Numerical integration of the stochastic Landau-Lifshitz-Gilbert equation in generic time-discretization schemes

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

Figure 1

A typical magnetization trajectory as a function of real time (in $\mu \mathrm{s}$) showing rapid fluctuations around 0 for $m_x$ (a) and $m_y$ (b) and telegraphic noise with sudden transitions between the up and the down magnetization configurations for $m_z$ (c). The initial condition is $\mathbf{m}=(0,0,1)$, $\alpha =0.5$, ${\eta }_0=0.005$, $\Delta \tau =0.5$, and ${\tau }_{\max }=2\times {10}^6$ that is equivalent to $t_{\max }=6.36$ $\mu \mathrm{s}$. In this and all other figures the working temperature is $T=300$ K.

Figure 2

Average value of the magnetization $z$ component as a function of $\tau$. Insets: $\tau$ dependence of the other two components, $\langle m_x\rangle$ and $\langle m_y\rangle$. $\alpha =0.5$, ${\eta }_0=0.005$, and $\Delta \tau =0.5$.

Figure 3

(a) $P\left(m_z\right)$, on a linear-log scale, obtained as explained in the text for the three values of ${\tau }_{\max }$ given in the key compared to the exact equilibrium law (solid line). Inset: The parameter $S$ defined in Eq. (26) as a function of ${\tau }_{\max }^{-1}$. The upper (dotted) curve was computed using the exact pdf $P_{\mathrm{eq}}\left(m_z\right)$, while the lower (solid) one, which gets closer to 0, was computed using a finite number of bins to approximate the exact $P_{\mathrm{eq}}\left(m_z\right)$. (b) $P\left(m_x\right)$, on a double-linear scale, for the same runs. $\alpha =0.5$, ${\eta }_0=0.005$, and $\Delta \tau =0.5$.

Figure 4

$P\left(m_z\right)$, on a linear-log scale, for the three values of ${\tau }_{\max }$ given in the key, compared to the exact equilibrium law (solid line). Insets: Averages of $\langle m_y\rangle$ and $\langle m_z\rangle$ as a function of $\tau$. The parameters are the same as in Fig. 2 but the initial condition is $\mathbf{m}=(0,1,0)$.

Figure 5

$\langle m_z\rangle$ as a function of $\tau$ for $\alpha =0.5$, ${\eta }_0=0.005$, and $\Delta \tau =0.5$, $0.05$, and $0.005$. Inset: $\langle m_z\rangle$ vs $\tau$ for $\alpha =0.0$ and the same $\Delta \tau$ with the same symbol code as in the figure. The dashed black line is a reference and corresponds to $\alpha =0.5$ and $\Delta \tau =0.005$. The initial condition is $\mathbf{m}=(0,0,1)$.

Figure 6

$P\left(m_z\right)$ for $\alpha =0.0$, ${\eta }_0=0.005$, and $\Delta \tau =0.5$, 0.05, and 0.005, compared to the exact equilibrium law (solid line). The initial condition is $\mathbf{m}=(0,1,0)$. Inset: Distributions $P\left(m_x\right)$ for the same runs and using the same symbol code as in the rest of the figure, compared to the limit (equilibrium) function shown in Fig. 3 for $\alpha =0.5$ and the longest ${\tau }_{\max }$.

Figure 7

(a) $\langle m_z\rangle$ as a function of $\tau$ for two values of the time increment, $\Delta \tau =0.5$ (open symbols) and 0.005 (filled symbols). $\alpha =0.5$ and several values of the damping coefficient ${\eta }_0$ (shown in different colors) as defined in the legend. (b) Ito calculus, $\alpha =0$, and ${\eta }_0=0.04$. Curves correspond to ${\tau }_{\max }={10}^5$ and different $\Delta \tau$ values: 0.005, 0.01, 0.02, 0.04, and 0.08 (from top to bottom). The solid black line displays $\langle m_z\rangle$ for $\alpha =0.5$, ${\eta }_0=0.04$, and $\Delta \tau =0.005$. Inset: The parameter $S$ defined in Eq. (28) for these curves, taking as a reference the curve for $\alpha =0.5$.