Journal of Magnetism and Magnetic Materials 324 (2012) 2434–2441

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Journal of Magnetism and Magnetic Materials

0304-88

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des Sys

Ismail,

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

Hysteresis loops and susceptibility of a transverse Ising nanowire

S. Bouhou a, I. Essaoudi a, A. Ainane a,b,c,n, M. Saber a,b, F. Dujardin c, J.J. de Miguel d

a Laboratoire de Physique des Materiaux et Modelisation, des Syst�emes, (LP2MS), Unite Associee au CNRST-URAC 08, University of Moulay Ismail,

Physics Department, Faculty of Sciences, B.P. 11201 Meknes, Moroccob Max-Planck-Institut fur Physik Complexer Systeme, Nothnitzer Str. 38, D-01187 Dresden, Germanyc Laboratoire de Physique des Milieux Denses (LPMD), Institut de Chimie, Physique et Materiaux (ICPM), 1 Bd. Arago, 57070 Metz, Franced Dpto. Fısica de la Materia Condensada, C-3 and Institut of Materials Sciences ‘‘Nicolas Cabrera’’, Univ. Autonoma, Cantoblanco, E-28049, Madrid, Spain

a r t i c l e i n f o

Article history:

Received 13 October 2011

Received in revised form

20 February 2012Available online 17 March 2012

Keywords:

Nanowire core/shell

Hysteresis loop

Susceptibility

53/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.jmmm.2012.02.104

esponding author at: Laboratoire de Physique d

t�emes, (LP2MS), Unite Associee au CNRST-UR

Physics Department, Faculty of Sciences, B.

12 535536808.

ail address: ainane@pks.mpg.de (A. Ainane).

a b s t r a c t

In this work, the magnetization, susceptibility, and hysteresis loops of a magnetic nanowire are

described by the transverse Ising model using the effective field theory within a probability distribution

technique. The effects of the exchange interaction between core/shell and the external fields on the

magnetization and the susceptibility of the system are examined. Some characteristic phenomena are

found in the thermal variations, depending on the ratios of the physical parameters in the shell and

the core.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, the synthesis of magnetic nanoparticles andmagnetic nanostructures with a fascinating variety of morpholo-gies, ranging from complex bulk structures to a wide diversity oflow-dimensional systems, has been intensely studied [1–3]. Infact, quasi-one-dimensional (quasi-1D) magnetic nanowires havedrawn a lot of research interest due to their physical propertiesand potential applications in magnetic recording, spintronics,optics, sensors and thermoelectronics devices, due to the follow-ing fact that much attraction is directed to their applications innanotechnology and their unique phenomena when the size of amagnetic material decreases to a nanometer scale [4,5]. Recently,it was reported that Ni nanowires can be used in bio-separationand have higher yields compared with magnetic polymer micro-spheres [6,7]. This provides a new chance for magnetic nanowiresapplied in biomedical fields. Among lots of preparation methods,porous anodic aluminium oxide (AAO) template is the mostcommon one due to the low-cost and convenient preparationprocess. So far, a lot of metal, alloy and multilayer magneticnanowires based on AAO templates have been successfullyfabricated. Some magnetic properties, such as anisotropic mag-netization, Giant Magnetoresistance effect and magneto-opticalproperties, have been widely investigated [8–10]. It was found

ll rights reserved.

es Materiaux et Modelisation,

AC 08, University of Moulay

P. 11201 Meknes, Morocco.

that these properties strongly depend on the diameter and theaspect ratio of nanowires. Theoretically, the core–shell modelhas been accepted to explain many characteristic phenomena innanoparticle magnetism [11–18]. The same concept has beenapplied to the investigations of magnetic nanowires and nano-tubes. In particular, the magnetic properties of a cubic Isingnanowire [19], which consists of a ferromagnetic spin-1/2 coreand a ferromagnetic spin-1 shell coupled with an antiferromag-netic interlayer coupling Jinter to the core, have been investigatedby the use of the Monte Carlo method. In a recent publication[20], the phase diagrams and the temperature dependences ofmagnetization in a Ising nanowire (or nanotube) with dilutedsurface have been examined using the core–shell concept and thetwo theoretical frameworks, namely the mean field approxima-tion (MFA) and the effective field theory (EFT) with correlation[21,22]. The EFT corresponds to the Zernike approximation [23],which is superior to the standard MFA, since some parts of spincorrelations are automatically included [22]. Somesimilaritiesbetween the nanowire and the nanotube in the phase diagramshave been discussed using the two theoretical frameworks.

