the transverse ising thin film with decorated ferrimagnetic surfaces

Journal of Magnetism and Magnetic Materials 233 (2001) 236–247

The transverse Ising thin film with decoratedferrimagnetic surfaces

N. El Aouad*, A. Moutie, M. Kerouad

D !epartement de Physique, Facult !e des Sciences, Universit !e Moulay Ismail, B.P. 4010, Mekn "es, Morocco

Received 4 October 2000; received in revised form 31 January 2001

Abstract

The phase diagrams and magnetization curves of the transverse Ising film with decorated spin-1 atoms on the surface,are investigated by the use of an effective field method within the framework of a single cluster theory. In particular, thecase that decorated atoms at the surfaces are ordered ferrimagnetically is examined. The effects of the interactions,

transverse field and film thickness are studied. A number of characteristic phenomena, such as the possibility ofcompensation points on the surfaces and in the film are found. r 2001 Elsevier Science B.V. All rights reserved.

Keywords: Decorated Ising model; Thin film; Ferrimagnetism; Compensation temperature; Phase transition

1. Introduction

Ferrimagnetism has been extensively investi-gated in the past, both experimentally andtheoretically, since important magnetic materialsfor technological applications, such as garnets andferrites, are ferrimagnetic. Ferrimagnets haveseveral sublattices with finite resultant momentsand show paramagnetic behavior above thetransition temperature Tc. In contrast with aferromagnet, there is an interesting possibility ofthe existence, under certain conditions, of com-pensation temperature TkðTkoTcÞ, at which theresultant magnetization vanishes [1,2]. The occur-rence of a compensation point is of greattechnological importance, since at this point onlya small driving field is required to change the sign

of the resultant magnetization. This property isvery useful in thermomagnetic recording.Decorated Ising models, which were originally

introduced in the literature by Syozi [3] have beenstudied many years ago as the models exhibitingferrimagnetism [4,5]. The arrangement of atomswas like that in the normal spinel. The system ismade up of two sublattices L1 and L2 wheresublattice L1 is occupied by spin-12 A atoms andsublattice L2, which is composed of one decoratedpoint on every bond of L1, is occupied by spin-1 Batoms. However, most of the decorated modelsstudied have been restricted to the effects of acrystal field on the phase diagrams [4–7]. Further-more, the surface magnetic properties of a semi-infinite spin-12 ferromagnet with a decorated sur-face have been studied [8].Phase transitions in Ising spin systems driven

entirely by quantum fluctuations have beenattracting a lot of interest recently [9]. The simplest

*Corresponding author.

E-mail address: [email protected] (N. El Aouad).

0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 1 5 3 - 6

of such systems is the Ising model in a transversefield. The spin-12 transverse Ising model wasoriginally introduced by De Gennis [10] as avaluable model for tunneling of the proton, inhydrogen-bonded ferroelectrics such as KH2PO4.It has been extensively studied by the use ofvarious techniques [11–14] including the effectivefield theory [15,16] based on a generalized butapproximated Callen-Suzuki relation derived byBarreto et al.On the other hand, surface magnetism is a rich

problem for study on both theoretical and experi-mental grounds, and has attracted considerableinterest in recent years on account of its variedapplication and relevance to problems of corro-sions, catalysis etc. General reviews of the topichave been provided, for example, by Binder [17],Diehl [18] and Kaneyoshi [19]. It is now wellestablished that if the surface exchange coupling isgreater than a critical value, the surface region canorder even when the bulk is paramagnetic and hasa transition temperature higher than the bulk one.Surface magnetism order higher than the bulk hasbeen detected experimentally [20–22]. More re-cently, great attention has been paid to the studyof magnetic properties of thin films. Because of therecent advances of modern vacuum science, suchas epitaxial growth techniques, it is now possibleto grow very thin magnetic films with arbitrarythickness and this has stimulated renewed interestin both experimental and theoretical film magnet-ism. The fabrication of thin magnetic films has ledto a surge of experimental activity, see forexample, Refs. [23–25] and references containedtherein. From the theoretical point of view, theIsing model has been frequently and successfullyadopted for the description and understanding ofmany characteristic features of thin magneticfilms. These theoretical works deal mainly withferromagnetic thin films [26–35]. Furthermore,ferrimagnetically ordered thin films have beenproduced experimentally for technologicalapplications such as, medium for magneto-opticmemories [36,37]. The existence of a compensa-tion point in such a thin film is important forrecording memories. Only a few theoretical workshave considered the study of ferrimagnetic films[38,39].

