Monte Carlo simulations of frozen metastable states in ordered systems of ultrafine magnetic particles

For studying the interplay of dipolar interaction and anisotropy energy in systems of ultrafine magnetic particles we consider simple cubic systems of magnetic dipoles with anisotropy axes pointing into the $z$ direction. Using Monte Carlo simulations we study the magnetic relaxation from several initial states. We show explicitly that, due to the combined influence of anisotropy energy and dipole interaction, magnetic chains are formed along the $z$ direction that organize themselves in frozen metastable domains of columnar antiferromagnetic order. We show that the domains depend explicitly on the history and relax only at extremely large time scales towards the ordered state. We consider this as an indication for the appearance of frozen metastable states also in real sytems, where the dipoles are located in a liquidlike fashion and the anisotropy axes point into random directions.

Figure 1

The order parameters (a) $O_{\ell }\left(t\right)$ and (b) $O_t\left(t\right)$ as a function of $t$ (number of Monte Carlo steps) are shown for the system sizes $L=6$ and $L=10$ when starting in the CAF state for several values of the reduced temperature $\tilde{T}=k_{B}T∕\left(2KV\right)=10$ (circles), 1 (squares), $2∕3$ (diamonds), $1∕2$ (triangles up), $1∕2.5$ (triangles left), and $1∕10$ (triangles right), where $T$ is the temperature, $k_B$ the Boltzmann constant, $K$ the anisotropy constant, and $V$ the particle volume. In (c) and (d) the plateau values $O_{\ell }^{*}$ and $O_t^{*}$ are shown for fixed $t={10}^4$ as a function of $\tilde{T}$.

Figure 2

The order parameters (a) $O_{\ell }\left(t\right)$ and (b) $O_t\left(t\right)$ are plotted versus the time $t$ (number of Monte Carlo steps) when starting in random configurations for the temperature $\tilde{T}=1∕10$ and for $L=4$ (circles), 6 (squares), 8 (diamonds), 10 (triangles up), and $L=12$ (triangles left). The arrows indicate the approximate crossover times $t_1$ for the system sizes (from left to right) $L=4$ to $L=12$.

Figure 3

The order parameters $O_t\left(t\right)$ are plotted versus the time $t$ (number of Monte Carlo steps) when starting in random configurations for several $L$ from $L=4$ to $L=12$ (same symbols as in Fig. 2) and for the temperature $\tilde{T}=1∕5$. The second crossover time $t_2$ is shown for $L=4$.

Figure 4

Visualization of the antiferromagnetic order in the $xy$ plane for systems at $\beta =10$ when starting in a random configuration. (a)–(d) One $L=10$ system at fixed $t$ (number of Monte Carlo steps), $t={10}^2$, ${10}^3$, ${10}^4$, and ${10}^5$, (e)–(h) one $L=4$ system after $t=20,{10}^2,2\times {10}^3$, and ${10}^5$. The complete chains are indicated by + or − signs, depending on the direction of the chain. Antiferromagnetic domains are shown in black and white. Domains of the same shade fit to each other and are allowed to emerge, whereas domain walls exist between clusters of different shades. Sites where chains have not yet been built are indicated by the gray shade. (e) The average relative size $N_{\max }∕L^2$ of the largest cluster at $t$ with $t_1<t<t_2$ is plotted versus the size $L$ for $\tilde{T}=1∕10$ (squares) and $\tilde{T}=1∕20$ (diamonds).