Phase diagrams of Ising multilayer systems with amorphous layers

ELSEVIER

Journal of Magnetism and Magnetic Materials 168 (1997)47-54

~ lewnalof n~dgnetlsm ~ i magnetic materials

Phase diagrams of Ising multilayer systems with amorphous layers M. Ja~ur*, T.

Kaneyoshi

Department of Natural Science lnJbrmatics, School of lnformatics and Sciences, Nagoya University, Chikusa-ku, 464-01 Nagoya, Japan

Received 8 March 1996;revised 18 September 1996

Abstract

A general formulation for studying magnetic properties of layered Ising-type systems A2B. composed of two amorphous and n crystalline layers is presented. The case that both A and B layers consist of the spins with magnitude ½is analysed in detail. Some interesting results such as the possibility of reentrant behaviour are reported. Keywords: Magnetic multilayers; Amorphous systems - multilayers; Ising model; Phase diagrams

I. Introduction

The study of various magnetic multilayers is currently of great interest because they are expected to have new and possibly useful properties for technological applications (see, for example, Ref. [1]). From the theoretical point of view, great interest has been paid to spin wave excitations as well as critical phenomena in two-component layered magnetic superlattices [2-9]. These works are based on the Heisenberg or Ising multilayered systems with coupling exchange constants of different magnitude within each layer. However, in these works some very important effects such as the existence of disordered or amorphous interfaces are completely neglected. On the other hand, some experimental works clearly indicate that the amorphous [ 10] or disordered [11] interfaces may appear in real multilayered systems. In fact, experimental data of Tb/Fe multilayers have exhibited that when the Fe layer thickness dFe decreases the crystallographic structure of Fe layer changes from BCC to amorphous at the critical thickness d ~ ' around 17-20 A [10]. Since the interface effects may play an important role in magnetic properties of multilayers it is very important to take account of this fact in theoretical studies. Here one should notice that recently the role of disordered interface in various multilayered systems has been investigated in many works [12], but multilayers with amorphous

* Corresponding author. Permanent address: Department of TheoreticalPhysicsand Geophysics,Faculty of Natural Sciences,P. J. Saffirik University,Moyzesova 16, 041 54 Ko~ice,Slovak Republic. 0304-8853/97/$17.00 © 1997ElsevierScienceB.V. All rights reserved PII S0304- 88 5 3(96)00669-5

M Jak~ur, T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 168 (1997) 47-54

48

interfaces have been only little examined [ 13]. Therefore, the main purpose of this work is to investigate how the existence of an amorphous interface may influence the magnetic properties of magnetic multilayers. The outline of this paper is as follows. In Section 2, the basic equations for studying multilayers consisting of two amorphous layers and n crystalline layers within the framework of an effective-field theory (EFT) are given. In Section 3 we present numerical results for the case that both amorphous and crystalline layers consist of ions with spin ½ and the number of crystalline layers is changed. Finally, concluding remarks are given in Section 4.

2. Formulation We consider a multilayered Ising system on the simple cubic lattice. The system consists of n alternating crystalline layers (B) with an arbitrary spin S separated by two amorphous layers (A) with spin-½, as depicted in Fig. 1. The Hamiltonian of the system is given by = _ 1 ~

z z _ DZ

2 ~.~ JqSi Sj

~ , . ( s ~ ) 2,

(1)

where the first summation is over all the nearest-neighbour pairs, and the spin variable S~ takes the values of _+½ for atoms in the A layer or the values of - S, - S + 1. . . . , S for the atoms in the B layer. D represents the crystal field parameter and 6 is the usual Kronecker symbol. The exchange interaction parameters J q take the value JBB = JB for the atoms in the B layer, while in the A layers and between A and B layers the Jgj are assumed to be randomly distributed according to the Handrich-Kaneyoshi distribution for amorphous magnets [14], namely,

(2)

P(Jq) = ½[6(Jq - J ~ - AJ~) + 6(Jq - J,~ + AJ:~],

where ~, 7 = A or B depends on the position of atoms. Now, we are interested in evaluating of the mean values ((Si~)),, where ( ... ) denotes the thermal average and ( . - - ) , means the random configurational average. Using the conventional decoupling of multispin

B

B

B

A

A

B

B

B

mA

mA

ml

mn_l m n

" ~ ~ ......(

m 1

m2

mn

Fig. 1. Part of two-dimensionalcross-sectionthrough the magneticmultilayersystemA2B,consistingof two amorphouslayers(A) and n crystallinelayers(B).SA= ½and SB= S. The exchangeparametersare labeledas JA, JB and JABcorrespondingto the bonds A-A, B-B and A-B, respectively.

