IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013 4951

Magnetic Behavior of Ternary Prussian Blue Analog inPresence Single-Ion Anisotropy

Ebru Kış Çam and Ekrem Aydıner

Department of Physics, Dokuz Eylül University, Tr-35160 İzmir, TurkeyDepartment of Physics, İstanbul University, Tr-34134 İstanbul, Turkey

In this work, with help of Monte Carlo simulation method, we have investigated the effects of single-ion anisotropy on magneticproperties of the Prussian blue analog consisting of three different Ising spins ,

, and . We have found that the critical temperature of this system nearly linear changes dependent upon theinteraction ratio for any mixing ratio value, and the critical interaction decreases for increasing values. On the other hand,we have shown that the critical and compensation temperature of the model smoothly increase for increasing values. In addition,we have demonstrated that the magnetic pole inversion can appear, and compensation temperature decreases for the increasingexternal magnetic field, dependent upon some values of the Hamiltonian parameters. As a result, we state that single-ion anisotropy canbe used as a control parameter like mixing rate to arrange the critical and compensation temperature of the Prussian blue analog

.

Index Terms—Compensation temperature, magnetic materials, magnetic recording, Monte Carlo methods.

I. INTRODUCTION

P RUSSIAN blue analogs which types of themodel are new molecular-based magnetic materials

(for more details, see [1]). They have received considerableattention recently since they have potential application intechnology such as magnetooptical recording media [2] orquantum computing devices [3]. Prussian blue analogs, forinstance, [4], which is aternary alloy consisting of three different Ising spins ,

, and . This ternary alloy has anlattice that consists of two interpenetrating face-centered cubic(fcc) sublattices, each one comprising sites. In this latticeformation, the Cr ions of one sublattice are alternately con-nected with the Fe or Mn ions randomly located on the othersublattice with the concentration or , respectively. Hence,it includes both ferromagnetic and antiferromag-netic super-exchange interactions between theneighboring metal ions through the cyanide bridging ligandsdue to their fcc structure.Magnetic Prussian blue analogs show unusual remarkable

properties, such as the magnetic pole inversion [4], [5], the pho-toinduced magnetization [6], [7], the inverted magnetic hys-teresis loop [8], and the multicompensation points [9], [10]. It isknown that the magnetic properties such as compensation tem-perature point and critical temperature point of the consideredPrussian blue analogs can be tuned during a synthesis processby changing the mixing ratio (i.e., concentration) of the dif-ferent incorporated metal ions. Up to now, in order to under-

Manuscript received November 12, 2012; revised February 06, 2013 andApril 01, 2013; accepted April 13, 2013. Date of publication April 18, 2013;date of current version August 21, 2013. Corresponding author: E. Kış Çam(e-mail: ebru.kis@deu.edu.tr).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2258932

stand the unusual magnetic and thermal behavior of the ternaryalloys, which contain three various kinds of magnetic ions withdifferent Ising spins, many models that correspond to ternaryalloys have been investigated by the use of a mean-field (MFT)[11]–[14] or an effective-field theory (EFT) [15], Monte Carlo(MC) simulations [16]–[20], and exact recursion relations onthe Bethe lattice [21], [22].Although many theoretical studies have been devoted to the

Prussian blue analogs, the effect of the single-ion anisotropyon the magnetic properties of these compounds has not beenstudied extensively. Therefore, in this study, we will investi-gate the effects of single-ion anisotropy on the magnetic prop-erties of the ternary alloy model , which is a type ofthe Prussian blue analogconsisting of three different Ising spins , ,

, with the help of the Monte Carlo simulation method.We will also discuss the concentration parameter , interactionparameter , and temperature dependence of thethree-dimensional ternary alloy model.

