Nonequilibrium Multiple Transitions in the Core-shell Ising Nanoparticles Driven by Randomly Varying Magnetic Fields
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Nonequilibrium Multiple Transitions in the Core-shellIsing Nanomagnets Driven by Randomly VaryingMagnetic Fields
Erol Vatansever , ∗ and Muktish Acharyya , † Department of Physics, Dokuz Eylul University, TR-35160, Izmir-Turkey Department of Physics, Presidency University, 86/1 College Street, Kolkata-700073, India
E-mail ∗ :[email protected] † :[email protected] Abstract:
The nonequilibrium behaviour of a core-shell nanomagnet has been studied by Monte-Carlo simulation. The core consists of Ising spins of σ = 1 / S = 1. The interactions within the core and in the shell are considered ferromagnetic but theinterfacial interaction between core and shell is antiferromagnetic. The nanomagnetic system iskept in open boundary conditions and is driven by randomly varying (in time but uniform overthe space) magnetic field. Depending on the width of the randomly varying field and the temper-ature of the system, the core, shell and total magnetization varies in such a manner that the timeaverages vanish for higher magnitude of the width of random field, exhibiting a dynamical symme-try breaking transitions. The susceptibilities get peaked at two different temperatures indicatingnonequilibrium multiple transitions. The phase boundaries of the nonequilibrium multiple transi-tions are drawn in the plane formed by the axes of temperature and the width of the randomlyvarying field. Keywords: Magnetic hyperthermia, Core/shell nanoparticles, Monte Carlo simu-lation, Dynamic phase transitions Introduction
The random field Ising model (RFIM) including quenched random magnetic field has attracteda considerable interest in the last four decades [1, 2]. Despite of its simplicity, many problems instatistical physics and condensed matter physics can be studied by means of RFIM. Experimentalexamples include diluted antiferromagnets Fe x Zn − x F [3, 4], Rb Co x Mg − x F [5, 6], Co x Zn − x F [6] in a magnetic field and colloid-polymer mixtures [7, 8]. From the theoretical point of view,thermal and magnetic phase transition properties of the static RFIM have been investigated bya wide variety of techniques such as Molecular Field Theory (MFT) [2, 9–13], Effective-FieldTheory (EFT) [14–20] and Monte Carlo (MC) simulations [21–28]. These theoretical studies showthat different random-field distributions may lead to different physical outcomes, and thereby theexistence of a quenched impurity in magnetic field has an important role in material science. Theyalso indicate that our understanding of equilibrium critical phenomena associated with the RFIMhas reached a point in which the satisfactory results are available. The readers may refer to [29]for a detailed review of recent developments in the RFIM. However, far less is known for thephysical mechanisms underlying the out of equilibrium phase transitions of the Ising systems inthe presence of a randomly varying magnetic field.Stationary state properties of a randomly driven Ising ferromagnet has been investigated bybenefiting from the Glauber dynamics [30]. Based on the MFT calculations, it is found that thesystem shows a first order phase transition related to dynamic freezing. Paula and Figueiredo [31]have attempted to study dynamical behavior of the Ising model in a quenched random magneticfield, with a bimodal distribution for the random fields. The dynamics of the system has beendefined in terms of Glauber type stochastic process. It is obtained that the magnetic field valuesleading to first order transitions are greater than the corresponding fields at equilibrium. Oneof us [32] has focused on the two dimensional Ising model in the presence of randomly varyingmagnetic field to understand how the randomly changing magnetic affects the physical propertiesof the system. By benefiting from both MFT and MC simulations, it is reported that the time-averaged magnetization disappears from a nonzero value depending upon the values of the widthof randomly varying field and the temperature. Nonequilibrium phase transition properties in athree-dimensional lattice system with random-flip kinetics have been elucidated in detail by usingMC simulations [33]. One of the remarkable findings is that the system displays a first order phasetransitions located at low temperature region and large disorder strengths, denoting a nonequi-librium tricritical point at a finite temperature. Moreover, nonequilibrium phase transitions andstationary-state solutions of a three-dimensional random-field Ising model under a time-dependentperiodic external field have been investigated within the framework of EFT with single-site corre-lations [34]. The amplitude of the external is chosen such that they will be according to bimodaland trimodal distribution functions. It is shown that the system explicits unusual and interest-ing behaviors depending on type of the magnetic field source. The readers may refer to [35] fora review of the dynamic phase transitions and hysteresis phenomena observed in different kindsof magnetic systems. To the best our knowledge, all of the studies mentioned above have beendedicated to bulk materials in the presence of randomly varying magnetic field. There are, how-ever, only a few studies regarding the random magnetic field effects on the core/shell nanoparticlesystems including surface and finite size effects [36–38]. It is a well known fact that when thephysical size of an interacting magnetic system reduces to a characteristic length, surface effectsbegin to show themselves on the system, and as a result of this, some interesting behaviours canbe observed, differing from those of bulk systems [39]. Two notable examples are Co / CoO [40, 41]2nd Mn / Mn O [42], where the physical properties of the nanomagnets sensitively depend on itsown chemical characters. In this paper, we will consider the core/shell ferrimagnetic nanoparticlesystem driven by a randomly varying magnetic field. More specifically, our motivation is to under-stand how randomly changing field affects the thermal and magnetic properties of a nanomagnet,by means of an extensive MC simulation. In a nutshell, our simulation results indicate that thepresent system exhibits multiple dynamic phase transitions depending on the chosen values of theexternal magnetic field width and the temperature of the system.The rest of the paper is organized as follows: In Sec. (2), we present the model and simulationdetails. The numerical findings and discussion are given in Sec. (3), finally Sec. (4) is dedicatedto a brief summary of our conclusion. We consider a cubic with thickness ( L = L c + L sh ) ferrimagnetic nanoparticle composed of a spin-1/2 ferromagnetic core which is surrounded by a spin-1 ferromagnetic shell layer. Here, L, L c and L sh are the total, core and shell thicknesses of the particle, respectively. At the interface, we definean antiferromagnetic interaction between core and shell spins. The nanoparticle is exposed to atime dependent randomly varying magnetic field. The Hamiltonian describing our system can bewritten as follows: H = − J c X h ij i σ i σ j − J sh X h kl i S k S l − J int X h ik i σ i S k − h ( t ) X i σ i + X k S k ! (1)here σ i = ± / S k = ± , J c and J sh denotes the ferromagnetic spin-spin interactions in the core andshell components of the system while J int is the antiferromagnetic interaction at the interface ofthe particle. The symbol h· · · i represents the nearest neighbor interactions in the system. h ( t ) isthe randomly varying magnetic field (in time but uniform in space). The time variation of h ( t )can be given as follows [32]: h ( t ) = ( h r ( t ) for t < t < t + τ r ( t ) is the random number which is distributed uniformly between -1/2 and 1/2. Thereby,the field h ( t ) varies randomly from − h / h / τ Z t + τt h ( t ) dt = 0 (3)For the sake of simplicity, J sh is fixed to unity throughout the simulations, and the remainingsystem parameters are normalized with J sh . In order to study thermal and magnetic properties ofthe system, we employ Monte Carlo simulation with single-site update Metropolis algorithm [43,44]on a L × L × L simple cubic lattice. We apply boundary conditions such that they are free in alldirections of the particle. We note that such type of a boundary condition is an appropriate choicefor considered finite small system. Let us briefly summarize the simulation protocol we follow here:The system is in contact with an isothermal heat bath at a reduced temperature k B T /J sh , where k B is the Boltzmann constant. Spin configurations were produced by selecting the spins randomly3hrough the lattice, and the single-site update Metropolis algorithm was used for each consideredspin. This process was repeated L times, which also defines a MC step per site.Using the above scheme we simulated the nanoparticle system. 100 independent initial config-urations have been generated to get a satisfactory statistics. For each initial spin configuration,the first 5 × MC steps have been discarded for thermalization process, and the numerical datawere measured during the following 5 × MC steps. Based on our detailed test calculations, itis possible to say that this number of transient steps is found to be enough for thermalization ofthe particle. We have verified that higher values of transient steps does not change the outcomesreported here. Error bars have been obtained using the jackknife method [43]. The main quantityof interest is the time-averaged magnetization, which is defined as follows: Q α = 1 τ Z t + τt m α ( t ) dt (4)where α = c, sh and T corresponding to the core and shell components of the particle and theoverall of the system. m α ( t ) is the time-dependent magnetization, which can be given as follows: m c ( t ) = 1 N c N c X i σ i , m sh ( t ) = 1 N sh N sh X i S i , m tot ( t ) = N c m c ( t ) + N sh m sh ( t ) N c + N sh (5)here N c = L c and N sh = L − L c denotes the total number of spins lying in the core and shellparts