Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

De Marchi, G.

International Centre for Theoretical Physics, Trieste (Italy)

Lo Russo, S.

Mattei, G.

Mazzoldi, P.

Saber, A.

Saber, A

Mazzoldi, P

De Marchi, G

Lo Russo, S

Mattei, G

CERN Document Server

Available at: http://www.ictp.trieste.it/~pub off IC/2001/111United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSPHASE TRANSITIONS OF A SPIN-ONE ISINGFERROMAGNETIC SUPERLATTICEA. Saber1Departement de Physique, Faculte des Sciences, Universite Moulay Ismail,B.P. 4010, Meknes Morocco,2INFM, Dipartimento di Fisica, Universita di Padova, Via Marzolo 8, 35131 Padova, ItalyandThe Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyP. Mazzoldi, G. De Marchi, S. Lo Russo and G. MatteiINFM, Dipartimento di Fisica, Universita di Padova, Via Marzolo 8, 35131 Padova, Italy.AbstractUsing the eective eld theory with a probability distribution technique, the magnetic prop-erties in an innite superlattice consisting of two dierent ferromagnets are studied in a spin-oneIsing model. The dependence of the Curie temperatures are calculated as a function of two slabsin one period and as a function of the intra- and interlayer exchange interactions. A criticalvalue of the exchange reduced interaction above which the interface magnetism appears is found.MIRAMARE { TRIESTESeptember 20011E-mail: Saber@padova.infm.it2Permanent address.1 INTRODUCTIONThe magnetic properties of the articially fabricated superlattices that consists of two or moreferromagnets materials have been widely studied over the years because their physics propertiesdier dramatically from simple solids formed from the same materials. The development of lmdeposition technique has aroused great interest in the synthesis and study of superlattices inother materials. In layered ferromagnetic materials, it has been found experimentally that onecan obtain a rich variety of magnetic behaviour depending on the materials, the thickness andthe number of slabs and of the applied eld [1-5]. A number of theoretical works have beendevoted to the magnetic and phase transition properties of superlattices formed from layers ofdierent materials [6-14].In the present paper, we will study the critical properties of an innite superlattices madeup of elementary units of two kinds of atoms. In each elementary unit, there are Laatomiclayers of magnetic ions of type A and Lbatomic layers of magnetic ions of type B, using theeective eld theory with the probability distribution technique in its simplest form [15,16].This technique is believed to give more exact results than those of the standard mean-eldapproximation. In section 2 we outline the formalism and derived the equation that determinethe transition temperature. Numerical results are discussed in section 3. A brief conclusion isgiven in section 4.2 FORMALISMConsider an innite superlattice consisting of two dierent ferromagnetic materials A and B. Forsimplicity, we restrict our attention to the case of the simple Ising-type structure. The periodiccondition suggest that we only have to consider one unit cell. The situation is shown in Fig. 1.The exchange coupling between the nearst-neighbor spins in A(B) is denoted by Jaa(Jbb), thenwe take Jaaas the unit of energy. Here, we consider the interface to be composed of two layers(Laand La+1). Jabstands for the exchange coupling between the nearst-neighbor spins acrossthe interface. The number of atomic layers in material A(B) is La(Lb) and the thickness of theunit cell is L = La+ Lb. The spin-one Ising Hamiltonian