Spin-wave eigenmodes in direct-write 3D nanovolcanoes

Extending nanomagnets into the third dimension has become a vibrant research avenue in modern magnetism.^1,2 It encompasses investigations of 3D frustrated systems,^3–6 topology- and curvature-induced effects in complex-shaped nano-architectures,^7–10 and the dynamics of spin waves in 3D magnonic systems.^11–15 In magnonics, which is concerned with the operations with data carried by spin waves, magnonic conduits have traditionally been made from 2D structures.^16–21 Extension of spin-wave circuits into the third dimension is required for the reduction of footprints of magnonic logic gates,²⁰ and it allows, e.g., steering of spin-wave beams in graded-index magnonics.^22–24 In addition, the height of nanomagnets offers an additional degree of freedom in the rapidly developing domain of inverse-design magnonics^25,26 in which a device functionality is first specified and then a feedback-based computational algorithm is used to obtain the device design. In the past, peculiarities of the lithographic process were used, e.g., for the formation of crowns on the tops of nanodisks.²⁷ However, lithographic techniques insufficiently suit the demands of 3D magnonics. This motivates the increasingly growing attention to additive manufacturing nanotechnologies.² 

In recent years, two-photon 3D lithography and 3D direct writing by focused electron and ion beam-induced deposition (FEBID and FIBID) have become the techniques of choice for the fabrication of complex-shaped nano-architectures in magnetism, superconductivity, and plasmonics.^{2–4,9,28–33} For magnonics, the propagation of spin waves in direct-write Fe- and Co-based conduits has recently been demonstrated, with a spin-wave decay length in the range of 3–6 μm.^34,35 Given FEBID's lateral resolution down to 10 nm and its versatility regarding the substrate material, FEBID appears as a promising nanofabrication technology for 3D magnonics.³⁶ 

Here, we investigate the spin-wave eigenmodes in individual direct-write Co-Fe nanovolcanoes by spin-wave resonance (SWR) spectroscopy^37,38 and analyze the experimental data with the aid of micromagnetic simulations. We reveal that the microwave response of the nanovolcanoes essentially differs from the sum of the microwave responses of their constituent 2D elements—nanorings and nanodisks. We demonstrate that the ring encircling the volcano crater leads to an effective confinement of the low-frequency eigenmodes under the volcano crater, while the higher-frequency eigenmodes are confined in the ring area. By varying the crater diameter by ±20 nm, we demonstrate the deliberate tuning of the higher eigenfrequencies by about ±2 GHz without essential variation of the lowest eigenfrequency. The presented nanovolcanoes can be viewed as multi-mode resonators and as 3D building blocks for nanomagnonics.

The 40 nm-thick Co-Fe nanovolcanoes with diameters down to 300 nm and crater diameters down to 180 nm were fabricated by FEBID (SEM: FEI Nova NanoLab 600) on top of a gold coplanar waveguide (CPW) (see Fig. 1). FEBID was performed with 5 kV/1.6 nA, a pitch of 20 nm, a dwell time of $1 μ$s, HCo₃Fe(CO)₁₂ as precursor gas,^39,40 and a serpentine scanning strategy.³⁸ For the experiments discussed in what follows, we selected, by atomic force microscopy (AFM) inspection, five nano-volcanoes with the outer and inner (crater) diameters, D and d, respectively, deviating by less than ±3 nm from the nominal values reported in Table I. We label the nanovolcanoes with their diameter ratios NVD/d (omitting the units “nm”) and use the prefixes ND and NR for the nanodisks and nanorings. These simpler objects, investigated extensively in the past,^41–47 can be naively viewed as constituent elements of the nanovolcanoes [see Fig. 1(c)]. All nanovolcanoes exhibit a flat morphology and a slightly trapezoidal cross-sectional profile, as revealed by atomic force microscopy [see Fig. 1(a)].

The CPWs were prepared by e-beam lithography from a 55 nm-thick Au film sputtered onto a Si/SiO₂ (200 nm) substrate with a 5 nm-thick Cr buffer layer. The CPWs were covered with a 5 nm-thick TiO₂ layer (e-beam lithography) for electrical insulation from the nanovolcanoes. The width and length of the narrowed middle part of the CPW signal line [see Fig. 1(b)] were equal to 2D and 4D of the nanovolcano under study, respectively [see Fig. 1(b)]. SWR measurements were taken in the frequency range of 4–32 GHz with a bias magnetic field in the range of 1.2–2 T, oriented perpendicular to the volcano plane. The field alignment error was less than $0.1°$⁠, which is too small to cause splitting of spin-wave modes in perpendicularly magnetized nanostructures.⁴⁸ The high-frequency ac excitation was provided by a microwave generator, and the transmitted signal was detected using a signal analyzer. The measurements were performed with a bias magnetic field modulation amplitude of 1 mT, a frequency of 15 Hz, and a phase-sensitive recording of the microwave transmission.

