Magnon spectrum of Bloch hopfion beyond ferromagnetic resonance

The resonant spectrum of any object can be seen, from the point of view of its energy states around equilibrium, as its characteristic signature. This holds for different types of systems and forms a ground for various spectroscopic techniques. In the context of the magnetic nanostructures,¹ the spin-wave modes are collective small-amplitude oscillations of the magnetization around its equilibrium orientation. They characterize the magnetic systems, reflect the existing symmetries, and provide information about basic magnetic properties.^2–5 

The spin-wave resonance spectrum in the homogeneous and uniformly magnetized ferromagnetic systems is quite well-known, and it is described by the Kittel formula.⁶ In thin films, the ferromagnetic resonance (FMR) spectrum is dependent on the magnetization orientation and may consist of a fundamental mode and perpendicular standing spin waves. In ferromagnetic rods, the spectrum strongly depends on the magnetization orientation. In homogeneously axially magnetized circular nanorods, the azimuthal and radial modes were identified.^7–9 Interesting spin-wave spectrum exists in planar ferromagnetic nanosystems. It depends on the element shape (e.g., dots, rings, and nanovolcanoes)^2,10–14 and magnetization configurations.^15,16 Nanostructures with topologically protected magnetization textures, like vortexes and skyrmions, are particularly interesting.^17–24 Here, the spectrum consists of the gyrotropic modes, usually at low-frequencies, but at thicker dots, higher-order gyrotropic modes at GHz frequencies can exist as well.²⁵ There is a family of the azimuthal and radial spin-wave excitations, which reflect a circular symmetry of the system, also, the breathing mode in the skyrmion and the curled modes in the vortex state²⁵ were identified. Interestingly, in various ferromagnetic nanostructures, the Dzialoshynski–Moriya interactions (DMI), but also dipolar or topological phase, can lift the degeneracy between clockwise and counter-clockwise azimuthal modes.^8,9,21,22,26

When the system becomes thick enough, the magnetization can also form stable, inhomogeneous configurations in the third dimension, providing further features to the spin-wave spectrum.^27,28 Recently, an interest in the magnetic research community focuses on three-dimensional topologically protected magnetic objects, such as hopfions, torons, hedgehogs, or Bloch points.^29–34 However, such objects are difficult for experimental realization mainly because of the requirement for the specific material and geometrical parameters.

The other difficulty is in the detection of the 3D magnetic objects and especially their dynamics. The techniques for the experimental detection of spin-wave modes in thin films and nanostructures are well developed. These are Brillouin light scattering spectroscopy in a microfocusing mode,³⁵ time-resolved magnetooptical Kerr microscopy,³⁶ time-resolved scanning x-ray microscopy,³⁷ Lorentz microscopy,³⁸ or NV-centers based method.³⁹ They are very well suited for imaging the dynamics of surfaces or thin films. However, they are not dedicated for imaging the magnetization objects in the ferromagnet interior and thus are not suitable to measure the spin-wave dynamics of 3D magnetic textures. Synchrotron radiation has only recently been successfully used to visualize the static magnetization configuration in 3D systems.⁴⁰ It also allows for the first demonstration of the hopfion in the thick multilayered Ir/Co/Pt structure.⁴¹ However, these methods are very complex, and they require high-scale research facility, which strongly limits the study of 3D magnetization objects.

The use of FMR measurements for hopfion requires deep knowledge of the spin-wave dynamics in such a system, in dependence on many parameters, details of the magnetization configuration, boundary effects, and knowledge about the selection rules for the spin-wave excitation by a microwave field as well as knowledge of the spin-wave spectra in other, alternative magnetization states.^11,37,42 The idea of using the spin-wave spectra to discriminate between the hopfion and the skyrmion lattice was proposed in Ref. 43. Here, the spin-wave spectra were analyzed in dependence on the external magnetic field oriented perpendicularly to the dot. The external magnetic field was used to transform Néel or Bloch type hopfion to a toron.^44,45 The only small changes in the associated spin-wave spectrum were observed but the study was limited to the in-plane polarized microwave excitation field.⁴⁴ In Ref. 45, the breathing and rotating modes in the hopfion state were calculated. The basic features of the spin-wave spectrum, which distinguish Bloch and Néel types of hopfions in dependence on the magnetic field, were also identified.⁴⁶ However, the full spin-wave spectrum of hopfion has not yet been presented.

