Effect of the dipolar interaction on the dynamic hysteresis properties of 2D-nanodisks: out of plane driving field case

We have systematically investigated the effect of dipolar interaction strength on the dynamical hysteresis behavior of the in-plane uniaxial anisotropic nanodisk system modeled by the classical Heisenberg model under the effect of the time-dependent external out-of-plane periodic magnetic field. Dynamical hysteresis loops, as well as hysteresis quantities (hysteresis loop area, coercive field, remanent magnetization), have been examined both in-plane and out-of-plane magnetization components by means of Monte Carlo simulation based on Metropolis Algorithm. Our simulation results suggest that the response of the in-plane and out-of-plane components of the magnetization have different hysteresis characteristics. For instance, while the out-of-plane component of the magnetization has ordinary dynamically disordered hysteresis curves, bowtie-shaped hysteresis loops have been obtained for the in-plane component of the magnetization. Disappeared dynamical order has been observed with the rising strength of the dipolar interaction.


Introduction
Magnetic nanoparticles are not only the smallest unit of materials, having many of the similar physical properties as bulk materials, but they additionally possess nonlinear size-related magnetic properties between bulk and atomic scales [1].These distinctive characteristics of nanoparticles have drawn attention to them in vast stretches of applications spanning spintronics, hyperthermia-based cancer therapy, magnetic information storage, along with environmental applications [2][3][4][5].From a theoretical point of view, the study of nanoparticles dates back to the pioneering research of Stoner and Wohlfarth [6], Néel [7] and Brown [8,9].After these studies, the interest of researchers in the magnetism of nanoparticles rapidly increased [10][11][12][13][14][15][16][17][18][19].
Although at the atomic scale in magnetic materials, the exchange interaction suppresses the dipolar interaction, at the nanoscale, unexpected and intricate properties for magnetic particles and clustered structures are primarily triggered by the dipolar interaction [20].With many modifications, the dipolar interactions can also decide the system's ground state owing to its long-ranged and anisotropic character [21].For example, it has been demonstrated that in two dimensions, the ground state configuration is ferromagnetic coupling in triangular lattice [22], whereas square lattice exhibits antiferromagnetic coupling [23].
Moreover Kechrakos has emphasized that dipolar interactions can lead to the anisotropic behavior of interested material between in-plane and out-of-plane magnetic properties such as magnetoresistance, blocking temperature, remanent magnetization, and coercive field [24].Yang et al have reported a study on the tunnel magnetoresistance for square anisotropic magnetic nanoparticles coupled through exchange and dipolar interactions [25].Their simulation results revealed that the coercivity value increases with the decreasing magnitude of dipolar interaction and increases with exchange interaction.The effect of the low field on blocking temperature at various sample conditions has been simulated by Lan et al [26].They have found that while there is an increase in blocking temperature at the low field for the diluted sample, there is a slight decrease for the concentrated sample.Two ferromagnetic nanoparticles through long-range dipolar interactions have been investigated by Weizenmann et al [27].Particle-particle and spin-spin computations are the two methods they used to study the effect of dipolar interactions.They have discovered that the outcomes of the two calculations are equivalent if the distance between particles is large enough.However, they have added that, in contrast to the particle-particle dipolar interaction, the spin-spin dipolar energy is high when the particles are sufficiently close.
The magnetic systems under the effect of the periodic time-dependent external magnetic field display many fascinating properties.The dynamical hysteresis loops and dynamically ordered disorders phases are two examples of these exciting properties.According to the relation between the relaxation time of the system and the period of the driving magnetic field, the system can be in the dynamically ordered or disordered phase.The order of the system is given by the dynamical order parameter Q, which is one period average of the magnetization over the driving field.The dynamically disordered phase order parameter is zero, while it is nonzero for the dynamically ordered phase.The distinction between these two phases depends on the ability to follow the magnetization of the system to the time-dependent magnetic field.This following process may have some delay in time.This is the primary cause of the dynamical hysteresis behavior.Dynamical hysteresis behavior has many important applications, such as magnetic hyperthermia treatment.The specific absorption rate (SAR) is this application's key concept, directly related to the hysteresis loop area.It is very important to fabricate magnetic nanoparticles with the desired SAR value.
The dipolar interaction in the magnetic nanoparticles driven by a time-dependent oscillating magnetic field gives rise to unexpected magnetic behaviors.The effect of dipolar interaction on the dynamical hysteresis behavior of a one-dimensional chain [28] and a two-dimensional array of magnetic nanostructure [29] have been studied under different circumstances, which involves dipolar interaction strength, temperature, and timevarying magnetic field by employing the kinetic Monte Carlo technique.Both studies found that hysteresis curves behave as the Stoner-Wohlfarth model without dipolar interaction and thermal fluctuations, as expected.Due to ferromagnetic coupling, the hysteresis loop area of the chain nanoparticle system extends as the dipolar interaction increases.When the dipolar interaction parameter is strong enough, they have discovered that the coercive field and remanence values in two-dimensional array systems exhibit weak temperature dependency.The effect of dipolar interaction on hysteresis behavior has been presented theoretically and experimentally for monodisperse ferromagnetic nanoparticle assemblies in [30].Their simulation results showed that the density of interparticle interactions causes a decrease in magnetic susceptibility and hysteresis losses, which are responsible for the diminished hyperthermia properties.Conde-Leboran et al have examined hyperthermia properties of single-domain nanoparticles with random anisotropy driven into alternating magnetic fields [31].They have indicated that the heating efficiency of magnetic nanoparticles can be explained in a single general scheme by considering the particles' intrinsic magnetic properties and experimental conditions.
Due to the motivations coming from the broad application areas, and theoretical questions about the effect of the dipolar interaction on the dynamical magnetic properties of the magnetic systems, the dynamical properties of the nanoparticle systems with dipolar interaction take attention.Although the theoretical literature on this subject has begun to emerge, it still needs to develop.From these motivations, this work aims to determine the effect of the dipolar interaction on the magnetic nanodisk system with in-plane magnetic anisotropy under the influence of the perpendicular driving sinusoidal magnetic field.For this aim, the paper is organized as follows: In section 2, we briefly present the model and formulation.The results and discussions are presented in section 3, and finally, section 4 contains our conclusions.