of the system is given byH =  Xn;n0Xr;r0Jnn0znrzn0r0; (1)where znrdenotes the z component of a quantum spin nrof magnitude nr= 1 at site (n; r);(n; n0), are plane indices and (r,r0) are dierent sites of the planes, and Jnn0is the strength of theferromagnetic exchange interaction which is only plane dependent. The statistical properties ofthe system are studied using an eective eld theory that employs the probability distributiontechnique, which based on a single-site cluster comprising just a single selected spin, labeled(n; r), and the neighbouring spins with which it directly interacts. To this end, the Hamiltonian2is split into two parts, H = Hnr+H0; where Hnris that part of the Hamiltonian containing thespin (n; r) ; namelyHnr=  0@Xn0; r0Jnn0zn0r01Aznr; (2)The Starting point of the eective eld theory is a set of formal identities of the typehh(znr)pici =Trnr[(znr)pexp ( Hnr)]Trnr[exp ( Hnr)](3)where h(znr)picdenotes the mean value of (znr)pfor a given conguration c of all other spins,h:::i denotes the average over all spin congurations zn0r0; T rnrmeans the trace performed over(znr)ponly,  = 1=kBT with kBthe Boltzmann constant and T the absolute temperature. Fora xed conguration of neighbouring spins of the site (n; r) the longitudinal and the transversemagnetizations and quadrupolar moments of any spin at site (n; r) are given by,mnrz= hhznrici = hf1z(A)i (4)qnrz=DD(znr)2EcE= hf2z(A)i (5)wheref1z(A) =2 sinh (A)1 + 2 cosh (A)(6)f2z(A) =2 cosh (A)1 + 2 cosh (A)(7)withA =Xn0Xr0Jnn0zn0r0; (8)where the rst and second sums run over all possible congurations of atoms environing or lyingon the (n; r) site, respectively. Each of these congurations can be characterized by numbers ofmagnetic atoms in the planes n  1; n; n+ 1. To perform thermal averaging on the right-handside of equations (4) and (5) one now follows the general approach described in [15,16]. Thuswith the use of the integral representation method of Dirac Æ distribution, equations (4) and(5) can be written in the formhhznrici =Zd!f1z(!;B)12Zdt exp (i!t)Yn0r0exp  itJn;n0zn0r0(9)DD(znr)2EcE=Zd!f2z(!;B)12Zdt exp (i!t)Yn0r0exp  itJn;n0zn0r0(10)In the derivation of the equations (9) and (10), the commonly used approximation has beenmade according to which the multi-spin correlation functions are decoupled into products of thespin averages (the simplest approximation of neglecting the correlations between dierent siteshas been made). That is< zj(zk)2: : : zl>< zj>< (zk)2> : : : < zl> for j 6= k : : : 6= l: (11)3Then, as hhznrici andDD(znr)2EcEare independent of r, we introduce the longitudinal magneti-zation and the longitudinal quadrupolar moment of the n  th layer, on the basis of equations(4) and (5), with the use of the probability distribution of the spin variables [15,16]P (znr) =12[(qnz mnz) Æ (znr+ 1) + 2 (1  qnz) Æ (znr) + (qnz+mnz) Æ (znr  1)] (12)Allowing for the site magnetizations and quadrupolar moments to take dierent values in eachatomic layer parallel to the surfaces of the superlattice, and labeling them in accordance withthe layer number in which they are situated, the application of Eqs. (4), (9) and (12) yields thefollowing set of equations for the layer longitudinal magnetizationsmnz= 2 N 2N0NX=0N X=0N0X1=0N0 1X1=0N0X2=0N0 2X2=02+1+2CNCN CN01CN0 11CN02CN0 22(1  2qnz)(qnz mnz)(qnz+mnz)N  (1  2qn 1;z)1(qn 1;z mn 1;z)1(qn 1;z+mn 1;z)N0 1 1(1  2qn+1;z)2(qn+1;z mn+1;z)2(qn+1;z+mn+1;z)N0 2 2f1z(yn) (13)whereyn= [Jn;n(N     2) + Jn;n 1(N0  1  21) + Jn;n+1(N0  2  22)] (14)N and N0are the numbers of nearest neighbours in the plane and between adjacent planesrespectively ( N = 4 and N0= 1 in the case of a simple cubic lattice which is considered here)and Clkare the binomial coeÆcients, Clk=l!k!