Elementary transitions and magnetic correlations in two-dimensional disordered nanoparticle ensembles

The magnetic relaxation processes in disordered two-dimensional ensembles of dipole-coupled magnetic nanoparticles are theoretically investigated by performing numerical simulations. The energy landscape of the system is explored by determining saddle points, adjacent local minima, energy barriers, and the associated minimum energy paths (MEPs) as functions of the structural disorder and particle density. The changes in the magnetic order of the nanostructure along the MEPs connecting adjacent minima are analyzed from a local perspective. In particular, we determine the extension of the correlated region where the directions of the particle magnetic moments vary significantly. It is shown that with increasing degree of disorder, the magnetic correlation range decreases, i.e., the elementary relaxation processes become more localized. The distribution of the energy barriers and their relation to the changes in the magnetic configurations are quantified. Finally, some implications for the long-time magnetic relaxation dynamics of nanostructures are discussed.

Figure 1

(Color online) Illustration of elementary transitions in dipole-coupled nanostructures having (a) a weak disorder with ${\sigma }_R=5\%$, and (b) a random particle distribution. In both cases the unit cell (dotted frame) contains $N=100$ particles and the coverage is $C=0.35$. The arrows show the directions of the magnetic moments corresponding to two adjacent local minima that are separated by a single energy barrier. The shading indicates the importance of the changes in the direction of the local moments. The dashed circle refers to the correlation range ${\lambda }_c$ where the angular variations are most significant.

Figure 2

(Color online) Average correlation range ${\overline{\lambda }}_c$ and standard deviation $\Delta {\lambda }_c$ as functions of the structural disorder ${\sigma }_R$. The results are obtained by sampling a large number of elementary transitions. In the top figure different coverages $C$ are considered, while in the bottom figure the number of particles $N$ in the unit cell is varied.

Figure 3

(Color online) Distribution of the energy barriers $\Delta E_{SM}$ corresponding to the elementary transitions in a random ensemble having a coverage $C=0.35$. The curve shows a fitted exponential law.

Figure 4

(Color online) Scatter plot of the energy barrier $\Delta E_{SM}$ as a function of the distance $D_{SM}$ between a saddle point and the adjacent local minima. As in Fig. 3 the particle setup is random and the coverage is $C=0.35$.

Figure 5

(Color online) Average distance ${\overline{D}}_{SM}$ between saddle points and adjacent minima, as well as the standard deviation $\Delta D_{SM}=\sqrt{⟨{\left(D_{SM}-{\overline{D}}_{SM}\right)}^2⟩}$ as functions of the structural disorder ${\sigma }_R$. The results are derived from a large number of elementary transitions for each different coverage $C$.

Figure 6

(Color online) Sketch of the eigenvector-following method to determine first-order saddle points.