Realization and Control of Bulk and Surface Modes in 3D Nanomagnonic Networks by Additive Manufacturing of Ferromagnets

The high‐density integration in information technology fuels the research on functional 3D nanodevices. Particularly ferromagnets promise multifunctional 3D devices for nonvolatile data storage, high‐speed data processing, and non‐charge‐based logic operations via spintronics and magnonics concepts. However, 3D nanofabrication of ferromagnets is extremely challenging. In this work, an additive manufacturing methodology is reported, and unprecedented 3D ferromagnetic nanonetworks with a woodpile‐structure unit cell are fabricated. The collective spin dynamics (magnons) at frequencies up to 25 GHz are investigated by Brillouin Light Scattering (BLS) microscopy and micromagnetic simulations. A clear discrepancy of about 10 GHz is found between the bulk and surface modes, which are engineered by different unit cell sizes in the Ni‐based nanonetworks. The angle‐ and spatially‐dependent modes demonstrate opportunities for multi‐frequency signal processing in 3D circuits via magnons. The developed synthesis route will allow one to create 3D magnonic crystals with chiral unit cells, which are a prerequisite toward surface modes with topologically protected properties.


Introduction
[3] Novel physical phenomena have been found in this field based on curvature-induced effects, [4][5][6] non-collinear spin textures including Bloch points, [7] and vortex domain walls, [8] as well as fascinating spin dynamics. [9,10]OI: 10.1002/adma.202303292Moreover, the 3D architecture brings advanced functionalities for spintronic and magnonic devices.In spintronics, it naturally increases the storage capacity such as the 3D racetrack memory. [11][15] Spin waves (magnons) are expected to advance substantially both on-chip GHz signal processing and chargefree computing schemes beyond the von Neumann architecture. [16,17]These perspectives are particularly true for 3D circuits which exploit the charge-free angular momentum flow of magnons.
However, one of the main bottlenecks for further exploration remains in a versatile 3D nanofabrication method which allows one to generate complex nanoarchitectures with ferromagnetic coatings of optimized spintronic and magnonic functionality.Besides the method of rolled-up thin films by strain engineering, [18] there are mainly two routes for 3D nanofabrication.On the one hand, there is focused electron beam induced deposition (FEBID) of metal-organic precursors. [19]his direct writing process of ferromagnets provides individual prototype 3D systems with resolutions of a few tens of nanometers.Currently, cobalt, iron, and cobalt-iron alloys are available in the materials tool box. [20,21]The sequential deposition method remains challenging in that the materials composition and properties can vary during the complex exposure pathways.Still, complex structures like 3D cobalt double helices have been prepared where the 3D geometry resulted in highly stable and robust locked domain wall pairs. [22]On the other hand, 3D nonmagnetic scaffolds are combined with magnetic materials prepared by a physical or chemical synthesis technique.[25][26][27][28][29][30][31][32][33] In particular, TPL allows for complex 3D nanoscaffolds and additive manufacturing by depositing a further functional material like Co for the artificial ferromagnetic buckyball in Ref. [26] The material inherits the TPLdefined topography.In Ref. [34], Gliga et al. used the chemically assisted atomic layer depositon (ALD) to first deposit a uniform conductive film of iridium on an individual buckyball nanotemplate produced by TPL and subsequently applied electroplating of Ni.Information about the structure's magnetic properties was not provided.We note that Ir is a heavy metal and detrimental for magnonic applications as it introduces significant spin-orbit coupling, additional interfacial anisotropy and spin pumping which produce potentially severe spin wave damping in metallic ferromagnets.The direct coating of Ni via ALD circumvents these detrimental effects.In Ref. [35], the authors Pip et al. performed electroless deposition of Ni-Fe alloys on TPL-produced nanotemplates such as a buckyball, a periodic 3D scaffold structure and a trefoil-knot.These scaffolds were immersed in an aqueous solution containing metals in the form of ions, a complexing agent and a reductant.The polymer surfaces required a pre-treatment involving activation, functionalization, and catalyzation.Structural morphologies were reported, but magnetic properties were missing in Ref. [35].In Ref. [36], the authors Porrati et al. demonstrated the fabrication of a conformal magnetic core-shell heterostructure.First, a scaffold PtC microbridge consisting of Pt nanograins embedded in a carbonaceous (C) matrix was produced by 3D FEBID.As a second step, postgrowth electron beam irradiation was carried out to reduce the resistivity of the PtC.Third, the authors applied an electrical current to the PtC microbridge (core).The local Joule heating led to thermal decomposition of a chemical precursor introduced into the scanning electron microscope (SEM) which resulted in site-selective chemical vapor deposition (CVD) of Co 3 Fe on the Pt-containing microbridge.Its anisotropic magnetoresistance (AMR) showed a magnetic hysteresis at 50 K with a relative effect of up to 0.06%.Pt is a heavy metal and, consequently, the core is expected to increase spin wave damping of metallic ferromagnets as well.The fabrication difficulties demonstrated by the previous works underline that a thin-film deposition technique allowing for direct conformal coating of ferromagnets on polymeric scaffolds is innovative.It would allow one to harvest the full versatility of TPL concerning complex nanoscaffolds and enable novel 3D nanomagnetic functionalities at room temperature.
May et al. fabricated a 3D diamond-bond lattice by TPL and evaporated a 50-nm-thick NiFe alloy (permalloy) directly onto the top surface.This physical synthesis technique led to interconnected nanomagnets with a crescent-shaped cross-section that showed intriguing artificial spin ice properties. [24]Due to shadowing effects the stacked nanomagnets were limited to four layers.A Au underlayer was added to enhance the thermal conductivity for optical microscopy at large laser intensity.Coherent spin wave modes in the GHz frequency regime were detected by Brillouin Light Scattering (BLS) spectroscopy. [25]Here, the laser spot diameter was about 40 μm.The spin waves were measured from almost the entire sample volume.The authors extracted two modes which depended on the strength of the magnetic field applied along the diagonal direction.Micromagnetic simulations addressed the four layers of stacked nanomagnets.They predicted an either localized or extended nature of spin wave modes with different mode quantization numbers.More modes were predicted than experimentally observed.Spatially resolved measurements of the spin dynamics were not conducted, and a discrimination between bulk and surface modes was not adequate.
A chemical synthesis technique such as ALD has the unprecedented advantage of conformality which avoids the shadowing effect.In addition, ALD is performed with an atomiclevel control of thickness.Recently developed ALD processes for Ni-based ferromagnets [23,37] have demonstrated excellent magnetic properties by coating vertically standing nanopillars.The pillars were up to 15 μm long when arranged in dense arrays and had high aspect ratios (length to diameter) of up to 30:1.Ferromagnetic tubes were achieved which exhibited a conformal coating with an excellent step coverage of ≥ 88 %. [23] The relative AMR was large and amounted to 1.4% at room temperature for ALD-grown Ni nanotubes. [38]The plasma-enhanced ALD process enabled an experimental study on the magnetochiral properties of magnons in individual permalloy nanotubes. [39][42][43] Here, nanotubular constituent magnets provide an especially promising platform for 3D magnonics.On the one hand, unlike nanowires, tubular structures can host stable flux-closure magnetic states. [44]hese vortex states avoid the Bloch point structure through the central axis.For the nanowires with a crescent-shaped crosssection the vortex state is not expected.On the other hand, nanotubes support nonreciprocal spin waves and magnetochiral characteristics. [39,45]All these aspects promise interesting physical properties in magnonic crystals.The realization of such 3D magnonic crystals on polymeric scaffolds has not yet been achieved however.A particular targeted advantage and valuable features consist in edge or surface magnon modes which are different from the bulk modes and promise topological or chiral properties in specifically engineered lattices.It is particularly expected that such properties are controlled via different magnetic configurations and fields. [13,46]n this work, we demonstrate TPL of 3D nanonetworks conformally coated with a Ni shell (Figure 1).To the best of our knowledge, this is an unprecedented artificial architecture in which interconnected ferromagnetic nanostructures are periodically arranged in all three spatial directions (over a macroscopic length of up to 17 μm in the vertical direction).For different 3D Ni nanonetworks with lattice constants down to 500 nm, we study the magnon modes locally by BLS with high spatial resolution (μ-BLS). [23,37]We base our 3D scaffolds on the woodpile structure. [47]t is a face-centered-cubic (fcc) structure which offers the highest packing density in 3D.Its valuable feature is the dense packing of elements which leads to a strong coupling via dipolar and exchange interaction between ferromagnetic segments in the lattice.The strong coupling is expected to support the formation of a magnonic band structure with minibands of relevant frequency band width and propagating magnon modes.We observe multiple modes which have different responses in the GHz frequency regime when altering the lattice period, magnetic configuration and field orientation.In contrast to previous studies, our 3D superstructures are fabricated in an additive manufacturing process which provides a homogeneous ferromagnetic material.The obtained results thereby reflect directly the nanoengineering of dynamic magnetic responses by the structural design without variations in materials quality.We identify characteristic frequency shifts between magnon modes in the first (top) and second layer of a 3D superstructure, attributed to surface and bulk modes, respectively.By means of micromagnetic simulation we visualize the microscopic nature of excited bulk magnon modes.Our work demonstrates that the additive manufacturing methodology enables engineered GHz responses in 3D ferromagnetic nanonetworks.It offers a technology platform for 3D magnonic devices with complex unit cells which might give rise to magnetochiral and topologically protected surface modes.

