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Heavy and Light Monopoles in Magnetic Reversion in Artificial Spin Ice

This work makes a theoretical study of the dynamics of emergent elemental excitations in artificial spin ice systems with hexagonal geometry during the magnetic reversion of the system. The magnetic and physical parameters of the nanoislands that form the array are considered as variables in the study. The parameters considered are: the energy barrier for the inversion of each nanoisland, the magnetic moment of the nanomagnets and the possible disorder in the sample. Our results show that the reversion dynamic presents two distinct mechanisms of magnetic reversion, with different elemental excitations for each mechanism. The first mechanism presents a reversion with the appearance of magnetic monopoles that do not move in the samples (heavy monopoles) and the absence of Dirac chains. In the other mechanism elemental magnetic excitations (light monopoles) appear that move great distances in the sample, giving rise to extensive Dirac chains during the magnetic reversion.


Introduction
Artificial spin ice systems have been intensely studied theoretically and experimentally in recent years [1][2][3][4][5][6][7][8][9] because they present magnetic charge defects with similar characteristics to the magnetic monopoles postulated by Dirac [10]. The main advantage of these systems lies in the fact that studies can be made of the dynamic of these elemental excitations at ambient temperature because the nanoislands have energy reversion barriers above the thermal fluctuations at this temperature. The work of Ladak et al. [7] showed the direct observation of these elemental excitations in a cobalt nanoisland array with hexagonal geometry. Another recent study [8] showed the experimental evidence of magnetic reversion through Dirac chains in an artificial spin ice system with hexagonal geometry. The work of Mengotti et al. [8] includes the statistics that demonstrate the decrease in the dimensionality of the system based on the propagation of Dirac chains, with mobile elemental excitations (monopoles) at their ends. Another recent experimental work [9] verified that magnetic reversion is produced independently in a square network for the two sub-networks that form the array. As well, the same study verified that the reversion is produced through unidimensional cascades of the parallel nanomagnets of the system. Some studies have been published on the effect of the size of the system in artificial spin ice arrays on energy minimization protocols [11] and the effect of the edges on vertice dynamics [12]. A recent study [13] was published on the effects of the size of finite systems, the aspect ratio of the system and the concentration of impurities on the dynamic of emergent monopoles in a hexagonal network during magnetic reversion. This work establishes that the density of mobile monopoles depends on the size, concentration of impurities and shape of the bidimensional system.
The current techniques of synthesizing artificial spin ice systems [1,[7][8][9]] allows for controlling the size and shape of the system and the magnetic properties of the individual nanoislands. However, there is still no fine control of the relationship between the magnetic moments of each nanoisland and the external magnetic field necessary to provoke magnetic reversion in isolated nanoislands, giving rise to the possibility of having a broad spectrum of combinations of these two variables. In this work we studied the dynamic of magnetic reversion in a hexagonal array of nanomagnets in function of the magnetic properties of the individual components of the array. Our results show that there are two different reversion mechanisms with two distinct types of emergent magnetic monopoles, depending on the energy barrier for the reversion of the nanomagnets and on their magnetic moment. In one case monopoles emerge and remain in repose during the reversion. We term this type of excitation "heavy monopoles". In the other case, elemental excitations emerge that move great distances through the sample generating extensive Dirac chains. We term this type of excitation "light monopoles".

Emergent Magnetic Monopoles
The system that we studied is a bidimensional array of magnetic nanoislands on a hexagonal network. Three nanomagnets converge in each vertice of the array (except on the edges). We assign a magnetic load value to each vertice of the array in function of the magnetic poles present in the vertice. The north pole of each nanoisland is associated with a q = +1 charge and the south pole is associated with a q = −1 charge. This implies that each vertice of the array has the following net charge values: and Q 3 = −3. If the system is submitted to a magnetic field along the x-axis, as shown in the upper part of Figure 1, the vertices in the array (except at the edges) acquire a charge of +1 or −1. If the samples are initially magnetized, to the left, for example, and we apply a magnetic field in the inverse sense, elemental excitations appear (magnetic monopoles) in the vertice. The monopoles are defined as in reference [8]. When the samples are totally Of the emergent monopoles only those that maintain at least one of the nanomagnets that converge in the vertice are mobile, with their component over the x-axis directed to the left.
We termed the total quantity of mobile monopoles for each site of the network the "density of mobile monopoles" with the designation σ M . When the conditions Q = ±2, are present in a vertice, but with all the nanomagnets, that converge in the vertice, with the component of the magnetic moment directed to the right, monopoles emerge that are termed "trapped monopoles" [8] and in this work are not considered as monopoles properly but rather as "charge defects". We termed the quantity of these defects per site of the hexagonal network as "defect density" and denote it by ρ. reference [14]. Likewise, the results of reference [13] validate our model using experimental data from the work of Mengotti et al. [8]. In the FCA, the automaton is updated in several steps (three in this case), minimizing the total energy total of the system, evaluated with dipolar interaction.

Heavy and light monopoles
The methodology used in our work consists of fixing the magnetic field H b , necessary to revert an isolated nanoisland and simulate magnetic reversions of the array for different values of m 0 . Just one simulation provided us with information of the hysteresis curve, the total density of impurities ρ and the density of mobile monopoles in function of the applied magnetic field. During the magnetic reversion of the system, the quantity of mobile monopoles increases, reaches a maximum, for a field value close to the coercivity field and then decreases to zero. As relevant data of the simulation, we use the maximum value reached of the density of the σ M mobile monopoles. We term this maximum value σ max . Because we simulated the presence of impurities in the system, with a random procedure we repeated the simulation for each experiment 100 times and determined σ max and the standard error of the measured samples.      Figure 3 shows the hysteresis curve, the maximum density of monopoles and the density of defects for the simulations, which is shown in Figure 4. We can note that coercivity is reached in the heavy monopole regime before it is in the light monopole regime. In the light monopole regime magnetization goes abruptly from a value of -1 al to 0.5. It is interesting to note that this characteristic could be desirable to use this type of system to propagate binary information [15]. A species of plateau is then produced for the three curves.  This behavior can be observed in Figure 5, which shows the magnetization for a simulation with the same parameters as shown in Figure 4, but in the limit case where there are no impurities (s = 0). The same figure shows the heavy monopole regime without impurities.
We can note that when the system reaches the coercitivity value, it remains in an extended plateau. During this plateau, the density of the mobile monopoles is at a maximum. There is then a second abrupt rise and a second plateau. In the final part of the magnetic reversion there is a third abrupt rise, reaching the maximum value of magnetization. It is only in the last rise that the density of defects goes from a low to a maximum value.

Conclusion
Magnetic reversion in artificial spin ice systems with hexagonal geometry presents two different dynamic mechanisms of elemental excitations, depending on the magnetic proper-9 ties of the individual nanoislands that form the array. In one case we can appreciate that reversion occurs with the appearance of a large number of monopoles that however remain static in the sample. In the other case, we can note the low number of monopoles that move great distances in the samples with very extensive Dirac chains. Impurities in the sample condition the number of these heavy and light monopoles. In the heavy monopole regime, the impurities decrease the number of monopoles and in the light monopole regime the opposite occurs, increasing the number of monopoles.