Magnetic hysteresis and magnetic relaxation are two relatedphenomena, present in all stages of the development of magnet-ism and different branches of technology. Ferromagnetic hyster-esis remains, however, a paradigm. There are many descriptionsof the phenomenon available [24]. The formalism developed in[25] is based on physical premises concerning domain wallmovement in soft magnetic materials. It is interesting from boththe theoretical and practical points of view. It offers some insightsinto the details of the magnetization processes. To the best of ourknowledge, whether triple hysteresis loops exist or not in such

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–2441 2435

a system has not been discussed. The triple hysteresis looppatterns may have potential applications in multi-state memorydevices.

The aim of this work is to study the magnetization, suscept-ibility, and hysteresis loops of a magnetic nanowire described bythe transverse Ising model using the EFT with a probabilitydistribution technique which takes into consideration the fluctua-tions of the effective field [26]. The effects of the core/shellexchange interaction and the external magnetic field on themagnetization of the system are examined. The outline of thispaper is as follows: in Section 2, we present the formulation of thesystem in terms of the EFT. The detailed numerical results anddiscussions are presented in Section 3 and finally we give a briefconclusion.

2. Model and formalism

We consider a magnetic nanowire consisting of surface shelland core where each site is occupied by an Ising spin as depictedin Fig. 1. The Hamiltonian of the system is given by

H¼�Js

X/ijS

szis

zj�Jc

X/nmS

szns

zm�Jcs

X/imS

szis

zm

�Os

Xi

sxi�Oc

Xm

sxm�h

Xi

szi þX

m

szm

!, ð1Þ

where szi and sx

i denote the z and x components of a quantumspin s! operator of magnitude s¼ 71=2 at site i. The first threesums are carried out only over nearest-neighbor pairs. Js, Jc and Jcs

are the exchange interaction constants between two nearest-neighbor magnetic spins in the surface shell, the core andbetween the core and surface shell respectively. Os and Oc

represent the transverse fields at the surface shell and in corerespectively and h is the longitudinal magnetic field. Each spin isconnected to the two nearest neighbor spins on above and belowsections.

In the single site cluster theory, attention is focused on thecluster comprising just a single selected spin and the neighboringspins with which it directly interacts. For the derivation, let usseparate the Hamiltonian into two parts one denoted by Hi which

Js

Jcs

Jc

Fig. 1. A transverse coup of nanowire, the black circles represent magnetic atoms

the surface shell. The empty circles are magnetic atoms constituting the core.

includes all contributions associated with the site i, the other(denoted by H0) does not depend on the site i. Then, we rewritethe Hamiltonian at site i with the following form:

H¼HiþH0, ð2Þ

where

Hi ¼�X

j

Jijszj þh

0@

1Asz

i�Oasxi ¼�Asz

i�Oasxi , ð3Þ

with

A¼X

j

Jijszj þh and Oa ¼Os or Oc: ð4Þ

For ‘‘classical systems’’ in which Hi commute with H0, thestarting point of single site cluster theory is a set of formalidentity of the type

/szi S¼

Tr½szi e�bHi �

Tr½e�bHi �

* +, ð5Þ

where the angular bracket / � � �S denotes a canonical thermalaverage, b¼ 1=kBT and T is the temperature. In this way, andusing the EFT with a probability distribution technique, thelongitudinal magnetization of the system is given by [26]

mzi ¼/sz

i S¼/f zðA,OaÞS, ð6Þ

with

f zðA,OaÞ ¼A

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2þO2

a

q tanhb2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2þO2

a

q� �: ð7Þ

To perform thermal averaging on the right-hand side of Eq. (6),we follow the general approach described in Ref. [26]. First ofall, in the spirit of the EFT, multispin-correlation functions areapproximated by products of single spin averages. We then takeadvantage of the integral representation of Dirac’s delta distribu-tion, in order to write Eq. (6) in the following form:

mzi ¼

Zdy f zðy,OaÞ

1

2p

Zdl exp ðiylÞ

Yj

/expðilJijszj þhÞS

24

35: ð8Þ

By introducing the probability distribution of the spin vari-ables (for details see Ref. [26])

mzi ¼

Xszj¼ 1=2

szj¼ �1=2

Pðszj Þf z

Xj

Jijszj þh,Oa

0@

1A, ð9Þ

with

Pðszj Þ ¼

12 ½ð1�2mz

i Þdðszj þ

12 Þþð1þ2mz

i Þdðszj�

12Þ�: ð10Þ

We get the following expressions of the longitudinal magne-tization for each site of the nanowire:

�

For the central spin c1

mzc1¼

1

2N2þN4

XN2

m1 ¼ 0

XN4

m2 ¼ 0

CN2m1

CN4m2ð1�mz

c1Þm1 ð1þmz

c1ÞN2�m1

�ð1�mzc2Þm2 ð1þmz

c2ÞN4�m2 f zðJcðN2þN4�2ðm1þm2ÞÞþh,OcÞ:

