Spin-wave eigenmodes in direct-write 3D nanovolcanoes
O. V. Dobrovolskiy, N. R. Vovk, A. V. Bondarenko, S. A. Bunyaev, S. Lamb-Camarena, N. Zenbaa, R. Sachser, S. Barth, K. Y. Guslienko, A. V. Chumak, M. Huth, G. N. Kakazei
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Spin-wave eigenmodes in direct-write 3D nanovolcanoes
O. V. Dobrovolskiy, a) N. R. Vovk,
2, 3
A. V. Bondarenko, S. A. Bunyaev, S. Lamb-Camarena, N. Zenbaa, R. Sachser, S. Barth, K. Y. Guslienko,
5, 6
A. V. Chumak, M. Huth, and G. N. Kakazei Faculty of Physics, University of Vienna, 1090 Vienna, Austria Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Física e Astronomia,Universidade do Porto, 4169-007 Porto, Portugal Department of Physics, V. N. Karazin Kharkiv National University, Svobody Sq. 4, Kharkiv 61022,Ukraine Physikalisches Institut, Goethe University, 60438 Frankfurt am Main, Germany Division de Fisica de Materiales, Depto. Polimeros y Materiales Avanzados: Fisica,Quimica y Tecnologia, Universidad del Pais Vasco, UPV/EHU, 20018 San Sebastian,Spain IKERBASQUE, the Basque Foundation for Science, 48009 Bilbao, Spain (Dated: 9 February 2021)
Extending nanostructures into the third dimension has become a major research avenue in modern magnetism, super-conductivity and spintronics, because of geometry-, curvature- and topology-induced phenomena. Here, we introduceCo-Fe nanovolcanoes—nanodisks overlaid by nanorings—as purpose-engineered 3D architectures for nanomagnonics,fabricated by focused electron beam induced deposition. We use both perpendicular spin-wave resonance measure-ments and micromagnetic simulations to demonstrate that the rings encircling the volcano craters harbor the highest-frequency eigenmodes, while the lower-frequency eigenmodes are concentrated within the volcano crater, due to thenon-uniformity of the internal magnetic field. By varying the crater diameter, we demonstrate the deliberate tuning ofhigher-frequency eigenmodes without affecting the lowest-frequency mode. Thereby, the extension of 2D nanodisksinto the third dimension allows one to engineer their lowest eigenfrequency by using 3D nanovolcanoes with 30%smaller footprints. The presented nanovolcanoes can be viewed as multi-mode microwave resonators and 3D buildingblocks for nanomagnonics.Extending nanomagnets into the third dimension has be-come a vibrant research avenue in modern magnetism .It encompasses investigations of 3D frustrated systems ,topology- and curvature-induced effects in complex-shapednano-architectures , and the dynamics of spin waves in3D magnonic systems . In magnonics, which is con-cerned with the operations with data carried by spin waves,magnonic conduits have traditionally been made from 2Dstructures . Extension of spin-wave circuits into thethird dimension is required for the reduction of footprintsof magnonic logic gates and it allows, e.g., steering ofspin-wave beams in graded-index magnonics . In addi-tion, the height of nanomagnets offers an additional degree offreedom in the rapidly developing domain of inverse-designmagnonics in which a device functionality is first speci-fied, then a feedback-based computational algorithm is used toobtain the device design. In the past, peculiarities of the litho-graphic process were used, e.g., for the formation of crownson the tops of nanodisks . However, lithographic techniquesinsufficiently suit the demands of 3D magnonics. This moti-vates the increasingly growing attention to additive manufac-turing nanotechnologies .In recent years, two-photon 3D lithography and 3D directwriting