Artificially fabricated three-dimensional magnetic nanostructures have recently emerged as a new type of magnetic material with the potential of displaying physical properties absent in thin-film geometries. Interconnected nanowire arrays yielding three-dimensional versions of artificial spin-ices are of particular interest within this material category. Despite growing interest in the topic, several properties of these systems are still unexplored. Here, we study, through micromagnetic simulations, the high-frequency dynamic modes developing in buckyball-type magnetic nanoarchitectures. We obtain a characteristic excitation spectrum and analyze the corresponding mode profiles and their magnetic field dependence. The magnetic resonances are localized at different geometric constituents of the structure and depend on the local magnetic configuration. These features foreshow the potential of such systems for reprogrammable magnonic device applications with geometrically tunable frequencies.

## I. INTRODUCTION

In recent years, the properties of three-dimensional (3D) magnetic nanostructures have emerged as a promising research topic in magnetism.^{1} After decades in which scientists have thoroughly studied the shape and size dependence of patterned magnetic thin-film elements,^{2,3} modern sample fabrication and analysis techniques have paved the way to a different category of magnetic nanostructures in the form of 3D sample geometries.^{4,5} The access to the third dimension is expected to result in the appearance of magnetic properties that do not unfold in planar structures.^{6} Ultimately, the discovery of specific effects developing only in 3D structures could lead to nanoscale functional materials with unique features.

One of the first geometries considered in this context was the magnetic buckyball—a spherical network of magnetic nanowires connecting nearest-neighbor vertices located at positions that correspond to the atomic structure of a C_{60} fullerene molecule.^{7–9} The buckyball’s iconic shape (“the most beautiful molecule”^{10}) makes this geometry a natural candidate to showcase features of the new class of magnetic materials provided by 3D nanoarchitectures. This geometry combines two- and three-dimensional aspects,^{11} making this type of nanostructure particularly suitable to study how the properties of a two-dimensional magnetic system change with a transition toward a 3D arrangement. On the one hand, the network of magnetic nanowires forming the buckyball is organized on a two-dimensional spherical surface. On the other hand, the nanowires’ varying orientation in 3D space introduces significant qualitative changes compared to a traditional arrangement on a planar surface typical for, e.g., two-dimensional artificial spin-ice (ASI)^{12,13} systems.

Like a two-dimensional ASI, the magnetic structure of a nanoscale buckyball architecture can display apparent spin-ice features if the magnetic nanowires of the network are sufficiently thin and elongated and if the material is magnetically soft. In that case, shape anisotropy enforces a magnetic alignment along the wires, and each nanowire represents an Ising-type macroscopic magnetic moment. Under these conditions, the magnetic buckyball can attain a quasi-continuum of nearly degenerate and frustrated magnetic states—a typical property of ASIs. These states can be characterized by the distribution of the different types of magnetic configurations developing at the intersection points (the vertices). The vertex configurations may include magnetic defect structures with Dirac-type monopole^{14,15} properties. A typical magnetic structure is illustrated in Fig. 1.

In a previous article, we investigated the static and hysteretic properties of magnetic buckyballs as a function of their size and discussed the magnetic structures forming at the vertices.^{11} Here, we study their intrinsic high-frequency magnetization dynamics, i.e., the magnetic modes. Such an analysis of the high-frequency oscillation spectrum is essential to characterize the magnetization state fully and plays a significant role in two- and three-dimensional ASI lattices.^{16–20} It is worth noting that high-frequency excitations have also been the subject of intense research in the case of real C_{60} molecules, whose optical absorption spectrum and related vibrational modes have been studied in detail.^{21,22}

## II. SIMULATION METHOD

We use micromagnetic finite-element simulations to study the high-frequency magnetic properties of buckyball-type nanostructures and how these magnetic excitations depend on the magnetic configuration. The simulations are carried out in a two-step process, each involving different simulation methods. First, we simulate the zero-field static equilibrium state. For this part of the simulation, we use our micromagnetic software tetmag,^{23} which is based on the numerical integration of the Landau–Lifshitz–Gilbert (LLG) equation^{24}

where *α* is the Gilbert damping constant, *γ* is the gyromagnetic ratio, $Ms=M$ is the material’s spontaneous magnetization, and *H*_{eff} is the micromagnetic effective field.^{25} Despite implementing several advanced features, such as graphics processing unit (GPU) acceleration and $H2$-type hierarchical matrix compression allowing us to treat large-scale problem with almost linear $O(N)$ complexity, the main operation principle of our tetmag software corresponds to that of traditional micromagnetic solvers. In this first part of the simulation, we aim at calculating equilibrium magnetization configurations *M*_{0}(** x**) under static external field

*H*_{0}, and we, therefore, use a high damping (

*α*= 0.5) to achieve fast convergence.