Structural and Quasistatic Magnetic Properties of 3D Nanonetworks
We fabricated 3D magnetic nanonetworks by depositing ferromagnetic Ni layers onto polymer templates (Figure 1a).The polymer template was a woodpile structure following Ref.[47].It was characterized by the lateral period a xy and vertical period a z .The structure consisted of alternating orthogonally oriented polymer rods with width w and height h.For the fcc woodpile structure investigated here we considered a z = √ 2a xy .3D polymer scaffolds were fabricated with different initial lattice periods a xy = 1.57and 1 μm on a fused-silica substrate by Nanoscribe Photonic Professonal GT+.Nanoscribe offered a resolution of about 200 nm in horizontal and 700 nm in vertical direction with IP-Dip photoresist.After development we obtained a 3D nanoscaffold consisting of interconnected polymeric nanorods (Figure 1b).The geometrical parameters were the same for all the nanorods.After development, we applied the heatinduced shrinking method of Liu et al. by heating the polymer scaffold to 450 °C for 12 min for selected samples. [47]We thereby reduced their initial lattice period of 1.57 μm to finally 0.5 μm.Using ALD, we first deposited a 5-nm-thick Al 2 O 3 layer onto the polymer scaffolds with and without heat-induced shrinking.Then, we applied plasma-enhanced ALD [23] to achieve a coating of a 10-nm-thick Ni shell in the same ALD chamber (Figure 1b; Figure S1, Supporting Information).The quality of the Ni thin film grown on the polymeric scaffolds (Figure 1c) was substantiated by in situ deposition on a planar reference substrate.Data are shown in Figure S2 (Supporting Information).In addition, we investigated the structure and chemical composition of ALDgrown shell on top of a polymer template by transmission electron microscopy (TEM) (Figure S3, Supporting Information).For the preparation of the relevant membrane (lamella), we produced separately a simple 3D geometry.Its polymer was conformally coated by a 5-nm-thick Al 2 O 3 shell followed by a 30-nm-thick Ni shell similar to Ref. [23].For stabilization, the sample was embedded in carbon before extracting the membrane for TEM via focussed ion beam etching.The polycrystalline Ni film was found to consist of grains exhibiting an fcc lattice (Figure S3a, Supporting Information).Using scanning TEM (STEM), the composition and elemental distribution of the sample were studied by energy-dispersive X-ray spectroscopy (EDS) (Figure S3b,c, Supporting Information).The characteristics of Ni on top of the polymer template were consistent with shells previously created on vertically standing semiconductor nanowires. [23]For the present manuscript, we have prepared and investigated multiple samples (see Table S1, Supporting Information, for their parameters).In the following, we focus on the samples which have provided the richest spectra and fine structure of well-resolved modes.
In Figure 1c, a colored SEM image of a 3D woodpile nanonetwork fully covered by the 10-nm-thick ferromagnetic Ni shell is presented.The total height of the scaffolds is 8.5 μm after the conducted pyrolysis and similar to the heights of the dense arrays of self-organized semiconductor nanowires coated in Ref. [23].It can be seen that the unit cells of the four bottom woodpile layers has not shrunk to the same extent as the ones on the top.We attribute this observation to the adhesion of the polymer to the substrate.The top eight unit cells which were far from the substrate have shrunk homogeneously.
Micromagnetic simulations of the full 3D ferromagnetic superstructure faced limitations by the computational power of state-of-the-art high-performance computing platforms.Note that ferromagnetic nanoscaffolds were up to 17 μm high, that is, their thickness was about three orders of magnitude larger than the one of typical thin-film magnonic crystals studied in the literature.Such large samples were not accessible via the stateof-the-art micromagnetic modeling in the available high performance computing center.Considering such restrictions, we simulated the static magnetic configurations of a 4-layer-high woodpile structure using the GPU-accelerated software MuMax 3 [48] (Figure 1d).In order to capture the spin texture of the full structure at a computation level consistent with the used hardware, we simulated four lateral unit cells with geometrical parameters a xy = 1000 nm, a z = √ 2a xy .The tubes had dimensions h = 700 and w = 250 nm with an additional 10-nm-thick uniform Ni coating.The samples were discretized into 512 × 512 × 320 cells with a volume of 5.8 × 5.8 × 6.7 nm 3 .We considered a saturation magnetization M s = 490 kA m −1 , [49] an exchange stiffness A ex = 8 pJ m −1 , and gyromagnetic ratio 1.1 × 176 rad GHz T −1 .A field μ 0 H of 50 mT was applied along the axis of one of the two tube lattices (x-direction).As a result, spins arranged in mainly two distinct patterns (Figure 1d), that is, an axially polarized state and an onion state [50] corresponding to the tube segments which were parallel and transverse to H, respectively.The simulated hysteresis loop and magnetic states of 3D nanonetworks with a lattice period of 500 nm are shown in Figure S4 (Supporting Information).

