Magnetism of antiferromagnetically coupled multilayers

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Journal of Magnetism and Magnetic Materials 136 (1994) 99-104

magnetic materials

Magnetism of antiferromagnetically coupled multilayers A . M o s c h e l *, K . D . U s a d e l Theoretische Tieftemperaturphysik, Universitiit Duisburg, Lotharstr. 1, 47048 Duisburg, Germany

Received 9 August 1993

Abstract

We have calculated, with a Green function method, the magnetization profiles of ferromagnetic multilayers consisting of two different ferromagnetic materials with antiferromagnetic couplings between the films. For strong coupling constants between the films we find at low temperatures a lowering of the layer magnetization at the interfaces. The magnetization M ( T ) of the whole system shows a strong dependence on the coupling constant Jz between the films and on the ratio J2/J1 of the coupling constants of the two ferromagnetic materials.

1. Introduction

The study of the behavior of magnetic multilayers is a very exciting subject of growing interest. In recent years, many different systems have been studied by theoretical and experimental methods [1,2]. Much work has been done to investigate multilayers consisting of different ferromagnetic materials coupled antiferromagnetically. For these systems many different phases were found by Camley et al. [3-5] with the help of a self-consistent mean-field calculation. Later, Le Page and Camley [6,7] investigated the spinwave spectrum of these systems. All these calculations were done with special parameters to describe the F e / G d system, where the ferromagnetic layers Fe and Gd are coupled antiferromagnetically. Recent experiments on F e / G d multi-

* Corresponding author. E-mail: [email protected].

layers show that all the predicted phases really exist [8-10]. An alternative method for studying the excitation spectrum of magnetic films and multilayers at low temperatures is the Green function method [11,12], which is also capable of reproducing the correct low-temperature behavior which is dominated by spin-wave excitations. Endo and Ayukawa [13] investigated ferromagnetic multilayers with ferromagnetic couplings between the layers with such a Green function method and calculated especially the magnetization profiles of these systems. In the case of an antiferromagnetic coupling between ferro- and antiferromagnetic materials, Green function calculations have shown that in the low-temperature region very interesting quantum effects appear [14]. The purpose of this paper is to show that in the low-temperature region the Green function method describes the qualitative behavior of ferromagnetic multilayers with antiferromagnetic couplings between the layers quite well. In partic-

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A. Moschel, K.D. Usadel /Journal of Magnetism and Magnetic Materials 136 (1994) 99-104

ular, we study the influence of the antiferromagnetic coupling constants between the layers and the influence of the coupling constant of the second ferromagnetic material on the magnetic behavior of the whole system. We discuss the temperature dependence of the magnetization, restricting ourselves to the zero-field behavior of the system. Thus, we are always in the region where only the aligned but not the twisted phases can occur, a problem of much greater complexity In Section 2 we describe the model and give a short overview of the Green function formalism for the layered systems used in this paper. Details of the method can be found in our previous publication [14] or in the work by Diep [11,12,15]. In Section 3 we present and discuss the results of our calculations.

interaction between the films is antiferromagnetic and labeled Jz. The Hamiltonian reads

,,~

(i,j)

We consider a model consisting of three ferromagnetic films with a simple cubic structure where each film consist of four layers. The first and the third films have the coupling constant J1 and the second film has the coupling constant -/2. The

I

I

(1)

(i,j)

where the sums are over distinct pairs of nearest-neighbour spins only, and Si denotes spin operators. Jiy(Dii) is equal to Jl(D1) if both spins are in the first or the third film; equal to J2(D2) if both spins are in the second film; and equal to Jz(Dz) if one spin is in the first or third film and the other is in the second film. JI(D 1) and J2(D2) are both positive corresponding to ferromagnetic ordering along the z-axis, which is assumed to be perpendicular to the films, while J~ and D, are both negative, corresponding to antiferromagnetic ordering along the z-axis. Following Zubarev [16] we define a doubletime Green function, ((Si+(t); ST(t'))) , where S~± are the usual spin lowering and raising operators and we restrict ourselves to S = 3. 1 After writing the equation of motion for the Green functions Gu(t, t') we obtain higher-order Green functions which are decoupled by the so-called Tyablikov decoupling scheme:

