compensation temperatures of ferrimagnetic bilayer systems

Journal of Magnetism and Magnetic Materials 118 (1993) 17-27 North-Holland

Compensation temperatures of ferrimagnetic bilayer systems

T. K a n e y o s h i a n d M. J a ~ u r

Department of Physics, Nagoya University, Nagoya 464-01, Japan

Received 31 March 1992

Compensation points of two ferrimagnetic bilayer systems consisting of two different magnetic layers (A and B) coupled together with a negative interlayer coupling are examined by the use of a new formulation based on both the Ising spin identities and differential operator technique. In particular, the effects of interlayer coupling and crystal-field constant on the compensation temperature are investigated by selecting the layer spins as S A = ½ and S B = 3. We have obtained a lot of new phenomena. Some of them may be related to the experimental works of rare-earth/transition-metal multilayer films.

1. I n t r o d u c t i o n

Some bilayer systems made up of alternating layers of transition metal (TM) and rare earth (RE) atoms are synthesized in recent years [1,2]. These materials are of great interest because they are new materials with new and possibly useful properties. In particular, multilayer films of RE(Tb, G d ) / T M ( C o , Fe) show a ferrimagnetic behavior with a compensat ion temperature , when the layer thickness is not so thick. On the other hand, most of the theoretical works related to the magnetic multilayered systems are restricted to Ising or Heisenberg systems consisting of only

1 spin ~ ions where each layer has coupling constants of different magnitudes [3-8]. Therefore, in order to treat a ferrimagnetic bilayer system, it is necessary to discuss the magnetic layers with different spins, such as R E / T M multiple-layer systems.

We study, in this article, the compensat ion tempera ture in the two ferrimagnetic bilayer systems consisting of two ferromagnetic layers (A and B) coupled with a negative interlayer coupling. In particular, the spin S A of the A layers is

Correspondence to: Dr. T. Kaneyoshi, Department of Physics, Nagoya University, Nagoya 464-01, Japan.

1 fixed a t S A = ~ and the spin S a of the B layers is taken as S a 3 = ~. The crystal-field interaction term is also included on the B layers and the effects of crystal-field constant D on the compensation tempera ture are examined by the use of the new effective-field theory superior to the standard mean-field theory. In section 2, the basic framework of the theory is given. In section 3, the formulation is applied to get the expressions of the transition temperature , compensation temperature and magnetizations in the two bilayer systems. In section 4, the compensation temperature and magnetizations are examined in detail by solving the coupled equations given in section 3 numerically. Our formulation corresponds to the Zernike approximation. We have found many interesting phenomena in this work. (i) The compensation tempera ture of the bilayer systems de- pends on the exchange interactions and crystal- field constant as well as the thickness of each layer. In the systems with D >_ 0, a critical interlayer interaction exists at which the compensation tempera ture may decrease or increase when it is plotted as a function of layer thickness. (ii) When D < 0, the compensation tempera ture of the bilayer systems may exhibit some outstanding features when it is plotted as a function of the interlayer coupling. In particular, for an appropriate system with a large negative value of D

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

18 T. Kaneyoshi, M. Ja~6ur / Compensation temperatures o f ferrimagnetic bilayer systems

m -

t7 --

m-

(7 --

m -

t7 --

a b with

. - - O" - - ~

~ - O" - -

' - - m -

~ - m - ~

_ o _

~ - o - -

) - -

~ m

~ B

>.__

Ei = J l ~ t Z z + a .-k J 3 g " S z , i+8 •

For the B layers, we have

m = ( S ~ ) = <fB(Em)>

Fig. 1. Parts of two-dimensional cross sections through the bilayer systems (a) and (b) consisting of the two magnetic

_~. layers A and B with S A = ~ and SB =

two compensation points may be possible in the total magnetization curve while the layer magnetization may take normal thermal variations.