II. MODEL AND SIMULATION METHOD

The Hamiltonian for the three-dimensional ferro-fer-rimagnetic ternary alloy model whichcorrespond to the Prussian blue analog of the type

can be given as

(1)

where , , andare spin operators which, respectively,

correspond to A, B, and C in the ternary alloy model in the

0018-9464 © 2013 IEEE

4952 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013

form . Also is a random variable that takes thevalue of unity if there is a spin or at the site ; otherwise,is zero. , , and are the numbers of sites occupied

by the A, B, and C ions, respectively. andequal to , where is the total number of sites of the threedimensional lattice. In this Hamiltonian, and are thenearest neighbor interaction parameters. In this study, we havechosen and so that the interaction in theHamiltonian (1) corresponds to ferro-ferrimagnetic the Prussianblue analog of the type .As it is known, and denote single-ion anisotropy andexternal magnetic field, respectively. Hamiltonian (1) repre-sents the ternary alloy model, which consist of twointerpenetrating cubic sublattices. Thus, one can see that firstcubic sublattice is filled by A (Cr) ions and the B (Fe) andthen C (Mn) ions are randomly distributed on the second cubicsublattice with the concentration and , respectively.In order to investigate the effect of the single-ion anisotropy

on the magnetic properties in three-dimensional ternary alloymodel, we simulate the Hamiltonian (1). To simulate thismodel, we employed the Metropolis Monte Carlo simulationalgorithm to the three-dimensional lattice withperiodic boundary conditions for . One of the cubicsublattice is fully decorated with spin , and spins andare randomly distributed on the other cubic sublattice with

the concentration or , respectively. All initial spin statesin the three-dimensional lattice are randomly as-signed. Configurations are generated by making single-spin-flipattempts, which were accepted or rejected according to theMetropolis algorithm. To calculate the averages, data, over20 different spin configuration, is obtained by using 50 000Monte Carlo steps per site after discarding 10 000 steps. Themagnetization per site for this model was computed using thefollowing relations:

(2)

where denotes the number of A ions onone sublattice, and denotes the number of B ions

, whilst represents the number of C ionson the other sublattice.

III. MONTE CARLO SIMULATION RESULTS

In this section, we have given the numerical results obtainedused by Monte Carlo simulation, and we have discussed theeffects of the single-ion anisotropy on the magnetic propertiesof the three-dimensional ferro-ferrimagnetic ternaryalloy. Numerical results and related discussions are presentedas follows.In Fig. 1, the interaction ratio dependence of critical

temperature and single-ion anisotropy dependence of the

Fig. 1. (a) Critical temperatures versus interaction ratio forand several values of concentration. (b) Critical interaction ratio versussingle-ion anisotropy .

critical ratio value of the three-dimensional ternary alloymodel given by Hamiltonian (1) are presented. Fig. 1(a), wehave plotted the critical temperature versus interaction ratiofor different values of the concentration and . As

can be seen from this figure, the critical temperature of themodel nearly linear change depending on the interaction ratio, and the slope of the curves decrease for the increasing value

of . Furthermore, all curves cross each others at the criticalpoint. Nearly linear curves present critical temperature of

the model for the parameters , , and . At critical , allcritical temperature values are equal for the different valuesat fixed value. We say that with variations of from 0 to 1,the system is turn from the ferrimagnetic mixed spin-3/2 andspin-5/2 binary system to the ferromagnetic mixed spin-3/2 andspin-2 binary system. The straight line with correspondsto the critical temperatures of the mixed spin systems with

, , which is not affected by changing the ratio. Throughout this line, the critical temperature value of the

mixed spin-3/2 and spin-2 Ising system is equal to the mixedspin-3/2 and spin-5/2 system at . The critical value of inter-action ratio obtained in this work is approximatelyfor , which is close to other results obtained using theMonte Carlo simulation in [18] and the MFT work

in [12]. To demonstrate and unify the depen-dence of the critical ratio on the single-ion anisotropy ,we plotted the interaction ratio versus single-ion anisotropyin Fig. 1(b). As can be seen from this figure, there is no