of the system, respectively. We select the number of core and shell spins as N c = 10 and N sh = 14 − , such that it allows us to create a core-shell nanocubic particle with shell thickness L sh = 2. We also define two additional order parameters for the core and shell layers of the particleas follows: O c = N c N c + N sh Q c , O sh = N c N c + N sh Q sh . (6)To estimate the pseudo-critical transition temperature for a finite-size system as a function ofthe external field, it is useful to focus on the scaled variances of the dynamic order parameters [45], χ α = N α (cid:0) h O α i − h| O α |i (cid:1) , (7)where N tot = N c + N sh is the total number of spins in the system. In addition to the Eq. (7), wealso measure the scaled variance of the total energy (which can be considered as heat-capacity inequilibrium system) of the particle including the cooperative part as follows: χ E = N tot (cid:0) h E i − h E i (cid:1) , (8)where E is the time-averaged energy of the particle per site, which is defined as follows: E = − τ N tot Z t + τt J c X h ij i σ i σ j + J sh X h kl i S k S l + J int X h ik i σ i S k dt. (9) Figure 1 shows the time series of core m c ( t ), shell m sh ( t ) and total m T ( t ) magnetizations at aconsidered value of temperature and for two different values of randomly field width h/J sh . All4 h ( t ) (a) m ( t ) t (MCS) m c (t) m sh (t) m T (t) -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2000 4000 6000 8000 10000-0.6-0.4-0.20.00.20.40.6 h ( t ) (b) m ( t ) t (MCS) m c (t) m sh (t) m T (t) Figure 1: (Color online) Monte Carlo results of the time variations of randomly changing mag-netic field and core, shell and total magnetizations. All results are obtained for the temperature k B T /J sh = 2 .
0. Magnetic field sources, (a) h/J sh = 1 . h/J sh = 4 .
0, are opened at t = 5000 MCs. This diagram demonstrates the dynamical symmetry breaking associated to thetransition.numerical results given here are obtained for k B T /J sh = 2 .
0. It is clear from the Figure 1(a)that when the external field width is relatively small (for example h/J sh = 1 .
0) time dependentmagnetizations can not give a response to changing magnetic field simultaneously. Thereby, thesystem remains in a dynamically symmetry broken phase where the instantaneous magnetizationsoscillate around a non-zero value. If h/J sh gets bigger, time dependent magnetizations tend to givean answer to randomly changing magnetic field. Therefore, the system remains in a dynamicallysymmetric phase where the magnetizations oscillate around zero value, as depicted in Figure 1(b).The results aforementioned here indicate that dynamics of the core/shell nanomagnet sensitivelydepends on the chosen external field width value. It should be noted that such type of an observa-tion has been reported for the bulk spin-1/2 Ising models driven by a randomly changing magneticfield in both two and three-dimensional space [32, 34].As displayed in Figure 2, in order to get a better understanding of the effect of randomly varyingmagnetic field width on the critical properties of the considered nanomagnet, we present thedynamic phase diagram plotted in a ( k B T C /J sh − h/J sh ) plane for a selected combination ofsystem parameters such as J int /J sh = − . J c /J sh = 0 .
75. Here, T C means the pseudo-criticaltemperature. Transition temperatures are extracted from the thermal variations of variances ofthe core, shell and total dynamic order parameters and internal energies. One of the outstandingresults is that there is a multiple phase transition region including two branches in the system upto a particular magnetic field width. In this region, when the temperature increases starting fromthe relatively lower values, it has been found that the core magnetization first exhibits a phasetransition. This corresponds to the first branch of the multiple transition line. Furthermore, if thetemperature is increased further, then the overall particle shows a transition between ordered anddisordered phases for a fixed value of h/J sh . Such kind of a transition corresponds to the secondbranch of multiple transition line. Our Monte Carlo simulation findings show that the multipletransitions observed here strongly depend on the chosen applied field width value. As explicitlyseen from the phase diagram, transition temperatures decrease when the h/J sh value gets bigger.This is because the energy contribution coming from the Zeeman term to the total energy increaseswith an increment in h/J sh . Thereby, the phase transition lines gets shrink. We note that all ofthe transitions found here are second-order transition, i.e., there is no first order phase transitionin the system, indicating a tricritical point. Similar kind of a multiple transition has also been5 .0 0.5 1.0 1.5 2.0 2.5012345678 Total particle Core partJ int /J sh =-1.5J c /J sh =0.75 k B T c /J sh h / J s h Figure 2: (Color online) Dynamic phase boundary of core-shell nanocubic system in a ( k B T c /J sh − h/J sh ) plane for selected values of J c /J sh = 0 .