ð11Þ

�

For spins around the center

mzc2¼

1

2N2þ2N1þN3

XN3

m1 ¼ 0

XN1

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

�CN3m1

CN1m2

CN1m3

CN2m4ð1�mz

c2Þm1 ð1þmz

c2ÞN3�m1

�ð1�mzc1Þm2 ð1þmz

c1ÞN1�m2 ð1�mz

s1Þm3 ð1þmz

s1ÞN1�m3

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–24412436

�ð1�mzs2Þm4 ð1þmz

s2ÞN2�m4 � f zðJcðN3þN1�2ðm1þm2ÞÞ

þ JcsððN1þN2�2ðm3þm4ÞÞþh,OcÞ: ð12Þ

�

For spins of type-1 surface shell nanowire

mzs1¼XN2

m1 ¼ 0

XN2

m2 ¼ 0

XN1

m3 ¼ 0

CN2m1

CN2m2

CN1m3

1

22N2þN1ð1�mz

s1Þm1 ð1þmz

s1ÞN2�m1

�ð1�mzs2Þm2 ð1þmz

s2ÞN2�m2 ð1�mz

c2Þm3 ð1þmz

c2ÞN1�m3

�f zðJsð2N2�2ðm1þm2ÞÞþ JcsðN1�2m3ÞÞþh,OsÞ: ð13Þ

�

Fig. 2. Temperature dependence of the magnetization and susceptibilities of

magnetic nanowire for a few values of the ratio constant rcs ¼ Jcs=Jc , rs ¼ Js=Jc ¼

0:5, os ¼ 2:0, oc ¼ 1:0 and h=Jc ¼ 0:0.

For spins of type-2 surface shell nanowire

mzs2¼

1

23N2

XN2

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

CN2m1

CN2m2

CN2m3ð1�mz

s2Þm1 ð1þmz

s2ÞN2�m1

�ð1�mzs1Þm2 ð1þmz

s2ÞN2�m2 ð1�mz

c2Þm3 ð1þmz

c2ÞN2�m3

�f zðJsð2N2�2ðm1þm2ÞÞþ JcsðN2�2m3Þþh,OsÞ: ð14Þ

With N1 ¼ 1, N2 ¼ 2, N3 ¼ 4 and N4 ¼ 6 denote respectivelythe coordination numbers and Ck

l are the binomial coefficientCl

k ¼ l!=k!ðl�kÞ!.The total longitudinal magnetization of the nanowire is

defined by

mt ¼1

19ð12mzsþ7mz

cÞ, ð15Þ

where msz and mc

z represent respectively the longitudinal magne-tizations of surface shell and core of the nanowire, which aregiven by

mzs ¼

12 ðm

zs1þmz

s2Þ and mz

c ¼17ðm

zc1þ6mz

c2Þ: ð16Þ

It is interesting to study the behavior of the longitudinalsusceptibility of the each site of the nanowire which is defined by

wt ¼@mt

@h

�h ¼ 0

¼1

19ð12wsþ7wcÞ, ð17Þ

where the susceptibilities of the core and the shell are given by

wc ¼17 ðwc1

þ6wc2Þ and ws ¼

12ðws1þws2Þ: ð18Þ

The details of the calculus of the each site longitudinalsusceptibility are given in the Appendix.

Fig. 3. Temperature dependence of the magnetization and susceptibilities of

magnetic nanowire for a few negative values of the ratio constant rcs ¼ Jcs=Jc ,

rs ¼ Js=Jcs ¼ 0:5, os ¼ 2:0, oc ¼ 1:0 and h=Jc ¼ 0:0.

3. Results and discussions

The model reveals some features of the magnetic nanowiresystem. In particular we investigate its transition and hysteresisbehavior for various values of its characteristic parameters. As weare interested with the calculation of the total magnetization andsusceptibility which depends on the core and the shell magneti-zations, we use the self consistent equations (10)–(14) that can besolved directly by numerical iteration method. In this model, wetake Jc as the unit of the energy and we introduce the reducedexchange interactions and transverse field (rcs ¼ Jcs=Jc , rs ¼ Js=Jc ,oc ¼Oc=Jc and os ¼Os=Jc).

In Figs. 2–7, we depict the temperature dependence of thetotal longitudinal magnetization and susceptibility in the absenceof the longitudinal magnetic field and with different values of rcs,rs, oc and os. Fig. 2 shows the effect of the ferrimagneticinteraction between the core and the surface shell spins (rcso0)on the magnetization and susceptibility when the parameters rcs,oc and os are fixed at rs ¼ 0:5, os ¼ 2:0 and oc ¼ 1:0. The criticaltemperature, which corresponds to the pic of the susceptibility,increases with the decrease of rcs. According to the Neel theory,the shape of the magnetization versus temperature curve canexhibit five characteristic features classified as P, Q, N, L and

M-types in the absence of the longitudinal magnetic field [27]. Inthis case, the total magnetization shows a compensation pointbelow its transition temperature and exhibits the typical N-typebehavior for rcs ¼�1:0 (Tcomp ¼ 3:39) and rcs ¼�0:5 (Tcomp ¼ 1:89).However, this behavior may change to the P-type for rcs ¼�0:25.Our results are in agreement with those reported in Ref. [18].

