The ferrimagnetic multilayer system with disordered interfaces

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4 77 2001 The Moroccan Statistical Physical and Condensed Matter Society The ferrimagnetic multilayer system with disordered interfaces

A. Moutie and M. Kerouad

Université Moulay Ismail - Faculté des Sciences Département de Physique - B.P. 4010 - Meknés - Maroc

The magnetic properties (Transition temperature and magnetizations), of aferrimagnetic multilayer system consisting of L layers of spin-1/2 A atoms, L layers of spin-1 B atoms and a disordered interface in between that characterized by a random arrangement of A and B atoms of ApB{1-p} type and a negative A-B coupling, are studied within the framework of aneffective-field theory. The effect of the disordered interface, interactions and different anisotropies on the magnetic properties are examined. The obtained results show a number of characteristic features, such as the possibility of two compensation points.

I. INTRODUCTION

In recent years, the study of magnetic multi-layer systems(superlattices, films ..) is of current interest because they are expected to have new and possibly useful properties for technological applications.

In particular, rare-earth (RE) /transition-metal (TM) ferrimagnetic multi-layers have been produced and extensively studied [1-3].

They can show a compensation point when their thickness are not very great [1,2]. A material with a compensation temperature slightly higher than room temperature may be a good condidate for magneto-optic storage media.

On the other hand, detailed analyses of the experimental data reveal that in the region between the two components of the Layered systems the two types of magnetic atoms are mixed randomly to give an alloy like disorderedinterfaces [2,4].

Furthermore, many experimental data indicate that the magnetic anisotrpy on the interface is usually different from that in the bulk multiple layers [5].

Therefore, it is raisonable to assume that the existence of a disordered interface and different anisotropies can modify the magnetic properties of the multi-layer systems. In order to explain the experimental data, the transition temperature TC and the magnetization of Tb/Fe ferrimagnetic multi-layer have been examined using mean field theory (M.F.T) [2,3] by including disordered interfaces of the type ApB{1-p} .

On the theoretical side, many efforts [6-9] have been directed to clarify the magnetic properties of Heisenberg

Some recent works [10-13] have investigated the role of disordered interfaces in the bilayer system consisting periodically of two magnetic layers A and B where A and B can possess different bulk properties. In fact, the magnetic properties of the bilayer Ising system with disordered interfaces have been studied when the spin SA of the A atoms is 1/2 and the spin SB, of the B atoms is a half integer (1/2 or 3/2).

The aim of this work is to investigate the magnetic multi-layer Ising system, consisting periodically of L layers of spin-1/2 A atoms, L layers of spin-1 B atoms and a disordered interface characterized by a random arrangement of A and B atoms like a tow-dimensional ApB{1-p} alloy. This study is done within the framework of an effective field theory based on the use of a probability distribution technique [14] which is superior to the mean field theory.

II. FORMALISM

We consider a superlattice consisting periodically of L layers of A atoms, a disordered interface layer ApB{1-p}

of A and B atoms randomly mixed, and then L layers of B atoms. For simplicity, we restrict our attention to the case of a simple cubic structure. Let the A and B atoms have different spins (SA=1/2 and S_B=1 respectively).

The two-dimensional cross section of the system is depicted in fig.1:

or Ising multi-layer systems, consisting of only spin-1/2 atoms, with different bulk properties.

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FIG.1: Part of the two dimensional cross-section through the magnetic multilayer system consisting of the two magnetic A and B layers (S_A=1/2 and S_B=1) and disordered interfaces of the type ApB1-p.

The Hamiltonian of the system is given by:

{ }

∑

⁺ ⁻ ⁻

−

=

) , (

) 1 )(

1 (

j i

j i z j z i BB z j z i

AA J S S

J

H σ σ ξ ξ

{ } _∑

∑

⁻ ⁻

−

i z i j

i

j i z i z i

AB S D S

J ²

) , (

0 ( )

) 1

( ξ

ξ σ

∑

−

i

i iB z

Si

D ⁽ ⁾²δ ξ

where the first sum runs over all pairs of nearest neighbours. J_AA, J_BB and J_AB are the exchange interactions between A-A, B-B and A-B pairs of atoms.

