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Dynamics of magnetization and easy axis of individual ferromagnetic nanoparticle subject to anisotropy and thermal fluctuations

In order to design magnetic nanoparticles that are optimized for biomedical applications such as hyperthermia treatment and magnetic particle imaging, their magnetization dynamics should be analyzed. Through conventional analytical methods related to magnetization dynamics such as susceptibility and the magnetization curve, the net properties of magnetization have been previously evaluated. However, an examination of net properties does not yield information related to the stochastic dynamics due to thermal fluctuations and systematic dynamics influenced by the anisotropy with respect to single particles. In this study, the time evolution of the magnetization and easy axis of an individual magnetic nanoparticle was numerically simulated using the Landau-Lifshiz-Gilbert equation in terms of magnetic nanoparticles with fixed and rotatable easy axes, solids and liquids, respectively. The superparamagnetic and ferromagnetic regimes of the Langevin dynamics were clearly discerned through evaluation of the effects of anisotropy and thermal fluctuation, and an index for determining the transition point between these regimes provided. In particular, effects related to the core volume and the anisotropy constant associated with the anisotropy energy were assessed using the Stoner-Wohlfarth model.


Introduction
Magnetization dynamics in magnetic nanoparticles (MNPs) were important to develop biomedical application. MNPs are used as a heat source for hyperthermia 1) and as a tracer for magnetic particle imaging (MPI) 2) . In general, with respect to hyperthermia, heat dissipation is derived from the energy losses such as hysteresis and magnetic relaxation losses. The hysteresis loss is due to the presence of the ferromagnetic regime of the MNPs, which in turn is derived from the anisotropy of the magnetization along the easy axis. The magnetic relaxation loss is related to the hysteresis associated with the time delay of the magnetization response from the applied field. This hysteresis loss that is due to anisotropic effects and magnetic relaxation also affects the harmonic response of the magnetization that is crucial for MPI.
In order to evaluate magnetization dynamics, the AC susceptibility 3)-6) and AC magnetization curve 7)-9) have been previously determined. In addition, the specific loss power and the intrinsic loss power 10)-13) for hyperthermia, the magnetic particle spectrometry signal 14) , and the harmonics of the magnetization 15)- 17) for MPI have been assessed. Theoretical and numerical simulations focusing on AC susceptibility and AC magnetization curves have also revealed the dynamics of magnetization 18)-21) and the easy axis 22), 23) with respect to magnetic relaxation.
Magnetic relaxation is divided into Néel and Brownian relaxations, referring to relaxations of the magnetization and the easy axis, respectively. In terms of conventional theory, magnetic relaxation occurs in accordance with the theory of the effective relaxation time τeff, given by 1/τeff= 1/τN + 1/τB, where τN and τB are the Néel and Brownian relaxation times, respectively 18) . This conventional theory indicated the dominance of one type of relaxation, with one relaxation time shorter than the other. However, the superimposition of the Néel and Brownian relaxations has been observed using numerical simulations 22) and magnetization measurements 24) . In our previous study, increases in magnetization derived from the Brownian relaxation in addition to the Néel relaxation were measured when applying a pulse field 25) . Moreover, the dynamics of the easy axis of MNPs in a liquid was estimated using the subtracted magnetization curve indicating the difference between the measured magnetization curve in a solid and that in a liquid 26) .
Through an analysis of susceptibility and the magnetization curve, their dynamics in terms of individual particles remain unclear, although the net properties of the magnetization and easy axis are evaluated. García-Palacios and Lázaro observed the trajectories of individual magnetization using the Langevin-dynamics approach, which indicated the presence of magnetization reversal properties associated with anisotropy and thermal fluctuation 27) . Lopez-Diaz et al. showed that the transition of magnetism depended on particle size, in terms of the switching responses of typical individual magnetizations 28) . In this study, using numerical simulation methods that employ the Landau-Lifshiz-Gilbert (LLG) equation, the dynamics of magnetization and the easy axis were evaluated by focusing on single particle trajectories in conditions with fixed and rotatable easy axes, i.e., a solid and a fluid, respectively. The effects of the uniaxial anisotropy along the easy axis and thermal fluctuation at the transition from the superparamagnetic regime represented by stochastic Langevin dynamics to the ferromagnetic regime given by the Stoner-Wohlfarth model 29) were assessed. This single particle model reveals the effects of anisotropy and thermal fluctuations in the superparamagnetic and ferromagnetic regimes.