Here, we perform an investigation of the spin-wave eigen-oscillations in the Bloch hopfion confined to the circular ferromagnetic nanodot. With the finite element method (FEM) in the frequency domain, we calculate the full spin-wave spectrum up to 20 GHz and analyze the modes. We identify four groups of modes based on the location of their amplitude. Interestingly, the modes from these groups differ not only in localization but also in the sense of rotation occurring on the in-plane cross section as well as on the vertical cross section of the dot. With the micromagnetic simulations in the time domain, we calculate the ferromagnetic resonance spectra for the different polarizations of the microwave magnetic field. We identify spin-wave eigen-oscillations of the hopfion in a wide range of frequencies and estimate their susceptibility to the microwave magnetic field. We have significantly deepened our knowledge of dynamics in 3D magnetic structures, which can help in the identification of the hopfions and ensure further development toward 3D magnonics.

We consider a ferromagnetic dot of 200 nm diameter and 70 nm thickness (see Fig. 1), i.e., the dimensions considered in Ref. 45. In the literature, FeGe is considered as potential materials for hosting hopfions.^45,46 Accordingly, we assume similar materials parameters in our simulations. These are the surface anisotropy constant on the top and bottom surface of the dot K_S = 5 mJ/m²,⁴⁷ the uniaxial anisotropy constant K_u = 0 mJ/m³, DMI constant D = 0.58 mJ/m², exchange constant A_ex = 3.25 pJ/m², and magnetization saturation M_S = 580 kA/m.⁴⁸ In all calculations, we assume γ = 176 rad GHz/T.

In our work, we use the two simulation methods. The first is the finite element method (FEM) with stabilization in the time domain and spin-wave spectra calculation in a frequency domain, both performed in the Comsol Multiphysics environment.⁴⁹ The Landau–Lifshitz–Gilbert equation $m⃗̇=−γμ0m⃗×H⃗+α/Msm⃗×m⃗̇$ is solved for the magnetization vector $m⃗$⁠, together with the Maxwell equation for the magnetic potential ϕ, taking into account dipolar, exchange, surface anisotropy, and bulk DMI contributions to the free energy E. The effective magnetic field $H⃗$ is calculated as $H⃗=μ0−1∂E/∂m⃗−∇⋅HF$⁠, where $HF=μ0−1∂E/∂(∂jmi)$ is a second-rank tensor and $n̂⋅HF=0$ defines boundary conditions for the Landau–Lifshitz–Gilbert equation. The power flux density of the spin-wave eigenmode is then defined as⁵⁰ 

where ω is the circular frequency of the mode and $B⃗$ is the magnetic flux density. The power flow through the surface $S⃗$ is then

The second method is micromagnetic simulations in the finite difference time domain (FDTD) carried out with Mumax3.⁵¹ Here, the system is discretized on a regular mesh 2 × 2 × 2 nm³ (along the x, y, and z axes, respectively). On top of the material parameters listed above, we define the attenuation constant in Landau–Lifshitz–Gilbert equation as α = 0.01. The surface anisotropy was introduced by assigning K_u = 2.5 MJ/m³ only to the upper and lower mesh layers in the (x, y) plane of thickness 2 nm. We do not assign any static external magnetic field in the system. For resonance spectrum calculations, we use a dynamic magnetic field as an excitation, which is defined as

where h₀ is the amplitude of the dynamic field and f_cutoff is the cutoff frequency. In the simulations, we apply this field along the x or z axis according to our needs. To calculate the resonance spectrum D(f) in a given case, we use the formula

where F is the fast Fourier transform (FFT) calculated on the magnetic response for a selected Cartesian component of the magnetization of the system averaged over the space.

We start simulations with the hopfion ansatz defined in Ref. 30 and relax the magnetic system. The relaxed hopfion is shown in Fig. 1 in the two mid-plane cross sections.

The hopfion radius R_h is defined as a distance from the dot center, the point (0, 0, 0), to the position along the radial direction on the mid-plane, where normalized to 1, m_z component of the magnetization reaches the value −1, while m_x, m_y = 0 is R_h = 37 nm. We also introduce the reference lines indicating the positions, where the $(ẑ×r̂)⋅m⃗=420$ kA/m, marked in Fig. 1 with black lines, which will be used for interpretation of the spin-wave modes in the further part of the paper.

It is worth mentioning that the static configuration as well as FMR frequency spectrum obtained in Comsol Multiphysics are slightly different than those obtained in Mumax3. The difference in the hopfion radius between both methods is 3 nm. The difference is attributed to the two auxiliary magnetic layers with strong perpendicular magnetic anisotropy above and below nanodisc used in Mumax3 to induce surface anisotropy at nanodisc surfaces. The magnetic moments of those two auxiliary layers couple dipolarly with the nanodisc, shifting the stabilization point and frequency spectrum. In Comsol Multiphysics, we use surface anisotropy defined by boundary conditions, thus avoiding the presence of additional magnetic layers. On the other hand, we checked that introducing auxiliary magnetic layers in Comsol Multiphysics allows us to obtain agreement between those two numerical methods.

