2 Ising superlattice in a transverse field

Physica A 269 (1999) 322–328

www.elsevier.com/locate/physa

The ferromagnet spin-1/2 Ising superlattice in a transverse ÿeld F. Dujardina; ∗ , B. StÃebÃea , A. Ainanea;b , M. Sabera;b

a Laboratoire

de ThÃeorie de la MatiÂere CondensÃee, UniversitÃe de Metz - Institut de Physique et d’ Electronique, 1 Boulevard Arago, 57078 Metz Cedex 3, France b Department of Physics, Faculty of Sciences, University Moulay Ismail, B. P. 4010, Meknes, Morocco Received 15 December 1998

Abstract The phase transitions of a ferromagnet spin-1/2 Ising superlattice consisting of two dierent materials are examined using the eective ÿeld theory. The critical temperature of the system is studied as a function of the thickness of the constituents in a unit cell, exchange interactions in each material and transverse ÿeld. A critical value of the interface exchange interaction above c 1999 Elsevier Science B.V. All rights which the interface magnetism appears is found. reserved. PACS: 75.10.−b; 75.40.Mg; 75.70.Cn Keywords: Magnetic superlattices; Phase transition; Critical temperature; Ising model

1. Introduction The magnetic properties of artiÿcially fabricated superlattices have been widely studied over the years because their physical properties dier dramatically from simple solids formed from the same materials (for a review see Ref. [1]). With the advances in modern vacuum science, in particular the epitaxial growth technique, it is possible to grow very thin ÿlms of predetermined thickness, even of a few monolayers. Ferromagnetic ordering in some of these monolayers has been reported and the dependence of the critical temperature on the thickness of the ÿlms has also been measured. E.g., some systems that have been studied are Fe on Pd [2], Ni on Cu [3,4], Co on Cu [5], Ni on W [6], Fe on W [7] and Gd on Cu [8]. ∗

E-mail address: [email protected] (F. Dujardin)

c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 0 3 - X

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In two earlier papers [9,10] two of the present authors have studied the eects of a uniform transverse ÿeld and surface magnetism on the critical behavior of an alternating ferromagnet spin-1/2 Ising superlattice. The eective ÿeld theory with a probability distribution technique [11] which gives more satisfactory results than the standard mean ÿeld theory has been employed and the results for the critical temperatures and the critical exchange couplings have been determined. We study in this paper the eects of a transverse ÿeld and exchange couplings on the critical temperature in the ferromagnet spin-1/2 Ising superlattice consisting of two ferromagnetic materials with dierent bulk properties. In particular, we consider the two constituents A and B with dierent bulk transition temperatures as a simple model, i.e. TcA 6= TcB . The interface is in general dierent in nature from both bulks even if the bulk critical temperatures are the same. We use the eective ÿeld theory [11] that accounts for the self-spin correlation function. In Section 2, we outline the eective ÿeld theory and derive the equation that determines the critical temperature. Numerical results are discussed in Section 3 where the existence of the interface magnetic phase transition is shown and the critical value of the interface coupling is determined. A brief conclusion is given in Section 4. 2. Eective ÿeld theory Let us consider a superlattice consisting of two dierent ferromagnetic materials A and B with a simple cubic structure. The periodic condition suggests that we only have to consider one unit cell which interacts with its nearest-neighbors via the interface coupling. The coupling strength between nearest neighboring spins in A (B) is denoted by Ja ; (Jb ), while Jab stands for the exchange coupling between the nearest neighbor spins across the interface. The number of atomic layers in A (B) is La (Lb ) and the thickness of the cell is L = La + Lb . The Hamiltonian of this system is given by X X Jij iz jz −

ix ; (1) H =− i

(i; j)

where the ÿrst sum runs over all nearest neighbor pairs and the second sum is taken → over all the spins, iz and ix denote the z and x components of a quantum spin i of magnitude  = 12 at site i; Jij stands for one of the three coupling constants depending on where the spin pair is located and represents the transverse ÿeld. The longitudinal and transverse magnetizations of the spin at any site of the nth layer are given by [9,10] * !+ X Jij jz ;

; (2) mn = hn i = f j

where  = z and x for the longitudinal and transverse magnetizations respectively and   y 1 1 2 2 1=2 tanh (y + ) ; (3) fz (y; ) = 2 (y2 + 2 )1=2 2 fx (y; ) = fz ( ; y) ;

(4)

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with = 1=kB T (we take kB = 1 for simplicity) and T is the temperature. In Eq. (2), h· · ·i indicates the usual canonical ensemble thermal average for a given conÿguration, and the sum runs over all nearest-neighbors of the spin at any site of the nth layer. To perform thermal averaging on the right hand side of Eq. (2), we follow the general approach described in Ref. [11]. First of all, in the spirit of the eective ÿeld theory, multispin-correlation functions are approximated by products of single spin averages. We then take advantage of the integral representation of the Dirac’s delta distribution, in order to write Eq. (2) in the form. #) Z " Z ( Y 1 dt exp(i!t) hexp(−it Jij jz )i d!f (!; ) : (5) mn = 2 j We now introduce the probability distribution of the spin variables (for details see Ref. [11])      1 1 1 (1 − 2mnz ) nz + + (1 + 2mnz ) nz − : (6) P(nz ) = 2 2 2 Using this expression and Eq. (5), we obtain the following equations for the magnetization of the layer n: mn = 2−N −2N0