2. Theory

0.55

- 2 E JuSi " Sj - 2 E DuSTS;,

=

(
S7(t'))). (2)

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1 o.35 0.3 0.25 0.2 0.15

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i Fig. I. Magnetization profile of the system with: J r = +I.0, ./2 = +0.2 and ./z = - 0 . 2 ( D I = D 2 = IDzl =0.01) at different temperatures: T = 0.1 (~), 0.2 ( + ) , 0.3 (D), 0.4 ( x ) and 0.5 ( A ) (all units in quantities of ./i).

A. Moschel, K.D. Usadel/Journal of Magnetism and Magnetic Materials 136 (1994) 99-104 0.55

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i Fig. 2. Magnetization profile of the same system as in Fig. 1, but with Jz = - 1.0 and additional at the temperatures T = 0.6 (*) and 0.7 (o).

In the following we assume that the expectation values ( S z ) are equal in the same layer, i.e. in the following we have to deal with 12 order parameters: (S/~) := mi

transform for the G r e e n functions Gij(t , t') which we insert into the equation of motion we obtain for each layer magnetization a set of equations which connect the G r e e n functions with the other order parameters of the system. To solve these sets of equations we need another set of equations which connects the G r e e n functions with the layer magnetizations. Using

(3)

where i = 1, 2 . . . . . 12 denotes the layer index. After introducing a two-dimensional Fourier

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T - - T

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r

T I T

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i Fig. 3. Magnetization profile of the system for different couplings Jz at T = 0.1:J1 = + 1.0, J2 = +0.2 and Jz = - 0 . 2 (<5), - 0 . 5 ( + ) , - 1.0 (D), - 1.5 ( x ) and - 2 . 0 (zx).

A. Moschel, K.D. Usadel /Journal of Magnetism and Magnetic Materials 136 (1994) 99-104

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temperatures is also in the region of 0.155 [3]. In Fig. 1 we ave calculated the low-coupling case (Jz = - 0 . 2 ) and in Fig. 2 we show the high-coupling case (J~ = - 1.0). It is not surprising that in the first case, due to the weak coupling between the films, the magnetization in the second film breaks down for temperatures around T = 0.5, and in the second case the magnetization goes to zero at around T = 0.7. More interesting is the fact that we find in the high-coupling case at the interfaces a significant lowering of the magnetization at low temperatures. This lowering is an effect due to the antiferromagnetic coupling between the films and it is not found in ferromagnetically coupled multilayers. In Fig. 3 we have plotted the magnetization for different Jz-values at T = 0.1 and find at the interfaces a very strong lowering of the magnetization for strong Jz-values. The reason for this strong lowering is the fact that with increasing the antiferromagnetic coupling between the spins at the interface the quantum fluctuations for these spins at low temperatures also increase [12]. Therefore we find for the expectation values of these spins smaller values than for weakly antiferromagnetic coupled spins. From a theoretical point of view the magneti-

the spectral theorem that relates the correlation function ($7S +) with the Green functions, and 1 considering that for S = ~ we can write for the layer magnetizations: m i = (S/z } =

i _

(SFS•)

(4)

'

we get a second set of equations. A more detailed outline of this formalism can be found in Refs. [11,12,14,15]. These systems of equations are solved in a self-consistent manner and the results are discussed in Section 3.