Simulated materials
Monodispersed single-domain MNPs were considered in the numerical simulation. The interparticle dipole interaction was ignored, and the hydrodynamic diameters dh of the MNPs was equal to their core diameters dc. The effective uniaxial anisotropy constants Ku were 8, 9.8, 12, 14.7, and 18 kJ/m 3 with dc of 11 nm. In addition, with respect to the case for which Ku was equal to 12 kJ/m 3 , dc of 9.61, 10.3, 11.8, and 12.6 nm were used. These parameters were considered for the evaluation of the magnetization dynamics and depended on the ratio of the anisotropy energy to the thermal energy, ξ= KuVM/kBT, where VM is the volume of a core particle, kB is the Boltzmann constant (1.38 × 10 −23 J/K), and T is the temperature, set to 300 K ( Table  1). The saturation magnetic moment Ms was 269 kA/m, which was taken from the information related to the material measured in Ref. 24).

Simulation methods
The dynamics of magnetization in the case of a rotatable easy axis such as the MNPs in a fluid are calculated using the modified LLG equation 21), 23) . In a fluid, the dynamics of the easy axis is considered in addition to the magnetization dynamics. The orientation of the easy axis is calculated using the unit vector and the angular velocity of the easy axis, n and ω, respectively, as follows: (1) The differential equation of the angular velocity of the easy axis is given by 21) (2) where Θ is the moment of inertia, μ0 is the permeability of free space, γ is the gyromagnetic ratio, m is the magnetic moment normalized by the saturation magnetic moment, Hex is the excitation field, Hth is thermally fluctuating field, is the friction coefficient, Γth is random torque due to thermal fluctuation. γ is estimated to be γ= μ0MsVM(1+α 2 )/(2ατNkBT) where τN= π 1/2 τ0exp(ξ)/2ξ 1/2 30) . is expressed as = 6ηVH where η is the medium viscosity, set to 0.89 mPa·s, and VH is the particle hydrodynamic volume. Hth and Γth have Gaussian distributions with a zero mean. The variance of Hth and Γth satisfied the following equations: In Eqs. (3) and (4), i and j are the Cartesian indices of different particles, and δij is the Kronecker delta function, and δ is the Dirac delta function.
Here, with respect to ω, the inertial effects can be ignored, and μ0MsVM/γ is extremely small. Thus, Eq. (2) is rewritten as the equation of ω, as follows: When the magnetization is entirely bound to the easy axis represented by Eq. (1), the dynamics of the magnetization is given by In addition to the term influenced by the rotation of the easy axis in Eq. (6), the magnetization dynamics is represented by the damping process according to the conventional LLG equation 31) , as follows: where α is damping parameter (α= 0.1), Heff is effective field composed of Hex and anisotropy field (Han). The anisotropy field is estimated to be Han= 2Ku(m·n)n/(μ0Ms). ω/γ indicates the field derived from the rotation of the easy axis.
In the case of the model with a fixed easy axis, the term including ω in Eq. (7) was omitted. For the numerical simulations, the differential equations were solved with the Runge-Kutta method. In the numerical simulations used for the analysis of the net magnetization dynamics, there were 29440 particles. The excitation AC field was applied along the x-axis with the amplitude Hmax of 8 kA/m and the frequency f of 10 kHz except for the frequency dependence analysis.