M. Jak~ur, T. Kaneyoshi/ Journalof MagnetismandMagneticMaterials168 (1997)47-54

49

correlations one obtains the approximation equivalent to Zernike approximation 1-16] which improves the standard mean-field theory. Then, following Refs. [15, 7, 8] we obtain the following set of equations for magnetizations of our system:

mA ~ [ P ( ~ V ; ~ - V ) - ~ x ~ tanh ( ~ )

2?nAR(@V;-~V)]5[p(~IJABV;~IAJABV)-}- D'll

x=o ,

(3)

ml = [P(~-~V; ~ - ~ V ) + 2mAR(~V; ~-~V)][P(qaJBV;O' + ~R(qxJBV;O'] a x [ P(tl2JaV;O) + -m2 - R(rlEJBV;0)]Fs(X)=o , t12 mi =

P(th- aJBV; 0) +

x [ P(tli+1JBV; 0) where fl =

t/i-1 -b

cosh(x) cosh(y)

P(t/iJBV; 0) + - r/i

mi+ln" /~tr/i+ 1JBV; 0) ]Fs(x)x=o , r/i+1

1/(kBT), V = ~/8x is a

P(x, y) =

R(th_ 1JBV; 0)

(4)

R(thJBV; O)

differential operator and the functions P(x, y) and

and

R(x, y) =

(5)

i /> 2,

sinh(x) cosh(y).

R(x, y) are

defined by (6)

The parameters ~/i are defined as rh ~<<(S~)2>>r and the equations for their evaluation have identical forms as Eqs. (4) and (5), while the function Fs(x) must be replaced by another function Gs(x). The functions Fs(x) and Gs(x) depend on the spin value S and they can be expressed as follows: =

~s= - s k

G x):

exp(flDk 2) sinh(flkx),

~s= - s k2 exp(flDk2) cosh(flkx)

(7) (8)

3. Phase diagram In order to obtain the transition temperature To, we expand the right-hand sides of Eqs. (3)(5) and consider only terms linear in mn and mi. Then, for the system with composition AaB, (n >~ 2) we obtain the following matrix equation:

(mA) ml

~/g m 2

\ran !

= 0.

(9)

5O

M. JakSur, T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 168 (1997) 47-54

The matrix J//is given by 10Ao- l

A1

0

0

..-

0

0

2K(o°)

4K(21)- 1

K(a2)

0

'-.

0

0

0

K~1)

4K(22 ) - 1

K~33)

-"

0

0

0

0

0

0

"'"

K(1i - z )

0

0

0

0

0

""

Jg =

4K~ ) -

0 0 1

Ktli- 1)( 1 + ~i~)

°

K(~+1)

I

4K(~)(1 + 6i,~) - 1

where coefficients Ai and K! j) depend on the exchange parameters, their fluctuations and temperature, and they are defined in the appendix. Moreover, from the symmetry of the system one can easily find that ( n + 1) - ~ ,

n ),=~

and

~ ( n + 1)/2 i=[n/2

ifnisodd, ifniseven.

(10)

The critical temperature is then calculated from the equation det(J#) = O,

(11)

in which, however, unknown parameters r/z are included. These parameters are simultaneously calculated from the equations (/]1) 2 = Q(01J(JAa, AJAa; Ja, AJB; ql,/'12, To),

(12)

(v/i)2 = Q~)(JB, AJa; ~/i-1, qz, ~/~+1, To),

(13)

i >~ 2,

where Q~) are also given in the appendix.