(l k)!: The periodic condition of the superlatticehas to be satised, namely m0z= mLz, mL+1;z= m1zand q0z= qLz, and qL+1;z= q1z: Theequations of the longitudinal the quadrupolar moments are obtained by substituting the functionf1zby f2zin the expression of the layer longitudinal magnetizations. This yieldsqnz= mnz[f1z(yn)! f2z(yn)] (15)In this work we are interested with the calculation of the ordering near the transition Curietemperature. The usual argument that mnztends to zero as the temperature approachs itscritical value, allows us to consider only terms linear in mnzbecause higher order terms tend tozero faster than mnzon approaching a Curie temperature. Consequently, all terms of the orderhigher than linear terms in eqs. (13) that give the expressions of mnzcan be neglected. Thisleads to the set of simultaneous equationsmnz= An;n 1mn 1;z+An;nmnz+An;n+1mn+1;z(16)orAmz=mz(17)4where mzis a vector of components (m1z;m2z; :::;mnz; :::;mLz) and the matrix A is symmetricand tridiagonal with elementsAi;j= Ai;iÆi;j+Ai;j(Æi;j 1+ Æi;j+1) (18)The system of eqs. (17) is of the formMmz= 0 (19)whereMi;j= (Ai;j  1) Æi;j+Ai;j(Æi;j 1+ Æi;j+1) (20)The only non zero elements of the matrix M are given byMn;n 1= 2 N 2N0NX=0N X=0N0X1=0N0 1X1=0N0X2=0N0 2X2=01Xi=0N0 (1+1)Xj=0( 1)i2+1+2Æ1;i+j(21)CNCN CN01CN0 11CN02CN0 22C1iCN0 (1+1)j(1  tn)(1  tn 1)1(1  tn+1)2tN nt(N0 1) (i+j)n 1tN0 2n+1f1z(yn)Mn;n= 2 N 2N0NX=0N X=0N0X1=0N0 1X1=0N0X2=0N0 2X2=0Xi=0N (+)Xj=0( 1)i2+1+2Æ1;i+jCNCN CN01CN0 11CN02CN0 22CiCN (+)j(1  tn)(1  tn 1)1(1  tn+1)2tN  (i+j)nt(N0 1)n 1tN0 2n+1f1z(yn)  1 (22)Mn;n+1= 2 N 2N0NX=0N X=0N0X1=0N0 1X1=0N0X2=0N0 2X2=02Xi=0N0 (2+2)Xj=0( 1)i2+1+2Æ1;i+jCNCN CN01CN0 11CN02CN0 22C1iCN0 (2+2)j(1  tn)(1  tn 1)1(1  tn+1)2tN nt(N0 1)n 1tN0 2 (i+j)n+1f1z(yn) (23)where the tnare the values of the qnzwhen mnz= 0 at the critical point which are given by.tn= 2 N 2N0NX=0N X=0N0X1=0N0 1X1=0N0X2=0N0 2X2=02+1+2CNCN CN01CN0 11CN02CN0 22(1  2tn)tN n(1  tn 1)1t(N0 1)n 1(1  2tn+1)2tN0 2n+1f2z(yn) (24)All the information about the Curie temperature of the system is contained in eq. (19). Upto know we did not dene the values of the exchange interactions; the terms in matrix (19)are general ones. In a general case, for arbitrary coupling constants and superlattice thicknessthe evaluation of the Curie temperature relies on the numerical solution of the system of linearequations (19). These equations are fullled if and only ifdetM = 0 (25)5This condition can be satised for L dierent values of the Curie temperature Tc: In this paper,we take Jaaas the unit of the energy, the length is measured in units of the lattice constant. Letus begin with the evaluation of the Curie temperature with an example: the Curie temperatureof the spin one Ising model for the simplest possible " bulk case" of a material A (i.e. N = 4;N0= 1; Ji;j= Jaa). Then we can reduce det M to the following formdetM =a b bb a bb a b::: ::: ::: ::: ::: :::: ::: ::: ::: ::: ::: :::b a b::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: :::b a bb a bb b a(L;L)(26)whose value isdetMbulk=LYk=1a+ 2b cos2 (k   1)L(27)where the elements in the above determinant are given bya =Mn;n(Jn;n= Jn;n 1= Jn;n+1= Jaa) (28)b =14(a+ 1) (29)and L in the "bulk" case is an arbitrary number. Now we obtain the Curie temperature fromthe condition given bydetMbulk= 0 (30)We apply the obtained formalism to a two component magnetic superlattice