Discrete Surface Magnon Modes in the Top-Layer of a 3D Ni Nanonetwork
The spin dynamics of the 3D ferromagnetic nanonetworks were investigated by μ-BLS.We applied the technique to the two uppermost structural layers of the interconnected Ni nanotube network and aimed at the exploration of its surface and bulk magnon modes.A 532-nm-wavelength green laser was focused on the network with a laser spot diameter of about 250 nm, as schematically illustrated in Figure 1c.The BLS utilized the energy and momentum conservation during the inelastic scattering process between photons and magnons.When the energy of the scattered photon was reduced, a magnon or phonon was created (Stokes scattering).For an increased photon energy, a magnon or phonon was annihilated (anti-Stokes scattering).Considering the long measurement times under ambient conditions and the assumed poor thermal conductivity of the network, a 0.25 mW laser power was chosen to avoid local overheating and maintain the structural integrity of the 3D magnetic nanonetworks.The energy shifts of the back reflected light were analyzed by a Fabry-Perot interferometer.The external magnetic field was applied in the plane of the top surface (substrate).After saturating the magnetic nanonetworks in a high magnetic field, the sample was measured at gradually decreasing external magnetic field values.We focus on Stokes scattering signals here.The resonance frequencies extracted from the fitted functions are displayed in Figure 2a (symbols).As H increases, several of the detected resonance modes in the top (1 st ) layer are found to move toward higher frequencies.Only one mode (black symbols) does not depend on the magnetic field.We speculate that this peak originates from a phonon mode of the 3D superstructure.Further BLS experiments were performed on a bare polymer woodpile structure, which showed the same mode independent of magnetic field.Hence, this mode does not have a magnetic origin.
In the following, we discuss the three field-dependent modes and compare them to magnon modes recently reported for straight and long nanotubes. [39]For this, we classified the extracted resonance frequencies into different branches (red, green, and purple symbols in Figure 2a).Their mode frequencies increased with increasing magnetic field.Interestingly, all the three modes were at significantly higher frequencies than both the resonance detected on the planar Ni film (Figure S5, Supporting Information) and the ones detected on individual Ni nanotubes in Ref. [23].The broken line in Figure 2a represents the fielddependent resonance frequency obtained on the Ni film when fitting the Kittel formula (Kittel fitting). [51]n Ref. [39], it has been shown that individual nanotubes with unintentional defects exhibit a multitude of resonant modes due to two discretization effects.On the one hand, spin waves undergo constructive interference along the azimuthal direction.The constructive interference condition reads n ×  = C, where C is the circumference of the tube,  is the magnon wavelength and n is an integer number (n = 0, 1, 2, …).The mode with n = 0 stands for uniform spin precession.For n ⩾ 1 wave vectors k n = 2/ = n × 2/C are non-zero in azimuthal direction.The corresponding wave vectors are orthogonal to the magnetization M in a high field applied parallel to the long axis of a nanotube, which reflects the Damon-Eshbach (DE) configuration.Consequently, increasing n suggests increasing mode frequencies.On the other hand, although there are no nanotroughs which set the tubes into segments (Figure S6c, Supporting Information), there may be standing waves inside a unit cell of the top-most tubes with wave vectors parallel to the long axis of a tubular segment.Such modes are confined within a lattice period a xy with k j = j/a xy , where j = 1, 2, 3… Here, wave vectors are parallel to the magnetization M in a high field applied parallel to the long axis of a nanotube, which corresponds to a backward volume magnetostatic spin wave (BVMSW) configuration. [51]For BVMSWs, frequencies first decrease with j.None of the two scenarios explains the high resonance frequencies observed on the 1 st layer

