Resonantly exited precession motion of three-dimensional vortex core in magnetic nanospheres

We found resonantly excited precession motions of a three-dimensional vortex core in soft magnetic nanospheres and controllable precession frequency with the sphere diameter 2R, as studied by micromagnetic numerical and analytical calculations. The precession angular frequency for an applied static field HDC is given as ωMV = γeffHDC, where γeff = γ〈mΓ〉 is the effective gyromagnetic ratio in collective vortex dynamics, with the gyromagnetic ratio γ and the average magnetization component 〈mΓ〉 of the ground-state vortex in the core direction. Fitting to the micromagnetic simulation data for 〈mΓ〉 yields a simple explicit form of 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)2.20±0.14, where lex is the exchange length of a given material. This dynamic behavior might serve as a foundation for potential bio-applications of size-specific resonant excitation of magnetic vortex-state nanoparticles, for example, magnetic particle resonance imaging.

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 ω L = 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 dynamics [1][2][3][4][5][6] . Such dynamic fundamentals have been widely utilized in a significant number of applications, including material analysis 1,4 , bio-medical imaging 7,8 , and information recording in magnetic media 9,10 .
In this paper, we report the discovery of resonantly excited precession motions of a magnetic vortex core in soft magnetic nanoparticles of spherical shape 11 , but with totally different underlying physics from those for vortex motions so far reported [12][13][14][15][16][17] . We also were able to identify sphere-size-controllable precession angular frequency ω MV 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 ω MV 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-applications 18 .

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, Ni 80 Fe 20 ) 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 m x (= M x /M s ) 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.
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 H DC 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 dots [12][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 m x averaged over the entire volume of the sphere, < m x > , 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 < m x > evolution for the different values of 2R and H DC (see Fig. 2a,b, respectively). In the cases of uniformly saturated particles (2R = 10, 20, or 30 nm), the . 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 f MV = (γ eff /2π)H DC , where γ eff (< γ) is the effective gyromagnetic ratio, which is variable with the sphere diameter.
In order to quantitatively elucidate the γ eff -versus-2R relation, we plotted the value of f/H DC as a function of 2R, as compared with the average magnetization component over the sphere volume in the vortex-core orientation,〈 m Γ 〉 , 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 〈 m Γ 〉 = 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 γ eff /γ = 〈 m Γ 〉 (for single-domain states, γ eff = γ, because of 〈 m Γ 〉 = 1). Therefore, the precession frequency of a vortex core in nanospheres can be expressed as 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 equation 12-17 . Analytical derivation of vortex-core precession in nanospheres. In order to gain physical insight into the f MV = (γ/2π)〈 m Γ 〉 H DC 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 = (m r ,m θ ,m φ ), 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  By inserting the m r distribution function of the vortex's spin configuration into Eqs. (1a) and (1b), we finally obtained the governing equation for vortex-core precession motion, 3 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, =Ĥ H z DC , can be expressed simply as (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 E H into Eq. (2) and assuming negligible damping, the precession frequency of a rigid vortex core can be given as ∂φ 0 /∂t = 2πf MV with f MV = (γ/2π)〈 m Γ 〉 H DC . Consequently, we obtained the effective gyromagnetic ratio of the motion of a vortex in a given nanosphere as γ eff = γ〈 m Γ 〉 . This analytic form provides a clear physical insight into 2R-dependent f MV , because 〈 m Γ 〉 , 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 R 12,15-17 . 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 〈m Γ 〉 on sphere's diameter and constituent material parameters. Next, it is necessary to quantify how 〈 m Γ 〉 varies with 2R. We estimated, from further micromagnetic numerical calculations, the quantitative relation between 〈 m Γ 〉 and 2R within the 2R = 50-200 nm range for the different material parameters of both M s and A ex . Figure 5 reveals that 〈 m Γ 〉 is given as This explicit form provides a simple and reliable estimation of 〈 m Γ 〉 for a given value of 2R and a given material of l ex , though there is yet no concrete model matching the form. We also note that, based on the single-domain states of 〈 m Γ 〉 = 1, the critical size for transition from a single domain to a vortex state 1,22 can be simply estimated as 2R c = 7.06 l ex . For example, the critical diameter, 2R c = 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 〈 m Γ 〉 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.

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 f MV of a sphere of a given diameter and material. In this modeling, an external AC field and a static field were given by  Figure 6a shows the oscillation of the core orientation θ 0 from the + z direction during the precession process for 2R = 60 nm (f MV = 95 MHz), as excited by f AC = 91, 95 and 99 MHz. The oscillation of θ 0 was hardly observable for the cases where f AC was far from f MV , whereas it was very large for the case of f AC = f MV , that is, at resonance. The resonantly excited precession leads even to vortex-core reversals between θ 0 = + 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 condition 3,23 . The oscillation of θ 0 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, Δ E 1 , as noted in Fig. 6a. Figure 6b plots Δ E 1 versus f AC for different sphere diameters 24 . For each diameter, the maximum peak height ∆E max in the Δ E 1 -versus-f AC 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 f MV 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.
In Fig. 6c are shown the Δ E max -versus-2R curves for comparison between the simulation data (solid circles) and the analytical form (lines) of the Zeeman energy, for single-domain states or 〈 m Γ 〉 ≈ (73.6 ± 3.4)(l ex /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 f AC 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 equation 20,21 . 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 errors 26 . The chosen material parameters corresponding to Py were as follows: saturation magnetization M s = 860 emu/cm 3 , exchange stiffness A ex = 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.