The full spin-wave spectrum of nanodot with hopfion, calculated with FEM, is presented in Fig. 2(a). There are a large number of eigen-oscillations, which cover a broad frequency range, starting from a few MHz to 20 GHz. To have a deeper insight into the types of oscillations, we have ordered them according to the position of the center of the mode amplitude along the radial direction, r_M,

where the integration is over the surface S: y = 0, for x > 0. In Fig. 2(a), we can distinguish the four groups of modes in dependence on the range of their center amplitude positions and their frequencies. These are

• (A)—the amplitude is centered at a distance larger than 75 nm from the dot center. These modes are concentrated primarily at the edge of the nanodot and cover the whole frequency range.

• (B)—the center of amplitude is at the distance from 50 to 75 nm. These modes are concentrated at the outer edge of the hopfion and cover the whole frequency range.

• (C)—the center of amplitude is between 30 and 40 nm from the disk center, and the frequency of the mode is below 10 GHz. These modes are concentrated at the inner edge of the hopfion.

• (D)—these modes are the hybrids of the elemental modes A, B, and C. Most of them are located around r_M = 50 nm and have a frequency mainly above 10 GHz, but this limit is very rough.

The provided limits for the groups are not sharp, and there are also modes between branches B and C as well as parts of branches A that drop in the region of branches B. This preliminary characterization of the spin-wave spectra of hopfion in nanodot can be extended by the evaluation of the mode profiles.

A brief analysis of the mode profiles from groups A, B, and C implies that most of the modes rotate along isolines defined in Fig. 1, and they are quantized along those lines. Thus, we introduced the measure of the modes’ quantization in the dotted plane (x, y) in the azimuthal direction, ϕ as

and the quantization along the dot thickness, i.e., in the plane (y, z) as

The values of ξ_XY and ξ_YZ are calculated along isolines K defined by $(ẑ×r̂)⋅m⃗=420$ kA/m and marked in Figs. 1, 3, 4, and 6. While Eq. (6) defines the quantization number ξ_XY strictly (assuming sinusoidal-like mode shape), ξ_YZ defined by Eq. (7) may be only considered as proportional to the quantization number in the (y, z) plane.

With these measures, we can come back to the deeper interpretation of the groups of modes in the full spectrum in Fig. 2(a). The color of each point indicates the ξ_YZ value, and the number at each point indicates the ξ_XY value. Additionally, in Fig. 2(b), we re-plot the spectrum providing the color of each point and the information about the power flow P, Eq. (2), across the half (y < 0) nanodot cross section at the surface x = 0.

The group of modes A (Fig. 3, upper part) are azimuthal modes localized at the outer area of the dot, covering the outer edge of hopfion and the dot edge. Here, we observe increasing frequency r_M with an increase in ξ_XY. These properties remind the whispering gallery modes, well known and used in photonics,⁵² and also recently demonstrated for spin waves in thin dots in the vortex state⁵³ as well as used in optomagnonic coupling.⁵⁴ The A modes are also inhomogeneous across the nanodot thickness [see, the bottom line in Fig. 3(a)]: the higher ξ_YZ, the higher frequency of the mode, at the same azimuthal number. However, the number ξ_YZ is only approximate, as the mode quantization across the thickness is not unambiguously defined (see, e.g., the mode A4). Interestingly, the family of modes with approximately the same ξ_YZ group form quasi-continuous bands with increasing ξ_XY, marked with the dashed-black lines. Surprisingly, all modes from the A group have positive signs of P, which means clockwise rotation of spin waves in the dotted plane.

The B modes (Fig. 3, lower part) concentrate the amplitude in the hopfion outer edge (e.g., B1). Similar to the A modes, the modes with the same ξ_YZ form the bands [see dashed-lines connecting the modes in Fig. 2(a)]. Interestingly, the modes from the B group have an opposite sense of rotation, as all have negative signs of P.

Modes C (Fig. 4, upper part) are azimuthal modes concentrated at the inner edge of the hopfion. They are also characterized by increasing numbers ξ_XY and ξ_YZ, but they do not form such clear bands as modes A and B in Fig. 2. The power flow P of modes C is positive, i.e., the sense of rotation is clockwise.

The modes D (Fig. 4, lower part) form a strongly inhomogeneous group of modes, with the amplitude concentrated in various parts of the hopfion and dot. Those are modes with a rather high quantization number ξ_YZ. Their rotation is also mixed; nevertheless, we can observe higher values of P in the group of D.