In a second step, we simulate the steady-state high-frequency magnetization dynamics that an externally applied sinusoidal field *δ*** H** superimposed to the static field

*H*_{0}(such that |

*δ*

**| ≪ |**

*H*

*H*_{0}|),

excites in the system. Exposed to such a field, the steady-state magnetization exhibits a forced small-angle precessional motion around the equilibrium configuration *M*_{0}(** x**) such that it can be decomposed into a static and a dynamic time-harmonic component,

Both spatial distribution and amplitude of the dynamic magnetization $\delta M\u0302(x)$ depend on the frequency *ω* of the ac field. By assuming $\delta M\u0302\u226aM0$, the LLG equation [Eq. (1)] can be appropriately linearized, which allows us to solve for the dynamic magnetization component $\delta M\u0302(x)$.

The linearized LLG equation can be written in the frequency domain as^{26,27}

where $L=L[M0,\omega ,\alpha ]$ is a linear operator accounting for the micromagnetic interactions in the system and $P$ is a projection operator. When discretized on a computational grid, the operator $L$ becomes a dense matrix with $O(N2)$ (*N* being the number of discretization points) dimension that makes the analysis of moderately large micromagnetic systems unfeasible. Nevertheless, by using an appropriate formalism developed by d’Aquino *et al.*,^{27,28} we have developed a frequency domain matrix-free micromagnetic linear response solver (MF-*μ*LRS), which allows us to simulate the high-frequency magnetization dynamics for large magnetic systems. The solution of Eq. (4) in combination with the ansatz (3) yields the steady-state dynamic magnetization *δ*** M**(

**,**

*x**t*) at a given field

*δ*

**with frequency**

*H**ω*. By solving Eq. (4) for different operators $L=L(\omega )$, we can numerically “sweep” the frequency within a user-defined range to obtain the frequency-dependent dynamic response of the magnetic system. In practice, we vary the frequency in steps of 50 MHz in a range between 0.1 and 20 GHz and solve Eq. (4) for each frequency value.

Due to space limitations, and in order to maintain the focus of this article on the physical properties of the studied system, we cannot explain here more details of the MF-*μ*LRS. This newly developed method, which we have thoroughly tested,^{29} is tailored to treat large-scale micromagnetic problems with millions of finite-elements. Its numerical aspects will be discussed in more detail in a subsequent article.

## III. MODEL SYSTEM

Our model system of a magnetic buckyball consists of cylindrical nanowires with length *L* = 100 nm and diameter *D* = 12 nm. To ensure smooth surfaces at the vertices, the wire intersections are modeled with spheres of diameter *S* slightly larger than *D*, in our case, *S* = 14 nm. The sample is discretized into about 101 000 irregularly shaped tetrahedral finite-elements with a maximum edge length of 5 nm. We use material parameters corresponding to those of Co obtained from focused-electron-beam-induced deposition (FEBID),^{30,31} i.e., a saturation magnetization of *μ*_{0}*M*_{s} = 1.2 T, zero magnetocrystalline anisotropy, and a ferromagnetic exchange constant *A* = 1.5 × 10^{−}^{11} J m^{−1}.^{32} The Gilbert damping is set to *α* = 0.01. We have studied the high-frequency dynamics of the magnetization in several buckyball geometries of varying parameters *L*, *D*, and *S*. However, according to our simulations, the high-frequency modes do not depend sensitively on the sample size. Their qualitative properties such as the mode profile, field dependence, and their order of frequency remain the same. It, therefore, suffices to illustrate the high-frequency modes on a single model system, where we chose *L* = 100 nm.

## IV. STATIC MAGNETIZATION STRUCTURE

The ensemble of the vertex configurations determines the magnetization state of the buckyball. Two qualitatively different types of vertex configurations can be distinguished, depending on their degree of local frustration. The statistically most frequent ones are the single-charge configurations, which appear in two variants: the “two-in/one-out” and “one-in/two-out” structures. These configurations obey the spin-ice-rule and display a minimum in local energy density and magnetostatic charge. The two variants differ by the sign of the magnetic charge density (the divergence of the magnetization) at the vertices. The other type of structure, the triple-charge configuration, occurs when the magnetization in each of the three branches points toward the vertex (three-in) or away from it (three-out). Due to time-inversion symmetry, the single-charge magnetic states with different charge signs are equivalent and display identical micromagnetic properties. The same holds for the two variants of the triple-charge structures.

