Introduction

The Larmor precession is a universal dynamic phenomenon in nature that represents the precession of a magnetic moment about a magnetic field at a characteristic Larmor frequency, which is expressed as ωL = γH, where γ is the gyromagnetic ratio and H, the static field strength. This type of precession plays very crucial roles in a rich variety of electron- or nuclei-spin-related dynamics such as electron-spin resonance, nuclear magnetic resonance, ferromagnetic resonance and related magnetization dynamics1,2,3,4,5,6. Such dynamic fundamentals have been widely utilized in a significant number of applications, including material analysis1,4, bio-medical imaging7,8 and information recording in magnetic media9,10.

In this paper, we report the discovery of resonantly excited precession motions of a magnetic vortex core in soft magnetic nanoparticles of spherical shape11, but with totally different underlying physics from those for vortex motions so far reported12,13,14,15,16,17. We also were able to identify sphere-size-controllable precession angular frequency ωMV and size-specific resonant excitations of nanoparticles bearing a magnetic vortex structure. We additionally determined, based on combined micromagnetic numerical and analytic calculations, that the size specificity of ωMV originates from the variable effective gyromagnetic ratio with the sphere size that modifies the vortex structure inside spheres. Our results could provide a potential means of implementing size-specific resonant excitation of nanoparticles in bio-applications18.

Results

Ground states of nanospheres

Figure 1a shows a nanosphere model of spherical symmetry. As described in Methods, we performed micromagnetic numerical calculations on Permalloy (Py, Ni80Fe20) nanoparticles of different diameters, 2R = 10 nm – 150 nm (see Methods). Figure 1b illustrates the ground states of the spheres obtained through relaxation from their saturated states in the +x direction. For the 2R < 40 nm cases, uniformly magnetized single-domain states were obtained, whereas for the 50 nm ≤ 2R ≤ 150 nm cases, single magnetic vortex states were well established. The vortex state of the 2R = 150 nm sphere, for example, was visualized by streamlines circulating around the vortex core oriented in the +x direction. We noted that the region of the vortex core aligned in the +x direction relative to the region of the in-plane circulating magnetizations varies markedly with 2R, as indicated by the x-component of the local magnetization mx (=Mx/Ms) profiles in Fig. 1c. This dramatic variation is the result of a strong competition between the long-range dipolar and short-range exchange interactions in those nanospheres of such varying size.

Figure 1
figure 1

Ground-state magnetization configurations of Py nanospheres according to the diameter.

(a) Finite-element sphere model for diameter 2R = 30 nm. (b) Ground-state magnetization configurations of Py nanospheres for different 2R values as indicated: upper, viewed from positive z-direction and sliced across x-y plane; lower, viewed from positive x-direction and sliced across y-z plane. The color represents the x-component of the local magnetizations, mx = Mx/Ms (see the color bar). The arrows inside the sphere of 2R = 150 nm represent the local curling magnetizations. (c) mx profiles along y axis for different diameters. The inset shows the mx profiles versus the normalized distance for each diameter.

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Resonantly excited precession motion of a vortex core in spheres

Since the spherical symmetry of nanospheres does not lead to any magnetic shape anisotropy, when a sizable static field HDC is applied in the +z direction, the vortex cores for 40 nm < 2R ≤ 150 nm start to reorient to the field direction, but with accompanying precession motions (see Supplementary Movie). This precession motion is different from the well-known gyration and even its higher-order modes of vortex cores in planar dots12,13,14,15,16,17. Although very weak spin waves are emitted inside the nanospheres, the vortex’s spin configurations are maintained as a whole structure, because the field strength is sufficiently small. In the relaxation process, the core orientation converges in the field direction (+z-direction), reflecting the fact that the mx averaged over the entire volume of the sphere, <mx>, undergoes decaying oscillation through its vortex-core precession (inset of Fig. 2a). The precession frequency was obtained by Fast Fourier Transformation (FFT) of the temporal <mx> evolution for the different values of 2R and HDC (see Fig. 2a,b, respectively). In the cases of uniformly saturated particles (2R = 10, 20, or 30 nm), the frequency was independent of 2R, as determined by the Larmor frequency fL = (γ/2π)HDC19. By contrast, for the vortex-state spheres (40 nm ≤ 2R ≤ 120 nm), the precession frequency of a vortex core showed a strong variation with 2R, as can be expressed by fMV = (γeff/2π)HDC, where γeff(<γ) is the effective gyromagnetic ratio, which is variable with the sphere diameter.