D0 and D are the uniaxial single-ion anisotropy constants on the B layers and disordered interfaces respectively.

σ

_i^z

is the spin-1/2 operator on the A atoms, Szi is the spin-1 operator on the B atoms, and the random variable

ξ

i takes the averages

< ξ

i

>

A

= 1

^,

< ξ

i

>

B

= 0

^and

i

> = p

< ξ

int in the A layers, B layers, and interface layers respectively.

The purpose of the paper is to study the phase diagrams of such a system, using the effective field theory based on the use of a probability distribution technique that correctly

accounts for the single site Kinematic relations [14,15].

This method without introducing mathematical and provide results which are superior to those obtained within the mean field theory approximation. For the system under consideration, the application of this method leads to the layer magnetizations:

>

=<

−

= − 1

0 0 0

1 (

) exp(

)

exp( F X

H Tr

β β

σ σ (1)

>

=<F(X2)

σ

n (2)

for

2 ≤ n ≤ L − 1

,

σ

1

σ =

L (3)

for the A layers.

>

=<F(X3)

σ

I (4)

>

=<G₁(X3)

m_I (5)

for the interface.

>

=< ₁( 4)

1 G X

m (6)

(3)

>

=<G₁(X5)

m_n (7)

for 2≤n≤L−1,

m

L

m

₁

=

(8)

for the B layer. With ) tanh(

2 ) 1

(X X

F = β

) exp(

) cosh(

2

) sinh(

) 2

1(

X X

G

β β

β

−

= +

) ) 1 (

( ₁₀ ₁₀

0

20 0 10 1

z i AB z i AA

z AA z

AA

S J

J N

J N NJ

X

ξ σ

ξ

σ σ

− +

+ +

=

z n AA z

n AA z

n

AA

N J N J

NJ

X

₂

= σ

_,₀

+

₀

σ

₋₁_,₀

+

₀

σ

₊₁_,₀

)

) 1 (

(

₀ ₀

10 0 0 0

3

z I i AB z I i AA

z AB z

L AA

S J

J N

J N J

N X

ξ σ

ξ

σ σ

− +

+ +

=

) ) 1 (

(

₀ ₀

0

20 0 0 4

z I i BB z I i AB

z BB z

L BB

S J

J N

J N NJ

X

ξ σ

ξ

σ σ

− +

+ +

=

z n BB z

n BB z

n

BB

N J N J

NJ

X

₅

= σ

_,₀

+

₀

σ

₋₁_,₀

+

₀

σ

₊₁_,₀

and

k

_B

T

= 1

β

^.

Where

σ

_i0^z ^and

S

i^z₀ are the components of spin-1/2 and spin-1 nearest neighbors respectively of the central site in the i-th layer.

For the B atoms ( S=1), we need also the quadrupolar moment defined as

q = ( S

_i^z

)

² , in our case, this quantity can be obtained easily by substituting the function G1 in equations (5-8 ) by G2. This yields

) (G₁ G₂ m

q_n^z = _n^z → (9)

where

) exp(

) cosh(

2

) cosh(

) 2

2(

X X

G β β

β

−

= +

To perform the averaging on the right hand sides of equations (1-8) we follow the general approach described in references [14,15] and obtain the following equations for the layer magnetizations:

∑

⁻ ⁻

=

−

=

= +

=

− ⁰ ³ ⁵

0 6 3 0

0 5 3

0 4 0

0 3 0

0 2 0 1 ) 2 ( 1

2

0

i i N

i i

i N

σ

N

3 0 3

5 5 3 0 6 3 0 5 3 4 0 3 0 1

1 _i^N _i^N _iⁱ _i^N ⁱ _i^N ⁱ ⁱ

2

ⁱ ⁱ

( 1 )

^N ⁱ

N

i

C C C C C p p

C

⁻ ⁻ ⁻

−

⁻

2 2 2

2 1

1 1

1

) ( 1 2 ) ( 1 2 ) ( 1 2 )

2 1

( − σ

ⁱ

+ σ

^N⁻ⁱ

− σ

ⁱ

+ σ

^N⁻ⁱ

6 5

4 3

4

( 1 2 ) ( 1 ) ( )

