3.3.2 Uniform mode approximation

To start the discussion, consider a uniformly magnetized Stoner-Wohlfarth (SW) particle, with easy axis along the $x$-axis of a cartesian reference system, characterized by the following normalized free energy $g(\textbf{{m}})$:

$$g(\textbf{{m}})=\frac{1}{2}(D_x-N_\bot)m_x^2-h_{ax}m_x-h_{ay}m_y \quad,$$ (3.42)

where $D_x=N_x-K_1/(\mu_0 M_s^2)$, $K_1$ is the uniaxial anisotropy constant, $N_x$ and $N_\bot$ are demagnetizing factors along and perpendicular to the $x$-axis, respectively. Equilibria and switching of this particles can be analyzed by SW model which leads to the well-known asteroid curve in the applied field plane (see section 2.2 and Fig. 3.16) which separates the regions characterized by two minima of free energy (inside the asteroid) and one minimum (outside the asteroid). Within the static SW theory switching is attainable only for field above $h_$SW$=N_\bot-D_x$. Nevertheless by using magnetization dynamics^3.6, it is possible to realize switching also below this limit. To investigate this circumstance, we will analyze magnetization dynamics by means of Landau-Lifshitz-Gilbert (LLG) equation:

$$\frac{d\textbf{{m}}}{dt} =-\textbf{{m}}\times\textbf{h}_{\text{eff}} +\alpha\left(\textbf{{m}}\times\frac{d\textbf{{m}}}{dt}\right) \quad,$$ (3.43)

where $\textbf{h}_{\text{eff}}=-\nabla_\textbf{{m}}g$.

Figure: Phase portrait of LLG equation in the stereographic plane: $M$ is a maximum, $S$ is the saddle point, $m$ is the target minimum. $N_\perp =0.5$, $D_x=-3.0159$, $h_$SW$=3.5159$.

In order to illustrate the main idea behind fast switching below SW limit, let us consider the conservative LLG equation, i.e. we put $\alpha=0$ in Eq. (3.43). The qualitative features of LLG dynamics are conveniently represented on the plane by using the stereographic transformation introduced in section 2.3. In Fig. 3.15 a portion of the conservative phase portrait of Eq. (3.43) is reported in the case of an external field with $h_a=0.43 \times h_$SW and applied at an angle 45$^\circ$ off the $x$ axis. The magnetization trajectories coincide with the contour lines of the energy function (3.42). Since $h_a<h_$SW, there is still a minimum of the free energy in the point $U$ near the initial state $I$ which is assumed to be in the position $(1,0)$ (positive easy axis direction). One can see that if the initial state is inside the homoclinic loop $\Gamma$ around the minimum, the dynamics remains trapped around $U$, conversely when the initial magnetization state is outside $\Gamma$, the precessional oscillation brings magnetization far from the minimum $U$ and this is instrumental to speed up the switching process. The dashed lines in Fig. 3.15 represent an example of these two possible trajectories. The critical case between the two behaviors occurs when the initial state $I$ exactly lies on the homoclinic loop $\Gamma$. This situation is realized when the applied field is on a line in the $h_{ax}-h_{ay}$ plane which is inside the SW asteroid (see Fig.3.16). This curve can be expressed, in parametric form [28,36] as:

$$h_{ax}=-h_{\text{SW}}\cos u\cos^2\frac{u}{2} \quad,\quad h_{ay}=h_{\text{SW}}\sin u\sin^2\frac{u}{2} ,$$ (3.44)

where $\vert u\vert<\pi/2$ is the parameter. It turns out that it is necessary that the applied field is inside the region $F$ between the dashed and the solid line to realize the switching. This region can be characterized by specifying for each angle $\theta_h$ the allowed applied field amplitude range $[h_{a1},h_{a2}]$ (emphasized in Fig. 3.16). The upper limit $h_{a2}$ is given by the SW theory [77]

$$h_{a2}=h_{\text{SW}}[(\sin\theta_h)^{2/3}+(\cos\theta_h)^{2/3}]^{-3/2} \quad,$$ (3.45)