Figure 2
figure 2

Precession frequency of Py nanospheres as functions of (a) 2R and (b) HDC applied in +z direction (perpendicularly to initial vortex-core orientation).

The inset in (a) shows the <mx> oscillation versus time, for a sphere of 2R = 80 nm. In (a), uniform single-domain (SD) and vortex states are distinguished at about 2R = 37 nm by the gray color. The symbols indicate the micromagnetic numerical calculations, with corresponding lines drawn by eye. In (b), the lines are the results of linear fits for the individual diameters, as indicated.

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In order to quantitatively elucidate the γeff-versus-2R relation, we plotted the value of f/HDC as a function of 2R, as compared with the average magnetization component over the sphere volume in the vortex-core orientation,〈mΓ〉, both of which were obtained from the micromagnetic simulations. As shown in Fig. 3, when γ/2π = 2.8 (MHz/Oe) on the left axis is scaled to 〈mΓ〉 = 1 on the right axis, both numerical values are in excellent agreement over the entire range of diameters studied, resulting in an explicit form of γeff/γ = 〈mΓ〉 (for single-domain states, γeff = γ, because of 〈mΓ〉 = 1). Therefore, the precession frequency of a vortex core in nanospheres can be expressed as fMV = (γ/2π)〈mΓ〉HDC. This precession frequency cannot be explained by the gyration mode (or even by higher-order modes) of vortex cores in thin or thick film dots and neither, consequently, by Thiele’s equation12,13,14,15,16,17.

Figure 3
figure 3

Precession frequency normalized by HDC (circles) and 〈mΓ〉 (crosses) obtained from micromagnetic numerical calculations.

The value of γ/2π = 2.8 (MHz/Oe) on the left axis is scaled to 〈mΓ〉 = 1 on the right axis. The different colors of the circle symbols indicate the numerical data for different sphere diameters, as indicated by the colors shown in Fig. 2a. The solid curve is the result of a numerical calculation of the analytical form of 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)2.20±0.14.

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Analytical derivation of vortex-core precession in nanospheres

In order to gain physical insight into the fMV = (γ/2π)〈mΓ〉HDC relation obtained from the micromagnetic simulations, we analytically derived vortex-core precession dynamics in nanospheres. In our modeling, a weak static field was applied in the +z direction, which field sustained the rigid vortex structure in a certain potential and thus allowed the initial ground-state vortex core to align in the +z direction through the precession around the field direction along with certain damping. We used the local spherical reference frame on infinitesimal segments of the surface, where the unit vector of local magnetizations is expressed as m = (mr,mθ,mϕ), r is the radial distance, θ is the polar angle and ϕ is the azimuthal angle, as shown in Fig. 4a. Time-variable vortex-core orientation can be defined as a unit vector , as illustrated in Fig. 4b. Following the rigid vortex Ansatz, which agreed with the micromagnetic simulation results, local magnetizations inside a given sphere could be expressed as and , where Φ is the azimuthal angle of the magnetization in the local spherical reference frame (inset of Fig. 4a). Here we assume some general shapes of mr that are restricted by the condition f(r,1) = −f(r,−1) = 1 for all r values. Since mr and Φ are canonically conjugated variables, the time evolution of the local magnetizations can be determined from the Landau-Lifshitz-Gilbert (LLG) equations20,21 as

Figure 4
figure 4

Model for analytical derivations.