2. Theory

We consider the two bilayer systems consisting of different ferromagnetic materials A and B coupled with a negative interlayer coupling. For simplicity, we restrict our attention to the case of simple cubic Ising-type structures. The two-dimensional cross sections of the two systems are depicted in fig. 1. The Hamiltonian of the systems is given by

I-I = - J , E _ j2 E s .s z ij mn

_ j 3 ~ . , i z i S m ~ z 2 z _ D E ( S I n ) , (1) / m m

where the first three sums are taken over all the 1 nearest-neighbor pairs only once, ~. = + 3 is the

usual Ising variable on the A layers, and S,~ is the 3 3 1 spin-3 operator on the B layers (S z = + 3, + 3).

D is the crystal-field interaction constant. Three coupling constants exist (J1 > 0, J2 > 0, J3 < 0) depending on where the spin pair is located.

To evaluate the mean values (tz z) and (S,~), we start with the exact Ising spin identities [9,10]; for the A layers, it is given by

= (lz~.) = (fA(E~)) (2)

(3)

(4)

with

Em = J2Esz+ + 4 (5) 8 3'

where the functions fa(x) and f a (x ) are defined by

1 .

fA(x) = ~tanla[ ~ ) (6)

and

1 3 sinh(3flx) + e -za° sinh(~x)

f s ( x ) = ~ cosh(3flx ) + e_2t3O cosh(2~x ) . (7)

Here, /3 = 1 / k a T , and 8 and 8' in eqs. (3) and (5) denote the nearest neighbors of sites i and m.

By the use of the differential operator technique [11], these identities can be transformed into the convenient forms for calculation:

o" = (e e' V)fg (x ) I x=0 (8)

and

m = (e Em v ) f n ( x ) l x = o , (9)

where V = a /ax is a differential operator. At this stage, for ~z = + ½, the exact Van der Waerden identity

exp(a~ z) --- cosh + 2/, z sinh (10)

can be used when rewriting the exponential function in eqs. (8) and (9), where a is a constant. For the spin 3 operator, it is possible to expand the exponential function in eqs. (8) and (9) exactly by using the corresponding Van der Waerden identity. From the practical point of view, however, next calculations become very tedious and com- plex even for the low-dimensional system (for instance, honeycomb lattice). For this reason, an approximate method has been introduced in ref.

T. Kaneyoshi, M. Ja~ur / Compensation temperatures of ferrimagnetic bilayer systems 19

[10], which enables us to treat this problem very On simply and gives reasonable results in comparison and

3 with the use of the exact spin -7 Van der Waer- den identity. The method is based on the follow- ing relation: tr--

exp(aS z) = c o s h ( - ~ a ) + S Z ~ s i n h ( - ~ a ) (11)

with

(_~12 z 2 q = = ( ( S i n ) ) , (12)

which is valid for any spin value. In particular, for the case of spin 1 3, eq. (11) reduces to the exact identity (101 because of (~7/a) 2 1 ~ .

Expanding the right-hand side of eqs. (8) and (9) with the help of identities (10) and (12), multi- spin correlation function may appear and hence they must be decoupled before a practical calculation can be made. Here, we follow the standard procedure [11] with the decoupling approximation

(/z~S~ • • • / ,~) -- (/z~)(S/,) • • • (/z~) (13)

for j ~ k ~ • • • ~= I. In this way, the magnetizations tr and m of the A and B layers in fig. la are then given by

O r =

j 4

2tr + sinh( )] ( ° ) ) 2

cosh J3 V + - - m sinh --J3 V fA(X) ]~=0 */

(141

and

m =

[ cosh -~-V + 2~ sinh V f B ( x ) I x=o.

(15)

the other hand, the magnetizations of the A B layers in fig. lb are given by

cosh ~-V + 2~ sinh V

+ - - m sinh J3 V fA(X)[ ,=0 (16) 7/

and

m = [ c o s h ( 7 / J a 2 V ) + ~ T a m s i n h ( - ~ J 2 V ) ] 5

J3 + 2 t r

(17)

For the evaluation of tr and m in each bilayer system of fig. 1, one has to calculate the parameter q defined by eq. (12). The parameter q can be derived in the same way as m by the use of the Ising spin identity and the decoupling approximation (13); For the B layers of fig. la, it is given by

~ cosh J2 V + - - m s i n h J2 V ~7

cosh V + 2~ sinh V gB(X) [x=O

(181

and for the B layers of fig. lb we have

q =

+

cosh ~-V + 2 t r sinh -~-V gB(X)lx=O,

(19)