consisting of atomsof type A and B which alternate as AAA...ABBB...B. The periodic condition suggests that weonly have to consider one unit cell which interacts with its nearest neighbours via the interlayercoupling. Let us consider a simple superlattice of L layers n = 1; 2; :::Laconsist of atoms oftype A, whereas layers n = La+1; :::L consist of atoms of type B. In this case we can representdetM asdetM = ca11 b11 c11::: ::: ::: ::: ::: ::: ::: ::: ::: :::1 c111 a1b1b2a211 c21::: ::: ::: ::: ::: ::: ::: ::: ::: :::1 c21b21 a2(L;L)(31)6where the elements in the determinant are given byc = 1M1;1!2 1M2;1!La 2 1MLa+1;La+1!2 1MLa+2;La+1!Lb 2(32)a1=M1;1=M1;2; a1=MLa+1;La+1=MLa+1;La+2b1=M1;1=M1;L; b2=MLa+Lb 1;La+1=MLa+1;La+2c1=M2;2=M2;1; c2=MLa+2;La+2=MLa+2;La+1(33)By solving eq. (31) numerically, we can obtain the critical properties of the superlattice.3 RESULTS AND DISCUSSIONFor the pure Ising model, we obtain the critical value of the temperature kBTBc=Jaa= 3:518 fromeq. (30) which is intermediate between the low-temperature series expansion result, TSEc=Jaa=3:194 [17], and the mean-eld theory result, kBTMFTc=Jaa= 4 [18] and is the same relsultreported by Fittipaldi et al [19] for the bulk media.From eq. (31), we can obtain the Curie temperature of the innite superlattice kBTc=Jaafor a given values of the coupling exchanges Jbband Jaband a xed number of layers of thetwo components. For the case of Jab= 0, the superlattice reduces to two slabs, so there existssepareted phase transitions in two slabs. But we are interested in the case of Jab6= 0. Westudy the dependence of phase transion temperature on the interface coupling Jab. We xthe layer-number of slab A (Lb= 2) and let the layer-number Lbof slab B changes. Suchresults are shown in Fig. 2. We plot the Curie temperature of the innite superlattice againstJab=Jaafor a xed values of Jbb=Jaa, Lband the layer-number of sab A changes. It is easy tosee from this gure that there exists a critical value of the reduced interface coupling strengthJab=Jaa)crit= 1:586 which is independant of the thickness Lbof slab B. As the layer-number Laof slabA increases, the Curie temperature of the superlattice decreases for Jab=Jaa)crit< Jab=Jaaand is approximatly lenear in good agreement with the result of the spin 1=2 case[13], butincreases for Jab=Jaa)crit> Jab=Jaa: At this critical point, the Curie temperature does notdepend on the layer-number Laof slab A and is equal to its bulk Curie temperature.Fig. 3 corresponds to Jbb=Jaa= 0:75 and Lb= 2. The horizonthal dotted line correspondsto the case of Jab=Jaa)crit= 1:586, such that kBTBc=Jaa= kBTc=Jaa= 3:518 and remainsconstant for any La. We see that kBTc(La)=Jaaincreases with the increase of Jab=Jaa. ForJab=Jaa)crit< Jab=Jaa= 2: the Curie temperature of the superlattice is higher than the bulkCurie temperature of slab A kBTBc=Jaaand kBTc(La)=Jaadecreases with the increase of Lato reach an asymptotic value. For Jab=Jaa)crit> Jab=Jaa= 1; the Curie temperature of thesuperlattice is smaller than the bulk Curie temperature kBTBc=Jaaand kBTc(La)=Jaaincreaseswith the increase of Lato reach asymptoticaly kBTBc=Jaafor large value of La.Fig. 4 shows the dependence of Jab=Jaa)criton Jbb=Jaafor xed value of Lb, the criticalcoupling exchange Jab=Jaais independent of the layer-number Laof slab A [see Fig. 4]. We7note that when Jbb=Jaadecreases the value of Jab=Jaa)critincreases and when Jbb=Jaa= 1,Jab=Jaa)crit= 1 as expected.In Fig. 5, we show the Curie temperature kBTc=Jaaagainst Jbb=Jaafor Lb= 2 xed, Jab=(Jaa+Jbb)=2 and for dierent values