by focused electron and ion beam-induced deposition(FEBID and FIBID) have become the techniques of choicefor the fabrication of complex-shaped nano-architectures inmagnetism, superconductivity and plasmonics . Formagnonics, the propagation of spin waves in direct-write Fe- a) Electronic mail: [email protected] and Co-based conduits has recently been demonstrated, witha spin-wave decay length in the range 3-6 µ m . GivenFEBID’s lateral resolution down to 10 nm and its versatilityregarding the substrate material, FEBID appears as a promis-ing nanofabrication technology for 3D magnonics .Here, we investigate the spin-wave eigenmodes in individ-ual direct-write Co-Fe nanovolcanoes by spin-wave resonance(SWR) spectroscopy and analyze the experimental datawith the aid of micromagnetic simulations. We reveal that themicrowave response of the nanovolcanoes essentially differsfrom the sum of the microwave responses of their constituent2D elements—nanorings and nanodisks. We demonstrate thatthe ring encircling the volcano crater leads to an effective con-finement of the low-frequency eigenmodes under the volcanocrater, while the higher-frequency eigenmodes are confinedin the ring area. By varying the crater diameter by ±
20 nm,we demonstrate the deliberate tuning of the higher eigenfre-quencies by about ± . µ s dwell time,HCo Fe(CO) as precursor gas and a serpentine scan-ning strategy . Five nanovolcanoes with different outer andinner (crater) diameters, D and d , respectively, were fabricated(see Table I). We label the nanovolcanoes with their diame-ter ratios NV D / d (nm) and use the prefixes ND and NR for
040 nm x yz H (a)(c) dDL nanoring nanodisk12 FIG. 1. Experimental geometry. (a) Atomic force microscopy imageof the nanovolcano NV300 /
200 with the outer diameter D =
300 nmand the crater diameter d =
200 nm. Inset: cross-sectional line scan,as indicated. (b) Active part of the coplanar waveguide (S: signal; G:ground) with a nanovolcano for microwave spectroscopy measure-ments (not to scale). (c) Possible ways of splitting a nanovolcanointo a combination of a disk and a ring for modeling. (d) Calcu-lated spatial dependence of the internal field H int for the 40 nm-thicknanovolcano NV300/200 and its individual basic elements (nanodiskND300L20 and nanoring NR300L20). the nanodisks and nanorings. These simpler objects, investi-gated extensively in the past , can be naively viewed asconstituent elements of the nanovolcanoes, see Fig. 1(c). Allnanovolcanoes exhibit a flat morphology and a slightly trape-zoidal cross-sectional profile, as revealed by atomic force mi- Sample D / d , L , M s , A , D1, D2, D3, R1, R2,nm nm kA/m pJ/m GHz GHz GHz GHz GHzNV1000 /
700 40 1125 16.7 10.19 11.70 13.11 20.22 21.05NV600 /
400 40 1030 16.1 14.57 17.04 19.04 26.93 28.58NV300 /
180 40 880 15.4 7.29 12.06 17.85 22.01 23.66NV300 /
200 40 880 15.4 7.32 11.67 17.01 21.95 25.16NV300 /
220 40 880 15.4 7.38 11.37 16.25 25.95 27.71NR300 /
200 20 880 15.4 – – – – 17.72NR300 /
200 40 880 15.4 – – – – 24.89ND200 20 880 15.4 10.47 15.83 21.68 – –ND300 20 880 15.4 7.92 11.37 15.05 – –ND340 20 880 15.4 7.32 10.44 13.65 – –TABLE I. Sample parameters and simulated SWR frequencies fornano-volcanoes, nano-rings and nano-disks. NV: nanovolcano; ND:nanodisk; NR: nanoring; D : outer diameter; d : inner diameter; L : thickness; M s : saturation magnetization; A : exchange stiffness. M s and A values correspond to those for equivalent diameter Co-Fenanodisks . The frequencies of the modes D1-D3, R1 and R2 arecalculated at 1 . /
700 and NV600 / .