Figures 2(a) and 2(b) present the experimentally measured microwave power absorption spectra for the nanovolcanoes NV300/200 and NV300/220 at 1.2 T. For both nanovolcanoes, the spectra contain the dominating low-frequency SWR modes (D1) at about 7.35 GHz and the pronounced higher-frequency SWR modes (R1 and R2). The increase in the crater diameter d from 180 nm to 220 nm leads to a blue shift of the R1 and R2 peaks by about 4 GHz (see also Table I). Interestingly, these changes in the high-frequency part of the spectrum are accompanied by the opposite (though much weaker) shifts of the lower eigenfrequencies (D₂, D₃) and a nearly constant eigenfrequency D₁, as shown in Fig. 2(c).

To identify the SWR modes associated with different parts of the nanovolcanoes, micromagnetic simulations were performed using the MuMax3 solver.⁴⁹ For all nanovolcanoes, the cell size was 2.5 × 2.5 × 2.5 nm³ and the damping parameter was $α=0.01$⁠. The simulations were performed in two stages. First, an equilibrium magnetic configuration of the system was reached by relaxing a random magnetic configuration for a given perpendicular bias field value. Subsequently, magnetization precession was excited by applying a small spatially uniform in-plane microwave field pulse. Finally, a fast Fourier transform was used to extract the normalized frequency spectra and the spatial dependences of the spin-wave eigenmodes. In the simulations, we used the saturation magnetization $Ms$ and exchange stiffness A values deduced for the disks with the same diameters and deposited with the same FEBID parameters³⁷ (Table I) and the assumed gyromagnetic ratio of $γ/2π=3.05$ MHz/Oe.⁵⁰ The decrease in M_s and A with the decrease in the disk diameter from $1 μ$m to 300 nm is associated with a decrease in the [Co₃Fe] content from 80 ± 3 at. % to 63 ± 3 at. % and an increase in the [C + O] content from 20 ± 3 at. % to 37 ± 3 at. % (as residues from the precursor in the FEBID process).³⁷ 

The calculated microwave absorption spectra for the nanovolcanoes NV300/200 and NV300/220 are shown by solid lines in Figs. 2(a) and 2(b). The calculated spectra fit well with the experimentally measured ones. With the increase in diameter d, the R1 and R2 modes are shifted by a few GHz toward higher frequencies. The increase in d has a very weak influence on the location of the SWR modes D1–D3. Observation of the enhancements afforded by the 3D extrusion in comparison to a pure nanodisk geometry is possible if one takes a naive view of a nanovolcano as a composite geometrical object composed of a 20 nm-thick nanodisk overlaid by a 20 nm-thick nanoring [geometry 1 in Fig. 1(c)]. The simulations predict the lowest-frequency mode for the nanodisk ND300 at a frequency $fres$ of 7.9 GHz [Fig. 2(d)], which is about 600 MHz above the experimentally measured value for the equivalent diameter nanovolcano, NV300/200. In contrast, the calculated $fres$ of the 20 nm-thick nanodisk ND340, which has a larger diameter of 340 nm, matches well with the experimentally measured $fres=7.32$ GHz of the NV300/200 nanovolcano. This is to say that the lowest-frequency eigenmode of a model nanovolcano, i.e., a 20 nm-thick disk (volcano basement) overlaid by a 20 nm-thick ring, can be obtained in simulations for a 20 nm-thick disk, which has a larger effective diameter $Deff=340$ nm. That is, extension of a 2D nanomagnet into the third dimension allows one to engineer the lowest eigenfrequency of a nanodisk by using a 3D nanovolcano, which has an about 30% smaller footprint. This phenomenon will be discussed below.

We have also calculated $fres$ for 20 nm- and 40 nm-thick nanorings NR300/200 at 1.2 T [Fig. 2(e)]. For the 20 nm-thick nanoring NR300/200 [geometry 1 in Fig. 1(c)], $fres=17.75$ GHz is very far away from modes R1 (22.02 GHz) and R2 (25.15 GHz) of the equivalently sized nanovolcano NV300/200. This is because of the essentially larger internal field in the volcano's ring area as compared to the 20 nm-thick ring [Fig. 1(d)]. By contrast, for the 40 nm-thick nanoring NR300/200 nm at 1.2 T [geometry 2 in Fig. 1(c)], $fres=24.9$ GHz is very close to the measured R2 eigenfrequency (25.1 GHz) of the equivalently sized nanovolcano NV300/200 nm. In this way, the R2 peak can be ascribed to the ring mode of a nanoring with the same thickness as the nanovolcano and the width equal to that of the ring encircling the nanovolcano crater [geometry 2 in Fig. 1(c)].