20 T. Kaneyoshi, M. Ja~gur / Compensation temperatures of ferriraagnetic bilayer systems

where the function gB(x) is defined by

1 9 cosh(3flx) + e -2D~ cosh(lflx)

gB(x) = ~ cosh(3flx ) + e_2Ot3 cosh(1/3x)

(20)

3. Transition temperature and compensation temperature

Expanding the right-hand side of the coupled equations in each bilayer system, namely eqs. (14), (15), (18) for fig. la and eqs. (16), (17), (19) for fig. lb, we obtain

o" =AIO" + Z 2 m +Z3(tr) 3 +A4(tr)2m

+h5 t r (m) 2 +h6( t r )4m +A7(o-)3(m) 3,

(21a)

m =Blm +B2tr+Ba(m) 3 + B40"(m) 2

+ Bs(tr)2m + n6(o')4m + B7(o')3(m) z, (21b)

q = C 1 'l- C2(m) 2 + C30"m + C4(or) 2 -1- Cs(m) 4

+ C60r(m) 3 + C7(0")2(m) 2 + C8(0")2(m) 4

(21c)

for the bilayer system of fig. la and

o" =Altr +.z~2m + h3(o') 3 + ~z~4(tr)2m

+ ~z~5 (tr) 5 + ~z~6( tr)4m (22a)

m =B1 m + B2tr + B3(m) 3 + B4tr(m) 2

-t- Bs(m) 5 + B6°'(m) 4, (22b)

q = t~ 1 + C2(m) 2 + C30rm + C4(m) 4

+ Cstr(m) 3 + C'6tr(m) 5, (22c)

for the bilayer system of fig. lb. The coefficients A v (or Av), B v (or By) and C~ (or C~) are easily derived from the coupled equations, namely eqs. (14), (15), (18) and eqs. (16), ~g7), (19). For in-

stance, the A1, A2, B, B 2 and C 1 in eqs. (21) are given by

A1 =8 cosh3(J1-~-~7) sinh(~-V)

X cosh2( r/J ~7)fA(X) [ 3 x=0,

A 2 = 2 ( ~ ) 4 [ J l ~ cos cosh(

= "/ sinh( ,, ") J3

"t

TV)gB(X)lx O. C1 = cosh4(-~J2 V)cosh2( J3

(23)

But, we shall not write the other coefficients explicitly. Also, they are easily calculated by using a mathematical relation

earl(x) = f ( x + a). (24)

When the temperature is higher than the transition temperature To, the whole system is demag- netized. Therefore, the transition temperature of the bilayer systems can be determined by requiring that the magnetizations tr and m approach zero continuously, since there is no tricritical behavior even for the systems with a large negative value of D [10]. Consequently, all terms of the order higher than linear in tr and m can be neglected. The transition temperature T c can be determined from

(A 1 - - 1)(B 1 - 1) =A2B 2 (25)

with

T. Kaneyoshi, M. Jaggur / Compensation temperatures of ferrimagnetic bilayer systems 21

for the bilayer system of fig. la and from

(/~1 -- 1)(nl - 1) =A2B2

with

(27)

(28)

for the bilayer system of fig. lb. Here, notice that the transition temperature T~ determined from eq. (25) (or (27)) is independent of the sign of J3, which can be easily understood from the coefficients (23).

On the other hand, the compensation temperature Tcomp of the bilayer systems can be evalu- ated by requiring the condition

tr + m = 0. (29)

That is to say, the Tcomp of the bilayer system of fig. la can be determined from the coupled equations

[ ( A 3 - A , +As) + { ( A 3 - A ` +As) 2

- 4 ( A 6 - A 7 ) ( 1 - A 1 +A2)} W2]

x [(A6-A )I = [ ( B 3 - B 4 +Bs) + { ( B 3 - B 4 +Bs) 2

- 4 ( B 6 -- B 7 ) ( 1 - g x - n 2 ) } 1/2]

X [( B 6 - B7) ] -1 , (30a )

