Size-selective resonant excitation of soft magnetic nano-spheres of threedimensional

Transcription

1 Size-selective resonant excitation of soft magnetic nano-spheres of threedimensional magnetic vortex Jehyun Lee,, Myoung-Woo Yoo, Dong-Soo Han, and Sang-Koog Kim * National Creative Research Initiative Center for Spin Dynamics and Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul National University, Seoul , South Korea Combined micromagnetic numerical and analytical calculations revealed strong resonant excitations of soft magnetic nano-spheres of a three-dimensional (3D) magnetic vortex, which excitations remarkably vary with the sphere diameter 2R. This function originates from the diameter-dependent frequency of precession of the 3D vortex core about the direction of an applied static field H DC. The characteristic angular frequency was found to be expressed as 3D mr ˆ H DC, where ˆ m r is the average magnetization component of the ground-state vortex structure in the core direction, and is the gyromagnetic ratio. Numerical calculations also revealed that m r ˆ is a strong function of 2R for a given material, as 2.2 mr ˆ ( ) lex 2R with the exchange length lex. Our results provide not only a new physical insight into 3D vortex dynamics but also their potential applications to sizeselective resonant excitation and/or possible detection of magnetic nano-particles

2 The Larmor precession is a universal dynamic phenomenon in nature, representing the precession of a magnetic moment about a magnetic field at a characteristic Larmor frequency. This type of precession plays very crucial roles in a rich variety of electron- or nuclei-spinrelated dynamics: there are novel dynamic phenomena such as electron-spin resonance, nuclear magnetic resonance, ferromagnetic resonance, and the related spin-wave dynamics [1 4]. These have been widely utilized in plenty of applications, including material analysis [1, 2], bio-medical imaging [5], and information recording in magnetic media [6]. For example, magnetic resonance imaging utilizes the Larmor frequency of an H + ion in order to activate local H + ions in a specific position using space-graded static and alternating fields [7]. The Larmor frequency is expressed very well as /2 ( /2 )H DC with the gyromagnetic ratio γ/2π = 2.8 MHz/Oe and the static field strength HDC [8]. In cases of magnetic nano-particles that are small enough to have uniform magnetizations as in a single domain, the Larmor frequency is independent of the particle size, but dependent only on HDC. However, magnetic particles that are larger than single-domain size but smaller than multiple-domain size can bear a three-dimensional (3D) magnetic vortex [9] and the unique 3D vortex structure can lead to novel spin-dynamic behaviors and relevant functionalities that are promising for practical applications, as will be addressed below. In this Letter, we report on our finding of novel 3D vortex dynamics in soft magnetic nano-spheres: the precession frequency /2 3D of the vortex core differs from the Larmor frequency of uniformly magnetized magnets. We also elucidate, using combined analytic and micromagnetic numerical calculations, the underlying physics of the size specificity of the 3D of nano-spheres. Our results pave the way towards practical applications of the size

3 selective resonant excitation and possible detection of magnetic nano-particles to biomedical technologies such as in-vitro diagnosis [10]. In the present study, we performed micromagnetic numerical calculations on Permalloy (Py, Ni80Fe20) nano-particles of spherical symmetry and of 2R diameter ranging from 10 nm to 150 nm. The surfaces of the model spheres were discretized into triangles of roughly equal area using Hierarchical Triangular Mesh (HTM), as shown in Fig.1(a), in order to prevent irregularity-incurred numerical errors [11]. The FEMME code (version 5.0.8) [12] was used to numerically calculate the magnetization motions of individual nodes (mesh size: 4 nm) interacting via exchange and dipolar coupling at the zero temperature, as based on the Landau- Lifshitz-Gilbert equation [13]. The chosen material parameters corresponding to Py were as follows: saturation magnetization Ms = A/m, exchange stiffness Aex = J/m, damping constant α = 0.01, gyromagnetic ratio γ/2π = 2.8 MHz/Oe [14], and zero magnetocrystalline anisotropy for a soft ferromagnet. The ground states of all of the spheres were obtained through relaxation from their saturated magnetization states in the +x direction. The results of the micromagnetic simulations are shown in Fig. 1(b). 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 note 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 the diameter, as shown in Fig. 1(b) and as indicated by the mx profiles in Fig. 1(c). This dramatic variation of the static 3D vortex structure with the diameter is the result of strong competition - 3 -