4. Numerical results In this work, we restrict ourselves to the simplest possible case taking SB = 1 (i.e. qi = ½, Vi). Before discussing the results, it is worth noticing that for JA ----JAa = 0.0 our system reduces to the n-layer spin-½ Ising thin film, while for JB = JAB = JA and 6AB ----6A = 0.0 it recovers the standard spin-½ Ising model on the simple cubic lattice. Since in all these cases we can obtain the results identical with those presented in Ref. [17], it proves the correctness of the present formulation. In the following, we will present the results obtained by solving Eq. (11) for many combinations of the exchange parameters, increasing the number of crystalline layers in the system. For practical calculations we have introduced two dimensionless parameters 6A = (AJA)/JA and 6AB = (AJAa)/JAa that measure the amount of fluctuations of the exchange parameters. At first, let us show the results for the system A 2 B 2. In Fig. 2a and 2b the results are depicted when the exchange interaction in the crystalline layers is weak (JB/J A --- 0 . 0 5 ) , while in Fig. 2c and 2d JB is assumed to be strong (JB/JA = 1.0). Here, one must notice that a common feature of Fig. 2a and 2c is the assumption t~AB ~ t~A, while in Fig. 2b and 2d we have selected 6AB = ~ g / 1 0 . The exchange interaction JAB is in all cases changed from weak to strong values. As expected, the magnetic properties of the A2B/system are strongly influenced by the existence of two amorphous layers• As we can see from Fig. 2a and 2b, for Ja = 0 . 0 5 J g the existence of amorphous layers become dominant in the behaviour of system. Accordingly, one can observe

M. Jak~ur, T. Kaneyoshi/ Journal of Magnetism and MagneticMaterials 168 (1997) 47-54 1.25

i

'

1.2

' Js/ ~ ' =0.05

~ "

51

1.0

1.00

0.8

~'< 0.75

A = 2 . 0

t-,Y 0.6 o.5o

1.0

0.4

0.25

(a)

0.2

0.00 0.0

I

I

I

I J l

0.5

i~

f

1.0

0.0 0.0

I

1.5

0,5

1.0

1.5

2,0

2.0

/

,

i

.

i

.

i

,

i

1.6

,

,

i

,

i

,

i

.

J

1.4 ~

1.5

~

0.0

I

0

~ J.~IJA =2.0

aAB/aA= o.1

oIo

,

1,0

JB/JA =

(c)

2.5

(b)

1.2 1.0

,

It

1

g

Ig

2

,

'~A

t"

3

,

I

4

,

o a

L,J

5

t

0 (d)

,

i

1

,

i

,

2

i

3

,

i

,

4

~A

Fig. 2. Phase diagram (To versus fA) of the multilayer system A2B 2 for (a) JB/JA= 0.05 and 6AB/rA ----"1.0, when the value of JAB/JAis changed; (b) JB/JA = 0.05 and 6Aa/fA = 0.1, when the value of JA•/JA is changed; (c) JB/JA = 1.0 and 0Aa~A = 1.0, when the value of JaB/JA is changed; and (d) JB/JA = 1.0 and 6Aa/fA = 0.1, when the value of JAS/JAis changed.

the appearance of the reentrant phenomena (or multiple critical temperatures) when the fluctuations in the system become strong enough (i.e. 6A > 1.0). The dependencies of critical temperature on 3A are very similar to those usually observed in other amorphous systems [14], except the one case in Fig. 2b where we have selected JAB/JA = 0.1 and 6A8 = 6A/10. In this case, the long-range order is never completely destroyed, because even for 6A >> 1 the existence of crystalline layers play an important role in the system. This behaviour is reasonable, since the influence of amorphous layers in the system is rapidly depressed due to the fact that both JAB and 6AB are very weak. On the other hand, as we can see from Fig. 2d, the reentrant behaviour cannot be observed for the system with a strong exchange interaction in the crystalline layers (JB/JA = 1.0) and a weak fluctuation of JAB (t~AB = 6A/10). Moreover, we have found that the phase boundaries for the case are only a little modified when the number of crystalline layers increases. Therefore, the case will not be analysed further. Now, let us show how the phase diagrams of Fig. 2a-2c are changed when the number of crystalline layers (i.e. n) increases. The results for the system A2B 3 are presented in Fig. 3a-3c, where the relevant parameters are fixed to the same values as those of Fig. 2. As seen from Fig. 3a and 3b, the reentrant behaviour also appears for the system A2B3 with a weak exchange interaction Ja. Moreover, comparing Fig. 3 with Fig. 2, one can find that the results show some common features, especially for weak fluctuations. On the other hand, an important difference can be seen in the region of strong fluctuations (rA >)> 1), although the

M Jak~ur, Z Kaneyoshi / Journal of Magnetism and Magnetic Materials 168 (1997) 47-54

52

1.25

1.2

-

~

JB/JA

= 0.05

1.00 0.9 ~

0.75 I-~ 0.6

0.50

(a)