) 2 1

( − σ

_Ii ⁱ

+ σ

_Ii ⁱ⁻ⁱ

− q

_I ⁱ

q

_I

− m

_I ⁱ +

−

+ ⁻ ⁻ ⁻ (( 2 )

( 2 )

( ⁰ ³ ⁵ ⁶ J N i₁

F m

q_I _I ^N ⁱ ⁱ ⁱ ^AA

)) 2 (

)) 2 ( ) 2

(N₀− i₂ + i₃− i₄ +J_AB N₀−i₃−i₅− i₆ (10)

))) 2 ( 2 ( ) 2 2 ((

(

) 2 1 ( ) 1 ( ) 2 1 ( ) 2 1 (

) 2 1 ( ) 2 1 ((

2

3 0 2 0 1

3 0 3

1 2 0 1 2

1

1 1

0 3 0 2 1 0

0 3 0

0 2 0 1 ) 0 2 (

i N i N i J N

F

C C C

AA

i N I i n i N n i

n

i N n i n N

i N i N i N

i N

i N N n

− +

−

+

− +

−

+

−

=

+ −

− −

−

=

= +

−

∑ ∑ ∑

σ σ

σ

(11)

σ

1

σ =

L (12)

))) 2 (

) 2 ((

)) 2 ( ) 2 2 ((

(

) (

) 1 (

) 2 1 ( ) 2 1 ( ) (

) ( ) 1 ( ) 2 1 ( ) 2 1 (

) 1 ( 2

2

7 6 4 3

2 0

5 4 1 0

7 6 4 7

6

5 4 5

3 2 0 1 1

3 1 1 2 1 1 0 1

4 4

6 2 6 4 7 4 6 4 5 4 2 0 3

0 2 0 1 6 4

0 7 4

0 6 4

0 5 0 4 2 0

0 3 0

0 2 0

0 1 ) 0 2 (

i i i N i i N J

i i i J N

F

m q m q q

m q

m q q

p p C

C C C C

C C

AB AA

i i i N I I i I I i I

i i I i

I i

i N

i i

i N L i

L

i N i

i i i i N i i N i i i N i i N i

N i N i i i N

i i N

i i

i N

i i N

i N

i N N I

−

− +

−

+

− +

−

+

−

+

− +

−

− +

−

=

−

− +

−

=

−

=

−

=

+

∑ ∑ ∑ ∑ ∑ ∑ ∑

σ σ

σ

(13)

))) 2 (

) 2 ((

)) 2 ( ) 2 2 ((

(

) (

) 1 (

) 2 1 ( ) 2 1 ( )

( ) (

) 1 ( ) 2 1 ( ) 2 1 ( ) 1 (

2 2

7 6 4 3

2 0

5 4 1 0 1

7 6 4 7

5

5 4 5

3 2 0 1 1 3 1 1

2 1 1 0 1

4

4 6 2 6 4 7 4 6 4 5 4 2 0 3 0 2

0 6 4

0 7 4

0 6 4

0 5 0 4 2 0

0 3 0

0 2 0

0 1 ) 0 2 (

i i i N i i N J

i i i J N

G

m q m q q

m q m q

q p

p C

C C C C C

C m

BB AA

i i I i

I i

i N i

i i

N L i

L i

N

i i i i i N i i N i i i N i i N i N i

N iI i i N

i i N

i i

i N

i i N

i N

i N N I

−

− +

−

+

− +

−

+

−

+

− +

−

− +

−

=

−

+

−

=

−

=

−

=

+

∑ ∑ ∑ ∑ ∑ ∑ ∑

σ σ

(14)

)) 2 2 (

)) 2 0

(

) 2 (

) 2 2 ( (

) (

) 1 (

) 2 1 ( ) 2 1 ( )