Model and formulation
We construct a circular nanodisk on a square lattice with a diameter D = 20.It has number of N = 305 classical spins which are denoted by  s i and components as ( ) s s s , , Let the distance between the neighbor lattice sites be unit magnitude.The Heisenberg Hamiltonian of the system is given by where  r ij is the vector from the lattice site i to j, J is the exchange interaction between the nearest neighbor spins, K x is the anisotropy, H d is the strength of the dipole-dipole interaction, and H(t) is the time-dependent magnetic field.Note that the direction of the magnetic field is z.The first summation is over the nearest neighbor sites, the second and the fourth summations are over all the lattice sites, and the third is over all the pairs in the system.Note, i > j is for avoiding double counting of the pairs.The time-dependent magnetic field is defined by where t is the time, ω = 2π/P is the frequency, P is the period of the magnetic field, and H 0 is the amplitude of the magnetic field.Note that time is measured in terms of the Monte Carlo step (MCS) unit.
In order to obtain observables of the system, we use the Monte Carlo simulation with the Metropolis algorithm [32].We construct the initial configuration by the Marsaglia algorithm [33].We generate a Markov chain about the system by using single spin-flip dynamics.One spin flip attempt for every spin in the system is defined as one MCS.After the transient regime, we collect the values of observables for calculating the averages.
At a specific instant of time, magnetization components (α = x, y, z) can be obtained via After the transient regime of the simulation ( > ¢ t t ), the averages of the magnetization components can be obtained by averaging the process over M periods, The dynamical order parameter defines the dynamical order of the system.Dynamical order parameters related to the magnetization components are one period average over the period of the magnetic field, and they can be calculated via Note that ¢ P is the upper time limit of the transient regime.The dynamical hysteresis loops can be obtained from equation (4) by sketching M α (t), H(t) in one period.The hysteresis loops of the system can be characterized by quantities, namely, the hysteresis loop area (HLA), remanent magnetization (RM), and the coercive field (CF).The HLA is the area that is covered by the hysteresis loop.This is the measure of the energy loss in one period.RM is the magnetization value of the system when there is no magnetic field.The last quantity CF is the field value needed to change the sign of the magnetization of the system.While RM gives the height of the hysteresis loop along the magnetization axis, CF gives the width along the magnetic field axis.

