Magnetic particle hyperthermia: Power losses under circularly polarized field in anisotropic nanoparticles

The deterministic Landau-Lifshitz-Gilbert equation has been used to investigate the nonlinear dynamics of magnetization and the specific power loss in magnetic nanoparticles with uniaxial anisotropy driven by a rotating magnetic field, generalizing the results obtained for the isotropic case found by P. F. de Châtel, I. Nándori, J. Hakl, S. Mészáros, and K. Vad [J. Phys. Condens. Matter 21, 124202 (2009)]. As opposed to many applications of magnetization reversal in single-domain ferromagnetic particles, where losses must be minimized, in this paper, we study the mechanisms of dissipation used in cancer therapy by hyperthermia, which requires the enhancement of energy losses. We show that for circularly polarized field, the energy loss per cycle is decreased by the anisotropy compared to the isotropic case when only dynamical effects are taken into account. Thus, in this case, in the low-frequency limit, a better heating efficiency can be achieved for isotropic nanoparticles. The possible role of thermal fluctuations is also discussed. Results obtained are compared to experimental data.

Figure 1

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] for the isotropic case (${\lambda }_{\mathrm{eff}}=0$) with the parameters ${\alpha }_N=0.1,\omega =0.01,{\omega }_L=0.2$. The circle indicates the repulsive fixed point at $\varphi =3.12$, $\theta =1.53$. The attractive fixed point is at $\varphi =0.02$, $\theta =1.61$.

Figure 2

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] in the limit of low frequency and small anisotropy for the parameters ${\alpha }_N=0.1,\omega =0.01,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=0.05$. It is similar but not identical to Fig. 1.

Figure 3

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] (medium frequency and small anisotropy) for the parameters ${\alpha }_N=0.1,\omega =0.05,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=0.05$. According to numerical results, the attractive fixed point is at $\varphi =0.097995$, $\theta =1.77908$, which gives $u_{y0}=-0.0957$.

Figure 4

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] in the limit of low frequency and large anisotropy for the parameters ${\alpha }_N=0.1,\omega =0.01,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=2.5$. Two attractive fixed points appear in the figure. The repulsive fixed point and the saddle point are indicated by a circle and a solid circle, respectively.

Figure 5

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] (low frequency, large anisotropy) for the parameters ${\alpha }_N=0.1,\omega =0.01,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=1.5$. The attractive fixed points are closer to the equator as compared to Fig. 4, where the anisotropy was larger.

Figure 6

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] in the limit of low frequency, slightly below the critical value of anisotropy. The parameters are ${\alpha }_N=0.1,\omega =0.01,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=1.175$. There is only a single attractive fixed point below the equator (large $\theta$). The other attractive fixed point (above the equator, small $\theta$) of the large anisotropic case and also the saddle point vanish in this regime of the parameter space.

Figure 7

The energy loss is plotted vs. the anisotropy ${\lambda }_{\mathrm{eff}}$. There are two scaling regimes: one is at small and the other is at large anisotropy, which are separated by the critical value ${\lambda }_{\mathrm{cr}}$. The solid line corresponds to energy loss obtained at the stable fixed point below the equator, and the dashed line represents energy loss at the fixed point above the equator, which is stable only for ${\lambda }_{\mathrm{eff}}>{\lambda }_{\mathrm{cr}}$.

Figure 8

Phase portrait in the rotating frame obtained by solving the LLG equation [Eq. (9)] in the limit of high frequency and large anisotropy for the parameters ${\alpha }_N=0.1,\omega =0.15,{\omega }_L=0.2$, and ${\lambda }_{\mathrm{eff}}=2.5$. There are two attractive fixed points in the figure similar to the low-frequency case (Fig. 4). Here, the attractive fixed points have different $y$ components; thus, the corresponding energy losses also differ from each other.