## V. ZERO-FIELD HIGH-FREQUENCY MAGNETIZATION DYNAMICS

Let us first describe the high-frequency modes of a randomly demagnetized buckyball structure at zero external field, with essentially zero remanence and which does not contain any triple-charge defect-vertex structures. This demagnetized state, labeled “single-charge state,” can occur in several nearly degenerate variants, as the positive and negatively charged vertex configurations can be scattered differently on the buckyball. Any realization of the demagnetized state contains an equal number of 30 “two-in/one-out” and 30 “one-in/two-out”-type vertices to preserve the system’s global charge neutrality. Here, we consider one version of the demagnetized state as a representative example, having confirmed that different variants of the state display negligible differences in their properties. The zero-field excitation spectrum is represented as a dark green line in Fig. 2.

Applying our previously described MF-*μ*LRS algorithm, we compute the power spectrum as the spatially averaged squared amplitude of the magnetization oscillation *δ**M*_{i} at each discretization point *i*,

where the sum extends over all discretization points, *V* is the total volume, and *V*_{i} is the volume attributed to the point *i* (*∑*_{i}*V*_{i} = *V*). Throughout the article, we use the magnitude |*δ*** M**| to calculate the power spectrum and identify the dynamically active regions of the magnetization.

^{33}We use a spatially homogeneous oscillating external field of amplitude $\delta H\u0302=$ 0.5 mT to generate high-frequency excitations of the system.

^{34}

One can clearly identify five modes in the spectrum of the defect-free buckyball. The two sharp peaks at 9.05 and 10.5 GHz are vertex modes, which, as our analysis shows, are due to the magnetic high-frequency oscillations localized at the vertices. We attribute the occurrence of two different vertex oscillation frequencies—a feature that we have seen in all variants of buckyball configurations—to variations among the vertices regarding the local effective field and the orientation of the magnetization with respect to the driving field direction.

The three broader peaks at 14, 17, and 20 GHz belong to nanowire modes, which can be interpreted as different orders of standing waves within the nanowires; see Fig. 3. The mode at 14 GHz can approximately be ascribed to a standing-wave mode in which predominantly the central part of the nanowires oscillates. The other nanowire modes, at 17 and 20 GHz, are more complicated and of higher order, containing one or two nodal planes within the wire. Although the mode profiles within the wires are not always clearly different at these three frequencies, one can, in general, identify these resonances as the appearance of first, second, and third-order modes within the wires. Contrary to the vertex modes, which exhibit a largely uniform intensity at different positions within the buckyball we find considerable variations in the intensity of the oscillations in different nanowires at a given frequency. This effect is particularly pronunced in the case of the mode at 20 GHz. These variations can be attributed to changes in the orientation of the nanowires with respect to the externally applied perturbation, an effect much less pronounced in the case of the vertices due to their more isotropic shape. Such differences among the nanowires occur for any choice of the direction of the (homogeneous) driving RF field because the quasi-continuous distribution of nanowire orientations in the buckyball structure invariantly leads to a broad distribution of angles enclosed between the applied field and the nanowire axis.

We can now compare the high-frequency modes found in the demagnetized, defect-free configuration to those developing in a high-remanence state at zero static field. The latter state is formed by magnetically saturating the sample in an external field and gradually reducing the field to zero. The resulting magnetic structure, despite the magnetization in the wires being aligned along their axis, retains a sizable magnetization component along the direction in which it was saturated and develops two triple-charge type defect states on diametrically opposite positions along the axis of the applied field. The structure, which we term the triple-charge state, is to some extent comparable to the well-known “onion state” developing in thin disks or rings after in-plane saturation.^{35} The defect structures in our buckyball correspond to the head-to-head and tail-to-tail configurations occurring at opposite sides of the disks in the onion state.