Results and discussion
We use scaled (dimensionless) quantities by using J as a unit of energy The anisotropy of the nanodisk is chosen as x-directed uniaxial anisotropy for simplicity.On the other hand, physical examples related to the anisotropy with controllable strength and orientation can be found [34][35][36].The anisotropy value is chosen as k x =0.5, which is moderate, and the magnetic field period as P = 100.
After the convergence tests of the simulation, we decided to use 200 000 total MCS, which is the first 100 000 MCS for the transient regime of the simulation.In other words, we take averages over the last 100 period of the simulation.
Note that, under the effect of the magnetic field along the z axis, the y component of the magnetization is not responding to the magnetic field.Because of the anisotropy along the x axis, x component of the magnetization, and because of the z directed magnetic field, z component of the magnetization change during the one period of the field.Thus, there are two hysteresis loops for each of the Hamiltonian parameter sets and the temperature.We investigate these two hysteresis loops separately below.

Hysteresis behavior in x direction
The typical hysteresis loops can be seen in figure 1, for selected values of the Hamiltonian parameters and the temperature.As seen in figure 1, the x component of the magnetization responds to the z directed magnetic field according to the values of the Hamiltonian parameters and the temperature.Rising temperature causes a decline in the value of the m x (compare the hysteresis curves related to the temperatures τ = 0.025 and τ = 0.125 figure 1(a)).This is due to rising thermal fluctuations.Secondly, the rising amplitude of the dynamic magnetic field also causes a decline of the value of the magnetization (compare the hysteresis curves related to the temperature τ = 0.025 in figures 1(a) and (c)).This fact comes from the rising amplitude of the z directed magnetic field that forces spins to align with the z direction.Thirdly, the effect of the dipolar interaction strength on the hysteresis is such that rising dipolar interaction raises the m x and curves to shift upper portions of the (h(t) − m x ) plane (compare the hysteresis curves related to the temperature τ = 0.025 in figures 1(a) and (b)).This is because an increment in dipolar interaction dictates the spins align in the nanodisk plane.On the other hand, the bowtie-shaped hysteresis curves take attention for nanodisks under the effect of the zero (as well as weak) dipolar interaction strength and low temperatures.When the strength of the dipolar interaction increases, these bowtie-shaped hysteresis curves give place to lines parallel to the h(t) axis.Our extensive investigation showed that all hysteresis curves of the x component of the magnetization are bowtie-shaped if a hysteresis curve is present.Also, there is no sign change in the m x in one period of the periodic z directed magnetic field.
To investigate hysteresis behavior, we calculate the hysteresis quantities (defined in the section 2) that characterize the loops.Since the sign of the m x is not changed in one period of the field, there is no CF for m x hysteresis loops.The first quantity is HLA; in figure 2, we depict the variation of the HLA with the temperature for selected values of h 0 and h d .While the temperature rises, the HLA increases; after a specific value of the temperature, the HLA suddenly drops to zero.The rate of increase of the HLA decreases with a rising magnitude of the dipolar interaction (e.g.compare curves related to the h d = 0.1 with h d = 0.4 in figure 2(b)).As seen in  figure 2, the maximum value of the HLA occurs at h d = 0.0 for h 0 = 1.0, h d = 0.1 for h 0 = 2.0 and h d = 0.2 for h 0 = 3.0.Increasing dipolar interaction strength lowers the value of the HLA; for higher values of h d , HLA values vanish (e.g.see curves related h d = 0.3, 0.4, 0.5 in figure 2(a)).Moreover, the HLA curves for h d > 0.5 are not depicted because of the same reason.This behavior is consistent with the behavior given in figure 1(a).Since rising h d causes the hysteresis loops from the bowtie-shaped to line parallel to the h(t) axis, it is expected the HLA will decrease.
The second quantity RM is seen in figure 3.As mentioned above, the response of the m x to the z directed field is either a bowtie-shaped hysteresis loop or a line parallel to the h(t) axis, i.e. no hysteresis.In both cases, only one RM value appears.As seen in figure 1, the bowtie-shaped hysteresis intersects at the zero field value.The m x value at this intersection point is just RM value.RM value is also the magnetization value at x direction.
As seen in figure 3, the RM values decreased for increasing temperature, as expected.This is because of the rising thermal fluctuations.For a fixed temperature, rising dipolar interaction strength enhances the value of RM.This is due to the fact that the rising dipolar interaction forces to spins align in the nanodisk plane, and this means enhancing the x component of the magnetization.As amplitude value increases, the ground state RM values decrease (e.g.compare curves related to h d = 0.3 in figures 3(a), (b), and (c)).However, the behavior of the RM curves with the changing h d values is non-monotonous.For instance, while for the h 0 = 2.0, the curves shift upper for rising h d , this is not valid for the h 0 = 1.0.At this value, the h d = 0.5 curve lies below the other curves.Besides, h d = 0.0, 0.1 and h d = 0.5 curves disappear for the amplitude value of h 0 = 3.0.
By comparing curves related to the same Hamiltonian parameters in figures 2 and 3 one can decide whether bowtie-shaped hysteresis occurs or m x does not respond to the field (i.e.lines parallel or on the h(t) axis).If HLA is not zero for selected values of the Hamiltonian parameters and the temperature, the magnetization response to the oscillating field is bowtie-shaped hysteresis.The zero value of HLA and the associated non-zero value of RM mean the magnetization response is just a line parallel to the h(t) axis.The last scenario is zero HLA, and zero RM corresponds to the line at the axis h(t), i.e. dynamically disordered phase.All curves in figure 3 are the sign of the dynamical order in the x direction.Although the time-dependent field along with z direction, the order in x direction comes from the anisotropy in x direction and dipolar interaction.