As far as we know, no investigation has beenmade of a transverse Ising film with decoratedsurfaces. So, in this work we are going to study themagnetic properties of an Ising film composed ofspin-12 atoms in bulk ferromagnetic layers and twosurfaces decorated by spin-1 atoms in a transversefield. This study is carried out within the frame-work of an effective field theory [40], which isbelieved to give more exact results than those ofthe standard mean-field approximation. In parti-cular, attention is paid to the surfaces in which thedecorated atoms are ordered ferrimagnetically.The outline of this paper is as follows. In Section

2, we describe briefly the effective field theory forthe transverse Ising film with decorated surfaces.In Section 3, we examine the magnetic propertiesof this system and discuss our results. A briefconclusion is given in Section 4.

2. Formulation

We consider a transverse spin-12 Ising ferromag-netic thin film with decorated spin-1 atomsordered ferrimagnetically on the surface. Thestructure has simple cubic symmetry composedof L atomic layers in the Z-direction. Thetwo-dimensional cross section is schematicallydepicted in Fig. 1. The Hamiltonian of the system

Fig. 1. The two-dimensional cross section of a simple cubic thin

film with decorated surfaces, where the white and black circles

represent, respectively, spin-12 atoms and spin-1 atoms.

N. El Aouad et al. / Journal of Magnetism and Magnetic Materials 233 (2001) 236–247 237

has the form:

H ¼@Xði;jÞ

Jijszi szj þ J

0S

Xi;m

szi Szm

@OXðiÞ

sxi@OXðiÞ

Sxi ; ð1Þ

where szi ; sxi denote the z and x components of a

quantum spin ~si of magnitude s ¼ 1=2 at site i(white circle), Szi ; S

xi are the z and x components

of a quantum spin ~Si of the decorated spin-1atoms on the two surfaces (black circle). Jij is thestrength of the interaction between spin-12 atnearest neighbor sites, Jij ¼ JS if both spins belongto surface layers and Jij ¼ JB otherwise. J 0S is theinteraction between spin-12 and spin-1 atoms at thesurfaces, (JS, J

0S, and JB are positive). O represents

the transverse field.The statistical properties of the system are

studied using an effective field theory whosestarting point is a generalized, but approximateCallen [41] relation derived by Barreto et al. [15]for the transverse Ising model. The longitudinalsite order parameters are approximately given by(for details, see Barreto and Fittipaldi [16])

sS ¼ sih ih i ¼ f 1=2Xj

Jijszj@J 0S

Xm

SZm;O

!* +;

ð2Þ

ms ¼ Sih ih i ¼ f 1 @J 0S

Xj

sZj ;O

!* +; ð3Þ

qS ¼ S2i� ��

¼ g1 @J 0S

Xj

sZj ;O

!* +; ð4Þ

for the two surfaces, and

sn ¼ sih ih i ¼ f 1=2 JBXj

szk;O

!* +: ð5Þ

For 1onoL, with s1 ¼ sL ¼ sS; where

f 1=2ðy;OÞ ¼1

2

y

ðy2 þ O2Þ1=2tanh

b2y2 þ O2� 1=2 �

;

ð6Þ

f 1ðy;OÞ ¼y

ðy2 þ O2Þ1=22 sinh b y2 þ O2

� 1=2� 2 cosh b y2 þ O2

� 1=2� þ 1

;