Magnon Modes in the First Versus Second Layer
The woodpile structure with a lattice period of 1 μm allowed us to focus the 250-nm-diameter laser spot onto the 2 nd layer of the 3D nanonetwork (Figure S7a, Supporting Information).This layer was inside the unit cell of the woodpile structure and consisted of interconnected nanotubes which were rotated by 90°with respect to the ones in the top layer.For the following BLS data H was applied parallel to the tubes of the 2 nd layer (that is, transverse to the tubes of the 1 st layer).Then the spectra were collected while reducing the external magnetic field in a step-wise manner.By this means, we explored the second-layer magnon modes in the same field configuration and field history as the first-layer magnon modes of Figure 2a.
Figure 2b summarizes the resonance modes extracted from the second layer (open symbols).Here, five distinct modes are seen.A field-independent branch (open triangles) resides around 16.4 GHz.Its frequency is close to the mode which we attributed to phonons in Figure 2a.For the remaining four branches in Figure 2b the frequency regime is completely different compared to Figure 2a.The branches reside at much lower frequencies and near the resonance frequency extracted on the Ni film (broken line in Figure 2a).Each of the modes shifts to larger frequency values f for increasing H.However, the slope df/dH varies depending on the branch.The lowest lying branch (open squares) has the smallest slope df/dH of all the magnon branches in Figure 2a,b.Some of the detected modes are lower in frequency than the resonance frequency extracted on the Ni film.Based on the findings of Figure 2a, the significant discrepancy in frequency regime is unexpected in Figure 2b as the relative orientation of the applied field as well as the geometrical and materials parameters are identical in the 1 st and 2 nd layer.The different resonance frequencies displayed in Figure 2a,b might therefore reflect surface and bulk modes, respectively, of a 3D magnonic crystal formed by the ferromagnetic woodpile structure.To further analyze the eigenfrequencies, dynamic micromagnetic simulations are presented in the following.We do not compare signal strengths, as the excitation mechanism and selection rules in the simulations and the BLS experiment are different (Experimental Section).For the analysis, we compare frequency values of modes.
The spectra displayed in Figure 2c were computed for different fields assuming a xy = 1 μm, a z = √ 2a xy .We simulated unit cells of the woodpile structure with periodic boundary conditions (PBCs) to model the bulk modes of a large 3D magnonic crystal at the Γ point.The tubes had dimensions h = 700 nm and w = 250 nm.We simulated 2 × 2 × 1 unit cells and applied PBC 6 × 6 × 12 in MuMax 3 .The samples were discretized into 320 × 320 × 256 cells with a volume of 6.3 × 6.3 × 5.5 nm 3 .To have a correct implementation of the periodic boundaries, a geometry with open ends was simulated.The simulated spectra of the coherently excited woodpile contain a large number of peaks in Figure 2c.The resonance frequencies of the two prominent peaks indicated by filled diamonds and upward triangles in Figure 2b,c show similar values and field dependencies like the high-frequency branches observed in the BLS experiment on the 2 nd layer.The largest peak (filled diamond) corresponds to a uniform-precession mode with n = 0.The higher lying modes are then attributed to azimuthal spin waves with n > 0 (filled upward triangle).The mode marked with a filled downward triangle in Figure 2c models the lowestlying branch of the BLS data with the smallest slope df/dH in Figure 2b.Modes of smaller frequency were not accessible by BLS.Compared to the experimental BLS data, the simulated spectra contain significantly better resolved modes.We assume that inhomogeneous line broadening due to defects and the higher Gilbert damping in the real sample washed out the fine-structure contained in the simulated spectra.
The frequency range covered by the simulated modes in Figure 2c agrees well with the experimentally accessible modes obtained on the 2 nd layer (Figure 2b).The high-frequency modes detected on the 1 st layer (Figure 2a) are not predicted by the simulations.We note that the PBCs set in the simulations are appropriate to model bulk modes, but they do not allow one to capture surface-attributed magnon modes.The PBCs suppress the symmetry breaking action of the top surface of the real 3D network and its effect on magnon modes.Strikingly, the modes observed on the 2 nd layer exhibit the relatively small frequencies predicted by the simulations in Figure 2c and are hence assumed to reflect bulk modes.They reside also close to the uniform spinprecession frequency of the planar Ni film.In Figure 2c, we attribute some of the lowest-frequency modes to reside in the nanotubes which are in a transverse configuration with respect to the applied field.We explain the small resonance frequencies by a correspondingly small effective magnetic field. [51]Such modes will be discussed later.We note that PBCs were necessary because even the computational power provided in the high performance computing center was not sufficient to model the real 3D network including their terminating top and side surfaces.