We excite magnetic oscillations with our MF-*μ*LRS algorithm by assuming an oscillating magnetic field oriented perpendicular to the direction of remenance. The impact of the defect states on the magnetic excitation spectrum is illustrated by the red line in Fig. 2. While all the modes discussed so far remain essentially the same, an additional low-frequency peak appears at 1.4 GHz. The magnetic activity of this mode is localized at the two diametrically opposite triple-charge defect-type vertices in the remanent state. Remarkably, the triple-charged vertex configurations giving rise to this oscillation can be generated and destroyed in a controlled way by means of external fields,^{11} which opens up a pathway to manipulate the magnonic properties of the sample. Conversely, the appearance of a low-frequency peak in the oscillation spectrum can be regarded as a “fingerprint” allowing for the indirect detection of these defect-type structures.

## VI. FIELD DEPENDENCE OF THE MAGNETIC MODES

Having established the frequencies and mode profiles of the principal zero-field resonances of the structure, both with and without defect configurations, we proceeded to investigate how the high-frequency spectrum is modified by an externally applied magnetic field. For this purpose, we performed exhaustive simulations. First, we simulated the equilibrium magnetic structure at different applied field strengths, quasistatically varied in step sizes Δ*H* = 25 mT from an initial field of 0 to ±250 mT for both the high-remanence state with the defect structures and the demagnetized, defect-free state. At each field value, using our MF-*μ*LRS software, we then determined the dynamic magnetization in a frequency range between 100 MHz and 25 GHz. The results are summarized in Fig. 4. Figure 4(a) refers to the high-remanence state that we previously compared to the “onion state” known from disks and rings. The external field is applied along the direction of remanence, parallel to the axis connecting the two diametrically opposite vertices with defect-type magnetization structures. Over a relatively large range, we observe a gradual increase in the frequency with positive fields and a decrease with negative fields. With increasing magnitude of the applied field, the resonance lines broaden slightly and the spectrum develops more complex features. Nevertheless, one can, in general, identify an almost linear shift of frequencies over a range of ±100 mT. At about −180 mT, the field-induced changes cease to be reversible as several nanowires switch in the field direction. This significant change in the magnetic structure of the buckyball entirely modifies the excitation spectrum.

For positive fields, the low-frequency defect mode at about 2 GHz depends weakly on the external field compared to the other modes. At negative fields, the triple-charge vertices dissolve by injecting a head-to-head domain wall in adjacent branches, whose oscillation leads to a weak low-frequency signal with a near-linear field-dependent frequency.

The field dependence of the modes in the case of the demagnetized and defect-free state, shown in Fig. 4(b), is more complex than in the case of the high-remanence state. As the field is increased, the modes—well-defined at zero-field—split up into a multitude of intersecting branches, leading to a complex broad-band absorption spectrum. This splitting occurs in a nearly symmetric fashion for positive and negative fields, which is consistent with the (on average) nearly isotropic magnetic structure of the demagnetized state. The external field lifts the degeneracy of various vertex and nanowire modes, leading to a pronounced Zeeman splitting of these resonances. In this case too, irreversible switching processes occur near 200 mT, which result in a sudden modification of the absorption spectrum.

## VII. HOLLOW BUCKYBALL STRUCTURE

So far, we have considered a magnetic buckyball structure composed of solid magnetic nanowires. Such a situation is consistent with fabrication techniques, such as FEBID, where the magnetic material is directly deposited on the desired position in space, conceptually similar to the operation of a three-dimensional printer. Other types of sample preparation techniques^{8,18,36–40} consist in depositing a magnetic layer on a non-magnetic three-dimensional shape. These techniques differ as they can lead to either a partial or a full coating of the non-magnetic sample with a magnetic layer. In the case of a full magnetic coating, covering the entire surface of a non-magnetic object, the result is a magnetically hollow structure.

Given that our study does not aim at reproducing a specific experimental situation, we simulated an idealized geometry to address such situations in a general manner, neglecting the potentially significant differences between samples with partial and full coating. Specifically, we conducted simulations on a hollow version of a buckyball structure (Fig. 5) and analyzed to which extent it behaves differently from a solid one. The hollow buckyball geometry has the same size and shape as the solid version discussed before. The magnetic layer thickness is 4 nm.