4 between the long-range dipolar and short-range exchange interactions in those nano-spheres of different sizes. We will address its quantitative interpretation below. Since the spherical symmetry of those model spheres does not lead to any shape anisotropy to the initial ground states of the given-diameter spheres, when a sizable static magnetic field HDC is applied in the + z direction, the 3D vortex cores (for 40 nm < 2R 150 nm) start to reorient to the field direction, without any resistance, but accompany precession of the vortex core about the field direction in the similar form of the single-spin Larmor precession. In this precession process, spin waves are very weakly generated inside the particle; however, as confirmed in our simulations, the 3D vortex configurations are maintained as a whole structure in a given sphere, because the field strength is sufficiently small. In the relaxation process, the core orientation converges in the field direction (herein, the +z-direction), reflecting the fact that the x-component of the local magnetizations <mx>, averaged over the entire volume of the sphere, undergoes decaying oscillation through its vortex-core precession, as shown in the inset of Fig. 2(a). The precession frequency was obtained by Fast Fourier Transformation (FFT) of the temporal <mx> evolution versus time for the different values of 2R and HDC (see Fig. 2(a) and 2(b), respectively). In cases of uniformly saturated particles (2R =10, 20, or 30 nm), the frequency was independent of the diameter, and was determined by the Larmor frequency L DC f 2 H. By contrast, for the 3D vortex (40 nm 2R 120 nm), the precession frequency showed a strong variation with 2R, but remained proportional to HDC. The precession frequency of the 3D vortex core in nano-spheres can be expressed as f 3D 3D DC 2 H with the constraint 3D, where the effective gyromagnetic ratio - 4 -

5 of a 3D vortex core 3D markedly varies with 2R and, thus, is controllable by it. In order to quantitatively elucidate the 3D -versus-2r relation, we used Thiele s equation [15] to describe the 3D vortex dynamics, which equation has been used to represent the eigenfrequency of a single vortex in two-dimensional (2D) soft magnetic nano-disks [16 18]. In our modeling a weak static magnetic field was applied in the +z direction, which field sustained the 3D vortex structure in a certain potential, and thus allowed the core orientation of the initial ground-state vortex structure to align in the +z direction. According to the collective-variable approach with the linearized Thiele s equation, the governing equation for the core motion can be given as G X/ t D X/ t W/ X 0, where X (, ) represents the orientation of a core, as defined by the position of the outward core magnetization on the surface of a sphere, and and are defined as the polar and azimuthal angles, respectively, as shown in Fig. 3(a). The first term in the equation is the gyroforce arising due to the presence of a 3D vortex core in the sphere. The gyroforce is proportional to the gyrovector, G Grˆ, where the gyrovector constant G can be numerically calculated from the ground state, using Guv guvd [15, 17]. The D X / t term represents the damping force, which, here, can be neglected, because it rarely influences the precession frequency of the 3D vortex in soft magnetic nano-particles. The total potential energy W is given as W W0 WH, where the Zeeman energy W ( X) H M H is a function of the deviation of X from the direction of HDC (here 0), and W 0 is the potential energy of the ground state at 0, considering the magnetostatic and exchange energies. For rigid vortex structures like those in our micromagnetic simulations, we assume W / 0 0 X and thus

6 W ( X ) M m H cos with the permeability of free space 0. Note that H 0 s rˆ DC m rˆ is defined as the average magnetization component in the unit direction of vortex-core rˆ, over the entire volume. Based on the spherical coordinate system, the governing equation of the 3D vortex motion for the zero-damping case becomes G ˆ ˆ 0Ms mrˆ HDC sin θ φ sin θˆ 0. Without damping, the vortex core precesses on the equipotential line of W on the sphere surface; accordingly, only varies with time under the = 0 condition. The solution of the equation indicates the precession frequency of the vortex core around the direction of HDC: f /2 ( /2 ) H, with M mr ˆ G. It should be noted that G is 3D 3D DC 3D 0 s / proportional to 0 M s [15, 17]; accordingly we obtain 3D mr ˆ with a proportional constant. The analytic form of f mr ( /2 ) H provides clear 3D ˆ DC physical insight into the observed behavior of the sphere-diameter dependence of f 3D : the variation of m r ˆ with 2R results in this novel dynamic behavior, because the other parameters are constant values. We note also that the relation f ~1/ 3D G agrees with the case of gyrations of a vortex core in a cylindrical nano-disk [16]. In order to confirm this analytical form and to explore how m r ˆ varies with 2R, we conducted numerical calculations of not only m r ˆ but also G and D [15, 17] for the ground states of spheres of different 2R values, which results are shown in Figs. 3(b), 3(c), and 3(d), respectively. Interestingly, the m r ˆ value is linearly proportional to (2R) -2.2, and the G and - 6 -