The calculated spatial dependences of the spin-wave eigenmodes for the nanovolcano NV300/200 [Fig. 2(b)] are shown in Fig. 3(a). At first glance, the eigenmodes in the nanovolcano resemble spin-wave “drum modes” known for nanodisks saturated in the perpendicular geometry.^37,41 These drum modes are approximately described by Bessel functions of the zeroth order⁴¹ and are shown in Fig. 3(b) for the 20 nm-thick nanodisk ND300 for comparison. This explains why the volcano low-frequency modes only weakly depend on the crater diameter [see Fig. 2(c)]. Indeed, a closer look at the D1–D3 mode profiles reveals that spin waves in the nanovolcano are confined under the volcano crater, which acts as a “concentrator” for spin waves. As it was mentioned earlier, the D1 mode frequency for NV300/200 is lower than that for ND300, while for higher frequency modes D2 and D3, the opposite behavior is observed. The following qualitative explanation of this effect can be proposed. (i) The internal field increases at the NV edge, and therefore, the D modes are pushed away from this region (i.e., toward the volcano center). As a result, the stronger confinement of the spin waves leads to an increase in their frequencies. This increase becomes more significant for higher mode numbers since the spin-wave frequency is approximately proportional to the square of the wavevector. (ii) In the NV300/200 center region, the internal field is slightly smaller than that in ND300, which results in a redshift of the D1 frequency. (iii) The effect of the reduced internal field is dominant for the D1 mode, resulting in its frequency redshift. On the contrary, the effect of confinement is dominant for the higher modes D2 and D3, resulting in their frequency blueshifts. The role of the ring overlying the volcano basement disk becomes more decisive for higher-frequency R eigenmodes. This can also be explained by the fact that the reduced effective width (proportional to the width $(D−d)/2$ of the ring) becomes more relevant for shorter wavelengths. The exchange interaction for the rings with widths $≃50$ nm essentially contributes to the R2 frequency (the exchange length⁵¹ $λ=2A/(μ0Ms2)$ in Co-Fe-FEBID is about 5 nm). Accordingly, the localization of the R2 mode in the ring is not a result of the reduced dipolar pinning,⁵² but is rather a result of the interplay of the exchange and internal magnetic fields.

Finally, with the increase in the nanovolcano diameter D, the number of SWR modes per given frequency interval increases (see Fig. 4). This can be understood as a reduction in the system sizes that leads to a stronger confinement of spin waves and the associated larger mode separation in the magnon frequency spectrum. With the increase in the magnetic field, the spectra are shifted toward higher frequencies as a whole, without qualitative changes in the structure of the spectra and the relative positions of the peaks. This can be understood on the basis of the nearly linear dependence $fres(H)≃(γ/2π)H$ for axially symmetric nanoelements magnetized along the symmetry axis.⁴¹ The slope of all straight lines $fres(H)$ in Fig. 4 is the same, and it is determined by the gyromagnetic ratio $γ/2π$⁠.

To summarize, we have introduced nanovolcanoes as on-demand engineered nano-architectures for 3D magnetism and magnon spintronics. The 40 nm-thick Co-Fe nanovolcanoes with diameters down to 300 nm were fabricated by the direct-write FEBID technique and studied by perpendicular SWR spectroscopy. The spin-wave eigenfrequencies of the nanovolcanoes have been demonstrated to notably differ from the eigenfrequencies of the nanodisks and nanorings they are built from, because of the strongly non-uniform internal magnetic field. The experimental findings were elucidated with the aid of micromagnetic simulations, which indicate that the rings encircling the volcano craters lead to an effective confinement of the lower-frequency eigenmodes under the volcano crater, while the highest-frequency eigenmodes are confined in the ring area. Accordingly, extension of 2D nanodisks into the third dimension allows one to engineer their lowest eigenfrequencies by using 3D nanovolcanoes having about 30% smaller footprints. By varying the crater diameter by ±20 nm, we have demonstrated a frequency tuning of about ±2 GHz for the ring modes without affecting the lowest spin-wave eigenfrequency. The presented nanovolcanoes can be viewed as multi-mode resonators with potential applications in telecom-frequency filters. In addition, the engineered spin-wave frequency spectra make them prospective platforms for 3D magnonics and inverse-design magnonic devices.

O.V.D. and S.L.C. acknowledge the Austrian Science Fund (FWF) for support through Grant No. I 4889 (CurviMag). The Portuguese team acknowledges the Network of Extreme Conditions Laboratories-NECL and the Portuguese Foundation of Science and Technology (FCT) support through Project Nos. NORTE-01-0145-FEDER-022096, PTDC/FIS-MAC/31302/2017, POCI-0145-FEDER-030085 (NOVAMAG), and EXPL/IF/00541/2015. N. Z. and A. V. C. acknowledge the Austrian Science Fund (FWF) for support through Grant No. I 4917. S. B. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG) through Grant Nos. BA 6595/2-1 and BA 6595/1-1. K. G. acknowledges support from IKERBASQUE (the Basque Foundation for Science). The work of K. G. was supported by the Spanish Ministry of Science and Innovation through Grant No. PID2019-108075RB-C33/AEI/10.13039/501100011033. M.H. acknowledges the DFG for support through Grant No. HU 752/16-1. Support through the Frankfurt Center of Electron Microscopy (FCEM) is gratefully acknowledged. Further, support of the European Cooperation in Science and Technology via COST Action No. CA16218 (NANOCOHYBRI) is acknowledged.