Journal of Magnetism and Magnetic Materials 329 (2013) 125–128

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Journal of Magnetism and Magnetic Materials

0304-88

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n Corr

E-m

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Review

Bimodal random crystal field distribution effects on the ferrimagnetic mixedspin-1

2 and spin-32 Blume–Capel model

Ali Yigit a,n, Erhan Albayrak b

a C- ankırı Karatekin University, Department of Physics, 18100 C- ankırı, Turkeyb Erciyes University, Department of Physics, 38039 Kayseri, Turkey

a r t i c l e i n f o

Article history:

Received 26 May 2012

Received in revised form

18 September 2012Available online 16 October 2012

Keywords:

Mixed spin model

Random crystal field

Effective-field theory

Compensation temperature

53/$ - see front matter Crown Copyright & 2

x.doi.org/10.1016/j.jmmm.2012.10.011

esponding author.

ail address: ayigit80@karatekin.edu.tr (A. Yigi

a b s t r a c t

The effects of bimodal random crystal field on the phase diagrams and magnetization curves of

ferrimagnetic mixed spin-1/2 and spin-3/2 Blume–Capel model are examined by using the effective

field theory with correlations for honeycomb lattice. The phase diagrams are obtained on the (D,kT=9J9),(D,Tcomp) and (p,kT=9J9) planes for given values of p and D, respectively. The model exhibits only the

second-order phase transitions as in the Blume–Capel model with constant crystal fields. In addition, it

was found that the model presents one or two compensation temperatures for appropriate values of

random crystal field for given probability in contrast to constant crystal field case. Therefore, it is

shown that the random crystal field considerably affects the thermal variations of net and sublattice

magnetizations.

Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction

In the last two decades, the mixed spin Ising models have beenstudied extensively. The mixed-spin Ising models in comparison tothe systems with one type of spins present less translationalsymmetry and may exhibit a new type of critical temperature calledas the compensation temperature (Tcomp) at which the total magne-tization vanishes below the critical temperatures [1]. Its existenceprovides interesting possibilities in technological applications suchas in the thermomagnetic recording and magneto-optical readoutapplications [2]. Furthermore, they have also been proposed aspossible models to describe a certain type of molecular-basedmagnetic materials that are studied experimentally [3].

On the other hand, the inclusion of a crystal field (CF) into themodel considerably affects its critical behavior. If it is strongenough, the energy difference between the split levels is large. Inthis case, it is energetically more favorable to put as many electronsinto the lower energy level before one starts to fill the higherenergy level. Filling all the orbitals in the lower level means thatthey have to be paired up (within each orbital) with opposite spins.The effect of pairing electrons with opposite spins makes noaddition to the total spin which results in a lower-spin state. Thusits competition with the bilinear interaction parameter leadsto first-order phase transitions. One usually considers a CF whichis constant throughout the lattice. There are only a few works about

012 Published by Elsevier B.V. All

t).

the spin-1/2 and spin-3/2 Blume–Capel (BC) model with constantCF in the literature: The phase diagrams and magnetization curvesof the model were investigated by using the effective field theory(EFT) with correlations [4], the critical properties of the model wasexamined in terms of recursion relations on the Bethe lattice [5],the transverse Ising model was studied within framework EFT withcorrelations on the honeycomb lattice [6] and on the square lattice[7,8] and the kinetic behaviors of the model were investigatedwithin mean field approach (MFA) [9]. Lately, the random crystalfield (RCF) with some probability distribution instead of a constantone, i.e. p¼1, become popular to investigate as well. The RCF effectsare considered, since the CF may be altered with some internal orexternal effects such as the lattice distortions, impurities, defects,etc. The impurities and defects are known to play important roles inthe existence of first-order phase transitions [10], therefore, themixed spin-1/2 and spin-3/2 BC model with RCF is going to bestudied in here. It should be mentioned that there are only twoworks with the mixed spin-1/2 and spin-3/2 model with RCF. Thefirst one analyzes the model in terms of recursion relations on theBethe lattice [11] and the other one approaches to the problemwithin the MFA [12]. The latter uses a different RCF distributionthen the one that we consider in here and from [11].

Therefore, the aim of the present work is to investigate the RCFeffects by using the EFT with correlations on the phase diagramsand magnetization curves of the ferrimagnetic mixed spin-1

2 andspin-3

2 BC model. The rest of this work is set up as follows: Thenext section is devoted to the explanation and formulation of themodel and the last section includes our illustrations and findingsin addition to a brief summary and conclusions.

rights reserved.

Fig. 1. The thermal variations of sublattice magnetizations for the mixed spin-1/2

and spin-3/2 BC model with p¼0.5 and D¼ 3:0, 0.0, �0.75, �1.0 and �3.0 for the

honeycomb lattice.