ð7Þ

g1ðy;OÞ ¼

1

ðy2 þ O2Þ1=22y2 þ O2�

cosh b y2 þ O2� 1=2�

þ O2

2 cosh b y2 þ O2� 1=2�

þ 1:

ð8Þ

si is the longitudinal magnetization at the sitei, ms, and qs, are the longitudinal magnetizationand quadrupolar moment of the spin-1 atoms atthe surfaces, b ¼ 1=kBT (we take kB ¼ 1 forsimplicity), ?h i indicates the usual canonicalensemble thermal average for a given configura-tion, and the sums runs over all nearest neighborsof site i.To proceed further, we use the effective field

theory in the same way as in Ref. [42], Eqs. (2)–(5)can be written in the form:

ss ¼1

2NþN0

XNm¼0

XMn¼0

XM@n

g¼0

XN0

b¼0

�

2n@MCNm CMn C

M@ng CN0

b

ð1@2ssÞmð1þ 2ssÞ

N@mð1@2s2Þbð1þ 2s2Þ

N0@b

ð1@qsÞnðqs þmsÞ

gðqs@msÞM@n@g

f 1=2Js2 ðN@2mÞ þ J 0sðM@n@2gÞ

þ JB2 ðN0@2bÞ; O

" #

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;;

ð9Þ

ms ¼1

2M0

XM0

j¼0

CM0

j ð1@2ssÞjð1

þ 2ssÞM0@j f 1

J 0s2ð2j@M0Þ;O

� �; ð10Þ

qs ¼1

2M0

XM0

j¼0

CM0

j ð1@2ssÞjð1

þ 2ssÞM0@jg1

J 0s2ð2j@M0Þ;O

� �; ð11Þ

N. El Aouad et al. / Journal of Magnetism and Magnetic Materials 233 (2001) 236–247238

sn ¼1

2Nþ2N0

XNm¼0

XN0

g¼0

XN0

b¼0

�

CNm CN0g C

N0

b

ð1@2snÞmð1þ 2snÞ

N@mð1@2snþ1Þb

ð1þ 2snþ1ÞN0@bð1@sn@1Þ

gð1þ 2sn@1ÞN0@g

f 1=2 JB2 ðN@2N0Þ þ 2m@2g@2bÞ;O� �

8>>>>><>>>>>:

9>>>>>=>>>>>;: ð12Þ

The magnetization ms and the quadrupolar mo-ment qs can be expressed in the form:

ms ¼ b10ss; ð13Þ

qs ¼ q10 þ q11s2s ; ð14Þ

with

b10 ¼1

2M0

XM0

j¼0

Xjk¼0

XM0@j

l¼0

CM0

j CjkCM0@jl

� ð@1Þl2kþldðkþ l; 1Þ

f 1a2ð2j@M0Þ;

OJB

� �; ð15Þ

q10 ¼1

2M0

XM0

j¼0

Xjk¼0

XM0@j

l¼0

CM0

j CjkCM0@jl


g1a2ð2j@M0Þ;

OJB

� �; ð16Þ

q11 ¼1

2M0

XM0

j¼0

Xjk¼0

XM0@j

l¼0

CM0

j CjkCM0@jl


g1a2ð2j@M0Þ;

OJB

� �: ð17Þ

We then substitute Eqs. (13) and (14) into Eq. (9).In these equations, we have introduced thenotation a ¼ J 0S=JB and Cji are the binomialcoefficients. N and N0 denote the coordinationnumbers on the parallel planes and interplanes.Mis the number of spin-1 atoms neighbors of a spin-12one and M0 is the number of spin-12 atomsneighbors of a spin-1 one. In this case,M ¼ N ¼ 4, N0 ¼ 1 and M0 ¼ 2.