Fig. 3 shows the effect of the ferromagnetic exchange interac-tion between the core and the surface shell spins on the long-itudinal magnetization and the longitudinal susceptibility for thesame parameters used in Fig. 2. The curves of the magnetization

Fig. 4. Temperature dependence of the magnetization and susceptibilities of

magnetic nanowire for a few values of the transverse field in surface shell os ,

rcs ¼ Jcs=Jc ¼ 1:0, rs ¼ Js=Jc ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 0.

Fig. 5. Temperature dependence of the magnetization and susceptibilities of

magnetic nanowire for a few values of the transverse field in surface shell os ,

rcs ¼ Jcs=Jc ¼�1:0, rs ¼ Js=Jc ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 0:0.

Fig. 6. Temperature dependence of the magnetization and susceptibilities of

magnetic nanowire for a few values of the transverse field in core oc , rcs ¼ Jcs=Jc ¼

�1:0, rs ¼ Js=Jc ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 0:0.

Fig. 7. Temperature dependence of the magnetization and susceptibilities

of magnetic nanowire for a few values of the transverse field in core oc , rcs ¼

Jcs=Jc ¼ 1:0, rs ¼ Js=Jc ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 0:0.

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–2441 2437

exhibits the Q-type behavior, while the total susceptibilitydiverge at the same critical temperature Tc in ferrimagnetic case(Fig. 2). But, wt in the ferrimagnetic result becomes narrower inwidth than the ferromagnetic case. The reason comes from thefollowing facts: in ferrimagnetic case the susceptibility wc of thecore diverges to the opposite direction at Tc, comparing with thesusceptibility ws of the surface shell.

The dependence of the total magnetization and susceptibilityon the transverse field in surface shell (os ¼ 0:0, 1.0, 2.0, 3.0and 4.0) is shown in Figs. 4 and 5 by selecting two values of rcs

(rcs ¼ 1:0 in Fig. 4 and rcs ¼�1:0 in Fig. 5) with oc ¼ 3:0 andrs ¼ 1:0. The critical temperature is independent of the sign of rcs.

It increases with decreasing the transverse field in surface shell. InFig. 4, we see the existence of the Q-type behavior for all values ofos and in Fig. 5 for os ¼ 0:0, 1.0 and 2.0. But for the values of theparameter os ¼ 3:0 and 4.0 the total magnetization exhibitthe N-type behavior, showing the compensation point whichincreases with the decrease of os (Tcomp ¼ 5:03 for os ¼ 3:0 andTcomp ¼ 4:22 for os ¼ 4:0). The effect of the transverse field in coreoc on the total magnetization and susceptibility is shown inFigs. 6 and 7 for selected values of the parameters oc ¼ 3:0 andrs ¼ 1:0 and for two values of rcs (rcs ¼�1:0 and 1.0). The influenceof the transverse field in the core on the total magnetization and

Fig. 9. Temperature dependence core, shell and total susceptibility of nanowire

for rcs ¼ Jcs=Jc ¼�1:0, rs ¼ Js=Jc ¼ 0:5, os ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 1:0.

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–24412438

susceptibility is similar to the effect of the transverse field in thesurface shell. In Figs. 4 and 5 the critical temperature is higherthan when the system was switched between os and oc inFigs. 6 and 7.

In Figs. 8 and 9 msðwsÞ, mcðwcÞ and mtðwtÞ are the magnetization(susceptibility) for the surface shell, the core and the nanowirerespectively. When we increase the temperature, the magnetiza-tion for the core increases at first, then it decreases, then itdecreases abruptly at a definite temperature, afterwards flippingto the opposite direction, approaching the remanent magnetiza-tion. On the other hand, the magnetization for surface shell ofnanowire initially decreases in a gradual manner, and it thenabruptly increases at the same definite temperature as wasdescribed before. Subsequently, the spins realign in the oppositedirection. On further increasing the temperature, it approachesthe same direction as the longitudinal magnetic field. The totalmagnetization of the nanowire has almost the same behavior ofthe magnetization of the core, then it behaves the same way tothe magnetization of the surface shell. It is clearly seen that all ofthe magnetic susceptibilities exhibit a singularity at the flippingpoint at the finite temperature given by kBT=JC4:12 where thecore magnetization, the surface shell magnetization and the totalmagnetization of the nanowire change directions. The magneticsusceptibility of the core and the total magnetization exhibit amaximum at the same temperature which the magnetization mc

and mt undergo abrupt change. While, the susceptibility of thesurface shell diverges to the opposite direction at the samedefinite temperature.

