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Journal of Magnetism and Magnetic Materials 321 (2009) 3396–3401

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

Technol

E-m

journal homepage: www.elsevier.com/locate/jmmm

A Monte Carlo study of the thermodynamic properties of a quasi-one-dimensional organic polymer ferromagnet

S.J. Luo a, Z.D. He a,b,�, Q.L. Jie b

a School of Science, Hubei University of Automotive Technology, Hubei 442002, Chinab College of Physical Science and Technology, Wuhan University, Wuhan 430074, China

a r t i c l e i n f o

Article history:

Received 25 September 2008

Received in revised form

21 May 2009Available online 13 June 2009

Keywords:

Thermodynamics

Monte Carlo

Organic ferromagnet

Heisenberg model

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.06.017

esponding author at: School of Science, Hub

ogy, Hubei 442002, China.

ail address: hezedong@tom.com (Z.D. He).

a b s t r a c t

The thermodynamic properties of a quasi-one-dimensional organic ferromagnet at different

temperatures and in different applied magnetic fields have been investigated by means of the

Heisenberg model combined with the Monte Carlo method. The results indicate that a peak in the

magnetic susceptibility is obtained at low temperatures. Furthermore, the effect of dimerization on the

magnetic properties has also been studied. We find that the dimerization suppresses the magnetization

in this model. The ferromagnetic couplings between the side-free radicals stabilize the ferromagnetism

and increase the apparent Curie temperature.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Since the discovery of macromolecule ferromagnetic (FM)materials in the 1980s, it is no longer believed that organicmaterials cannot be ferromagnetic, and the interest in studyingorganic ferromagnets, especially in pure organic ferromagnetswhich have only s/p electrons, has been aroused [1–5]. Severalorganic ferromagnets have been successfully synthesized, suchas poly-BIPO [1,4-bis(2,2,6,6-tetramethyl-4-piperidyl-1-oxyl)],m-PDPC (m-polydiphenyl-carbene), p-NPNN (p-nitrophenyl ni-tronyl nitroxide) and so on. However, until now, there has notbeen a clear understanding on the ferromagnetism mechanism inthese organic ferromagnetic materials.

Ovchinnikov and Spector [6] previously proposed a simplifiedmodel to describe a quasi-one-dimensional organic polymerferromagnet as shown in Fig. 1. The model consists of a mainchain with side-free radicals connected to it, where the mainchain consists of carbon atoms each with a p-electron, and Rdenotes a side-free radical that contains an unpaired electron.Ovchinnikov treated the unpaired electrons of the system aslocalized spins. Antiferromagnetic (AFM) couplings between allthe nearest neighbor p-electrons and the unpaired electrons atside-radicals exist. The electronic spin structure is shown in Fig.1(b). Because of the AFM interactions between all the nearestneighbor p-electrons and the unpaired electrons, the unpairedelectrons at side-radicals will be in an arrangement parallel to

ll rights reserved.

ei University of Automotive

each other to form a ferromagnetic chain. In the model, R can beorganic side-free radicals of different chemical structures. Hence,it is an important model that describes one kind of pure organicpolymer ferromagnet. The magnetic properties and electronicstructure of the ground state of the organic ferromagnets havebeen extensively studied. Rapid advancements have been made inthis field, and have further stimulated us to study the excited stateof an organic magnetic system.

In this article, we utilize the Monte Carlo (MC) method to studythe spin structure, the magnetization and the magnetic suscept-ibility at different temperatures and in different applied fields,the effect of the dimerization and the interactions betweenthe side-free radicals upon the magnetic properties of a quasi-one-dimensional chain. The remainder of this paper is organizedas follows: the model Hamiltonian and computational methodare given in Section 2. In Section 3, the spin structure, themagnetization, the magnetic susceptibility, the thermodynamicproperties and the dimerization are discussed. Conclusions aregiven in Section 4.

2. Model and method

The trans-polyacetylene with side-radicals is used as a modelof a quasi-one-dimensional ferromagnet as illustrated in Fig. 2.The most simple Hamiltonian which can be applied to explain ourresults is H ¼ �

PijJijSi Sj � h

PiS

zi , where the first term expresses

the exchange interaction energy, Si and Sj are the spins of thelattice site i and j, respectively, and Jij denotes the exchangeintegral between the nearest sites i and j. The second term

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J3

J1J2

J4Z

Y X

Fig. 2. The diagram of the quasi-one-dimensional organic ferromagnet model. J1

(J2), J3 and J4 are the couplings between the nearest lattice sites of the main chain,

the lattice site and the side-free radical connected to the site, and the nearest free-

radicals, respectively.