N0 X N0  N X X

CN CN10 CN20 (1 − 2mnz ) (1 + 2mnz ) N −

=0 1 =0 2 =0

×(1 − 2mn−1; z )1 (1 + 2mn−1; z ) N0 −1 ×(1 − 2mn+1; z )2 (1 + 2mn+1; z ) N0 −2  1 [Jnn (N − 2) + Jn; n−1 (N0 − 21 ) ×f 2  : + Jn; n+1 (N0 − 22 )];

(7)

In this equation, N and N0 denote, respectively the numbers of nearest neighbors in the plane and between adjacent planes. For the simple cubic lattice which we will consider l! . here, one has N = 4 and N0 = 1: Ckl are the binomial coecients, Ckl = k!(l−k)! We have thus obtained the self-consistent equations (7) for the layer longitudinal and transverse magnetization that can be solved directly by numerical iteration. However, since we are interested in the calculation of the longitudinal order near the critical temperature, the usual argument that the layer longitudinal magnetizations mnz tend to zero as the temperature approaches its critical value, allows us to consider only terms linear in mnz , because higher order terms tend to zero faster than mnz on approaching the critical temperature. Consequently, all terms of order higher than linear in Eq. (7) can be neglected. This leads to the following system of equations: mnz = An; n−1 mn−1; z + An; n mnz + An; n+1 mn+1; z ;

(8)

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which can be written as Mmnz = 0 ;

(9)

where the elements of the matrix M are given by Mij = (Ai; i − 1) i; j + Ai; j (i; j−1 + i; j+1 ) :

(10)

The only non-zero elements of the matrix M are given by  N −  N0 X N0 X N X X X −N −2N0 (−1)i 2i+j 1; i+j CN CN10 CN10 Ci CjN − Mn; n = 2 =0 1 =0 2 =0 i=0 j=0



1 [Jnn (N − 2) + Jn; n−1 (N0 − 21 ) 2  −1; + Jn; n+1 (N0 − 22 )];

×fz

Mn; n−1 = 2−N −2N0 

(11)

1 N0 −1  N0 X N0 X N X X X

(−1)i 2i+j 1; i+j CN CN10 CN20 Ci1 CjN0 −1

=0 1 =0 2 =0 i=0

j=0

1 [Jnn (N − 2) + Jn; n−1 (N0 − 21 ) 2  ; + Jn; n+1 (N0 − 22 )];

×fz

−N −2N0

Mn; n+1 = 2



(12)

2 N0 −2  N0 X N0 X N X X X

(−1)i 2i+j 1; i+j CN CN10 CN20 Ci2 CjN0 −2

=0 1 =0 2 =0 i=0

j=0

1 [Jnn (N − 2) + Jn; n−1 (N0 − 21 ) ×fz 2  ; + Jn; n+1 (N0 − 22 )];

(13)

with the periodic boundary conditions M1;0 = M1; L and ML; L+1 = ML; 1 . All the information about the critical temperature of the system is contained in Eq. (9). Up to now, we did not assign precise values to the coupling constants, transverse ÿeld and unit cell width: the terms in matrix (9) are general ones. In a general case, for arbitrary coupling constants, transverse ÿeld and unit cell width, the evaluation of the critical temperature relies on numerical solution of the system of linear equations (9). These equations can be satisÿed by non-zero magnetization vectors mnz only if det M = 0 :

(14)

In general, the system of linear equations (9) can be satisÿed for L = La + Lb dierent values of the critical temperature from which we choose the one corresponding to the highest possible transition temperature. This value of the critical temperature

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corresponds to a solution where m1z ; m2z ; : : : ; mLz are all positive, which is compatible with a ferromagnetic longitudinal ordering. The other solutions correspond in principle to other types of ordering that usually do not occur here. 3. Results and discussion Throughout this paper, we take Ja as the unit of energy, we introduce the reduced coupling constants R1 = Jb =Ja ; R2 = Jab =Ja and in our numerical calculation, length is measured in units of the lattice constant. We note that when Jab = 0 (R2 = 0); the superlattice reduces to two sets of simple ÿlms so there exists separated phase transitions in two slabs. But here we are interested in the case of Jab 6= 0 (R2 6= 0). Therefore without loss of generality, we assume the critical temperature of slab A higher than that of slab B, that is Ja ¿Jb (R1 61), and hence TcA =Ja ¿TcB =Ja where TcA =Ja is the bulk critical temperature of a uniform lattice of material A and TcB =Ja = R1 TcA =Ja . We ÿrst show the critical temperature of the superlattice Tc =Ja as a function of the unit cell width L in Fig. 1. The solid and dotted lines correspond, respectively, to the values of the transverse ÿeld =Ja = 0 and 1:5: Three cases of dierent Lb values are plotted for ÿxed R1 = 0:4 and R2 = 0:6: We ÿrst note that for a given L; Tc =Ja decreases with the increase of the transverse ÿeld. It also decreases with the increase

Fig. 1. Dependence of the critical temperature Tc =Ja on the thickness L when R1 = 0:4 and R2 = 0:6. The solid and dotted lines correspond to =Ja = 0, and 1.5, respectively.