In order to study the influence of the transverse field in thecore oc and in the surface shell os on the magnetizationprocesses, we represent the hysteresis loops at the temperaturekBT=J¼ 0:2. Fig. 10 shows examples of the hysteresis loops forvarious values of oc from 0.0 to 3.0. The parameters used arercs ¼�1:0 and rs ¼ 1:0, triple hysteresis loop patterns are observed

Fig. 8. Temperature dependence of core, shell and total magnetization of nano-

wire for rcs ¼ Jcs=Jc ¼�1:0, rs ¼ Js=Jc ¼ 0:5, os ¼ 1:0, oc ¼ 3:0 and h=Jc ¼ 1:0.

Fig. 10. Hysteresis loops of magnetic nanowire for few values of the transverse

field in the core.

Fig. 11. Hysteresis loops of magnetic nanowire for few values of the transverse

field in the surface shell. Fig. 12. Hysteresis loops of the nanowire for a few values of the exchange

interaction core/shell.

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–2441 2439

when the exchange interaction between core and shell is nega-tive. We remark that the central loop is almost unchanged and theshape of the outer loops becomes narrower with increasing oc.The curves saturate towards high longitudinal magnetic fields.Fig. 11 shows the hysteresis loops for various values of os from0.0 to 9.0. For os ¼ 0:0 we observed a large hysteresis loop andwill become narrower by increasing the transverse field in surfaceshell in other words, the coercive field decreases with increasingthe transverse magnetic field in surface shell, but for the value ofos ¼ 3:0 we see a triple hysteresis loops. The central loopbecomes narrower and the outer loops disappear by increasingthe transverse field in the shell.

In Fig. 12, for the same values of the transverse field in coreand surface shell (os ¼oc ¼ 2:0) and for fixed value of theexchange interaction in surface shell rs ¼ 0:5, we find that, whenthe exchange interaction core/shell decreases from �0.5 to �1.0triple hysteresis loop patterns are observed. Also, the central loopbecomes more important and the shape of the outer loops areunchanged for the large values of the longitudinal magnetic field.

The effect of the temperature on the hysteresis loops of thenanowire is shown in Fig. 13. We find that for the parametersrcs ¼�1:0, rs ¼ 0:5, oc ¼os ¼ 2:0 and kBT=J change from 1.0 to 6.0,the central loop becomes narrower and disappears at the transi-tion temperature kBTc=J¼ 5:14.

Fig. 13. Hysteresis loops of the nanowire for a few values of the exchange

interaction core/shell.

4. Conclusion

In this work, we have used the EFT with a probabilitydistribution technique to investigate the magnetic properties ofa magnetic nanowire in the presence of both the longitudinal andthe transverse fields. Our results show that the system exhibits alot of characteristic features in the longitudinal magnetizationand susceptibility.

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–24412440

In ferrimagnetic case, the critical temperature increases whenthe exchange interaction between the core and the shell (rcs)decreases and the shape of the magnetization versus temperatureexhibits the typical N-type behavior which can change to theP-type with the decrease of rcs. However, in ferromagnetic case,the shape of the magnetization exhibits the Q-type behavior. Thelongitudinal susceptibility in the ferrimagnetic case becomesnarrower in width than the ferromagnetic case because the corepart of the susceptibility in ferrimagnetic case diverges to theopposite direction at the critical temperature comparing with theshell part of the susceptibility.

For a positive value of rcs, the critical temperature increases withthe decrease of the transverse field in surface shell and the long-itudinal magnetization exhibits the Q-type behavior. However, for anegative value of rcs, the longitudinal magnetization exhibits achange of the behavior from Q to N-type with the increase of thetransverse field in surface shell. Finally, we have examined thehysteresis loops in the presence of the transverse fields in core andin surface shell. Triple hysteresis loop patterns are observed whichmay have potential in producing a multi-state memory. Thesemagnetic properties of the studied system offer the opportunitiesto develop nanomaterials with new functions for magnetic record-ing. Our theoretical predictions may be a reference for futureexperimental studies on the magnetic nanowire.

Acknowledgment

This work has been initiated with the support of URAC: 08 andthe Swedish Research Links programme dnr-348-2011-7264 dur-ing a visit of A.A. and M.S. at the Max Planck Institut fur PhysikKomplexer Systeme Dresden, Germany, and completed during avisit of A.A. and I.E. to the Departamento de Fısica de la MateriaCondensada, Universidad Autonoma, Madrid, Spain, in the frameof the project No: A/030519/10 financed by A.E.C.I. The authorswould like to thank all the organizations.