symmetry in the dependent upon ; moreover, the valueof the critical interaction ratio for the model decreases forincreasing negative and positive values as to nearly exponen-tial form. The interesting behavior in Fig. 1(b) may probablyemerge from the trend of the contribution of the single-ionanisotropy into the critical temperature of the model for alldifferent values. We can conclude that the critical tempera-ture of the model is mainly determined by interaction betweenspins; however, when the single-ion anisotropy is added intoHamiltonian, critical temperature curves shift up or down forany value. Therefore, the intersection point of curves for allvalues slides left or slides right systematically. Therefore,

when is decreased, the interaction ratio increases; eventhis increasing continues for decreasing in negative region.The concentration ratio and single-ion anisotropy de-

pendence of critical temperature of the three-dimensionalternary alloy model given by Hamiltonian (1) are presented

KIŞ ÇAM AND AYDINER: MAGNETIC BEHAVIOR OF TERNARY PRUSSIAN BLUE ANALOG IN PRESENCE SINGLE-ION ANISOTROPY 4953

Fig. 2. (a) Critical temperatures versus concentration for anddifferent values of interaction ratio . (b) Critical temperaturesversus single-ion anisotropy for different values.

[Fig. 2(a) and (b), respectively]. As can be seen from Fig. 2(a),all critical temperature curves come together at as lin-early increasing for and as linearly decreasing for

. This figure shows that the critical temperature valuesare independent from interaction value at in the case

. Indeed, we know that the interaction rate deter-mines the interaction type in the model Hamiltonian (1). As canbe seen from Fig. 2(b), the critical temperature of the systemgoes to a saturation value for different when the value isincreased from negative to positive values. The other impor-tant result is that the critical temperature curves are systemat-ically shifted dependent upon the concentration. These resultsare compatible with results in Fig. 1. When , the typeof ternary alloy model reduces spin-3/2 and spin-2mixed spin system. This binary system has been investigated inprevious works [23]–[25] in which variation of critical temper-ature with single-ion anisotropy is consistent with present work.In order to demonstrate the temperature dependence of

the total magnetization and to show critical points of thethree-dimensional ternary alloy model, magnetization profilesare given in Fig. 3 for various and fixed values. As canbe seen, the critical and compensation temperature of themodel appear depending on amount of concentration andinteraction ratio . For example, for in Fig. 3(a),the compensation temperature points appear in the interval of

at low temperature values. However, forin Fig. 3(b), only one compensation temperature

point appears in the interval of . Furthermore,Fig. 3(a) shows that the value of the compensation temperaturedecreases for increasing ; however, the value of the criticaltemperature increases for the same values in the case of

. On the other hand, Fig. 3(b) also shows that thevalue of the compensation temperature increases for increasing; however, the value of the critical temperature decreases forthe same values, unlike Fig. 3(a) in the case of .As a result, we can conclude that the total magnetization profileand the compensation and critical temperature points in thisprofile change dependent upon concentration and interactionratio .We know that the Prussian blue analog

is a complexmaterial. The concentration ratio determines magneticproperties of this complex material. In the limit of ,

Fig. 3. Total magnetization versus temperature for several values of concen-tration in the absence of single-ion anisotropy for (a) and (b)

.

Fig. 4. Magnetization versus temperature for andfor and several values of anisotropy .

ternary alloy model reduces to a relatively moresimple AC model; however, in the limit of , it reducesto another simple AB model. It means that in the limit of

, Fe atoms do not appear in the complex material;hence, the Prussian blue analog reduces to a binary system