As we are interested in the calculations of thelongitudinal ordering near the transition tempera-ture, the usual arguments that the layer long-itudinal magnetizations sn tend to zero as thetemperature approaches a critical temperature,allows us to consider only terms linear in a sn,because higher order terms tends to zero fasterthan sn on approaching a critical temperature [43].Consequently, all terms of an order higher thanlinear terms in Eqs. (9) and (12) can be neglected.From this procedure, we can obtain the followingset of simultaneous equations:

s1 ¼ A11s1 þ A12s2; ð18Þ

sn ¼ An;n@1sn@1 þ Annsn þ An;nþ1snþ1; ð19Þ

sL ¼ AL;L@1sL@1 þ AL;LsL; ð20Þ

where AL;L ¼ A11;AL;L@1 ¼ A12 and An;n@l ¼An;nþ1. The system of Eqs. (18), (19) and (20)can be written as

M

s1^

sn^

sL

0BBBBBB@

1CCCCCCA

¼ 0; ð21Þ

where Mij ¼ ð1@AijÞdij@Aijðdi; j@1 þ di; jþ1Þ. Allthe information about the critical temperature ofthe system is contained in Eq. (21). The phasetransition temperature Tc=JB is obtained from theequation:

detðMÞ ¼ 0: ð22Þ

Tc=JB depends on R ¼ JS=JB, a ¼ J 0S=JB, O=JBand the thickness film L. From the many solutionsof Eq. (22), we choose the one corresponding tothe highest possible transition temperature asdiscussed in Ref. [44].The total magnetization MT of the system

composed of L layers is

MT

NA¼ 2Ms þ

XL@1

n¼2

sn; ð23Þ

with

Ms ¼ ss þ 2ms: ð24Þ


where NA is the number of spin-12 atoms on eachlayer and the total magnetization on the surface isgiven by NAMs. As we have seen, at the surface,the sublattice magnetization ms, can be evaluatedby Eq. (13). Thus, the magnetization Ms at thesurface is given by

Ms ¼ ssð1þ 2b10Þ: ð25Þ

The Eq. (25) indicates an important fact: theremay exist a compensation point at which themagnetization Ms reduces to zero even when thetemperature is below the transition temperature Tc

of the system. If it exists in the temperaturedependence of Ms, the compensation temperatureT sk at the surfaces can be determined from

1þ 2b10¼ 0; for T skoTc: ð26Þ

3. Results and discussion

We are now able to study the magnetic proper-ties (compensation and transition temperaturesand magnetization curves) of the transverse Isingfilm with decorated ferrimagnetic surfaces. Inaddition, we are also going to give the results of

the semi-infinite system as a limit of those of thethin film when the number of layers L is very large(L-N). It is well known, that if the ratio of thesurface interaction to the bulk one R ¼ Js=JB isgreater than a critical value Rc, then there are twokinds of transition on the semi-infinite system, thesurface transition (T s

c=JB) and the bulk transition(Tb

c =JB). To derive the results for the semi-infinitesystem, the six-layer approximation has beenintroduced: we have assumed that sn ¼ sb forn ¼ 6. Now we calculate the phase diagrams(Tc=JB versus a plots) for different number oflayers with two values of O=JB, namely, O=JB ¼1:0 and 2.0 and for R ¼ Js=JB ¼ 1:0. The resultsare shown in Fig. 2. We can see that, as in thesemi-infinite system, there is a critical value of afor the film, at which the film critical temperatureTc=JB is independent of film thickness. Thenumerical values of ac of the film are exactly thesame as those of the semi-infinite system evenwhen we change O=JB and R. We can also seethat for a aoac, the critical temperature Tc=JBof the film is smaller than the bulk one Tb

c =JBand approaches it when we increase L. Whena > ac; Tc=JB is greater than both Tb

c =JB andT sc=JB of the corresponding semi-infinite system

Fig. 2. The phase diagrams (T sk=JB and Tc=JB) in the (T=JB, a) plane for the semi-infinite system and thin film with decorated spin-1

atoms on the surfaces for R ¼ 1. The solid and dotted lines correspond to O=JB ¼ 1 and 2, respectively. The number accompanying

each curve is the value of L. The broken lines are the corresponding semi-infinite transition temperature.