01 5 10 15 20 25 3001
15 20 2501
180 200 220
ND340ND300 (e)
D4D3D2D1 A b s . po w . ( a . u . ) Frequency, f (GHz) nanodisks
L20 simulation nanovolcanoNV300/200experiment R2R1D2 D3 A b s o r b . po w e r ( a . u . ) D1 (c)(b) nanovolcanoNV300/220simulationexperiment R2R1D4D3D2D1 A b s o r b . po w e r ( a . u . ) Frequency, f (GHz) (a) (d) L40L20 R2R2 nanorings
NR300/200 A b s . po w . ( a . u . ) Frequency, f (GHz)
NV300 R2R1D3D2D1 f r e s ( G H z ) Crater diameter, d (nm)
FIG. 2. Measured (symbols) and calculated (lines) power absorptionspectra for the nanovolcanoes (a) NV300 /
200 and (b) NV300 / f res vs craterdiameter d for the nanovolcanoes with D =
300 nm. Calculatedpower absorption spectra for (d) the nanodisks with thickness L =
20 nm and diameters D =
300 nm and 340 nm, and (e) the nanoringsNR300 /
200 with thicknesses L =
20 nm and 40 nm. In all panels, H = . croscopy, see Fig. 1(a).The CPWs were prepared by e-beam lithography from a55 nm-thick Au film sputtered onto a Si/SiO (200 nm) sub-strate with a 5 nm-thick Cr buffer layer. The CPWs werecovered with a 5 nm-thick TiO layer, fabricated by e-beamlithography, for electrical insulation from the nanovolcanoes.The width and length of the active part of the CPWs wereequal to 2 D and 4 D of the nanovolcano under study, respec-tively, see Fig. 1(b). SWR measurements were taken in thefrequency range 4-32 GHz with a bias magnetic field in therange 1 . . ◦ , which is toosmall to cause splitting of spin-wave modes in perpendicu-larly magnetized nanostructures . The high-frequency ac ex-citation was provided by a microwave generator and the trans-mitted signal was detected by a signal analyzer. The measure-ments were done with a bias magnetic field modulation ampli-tude of 1 mT and a frequency of 15 Hz, and a phase-sensitiverecording of the microwave transmission.Figure 2(a) and (b) present the experimentally measuredmicrowave power absorption spectra for the nanovolcanoesNV300 /
200 and NV300 /
220 at 1 . .
35 GHz and the pronounced higher-frequency SWR modes (R1 and R2). The increase of thecrater diameter d from 180 nm to 220 nm leads to a blueshift of the R1 and R2 peaks by about 4 GHz (see also TableI). Interestingly, these changes in the high-frequency part of nano v o l c . N V / i sk ND s i de v i e w t op v i e w t op v i e w s . v i e w (a) (b) nanoring (c) D1 D2 D3 R1 R2D1 D2 D3 D4 D5 R1spin-waveamplitude0 0.5 1
FIG. 3. Calculated spatial depen-dences of the spin-wave eigenfunctionsfor (a) the 40 nm-thick nanovolcanoNV300 / /
200 at the out-of-plane bias magnetic field 1 . the spectrum are accompanied by the opposite (though muchweaker) shifts of the lower eigenfrequencies ( D , D ), and anearly constant eigenfrequency D , as shown in Fig. 2(c).To identify the SWR modes associated with different partsof the nanovolcanoes, micromagnetic simulations were per-formed using the MuMax3 solver . For all nanovolcanoesthe cell size was 2.5 × × and the damping param-eter was α = .
01. The simulations were performed in twostages. Firstly, an equilibrium magnetic configuration of thesystem was reached by relaxing a random magnetic config-uration for a given perpendicular bias field value. Subse-quently, magnetization precession was excited by applying asmall spatially uniform in-plane microwave field pulse. Fi-nally, a fast Fourier transform was used to extract the nor-malized frequency spectra and the spatial dependences of thespin-wave eigenmodes. In the simulations, we used the satu-ration magnetization M s and exchange stiffness A values de-duced for the disks with the same diameters and depositedwith the same FEBID parameters (Table I) and the assumedgyromagnetic ratio of γ / π = .
05 MHz/Oe . The decreaseof M s and A with the decrease of the disk diameter from 1 µ mto 300 nm is associated with a decrease of the [Co+Fe] con-tent from 80 ± ± ± ± .The calculated microwave absorption spectra for the nano-volcanoes NV300 /
200 and NV300 /
220 are shown by solidlines in Fig. 2(a) and (b). The calculated spectra fit well withthe experimentally measured ones. With increase of the diam-eter d , the R1 and R2 modes are shifted by a few GHz towardhigher frequencies. The increase of d has a very weak influ-ence on the location of the SWR modes D1–D3. Observationof the enhancements afforded by the 3D extrusion in compar-ison to a pure nanodisk geometry is possible if one takes anaive view of a nanovolcano as a composite geometrical ob-ject composed of a 20 nm-thick nanodisk overlaid by a 20 nm-thick nanoring (geometry 1 in Fig. 1(c)). The simulations pre-dict the lowest-frequency mode for the nanodisk ND300 at afrequency f res of 7 . f res of the 20 nm-thick nanodisk ND340, which has alarger diameter of 340 nm, matches well with the experimen-tally measured f res = .