. However, in the limit of ,Mn atoms disappear in host material; hence, it reduces toanother binary system . It is expectedthat both binary systems show different magnetic behavior.In order to see the behavior of total magnetization of theternary alloy model in the limit values of concentrationdependent upon single-ion anisotropy, we have plotted themagnetization in the limit and in Fig. 4 forvarious single-ion anisotropy values . As can be seen fromFig. 4, the model of the Prussian blue analog showscompletely different magnetic behavior in the limitand . First of all, critical temperature values and themagnitude of the magnetization of both systems are differentfor the same value of the anisotropy at a fixed value of . Thisfigure clearly shows that the critical temperature of each binarysystem increases for increasing single-ion anisotropy values.These results are compatible with the results in Fig. 2(b).It is known that the compensation temperature points of

the ternary alloy model appear depending on and values.However, we show in Fig. 3(a) and (b) that the system maynot have a compensation point for all combinations of and. Since in this study we focus on the effects of the single-ion

4954 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 9, SEPTEMBER 2013

Fig. 5. (a) Total magnetization versus temperature for fixed ,, and different values. (b) Total magnetization versus temperature for

fixed , , and different values. (c) Dependence ofcompensation and critical temperatures on single-ion anisotropy for fixed

and values.

anisotropy on the system, we have discussed in the Fig. 5 thebehavior of the compensation temperature depending on forvarious values of and . Therefore, in order to demonstratethe single-ion anisotropy dependence of the compensationtemperature of the model, we have plotted total magnetiza-tion versus temperature for different , , and values inFig. 5(a) and (b). As can be seen from these figures, whileand are responsible for appearing in different magnetizationprofiles, as in Fig. 3(a) and (b), then the single-ion anisotropyis responsible for determining the values of the compensationtemperature points. Indeed, single-ion anisotropy in the Hamil-tonian (1) affects the position of the compensation point aswell as in the case of the critical temperature. In fact, one cansee from Fig. 5(c) that the compensation temperature increasesfor example at fixed and when the valueof the is increased. Similar behavior can be observed in thecritical temperature for the same values of the parameter and. In Fig. 5(c), behavior of the compensation (and the critical)

temperature depending on single-ion anisotropy are consistentwith the results in Figs. 1(b), 2(a), and 4. Fig. 5 also showsthat system has two compensation points for the same negativevalues at fixed and . In addition, we

note that the magnetic pole inversion can be observed in thisPrussian blue analog, as seen in Fig. 5(b), dependent uponsome parameter values. For example, this inversion appears for

and values when around .Finally, we have also considered the external magnetic field

on the critical behavior of the three-dimensional ternary modelin Fig. 6 for arbitrary fixed values of and in the case of

. As expected, the values of the critical temperature forand in Fig. 6(a) and for andin Fig. 6(b) increase for increasing values of the ex-

ternal field. Besides, the critical temperature point disappearsin the temperature axis for a large value of the magnetic field

Fig. 6. (a) Effects of magnetic field on compensation and critical temperaturesfor and values. (b) Effects of magnetic field on compensa-tion and critical temperatures for and values. (c) Compen-sation point versus magnetic field for and values.

as in Fig. 6(b). Similar behavior has been found in [26]. How-ever, unlike critical temperature, the values of the compensa-tion temperature points decrease for increasing magnetic fieldsin both figures for different values of the parameters. Further-more, the multicompensation point does not appear when themagnetic field is applied. This behavior is presented in Fig. 6(c)for and .

IV. CONCLUSION

In this work, by employing the Monte Carlo simulationmethod to the three-dimensional ferro-ferrimagneticternary alloy model, which is represented by Hamiltonian (1),we have investigated the magnetic properties and effects of thesingle-ion anisotropy on the magnetic properties of the Prussianblue analog consisting ofthree different Ising spins , , and .We have found that the critical temperature of the modellinearly changes dependent upon the interaction ratio forany concentration value, and the critical interaction ratiodecreases for increasing values. On the other hand, we haveshown that the critical and compensation temperature of themodel smoothly increase for increasing values. In addition,we have demonstrated that the magnetic pole inversion canappear and compensation temperature can decrease forincreasing external magnetic dependent upon some values ofthe Hamiltonian parameters. As a result, we have concludedthat single-ion anisotropy may play an important role as acontrol parameter like mixing ratio to arrange the critical andcompensation temperature points of the Prussian blue analog

.

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