A. Yigit, E. Albayrak / Journal of Magnetism and Magnetic Materials 329 (2013) 125–128126

2. Formulation

The Hamiltonian of the mixed spin-1/2 and spin-3/2 BC modelis given as

H¼�JX/i,jS

Sisj�X

j

Djs2j , ð1Þ

where each Si located at site i represents a spin-12 which takes the

discrete values 7 12 and each sj located at site j represents a spin-3

2

which takes the discrete values 7 32, 7 1

2. The first sum runs overall pairs of nearest-neighbor (NN) sites, J is the bilinear interactionparameter between the NN spin pairs and which is taken Jo0 forthe ferrimagnetic case and Dj is the site dependent CF. The latteris distributed in a bimodal fashion according to

PðDjÞ ¼ pdðDj�DÞþð1�pÞdðDjÞ, ð2Þ

where Dj ¼Dj=9J9. This random distribution of the CF either turnson or turns off the CF randomly for given probabilities p and 1�p,respectively, on the sites of lattice.

The sublattice magnetizations of mixed spin-1/2 and spin-3/2BC model is obtained in here in terms of the EFT with correlationswhich is widely used in the study of the Ising models. It was firstintroduced by Honmura and Kaneyoshi [13] and Kaneyoshi et al.[14]. In this work, the magnetizations of spin Si and spin sj forhoneycomb lattice are obtained within the framework of EFT withcorrelations and with the usage of the general but approximatevan der Waerden identity [15]. Therefore, the magnetizations ofthe sublattices are given as,

Ma ¼/SiS

¼ coshðJZrÞþMb

ZsinhðJZrÞ

� �3

f AðxÞ9x ¼ 0 ð3Þ

and

Mb ¼/sjS

¼ cosh 12 Jr� �

þ2Masinh 12 Jr� �� �3

f BðxÞ9x ¼ 0: ð4Þ

In addition a parameter q is defined as

q¼ Z2 ¼/ðSiÞ2S

¼ cosh 12 Jr� �

þ2Masinh 12 Jr� �� �3

GðxÞ9x ¼ 0, ð5Þ

where r¼ @=@x is the differential operator and the functions fA(x)for spin-1/2 and, fB(x) and G(x) for spin-3/2 are found as

f AðxÞ ¼1

2tanh

b2

x

� , ð6Þ

f BðxÞ ¼3 sinhð32 bxÞþsinhð12bxÞ expð�2bDiÞ

2 coshð32 bxÞþ2 coshð12bxÞ expð�2bDiÞ, ð7Þ

GðxÞ ¼9 cosh 3

2bx� �

þcosh 12bx� �

expð�2bDiÞ

4 cosh 32bx� �

þ4 cosh 12bx� �

expð�2bDiÞ, ð8Þ

where b¼ 1=kBT, kB is the Boltzmann constant, and T is theabsolute temperature.

We should note that after expanding the right-hand sides ofEqs. (3)–(5), they are solved numerically within an iterationscheme. The readers can find all the details of the calculationsin [6,16].

Lastly, the compensation temperatures Tcomp is the tempera-ture at which total magnetization vanishes and then given by

MTotal ¼ðMaþMbÞ

2� ð9Þ

In the next section, the topologically distinct phase diagramsand our findings are presented which are obtained by studyingthe thermal variations of magnetizations for given values of ourmodel parameters.

3. The thermal variations, phase diagrams and conclusions

In this section, we present the thermal variations of magneti-zations, i.e. Ma, Mb and Mt, therefore, the phase diagrams of themodel. The latter are obtained on the three possible planes for thehoneycomb lattice, i.e., on the (D,kT=9J9), (D,Tcomp) planes forgiven values of probability p between 0 and 1 and on the(p,kT=9J9) plane for given values of D. The solid lines are thesecond-order phase transition lines which separate the ferromag-netic (F) regions from the paramagnetic (P) ones.

In Fig. 1, we illustrate the temperature changes of Ma ¼M1=2 andMb ¼M3=2. It is obtained for various values of D when p¼0.5. At thesecond-order phase transition temperatures, these magnetizationcurves combine at temperatures labeled with Tc for each D. It isclear from the figure that, the ground-state (GS) values of M3=2 arethe usual ones, i.e., 1/2 for D¼�1:0 and �3.0 and 3/2 for D¼ 0:0and 3.0, in addition to the unusual one, i.e. 1 for D¼�0:75 (see alsoFig. 2 of [4]). Especially, at the critical value D¼�0:75, M3=2 ¼ 1,which indicates that in the GS the spin configuration of sj in thesystem consist of the mixed phase; the sj are randomly in thesj ¼73=2 or sj ¼ 71=2 state with equal probability. It is alsoobvious that the D drives the spins to the lower spin values forsome values of p as it becomes increasingly negative.