Hysteresis behavior in z direction
The magnetization of the nanodisk changes in the z direction due to the z directed time-dependent periodic field.This creates hysteresis loops of m z .
To illustrate the impact of temperature, dipolar interaction, and external field amplitude on the hysteresis behavior of magnetization in the z direction, we depicted figure 4 typical hysteresis loops.Due to the increase in thermal fluctuations as the temperature increases, all hysteresis loops tend to decrease in size, which is typical for all magnetic models.As the value of the dipolar interaction parameter increased, the HLA decreased (e.g.compare curves related to the τ = 0.125 in figures 4(a) and (b)).The rising amplitude leads to broadening the hysteresis loop (e.g.compare curves related to the τ = 1.125 in figures 4(b) and (b)).
By comparing hysteresis loops in figures 1 and 4, one can say that, while the x component of the magnetization is always positive (dynamically ordered phase) or zero (dynamically disordered phase), the z component of the magnetization is subject to the magnetization reversal process under the effect of the sinusoidal field.In other words, all curves in figure 1 (except lines at h(t)) dynamically ordered phase, whereas all curves in figure 4 correspond to the dynamically disordered phase.Besides, all hysteresis loops are bowtieshaped in x direction, no bowtie-shaped hysteresis loops for z component of the magnetization.
To obtain general conclusions about the hysteresis characteristics, we depict the variation of the hysteresis quantities with the temperature for selected values of the Hamiltonian parameters.The variation of the HLA with the temperature can be seen in figure 5. Two typical behaviors can be seen in these curves: increment behavior followed by the decrementing behavior (e.g.all curves in figure 5(a)) and starting at the maximum value followed by decrement behavior (e.g.all curves in figure 5(c)).In contrast to the behavior of x HLA, in this case, HLA decreases with rising dipolar interaction.Maximum HLA value obtained for the absence of the dipolar interaction.Increasing amplitude enhances HLA (e.g.compare curves related to h d = 0.2 in figures 5(b) and (c)).Besides, the temperature that peak appears in the curves shifts towards the lower values of the temperatures when h d increases (e.g.compare curves related to h d = 0.0 by the h d = 1.0 in figure 5(a)).
As a function of temperature and the dipolar interaction parameter, the RM value for three distinct applied external field values is shown In figure 6.It is visible that raising the dipolar interaction parameter results in a reduction in the RM value.When the dipolar interaction parameter is equal to zero, RM has the highest value for  the chosen field values, and as the temperature rises, the maximum RM values move nearer to the lowtemperature value.The symmetric character of the two branches of the RM curves about the temperature axis indicates that the nanodisk is in a dynamically disordered phase in the manner of m z .
Finally, variation of the CF with the temperature for selected values of amplitude and the dipolar interaction can be seen in figure 7. The z component of the magnetization reverses its sign in one period of the z-directed magnetic field, in contrast to the x hysteresis loops.As seen in figure 7, the CF value decreases with increasing temperatures.While CF reaches its maximum value in the absence of dipolar interaction, the value of CF decreases significantly as dipolar interaction increases.(e.g.compare curves related to h d = 0.0 with h d = 1.0 in figure 7(a)).This is because increasing the dipolar interaction forces spins to align in the plane of the disk; as a result, changing the sign of the z component of magnetization only requires a small increase in the directed magnetic field's z value.The plot also shows that a stronger CF is required to change the sign of the z component of magnetization of the system as the applied external field amplitude becomes greater.
To relate the hysteresis curves with the spin configurations of the nanodisk we depict two representative snapshots of the spin configurations in figures 8 and 9.In figure 8