and tends to the latter for large values of L. Thevalue of ac depends on O=JB and R (see Fig. 3).For a fixed R; ac, increases slightly with increasingO=JB, for example, for R ¼ 1:0, it increases fromac ¼ 0:5874 for O ¼0.0 to ac ¼0.6772 forOc=JB ¼ 2:3528. On the other hand, ac decreaseswith increasing R and vanishes at the critical valueRc of the ratio R. We have also plotted in Fig. 2,the solutions of Eq. (26), which gives the compen-sation temperature T s

k=JB at the surfaces. It is seenthat T s

k=JB increases with increasing a anddecreases with O=JB. This compensation tempera-ture exists only in an interval of a, this intervaldepending on O=JB, R and the thickness of the filmL. For example, for O=JB ¼ 1:0 and R ¼ 10, theinterval is [0.2582, 0.4928] for L ¼ 3 and [0.2582,0.5082] for L ¼ 16. To confirm this predictions, wehave plotted the surface magnetization Ms forO=JB ¼ 1:0, R ¼ 1:0 and for two values of a,namely a ¼ 0:4 and 0.493 (Fig. 4). It is seen thatfor a ¼ 0:4 the Ms curves present a compensationpoint for L ¼ 3 and 16, and for a ¼ 0:493, only theMs for L ¼ 16 presents a compensation point (seethe inside figure with an enlarged scale). Theseresults are consistent with those predicted fromFig. 2. The phase diagrams in the ðTc=JB;RÞ planehave the same topology as those in the ðTc=JB; aÞ

plane, and so we can draw the same conclusions asabove if we permute a and R. The only difference isthat here T s

k=JB is independent of R.In Fig. 5, we present the critical and surface

compensation temperatures of the film and semi-infinite system as a function of the strength of thetransverse field O=JB, for R ¼ 1:0 and a ¼ 0:4 andfor different thicknesses L. The presence of atransverse field, of course, causes a reduction inthe critical temperatures of the film and semi-infinite systems. We find that the ðTc=JB; O=JBÞcurves and the O=JB axis intersect at some criticalpoint, and the value of O=JB corresponding to thispoint is called the critical transverse field Oc=JB.The surface compensation temperature decreasesalso with O=JB and vanishes at some critical valueof O=JB, called the surface compensation fieldOsk=JB. It is seen that the Oc=JB of the film is less

than that of the semi-infinite system Obc=JB and

tends to it when we increase L (here RoRc there isno surface transition).Fig. 6 shows the phase diagrams in the

(Oc=JB; a) plane at T ¼ 0K for R ¼ 1:0 and fordifferent values of L. The phase diagrams have thesame topologies as those in the (Tc=JB; a) plane.That is, there exists a critical value ac of a, at whichthe critical field Oc=JB is independent of film

Fig. 3. The variation of aC with O=JB. The number accompanying each curve is the value of R.


thickness. Below ac;Oc=JB increases with L andtends to Ob

c=JB of the semi-infinite system when Lbecomes large. Above ac, Oc=JB decreases with Land tends to Os

c=JB of the semi-infinite system

when we increase L. ac decreases when we increasethe ratio R and vanishes at Rc. We have alsoplotted in the same figure the variation with a ofthe compensation field Os

k=JB at the surfaces for

Fig. 4. Temperature dependence of the total magnetization on the surface in the Ising ferromagnetic thin film with decorated spin-1

atoms on the surfaces, for a ¼ 0:4 (solid line) and a ¼ 0:493 (dotted line) and for L ¼ 3, 16. For a ¼ 0:493, we also show the

temperature region in an enlarged scale where we have a compensation point for L ¼ 16.