Magnon Modes in Longitudinally and Transversely Magnetized Nanonetwork Segments
Considering the geometry of the woodpile structure, tubes of one layer were parallel to H, while the tubes of the adjacent layers were transverse to H.In the following we study the angular dependencies of magnon modes.In Figure 3a, we replot the resonance frequencies of Figure 2a obtained on the top-layer for which the nanotubes were in a longitudinally applied magnetic field, that is, parallel to H.The woodpile structure had a lattice constant of 1 μm. Figure 3b shows the frequency peaks in a configuration for which H was transverse to the top-layer nanotubes.In this case, the frequencies of all the detected modes decreased while increasing H.This slope df/dH < 0 is opposite to the parallel-field configuration shown in Figure 3a.We attribute the changes in resonance frequency and slope df/dH to both the inhomogeneous demagnetization factors of individual tubular ferromagnets and the magnetic anisotropy of the 3D superstructure.Particularly, the negative slope df/dH is consistent with a magnetic field applied along the hard-axis direction of a nanomagnet.The data show that a field up to 250 mT does not fully saturate the magnetic moments of top-layer tubes in the transverse configuration.The shape-anisotropy field is hence larger than 250 mT, corresponding to a relatively large demagnetization factor in transverse direction.
In Figure 3c,d, we display mode frequencies measured on nanotubes in the top layer of a woodpile structure with a smaller lattice constant of a xy = 0.5 μm (Figure S7b, Supporting Information).We observe a significantly different behavior.For both field configurations, the detected modes reside at smaller frequencies compared to Figure 3a,b, respectively.The change of the lattice parameter modifies the exact mode frequencies and field dependencies detected on the 3D Ni nanonetworks.In Figure 3c, for field values ⩾150 mT we extract only one resonance frequency from the relatively broad spectra.For these fields, we do not observe a fine-structure in the 3D superstructure with the small lattice constant.Strikingly, in a transverse field (Figure 3d) the slope df/dH of eigenfrequencies is positive for fields larger than 100 mT.For small a xy , transversely applied fields >100 mT are hence sufficient to saturate the ferromagnetic nanotubes.Such an observation suggests that the smaller lattice constant and more compact 3D arrangement of nanotubes reduce the value of the relevant demagnetization factor and make the superstructure magnetically more isotropic.The geometrical parameters are found to control the mode frequencies of the 3D Ni nanonetworks prepared by our additive manufacturing route.We note that the mode frequencies obtained on the 1 st layer of the smalllattice-constant nanonetwork in Figure 3c continue to be clearly higher than the resonance frequencies extracted on the planar Ni film (broken line in Figure 2a).

Spatially Resolved Spectroscopy of the Surface Magnons
In Figure 4, we present resonance frequencies which we detected when varying the relative orientation of an applied field of 150 mT in the plane of the top surface of a 3D nanonetwork.We explored a superstructure with a xy = 0.5 μm and took spectra on three different positions within the top layer for each field orientation, as shown in Figure 4a.The resonance frequency observed on a nanotube in the central region of the top surface (circle) in Figure 4b decreased monotonously with increasing the angle  between the long axis of this top-layer nanotube and the field direction.Such a behavior is consistent with an easy axis along the  = 0°direction as expected for a nanotube (and a hard axis at  = 90°, see results above).
It is instructive to discuss the resonances at further positions in more detail.The mode in the corner of the top layer (square) displays the same tendency as the center mode, however, its frequency variation is smaller.We resolve two closely spaced modes for the corner at  = 26°and 90°.At the tube ends (triangle), two well separated (closely spaced) modes are observed for  = 45°(  = 90°).It is noted that at  = 45°one of the modes has an even greater frequency value than the tube-end mode detected for  = 0°.For  = 0°, the main resonances at the three investigated positions are well separated in frequency.Apparently, a clearly nonuniform internal field distribution exists in the top layer of the 3D Ni nanonetwork.One surprising observation is that for  = 90°the mode frequencies at the three measured positions are much closer than for 0°.The internal field inhomogeneity is found to be much less pronounced at  = 90°than 0°.A central top-layer nanotube in the transverse-field configuration exhibits almost the same internal field like tube ends and the corner.We find a nearly degenerate eigenfrequency at the three   different positions (upper three symbols at  = 90°).We speculate that a macroscopically coherent magnon state on the top surface might exist.Further investigations are needed to fully understand the angular dependence of the dynamic response and the collective behavior.