= 2.0

0.3

0.25 0.00 0.0

~AS/JA

0.5

1.0

1.5

0.0 0.0

2.0

0.5

8A 2.0

~ J

1.0

1.5

2.0

2.5

3.0

8A

(b)

i JB/JA=I'.O , i

1.5

JABIJA

= 0.1

~

1.0

5.0

0.5

0.0

(c)

1

2

3

4

~A

Fig. 3. (a) The same as in Fig. 2a but for the system A2B3; (b) same as in Fig. 2b but for the system AzB3; and (c) same as in Fig. 2c but for the system A2B 3.

long-range order is never completely broken for the system AzB3. Next, comparing Fig. 3c with Fig. 2c we can find that for the case of a strong exchange interaction JB, the results of the A2B2 and A2B3 systems differ both qualitatively and quantitatively. In particular, the reentrant behaviour does not appear in the A2B3 system. Of course, we have studied a number of other systems by increasing n, namely n = 4, 5. . . . . 10. However, the numerical results are qualitatively the same as those for the system A2B3, therefore they are not presented in this work. In general, our results indicate that the reentrant behaviour can be found in the multilayered system AzB, for an arbitrary n when the exchange interaction JB in crystalline layers is weak enough. Physically, the reentrant behaviour has the origin in the competition between ferromagnetic (Ju > 0) and antiferromagnetic (Ju < 0) exchange interactions that appear in the system A2B, for 6A > 1.

5. Conclusion

In this work, we have studied the phase diagrams of a ferromagnetic multilayered Ising system on the simple cubic lattice. The system consists from n alternating crystalline layers with an arbitrary spin, separated by two amorphous layers with spin 1. In Section 2, the general equations for evaluating phase diagrams within the framework of an effective-field theory are presented. The numerical results are obtained in Section 3 for the simplest case, assuming that the crystalline layers consist of atoms with spin ½. Our results

M. Jak~ur, T. Kaneyoshi / Journal o f Magnetism and Magnetic Materials 168 (1997) 47-54

53

allow to conclude that the reentrant behaviour can appear in the A2B . multilayered system for an arbitrary number of crystalline layers, if the exchange interaction JH within these layer is weak enough. In particular, as far as we know, the reentrant phenomenon has not been reported in the studies of magnetic multilayer systems. Finally, the formalism presented in this paper can be easily applied to the systems with higher spin values. In general, the increasing spin value leads to the increase of the critical temperature and therefore we assume that some new interesting results can be discovered in the layered higher spin systems, and we hope to investigate the problem in future.

Acknowledgements

M.J. wishes to thank the Japan Society for the Promotion of Science and the Charta 77 Foundation of Slovakia for their financial support.

Appendix The coefficients Ai, K/j) and Q o) are given by Ao = sinh(x) cosh4(x) coshS(y) cosh(z) cosh(u) ½tanh (fix/2), 1

A1 = - - coshS(x) cosh(y) 5 sinh(z) cosh(u) ½ tanh (fix/2), th where x = ½JAY,

y = ½ (AJA)V,

z = ~IJABV,

U = ~I(AJAs)V.

(A.I)

K~o°) = cosh4(x) cosh(y) sinh(z) cosh(u)Fs(fX), K(21) =

1 sinh(x) cosh(x) 3 cosh(y) cosh(z) cosh(u)Fs(flx), t/1

K(32) = 1 cosh(x)4 sinh(y) cosh(z) cosh(u)Fs(flx), q2 Q?) = cosh4(x) cosh(y) cosh(z)

cosh(u)Gs(flx),

where y

X = ~ I J B V,

1

K~Ii - l ) = - -

~/i- 1

=

~2JBV, z = ½JAaV, u = ~AJAa)V.

cosh(x) 4 sinh(y) cosh(z)Fs(flx),

K~ ) = 1 sinh(x) cosh(x) a cosh (y) cosh(z)Fs(flx),

qi

(k.21

54

M. Jak~ur, 7~ Kaneyoshi / Journal of Magnetism and Magnetic Materials 168 (1997) 47-54

1

K~ + t) _

cosh(x) 4 cosh(y)

sinh(z)Fs(flx),

qi+ 1

Q~) = cosh(x) 4 cosh(y)

cosh(z)Gs(flx),

where X = rhJaV,

Y = qI-1JBV,

z = qi+ xJBV.

(A.3)

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17]

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