32 GHz of the NV300/200 nanovol-cano. This is to say that the lowest-frequency eigenmode ofa model nanovolcano, i.e. a 20 nm-thick disk (volcano base-ment) overlaid by a 20 nm-thick ring, can be obtained in sim-ulations for a 20 nm-thick disk which has a larger effectivediameter D eff =
340 nm. That is, extension of a 2D nanomag-net into the third dimension allows one to engineer the low-est eigenfrequency of a nanodisk by using a 3D nanovolcanowhich has an about 30% smaller footprint . This phenomenoncan be explained by the complex internal field H int distributionin the 3D nanovolcanoes (Fig. 1(d)), which is responsible forthe dynamic demagnetization tensor resulting in a decrease ofthe effective spin pinning. Thus, while the spin precession an-gle in thin disks is zero at the disk edges , the spin precessionangle is non-zero in the case of nanovolcanoes.We have also calculated f res for 20 nm- and 40 nm-thicknanorings NR300 /
200 at 1 . /
200 (geometry 1 in Fig. 1(c)) f res = .
75 GHz is very far away from the modes R1 (22 .
02 GHz)and R2 (25 .
15 GHz) of the equivalently sized nanovolcanoNV300 / /
200 nm at 1 . f res = . . /
200 nm. In this way, the R2 peak can be ascribed tothe ring mode of a nanoring with the same thickness as thenanovolcano and the width equal to that of the ring encirclingthe nanovolcano crater (geometry 2 in Fig. 1(c)).The calculated spatial dependences of the spin-wave eigen-modes for the nanovolcano NV300/200 (Fig. 2(b)) are shownin Fig. 3(a). At first glance, the eigenmodes in the nanovol-cano resemble spin-wave “drum modes” known for nanodiskssaturated in the perpendicular geometry . These “drummodes” are approximately described by Bessel functions ofthe zeroth order and are shown in Fig. 3(b) for the 20 nm-thick nanodisk ND300 for comparison. Such interpretationexplains why the volcano low-frequency modes only weaklydepend on the crater diameter, see Fig. 2(c). Indeed, a closerlook at the D1-D3 mode profiles reveals that spin waves in thenanovolcano are confined under the volcano crater which actsas a “concentrator” for spin waves. The role of the ring over-lying the volcano basement disk becomes more decisive forhigher-frequency eigenmodes. Specifically, the R1 mode hasthe maximum amplitude at the bottom surface of the nano-volcano while the R2 mode has the maximal amplitude in theupper part of the ring around the crater. This elucidates whythe location of the R1 and R2 modes is very sensitive to thewidth ( D − d ) / ≃
50 nm essen-tially contributes to the R2 frequency (the exchange length λ = p A / ( µ M ) in Co-Fe-FEBID is about 5 nm). Accord-ingly, the localization of the R2 mode in the ring is not a resultof the reduced dipolar pinning , but is rather a result of theinterplay of the exchange and internal magnetic fields.Finally, with increase of the nanovolcano diameter D , thenumber of the SWR modes per given frequency interval in-creases, see Fig. 4. This can be understood as a reduction ofthe system sizes leads to a stronger confinement of spin wavesand the associated larger mode separation in the magnon fre-quency spectrum. With increase of the magnetic field, thespectra are shifted towards higher frequencies as a whole,without qualitative changes in the structure of the spectra andthe relative positions of the peaks. This can be understood onthe basis of the nearly linear dependence f res ( H ) ≃ ( γ / π ) H for axially symmetric nanoelements magnetized along thesymmetry axis . The slope of all straight lines f res ( H ) inFig. 4 is the same and it is determined by the gyromagneticratio γ / π .To summarize, we have introduced nanovolcanoes ason-demand engineered nano-architectures for 3D magnetismand magnon spintronics. The 40 nm-thick Co-Fe nanovolca-noes with diameters down to 300 nm were fabricated by thedirect-write FEBID technique and studied by perpendicularSWR spectroscopy. The spin-wave eigenfrequencies of thenanovolcanoes have been demonstrated to notably differ fromthe eigenfrequencies of the nanodisks and nanorings theyare built from, because of the strongly non-uniform internalmagnetic field. The experimental findings were elucidatedwith the aid of micromagnetic simulations which indicate thatthe rings encircling the volcano craters lead to an effectiveconfinement of the lower-frequency eigenmodes under thevolcano crater while the highest-frequency eigenmodes areconfined in the ring area. Accordingly, extension of 2Dnanodisks into the third dimension allows one to engineertheir lowest eigenfrequencies by using 3D nanovolcanoeshaving about 30% smaller footprints. By varying the craterdiameter by ±
20 nm, we have demonstrated frequency tuningof about ± ND340L20 nmcalc.exp.
