Theory of the collective behavior of two-dimensional periodic ensembles of dipole-coupled magnetic nanoparticles

The magnetic behavior of two-dimensional (2D) periodic ensembles of dipole-coupled nanoparticles (NPs) is investigated theoretically by considering all possible structural organizations in 2D Bravais lattices, namely, square, triangular, rectangular, rhombic, and oblique lattices. The different interaction-energy landscapes (ELs) are characterized by determining the local minima that define the stable and metastable magnetic configurations as well as the transition states connecting them. The topology of the resulting ergodic ensemble of stationary states is analyzed from both local and energy perspectives by calculating the corresponding kinetic networks and disconnectivity graphs. For all lattices, the magnetic orders of the ground-state and low-lying metastable configurations are identified, including the elementary relaxation processes between them. A remarkable profound dependence of the collective magnetic behavior on the structural arrangement of the NPs is revealed. Square and triangular nanostructures are extremely good structure seekers, showing starlike kinetic networks and disconnectivity graphs with palm-tree form. Their continuously degenerate ground states are throughout-reaching hubs connected to nearly all excited magnetic configurations over a single first-order saddle point. Rhombic ensembles have double-funnel ELs in which the time-inversion-related ground states play the role of hubs with exceedingly high connectivity densities. These systems need to undergo very few elementary transitions with small downward energy barriers to relax from any configuration towards one of the ground states. However, the energy barriers between the two funnels are quite large and, furthermore, they increase as the number of particles in the unit cell is increased. Therefore, ergodicity breaking is expected in the thermodynamic limit. Finally, the rectangular and oblique lattices show contrasting ground-state orders, the former consists of an antiferromagnetic alternation of head-to-tail chains of spins while the latter is ferromagnetic. Nevertheless, the ELs of these nanostructures are qualitatively very similar. In both cases, all excited magnetic configurations consist of independent flipping of the chains of spins that are formed along the direction of the shorter Bravais vector. The whole energy spectrum of metastable magnetic configurations can thus be mapped to a one-dimensional Ising model. As a result, the kinetic network of stationary points is latticelike and the disconnectivity graphs have a willow-tree form. In conclusion, the collective magnetic behaviors of nanostructures having different point-group symmetries are contrasted and related by varying the parameters of the lattices that interpolate between them.

Figure 1

Illustration of the five different types of two-dimensional Bravais lattices. In (a) the basis vectors are characterized by their lengths ${\lambda }_x$ and ${\lambda }_y$ and by the angle $\alpha$ between them. The longer and shorter diagonals $d_l$ and $d_s$ are also indicated here by dotted lines. For ${\lambda }_x={\lambda }_y$ we have (b) the square lattice if $\alpha =\pi /2$, (c) the triangular lattice if $\alpha =\pi /3$, and (d) a rhombic lattice otherwise. For ${\lambda }_x\ne {\lambda }_y$ one obtains (e) a rectangular lattice if $\alpha =\pi /2$ and (f) an oblique lattice if $\alpha \ne \pi /2$.

Figure 2

Lowest-energy metastable magnetic configurations of a periodic square lattice with $N=36$ dipole-coupled NPs in the unit cell. The circles indicate the position of the particles while the arrows indicate the orientation of the magnetic moments within the $xy$ plane. Examples of the configurations of the continuously degenerate ground state are shown in (a) for ${\eta }_{\text{MV}}=0$ and (b) for ${\eta }_{\text{MV}}=\pi /4$ [see Eq. (5)]. The configurations of the first and second excited states are shown in (c) and (d), respectively.

Figure 3

(a) Kinetic network of local minima (nodes) and transition states (edges) and (b) disconnectivity graph of the energy landscape of a periodic square lattice with $N=36$ dipole-coupled NPs in the unit cell. In (b) degenerate states are grouped together and the degeneracy is indicated at the corresponding branch. Energies are given in units of ${\varepsilon }_{DD}$ [Eq. (4)]. The ground state shows a continuous degeneracy with respect to the microvortex angle. See also Fig. 2 and Eq. (5).

Figure 4

Lowest-energy magnetic configurations of dipole-coupled NPs on a periodic triangular lattice with $N=36$ particles in the unit cell. The circles indicate the position of the particles while the arrows indicate the orientation of the magnetic moments within the $xy$ plane. One of the magnetic configurations of the degenerate ground state (a), first excited state (b), and second excited state (c) are shown.

Figure 5

(a) Kinetic network of local minima (nodes) and transition states (edges) and (b) disconnectivity graph of the energy landscape of a periodic triangular lattice with $N=36$ dipole-coupled magnetic moments. In (b) the degenerate states are grouped together and the degeneracy is indicated at the corresponding branch. The ferromagnetic ground state shows a continuous degeneracy with respect to the angle defining the direction of the magnetization (see also Fig. 4).

Figure 6

Magnetic configurations of (a) the ground state, (b) the first excited state, (c) the second excited state, and (d) the third excited state of a rectangular ensemble having $N=36$ magnetic moments in the unit cell with periodic boundary conditions. The ratio between the longer and shorter NN distances is ${\lambda }_x/{\lambda }_y=1.05$ (see Fig. 1).

Figure 7

(a) Kinetic network of local minima (nodes) and transition states (edges) and (b) disconnectivity graph of the energy landscape of a periodic rectangular arrangement of magnetic NPs with ${\lambda }_x/{\lambda }_y=1.05, N=36$ dipole-coupled magnetic moments in the unit cell and periodic boundary conditions (see also Fig. 6).

Figure 8

Magnetic configurations of (a) the ground state, (b) the first excited state, (c) the second excited state, and (d) a higher-energy antiferromagnetic configuration of the rhombic ensemble having $\alpha =5\pi /12$ and $N=36$ NPs in the unit cell (see Fig. 1).

Figure 9

(a) Kinetic network of local minima (nodes) and transition states (gray edges) and (b) disconnectivity graph of the energy landscape of a periodic rhombic lattice having $\alpha =5\pi /12$ and $N=36$ dipole-coupled magnetic moments in the unit cell (see also Fig. 8). In (a) the ferromagnetic ground states are indicated by the red circles. Furthermore, the edges in black highlight the downhill transition with the lowest-energy barriers. In (b) degenerate excited LM are grouped and the corresponding degeneracies are indicated.

Figure 10

Magnetic configurations of (a) the ground state, (b) the first excited state, and (c) the second excited state of an oblique-lattice ensemble having $N=36$ magnetic moments in the unit cell with periodic boundary conditions. The ratio between the NN distances is ${\lambda }_x/{\lambda }_y=1.05$ and the angle between the Bravais vectors is $\alpha =7\pi /18$. See Fig. 1.

Figure 11

(a) Kinetic network of local minima (nodes) and transition states (edges) and (b) disconnectivity graph of the energy landscape of a periodic oblique lattice having ${\lambda }_x/{\lambda }_y=1.05, \alpha =7\pi /18$, and $N=36$ dipole-coupled magnetic moments in the unit cell. In (a) the ground-state nodes are highlighted in red. See also Figs. 1 and 10.