3.2.5 Precessional switching: dependance on the anisotropy and switching time tolerance window

In this section we will demonstrate that the agreement with the analytical prediction increases for increasing values of the anisotropy constant. In this respect, we will verify that the time window tolerance $(t_1,t_2)$ computed by Eqs. (3.31)-(3.32) gives very accurate information on the reliability of the switching.

Figure: Plot of switching times $T_s$ (solid line), $t_1$ and $t_2$ (dashed lines). $h_a=D_y-D_x$, $D_x=N_x-2K_1/(\mu_0 M_s^2)$, $D_y=0.0175$, $D_z=0.9763$.

In Fig. 3.11 the plot of the time instants $t_1$, $T_s$, $t_2$ is reported as a function of the anisotropy constant $K_1$ when the applied field amplitude is chosen $H_a=M_s h_$SW$=(D_y-D_x)M_s$. The normalized applied field $h_a=D_y-D_x$ is related to $K_1$ through Eq. (3.29). It is important to underline that the time window for switching the field off is reasonably wide because, in the analyzed interval of $K_1\in[10^4,10^5]$ J/m$^3$ (moderately soft materials used in magnetic recording technology), is $t_1<0.75\times T_s$ and $t_2>1.25\times T_s$, that is, a tolerance of at least $\pm 25\%$ on the switching pulse is allowed.

Figure: Micromagnetic simulations: plot of $m_x$ vs time; (a) $K_1=10^4$ J/m$^3$; (b) $K_1=2.5\times 10^4$ J/m$^3$; (c) $K_1=5\times10^4$ J/m$^3$; (d) $K_1=7.5\times10^4$ J/m$^3$; (e) $K_1=10\times 10^4$ J/m$^3$. The switching time $T_s$ is marked with a vertical line.

On the basis of the above analysis we performed a set of micromagnetic simulations of precessional switching experiments for the $500\times 250 \times 5$ nm thin-film described at the beginning of section 3.2. Initially, the thin-film is saturated along the positive $x$-axis, then it is relaxed to the remanent state. At time $t=0$ the rectangular external field pulse is applied $H_a=(D_y-D_x)M_s$ until time $t=T_s$ at which the field is switched off and the magnetization relaxes towards equilibrium. We performed different simulations for different values of $K_1$, reported in Table 3.2.

Table: Values of the parameters used in micromagnetic simulations ($M_s=795$ kA/m, $A=1.3\times 10^{-11}$ J/m, $\alpha =0.02$).

$K_1$ [$10^4$ J/m$^3$]	$1.0$	$2.5$	$5.0$	$7.5$	$10$
$H_a$ [kA/m]	30.88	60.88	110.88	160.88	210.88
$T_s$ [ps]	124.3	86.6	62.0	49.8	42.1
$t_1$ [ps]	92.9	64.6	46.0	36.7	30.9
$t_2$ [ps]	155.6	108.7	78.0	62.9	53.3

The results are reported in Fig. 3.12. One can clearly see that for moderately low values of $K_1$ (Fig. 3.12a) at $t=T_s$ magnetization is not exactly close to the reversed state, but micromagnetic simulations show that the higher the applied field strength is, the better is the agreement with the uniform mode theory. By moderately increasing the value of the anisotropy constant there is a very good agreement with the above prediction and the remaining oscillation after $t=T_s$ tends to be very close to the magnetization reversed state [Fig. 3.12(b)-(e)].

Figure: Micromagnetic simulations: plot of $m_x$ vs time. $K_1=2.5\times 10^4$ J/m$^3$, $H_a=(D_y-D_x)M_s=60.88$ kA/m. The field is switched off at time (a) $t=108$ ps; (b) $t=120$ ps; (c) $t=65$ ps; (d) $t=63$ ps.

Next, we chose to verify the prediction of the uniform mode theory regarding the time window for switching the field off. We analyze, for sake of brevity, the case of anisotropy constant $K_1=2.5\times 10^4$ J/m$^3$. The applied field is $H_a=(D_y-D_x)M_s=60.88$ kA/m. The results (Fig. 3.13) show the accuracy of the uniform mode theory prediction. In fact, switching the applied field off just few picoseconds after time $t=t_2$ (Fig. 3.13b) or just a few picoseconds before time $t=t_1$ (Fig. 3.13d) leads to non-successful switching, while switching the applied field off just few picoseconds before time $t=t_2$ (Fig. 3.13a) or just a few picoseconds after time $t=t_1$ (Fig. 3.13c) leads to successful switching. Thus, we can conclude that, in precessional switching experiments on thin-film media constituted of moderately soft materials, the time window for switching the applied field off can be derived by using the uniform mode theory with a very high accuracy. Moreover, the knowledge of the time window $[t_1,t_2]$ can be used to find the switching diagrams proposed in Ref. [34] to design MRAM storage cells, in the case of short (rectangular) field pulse durations, without performing numerical simulations.