(a) Definition of spherical coordinates and local spherical reference frame (colored surface) for local magnetization m. (b) Schematic of model sphere wherein single rigid vortex core is pointed in direction of θ0 and ϕ0, as defined by the polar and azimuthal angle coordinates.

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By inserting the mr distribution function of the vortex’s spin configuration into Eqs. (1a) and (1b), we finally obtained the governing equation for vortex-core precession motion,

where E is the total magnetic energy, F is a dissipative functional and is the sphere volume. The first, second and third terms in Eq. (2) correspond to the gyrotropic, potential energy and damping terms, respectively. The total energy E under a weak magnetic field applied along the z-axis, , can be expressed simply as , where 〈mΓ〉 is rewritten as . Eq. (2) expresses the precession motion of vortex cores in collective spin dynamics; it differs from Thiele’s equation to describe the gyration of vortex cores in planar dot systems.

By inserting EH into Eq. (2) and assuming negligible damping, the precession frequency of a rigid vortex core can be given as ∂ϕ0/∂t = 2πfMV with fMV = (γ/2π)〈mΓ〉HDC. Consequently, we obtained the effective gyromagnetic ratio of the motion of a vortex in a given nanosphere as γeff = γ〈mΓ〉. This analytic form provides a clear physical insight into 2R-dependent fMV, because 〈mΓ〉, as indicated in the micromagnetic simulation results, varies with 2R. Here we note that the eigenfrequency of a single vortex in cylindrical dots is known to vary with the aspect ratio of thickness L to R12,15,16,17. However, the underlying physics of the size-dependent change in the precession frequency of the vortex core in nanospheres is totally different from that of the vortex gyration in planar disks, though both apparently show core-oscillation phenomena.

Dependence of 〈mΓ〉 on sphere’s diameter and constituent material parameters

Next, it is necessary to quantify how 〈mΓ〉 varies with 2R. We estimated, from further micromagnetic numerical calculations, the quantitative relation between 〈mΓ〉 and 2R within the 2R = 50–200 nm range for the different material parameters of both Ms and Aex. Figure 5 reveals that 〈mΓ〉 is given as with η = 73.6 ± 3.4. According to the relation 1, 〈mΓ〉 can be simplified as 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)2.20±0.14. This explicit form provides a simple and reliable estimation of 〈mΓ〉 for a given value of 2R and a given material of lex, though there is yet no concrete model matching the form. We also note that, based on the single-domain states of 〈mΓ〉 = 1, the critical size for transition from a single domain to a vortex state1,22 can be simply estimated as 2Rc = 7.06 lex. For example, the critical diameter, 2Rc = 37.3 nm for Py, was in good agreement with that obtained from the simulation results shown in Fig. 1. As quantitatively interpreted, the strong variation of 〈mΓ〉 versus 2R for a given material is related to the competition between the short-range, strong exchange interaction and long-range, but relatively weak dipolar interaction in nanospheres of given dimensions.

Figure 5
figure 5

Calculation of 〈mΓ〉 for indicated values of 2R, Aex/Aex,Py and Ms/Ms,Py.

For each graph, one parameter is fixed: (a) Aex/Aex,Py = 1, (b) Ms/Ms,Py = 1, (c) and (d) 2R = 100 nm. In both (a) and (b), 2R is in the 50–200 nm range. All of the symbols were obtained from the micromagnetic simulation results. The lines indicate linear fits.

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Size-specific resonant excitations