3. Results and discussion

We have studied the influence of the coupling constant J~ between the films and the influence of the coupling constant J2 of the second ferromagnetic film on the magnetic behavior of the whole system. In Figs. 1 and 2 we show two magnetization profiles of the system for two different coupling constants Jz. The ratio of the coupling constants J2/Jt in both cases is 0.2. The critical temperature Tc of the second ferromagnetic film is therefore very low compared with those of the two outer ferromagnetic films. Note that in F e / G d multilayers the ratio of the critical

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0.2 0.19 o.18 0.17 0.16

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T Fig. 4. M a g n e t i z a t i o n

M(T) o f t h e s y s t e m with: J1 = 1.0, J2 = 0.2 a n d Jz = - 0.5 ( ~ ) , - 1.0 ( + ), - 1.5 ( [] ) a n d - 2.0 ( × ).

A. Moschel, KD. Usadel /Journal o f Magnetism and Magnetic Materials 136 (1994) 99-104 0.25

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T Fig. 5. Magnetization M(T) of the system with: J1 = 1.0, Jz = - 0.5 and J2 = 0.2 (O), 0.35 ( + ), 0.5 ( [] ) and 0.75 ( x ).

zation profiles give very good insight into the magnetic behavior of the system. From an experimental point of view, however, the whole magnetization of the system is more interesting. In Figs. 4 and 5 we have plotted the magnetization M = l•im i as function of temperature T. In Fig. 4 the coupling constant Jz is varied and J2/J~ = 0.2 is fixed. We find a decrease in the magnetization M(T) with increasing Jz. This behavior becomes clear if we look at the magnetization profiles of the system (Figs. 1 and 2). For strong Jz values the influence of the second ferromagnetic film, which is antiferromagnetically coupled to the other two films, on the magnetization M(T) is stronger than for weak Jz values because the critical temperature is shifted to higher values and therefore we have a stronger influence of the negative layer magnetizations in the second film on M(T). The quantum effect that is seen in Fig. 3 is not visible in the magnetization M(T) of the system because the magnetizations of these layers neighboring the interfaces cancel each other due to the antiferromagnetic coupling. Only by local magnetic measurements may this effect be seen. In the experiments the curvature of the magnetization curves depends on the thickness of the different films in the multilayer. The same effect can be obtained by varying the coupling constant J2 of the second film. In Fig. 5 we have plotted

the magnetization M(T) with Jz = -0.5 but for different J2 values. In case of a weak J2 coupling (or equivalent small films) the influence of the antiferromagnetically coupled film on the magnetization M(T) is very weak and therefore M(T) increases for higher temperatures. For strong J2 values (or thick films) the influence of the negative layer magnetizations on M(T) is stronger than in the first case, and therefore M(T) is smaller. In summary, we have shown that we can qualitatively describe the behavior of ferromagnetic multilayers with antiferromagnetic coupling at the interfaces with a Green function method. At low temperatures we find a lowering of the layer magnetization at the interfaces due to quantum fluctuations in the case of antiferromagnetic coupling. We have shown that the different curvatures of the M(T) curves seen in the experiment depend strongly on the different coupling constants -/2 and Jz.

Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 166.

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[8] H. Fujimori, Y. Kamiguchi and Y. Hayakawa, J. Appl. Phys. 67 (1990) 5716. [9] K. Cherifi, C. Dufour, Ph. Bauer, G. Marchal and Ph. Mangin, Phys. Rev. B44 (1991) 7733. [10] K. Cherifi, C. Dufour, G. Marchal, Ph. Mangin and J. Hubsch, J. Magn. Magn. Mater. 104-107 (1992) 1833. [11] H.T. Diep, Phys. Rev. B40 (1989) 4818. [12] H.T. Diep, Phys. Rev. B43 (1991) 8509. [13] Y. Endo and T. Ayukawa, Phys. Rev. 1341 (1990) 6777. [14] A. Moschel, K.D. Usadel and A. Hucht, Phys. Rev. B47 (1993) 8676. [15] D.T. Hung, J.C.S. Levy and O. Nagai, Phys. Stat. Solidi (b) 93 (1979) 351. [16] D.N. Zuharev, Sov. Phys. Usp. 3 (1960) 320.