7 D values to (2R) 3, with G >> D. From the numerical values of m r ˆ and G as a function of 2R, is determined to be unity. The simulation results of f3d / H DC versus 2R were in perfect quantitative agreement with the numerical calculation of the analytical form of 3D mr, as shown in Fig. 4, where the value of 2 = 2.8 (MHz/Oe) is scaled to ˆ m = 1. As confirmed by the combined analytical and micromagnetic numerical ˆr calculations, the explicit relation, 3D mr is acceptable over the entire range of ˆ diameters studied here, as well as for the single-domain (SD) state, SD because of. m r ˆ 1. Therefore, the precession frequency of a magnetic nano-sphere can be expressed as f m H 3D r ˆ ( /2 ) DC for a 3D vortex and fsd fl 2 HDC for a single-domain state. Next, the quantitative relations between m rˆ and 2R in the 2R = 50 ~ 200 nm range were estimated from further micromagnetic numerical calculations for different material parameters of both Ms and Aex, for generality. The virtual values of Ms and Aex relative to those of Py varied within the 0.6 ~ 1.4 range. A summary of these calculations is provided in Figs. 5(a-d), which show that the m rˆ values were found to be proportional to R -2.2, Aex 1.1 and Ms -2.2 for nano-spheres with a 3D vortex structure. From all of those results, we derived the following relation: mr [2 A /( M )] (2 R) with a constant value of η = ˆ ex 0 s 2 ± 3.4. According to the relation of lex 2 Aex / 0Ms [1], m r ˆ can be simply rewritten as 2.2 m ( ) l / 2R r ˆ ex. The form of 3D rˆ DC r ˆ f ( 2 ) m H m f L

8 indicates that the 3D vortex structure, despite its complicated internal spin configuration, can be treated as a rigid giant spin [19]. Additionally, we note that a critical size for the transition from a single domain to a vortex state [1, 9, 20] for a given material can be estimated from 2Rc = 7.06 lex using the criterion of mˆ 1. For example, for Py, we have 2Rc = 37.3 nm, which r value is in good agreement with that obtained from the simulations, as shown in Fig. 1. As quantitatively interpreted, the strong variation of m rˆ versus 2R for a given material is related to the competition between the long-range dipolar and short-range exchange interactions in nano-particles of different sizes. On the basis of the fundamental physics we found, as noted above, that one can implement soft magnetic nano-particles of a single 3D vortex in size-selective resonant excitations by tuning the frequency of an applied AC field to the f3d value of a given diameter and material. Here, we present an example of the size-selective (or specific) activation of magnetic nano-particles. To that end, an external AC field H H sin 2 f t AC AC AC ˆ y is applied perpendicularly to the direction of the static field, H H DC DCˆ z, with HAC = 10 Oe and HDC = 100 Oe, which are sufficiently weak to avoid deformation of the initial vortex structures in the Py spheres. The fac used in this study is within the 25 ~ 310 MHz range. The 3D vortex excitation is strongly dependent on fac relative to f3d for a given-diameter sphere. Figure 6(a) shows the oscillation of Θ from the +z direction through the core precession for 2R = 60 nm (f3d = 95 MHz for this diameter), as excited by fac = 91, 95 and 99 MHz. As observed, the oscillation of Θ is hardly observed for the case where fac is far away from f3d, whereas it is very large for the case of fac = f3d. Such a large excitation leads even to core reversals between Θ = +π and 0, as vortex-core reversals in 2D disks occur periodically by linearly oscillating - 8 -

9 fields or currents applied on the disks plane under the resonance condition [4, 16, 21]. The oscillation of Θ 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 (i.e. heat [22] or microwave [23]). The maximum energy absorption can be defined by the first maximum energy increment, E1, as noted in Fig. 6(a). Figure 6(b) plots E1 versus f AC for different diameters [24]. For each diameter, the maximum peak height Emax in the E1-versus- f AC plot is obtained at the resonance condition, i.e., fac f3d. All of the curves are well distinguished from each other, indicating reliable size-selective excitation of magnetic particles by tuning fac to the f 3D of a given-diameter particle. For example, the difference in f3d between 50 and 60 nm particles is about 50 MHz, which is sufficiently large compared with the full width at half maximum values of both particles, 6.6 and 9.9 MHz, respectively. Figure 6(c) plots the Emax value versus 2R for the simulation data (solid circles) and the analytical form (lines) of 3 E M mr H with 4/3 R and max 2 0 s ˆ DC m ˆ 1 r for single-domain states or m 2.2 ˆ ( ) l / 2R r ex for 3D vortex states. The analytical relation of Emax with 2R clearly shows that the magnetic energy absorption varies with 3 2R and 0.8 2R for the single-domain and 3D vortex states, respectively. Both the simulation and analytical calculations agree very well. 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 makes possible size-selective excitations of magnetic nanoparticles of a 3D vortex through the size-specific resonant responses of different-size particles - 9 -

10 to f AC. To conclude, we discovered, by analytic and micromagnetic numerical calculations, not only the size-dependent precession frequency of magnetic spherical-shape nano-particles with a 3D vortex, but also its physical origin. The characteristic frequency was found to be f m H 3D r with 2.2 ( 2 ) ˆ DC m ( ) l / 2R rˆ ex for a given material. This finding paves the way to size-selective excitation and/or possible detection of magnetic nanoparticles by application of AC fields that are tuned to the resonant frequency of a given- diameter particle and of extremely low (i.e., a few Oe) field strength. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (grant no )

