Interaction-energy landscapes of kagome and honeycomb lattices of dipole-coupled magnetic nanoparticles

The collective magnetic behaviors of dipole-coupled magnetic nanoparticles organized in periodic two-dimensional kagome and honeycomb lattices are investigated theoretically. The extended nanostructures are modeled by considering finite unit cells containing nanoparticles with periodic boundary conditions. The energy landscapes (ELs) of the ensembles are systematically explored as a function of the orientation of all the NP moments by calculating the local minima and first-order saddle points connecting them. The low-lying magnetic orders and elementary reorientation transitions are identified. The thus obtained kinetic networks and disconnectivity graphs of the ELs reveal profound differences in the topology of the networks of stationary states. One observes that the honeycomb nanostructures are typically good structure seekers with starlike kinetic networks and palm-tree-like disconnectivity graphs. They have a continuously degenerate ground state, which is the center of the network and directly connected to all excited metastable states. In contrast, kagome nanostructures show a particular form of bad structure seeker, which is known as the latticelike stationary-point network. In this case, from a local perspective, the degree distribution among the LM is extremely homogeneous, with no centers or hubs through which relaxation can be funneled. Furthermore, from an energy perspective, no hierarchy among the low-energy metastable states can be identified.

Figure 1

Low-lying metastable magnetic configurations of a periodic honeycomb lattice with $N=54$ dipole-coupled NPs in the unit cell. The circles indicate the position of the particles and the arrows the orientations of the magnetic moments within the $xy$ plane. In panels (a) and (b), examples of the continuously degenerate ground state are shown with microvortex angles (a) ${\eta }_{\mathrm{MV}}=0$ and (b) ${\eta }_{\mathrm{MV}}=\pi /2$. The magnetic configurations of the first- and second-excited states are shown in panels (c) and (d), respectively.

Figure 2

(a) Kinetic network of a periodic honeycomb lattice with $N=54$ dipole-coupled NPs in the unit cell. The degenerate ground state is represented by a red circle. The $n\mathrm{th}$ level of degenerate metastable states are grouped into the gray circles labeled $E_n$. The transitions towards the ground state are highlighted in black, whereas the transitions among the excited states are indicated in gray. (b) Corresponding disconnectivity graph, where the degenerate states are also grouped into a single branch and the degeneracies are given by the numbers at the end of each branch. Notice that the energies are measured in units of ${\varepsilon }_{DD}$ [see Eq. (4)].

Figure 3

Metastable magnetic configurations of a periodic kagome lattice with $N=27$ 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 6-fold degenerate ground state are shown in panels (a) and (d). The other subfigures illustrate different degenerate magnetic configurations of the first-excited state, in which the changes with respect to the ground states are highlighted in gray.

Figure 4

Kinetic network of a periodic kagome lattice with $N=27$ dipole-coupled NPs in the unit cell. The 6-fold degenerate ground states are highlighted by red dots, the first-excited states by blue dots, and the second-excited states by green dots. The local minima having higher energies are represented by the smaller black dots. The edges connecting the dots indicate the elementary transitions between the LM across first-order saddle points. Notice that the degree distribution among the nodes is very homogeneous and that the network lacks any hubs or centers through which relaxation could be funneled.

Figure 5

Disconnectivity graph of the energy landscape of a periodic kagome lattice with $N=27$ dipole-coupled NPs in the unit cell, which contains $N_{\mathrm{LM}}=298$ local minima. The colors indicate the ground states (red), first-excited states (blue), second-excited states (green), and higher-energy LM (black), as in Fig. 4. The energies on the $y$ axis are given in units of ${\varepsilon }_{DD}$ [see Eq. (1)].