Energy dissipation in single-domain ferromagnetic nanoparticles: Dynamical approach

We study, both analytically and numerically, the phenomenon of energy dissipation in single-domain ferromagnetic nanoparticles driven by an alternating magnetic field. Our interest is focused on the power loss resulting from the Landau-Lifshitz-Gilbert equation, which describes the precessional motion of the nanoparticle magnetic moment. We determine the power loss as a function of the field amplitude and frequency and analyze its dependence on different regimes of forced precession induced by circularly and linearly polarized magnetic fields. The conditions to maximize the nanoparticle heating are also analyzed.

Figure 1

Reduced power loss $\tilde{Q}$ as a function of the reduced frequency $\tilde{\omega }$ in the small amplitude approximation. The parameters used in Eqs. (3.6) and (3.15) are $\tilde{h}=0.1,\tilde{H}=0,\sigma =+1,\rho =+1$, and $\alpha =0.1$ (a) and $\alpha =2$ (b).

Figure 2

Reduced power loss $\tilde{Q}$ as a function of the reduced amplitude $\tilde{h}$ for circularly $\left(\rho =+1\right)$ and linearly $\left(\rho =0\right)$ polarized magnetic fields. The solid and dashed lines represent the theoretical results obtained from Eqs. (3.6) and (3.15), respectively, and the symbols show the numerical results obtained from Eq. (4.1). It is assumed that $\sigma =+1,\alpha =0.1$, and $\tilde{\omega }=0.8$.

Figure 3

Color map for the reduced power loss $\tilde{Q}$ and the diagram for the steady-state regimes of precession of the magnetic moment driven by the circularly polarized magnetic field. The regions with different periodic regimes of precession are indicated by numbers 1–3, and the region with the quasiperiodic regime of precession by number 4. The numerical results are obtained for $\rho =+1,\tilde{H}=0$, and $\alpha =0.1$.

Figure 4

Examples of steady-state trajectories of the magnetic moment driven by the circularly polarized magnetic field. The simulation parameters are chosen to be $\rho =+1,\tilde{H}=0,\alpha =0.1$, and $\tilde{\omega }=0.8$. Trajectories for periodic regimes in regions $1 \left(\tilde{h}=0.05\right),2 \left(\tilde{h}=0.1\right)$, and $3 \left(\tilde{h}=0.4\right)$, and for the quasiperiodic regime in region $4 \left(\tilde{h}=0.26\right)$ are shown in (a) and (b), respectively.

Figure 5

Reduced power loss $\tilde{Q}$ as a function of the reduced amplitude $\tilde{h}$ of circularly polarized magnetic field for different values of the reduced magnetic field $\tilde{H}$. The parameters $\rho ,\alpha$, and $\tilde{\omega }$ are the same as in Fig. 4.

Figure 6

Examples of regular (a) and chaotic (b) trajectories of the magnetic moment driven by the linearly polarized magnetic field. The simulation parameters are $\rho =0,\tilde{H}=0,\alpha =0.1,\tilde{\omega }=0.8$, and $\tilde{h}=0.4$ (a) and $\tilde{h}=0.46$ (b).

Figure 7

Color map for the reduced power loss $\tilde{Q}$ and the regions of regular (1) and chaotic (2) precession of the magnetic moment driven by the linearly polarized magnetic field. The change of sign of the largest Lyapunov exponent is indicated by white lines. The parameters $\rho ,\tilde{H}$, and $\alpha$ are the same as in Fig. 6.