Fig. 5. The phase diagrams (T sk=JB and Tc=JB) in the (T=JB; O=JB) plane for to the semi-infinite system and thin film with decorated

spin-1 atoms on the surfaces for R ¼ 1 and a ¼ 0:4. The dashed line corresponds to the semi-infinite system. The dotted line is the

surface compensation temperature (solution of Eq. (28)).


R ¼ 1:0; these variations are slightly linear. It isseen that a compensation point at the surfacesexists only in a region of a which depends on L; forexample, the region is [0, 0.596] for L ¼ 4 and [0,0.604] for L ¼ 16.In the previous section, we have studied the

surface compensation temperature and transversefield T s

k=JB and Osk=JB obtained by resolving

Eq. (26). Let us now examine the compensationtemperature and transverse field TF

k =JB and OFk=JB

of the thin film. TFk =JB (O

Fk=JB) if they exist, can be

determined by solving the coupled Eqs. (9) and(12) numerically and, introducing the conditionthat MT ¼ 0 for ToTcðOoOcÞ into Eq. (23), wehave first studied the case O=JB ¼ 0. In Fig. 7, wegive the thermal variation of the total magnetiza-tion jMTj=LNA for R ¼ 1, a ¼ 0:4 and for differentL, a compensation point is found in theMT curvesfor 2pLo8. The compensation temperaturedecreases with the increase of L to vanish forLX8. For the system with L ¼ 8, the saturationvalue of MT is given by MT ¼ 0, while it takes afinite value for the system with L > 8, as shown forL ¼ 12. These characteristic features are due to theexistence of decorated ferrimagnetic surfaces. Ingeneral, the surface magnetization ms on the

surface decorated with spin-1 atoms is given byms ¼ @1 at T ¼ 0K and O ¼ 0. Accordingly, thesaturation value ofMs ¼ @3

2. Then the saturationmagnetization of Eq. (23) becomes

MT

NA¼L

2@4 at T ¼ 0 K; ð27Þ

from which the condition MT ¼ 0 is satisfied atL ¼ 8. This indicates an important fact that theexistence of the critical value L ¼ 8 at which thesaturation value of MT reduces to zero is anecessary condition for finding a compensationpoint. Here, one should notice that the magneticmoments of each layer in the bulk (sn for2pnpL@1) are aligned ferromagnetically, whilethe decorated surfaces are ordered ferrimagneti-cally. Thus, the appearance of a compensationpoint in the MT@T curve of the thin film forLo8 is due to the effect of the ferrimagneticallyordered surfaces. We have also studied thevariation of TF

k =JB with a for R ¼ 1 and O ¼0(see Fig. 8). It is seen that for a given L, TF

k =JBincreases linearly with a and disappeaxs for agreater than a critical value of a for which thecompensation temperature is equal to Tc=JB.TFk =JB also decreases when we increase L to

Fig. 6. The variation of the critical transverse field with a for the semi-infinite system (dotted line) and thin film (solid line) with

decorated spin-1 atoms on the surface for R ¼ 1. The dashed line represents the surface compensation transverse field Osk=JB.


vanish for LX8. The variation of TFk =JB with R

for a ¼ 0:4 and O ¼ 0 is plotted in Fig. 9 fordifferent L. We can see that TF

k =JB decreaseswith L and that, for ultrathin film (forexample L ¼ 3) increases with R and reaches asaturation value which depends on R and for thin

film (L ¼ 6 for example) TFk =JB is a quasi-

constant.Let us now turn to the case when O=JBa0. We

have plotted in Fig. 10 the variation of TFk =JB with

O=JB for R ¼ 1, a ¼ 0:4 and for different L. Thepresence of the transverse field causes a reduction

Fig. 7. Temperature dependence of jMTj=LNA in the Ising ferromagnetic thin film with decorated spin-1 atoms on the surface. When

the value of O=JB ¼ 0, a ¼0.4 and R ¼ 1. The value of L is changed from L ¼ 2 to 12.