As an application of the aforementioned fundamental dynamics, we could activate magnetic nanoparticles of a specific size by tuning the frequency of an applied AC field to the fMV of a sphere of a given diameter and material. In this modeling, an external AC field and a static field were given by and , respectively, with sufficiently small values of HAC = 10 Oe and HDC = 100 Oe to avoid deformation of the initial vortex structures in the Py spheres. Figure 6a shows the oscillation of the core orientation θ0 from the +z direction during the precession process for 2R = 60 nm (fMV = 95 MHz), as excited by fAC = 91, 95 and 99 MHz. The oscillation of θ0 was hardly observable for the cases where fAC was far from fMV, whereas it was very large for the case of fAC = fMV, that is, at resonance. The resonantly excited precession leads even to vortex-core reversals between θ0 = +π and 0, as such reversals in planar disks occur periodically by linearly oscillating fields or currents applied on the disks’ plane under the resonance condition3,23. The oscillation of θ0 represents a transfer of the external magnetic field to a magnetic sphere via the absorption of the Zeeman energy and subsequent emission to another form. The maximum energy absorption can be defined by the first maximum energy increment, ΔE1, as noted in Fig. 6a. Figure 6b plots ΔE1 versus fAC for different sphere diameters24. For each diameter, the maximum peak height in the ΔE1-versus-fAC curves was obtained under the corresponding resonance condition. All of the curves were well separated from each other, indicating reliable size-specific excitation of the magnetic particles. For example, the difference in fMV between the 50 and 60 nm particles was about 50 MHz, which is sufficiently large compared with the full width at half maximums of both particles, 6.6 and 9.9 MHz, respectively.

Figure 6
figure 6

Total magnetic energy variation.

(a) Total magnetic energy and polar angle of core orientation θ0 during excitations of vortex core in sphere of 2R = 60 nm by oscillating fields of HAC = 10 Oe with different field frequencies (fAC = 91, 95 and 99 MHz) under static field of HDC = 100 Oe applied on + z-axis. (b) Plot of ΔE1 versus fAC in fAC = 25–310 MHz range. (c) Maximum absorption energy ΔEmax versus 2R, calculated from micromagnetic simulations (solid circles) and analytical form (lines) of ΔEmax described in text.

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In Fig. 6c are shown the ΔEmax-versus-2R curves for comparison between the simulation data (solid circles) and the analytical form (lines) of the Zeeman energy, , where 〈mΓ〉 = 1 for single-domain states or 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)2.20±0.14 for vortex states. The simulation and analytical calculation agreed very well, as can be seen. The analytical calculation clearly shows that the magnetic energy absorption varies with (2R)3 and (2R)0.8 for the single-domain and vortex states, respectively. These results suggest that the magnetic energy absorption can be maximized by tuning fAC to the resonance frequency of a given-diameter particle. This effect is made possible through size-specific resonance, size-selective activation and corresponding detection of the magnetic nanoparticles of a vortex state.

Discussion

We discovered, by micromagnetic numerical calculations, not only the resonantly excited precession motion of a vortex core in nanospheres and its size-dependent precession frequency, but also its physical origin, based on the size effect on the effective gyromagnetic ratio in collective spin dynamics analytically derived. This finding paves the way for size-selective activation and/or possible detection of magnetic nanoparticles by application of extremely low-strength AC fields tuned to the resonant frequency of a given diameter and material. These results, notably, would be applicable to magnetic particle resonance imaging (MPRI) and bio-applications.

Methods

In our micromagnetic numerical calculations, the FEMME code (version 5.0.8)25 was used to numerically calculate the motions of the magnetizations of individual nodes (mesh size: ≤4 nm) interacting with each other via exchange and dipolar interactions at the zero temperature, as based on the LLG equation20,21. The surfaces of the model spheres were discretized into triangles of roughly equal area using Hierarchical Triangular Mesh (HTM), as shown in Fig. 1a, in order to prevent irregularity-incurred numerical errors26. The chosen material parameters corresponding to Py were as follows: saturation magnetization Ms = 860 emu/cm3, exchange stiffness Aex = 1.3 × 10−6 erg/cm, damping constant α = 0.01, γ/2π = 2.8 MHz/Oe and zero magnetocrystalline anisotropy for the soft ferromagnetic Py material.

Additional Information

How to cite this article: Kim, S.-K. et al. Resonantly excited precession motion of three-dimensional vortex core in magnetic nanospheres. Sci. Rep. 5, 11370; doi: 10.1038/srep11370 (2015).