of La= 3; 4 and 5. For Jbb=Jaa> 1; the Curie temperatureincreases with the increase of Laand is always larger than the bulk Curie temperature of theslab A which is kBTBc=Jaa= 3:518. For Jbb=Jaa< 1; we have just the opposit situation. Noticethat for small values of Jbb=Jaa, the Curie temperature is insensitive to La. For Jbb=Jaa= 1,the Curie temperature is independant of Lalayer-number of the slab A of the two componentssuperlattice and equal to kBTBc=Jaaas expected.4 CONCLUSIONIn conclusion, the properties of the phase transition, a ferromagnetic innite superlattices de-scribed the spin-one Ising model in eective eld theory, have been discussed in this paper.The dependence of the Curie temperature on the inter- and intra-layer strength coupling of thesimplest case when the superlattice is innite has been obtained.AcknowledgmentsThis work was initiated during a visit of the author, A. Saber at the 'Universita di Padova,Italy' and was completed during a visit of the same author to the AS-ICTP, Trieste, Italy in theframe of " ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy".8References[1] Kwo, J., Georgy, E.M., McWhan, D.B., Hong, M., Dsalvo, F.J., Vettier, C. and Wer,J.F., Phys. Rev. Lett. 55, 1985, 1402.[2] Majkrzak, C.F., Cable, J.W. and Vettier, C., Phys. Rev. Lett., 1986, 2700.[3] Camley, R.E. and Barnes, J., Phys. Rev. Lett., 63, 1986, 664.[4] Krebs, J.J., Lubirtz, P., Chaiken, A. and Prinz, G.A., Phys. Rev. Lett., 63, 1989, 1645.[5] Camiey, R.E., Kwo, J., Hong, M. and Chien, C.L., Phys. Rev. Lett., 64, 1990, 2703.[6] Hinchey, L.L. and Mills, D.L., Phys. Rev., B 33, 1986, 3329.[7] Qu, B.D., Zhong, W.L., Zhang, P.L., Physics Letters, A 189, 1994, 419.[8] Camley, R.E. and Tilley, D.R., Phys. Rev., B 37, 1988, 3413.[9] Fisman, F., Schwabl, F., and Schwenk, D., Phys. Rev. Lett., A 121, 1987, 192.[10] Gritz-Saaverda, J., Aguilra-Grania, F. and Moran-Lopez, J.L., Solid State Commun.,82, 1992, 5891.[11] Sy, H.K. and Ow, M.H., J. Phys. Condens. Matter, 4, 1992, 5891.[12] A. Saber, I. Essaoudi, A. Ainane, M. Saber, Physica status Solidi (b) 206, 161 (1998)[13] A. Saber, A. Ainane, M. Saber, I. Essaoudi, F. Dujardin, and B. Stebe, Phys. Rev. B60 4149 (1999).[14] A. Saber, G. Mattei and X. Zhang Eur. Phys. J. B 19, 393-399 (2001).[15] J.W. Tucker, M. Saber and L. Peliti 1994 Physica A 206, 497.[16] Elkouraychi A, Saber M and Tucker J. W. 1995 Physica A 213, 576.[17] P.F. Fox, A. J. Guttmann, J. Phys. C 6 (1973) 913.[18] Smart JS 1966 Eective Field Theories of Magnetism (New York: W B Saunders).[19] I. P. Fittipaldi, E. F. Sarmento, and T. Kaneyoshi, Physica A 186, 591 (1992).9Figure CaptionsFig. 1. Sketch of a unit cell of the superlattice.Fig. 2. The Curie temperature kBTc=Jaaversus Jab=Jaafor a xed values of Lb= 2 andJbb=Jaa= 0:75. The number accompanying each curve denotes the value of La.Fig. 3. The dependence of the Curie temperature kBTc(La)=Jaaon the layer-number Laofthe slab A for a xed values of Lb= 2 and Jbb=Jaa= 0:75. The number accompanying eachcurve denotes the value of Jab=Jaa.Fig. 4. The reduced critical interface exchange interaction Jab=Jaa)criton the reducedexchange interaction Jab=Jaa.Fig. 5. The dependence of the Curie temperature on Jbb=Jaafor xed value of Lb= 2,Jab= (Jaa+ Jbb)=2 with Jaa= 1 and dierent values of La= 3, 4 and 5 denoted by numbers.10

Phase transitions of a spin-one Ising ferromagnetic superlattice

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Phase transitions of a spin-one Ising ferromagnetic superlattice

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