Appendix

Susceptibility of the nanowire

@mzc1

@h

�h ¼ 0

¼ A1;1

@mzc1

@h

�h ¼ 0

þA1;2

@mzc2

@h

�h ¼ 0

þB1, ðA:1Þ

@mzc2

@h

�h ¼ 0

¼ A2;1

@mzc1

@h

�h ¼ 0

þA2;2

@mzc2

@h

�h ¼ 0

þA2;3

@mzs1

@h

�h ¼ 0

þA2;4

@mzs2

@h

�h ¼ 0

þB2, ðA:2Þ

@mzs1

@h

�h ¼ 0

¼ A3;2

@mzc2

@h

�h ¼ 0

þA3;3

@mzs1

@h

�h ¼ 0

þA3;4

@mzs2

@h

�h ¼ 0

þB3,

ðA:3Þ

@mzs2

@h

�h ¼ 0

¼ A4;2

@mzc2

@h

�h ¼ 0

þA4;3

@mzs1

@h

�h ¼ 0

þA4;4

@mzs2

@h

�h ¼ 0

þB4:

ðA:4Þ

Elements Ai,j and Bi,j are given by

A1;1 ¼ 2�ðN2þN4ÞXN2

m1 ¼ 0

XN4

m2 ¼ 0

Xm1

i ¼ 0

XN2�m1

j ¼ 0

�CN2m1

CN4m2

Cm1

i CN2�m1

j ð�1Þiðiþ jÞðmzc1Þiþ j�1ð1�mz

c2Þm2

�ð1þmzc2ÞN4�m2 f zðJcðN2þN4�2ðm1þm2ÞÞþh,OcÞ, ðA:5Þ

A1;2 ¼ 2�ðN2þN4ÞXN2

m1 ¼ 0

XN4

m2 ¼ 0

Xm2

i ¼ 0

XN4�m2

j ¼ 0

�CN2m1

CN4m2

Cm2

i CN4�m2

j ð�1Þiðiþ jÞ

�ðmzc2Þiþ j�1ð1�mz

c1Þm1 ð1þmz

c1ÞN2�m1

�f zðJcðN4þN2�2ðm1þm2ÞÞþh,OcÞ, ðA:6Þ

B1 ¼ 2�ðN2þN4ÞXN2

m1 ¼ 0

XN4

m2 ¼ 0

�CN2m1

CN4m2ð1�mz

c1Þm1 ð1þmz

c1ÞN2�m1 ð1�mz

c2Þm2 ð1þmz

c2ÞN4�m2

�@f z

@hðJcðN2þN4�2ðm1þm2ÞÞþh,OcÞ, ðA:7Þ

A2;1 ¼ 2�ð2N1þN2þN3ÞXN1

m1 ¼ 0

XN3

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

Xm1

i ¼ 0

XN1�m1

j ¼ 0

�CN1m1

CN3m2

CN1m3

CN2m4

Cm1

i CN1�m1

j ð�1Þiðiþ jÞðmzc1Þiþ j�1

�ð1�mzc2Þm2 ð1þmz

c2ÞN3�m2 ð1�mz

s1Þm3

�ð1þmzs1ÞN1�m3 ð1�mz

s2Þm4 ð1þmz

s2ÞN2�m4

�f zðJcðN3þN1�2ðm1þm2ÞÞþ JcsðN1þN2�2ðm3þm4ÞÞþh,OcÞ,

ðA:8Þ

A2;2 ¼ 2�ð2N1þN2þN3ÞXN1

m1 ¼ 0

XN3

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

Xm2

i ¼ 0

XN3�m2

j ¼ 0

�CN1m1

CN3m2

CN1m3

CN2m4

Cm2

i CN3�m2

j ð�1Þiðiþ jÞðmzc2Þiþ j�1

�ð1�mzc1Þm1 ð1þmz

c1ÞN1�m1 ð1�mz

s1Þm3 ð1þmz

s1ÞN1�m3

�ð1�mzs2Þm4 ð1þmz

s2ÞN2�m4 f zðJcðN3þN1�2ðm1þm2ÞÞ

þ JcsðN1þN2�2ðm3þm4ÞÞþh,OcÞ, ðA:9Þ

A2;3 ¼ 2�ð2N1þN2þN3ÞXN1

m1 ¼ 0

XN3

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

Xm3

i ¼ 0

XN1�m3

j ¼ 0

�CN1m1

CN3m2

CN1m3

CN2m4

Cm3

i CN1�m3

j ð�1Þiðiþ jÞðmzs1Þiþ j�1

�ð1�mzc1Þm1 ð1þmz

c1ÞN1�m1 ð1�mz

c2Þm2 ð1þmz

c2ÞN3�m2

�ð1�mzs2Þm4 ð1þmz

s2ÞN2�m4 f zðJcðN3þN1�2ðm1þm2ÞÞ

þ JcsðN1þN2�2ðm3þm4ÞÞþh,OcÞ, ðA:10Þ

A2;4 ¼ 2�ð2N1þN2þN3ÞXN1

m1 ¼ 0

XN3

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

Xm4

i ¼ 0

XN2�m4

j ¼ 0

�CN1m1

CN3m2

CN1m3

CN2m4

Cm4

i CN2�m4

j ð�1Þiðiþ jÞðmzs2Þiþ j�1

�ð1�mzc1Þm1 ð1þmz

c1ÞN1�m1 ð1�mz

s1Þm3 ð1þmz

s1ÞN1�m3

�ð1�mzc2Þm2 ð1þmz