Fig. 8. Variations of the compensation temperature TFk =JB with a in the Ising ferromagnetic thin film with decorated spin-1 atoms on

the surfaces, when O=JB, R ¼ 1 and the value of L is changed from L ¼ 3 to 7.


in the compensation temperatures of the film andthis compensation temperature decreases andvanishes at some critical value of O=JB called thecompensation field of the film OF

k=JB. In Fig. 11,we show the variations of this compensation field

With a for R ¼ 1 and for different L, the curveshaving the same shape as those of (TF

k =JB; a) thatis, OF

k=JB decreases when we increase L. For agiven L, OF

k=JB increases with a and disappearswhen a is greater than a critical value. Concerning

Fig. 9. Variations of the compensation temperature TFk =JB with R in the Ising ferromagnetic thin film with decorated spin-1 atoms on

the surfaces, when O=JB, R ¼ 1 and the value of L is changed from L ¼ 3 to 7.

Fig. 10. Variations of the compensation temperature TFk =JB with O=JB in the Ising ferromagnetic thin film with decorated spin-1

atoms on the surfaces, when a ¼ 0:4, R ¼ 1 and the value of L is changed from L ¼ 3 to 7.


the variations OFk=JB with R, the same conclusions

as for (TFk =JB; R) curves can be given here.

4. Conclusion

In this work, we have studied the phasediagrams and magnetization curves of both spin-12 Ising film and semi-infinite spin-12 Ising systemwith decorated spin-1 atoms on the surfaces withinthe framework of the effective field theory with aprobability distribution technique. We have shownthat the systems exhibit many interesting phenom-ena, such as the possibility of the existence ofcompensation points on the surface and in thefilm. These compensation temperatures exist onlyin an interval of a, which depends on O=JB, R, andthe thickness on the film. In particular, thecompensation temperature TF

k =JB of the film existsonly if 2pLp7. We have also shown that thesystem exhibits a compensation transverse field atT ¼ 0K. The existence of compensation trans-verse fields depends on R and on the thickness ofthe film L. The numerical values of the compensa-tion temperature and transverse field on thesurface T s

k=JB and Osk=JB are independent of L.

However, TFk =JB and OF

k=JB for the film dependstrongly on L.

Acknowledgements

This work was supported by the PARS GrantNo. Physique 22.

References

[1] L. Neel, Ann. Phys. 3 (1948) 137.

[2] A. Herpin, Theorie du Magnetisme, Presse universitaire de

France, Saclay, 1968.

[3] I. Syozi, in: C. Domb, M.S. Green (Eds.), Phase

Transitions and Critical Phenomena, Vol. 1, Academic

Press, New York, 1972.

[4] I. Syozi, H. Nakano, Prog. Theor. phys. 13 (1955) 69.

[5] M. Hattori, Prog. Theor. phys. 35 (1966) 600.

[6] T. Kaneyoshi, Physica A (229) (1996) 166.

[7] T. Kaneyoshi, Physica A (237) (1997) 554.

[8] T. Kaneyoshi, Phys. Rev. B (55) (1997) 12 497.

[9] B.K. Chakrabarty, A. Dutta, P. Sen, Quantum Ising

phases and transitions in transverse Ising models, Lecture

Notes in Physics, Vol. M41, Spinger, Verlag, Heidelberg,

1996.

[10] P.G. De Gennes, Solid State Commun. (1) (1963) 132.

[11] P. Pfenty, Ann. Phys. (NY) 57 (1670) 79.

Fig. 11. The variation of the compensation transverse field OFk =JB with a for the Ising ferromagnetic thin film with decorated spin-1

atoms on the surfaces, for R ¼ 1. The number accompanying each curve is the value of L.


[12] R.J. Elliot, I.D. Saville, J. Phys. C 7 (1974) 3145.

[13] R.B. Stinshcombe, J. Phys. C 14 (1981) L263.

[14] T. Yokota, J. Phys. C 21 (1988) 5987.