Microscopic Nature of Bulk Magnon Modes
Considering the good quantitative agreement between eigenfrequencies of 2 nd -layer nanotubes and the simulations performed with PBCs in Figure 2b,c, we performed further dynamic simulations on a woodpile structure with smaller geometrical parameters a xy = 500 nm, a z = √ 2a xy , h = 350 nm, and w = 125 nm (Figure 5).The Ni shell was 10 nm thick.We simulated 2 × 2 × 1 unit cells and applied PBCs (PBC 6 × 6 × 12).The sample was discretized into 160 × 160 × 128 cells with a volume of 6.3 × 6.3 × 5.5 nm 3 .Figure 5a shows the normalized spectrum (simulated power spectral density).It contains a prominent peak at a frequency of 11 GHz (main resonance labeled with m 3 ).The main branch experimentally detected for a xy = 500 nm (Figure 3c) resides clearly at a higher frequency.Again, the measured top surface resonance is at a larger frequency than resonances simulated with PBCs.
The lowest-frequency mode indicated as m 1 in Figure 5a corresponds to an in-phase precession of spins in specific regions of the transversely magnetized tubes (Figure 5b).The complex dynamic magnetization at the resonance frequencies was visualized as described in Experimental Section using Paraview. [52]e to the inhomogeneous internal field along the circumference, the spin precession of this low frequency mode localizes along the sides of the short axis of the tube where the internal field is low. [51]The modes between m 1 and m 2 are all localized on the transversely magnetized tubes and show complicated nodal patterns along the azimuthal and radial directions.We note that mode m 2 is at a sufficiently high frequency to extend over the transverse tubes (Figure 5c).The quantization considers both the azimuthal and longitudinal directions.The most prominent mode m 3 (Figure 5d) corresponds to the uniform in-phase precession with n = 0 of the longitudinally magnetized tubes.Mode m 4 (Figure 5e) is a higher order mode on these tubes discretized along the azimuthal direction.All these simulated bulk modes have eigenfrequencies smaller than the modes in BLS spectra that were taken on the top surface of the Ni woodpile with the small lattice constant (Figure 3c).This finding is consistent with the top-layer modes of the woodpile structure with a large lattice constant in Figure 2a which are also at higher frequencies than the modes extracted from the relevant simulations.
Comparing with the experimental data obtained on the evaporated Py nanomagnet lattice of Ref. [25], the spin wave modes of our ALD-grown Ni nanonetworks shown in Figure 2a are higher in frequency by up to 10 GHz under the same magnetic field.This discrepancy is surprisingly high and counterintuitive as Py exhibits a considerably higher saturation magnetization.It enters the equation of motion and suggests higher frequencies for Py nanomagnets compared to Ni. [51] Using spatially resolved BLS, we discriminated between magnon modes in the topmost and second layer.In addition, the local dynamic response was explored while applying a magnetic field in different directions.For the simulation results, we considered the tubular geometry achieved by conformal coating of the TPL lattice in contrast to the crescent-shaped nanowires obtained in Ref. [25].The woodpile structure gives rise to an additional segmentation of nanotubes and magnon modes become confined in half-tube segment on the top and bottom of TPL-produced nanorods.As a consequence a quantization in azimuthal direction similar to Ref. [25] plays a role which is different from the complete tubular shells studied in Ref. [39].Via the nanorod intersections (junctions), the conformally-coated 3D nanonetwork system exhibits mode quantization within a unit cell also in longitudinal direction.
Future research should explore further the magnon properties of the peculiar surface modes by local microwave excitation.This can be achieved via integration of individual woodpile structures into broadband coplanar waveguides (CPWs), [53] which provide radiofrequency (RF) signals for multi-frequency signal processing.For fields up to 200 mT, RF signals up to a frequency of 10 GHz excite directly bulk magnon modes (Figure 2b).These modes would give rise to absorption in the CPW and thereby RF signal filtering at the multiple magnon frequencies of Figure 2b.The large magnetic volume of a 3D magnonic crystal enhances the absorption (filtering) effect compared to thinfilm (2D) magnonic crystals explored so far.In the following we focus on the new opportunities which we expect from the peculiar resonances found in Figure 2a.For frequencies beyond 10 GHz the RF signal of a nearby CPW would excite the observed high-frequency resonances in the outermost layer (surface modes).The woodpile structure is an fcc lattice which exhibits a large packing density of constituent elements.Considering both the pioneering work on nanostructured microwave-tomagnon transducers [54] and the resonant magnonic grating coupler effect, [55] we expect the surface resonances to couple dynamically to the spins in the inner part and excite short-wave magnons in the assumed band structure of the bulk 3D magnonic crystal.The vision is an application in which these magnons propagate across the 3D lattice and are then controlled by its band structure, giving rise to data processing with magnons in a 3D architecture.Furthermore, advancements in the efficiency of micromagnetic codes are important to model the complete 3D magnonic nanoarchitectures. [56]

Conclusion
In summary, we have presented an additive manufacturing method by which we fabricated a 3D ferromagnetic nanonetwork.We combined TPL with ALD.The combination offered an unprecedented possibility to create complex 3D polymer nanoscaffolds conformally coated by a ferromagnet.In the 3D Ni nanonetworks, we found rich magnon spectra over a broad range of GHz frequencies.Simulations suggested spin-precessional motion with quantization patterns engineered via the geometric parameters.Different positions of the structures provided different responses when we varied the applied magnetic field direction.This was attributed to an inhomogeneous internal field.Magnon spectra obtained on the first and second layer of the woodpile structure showed significantly different eigenfrequencies.We assigned this to a difference between surface and bulk modes, respectively.The presented methodology and results are promising in view of on-chip microwave signal processing via 3D magnonic crystals which contain chiral unit cells giving rise to magnon modes with potentially topological properties.

