Disorder-Induced Transformation of the Energy Landscapes and Magnetization Dynamics in Two-Dimensional Ensembles of Dipole-Coupled Magnetic Nanoparticles

The interaction-energy landscapes (ELs) and magnetization dynamics of two-dimensional ensembles of dipole-coupled magnetic nanoparticles are theoretically investigated. Extended nanostructures are modeled by considering nonoverlapping nanoparticles (NPs) in a square unit cell with periodic boundary conditions. The local minima and connecting transition states of the EL are determined systematically for representative NP arrangements having different degrees of disorder. The topology of the ergodic networks of stationary points is analyzed from both local and energy perspectives by using kinetic networks and disconnectivity graphs. We show that increasing the degree of disorder not only increases, most significantly, the number of local minima and transition states but also changes the shape of the EL in a very profound way. While slightly disordered ensembles correspond to good structure seekers, which are funneled towards the global minima, strongly disordered systems show very rough landscapes with multiple low-energy local minima separated by relatively large energy barriers. The consequences of this transition on the long-time Markovian dynamics of the nanostructures are quantified by calculating the field-free magnetic relaxation after saturation and after quenching. The simulations indicate that the relaxation of weakly disordered systems follows a slightly stretched exponential law, with a single characteristic timescale for a wide range of temperatures. In contrast, strongly disordered systems show a much more complicated relaxation dynamics involving multiple timescales, slowing down and trapping, which is reminiscent of spin glasses.

Popular Summary

Magnetic nanoparticle ensembles are the focus of remarkable research activity, which is driven by their fundamental interest and technological possibilities. A crucial aspect of any artificial nanostructure realization is the practical difficulty in controlling the size and position of the particles with arbitrary precision. Therefore, a variable inherent degree of disorder becomes unavoidable and its physical consequences are not properly understood. This work reveals how disorder thoroughly reshapes the static and dynamic magnetic behavior of 2D nanoparticle ensembles.

We investigate different structural arrangements and degrees of disorder of 2D ensembles of magnetic nanoparticles as a function of the orientations of the magnetic moments of all nanoparticles. A detailed theoretical analysis shows that disorder changes the nature and energy of the magnetic configurations in a most profound way. While weakly disordered ensembles are funneled toward their most stable magnetic order, strongly disordered systems show very rough energy landscapes with multiple low-energy local minima separated by relatively large energy barriers. These striking transformations are at the origin of qualitative changes in the collective magnetic relaxation dynamics. As disorder increases, rapidly relaxing systems are transformed into trapped glassy ones.

We have proposed a new theoretical perspective to magnetic nanostructure that opens a number of challenging research directions in which disorder is expected to play a central role. Such directions include the manipulation of the magnetic order with external fields, the dependence on nanoparticle arrangement in different dimensions, or the development of applications in high-density recording and magnetic-based disease treatments.

Figure 1

Ground-state magnetic configurations of representative two-dimensional ensembles of nanoparticles: (a) weakly disordered square structure (${\sigma }_r/r_0=0.05$), (b) weakly disordered triangular structure (${\sigma }_r/r_0=0.05$), and (c) nonoverlapping randomly distributed particles. The unit cells are shown, on which periodic boundary conditions apply.

Figure 2

Undirected kinetic network of the energy landscape of the weakly disordered square-lattice ensemble illustrated in Fig. 1. Each grey segment represents an elementary transition between two local minima through a first-order saddle point. The barrier heights and the lengths of the minimum energy paths connecting the minima are disregarded at this stage. The two degenerate ground states are indicated by red circles. Additional hubs (${\rho }_c=0.33$) of the kinetic network are indicated by orange circles. The thick black segments highlight the transitions that, starting from a given minimum, lead to a minimum having a lower or equal energy involving the smallest energy barrier [95].

Figure 3

Undirected kinetic network of the energy landscape of the weakly disordered, triangular NP ensemble illustrated in Fig. 1. The two degenerate ground states are indicated by the red circles. Additional hubs (${\rho }_c=0.3$) are indicated by orange circles. The thick black segments highlight the transitions that, starting from a given minimum, lead to a lower or equal energy involving the smallest energy barrier [95].

Figure 4

Undirected kinetic network of the energy landscape of the strongly disordered ensemble illustrated in Fig. 1. The two degenerate ground states are indicated by the red circles. The thick black segments highlight the transitions that, starting from a given minimum, lead to a lower or equal energy involving the smallest energy barrier. Note that even the ground states do not qualify as hubs (${\rho }_c={10}^{-3}$) [95].

Figure 5

Disconnectivity graph of the weakly disordered, square-lattice ensemble illustrated in Fig. 1. Notice that the degenerate global minima (red circles) and hubs (orange circles) are distinctly more stable than all other local minima. The insets illustrate the magnetic configuration at one of the global minima ($E_0$) and the closest hub ($E_1$). The twofold degeneracy is a consequence of time-inversion symmetry. See also Fig. 2 [95].

Figure 6

Disconnectivity graph of the weakly disordered, triangular-lattice ensemble illustrated in Fig. 1. As in the square lattice (Fig. 5), the doubly degenerate global minima (red circles) and hubs (orange circles) can be clearly distinguished from the remaining higher-lying local minima, which show particularly small energy barriers towards the ground states. The insets illustrate the magnetic configuration at a global minimum ($E_0$) and its closest hub ($E_1$). See also Fig. 3 [95].

Figure 7

Disconnectivity graph of the strongly disordered ensemble of nonoverlapping NPs illustrated in Fig. 1. In graph (a), all calculated LM are taken into account, whereas graph (b) focuses on the lowest 400 LM. Notice the numerous low-lying local minima, which are separated by important energy barriers that are often much larger than the energy differences between the LM. The energy barriers from the metastable minima towards the low-lying superbasins tend to decrease as the excitation energy increases. The insets illustrate the magnetic configuration in the ground state ($E_0$) as well as the second ($E_2$) and third ($E_3$) excited configurations. The first excited configuration (not shown) is very similar to the ground state [95].

Figure 8

Time dependence of the MV order parameter ${\eta }_{\mathrm{MV}}$ and configurational entropy $S$ in the weakly disordered square-lattice ensemble illustrated in Fig. 1. The simulations describe the RAQ or the RAS at different temperatures $T$ as indicated.

Figure 9

Time dependence of the FM order parameter ${\eta }_{\mathrm{FM}}$ and configurational entropy $S$ in the weakly disordered, triangular-lattice ensemble illustrated in Fig. 1. The simulations correspond to the RAQ or to the RAS at different temperatures $T$ as indicated.

Figure 10

Time dependence of the FM order parameter ${\eta }_{\mathrm{FM}}$ and configurational entropy $S$ in the random ensemble of nonoverlapping NPs illustrated in Fig. 1. Results are given for the RAQ and for the RAS at different temperatures $T$.