3.2.4 Numerical results

We performed a set of micromagnetic numerical simulations of the precessional switching process for the values of $H_a$ and $T_s$ specified in the table 3.1. This table reports the switching time $T_s$, analytically computed using Eq. (3.33), for different values of $H_a$.

Table: Values of the switching times $T_s$, analytically computed with formula (3.33) and used in micromagnetic simulations ($M_s=795$ kA/m, $A=1.3\times 10^{-11}$ J/m, $\alpha =0.02$).

$h_a/h_$SW	1.0	1.1	1.2	1.3	1.4	1.5
$H_a$ [kA/m]	13.01	14.31	15.61	16.91	18.21	19.51
$T_s$ [ns]	0.194	0.181	0.171	0.162	0.155	0.149

Figure: Analytical and numerical solutions of Landau-Lifshitz-Gilbert equation. Plot of $<m_x>$ vs time. $h_a=1.5\times h_{SW}$, $D_x=1.2\times 10^{-3}$, $D_y=0.0175$, $D_z=0.9763$.

The simulations were started from both initial magnetization configurations which can be observed in the experiments on thin-film media: the S-state and C-state (see figure 3.4).

Figure: Numerically computed $<m_x>$ as a function of time. S-state (a), C-state (b) initial condition. (In both figures) symbol ``$\Box$'' for $h_a=h_$SW; ``O'' for $h_a=1.1\times h_$SW; ``X'' for $h_a=1.2\times h_$SW; ``$\nabla$'' for $h_a=1.3\times h_$SW; ``$\diamond$'' for $h_a=1.4\times h_$SW; ``$\triangle$'' for $h_a=1.5\times h_$SW.

In figure 3.9 a comparison between the analytical solution of LLG Eq. (2.18) with $\alpha=0$, the numerical solution of LLG Eq. (2.18) with $\alpha =0.02$ for a uniformly magnetized thin-film shaped ellipsoidal particle (macro-spin model) and the finite element solution of Eq. (3.20) is reported for an applied field strength $h_a=1.5\times h_$SW. In the undamped case, at time $t=T_s$ the magnetization is exactly in the reversed position. Therefore, when the external field is switched off, it remains definitely in this state. If the damping term is introduced, one can see that after $t=T_s$ there is a small oscillation of $<m_x>$ because the system is not yet in the minimum energy state. In the general nonuniform case one can easily see that the uniform mode theory provide anyway a reasonably good information about the duration of the field pulse, but the presence of nonuniform modes produces an oscillation that can bring magnetization back to the initial state as one can see in Figs. 3.10(a)-(b). For this reason, a field strength $h_a=1.5\times h_$SW is required to achieve successful switching starting from either an S-state or a C-state. We observe that this value is moderately larger than the critical value provided by uniform mode theory, $h_c=h_$SW$/2$.