Resonant suppression of thermal stability of the nanoparticle magnetization by a rotating magnetic field

We study the thermal stability of the periodic (P) and quasiperiodic (Q) precessional modes of the nanoparticle magnetic moment induced by a rotating magnetic field. An analytical method for determining the lifetime of the P mode, in the case of high anisotropy barrier and small amplitudes of the rotating field, is developed within the Fokker-Planck formalism. In general case, the thermal stability of both P and Q modes is investigated by numerical simulation of the stochastic Landau-Lifshitz equation. We show analytically and numerically that the lifetime is a nonmonotonic function of the rotating field frequency, which, depending on the direction of field rotation, has either a pronounced maximum or a deep minimum near the Larmor frequency.

Figure 1

Diagram of the precessional modes for $\sigma \rho =+1$. The regions in the $\tilde{h}$-$\tilde{\omega }$ plane where different P modes exist at $\sigma \rho =+1$ are denoted as ${\mathrm{P}}_{+1}$ (white) and ${\mathrm{P}}_{+1}^{†}$ (light-green). The Q mode is realized in the white shaded region. In the region denoted as ${\mathrm{P}}_{-1}$ (blue), the stable precessional modes with $\sigma \rho =+1$ do not exist. Here, only the P mode with $\sigma \rho =-1$ is realized. The vertical dotted lines (a), (b), (c), and (d) correspond to $\tilde{h}=0.05,0.1,0.18$, and $0.25$, respectively.

Figure 2

Frequency dependence of the precession angle ${\Theta }_{+1}$ for different modes that exist at $\tilde{h}=0.25$. The frequencies ${\tilde{\omega }}_1=0.49$, ${\tilde{\omega }}_2=0.70$, and ${\tilde{\omega }}_3=0.89$ are the coordinates of the points in which the vertical dotted line (d), see Fig. 1, crosses the boundaries of the diagram(${\tilde{\omega }}_1$, ${\tilde{\omega }}_2$, and ${\tilde{\omega }}_3$ depend on $\tilde{h}$). The green line (with squares) and brown line (with triangles) show the frequency dependence of $\max {\Theta }_{+1}\left(\tilde{t}\right)$ and $\min {\Theta }_{+1}\left(\tilde{t}\right)$, respectively, in the case of Q mode.

Figure 3

Time dependence of the precession angle ${\Theta }_{+1}\left(\tilde{t}\right)$ in the case of Q mode. The parameters of the rotating field are as follows: $\rho =+1$, $\tilde{\omega }=0.725$, and $\tilde{h}=0.25$.

Figure 4

Time dependence of the difference of phases ${\Psi }_{+1}\left(\tilde{t}\right)=-\nu \tilde{t}+{\Phi }_{+1}\left(\tilde{t}\right)$ in the case of Q mode. Insert: time dependence of the function ${\Phi }_{+1}\left(\tilde{t}\right)$. The parameters of the rotating field are the same as in Fig. 3 and $\nu =0.38$.

Figure 5

Frequency dependence of the period ${\tilde{T}}_Q$ of the precession angle ${\Theta }_{+1}\left(\tilde{t}\right)$ in the case of Q mode. The rotating field parameters are taken as $\rho =+1$ and $\tilde{h}=0.25$.

Figure 6

Schematic time dependence of the polar angle $\theta \left(\tilde{t}\right)$ in the regions ${\mathrm{P}}_{+1}$ and ${\mathrm{P}}_{+1}^{†}$ shown in Fig. 1. The change of the magnetic moment state $\sigma$ from $+1$ to $-1$ occurs at $\tilde{t}={\tilde{t}}_0+{\tilde{t}}_{\mathrm{rel}}+{\tilde{t}}_{+1}$, when $\theta \left(\tilde{t}\right)$ reaches the angle ${\theta }_0=0.8\pi$ (the horizontal dashed line) for the first time. For a given trajectory $\theta \left(\tilde{t}\right)$, the lifetime of the P mode in the state $\sigma =+1$ is equal to ${\tilde{t}}_{+1}$. Running $N\gg 1$ trajectories, the mean lifetime can be evaluated as ${\mathcal{T}}_{+1}=(1/N){\sum }_{i=1}^N{\tilde{t}}_{+1}^{\left(i\right)}$.

Figure 7

Frequency dependencies of the lifetime of the precessional modes induced by the rotating field with $\rho =+1$ in the up state ($\sigma =+1$) of the magnetic moment.

Figure 8

Dependence of the resonant frequency of the lifetime ${\mathcal{T}}_{+1}$ on the rotating field amplitude.

Figure 9

Probability density functions of the lifetime of the Q mode for two values of ${\tilde{t}}_{\mathrm{st}}$. The blue solid and red dashed lines correspond to such ${\tilde{t}}_{\mathrm{st}}$ that ${\Theta }_{+1}\left({\tilde{t}}_{\mathrm{st}}\right)=\max {\Theta }_{+1}\left(\tilde{t}\right)$ and ${\Theta }_{+1}\left({\tilde{t}}_{\mathrm{st}}\right)=\min {\Theta }_{+1}\left(\tilde{t}\right)$, respectively. Insert: the same density functions in a larger time scale. The parameters of the rotating field are as follows: $\rho =+1$, $\tilde{\omega }=0.75$, and $\tilde{h}=0.25$.

Figure 10

Frequency dependencies of the lifetime of the precessional modes in the up state ($\sigma =+1$) of the magnetic moment. It is assumed that $\tilde{h}=0.25$ for both clockwise ($\rho =-1$) and counterclockwise ($\rho =+1$) rotation of the magnetic field.