75 and J int /J sh = − .
5. The magnetic field variesrandomly between − h/ J sh and h/ J sh . Transition points are obtained from the peak positions ofthe variances of the core, shell and total magnetizations and specific heat curve of the nanoparticleas a function of temperature.reported in Ref. [46] where thermal and magnetic properties of a classical anisotropic Heisenbergmodel driven by a polarized magnetic field are investigated by means of MC simulations.In Figures 3(a-c), we depict the effect of randomly changing magnetic field width on the core,shell and total dynamic order parameters as functions of the temperature, corresponding to thedynamic phase diagram given in Figure 2. As an interesting observation, we can see from totalmagnetization curves plotted in Figure 3(c), they exhibit a temperature induced maximum, whichis strongly depend on the chosen applied field width. At this point we note that in the bulkferrimagnetism of N´eel [47, 48], it is possible to classify the magnetization profiles based on thetotal magnetization behaviors in certain categories. According to this nomenclature, the systemshows a P-type behavior for some selected values of h/J sh such as 1 . .
0. It also meansthat the results given here are in common at various types of magnetic materials, which are notonly dependent on their size. As the h/J sh gets bigger, for example h/J sh = 7 .
0, P-type behaviorobserved here tends to disappear, leading to a Q-type behavior. In Figures 3(d-f), we presentthe thermal variations of the variances of core, shell and total dynamic order parameters forthe same system parameters used in Figures 3(a-c). As clearly seen from the figures that whenthe temperature reaches to the transition temperature, the related variances give rise to show apeak behavior, which is also a fingerprint of a continuous transition. Peak positions in x -axisare sensitively dependent on the studied external field width such that they tend to decreasewith an increment in h/J sh . Moreover, it is obvious from the Figure 3(d) that the variance ofcore magnetization shows two peaks for some selected values of h/J sh such as 1 . .
0. Thefirst peak located at the relatively lower temperature region originates from the order-disordertransition of the core part of the nanomagnet whereas the second one is a result of the strongcoupling between core and shell layers.In Figure 4, we give the thermal variations of the internal energy and corresponding variancesfor the same applied field width values used in Figure 3. When the temperature begins to decrease6 .000.050.100.150.20 0.00.10.20.30.40.50.60.7 0.00.10.20.30.40.50.60.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51E-40.0010.010.11 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51E-40.0010.010.1110100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51E-40.0010.010.1110100 h/J sh = 1.0 4.0 7.0 | O c | (a) (b) | O s h | (c) | O T | (d)k B T/J sh c c (e)k B T/J sh c s h (f)k B T/J sh c T Figure 3: (Color online) Effects of the randomly changing magnetic field width on the core (a),shell (b) and the total magnetizations (c), and their corresponding variances (d), (e) and (f),respectively. Here, different symbols denote the varying magnetic values.starting relatively higher values, numerical values of the internal energies are decreased for allvalues of h/J sh . In addition to this general trend, as shown in Figure 4(a), it is possible to saythat a sudden decrement is observed when the temperature reaches to the relevant critical point,as in the case of equilibrium phase transitions [43, 44]. More specifically, figure 4(b) displays thevariances of the internal energies as functions of the temperatures. As in the case of the variances ofthe dynamic order parameters discussed above, when the temperature reaches to the critical point,they tend to show a clear peak behavior, indicating a second order phase transition. Their positionson the x -axis are sensitively dependent on the considered applied field width values. In accordancewith the previously observed results from the variances of the dynamic order parameters, thepseudo-critical temperatures start to shift to the lower regions with increasing h/J sh values. Inthe inset, we also plot the low temperature behavior of the χ E for two considered applied fieldwidth h/J sh such as 1 . .
0. It is clear from the figure that, it shows a clear peak treatmentsupporting the transition of the nanomagnet belonging to only the core part, in the vicinity of therelated transition point.As a final investigation, we represent the spin configurations of the midplane cross-sections ofthe nanoparticle in the x − y plane in Figure 5. All snapshots are captured for the h/J sh = 1 .
0. Bybenefiting from the calculations mentioned above, the pseudo-critical is obtained as k B T C /J sh ≈ .
36 for h/J sh = 1 .