Time evolution of magnetization and easy axis orientations
The magnetization and easy axis orientations were observed during the half cycle for a phase of 0-π in an applied AC field, Hmaxcos(2πft), where t is the time. The chosen time interval was 100 ns. Because the dynamics of the individual magnetization and easy axis are subject to fluctuations caused by thermal disturbances and therefore show variance, it is necessary to confirm the statistical accuracy of these simulations. With respect to the statistical analysis, the magnetization and easy axis orientations were evaluated in one cycle.
The number of particles N was 100.  Figure 1 shows the time evolutions of the individual magnetization orientation in the three-dimensional model with respect to a MNP of Ku= 12 kJ/m 3 and dc= 11 nm. The easy axis was oriented so that the angle between the easy axis and the applied field was θ= 0, π/3, and π/2 rad in the x-z plane in Figs. 1(a-i)-(a-iii), respectively. On the basis of the Stoner-Wohlfarth model, the magnetization was distributed around the easy axis and along the direction of the applied field because of low potential energy 20), 29) . Figure 1(b) shows the magnetization curve of the MNP with Ku= 12 kJ/m 3 and dc= 11 nm. A slight coercivity indicates ferromagnetism. The distribution of the magnetization along the same direction as the applied field is evidenced in Fig. 1(c). With decreasing θ, the magnetization distribution was strongly aligned along the direction of the applied field, owing to the reduced anisotropy energy barrier 9),32), 33) . Two nodes connected by a line show the continuous trajectory of the magnetization in Fig. 1(a). The lines over the y-z plane perpendicular to the applied field indicate the magnetization reversal, while other lines represent the rotation of the magnetization. Random oscillations in the magnetization including frequent reversals and rotations were due to thermal fluctuations resulting from the superparamagnetism.
3.1.2 Dependence of the anisotropy constant and core diameter Figure 2 shows the time evolutions of the individual magnetization depending on Ku, with dc= 11 nm. The magnetization reversals that are due to thermal fluctuations were suppressed by the large anisotropy energy. Moreover, in contrast to the spreading of the magnetization along the spherical surface in the small-Ku regime, the distribution of the magnetization was concentrated on the direction of the easy axis for large ξ. The magnetization was strongly bound by the anisotropy. The ratio of the magnetization reversal in the trajectories was confirmed in Fig. 3. According to the time evolution shown in Fig. 2, the magnetization reversal that was due to the thermal noise was suppressed by the large anisotropy energy. When dc was increased with Ku= 12 kJ/m 3 , the magnetization reversal ratio decreased because of the increase in the anisotropy energy, which is similar to the case in which Ku increased with dc= 11 nm. However, the magnetization reversal ratios for each ξ were statistically larger than for the case of large dc despite the same ξ values ( Table 1). The p-values were p < 0.01 in ξ ≤ 2.47 and p < 0.1 in ξ= 3.03. The potential energy Ep is given by (8) where φ is the angle between the magnetization and the applied field 29) . The first and second terms in Eq. (8) are the anisotropy and Zeeman energies, respectively.  Zeeman energy is similar with that of |Hex| in the Stoner-Wohlfarth model 34) . Thus, the effectively large |Hex| in addition to the large anisotropy inhibited magnetization reversal due to thermal noise in the case of a large dc for the same ξ. The magnetization reversal along the z-axis was also evaluated in Fig. 3. However, the reversal ratio was independent of the anisotropy energy because the easy axis associated with the uniaxial anisotropy and the applied field were perpendicular to the z-axis, and only random thermal fluctuations affected the magnetization oscillation. Figures 4(a)