Figure 3.16: The solid line represents SW astroid, the dashed line represents the parametric curve defined by Eqs. (3.44).

while $h_{a1}$ can be found using Eqs. (3.44) once the applied field angle $\theta_h$ is connected with the corresponding parameter $u^*$, through the equation

$$h_{ay}(u^*)/h_{ax}(u^*)=-\tan\theta_h \quad.$$ (3.46)

Next we want to use the above uniform mode theory to study an ensemble of $N$ noninteracting particles to take into account dispersion of anisotropy axis and initial magnetization. We assume that the nominal (average) easy axis direction of the media is along the unit vector $\mathbf{e}$ which is aligned with the $x$ axis. The $j$-th particle is characterized by a magnetization direction $\textbf{{m}}_j$ and an easy axis direction $\mathbf{e}_j$ forming an angle $\beta_j$ with $\mathbf{e}$. We assume that the maximum $\beta_j$ is less than a prescribed angle $\beta$. We indicate by $\theta_{h,j}$ the angle between $\textbf{h}_a$ and $-\mathbf{e}_j$ which satisfies the inequality

$$\theta_h-\beta<\theta_{h,j}<\theta_h+\beta \quad.$$ (3.47)

By assuming that the initial states are given by $\textbf{{m}}_j=\mathbf{e}_j$, we find a family of applied field intervals $S_j=[h_{a1,j},h_{a2,j}]$ which corresponds to the fast switching of each particle. Thus, we can determine the applied field interval $S$ that allows the fast switching of all the particles by taking the intersection of the $S_j$:

$$S=\bigcap_{j=1}^N S_j \quad.$$ (3.48)

We computed this applied field interval for $\theta_h=45^\circ$ and $\beta=5^\circ$ and the result is $S=[0.401\times h_$SW$,0.47 \times h_$SW$]$.

Figure: Up: phase portrait of LLG equation for $\alpha =0.02$, $\theta_h=45^\circ$. Down: plots of the deviation angle $\delta$ as a function of $h_a$.

The analysis above has been carried out by using the conservative dynamics. In the case of the actual dynamics with damping the study of switching is more complicated and a careful analysis of the phase portrait is required. Magnetization dynamics is now described by Eq. (3.43) with $\alpha >0$. In Fig. 3.17 (up) one can see a portion of the phase portrait of the dynamical system for $\alpha =0.02$, $\theta_h=45^\circ$, $h_a=0.43 \times h_$SW. The shaded region $B$, enclosed in the separatrices $\Gamma_1$ and $\Gamma_2$, is the basin of attraction of the undesired minimum $U$. Magnetization trajectories spiral toward minima of the free energy and the phase portrait (see Fig. 3.17) is divided into the basins of attraction of the two minima $m$ and $U$. A necessary condition for switching is that the initial position $I$ is in the basin of attraction of the target minimum $m$. In addition, it is desirable that a whole neighbor of the initial position is inside this basin of attraction. Indeed, in real granular materials the different grains, due to exchange and strayfield interactions with other grains, will have slightly different initial conditions with certain deviation from the easy axis. In this respect, by analyzing the phase portrait in Fig. 3.17 we can determine quantitatively what is the allowed deviation $\delta_j$ of the $j$-th particle magnetization $\textbf{{m}}_j$ from the its easy axis $\mathbf{e}_j$, compatible with the requirement that the magnetization state is within the basin of attraction of the target minimum. This can be done by considering for each particle the circle of allowed initial conditions (see the dashed circle in Fig. 3.17). The analysis must be carried out for each particle in the ensemble and lead to the estimate of the maximum angular deviation $\delta$. The results are summarized in Fig. 3.17 (down).

Figure 3.18: Comparison of the switching time for a perpendicular recording media and a tilted media.