Experimental Section
Sample Fabrication: The 3D polymer scaffolds were fabricated by the Photonic Professional GT+ (Nanoscribe Inc., Germany) in CMi (EPFL).A negative photoresist IP-Dip with a refractive index n ≈ 1.511 was used for two-photon lithography (TPL).During a fabrication process, a droplet of IP-Dip photoresist was dropped onto the surface of a fused-silica substrate (25 × 25 mm 2 square with a thickness of 0.7 mm).The microscope objective (63 ×, NA = 1.4) dipped in the liquid resist in a dip-in laser lithography (DILL) configuration.An infrared (780 nm) femtosecond laser with a laser power of 20 mW was exposed to the resist.The samples were written in a GalvoScan mode.For the development, the whole substrate was immersed in propylene glycol monomethyl ether acetate (PGMEA) for 20 min and isopropyl alcohol (IPA) for another 5 min.Then the samples were taken out to dry in the ambient condition.A pyrolysis process was performed under the constant N 2 flow in a bench top Rapid Thermal Process tool (JetFirst 200) in CMi.The process includes three stages: 1.The chamber was heated from room temperature to 450 °C at a ramp rate of 10 °C min −1 .2. The temperature was kept at 450 °C for 12 min.3. The chamber was naturally cooling down to room temperature.The film deposition experiments were processed in a hot wall Beneq TFS200 ALD system.The 3D polymer nanoscaffolds with/without shrinking on the fusedsilica substrate were positioned in the center of the ALD chamber.They were coated with a 5-nm-thick Al 2 O 3 layer followed by a 10-nm-thick Ni layer as previously described.The 3D polymer nanoscaffold to which we applied pyrolysis was investigated in the scanning electron microscope (SEM).To avoid charging and to improve the contrast for SEM imaging we sputtered a 5-nm-thick Au layer onto the surface of the sample before depositing Al 2 O 3 (5 nm) and Ni (10 nm) by ALD.We investigated the plane Ni film which contained the same Au underlayer underneath the Al 2 O 3 (5 nm) film by means of BLS.It showed a consistent magnetic resonance as the Ni film without the Au underlayer.We deposited Al 2 O 3 (5 nm) and Ni (30 nm) by ALD on a Si(100) wafer for reference broadband spectroscopy measurements.
Scanning Electron Microscope: The morphology of the woodpile structures were investigated by SEM MERLIN from Zeiss.
Transmission Electron Microscope: The microstructure of Ni on a polymer was investigated by a transmission electron microscope FEI Tecnai Osiris.The element analysis and distribution were studied by EDS in the STEM mode.
Micro-BLS Microscopy: The spin-wave eigenmodes were imaged by micro-focus Brillouin Light Scattering microscopy (μ-BLS) at room temperature.A 532-nm wavelength green laser was focused on the top or second layer surface of the 3D magnetic nanostructure by a 100 × objective lens with a numerical aperture of 0.75.The laser spot was around 250 nm and laser power was set to 0.25 mW.The samples were mounted on a piezo stage whose position could be precisely controlled by computer in x, y, and z axis.A magnetic field was applied with varied angles to the top layer tubes via a permanent magnet.The fitting to extract resonance frequencies of modes was done in the following way: For the field-dependent spectra of each sample, the fitting region starts from the end of the tail of the laser side peak under the lowest magnetic field (50 mT).The spectra were fitted as the Lorenzian-function in Origin.
Micromagnetic Simulations: The micromagnetic simulations were performed with Mumax 3 . [48]The geometry was initialized using built-in elementary geometric shapes and logic operations.The magnetic parameters were chosen to correspond to bulk Ni: saturation magnetization M s = 490 kA m −1 , exchange stiffness A ex = 8 pJ m −1 , gyromagnetic ratio = 1.1 × 176 rad GHz T −1 , and Gilbert damping  = 0.01.We obtained the static profile at various fields by first randomly initializing the magnetization.Then, a large field of 1 T was applied to prevent the state from being in a minor loop.The field was applied along the x-direction with a slight misalignment of 2°in the x,y-plane in order to prevent the system from reaching an artificial energy minima induced by the spatial discretization.After relaxing to the groundstate using the built-in relax() function of Mumax 3 , the field was swept from +400 to −400 mT and subsequently from −400 to +400 mT.For the dynamic simulations, the static profile obtained while sweeping from the saturated state was used as initial conditions.Additionally, periodic boundary conditions were applied (PBC 6 × 6 × 12).To obtain the spectrum, the system was excited with a sinc pulse h mag = h mag, 0 sinc(2f c (t − T/2)) with amplitude μ 0 h mag, 0 = 2 mT, cut-off frequency f c = 25 GHz and running time 20 ns.We note that the dynamic simulations were performed in that spin precession was induced by means of a homogeneous magnetic field pulse applied to all simulation cells.Hence simulations provided coherently excited eigenmodes which we compared to the eigenmodes excited incoherently by thermal fluctuations as studied experimentally by μ-BLS.

