Ising-like dynamics and frozen states in systems of ultrafine magnetic particles

We use Monte Carlo simulations to study aging phenomena and the occurence of spinglass phases in systems of single-domain ferromagnetic nanoparticles under the combined influence of dipolar interaction and anisotropy energy for different combinations of positional and orientational disorder. We find that the magnetic moments orient themselves preferably parallel to their anisotropy axes and changes of the total magnetization are solely achieved by $180\text{degree}$ flips of the magnetic moments, as in Ising systems. Since the dipolar interaction favorizes the formation of antiparallel chain-like structures, antiparallel chain-like patterns are frozen in at low temperatures, leading to aging phenomena characteristic for spin-glasses. Contrary to the intuition, these aging effects are more pronounced in ordered than in disordered structures.

Figure 1

Two-dimensional sketches of the geometries considered in this paper: (a) cubic arrangement of the particles and all anisotropy axes aligned into the $z$-direction, (b) liquid-like arrangement and all axes arranged, (c) cubic arrangement and all axes randomly oriented, and (d) liquid-like arrangement and all axes randomly oriented. In the simulations, the systems were three-dimensional (64 particles per cube).

Figure 2

(Color online) The magnetization $m\left(\tau \right)$ after waiting times $t_w=0$ (filled symbols) and $t_w=10000$ Monte Carlo steps (open symbols) is plotted versus $\tau$ (number of Monte Carlo steps with applied external field) for (a) cubic lattice and aligned axes, (b) liquid-like system and aligned axes, (c) cubic system and random axes, and (d) liquid-like systems and random axes for the reduced temperatures $\tilde{T}=k_{B}T∕\left(2KV\right)=5$ (black symbols, circles), $\tilde{T}=1∕10$ (red symbols, squares), and $\tilde{T}=1∕40$ (blue symbols, diamonds).

Figure 3

(Color online) The order parameter $O_{\mu }$ after a waiting time $t_w=0$ (filled symbols) and $t_w=10000$ (open symbols) is plotted versus $\tau$ (number of Monte Carlo steps) for the same geometries, temperatures, system parameters, and symbols and colors as in Fig. 2.

Figure 4

(Color online) The percentage $N_{\mathrm{up}}$ of particles per system pointing upwards after waiting times $t_w=0$ (filled symbols) and $t_w=10000$ (open symbols) is plotted versus $\tau$ (number of Monte Carlo steps) for the same geometries, temperatures, system parameters and symbols and colors as in Fig. 2.

Figure 5

Visualization of the chains perpendicular to the $xy$-plane in the cubic system with aligned anisotropy axes at $\tilde{T}=1∕5$ for one typical system. The complete chains are indicated by white sites and by + or − signs, depending on the direction of the chain. Sites, where chains have not (yet) been built are indicated by the gray shade. (a–c) System with waiting time $t_w=10000$, i.e., (a) after the cooling process and $t_w=10000$ (b,c) after an external magnetic field in the + direction has been applied for (b) 1000 and (c) 10000 Monte Carlo steps. (d-f) System without waiting time $(t_w=0)$, i.e., (d) after the cooling process (and $t_w=0$) (e,f) after an external magnetic field in the + direction has been applied for (e) 1000 and (f) 10000 Monte Carlo steps.

Figure 6

(Color online) The magnetization $m\left(\tau \right)$ after a waiting time $t_w=0$ (filled symbols) and $t_w=10000$ (open symbols) for $\tilde{T}=1∕10$ [red (light gray) symbols] and $\tilde{T}=1∕40$ [blue (dark gray) symbols] of systems with aligned and randomly oriented anisotropy axes (red circles and diamonds, respectively, for $\tilde{T}=1∕10$ and blue squares and triangles, respectively, for $\tilde{T}=1∕40$) are plotted versus $\tau$ (number of Monte Carlo steps) for systems without dipole-interaction.