Regarding the static magnetic structure and the configurations at the vertices, the hollow version shows properties largely identical to those found in the solid geometry. The high-frequency modes, however, display some changes. We simulated the absorption spectrum of the hollow buckyball at zero static field, both in the high-remanence “triple-charge” and in a demagnetized “single-charge” state. As shown in Fig. 6, the spectra put into evidence a few differences with respect to the solid case. The defect-type mode occurs here too, as the lowest-frequency excitation. Its frequency, however, is shifted toward a considerably higher value compared to the solid case (about 2.7 vs 1.4 GHz). Moreover, upon close inspection, one can notice that this low-frequency mode splits into a doublet in the hollow case, as discussed in Fig. 7. The hollow version also features a weakly pronounced “satellite” mode of the defect-vertex oscillation at 6.2 GHz. Despite being localized at the defect-type vertices, this mode does not appear to be a simple higher-order oscillation of the low-frequency modes at 2.6 and 2.8 GHz. As can be seen in Fig. 7(c), within the spherical vertex junction, this higher-frequency mode is most pronounced on the outer surface of the buckyball, whereas the oscillation in the low-frequency modes, shown in Figs. 7(a) and 7(b), is primarily active in the inner part of the buckyball.

Apart from quantitative differences in the frequencies, the profile of the main modes and their sequence in the spectrum are quite similar for the hollow and the solid version. The oscillations of the single-charge vertices are found in the hollow buckyball too, albeit at clearly *lower* frequencies. The vertex mode profiles of the hollow sample and their sequence in the power spectrum are largely similar to those of the solid one. A clear difference between the solid and the hollow versions is the appearance of two major peaks related to the wire modes in the hollow case instead of three in the solid one. The mode profile corresponding to the major peak at 15.40 GHz is similar to that shown in Fig. 3(b), with a high-frequency magnetic activity distributed over large regions within the sample. Its high intensity could be interpreted as the result of a change of frequency in the hollow structure leading to a superposition of the two peaks appearing at 14 and 17 GHz in the solid buckyball. At frequencies higher than about 17 GHz, the background signal is due to the hollow nanowires (nanotubes), which remain active, albeit at significantly lower amplitudes. At 23.4 GHz, one can identify a weakly pronounced additional peak, which, however, results in such a small increase in intensity that it hardly constitutes a mode that would be of practical significance.

Our study of the driven magnetic oscillations in the hollow structure indicates that, despite many striking similarities in the static and quasistatic properties, quantitative and qualitative differences can be expected in the high-frequency properties of magnetic nanoarchitectures consisting of interconnected nanowires, depending on whether their fabrication method results in magnetically solid or hollow samples.

## VIII. CONCLUSION

The advent of three-dimensional magnetic nanopatterning techniques has opened access to a new and largely unexplored class of artificially structured materials, in which arrays of interconnected nanowires play a significant role. With its iconic buckyball shape, the nanoarchitecture discussed in this study serves as a model system to study the interplay of geometry, magnetic structure, and magnetization dynamics in complex three-dimensional magnetic systems of this type. We analyzed the high-frequency magnetic modes unfolding in this system by using a novel large-scale frequency domain method (MF-*μ*LRS) to simulate the small-angle magnetic oscillations in a linearized model. The characteristic resonances found in the spectrum show a clear dependence on the geometric constituents of the sample, i.e., the nanowires and the vertices at which they meet, as well as on the magnetic configuration. This dependence highlights the potential of such artificially structured materials for reprogrammable magnonic applications whose properties can be modified through variations in geometry and magnetic structure.

## ACKNOWLEDGMENTS

The authors acknowledge the Interdisciplinary Thematic Institute QMat (ANR-17-EURE-0024), as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS ans INSERM, supported by the IdEx Unistra (ANR-10-IDEX-0002) and SFRI STRAT’US (ANR-20-SFRI-0012) through the French Programme d’Investissement d’Avenir. The authors acknowledge the High Performance Computing Center of the University of Strasbourg for supporting this work by providing access to computing resources.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Rajgowrav Cheenikundil**: Formal analysis (lead); Investigation (equal); Visualization (lead). **Julien Bauer**: Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal). **Mehrdad Goharyan**: Investigation (supporting); Visualization (supporting). **Massimiliano d’Aquino**: Methodology (lead); Software (lead); Writing – review & editing (equal). **Riccardo Hertel**: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{60}

^{2}-matrices

Alternatively, one could use the oscillations of individual dynamic magnetization components *δ**M*_{j} with *j* = *x*, *y*, *z*, but the results would change only insignificantly. Moreover, given the three-dimensional and nearly isotropic sample shape, the magnitude of the magnetic fluctuations is a more suitable choice to detect frequencies and locations of increased magnetization oscillations than their decomposition into Cartesian components.

The amplitude of the external RF field is in our case always oriented along the *y* direction. However, in view of the nearly isotropic shape of the buckyball, the system behaves practically in the same way irrespective of the direction chosen for the oscillating field’s amplitude.