An important explanation is now in order: If one uses the EFTwith van der Waerden identity such as in Ref. [17] which actuallyleads to MFA [18] since it neglects all the correlations betweenthe spins and in which case one cannot observe the GS with 1 [19](call it as I. Approximation). But when one introduces thegeneralized van der Waerden identity such as in [6,16] (call itas II. Approximation) then the correlations are somewhat intro-duced into the model which leads to GS value of 1 as in this work.

The model displays one or two compensation temperatures,TcompoTc , as shown in Fig. 2. Only one Tcomp ¼ 0:447 is seen whenp¼0.9 and D¼�2:5. But the model displays two Tcomp’s which arefound at Tcomp1 ¼ 0:493 and Tcomp2 ¼ 0:79 for p¼0.4 and D¼�1:53. We have only found the compensation temperatures whenthe sublattice magnetizations have the same GS’s that is in this case is1/2. Our obtained Fig. 2 are compatible with Fig. 6(b) and (c) of [20].We also note the compensation temperature is not observed in theconstant CF case [4] in opposition to our RCF work.

The first phase diagrams of the model is given on the (D,kT=J)plane for given p values with the intervals of 0.1 as illustrated inFig. 3. The p¼0 case has the same effect as D¼ 0:0, since the

Fig. 2. The thermal variations of net magnetizations for p¼0.4 and D¼�1:53, and

in the inset for p¼0.9 and D¼�2:5 for the honeycomb lattice.

Fig. 3. The phase diagrams on the ðD,kT=9J9Þ planes for given values of p for the

honeycomb lattice.

Fig. 4. The phase diagrams on the ðp,kT=9J9Þ planes for given values of D for the

honeycomb lattice.

Fig. 5. The phase diagrams on the ðD,TcompÞ planes for given values of p for the

honeycomb lattice.

A. Yigit, E. Albayrak / Journal of Magnetism and Magnetic Materials 329 (2013) 125–128 127

Hamiltonian is left with only the interaction parameter J whichenters into the formulation as a scaling parameter. This leads to astraight Tc-line as expected. The p¼1 case corresponds to mixedspin-1/2 and 3/2 Ising model with all the spin-3/2 sites are underthe influence of the same crystal field [4–6], i.e., the constant CFcase. The model is the RCF model for the intermediate values ofthe probability, i.e. for 0opo1. As seen from the figure, the Tc-lines start from higher temperatures at lower negative D for lowerp, i.e. the lowest temperature is seen for p¼1. Then, as D increasestowards zero the Tc-lines increase monotonically at lower p’s butthe increase gets sharper for higher p’s. This is expected asexplained above for the cases of p¼0 and p¼1. All the Tc-linesfor all p combine at D¼ 0 as expected which corresponds to p¼0case. The Tc-lines start spreading up with the further increase of Dand they are seen at higher temperatures for higher p for D40:0.As we mentioned above, the temperatures of the Tc-lines becomeconstant as D-�1 and D-1 at lower and higher temperatures,respectively, for pa0. The largest temperature difference isseen for the p¼1 case and there is no temperature differencefor the p¼0 case. We should mention that this II. Approximationdoes not lead to any first-order phase transitions in disagreementwith I. Approximation, which also leads to higher Tc’s.The ordered-phases ð�1=2;3=2Þ for D4�0:75 and ð�1=2;1=2Þ

Do �0:75 are seen at T¼0 K. When D¼�0:75, M3=2 ¼ 3=2changes to the unusual phase M3=2 ¼ 1. One may expect thatthere has to be some transition lines, either first- or second-order,separating these two phases at nonzero temperatures. We shouldnote that in literature, especially in spin systems, there may occurcases without any transition lines between the two differentphases as also given in Refs. [4,21,22].

The other phase diagrams are obtained on the (p,kT=J) planesfor given values of D as shown in Fig. 4. p¼0 and D¼ 0 cases havethe same temperature, i.e. all the lines start from the sametemperature 1.252 which is less than the temperature 1.546 of I.Approximation [19]. The Tc-lines below D¼ 0:0 decrease in tem-perature with decreasing negative crystal field values and aboveD¼ 0:0 the lines increase in temperature with the increasingpositive crystal field values for higher pa0’s. The Tc-lines of II.Approximation for lower negative D values are much smoother incomparison to the I. one where they are a little wiggly.

