Field sweep-rate dependence of magnetic domain patterns: Numerical simulations for a simple Ising-like model

We study magnetic domain patterns in ferromagnetic thin films by numerical simulations for a simple Ising-like model. Magnetic domain patterns after quench demonstrate various types of patterns depending on the field sweep rate and parameters of the model. How the domain patterns are formed is shown with the use of the number of domains, the domain area, and domain-area distributions, as well as snapshots of domain patterns. Considering the proper time scale of the system, we propose a criterion for the structure of domain patterns.

Figure 1

Domain patterns under zero field after a fast quench $(v={10}^{-2})$, which display the dependence on the parameter $\alpha$: [(a) and (b)] $\alpha =1.5$, (c) $\alpha =2.0$, (d) $\alpha =2.5$, (e) $\alpha =3.0$, and [(f) and (g)] $\alpha =3.5$. They are the patterns at $t=2t_0$ except for (a) and (f); these two are the ones at $t=t_0$. The size of the snapshots is $96\times 96$, although the system size is $512\times 512$. The value of $\varphi \left(\mathit{r}\right)$ is positive and negative in the white and black areas, respectively.

Figure 2

(Color online) The field dependence of (a) the total number of domains $N_{\mathrm{tot}}$ and (b) the average domain area $A_{\mathrm{av}}$; (c) the $A_{\mathrm{av}}-N_{\mathrm{tot}}$ graph. The broken line in (c) expresses the curve of $A_{\mathrm{av}}{N}_{\mathrm{tot}}={512}^2∕2$, where the total areas of black and white domains are equal. The calculation was performed under a rapidly decreasing field $(v={10}^{-2})$ until $t=t_0$ and then under zero field until $t=2t_0$. The number of domains means the number of negative-$\varphi \left(\mathit{r}\right)$ (black) domains. The domain area is the number of grid points that have negative values of $\varphi \left(\mathit{r}\right)$.

Figure 3

(Color online) Domain-area distributions at $t=t_0$ and $t=2t_0$. The left, center, and right histograms at each time are for $\alpha =1.5$, $\alpha =2.5$, and $\alpha =3.5$, respectively. The domain area is the number of grid points that have negative values of $\varphi \left(\mathit{r}\right)$.

Figure 4

Domain patterns under zero field after a slow quench $(v={10}^{-4})$, which display the dependence on the parameter $\alpha$: [(a) and (b)] $\alpha =1.5$, (c) $\alpha =2.0$, (d) $\alpha =2.5$, (e) $\alpha =3.0$, and (f) $\alpha =3.5$. They are the patterns at $t=2t_0$ except for (a) at $t=t_0$.

Figure 5

(Color online) The field dependence of (a) the total number of domains $N_{\mathrm{tot}}$ and (b) the average domain area $A_{\mathrm{av}}$; (c) the $A_{\mathrm{av}}-N_{\mathrm{tot}}$ graph. The calculation was performed under a slowly decreasing field $(v={10}^{-4})$ until $t=t_0$ and then under zero field until $t=2t_0$. The data are displayed on a semilogarithmic scale in (a) and (b) and on a log-log scale in (c), while the inset of (b) shows the same data as its main panel on a normal scale. The broken line in (c) expresses the curve of $A_{\mathrm{av}}{N}_{\mathrm{tot}}={512}^2∕2$.

Figure 6

Dependence of the average domain length at $t=2t_0$ on the field sweep rate $v$. The average domain length is calculated from the average domain area and the average domain width (see text). The domain pattern looks like labyrinth and sea-island structures above and below the gray patched area, respectively.

Figure 7

Schematic picture about the characteristic times in the nucleation process. See text for the details.