0. In order to show the influences of the temperature on the spin-snapshots,three different temperature values are taken into consideration, being smaller (a), equal (b) andlarger than the critical temperature. As seen clearly from the figure 5(a) that the spins in the coreand shell parts of the system are opposite to each other due the existence of an antiferromagneticexchange coupling betwen core and shell layers of the nanomagnet. However, when taking a look7 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.2 E k B T/J sh h/J sh = 1.0 4.0 7.0(a) (b) c E k B T/J sh c E k B T/J sh Figure 4: (Color online) Thermal variations of the internal energy (a) and the correspondingvariances (b). The curves are obtained for three different values of the randomly changing magneticfield: h/J sh = 1 . , . .
0. Inset given here represents the low temperature regions of thevariances of the internal energies for two considered values of h/J sh : 1 . . k B T c /J sh , the spins in the system try to follow randomlychanging magnetic field. Therefore, they are aligned with respect to each other almost randomly[see Figure 5(c)]. y (a) y 2 4 6 8 10 12 14 y -0.5 0.5 Figure 5: (Color online) The spin configurations of the midplane cross-sections of the nanoparticlein the x − y plane. They are taken for three different temperature values: (a) k B T /J sh = 0 . .
36 (in the vicinity of transition point) and (c) 4.0 (disordered region) with h/J sh = 1 .
0. All snapshots are captured at t = 75 × MCs. Here, yellow, black, red, green andcyan colors correspond to the spin values of 0 . , − . , , −
1, respectively.
The equilibrium behaviours of the core-shell nanomagnets have been investigated widely and wellunderstood. However, the nonequilibrium responses of the core-shell nanomagnets have not yet8een investigated. Here we have studied the nonequilibrium responses of Ising core-shell nanomag-nets driven by randomly varying (in time but uniform over the space) magnetic field. The timedependent magnetic field keeps the core-shell nanomagnetic particle far from equilibrium. Hence,the responses of the system is truely of nonequilibrium type. The ferromagnetic Ising (spin-1/2)core is covered by ferromagnetc Ising (spin-1) shell. However, the interfacial interaction betweencore and shell is being considered as antiferromagnetic. We have calculated the time averaged coremagnetization, shell magnetization and the total magnetization and their variances (susceptibilityin a sense) and studied those as functions of the temperature with field as parameter. We haveobserved, for relatively higher value of the temperature and the width of the randomly varyingmagnetic field, the system shows a dynamical symmetric behaviour. In this case, the instantaneusmagnetizations are found to fluctuate symetrically around zero value. On the other hand, for rel-atively lower values of the temperature and the field width a dynamically symmetry broken phaseis observed. Where the instantaneus magnetizations fluctuates asymmetrically around zero value.These are demonstrated in figure-1. Obviously, a nonequilibrium phase transition is observed inassociation with a dynamical symmetry breaking. In the above mentioned dynamically symmetricphase, the time averaged magnetizations (serving as the order parameters in the present study)vanishes, indicating a nonequilibrium disordered phase. Whereas, in the dynamically asymmetricphase, the time averaged magnetizations, acquires nonzero values, indicating the nonequilibriumordered phase. The transition temperatures were estimated from the value of the temperatureswhere the variances of those magnetizations get peaked (believed to be diverging in the thermo-dynamic limit). It was observed, that the variance of total magnetization gets peaked at highertemperature and that of core magnetization gets peaked at some lower temperature. This is a sig-nature of nonequilibrium multiple phase transition observed in the driven core-shell nanomagneticparticle.Collecting all these nonequilibrium transition temperatures the comprehensive nonequilibriumphase diagarm is drawn in the plane consists with the axes of the temperature and the width ofthe randomly varying magnetic field.It would be quite interesting to study such behaviours in the case of Ising core-shell nanomag-nets where the values, of spins in the core and that in the shell, differes largely. That will affectcertainly on the difference in the region bounded by core-transition line and the shell-transitionline in the comprehensive phase diagram. What will be the effects of dilution here ? If the coreand shell are diluted by nonmagnetic impurities (with different concentrations in core and shell),how does it affect the phase boundaries of the multiple phase diagram. Last but not the least, howsensitive the nature of the distribution (i.e.,uniform, bimodal, normal) of the randomly varyingmagnetic fields, would also be an interesting matter to be studied. We have plans to study theseand the results will be reported elsewhere.
Acknowledgements
The numerical calculations reported in this paper were performed at TUBITAK ULAKBIM HighPerformance and Grid Computing Center (TR-Grid e-Infrastructure).
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