Transition in the magnetic regime
As shown in Fig. 2, the change in the anisotropy energy induces the transition from the superparamagnetic regime, the stochastic aspect of the Langevin dynamics, to the ferromagnetic regime, their energetically systematic aspect shown by the Stoner-Wohlfarth model 28) . The Langevin equation describing the system of time evolution for the magnetization is generalized as (9) where the function A(m(t)) represents the systematic term that represents the potential energy and the function F(t) is the term that describes fluctuations that are due to thermal disturbance 27), 35) . Where the fluctuating term dominates, the magnetization dynamics clearly place the system in the superparamagnetic regime. Figure 5 shows the dependence of the energy ratio Ep/kBT on φ by Eq. (8). The large anisotropy in the superparamagnetic regime prevents the most energetically stable magnetization from reversion to the metastable condition by overcoming the anisotropy energy barrier through thermal fluctuations. Here, the � � � �� � �� � �� �,  Table 1. The easy axis was fixed on the x-axis. The statistical analysis was conducted within one cycle of the applied field with a time interval of 100 ns (N= 100).    5). When dc was increased, the magnetic moment was monotonically increased in ξ ≤ 3.03 because energetic stability was enhanced and reduced in the most stable and metastable conditions, respectively ( Fig.  4(b)). The inverse reversal into the metastable condition that was due to the thermal disturbance in the superparamagnetic regime was suppressed by the high energy barrier, which led to an increased probability of bound magnetization in the most stable condition. By contrast, in the ferromagnetic regime at large Ku, the high energy barrier inhibits magnetization reversal from the metastable to the most stable conditions because both the minimal and secondary minimal potential energies were unchanged, which results in a reduced Mmax. Thus, the plotted dependence of Mmax on Ku showed a peak at Ku= 12 kJ/m 3 (Fig. 4(b)). The value of ξ at the Mmax peak is the index of the energetic threshold of the transition from the superparamagnetic regime to the ferromagnetic regime in the Langevin dynamics described by Eq. (9). Figure 6 shows the time evolutions of the individual magnetization and the rotatable easy axis depending on Ku for dc= 11 nm. For Ku ≤ 12 kJ/m 3 (ξ ≤ 2.02), as an effect of thermal fluctuations, the easy axis was randomly rotated along the spherical surface, while fluctuating magnetization reversal occurred. For Ku ≥ 14.7 kJ/m 3 (ξ ≥ 2.47), both the magnetization and the easy axis were rotated along the spherical surface.

Magnetization dynamics with a rotatable easy axis 3.2.1 Trajectories of individual magnetization and easy axis
The anisotropy energy bound the magnetization to the rotatable easy axis, as in the case of the fixed easy axis. The angle between the magnetization and the easy axis ψ was evaluated for the individual particles. ψ was calculated from the vector product between the unit vector of the magnetization and the easy axis, |m × n|. Origin symmetry is applicable for the easy axis (Eq. (8)).
The averaged value of ψ shown in Fig. 7 indicates that ψ was reduced with a large anisotropy energy with the increase of both Ku and dc. Therefore, considering the magnetization dynamics in a fluid, the reduction in the anisotropy energy barrier implies that the particles exhibit superparamagnetic behavior in the ferromagnetic regime, which is empirically confirmed by the reduced coercivity in a fluid compared with that in a solid on the DC magnetization curve 24) . This may be distinguished from the superparamagnetic regime as the effect of thermal fluctuations is reduced in the case of MNPs due to the large anisotropy energy.  Table 1. The easy axis was rotatable. ψ was calculated using the expression |m × n|. The statistical analysis was conducted in one cycle of the applied field with a time interval of 100 ns (N= 100).