c2ÞN3�m2 f zðJcðN3þN1�2ðm1þm2ÞÞ

þ JcsðN1þN2�2ðm3þm4ÞÞþh,OcÞ, ðA:11Þ

B2 ¼ 2�ð2N1þN2þN3ÞXN1

m1 ¼ 0

XN3

m2 ¼ 0

XN1

m3 ¼ 0

XN2

m4 ¼ 0

CN1m1

�CN3m2

CN1m3

CN2m4ð1�mz

c1Þm1 ð1þmz

c1ÞN1�m1 ð1�mz

c2Þm2

�ð1þmzc2ÞN3�m2 ð1�mz

s1Þm3 ð1þmz

s1ÞN1�m3 ð1�mz

s2Þm4

�ð1þmzs2ÞN2�m4

@f z

@hðJcðN3þN1�2ðm1þm2ÞÞ

þ JcsðN1þN2�2ðm3þm4ÞÞþh,OcÞ, ðA:12Þ

A3;2 ¼ 2�ðN1þ2N2ÞXN1

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm1

i ¼ 0

XN1�m1

j ¼ 0

CN1m1

�CN2m2

CN2m3

Cm1

i CN1�m1

j ð�1Þiðiþ jÞðmzc2Þiþ j�1ð1�mz

s1Þm2

�ð1þmzs1ÞN2�m2 ð1�mz

s2Þm3 ð1þmz

s2ÞN2�m3

�f zðJsð2N2�2ðm2þm3ÞÞþ JcsðN1�2m1Þþh,OsÞ, ðA:13Þ

S. Bouhou et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2434–2441 2441

A3;3 ¼ 2�ðN1þ2N2ÞXN1

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm2

i ¼ 0

XN2�m2

j ¼ 0

�CN1m1

CN2m2

CN2m3

Cm2

i CN2�m2

j ð�1Þiðiþ jÞðmzs1Þiþ j�1ð1�mz

c2Þm1

�ð1þmzc2ÞN1�m1 ð1�mz

s2Þm3 ð1þmz

s2ÞN2�m3

�f zðJsð2N2�2ðm2þm3ÞÞþ JcsðN1�2m1Þþh,OsÞ, ðA:14Þ

A3;4 ¼ 2�ðN1þ2N2ÞXN1

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm3

i ¼ 0

XN2�m3

j ¼ 0

�CN1m1

CN2m2

CN2m3

Cm3

i CN2�m3

j ð�1Þiðiþ jÞðmzs2Þiþ j�1ð1�mz

s1Þm2

�ð1þmzs1ÞN2�m2 ð1�mz

c2Þm1 ð1þmz

c2ÞN1�m1

�f zðJsð2N2�2ðm2þm3ÞÞþ JcsðN1�2m1Þþh,OsÞ, ðA:15Þ

B3 ¼ 2�ðN1þ2N2ÞXN1

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

CN1m1

CN2m2

CN2m3

�ð1�mzc2Þm1 ð1þmz

c2ÞN1�m1 ð1�mz

s1Þm2 ð1þmz

s1ÞN2�m2

�ð1�mzs2Þm3 ð1þmz

s2ÞN2�m3

�@f z

@hðJsð2N2�2ðm2þm3ÞÞþ JcsðN1�2m1Þþh,OsÞ, ðA:16Þ

A4;2 ¼ 2�3N2XN2

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm1

i ¼ 0

XN2�m1

j ¼ 0

CN2m1

�CN2m2

CN2m3

Cm1

i CN2�m1

j ð�1Þiðiþ jÞðmzc2Þiþ j�1ð1�mz

s1Þm2

�ð1þmzs1ÞN2�m2 ð1�mz

s2Þm3 ð1þmz

s2ÞN2�m3

�f zðJsð2N2�2ðm2þm3ÞÞþ JcsðN2�2m1Þþh,OsÞ, ðA:17Þ

A4;3 ¼ 2�3N2XN2

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm2

i ¼ 0

XN2�m2

j ¼ 0

�CN2m1

CN2m2

CN2m3

Cm2

i CN2�m2

j ð�1Þiðiþ jÞðmzs1Þiþ j�1ð1�mz

c2Þm1

�ð1þmzc2ÞN2�m1 ð1�mz

s2Þm3 ð1þmz

s2ÞN2�m3

�f zðJsð2N2�2ðm2þm3ÞÞþ JcsðN2�2m1Þþh,OsÞ, ðA:18Þ

A4;4 ¼ 2�3N2XN2

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

Xm3

i ¼ 0

XN2�m3

j ¼ 0

�CN2m1

CN2m2

CN2m3

Cm3

i CN2�m3

j ð�1Þiðiþ jÞðmzs2Þiþ j�1ð1�mz

s1Þm2

�ð1þmzs1ÞN2�m2 ð1�mz

c2Þm1 ð1þmz

c2ÞN2�m1 f zðJsð2N2�2ðm2þm3ÞÞ

þ JcsðN2�2m1Þþh,OsÞÞ, ðA:19Þ

B4 ¼ 2�ð3N2ÞXN2

m1 ¼ 0

XN2

m2 ¼ 0

XN2

m3 ¼ 0

�CN2m1