In the broad-band FMR measurements, the microwave magnetic field carrying an excitation signal is usually linearly polarized. With the sample placed on the microstrip or coplanar waveguide (CPW), the oscillating magnetic field is in the sample plane and perpendicular to the signal line,^12,55,56 while for the ring resonators^53,55 or when the sample is placed in the gap between the signal and ground line of CPW,⁵⁷ the field is polarized perpendicular to the dot plane. To calculate the spin-wave spectra of the nanodot with hopfion in these two configurations, we perform micromagnetic simulations in the time domain using a sinc excitation with f_cutoff = 20 GHz, see, Eq. (3). The results are shown in Fig. 5 with the excitation field in the dot plane (a) and along the out-of-plane direction (b).⁵⁸ The spectra excited by the B_x are much richer than the one obtained using B_z, but at high frequencies, the only excitation at 15 GHz is visible for the B_z field. The magnetization components orthogonal to the polarization of the microwave field are almost insensitive to the excitation, apart from some modes at very low frequency, i.e., below 1 GHz. The resonances observed in FMR spectra are composed of a number of eigen-oscillations (also resolved by FEM calculations in Fig. 2, see the next paragraph), resulting in the broadening of the peaks.

We also marked the most intensive modes obtained from FEM in Figs. 2(a) and 2(b) with bold squares and crosses for the B_z and B_x excitation field, respectively. The modes with intensity larger than 5% of the most intensive ones are marked. It is seen that only the modes with ξ_XY = 0 are sensitive to the out-of-plane excitation, while with the in-plane excitation, only the modes with ξ_XY = 1 are visible. Also, none of the modes from groups A, B, and C are FMR-sensitive.

For the in-plane microwave field [Fig. 5(a)], the resonances of highest intensity are at 1.96, 5.88, 7.26 (the most intensive), and 9.87 GHz. We can identify them with the help of the amplitude distribution shown in Figs. 6(a)–6(d) as a mode confined to the dot center, two modes with the amplitude in the center and the contribution from the nanodisc edge, and the mode confined to the hopfion internal edge, respectively.

The microwave field oriented perpendicularly to the disk plane excites the two modes with high efficiency at 8.50 and 14.71 GHz, and the two modes at 5.61 and 0.68 GHz with smaller intensity [Figs. 6(e)–6(h)]. We do not observe the phase oscillations of m_z in the azimuthal direction; thus, we can identify them as breathing modes of the hopfion.

The mode profiles of the most intensive FMR lines that we obtained here are very similar to those presented in Ref. 45 (note the different format of the presented results here), despite differences in M_S and surface anisotropy. Interestingly, the resonances at 5.61 (f) and 8.50 GHz (g) are very similar but with different phase relationships between the dot edge and the inner part of the hopfion, in-phase and out-of-phase oscillations, respectively. This indicates that these oscillations are similar to excitations in other magnetic systems, like pair of magnetic vortices or coupled two domain walls.^59,60

With FEM and FDTD micromagnetic simulations, we have investigated spin-wave dynamics in hopfion confined to the ferromagnetic dot. The full spectrum obtained from the solving eigenproblem for spin-wave frequencies shows a large number of eigen-oscillations. They cover the whole considered frequency range up to 20 GHz. We distinguished four groups of modes according to their average amplitude localization along the radial direction and frequency. These are the modes concentrated in the outer region of the disk (A), at the external edge of the hopfion (B), at the inner edge of the hopfion (C), and high frequency modes without particular localization (D). Furthermore, the modes from a given group A, B, or C form bands with modes of similar quantization in the vertical cross section of the disk. The modes from the given band increase the frequency with increasing azimuthal number and shift localization to the outer direction. Interestingly, by analyzing energy flux in the disk cross section, we identified that the modes from the A and C groups have opposite circulation to the modes from group B. Modes from the D group possess rather high quantization in the vertical plane of the disk.

Hopfion also possesses interesting FMR spectra, clearly different depending on the polarization of the used microwave magnetic field, in-plane or out-of-plane. According to simulations, the broad peaks visible in the resonance spectra at high frequencies consist of many modes from the D group.

The full spectra of spin waves provide deep knowledge of the richness and complexity of the spin-wave dynamics in the 3D topological magnetic solitons. It indicates the inherent time-reversal symmetry breaking with the separation of the modes with an opposite sense of rotation. The findings point to the possibility of selected mode detection. Moreover, by analogy to the magnetic vortexes, skyrmions, whispering gallery modes, and nonlinear effects, our study indicates interesting physics and potential applications of dynamics in hopfion but now with 3D magnetic objects.

The research leading to these results received funding from Norwegian Financial Mechanism 2014–2021 under Project No. UMO-2020/37/K/ST3/02450 and from the National Science Center in Poland Sonata No. 2019/35/D/ST3/03729 and Sonatina No. 2018/28/C/ST3/00052. Simulations were partially performed at the Poznan Supercomputing and Networking Center (Grant No. 398).

The authors have no conflicts to disclose.

K. Sobucki: Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal). M. Krawczyk: Conceptualization (equal); Funding acquisition (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal). O. Tartakivska: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (equal). P. Graczyk: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Methodology (lead); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

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