The last phase diagrams, i.e. Fig. 5, are calculated on (D,Tcomp)planes for some values of p. First, we note that the model does notpresent any compensation behavior when p¼1 which correspondto non-random case [4,5]. All the Tcomp lines emerge from thecorresponding Tc lines then they go down and bending towards leftwith the decreasing negatively D values. They are monotonically

A. Yigit, E. Albayrak / Journal of Magnetism and Magnetic Materials 329 (2013) 125–128128

increasing at higher p’s and the increase become sharper as the Dincreases positively. Lastly, we note that these lines terminate atlower temperatures for lower p’s.

In this work, the effects of bimodal random crystal field on thephase diagrams and magnetization curves of ferrimagnetic mixedspin-1/2 and spin-3/2 Blume–Capel model are examined by usingthe EFT with correlations for honeycomb lattice. The phasediagrams are obtained on the (D,kT=9J9), (D,Tcomp) and (D,kT=9J9)planes for given values of p and D, respectively. It was found thatthe model exhibits only Tc-lines as in the Blume–Capel modelwith constant CF. One or two compensation temperatures areobserved for appropriate values of random crystal field for givenprobability in contrast to constant crystal field case. Therefore,the random crystal field considerably affects the thermal varia-tions of net and sublattice magnetizations. We have also indicatedthe possible differences of the usage of EFT with two differentapproximations, i.e. I. and II. Approximations. Finally, we can alsosummarize a few of our findings as follows: (i) We have not foundTcomp’s for po0:39. (ii) The critical lines separating the F phasefrom P phase consist only from second-order phase transitionswhich contradicts with the results of [11]. (iii) Randomness donot have any effects on the degree of phase transitions in thiswork which again contradicts with [11]. Finally, (iv) the random-ness of CF does not induce any tricritical points which is samewith the non-random CF case.

References

[1] L. Neel, Annals of Physics 3 (1948) 137.[2] M. Mansuripur, Journal of Applied Physics 61 (1987) 1580.

[3] O. Khan, Molecular Magnetism, VCH Publishers, New York, 1993;T. Mallah, S. Thie�baut, M. Vardegauer, P. Veillet, Science 262 (1993) 1554;H. Okawa, N. Matsumoto, H. Tamaki, M. Ohba, Molecular Crystals and LiquidCrystals 233 (1993) 257;C. Mathonie�re, C.J. Nutall, S.G. Carling, P. Day, Inorganic Chemistry 35 (1996)1201;M. Drillon, E. Coronado, D. Beltran, R. Georges, Journal of Chemical Physics 79(1983) 449.

[4] T. Kaneyoshi, M. Jascur, P. Tomczak, Journal of Physics: Condensed Matter 4(1992) L653.

[5] E. Albayrak, A. Alc- ı, Physica A 345 (2005) 48.[6] W. Jiang, G-Z. Wei, Z-H. Xin, Physica A 293 (2001) 455.[7] N. Beneyad, A. Dakhama, Klumper, J. Zitttartz, Annalen der Physik 5 (1996) 387.[8] A. Bobak, M. Jurcisin, Journal de Physique IV France 7 (1997) C1.[9] B. Deviren, M. Keskin, O. Canko, Journal of Magnetism and Magnetic Materials

321 (2000) 458.[10] G.-P. Zheng, M. Li, Physical Review B 66 (2002) 054406.[11] E. Albayrak, Chinese Physics B 21 (2012) 067501.[12] L. Bahmad, M.R. Beneyad, A. Benyoussef, A.El. Kenz, Acta Physica Polanica A

119 (2011) 740.[13] R. Honmura, T. Kaneyoshi, Journal of Physics C: Solid State Physics 12 (1979)

3979.[14] T. Kaneyoshi, I.P. Fittipaldi, R. Honmura, T. Manabe, Physical Review B 29

(1981) 481.[15] T. Kaneyoshi, Zeitschrift fuer Physik B 71 (1988) 109.[16] Y.-Q. Liang, G.-Z. Wei, G.-L. Song, Physica Status Solidi B 241 (2004) 1916.[17] M. Ertas, B. Deviren, M. Keskin, Journal of Magnetism and Magnetic Materials

324 (2012) 704.[18] S. Mukhopadhyay, I. Chatterjee, Journal of Magnetism and Magnetic Materi-

als 270 (2004) 247.[19] A. Yigit, E. Albayrak, submitted for publication.[20] B. Deviren, M. Keskin, O. Canko, Physica A 388 (2009) 1835.[21] A. Bobak, Z. Feckova, M. Zukovic, Journal of Magnetism and Magnetic

Materials 323 (2011) 813.[22] X. Zhang, X.-M. Kong, Physica A 369 (2006) 589.