R2 R1D3 R e s onan c e f r equen cy , f r e s ( G H z ) NV300/200 nm (a) D1 exp.calc. R1R2 D1D5 (c)
NV600/400 nm exp.calc.
R1R2 D7 D1 (d)
NV1000/700 nm
Magnetic field, H (T) calc. (b) D5 D1
FIG. 4. Resonance frequencies versus bias magnetic field as de-duced from SWR measurements (circles) and calculated numeri-cally (squares) for (a) the nanovolcano NV300 /
220 nm, (b) nanodiskND340L20 nm, (c) nanovolcano NV600 /
400 nm, (d) and nanovol-cano NV1000 /
700 nm. The dominating modes are indicated withlarger symbols and thicker lines. The slope of straight lines is deter-mined by γ / π = .
05 MHz/Oe. gineered spin-wave frequency spectra make them prospectiveplatforms for 3D magnonics and inverse-design magnonicdevices.
OVD and SLC acknowledge the Austrian Science Fund (FWF) for supportthrough Grant No. I 4889 (CurviMag). The Portuguese team acknowledgesthe Network of Extreme Conditions Laboratories-NECL and PortugueseFoundation of Science and Technology (FCT) support through ProjectNos. NORTE-01-0145-FEDER-022096, POCI-0145-FEDER-030085 (NO-VAMAG), and EXPL/IF/00541/2015. NZ and AVC acknowledge the Aus-trian Science Fund (FWF) for support through Grant No. I 4917. KGacknowledges support by IKERBASQUE (the Basque Foundation for Sci-ence). The work of KG was supported by the Spanish Ministerio de Ciencia,Innovacion y Universidades grant FIS2016-78591-C3-3-R. Support throughthe Frankfurt Center of Electron Microscopy (FCEM) is gratefully acknowl-edged. Further, support by the European Cooperation in Science and Tech-nology via COST Action CA16218 (NANOCOHYBRI) is acknowledged.The data that supports the findings of this study are available within the arti-cle. A. Fernández-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, andR. P. Cowburn, Nat. Comms. , 15756 EP (2017), review Article. A. Fernandez-Pacheco, L. Skoric, J. De Teresa, J. Pablo-Navarro, M. Huth,and O. V. Dobrovolskiy, Mater. , 3774 (2020). L. Keller, M. K. I. Al Mamoori, J. Pieper, C. Gspan, I. Stockem,C. Schröder, S. Barth, R. Winkler, H. Plank, M. Pohlit, J. Müller, andM. Huth, Sci. Rep. , 6160 (2018). A. May, M. Hunt, A. Van Den Berg, A. Hejazi, and S. Ladak, Commun.Phys. , 13 (2019). S. Gliga, E. Iacocca, and O. G. Heinonen, APL Mater. , 040911 (2020). S. H. Skjærvø, C. H. Marrows, R. L. Stamps, and L. J. Heyderman, Nat.Rev. Phys. , 13 (2020). R. Streubel, P. Fischer, F. Kronast, V. P. Kravchuk, D. D. Sheka, Y. Gaididei,O. G. Schmidt, and D. Makarov, J. Phys. D , 363001 (2016). O. M. Volkov, A. Kákay, F. Kronast, I. Mönch, M.-A. Mawass, J. Fassben-der, and D. Makarov, Phys. Rev. Lett. , 077201 (2019). D. Sanz-Hernández, A. Hierro-Rodriguez, C. Donnelly, J. Pablo-Navarro,A. Sorrentino, E. Pereiro, C. Magén, S. McVitie, J. M. de Teresa, S. Ferrer,P. Fischer, and A. Fernández-Pacheco, ACS Nano , 8084 (2020). D. D. Sheka, O. V. Pylypovskyi, P. Landeros, Y. Gaididei, A. Kákay, andD. Makarov, Commun. Phys. , 128 (2020). M. Krawczyk and H. Puszkarski, Phys. Rev. B , 054437 (2008). M. Yan, C. Andreas, A. Kakay, F. Garcia-Sanchez, and R. Hertel, Appl.Phys. Lett. , 122505 (2011). J. A. Otálora, M. Yan, H. Schultheiss, R. Hertel, and A. Kákay, Phys. Rev.Lett. , 227203 (2016). G. Gubbiotti, ed.,
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