4.8.1 Definition of the problem

We apply the above numerical technique to solve the $\mu$-mag standard problem n. 4 (see Ref. [79]). This problem concerns the study of magnetization reversal dynamics in a thin-film subject to a constant and spatially uniform external field, applied almost antiparallel to the initial magnetization. The geometry of the medium is sketched in Fig. 4.1. The material parameters are $A=1.3\times 10^{-11}$ J/m, $M_s=8.0\times10^5$ A/m, $K_1=0$ J/m$^3$ and $\alpha =0.02$ (permalloy). The initial state is an equilibrium s-state (see Fig. 4.1, right) such as is obtained after applying and slowly reducing a saturating field along the $[1,1,1]$ direction to zero. In all the numerical simulations the equilibrium condition has been chosen such that:

$$\max_{l=1\ldots N} \left\vert \frac{\textbf{{m}}_l^{n+1} - \textbf{{m}}_l^{n}}{ \Delta t}\right\vert <10^{-5} \quad,$$ (4.61)

i.e. the maximum of the (normalized) torque across the body has been checked for equilibrium. Moreover, the stopping criterion of the quasi-Newton iterative procedure has been chosen

$$\max\vert _q(\mathbf{y}_k)\vert<10^{-14} \quad,\quad q=1,\ldots,3N \quad.$$ (4.62)

where F$_q(\mathbf{y}_k)$ is the $q$-th components of the vector $\mathbf{F}(\mathbf{y}_k)$, and the index $k$ indicates the number of quasi-Newton iterations. Two switching events will be calculated using fields applied in the x-y plane of different magnitude and direction. In the first case the external field is applied at an angle of $170^\circ$ off the $x$ axis with $x-y$ components such that $\mu_0$   M$_$s   h$_{ax}=-24.6$ mT, $\mu_0$   M$_$sh$_{ay}=4.3$ mT and $\mu_0$   h$_a=25$ mT. In the second case the external field is applied at an angle of $190^\circ$ off the $x$ axis with $x-y$ components such that $\mu_0$   M$_$sh$_{ax}=-35.5$ mT, $\mu_0$   M$_$s   h$_{ay}=-6.3$ mT, and $\mu_0$   M$_$sh$_a=36$ mT. In both cases the cell edges are $d_x=3.125$ nm, $d_y=3.125$ nm, $d_z=3$nm and therefore the number of cells is $N=160\times 40 \times 1=6400$.