Fig. 3. (a) The spin structure at different temperatures. The arrows indicate the

classical directions of the spins on the 40 unit cells system with side-free radicals

at three different temperatures. The first, second and third row arrows simulate

the spins of the carbon atoms without and with the side-free radicals connected to

them, and the side-free radicals, respectively. (b) The magnetization and magnetic

susceptibility curves versus the reduced temperature kBT/J0 for h ¼ 0.1,

J1 ¼ J2 ¼ J3 ¼ �1.

R R R

π -electrons

Unpaired electrons at side radicals

The simplified structure

The spin structure

Fig. 1. The simplified structure and spin structure of the quasi-one-dimensional

organic ferromagnet.

S.J. Luo et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3396–3401 3397

represents the Zeeman energy, where h is an applied magneticfield whose direction is fixed along the Z-axis. In the presentsimulations, we only consider nearest neighbor interactions anduse periodic boundary conditions to erase the influence of theboundary effect upon the simulated calculations.

The spins of the carbon atoms on the main chain and the side-free radicals are supposed to be S ¼ 1

2. In the simulation, we useclassical spins to substitute for the spins 1

2, the classical spins arerepresented by vectors. The exchange interactions along the mainchain are AFM J1o0 and J2o0, and the exchange interactionsbetween the carbon atom of the main chain and its nearestneighbor free radical are also AFM J3o0, but the exchangeinteractions between the side-free radicals are ferromagneticJ440. We consider a periodic chain of 40 carbon atoms each witha p-electron. The side-free radicals, each containing an unpairedelectron, connect with the odd carbon atoms. Additionally, thespins of the p-electrons and the side-free radicals take newdirections with the aid of Metropolis importance sampling [7,8],and the accepting probability of the new structure can be written

as: p ¼1 DEo0

e�DE=kBT DE40

(, where DE denotes the energy differ-

ence between the new and old states of the system. T is thetemperature, and kB the Boltzmann constant.

We adopt the Metropolis method to sample. This methodensures that the microscopic state that contributes the mostto the statistic mean value will not be omitted. The equilibriumstate of the system is achieved by using a Markov process. Thepresent calculations show that the system has achieved anequilibrium state after much less than 50,000 MC steps, so weabandon the first 50,000 MC steps, then calculate the statisticmean value of the last 100,000 MC steps using the relations asfollows:

MðTÞ ¼/P

iSzSN

XðTÞ ¼/M2S�/MS2

NT;

where M is the magnetization of the system, and Sz the projectionon the Z-axis of the spin moment and X the magnetic susceptibility.

3. Results and discussion

3.1. The spin structure at different temperatures

Because of the fluctuation of the spins in the numericalcalculations, the calculation of the magnetization without anapplied magnetic field is difficult. For this reason, in the presentcalculations, weak applied magnetic fields are added to thesystem. We choose J1 ¼ J2 ¼ J3 ¼ �1, and neglect the exchangeinteractions between the nearest side-free radicals, that is J4 ¼ 0.In the simulations, we adopt the unit exchange interaction J0 asthe reduced unit for energy and applied magnetic field, and kBT/J0

as the dimensionless temperature. The spin structure of the quasi-one-dimensional organic ferromagnet at different temperatures issimulated for h ¼ 0.1 as illustrated in Fig. 3(a). In Fig. 3(a), thearrows of the first, second and third row simulate the spinsof the carbon atoms without and with the side-free radicalsconnected to them, and the side-free radicals, respectively. Itshows that the spin structure of the quasi-one-dimensionalferromagnet exhibits behavior ranging from order to disorderwith increasing temperature. At very low temperatures, the spindirections tend to the ordered arrangement. It begins to becomedisordered as the temperature rises due to the strenuous thermal

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motion. As the temperature increases above a certain criticaltemperature, the spin structure of the system becomes completelydisordered, and the magnetization reduces quickly. Theoretically,the temperature corresponding to the peak of the magneticsusceptibility is the phase-transition temperature. It is disputablewhether one- and two-dimensional systems exhibit phasetransitions [9]. So we use the word ‘‘apparent’’ to refer to thephase transition and Curie temperature in future references. FromFig. 3(b), we find that the peak in the magnetic susceptibility atT ¼ 0.5 is suggestive of magnetic ordering occurring below thistemperature.