11 REFERENCES These authors contributed equally to this work. *To whom all correspondence should be addressed: Present address: Advanced Materials Research Center, Samsung Advanced Institute of Technology, Yongin-si, Republic of Korea [1] R. C. O'Handley, Modern Magnetic Materials - Principles and Applications (John Wiley & Sons, 2000). [2] J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, 2010). [3] B. Hillebrands, and A. Thiaville, Spin Dynamics in Confined Magnetic Structures III (Springer, Berlin, 2006); S. Choi, K.-S. Lee, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. Lett. 98, (2007); M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A. ansteenkiste, B. an Waeyenberge, H. Stoll, G. Woltersdorf, C. H. Back, and G. Schuetz, Nat. Commun. 2, 279 (2011); M.-W. Yoo, J. Lee, and S.-K. Kim, Appl. Phys. Lett. 100, (2012). [4] K.-S. Lee, K. Y. Guslienko, J.-Y. Lee, and S.-K. Kim, Phys. Rev. B 76, (2007). [5] A. Trabesinger, Nature 435, 1173 (2005). [6] T. Devolder, and C. Chappert, The European Physical Journal B - Condensed Matter and Complex Systems 36, 57 (2003); S.-K. Kim, K.-S. Lee, Y.-S. Yu, and Y.-S. Choi, Appl. Phys. Lett. 92, (2008)

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15 FIGURE 1. FIG. 1 (color online). (a) Finite-element sphere model for diameter 2R = 30 nm. (b) Groundstate magnetization configurations of Py nano-spheres for different diameters 2R as indicated: upper, viewed from the positive z-direction and sliced across the x-y plane, lower, viewed from the positive x-direction and sliced across the y-z plane. The color represents the x-component of the local magnetizations (see the color bar). The arrows inside the sphere of 2R = 150 nm represent the local magnetization orientations around the vortex core. (c) mx profiles along the y axis for different diameters. The inset shows the mx profiles versus the normalized distance for each diameter

16 FIGURE 2. FIG. 2 (color online). Precession frequency of Py nano-spheres as functions of (a) diameter 2R and (b) strength of static magnetic field HDC applied in +z direction (perpendicularly to the 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 and 3D vortex states are distinguished at about 2R = 37 nm by the gray color. The symbols indicate the numerical calculations, with corresponding lines drawn by eye. In (b), the lines are the results of linear fits for the individual diameters, as indicated

17 FIGURE 3. FIG. 3 (color online). (a) Schematic of model sphere wherein single rigid vortex core is pointed to X= (Θ, Φ), as defined by the polar and azimuthal angle coordinates. (b) Magnetizations projected in direction of ground-state vortex core, averaged over entire sphere volume, m ˆr, and plotted as function of (2R) (c) Gyrovector constant G and (d) damping constant D versus (2R) 3, as obtained from numerical calculations of spheres of 3D vortex ground states for 2R = 50 ~ 150 nm

18 FIGURE 4. FIG. 4 (color online). Precession frequency normalized by H DC, corresponding to the effective gyromagnetic ratio of Py spheres, as obtained from analytic derivation (lines) and micromagnetic numerical calculation (circles). The line thickness (yellow) of the analytical calculation denotes the error range of the results obtained using Thiele s equation. The different colors of the symbols indicate the numerical data for different particle diameters, as indicated by the colors shown in Fig. 2(a). mˆr in the ground states of the individual particles are displayed together for comparison, where m ˆr = 1 is scaled to the value of 2 = 2.8 (MHz/Oe)

19 FIGURE 5. FIG. 5 (color online). Calculation of m ˆr for indicated diameters 2R, exchange stiffnesses Aex, and saturation magnetizations Ms, where Aex,Py = J/m and Ms, Py = A/m. For each graph, one parameter is fixed: (a) Aex/Aex,Py = 1, (b) Ms/ Ms,Py = 1, (c, d) 2R = 100 nm. In (a) and (b), 2R was varied between 25 nm and 100 nm. All of the symbols were obtained from the micromagnetic simulation results, whereas the lines display the linear fittings

20 FIGURE 6. FIG. 6 (color online). (a) Total magnetic energy of sphere of 2R = 60 nm and polar angle of core orientation Θ during excitation of 3D vortex by oscillating fields of HAC = 10 Oe for fac = 91, 95 and 99 MHz, under static field of HDC = 100 Oe applied in the +z-axis. (b) Plot of E1 versus fac in a range of fac = 25 ~ 310 MHz. (c) Maximum absorption energy ΔEmax versus 2R, calculated from micromagnetic simulations (solid circles) and analytical form (lines) of ΔEmax described in text