Properties of net magnetization Figures 8(a) and (b) show magnetization curves, and
their corresponding Mmax and Hc, which increased with increasing ξ. High dc values corresponded to a high Mmax, for the same ξ values. This was indicated by the large Zeeman energy that corresponded to large dc, as in the case of the fixed easy axis (Eq. (8)). In the case of the rotatable easy axis, because the MNPs exhibit superparamagnetic behavior, the Hc derived from the hysteresis that was due to the anisotropy energy barrier is marginal. At high frequencies, the increase in Hc is associated with the phase delay derived from the magnetic relaxations. The magnetic torques ΓM= μ0MsVM|m×Hex| that were included in the first term of Eq. (5) depended on both Ku and dc. Figure 9 shows the ratio of the maximum value of the magnetic torque ΓM,max to the thermal energy, ΓM,max/kBT. ΓM,max increased with increasing Ku (ξ ≤ 2.02) and saturated in 2.02 ≤ ξ ≤ 3.03. The dependence of magnetic torque on Ku has been confirmed by Carrey and Hallali 36) . ΓM is proportional to VM and increases for large dc.  37) . Mamiya and Jeyadevan also indicated the effect of magnetic torque on the Brownian relaxation time 22) . According to these conventional studies, the Brownian relaxation time is reduced due to the large magnetic torque, which induces a small Hc.
Usadel also showed that the phase delay of the magnetization in a viscous liquid was reduced in the case of a large field amplitude 21) . In Fig. 8 3.3 Frequency dependence of magnetization and easy axis dynamics Figure 10(a) shows the time evolutions of the individual magnetization at f= 1 kHz and 100 kHz. The time evolution at f= 10 kHz is shown in Fig. 1(a-i). The magnetization reversals that are due to thermal fluctuations was suppressed with the increasing field frequency, which is indicated in Fig. 10(b). The frequency dependence of the magnetization curves in Fig. 10(c) shows the increasing Hc with increasing field frequency due to the phase delay of the magnetization associated with the Néel relaxation. In the high frequency, the magnetization dynamics correspond to the ferromagnetic regime. The remanent magnetizations without overcoming the energy barrier along the easy axis in the high frequency are represented as the increases of the phase delay and Hc in the net magnetization properties.
In the case of the rotatable easy axis, the time evolutions of the individual magnetization and easy axis at f= 1 kHz and 100 kHz are shown in Fig. 11(a). The time evolution at f= 10 kHz is shown in Fig. 6(b, f). The slight Hc confirmed at 10 kHz was derived from the Néel relaxation because this Hc value was similar to the Hc in the case of the fixed easy axis. At 100 kHz, Hc in the case of the rotatable easy axis was larger than for the case of the fixed easy axis, which is described as the phase delay derived from the Brownian relaxation. In particular, the minimum value of <cos θ> at 100 kHz was higher than 0.5, which indicates that the easy axis was steadily oriented along the applied field direction 23) . The time evolution of the easy axis also illustrated the rotation with the orientation toward the specific direction at 100 kHz, whereas the easy axis was    Table 1.
randomly orientated over the spherical surface at 1 kHz and 10 kHz. At 100 kHz, the magnetization was oriented along the easy axis toward the specific direction and was rotated on the spherical surface, which is similar to the case of the large ξ at 10 kHz in Fig. 6. The transition from the superparamagnetic behavior in the low frequency to the ferromagnetic behavior in the high frequency was observed in the case of the rotatable easy axis. Though ψ increased with increasing frequency, the increased value was slight in Fig. 11(c), which indicates that the magnetization dynamics show the ferromagnetic regime regardless of the field frequency at Ku= 12 kJ/m 3 and dc= 11 nm. It is observed that the frequency no longer affected the magnetic regime whereas the magnetic behavior was influenced by the frequency.

Conclusion
The dynamics of individual particle magnetization and the easy axis were evaluated for the three-dimensional trajectories of a single MNP using numerical simulation. In this paper, the effects of the core size and anisotropy of MNPs on the magnetization and easy axis dynamics were revealed in the same anisotropy energy. For the MNP in a solid (fixed easy axis) model, the ratio of the anisotropy energy to the thermal energy acts as the index of the threshold for the transition from the superparamagnetic to the ferromagnetic regime, considering the effects of stochastic thermal fluctuations and the potential energy of the MNP based on the Stoner-Wohlfarth model. In a fluid (rotatable easy axis), depending on the anisotropy energy, the dynamics of the magnetization and the easy axis changed from that of steep reversal to rotation on the spherical surface, which was associated with the transition from the superparamagnetic to the ferromagnetic regime. The net magnetization shown in the magnetization curves indicates superparamagnetic behavior in the case of the rotatable easy axis, while the ferromagnetic regime was clearly observed when examining the trajectories of the  single MNP. The specific magnetic behavior in a fluid was clearly distinguished from the magnetic regime by the observation of the individual MNP. The magnetic regime in a fluid was illustrated as the difference between the reversal and rotation in the magnetization dynamics. Moreover, it is also statistically revealed that the traceability of the easy axis to the magnetization in a fluid is enhanced in large anisotropy energy. The transition of the magnetic behavior depending on the applied field frequency was also confirmed. However, it is of note that the magnetic regime based on the potential energy was no longer influenced by the frequency. This evaluation of the dominance of superparamagnetic and ferromagnetic regimes in the stochastic Langevin dynamics of the system that depends on the anisotropy energy and thermal fluctuations contributes to the optimal material design of MNPs for use in biomedical applications such as hyperthermia treatment and MPI.