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Fig. 2. Tc =Ja versus L for Lb = 5; when R1 = 0:8 but varying R2 , which is written beside each curve. The solid, dotted and dashed lines correspond to =Ja = 0; 1:5; 2:2, respectively.

of Lb for a given =Ja : This is because we have assumed Jb ¡ Ja (R1 ¡ 1) in our calculation. For given =Ja and Lb , it is observed that Tc =Ja increases steadily as L increases indeÿnitely. All the curves approach the bulk critical temperature of material A; TcA =Ja ( =Ja ). To study the eects of the interface exchange coupling R2 and the transverse ÿeld

=Ja on the critical temperature of the superlattice, Tc =Ja is calculated for various unit cell widths L, and transverse ÿelds =Ja but ÿxed R1 = 0:8 and Lb = 5 (Fig. 2). It is interesting to note that, for every choice of L, the critical temperature decreases with the increase of the transverse ÿeld as expected. Tc =Ja increases with the increase of R2 for given values of L and =Ja . We note that, for any given value of =Ja ; there exists a critical value of the interface exchange coupling Rc2 such that, when R2 ¿Rc2 and consequently Tc =Ja ¿TcA =Ja ; TcB =Ja the system may order in the interface layers before the interlayer ordering, i.e. the interface magnetism dominates. For R2 6Rc2 ; Tc =Ja 6TcA =Ja ; TcB =Ja we have the contrary situation. Initially, it has a place intralayer ordering, i.e. the intralayer magnetism dominates and the system behaves like metamagnets. From the plots in Fig. 2, we obtain the following critical values of Rc2 : for =Ja = 0; Rc2 = 1:8023; for =Ja = 1:5; Rc2 = 1:8295, and for =Ja = 2:2; Rc2 = 1:8949. Rc2 increases with the increase of =Ja .

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4. Conclusions The critical behavior of a ferromagnet spin-1/2 Ising superlattice consisting of two materials in a transverse ÿeld is examined using the eective ÿeld theory. It has been found that there exists a critical value Rc2 of the reduced interface exchange coupling R2 such that when R2 ¿Rc2 the interface magnetism appears, and Tc =Ja for the superlattice is higher than either TcA =Ja or TcB =Ja and when R2 6Rc2 , Tc =Ja 6TcA =Ja ; TcB =Ja . The formalism of the critical temperature derivation obtained above is universal and can be used for the study of a superlattice of various thicknesses and structures. Although we have considered a superlattice with only ferromagnet exchange coupling, (all Jij ¿ 0), the formulation is also applicable for antiferromagnetic coupling (some or all Jij ¡ 0). Acknowledgements This work was done during a visit of A.A. and S.M. to the UniversitÃe de Metz, France. The visits of A.A. and M.S. were made possible in the frame of the Action intÃegrÃee Franco-Marocaine No. 46=SM=97. The authors would like to thank the governments concerned. References [1] M.G. Cotton, D.R. Tilley (Eds.), Introduction to Surface and Superlattice Excitations, Cambridge University Press, Cambridge, 1989. [2] C. Liu, S.D. Bader, J. Appl. Phys. 67 (1990) 5758. [3] L.H. Tjeng, Y.U. Idzerda, P. Rudolf, F. Sette, C.T. Chen, J. Magn. Magn. Mater. 109 (1992) 288. [4] C.A. Ballentine, R.L. Fink, J. Araya-Pochet, J.L. Erskine, Phys. Rev. B 44 (1990) 2631. [5] C.M. Schneider, P. Bressler, P. Schuster, J. Kirschner, J.J. de Miquel, R. Miranda, Phys. Rev. Lett. 64 (1990) 1059. [6] Yi Li, K. Baberschke, Phys. Rev. Lett. 68 (1992) 1208. [7] R. Bergholz, U. Gradmann, J. Magn. Magn. Mater. 45 (1984) 389. [8] M. Farle, K. Baberschke, U. Stetter, A. Aspelmeier, F. Gerhardter, Phys. Rev. B 47 (1993) 11571. [9] A. Saber, I. Essaoudi, A. Ainane, M. Saber, Phys. Stat. Sol (b) 209 (1998) 161–171. [10] I. Essaoudi, A. Saber, A. Ainane, M. Saber, Phys. Scr. 59 (1999) 168–173. [11] M. Saber, Chin. J. Phys. 35 (1997) 577.