3.2. The effect of the side-free radicals on the magnetic properties

When manufacturing this kind of organic ferromagnet,the side-free radicals can be different chemical structures. Theintensity of AFM interactions J3 between the side-free radicals andthe main chain is dependent on the side-free radicals. The effectof the side-free radicals upon the magnetic properties and thephase transition is very important for the molecular designof the ferromagnetic material. In our calculations, assuming that

Fig. 4. (a) The reduced Energy E/J0 curves versus the reduced temperature kBT/J0 with th

with the different J3. (c) The magnetic susceptibility curves versus the reduced temper

changing the free radical R does not affect the bond lengthbetween the carbon atoms along the main chain, we chooseJ1 ¼ J2 ¼ �1, h ¼ 0.2, and J3 is set as �0.1, �0.5, �1, �2 and �3,respectively. As shown in Fig. 4(a), the total energy of the systemincreases as the temperature rises. The total energy decreasesabruptly with the decreasing temperature. This phenomenonshows that the system becomes ordered below the apparent Curietemperature which is defined as the center of the peak in themagnetic susceptibility. The temperature dependence of themagnetization for the different AFM interactions J3 is shown inFig. 4(b). When T ¼ 0 K, the magnetization M ¼ 0.5 indicates thatthe system becomes ferromagnetically ordered, and all of thespins of the side-free radicals have the same direction. FromFig. 4(c) we can find that with increasing |J3|, the magneticsusceptibility of the system reduces in the whole temperatureregion, and the susceptibility peak temperature correspond-ing to the magnetic susceptibility maximum moves towardshigher temperatures. This reveals that the larger the |J3| is, thehigher the apparent Curie temperature becomes. Therefore, theantiferromagnetic interactions between the side-free radicalsand the main chain are of great significance in obtaining high-temperature organic ferromagnets.

e different J3. (b) The magnetization curves versus the reduced temperature kBT/J0

ature kBT/J0 with the different J3.

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3.3. The effect of applied magnetic fields upon the magnetic

properties

The properties of the magnetic material in the appliedmagnetic fields are always topics of concern in both the magnetictheory and practical application. Now, we turn our attention to theeffect of the applied magnetic fields upon the magnetic proper-ties. We plot the magnetization and the magnetic susceptibilityagainst temperature in Fig. 5(a) and Fig. 5(b), respectively. In ourcalculations, we choose J1 ¼ J2 ¼ J3 ¼ �1, h ¼ 0, 0.1, 0.2, 0.5 and 1.As shown from Fig. 5(a), the magnetization decreases abruptly inthe low-temperature region and the weak applied magnetic fields,and drops slowly with increasing applied magnetic field. This can

Fig. 5. (a) The magnetization curves versus the reduced temperature kBT/J0 in different

temperature kBT/J0 in different applied magnetic fields.

Fig. 6. (a) The magnetization curves versus the reduced temperature kBT/J0 when the

temperature kBT/J0 when the dimerization occurs.

be attributed to the splitting of the Zeeman energy that can leadto increasing magnetization.

The effect of the applied magnetic fields upon the magneticsusceptibility is illustrated in Fig. 5(b). This figure shows that whenthe applied magnetic field h ¼ 0, the magnetic susceptibilityapproaches infinity as the temperature approaches 0.5. Whenapplying a magnetic field to the system, the magnetic susceptibilityhas a maximum because of the finite energy gap between theground state and the excited state caused by the applied magneticfield. Furthermore, with increasing h, the maximums of themagnetic susceptibility reduce and the corresponding apparentphase-transition temperature shifts to the high temperature, whichindicates that the system is ferromagnetic.

applied magnetic fields. (b) The magnetic susceptibility curves versus the reduced

dimerization occurs. (b) The magnetic susceptibility curves versus the reduced

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Fig. 7. (a) Energy curves versus the reduced temperature kBT/J0 with the different J4. (b) The magnetization curves versus the reduced temperature kBT/J0 with the different

J4. (c) The magnetic susceptibility curves versus the reduced temperature kBT/J0 with the different J4.