(

) (

) 1 ( ) (

) (

) 1 ( ) 1 ( 2

2

6 5 8

7 5

4 3 0 2 1 1

8 7 5 0 8

7

6 5 6

4 3 0 2 2

4 2 2 3 2 2 1 1 1 2 1 1

1 1 5 0 5

7 3 1 7 5 0 8

5 0 7 5 6 0 5 3 0 4 0 3 1 2 1 7 5 0

0 8

5 0

0 7 5

0 6 0

0 5 3 0

0 4 0

0 3 1

0 2 0 1 ) 0 2 ( 1

i J i

i i i N

i i N i i N J G

m q m q q

m q

m q q m

q m q

q p

p C

C C C C C C C m

AB BB

i i I i

I i

i N

i i

i i N i

i i

N i

i i i i i N i

i N i i i N i i N i N i i N i N i i i N

i

i N

i i

i N

i i N

i N

i i N

i N

i N N

− +

−

+

−

− +

−

+

−

+

− +

−

− +

−

=

−

− +

+

−

− −

−

=

−

=

−

=

−

=

− +

∑

σ σ

(15)

(4)

))) 2 (

) 2 (

) 2 ((

( )

(

) (

) 1 ( ) (

) (

) 1 ( ) (

) (

) 1 ( 2 2

6 5 0 4 3 0

2 1 1

6 5 0 1 1

6 1 1 5 1 4

3 0 1 1

4 1 1 3 1 2

1 2

1 5

3 1 5 0 6 0 5 3 0 4 0 3 1 2

1 5 0

0 6 0

0 5 3 0

0 4 0

0 3 1

0 2 0

0 1 ) 0 2 (

i i N i i N

i i N J G m

q

m q q m

q

m q q m

q m q

q C

C C C C

C m

BB i

i N n n

i n n i n i i N n n

i n n i n i i N n n i n n

i n i i i i N i N i i N i N i i N i

N i i N

i N

i i N

i N

i i N

i N

i N N n

−

− +

−

+

−

− +

−

− +

−

− +

−

=

− + − +

+ +

− +

− −

−

− −

−

+ +

−

=

−

=

−

=

− +

∑ ∑ ∑ ∑ ∑ ∑

(16)

m

1

m

_L

=

(17) with N and N0 are the numbers of the nearest neighbors in the plane and between adjacent planes respectively (N=4 and N0=1 in the case of a simple cubic lattice).

The quadrupolar moments are given by:

) (G₁ G₂ m

q_i = _i → (18) The averaged magnetization mT per site in the interface is consequently given by:

I I

T

p p m

m = σ + ( 1 − )

(19) Thus, the total magnetization in the system is :

∑

= =

+ + + =

L

i i L

i

T i A

m N m

L M

1 1

) 2 2 2

( σ (20)

where N_A is the number of magnetic atoms in each layer.

In the vicinity of the transition temperature T_C, the layer quadrupolarmoments q_i^z →q₀^z_i such as

q

₀^z_i is the solution of equation (18 ) for σi^z →0 and mi^z →0. To obtain the critical temperature TC , we expand the right hand-side of equations ( 10-17) and considering only linear terms, this leads to a matrix equation of thetype:













=

























−

I I L L

I I L L I I L L

m m m

m m m m m m

dd cc bb aa

d c b a

d c a b

b a b b a b

b a d c

bb dd cc aa

b d c a

d c a b

b a b b a b

d c b

a

σ σ σ σ

σ σ

σ σ σ σ

σ σ

1 2 1 1 2 1

3 3 3 3

4 4 4 4 4 4

3 3 3 3

1 1 1 1

2 2 2 2 2 2

1 1 1

1

. .

.

0 . . . . . 0 . . . . 0

0 . . . . . . . . . . 0

0 0 .

. . . . . . . . . 0

. . . . . . . . . . . . . . . .

0 . . . . 0

. . . . . . 0

. . . . . 0 0

. . . . .

. . . . . . 0 0

. . . .

. . . . . . 0 0

. . . .

. . . . . . . 0 0

. . .

0 . . . . . . . . 0 .

. 0

. . . . . . . . . . . . . . . .

0 0 . . . . . . . . . . .

0 . . 0 . . . . . . . 0

The phase transition temperature T_C/J_AA is obtained from

equation: det(A-I)=0 (21) It depends on R1=JBB/J_AA, R2= JAB/J_AA, D/ J_AA , D0/ JAA and L. From many solution of this equation, we choose one corresponding to the highest T_C/J_AA [16]. This value of T_C/J_AA corresponds toa solution having

z

σ

i ^,

m

_i^z

positive, which is compatible with a ferromagnetic ordering. The other formal solutions correspond in principle to other types of ordering that usually do not occur [17].