CN2m2

CN2m3ð1�mz

c2Þm1 ð1þmz

c2ÞN2�m1 ð1�mz

s1Þm2 ð1þmz

s1ÞN2�m2

�ð1�mzs2Þm3 ð1þmz

s2ÞN2�m3

@f z

@hðJsð2N2�2ðm2þm3ÞÞ

þ JcsðN2�2m1Þþh,OsÞ, ðA:20Þ

@f z

@h

�h ¼ 0

¼O2

ðA2þO2Þ3=2

tanh ðbðA2þO2Þ1=2

þbA2

ðA2þO2Þð1�tanh2

ðbðA2þO2Þ1=2ÞÞ: ðA:21Þ

References

[1] F. Michael, C. Gonzalez, V. Mujica, M. Marquez, M.A. Ratner, Physical ReviewB 76 (2007) 224409.

[2] R.H. Kodama, A.E. Berkowitz, Physical Review B 59 (1999) 6321.[3] A. Zaim, M. Kerouad, Physica A 389 (2010) 3435.[4] A. Fert, L. Piraux, Journal of Magnetism and Magnetic Materials 200 (1999)

338.[5] J. Maller, K.Y. Zhang, C.L. Chien, T.S. Eagleton, P.C. Searson, Applied Physics

Letters 84 (2004) 3900.[6] A. Hultgren, M. Tanase, C.S. Chen, D.H. Reich, IEEE Transactions on Magnetics

40 (2004) 2988.[7] A. Hultgren, M. Tanase, E.J. Felton, K. Bhadriraju, A.K. Salem, C.S. Chen,

D.H. Reich, Biotechnology Progress 21 (2005) 509.[8] Q.F. Zhan, W. He, X. Ma, Y.Q. Liang, Z.Q. Kou, N.L. Di, Z.H. Cheng, Applied

Physics Letters 85 (2004) 4690.[9] S.H. Ge, C. Li, X. Ma, W. Li, C.X. Li, Journal of Applied Physics 90 (2001) 509.

[10] N. Cordente, M. Respaud, F. Senocq, M. Casanove, C. Amiens, B. Chaudret,Nano Letters 1 (2001) 565.

[11] V.S. Leite, W. Figueiredo, Physica A 350 (2005) 379.[12] A.F. Bakuzis, P.C. Morais, Journal of Magnetism and Magnetic Materials 285

(2005) 145.[13] T. Kaneyoshi, Physica Status Solidi (b) 242 (2005) 2938.[14] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 321 (2009) 3430.[15] M. Vasilakaki, K.N. Trohidou, Physical Review B 79 (2009) 144402.[16] T. Kaneyoshi, Physica Status Solidi (b) 248 (2011) 250.[17] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 322 (2010) 3014.[18] T. Kaneyoshi, Journal of Magnetism and Magnetic Materials 322 (2010) 3410.[19] A. Zaim, M. Kerouad, Y. El Amraoui, Journal of Magnetism and Magnetic

Materials 321 (2009) 1077.[20] T. Kaneyoshi, Physica Status Solidi (b) 248 (2011) 250.[21] R. Honmura, T. Kaneyoshi, Journal of Physics C12 (1979) 3979.[22] T. Kaneyoshi, Acta Physica Polonica A 83 (1993) 703.[23] F. Zernike, Physica 7 (1940) 565.[24] J.W. Macki, P. Nistri, P. Zecca, SIAM Review 35 (1993) 94.[25] D.C. Jiles, D.L. Atherton, Journal of Applied Physics 55 (1984) 2115.[26] M. Saber, Chinese Journal of Physics 35 (1997) 577.[27] L. Neel, Annales de Physique (Paris) 3 (1948) 137.