S.J. Luo et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3396–34013400

3.4. The effect of dimerization upon the magnetic properties of the

system

It is well known that the equal site separation quasi-one-dimensional crystal is energetically unstable at low temperaturesand that dimerization will occur due to strong electron–phononinteractions. The electron–phonon interactions are not included inthis model. However we attempt to simulate their effects on themagnetic properties by splitting the Heisenberg coupling from thedifferent J1 and J2. Perfect dimerization leads to alternating‘‘single’’ (long) and ‘‘double’’ (short) bonds. We use different J1

and J2 to indicate the occurrence of dimerization of the system,where the larger J indicates the shorter bond. To show the effect ofthe dimerization degree upon the magnetic properties clearly,we fix h ¼ 0.2, J3 ¼ �1 and change J1 and J2 in the simulations.As shown in Fig. 6(a), with increasing dimerization themagnetization in the low-temperature region monotonicallydecreases and becomes steeper with a corresponding increase inthe difference between J1 and J2. However, the magnetization isnegligibly affected in the high-temperature region because thetemperature is above the apparent Curie temperature. On the

other hand, Fig. 6(b) shows that with increasing dimerization, themagnetic susceptibility peak gradually increases, and thecorresponding apparent phase-transition temperature decreaseswith an increase in the difference between J1 and J2. From Fig. 6(a)it is found that dimerization suppresses the magnetization.

In the above calculations, the couplings between side-freeradicals have been neglected for J4 ¼ 0. We furthermore considerthe ferromagnetic couplings between the side-free radicalsfor J4 ¼ 0.2, and repeat the above calculations. In the end, thecalculations give the same conclusions.

3.5. The effect of couplings between the side-free radicals on the

ferromagnetism of the system

Although the couplings between the side-free radicals areweak, they always exist in a practical one-dimensional material.The ferromagnetism of the system is mainly due to the parallelarrangement of spins of the side-free radicals with unpairedelectrons. These ferromagnetic couplings may play an importantrole in the ferromagnetic ground state of the system. Thus, we

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consider the effect of the couplings between the side-free radicalsupon the magnetic properties of the system. In the calculations,we set J1 ¼ J2 ¼ J3 ¼ �1, h ¼ 0.2, J4 ¼ 0, 0.2, 0.6, 1.0 and 2.0,respectively.

As shown in Fig. 7(a), the ground-state energy of the systemdecreases with increasing J4, so the coupling J4 is beneficial to theferromagnetic ground state of the system. Moreover, it is shownin Fig. 7(b) that the magnetization has been enhanced by theferromagnetic coupling J4. Fig. 7(c) shows the temperaturedependence of the magnetic susceptibility for different J4. Thetemperature dependence of the magnetic susceptibility curveshifts to higher temperatures with increasing J4. It is well knownthat the temperature corresponding to the peak of magneticsusceptibility is the apparent phase-transition temperature. It canbe concluded that the ferromagnetic interactions between theside-free radicals stabilize the ferromagnetic ground state. Hence,they effectively enhance the ferromagnetism of the system andcause the apparent Curie temperature to increase, which is ingood agreement with the results of Refs. [10,11].

4. Conclusions

In this paper, we have studied a quasi-one-dimensional organicferromagnet with the MC method. We showed that the systemexhibits apparent spontaneous magnetization and ordered spinstructure at low temperatures. The strong antiferromagneticinteractions between the side-free radicals and the site on themain chain are beneficial and lead to an increase in the apparentCurie temperature. Dimerization makes the apparent Curietemperature of the system decrease. The ferromagnetic interac-tions between the side-free radicals effectively stabilize the

ground state of the system. The ferromagnetic couplings betweenthe side-free radicals enhance the ferromagnetism of the systemand increase the apparent Curie temperature. All these conclu-sions can aid in the optimal design of molecular magneticmaterials. Additionally, we have also studied the dimensionaleffect by considering the interchain couplings. Because theinclusion of the dimensional effect did not lead to any newconclusions gained using this model, we did not include thedetailed calculations for the dimensional effect in this paper.

Acknowledgements

The authors acknowledge the support from the ExcellentMiddle Age and Youth People Science and Technology CreativeTeam Foundation of the Educational Department of the HubeiProvince no. T200805.

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