On the other hand, the compensation temperature T_K/J_AA, if it does exist in the system with J_AB < 0, can be obtained by introducing the condition: M=0 into equation (20).

III. RESULTS AND DISCUSSION

We are now able to study the magnetic properties (compensation and transition temperatures)of the ferrimagnetic multilayer system with disordered interfaces.

We have first plotted the phase diagrams in the (T/J_AA, p) plane for D=D0=0, R2=-1and for different values of R1 (fig.2),

FIG.2: The phase diagram T_C/J_AA (solid line) and TK/JAA (doted line) versus p of the system with D=D0=0 and R₂=-1 when R1 is changed: (a) R1=0.2, (b) R1=0.1, (c) R₁=0.0.

It is seen that the critical temperature and the compensation temperature increase when we increase R1.

We can also see that for high values of R₁, the

compensation temperature does not exist for any value of p as for little R1. When it exist, the compensation

temperature is minimum for p=1.

In fig.3, we show the phase diagrams of the system in the same plane for R2=-1.5, R1=0.05 and for different values of the anisotropies D and D₀.

From this figure we can see that TC/JAA is very influenced by the change of the anisotropy D in the interface ( curves (a) and (c) ) for example and is less influenced by the

0,0 0,2 0,4 0,6 0,8 1,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

c b

a c

b a

T/JAA

p

(5)

change in the bulk anisotropy D₀ (curves(a) and (b) ) for example. The variations of T_C/J_AA induced by the interface anisotropy are less pronounced for p=1, for those induced by the bulk one we have the reverse of the situation. On the other hand, concerning the variations of TK/JAA, it is seen that any change in both anisotropies induces variations in TK/JAA curves (dotted lines).As we can see, expect the case (D/ JAA, D0/ JAA )=(-3,\,0.5), TK/JAA does not exist for any value of p and for the case (-3,-3) there is no compensation temperature for the system.

FIG.3: The phase diagram TC/JAA (solid line) and TK/JAA (doted line) versus p of the system with R2 = -1.5 and R1=0.05 when the value of (D/JAA,D0/ JAA) are changed: (a) ( 0.5,0), (b) (0.5,-3), (c) (-3,0.5), (d) (-3,-3).

In order to show that the system may exhibit two compensation points, we present in fig.4 the phase diagram in the (T/J_AA,|R₂|) plane for R₁=0.4, p=0.995 and _D=D0=0.

FIG.4: The phase diagram TC/JAA (solid line ) and TK/JAA (doted line ) versus |R2| for D=D0=0, R1=0.4 and p=0.995

It is shown that TC/JAA ( solid line ) increase linearly with

|R₂|. It is also seen that the compensation temperature

(doted line) exist only if |R2|<1.0203, and that for

|R2|<0.237 we have one compensation point, and for 0.237

<|R2|<1.0203 we have two compensation temperatures, for example for R2=-0.5, TK1/J_AA =0.51 and T_K2/J_AA

=1.014. To confirm these results we have plotted in fig.5, the total magnetization |M| of the system versus the temperature, for different values of R2.

FIG.5: The temperature dependence of | M| for D=D0=0, R1=0.4 and p=0.995 when R2 is changed: (a) R2=-2, (b) R2=-0.5,(c) R2=-0.22

It is clear that when |R₂|<0.237, we have one compensation point (see curve c) and when 0.237<|R₂|<1.0203 we have two compensation temperatures (see curve b).

We have also investigated the influence of the anisotropies on the critical temperature. In fig.6 ( fig.7), we present the variations of T_C/J_AA with p for different values of the anisotropy D₀ ( D) and for R₂=-1 and R₁=0.5.

FIG.6: The phase diagram of the system TC/JAA versus p for D=0, R2=-1, R1=0.5 and for different value of D0/JAA: (a) D0/JAA=2, (b) D0/JAA=0.5, (c) D₀/J_AA=-2.