[15] F.C.S. Baretto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39

(1981) 1103.

[16] F.C.S. Baretto, I.P. Fittipaldi, Physica A 129 (1985) 360.

[17] K. Binder, in: C. Domb, J.L. Lebowitz (Eds.), Phase

transition and critical phenomena, Vol. 8, Academic Press,

New York, 1983, p. 2.

[18] H.W. Diehl, in: C. Domb, J.L. Lebowitz (Eds.), Phase

Transition and Critical Phenomena, Vol. 10, Academic

Press, New York, 1986, p. 76.

[19] T. Kaneyoshi, Introduction to Surface Magnetism, CRC

Press, Boca Raton, 1990.

[20] C. Ran, C. Jin, M. Roberts, J. Appl. Phys. 63 (1988) 3667.

[21] M. Polak, L. Rubinovich, J. Deng, Phys. Rev. Lett. 74

(1995) 4059.

[22] H. Tang, Phys. Rev. Lett. 71 (1985) 444.

[23] W. Durr, M. Taborelli, O. Paul, R. Germar, W. Gudat, D.

Pescia, M. Landolt, Phys. Rev. Lett. 62 (1989) 206.

[24] Z.Q. Qiu, S.H. Mayer, C.J. Gutierrez, H. Tang, J.C.

Walwer, Phys. Rev. Lett. 63 (1989) 1649.

[25] D.P. Pappas, K.P. Kamper, B.P. Miller, H. Hopster, D.E.

Fowler, C.R. Brundle, A.C. Luntz, Z.X. Shen, Phys. Rev.

Lett. 66 (1991) 504.

[26] T. Balcerzak, J. Miejnicki, G. Wiatrowski, A. Urbaniak-

Kucharczyk, J. Phys. Condens. Matter 2 (1990) 3955.

[27] G. Wiatrowski, J. Miejnicki, T. Balcerzak, Phys. Stat. Sol.

B 164 (1991) 299.

[28] Q. Hong, Phys. Rev. B 41 (1990) 9621.

[29] X.Z. Wang, X.Y. Jiao, J.J. Wang, J. Phys. Condens.

Matter. 4 (1992) 3651.

[30] C. Jia, X.Z. Wang, Phys. Rev. B 51 (1995) 6715.

[31] J.W. Tucker, E.F. Sarmento, C. Cressoni, J. Magn. Magn.

Mater. 147 (1995) 24.

[32] T. Kaneyoshi, T. Balcerzak, Physica A 197 (1993) 667.

[33] T. Balcerzak, T. Kaneyoshi, Physica A 206 (1994) 176.

[34] S.C. Lii, X.Z. Wang, Physica A 232 (1996) 315.

[35] M. Saber, A. Ainane, F. Dujardin, B. Stb, Phys. Rev. B 59

(1999) 6908.

[36] H. Wan, A. Tsoukatos, G.C. Hadjipanayis, J. Magn.

Magn. Mater. 125 (1993) 157.

[37] G. Srinivasan, B. Uma, M. Rao, J. Zhao, M. Sachra, Appl.

Phys. Lett. 59 (1991) 372.

[38] M. Jascur, T. Kaneyoshi, Phys. Rev. B 54 (1996) 9232.

[39] T. Kaneyoshi, Physica B 245 (1998) 233.

[40] J.W. Tucker, M. Saber, L. Peliti, Physica A 206 (1994) 497.

[41] H.B. Callen, Phys. Lett. 4 (1963) 161.

[42] A. Elkouraichi, M. Saber, J.W. Tucker, Physica A 213

(1972) 576.

[43] I. Tamura, E.F. Sarmento, T. Kaneyoshi, J. Phys. C 17

(1984) 3207.

[44] A. Corciovei, G. Costache, D. Vamanu, Solid State Phys.

27 (1972) 237.


the transverse ising thin film with decorated ferrimagnetic surfaces

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