9.3 Time step dependence

The time step dependence of the numerical integration schemes has been investigated by simulating a single rigid magnetic moment. The material parameters were chosen as $M_\mathrm{s}=1281197~\frac{\mathrm{A}}{\mathrm{m}}$, $K_1=6.9\times 10^{6}~\frac{\mathrm{J}}{\mathrm{m^3}}$, $\alpha=0.1$, and $V=1~\mathrm{nm}^3$. The effective field, which is just the anisotropy field, is then given by

$$H_\mathrm{ani} = \frac{2 K_1}{\mu_0 M_\mathrm{s}} = 8571~\frac{\mathrm{kA}}{\mathrm{m}} \quad.$$

For the time for one full precession of the magnetization vector we obtain

$$T=\frac{1}{f}= \frac{2 \pi}{\omega}= \frac{2 \pi}{\gamma H_\mathrm{ani}}= 3.32~\mathrm{ps} \quad,$$

where $\omega=\gamma H$ is the Larmor frequency For the average magnetization in thermal equilibrium we find with ()

Temperature	$\langle {M_z} \rangle / M_\mathrm{s}$
10 K	$0.98979$
50 K	$0.94268$
200 K	$0.71976$

Figure: Time step dependence of numerical integration schemes

[Average magnetization at 10 K]
[ $\langle {M_z^2} \rangle$ at 10 K]

Figure: Time step dependence at higher temperatures

[Average magnetization at 50 K]
[Average magnetization at 200 K]

With the Milshtein scheme we find the correct values for time steps smaller than $0.01$ ps, which is about $1/300$ of the precession time. The Heun scheme is also suitable for time steps, which are ten times larger, because it has a higher order of convergence. As a rule, the discretization time step should be at most $1/30$th of the precession time of the magnetization vector in the effective field. This behaviour is verified at higher temperatures of 50 and 200 K. The results shown in figure  confirm, that the same rules apply for higher temperatures and therefore larger thermal fluctuations.