Figure 1 .
Figure 1.Schematic of a TPL-produced 3D nanonetwork conformally coated with a Ni shell.a) The schematic of the photoresist exposure of TPL fabrication.b) The illustration of the step in which the Ni precursor conformally coats the 3D polymer scaffold step during the ALD process.The molecule stands for the Ni precursor nickelocene.The full ALD process is in Figure S1 (Supporting Information).c) The colored SEM image of the 3D Ni nanonetwork after heat-induced shrinkage.The green laser and lens represent the -BLS measurement configuration.The scale bar is 2 μm.d) The static magnetic state of the simulated woodpile structure under a 50 mT bias magnetic field.e) Spectra of thermally excited magnons detected by μ-BLS under different external magnetic fields.

Figure
Figure1ereports BLS spectra obtained for μ 0 H varied from 250 to 50 mT with a step of −50 mT.The signal below 5 GHz originates from elastically scattered laser light.The small peaks at, for example, 9.5 GHz and 13.4 GHz are attributed to elastically scattered side peaks of the BLS laser.The red solid line represents the fitting of the remaining spectrum with a Lorentzian function assuming up to four resonances depending on the field.The 3D nanonetwork hence gives rise to a multitude of resonant modes.The Ni thin film fabricated via the same ALD process (FigureS5, Supporting Information) contains only one main peak at each applied magnetic field.The resonance of the polycrystalline Ni film did not show a magnetic anisotropy for H rotated in the plane.The resonance frequencies extracted from the fitted functions are displayed in Figure2a(symbols).As H increases, several of the detected resonance modes in the top (1 st ) layer are found to move toward higher frequencies.Only one mode (black symbols) does not depend on the magnetic field.We speculate that this peak originates from a phonon mode of the 3D superstructure.Further BLS experiments were performed on a bare polymer woodpile structure, which showed the same mode independent of magnetic field.Hence, this mode does not have a magnetic origin.In the following, we discuss the three field-dependent modes and compare them to magnon modes recently reported for straight and long nanotubes.[39]For this, we classified the extracted resonance frequencies into different branches (red, green, and purple symbols in Figure2a).Their mode frequencies increased with increasing magnetic field.Interestingly, all the three modes were at significantly higher frequencies than both the resonance detected on the planar Ni film (FigureS5, Supporting Information) and the ones detected on individual Ni nanotubes in Ref.[23].The broken line in Figure2arepresents the fielddependent resonance frequency obtained on the Ni film when fitting the Kittel formula (Kittel fitting).[51]In Ref.[39], it has been shown that individual nanotubes with unintentional defects exhibit a multitude of resonant modes due to two discretization effects.On the one hand, spin waves undergo constructive interference along the azimuthal direction.The constructive interference condition reads n ×  = C, where C is the circumference of the tube,  is the magnon wavelength and n is an integer number (n = 0, 1, 2, …).The mode with n = 0 stands for uniform spin precession.For n ⩾ 1 wave vectors k n = 2/ = n × 2/C are non-zero in azimuthal direction.The corresponding wave vectors are orthogonal to the magnetization M in a high field applied parallel to the long axis of a nanotube, which reflects the Damon-Eshbach (DE) configuration.Consequently, increasing n suggests increasing mode frequencies.On the other hand, although there are no nanotroughs which set the tubes into segments (FigureS6c, Supporting Information), there may be standing waves inside a unit cell of the top-most tubes with wave vectors parallel to the long axis of a tubular segment.Such modes are confined within a lattice period a xy with k j = j/a xy , where j = 1, 2, 3… Here, wave vectors are parallel to the magnetization M in a high field applied parallel to the long axis of a nanotube, which corresponds to a backward volume magnetostatic spin wave (BVMSW) configuration.[51]For BVMSWs, frequencies first decrease with j.None of the two scenarios explains the high resonance frequencies observed on the 1 st layer

Figure 2 .
Figure 2. Magnon modes detected on a 3D Ni nanonetwork with a lattice period 1 μm.Extracted resonance mode in dependence of external magnetic field when the magnetic field is parallel to the tubes of 1 st (a) and 2 nd layer (b).c) Simulated spectra of the coherently excited 3D Ni nanonetwork under different bias magnetic fields

Figure 3 .
Figure 3. Magnon modes of 3D Ni nanonetworks with different lattice constants for different magnetic configurations.Extracted resonance modes when the magnetic field is a) parallel or b) transverse to the tubes of the top layer in a 1 μm lattice constant nanonetwork.Extracted resonance modes when the magnetic field is c) parallel or d) transverse to the tubes of the top layer in a 0.5 μm lattice constant nanonetwork.

Figure 4 .
Figure 4. Spatially and angle-resolved magnon modes of a 3D Ni nanonetwork.a) The schematic of the measurement configuration. is the angle between the magnetic field H and the tubes of the top layer.The lattice constant a xy was 0.5 μm and the applied field was 150 mT.b) Resonance frequencies detected at three different positions when altering the angle  between the magnetic field and the 3D Ni nanonetwork.

Figure 5 .
Figure 5. Simulated dynamic response and visualized modes on 3D Ni nanonetworks.a) Spectrum obtained on a woodpile with PBCs at a bias field of 250 mT with labeled peaks visualized as follows: b) mode m 1 corresponding to a localized in-phase precession and c) higher-order mode m 2 of transversally magnetized tube segments.d) Uniform spin precession (mode m 3 ) and e) higher-order azimuthal mode m 4 around longitudinally magnetized tube segments consistent with n = 0 and n > 0, respectively.The lattice constant a xy was 0.5 μm.