubitz and A. Jansen for their help with the FMR measurements. The support by the Structural Characterization and Nanofabrication Facilities Rossendorfat IBC and the HZDR Department of Information Services and Computing is gratefully ac- FIG. 10. (Color online) (a)–(d) Fitted spin-wave resonance knowledged. This work was supported by the Centers spectra (colored circles) for different modulation heights ∆d of Excellence with Basal/CONICYT financing (grant together with the quasi-analytical calculations (PWM) plot- no. FB0807), CONICYT PAI/ACADEMIA 79140033, ted as orange lines. 10 height of approximately 9 nm the agreement of theory and experiment is striking. However, for higher modu- lation sizes significant differences between measurement and theory are visible and are likely due to thickness- related approximations within the theory.70 It shall be noted that up to now, such systems were theoretically handled using perturbation theory,22,24 which cannot be employed for such high levels of surface modulation. The firm quantitative agreement between the numer- ically calculated SW resonances and the ones obtained by PWM also can be confirmed by a comparison of the mode profiles. This was already done and published elsewhere58 confirming averyhighlevelofagreementup to a modulation height of ∆d=10 nm. The provenaccuracy of the theory for moderate mod- ulation heights justifies its application to calculate the spin-wave dispersion and mode profiles in the k-space. FIG.11. (Coloronline)Susceptibilityofthemodesabovethe For MCs of 300 nm periodicity or less, the measurement largefrequencygap(blue)andbelowthegap(orange)depen- oftheSWdispersionischallengingduetothelimitations dentfromthemodulationheight∆d. (a),(b)Thesusceptibil- ity at 150 mT and (c),(d) the susceptibility at 400 mT. The associatedwiththemaximumk-valuesaccessibleinBril- measured data is shown on the left side accomplished with louin light scattering (BLS). That makes PWM an ideal thesimulation data on theright side. toolregardingband-gapandband-structureanalysesand optimizations for such surface-modulated MCs. (i) Resonance Amplitude Analysis Figure 5 re- veals an enriched spin-wave spectrum of SMMCs com- VII. APPENDIX pared to the ‘full’ MC presented in Fig. 3. The dom- inance of the quasi-uniform trench mode for small ∆d valuesandits vanishingfor largevalues indicatesthe ex- (ii) Validation of the Quasi-Analytical Model istence of a threshold where both fundamental modes, Quasi-analytical theory based on the PWM42,67–70 was the trench and the wire mode, exhibit the same ampli- used as well in order to understand the transition of tude. In order to identify this threshold, an analysis of the SW modes in backward-volumegeometry. Figure 10 the measured and simulated mode amplitudes was car- shows the results from PWM (orange lines) in compari- ried out. Figure 11 depicts the resulting ∆d-dependent sonto the measurement(coloredcircles). The structural amplitudes of both, measurement (11(a),(c)) and simu- and magnetic parameters for the PWM-based calcula- lation (11(b),(d)) for two different field values, 150 mT tionswereusedaccordingtotheonesselectedforthemi- (11(a),(c)) and 400 mT (11(a),(c)). In the measurement cromagnetic simulations (see beginning of Sec. III). The a thresholdwasfound atapproximately∆d/d=0.26for only minor difference is that the precise ∆d values listed H = 150 mT and ∆d/d = 0.29 for H = 400 mT. The 0 0 inFig.10wereemployed. Inthesimulations,thesevalues data differs quite significantly compared to the simula- wereabstractedduetothenecessitytomodeltheSMMC tion. Here, the threshold was obtained at ∆d/d = 0.33 by a natural number of cells. Up to the modulation for both external field values. 1 J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani and 8 A.V.Chumak,V.I.Vasyuchka,A.A.SergaandB.Hille- H. Puszkarski, Phys.Rev. B 54, 1043 (1996). brands, Nat.Phys. 11, 453 (2015). 2 S. Nikitov, P. Tailhades and C. Tsai, J. Magn. Magn. 9 F. S. Ma, H. S. Lim, V. L. Zhang, S. C. Ng and M. H. Mater. 236, 320 (2001). Kuok,Nanoscale Res. Lett. 7, 1 (2012). 3 S.NeusserandD.Grundler,Adv.Mater.21,2927(2009). 10 S. Tacchi, G. Duerr, J. W. K los, M. Madami, S. Neusser, 4 V. V. 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