Conclusion
We have investigated the effect of dipolar interaction strength on dynamic hysteresis behaviors of in-plane uniaxial nanodisks driven into the time-dependent external magnetic field along with z direction.We have obtained our results as a function of dipolar interaction strength, magnetic field amplitude, and temperature, both m x component and m z component by employing Monte Carlo simulation with Metropolis Algorithm.After the typical hysteresis loops depicted both for m x and m z , the quantities that characterize the hysteresis loops have been investigated in detail, namely HLA, CF, and RM.
The moderate anisotropy in the x direction of the nanostructure is the primary cause of dynamic order in this direction at low temperatures and low h d values.When the temperature and amplitude increase, the spin components are compelled to align with the field in z-direction, as expected.Dynamical ordered phase in  x-direction diminished owing to increment in temperature.Our extensive investigation shows that while the response of the m x to the out-of-plane dynamical field is either bowtie-shaped hysteresis or line with respect to h(t) axis, m z conforms to dynamic disordered phase hysteresis behavior.
In terms of dynamic hysteresis characteristics, the variation of the HLA, RM, and CF related to m z with the temperature exhibits typical dynamically disordered phase behavior, whereas the m x component displays dynamically ordered phase characteristics.The dipolar interaction parameter h d supports the dynamic order of the m x component, compared to the m z component, wherein RM, HLA, and CF values decline with h d .
Our theoretical research, which is presented in this work, may serve as a valuable resource for the effectiveness of applications involving nanoparticles, such as magnetic hyperthermia and data storage.
spin configurations related to the hysteresis curves (for the temperature value of τ = 0.125) in figures 1(a) (for x components of the magnetization) and 4(a) (for z components of the magnetization) can be seen.As seen in figure8(a) almost all of the spins have positive z components and x components, which yield positive m z and m x value for the magnetic field H(P/4) = 0.For the value of t = P/2 (i.e.figure 8(b)) z component of the magnetization is negative while x component is still positive.When the magnetic field gets the value of H(3P/4) = 0 again, alignments almost all of the spins have negative z component (see figure8(c)).Finally, for t = P (i.e.H(P) = 0 figure 8(d)) the system again reaches positive m z value, and m x is again positive.These snapshots are completely consistent with the hysteresis curves seen in figures 1(a) and 4(a).Very weak m x value and weak m z values on the hysteresis curves depicted in figures 1(d) and 4(d) have snapshots given in figures 9(a)-(d).We can conclude that these snapshots are again consistent with the related hysteresis curves.The vortex structure of the m x component (seen figures 9(a)-(d)) is responsible for the almost zero m x on the hysteresis curve seen in figure1(d).By comparing figures 9 with 8 we can conclude that rising dipolar interaction gives rise to vortex behavior in m x .

Figure 8 .
Figure 8. Snapshots of the nanodisk related to the hysteresis curve in figure 1(a) and figure 4(a) at the temperature τ = 0.125 for the time values of (a) t = P/4, (b) t = P/2, (a) t = 3P/4 and (d) t = P.

Figure 9 .
Figure 9. Snapshots of the nanodisk related to the hysteresis curve in figures 1(d) and 4(d) at the temperature τ = 0.125 for the time values of (a) t = P/4, (b) t = P/2, (a) t = 3P/4 and (d) t = P.