0,0 0,2 0,4 0,6 0,8 1,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

b

a

b

c

a

d c

T/JAA

p

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045

c b

a

|M|

T /J_a

0,0 0,2 0,4 0,6 0,8 1,0

1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0

a

b

c

Tc/JAA

p

0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0 1 ,2 1 ,4

0 ,4 0 ,6 0 ,8 1 ,0 1 ,2 1 ,4 1 ,6

Tk/JA A

Tc/JA A

T/JAA

|R₂|

(6)

This figure clearly exhibit that the T_C/J_AA value depends on the value of the anisotropies D and D₀. The critical temperature decreases when we decrease both D or D₀. Fig.7 shows also that the variations of T_C/J_AA with the interface anisotropy is important. But, for p=1, the T_C/J_AA value is independent of the D value, since for p=1 the interfaces are occupied only by spin-1/2 atom.

FIG.7: The phase diagram of the system TC/JAA versus p for D0=0, R2=-1, R1=0.5 and for different value of D/JAA: (a) D/J_AA=2, (b) D/_JAA=-0.5, (c) D/J_AA=-2.

Let us now turn to see the influence of the thickness of the two layers on the critical behaviour of the system. The phase diagrams of the system in the (T_C/J_AA,R₁) plane for different values of L can have three kind of topologies depending on the parameters D, D₀, R₂ and p. The first kind have the same features as that of an Ising film. That is, it exists a critical value R1C of R1 for which TC/J_AA is independent of the thickness L. We have found also that we can have two or zero values of R1C for the second and third kind respectively. In fig.8, we give the phase diagrams of the system for R2=-1, D=D0=1, p=0.5 and for different values of L. In this case it is seen that we have two values of R1C (R1C =0.033 and R1C =0.441). We have also remarked that the value of R1C depends on D, .D0, R2 and p. In fig.9, we show the R1C with R2 for D/J_AA = D0/JAA =1 and for different values of p.

It is clear that there exist a certain range of R2 for which the system may exhibit two R1C and that this interval depends on the concentration p. When we decrease p the interval becomes larger. It is also seen that for a given R2, R1C increases when we decrease p.

FIG.8: The phase diagram T_C/J_AA ( solid line ) and T_K/J_AA ( doted line ) versus R₁ for D/J_AA=D0/JAA=1, R2=-1 and p=0.5. The number accompanying each curve is the value of L.

FIG.9: The variation of the critical value R1C With |R2|

for D/JAA =D0/JAA=1 and different values of p: (a) p=0.25, (b) p=0.5, (c) p=0.75

Concerning the variations of the compensation temperature of the system with L, we present in fig.10 the variation of TK/JAA with L for different values of p for D/JAA=-0.5, D0/ JAA =0.5, R2=-1.5 and R1=0.05. It is seen that, TK/JAA decreases with L to reach a saturation value, which is independent of p, for large values of L, T_K/J_AA decreases also with p. We have remarked that depending on the parameters D and R₂ the behaviour of T_K/J_AA can change.

0,0 0,2 0,4 0,6 0,8 1,0

1,4 1,5 1,6 1,7 1,8 1,9

b

c a

Tc/JAA

p

0,0 0,2 0,4 0,6 0,8

0,0 0,5 1,0 1,5 2,0 2,5

6 4 2

R1c 2=0.441 R_1c¹=0.033

6 4

2

T/JAA

R₁

0,4 0,6 0,8 1,0 1,2 1,4 1,6

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

b a

c

R1C

|R₂|

(7)

FIG.10: The phase diagram TK/JAA versus L for D/JAA=-0.5, D0/J_AA=0.5, R2=-1.5 and R1=0.05. The number accompanying each curve is the value of p.

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For example when D=-3, we remark that T_K/J_AA increases with p for a given L, and when D=-3 and

R2=-0.05, TK/J_AA increases when we increase the concentration L to reach a saturation value, which is independent of p, for large values of L.

IV. CONCLUSION

In this work, we have investigated the magnetic properties of a ferrimagnetic multilayer system with a disordered interface by the use of the effective-field theory with a probability distribution technique. We have found that the system can acquire the possibility of two compensation temperatures for certain values of p, R₁, R_2, D and D₀. We have also shown that the system can have two critical values of R1, at which the critical temperature is independent of the thickness L. These critical values depends on p, R₂, D and D0.

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2 4 6 8 10 12 14 16 18 20

0,2 0,4 0,6 0,8 1,0 1,2

0.75 0.5

p=0.25

Tk/JAA

L