( ff-~)2=Cl + (C2-C 3 + C4)(m)2

+ ( C 5 - C 6 + C 7 ) ( m ) 4 + C s ( m ) 6

(30b)

with

(m) [(A -A4 +A,) + {(A3-A4 +A,)

- 4 ( A 6 - A T ) ( 1 - A i -l-A2)} 1/2]

× [2(A 6 _A7) ] -1. (30c)

The T~omp for the bilayer system of fig. lb can

be also determined from eqs. (22) in the same way as the case of eqs. (3). Here, notice that the compensation temperature Tcomp must be smaller than the transition temperature T~. In fact, when the numerical evaluation of T~o is performed by using the above formulations,m(Pm) 2 (for exam- ple eq. (30c)) actually reduces to zero at T~omp = T~ and becomes negative (or unphysical) for Zcomp >

4. Numerical results

In this section, let us examine the compensation temperature Tcomp in the two bilayer systems of fig. 1 by solving the formulations in section 3 numerically. However, we have four parameters (Jl, J2, J3, D) for the numerical evaluations. In order to relate our results with some experimental data of R E / T M multilayer systems, let us take J1 > J2 > 0; We consider that A layers con- sist of the transition-metal atoms and B layers are made up of the rare-earth atoms. Then, the exchange interaction (J1) between the A atom pair results from the direct interaction and the interaction (J2) between the B atom pair is considered due to the indirect interaction. For the B atoms (or RE atoms), furthermore, the single-ion uniax- ial anisotropy (or D) is taken into account. The effects of J3 and D on the compensation temperature are examined in detail for the systems with J1 > J2 > 0. In the R E / T M bilayer system, the 3d-4f indirect interaction (or J3) is considered to be negative for heavy RE, while it is positive for light RE.

4.1. Bilayer system of fig. la

In fig. 2, we plot the variation of compensation temperature versus I J3 I in the bilayer system of fig. la with J2/J1 = 0.05, changing the value of D. In the figure, the solid lines represent the compensation temperature and the dashed lines are the variation of T c. When J3 = 0.0, the bilayer system (a) decomposes into the two independent monolayers. The exchange interaction J1 in the A monolayer is stronger than that (0.05J 1) of the B monolayer, so that the transition temperature of

22 T Kaneyoshi, M. Ja~ur / Compensation temperatures offerrimagnetic bilayer systems

the decomposed (J3 = 0.0)bilayer system is deter- 0.5 mined by that of the A monolayer. That is to say, IMk it is given by 4kBTc/J 1 --3.09, as shown by the dashed lines in fig. 2. 4kBTWJ 1 -- 3.09 is nothing but the transition temperature of the spin ~ bulk Ising square lattice in the Zernike approximation [4,12] superior to the standard mean-field result o.25 (kBTWJ , = 1.0).

At this place, the compensation point can be obtained in the bilayer system when the condition of J1 > J2 > 0 is satisfied. The solid line labeled d in fig. 2 is obtained for D f J 1 = 1.0. Increasing the value of D from D = J1, it is gradually unable to

0 find the compensation temperature. On the other hand, the solid lines labeled b and a are obtained Fig. 3. The for the two negative values of D (D/J , = - 0 . 2 and -0 .4) . The T~omp reduces rapidly to zero at the critical value ( I J3 I 1 J1 = 0.1 or 0.3). Here, the critical value may express that the ground state of

1 the bilayer system is in the spin- 1 state (S z = + for B layers) for the left-hand side of each critical

3 3 value and is in the spin ~ state (S z = + ~ for B layers) for the right-hand side. In fact, comparing

1 3 the ground state energies of S,~ = + ~ and + state in the bilayer system, one can obtain the

k BT I ' ' ' .~'J" I~

0.8

0.4 i

0 0:2 0.4 IJ31 i i

J1 Fig. 2. Phase diagram (Tcomp and T c vs. I -/3 I) of the bilayer system (a) in fig. 1, when the value of 'I2/')'1 is fixed at 0.05 and the value of D is changed; D/J1 = -0 .4 (curve a), -0 .2 (curve b), 0.0 (curve c) and 1.0 (curve d). The solid lines represent the compensation temperature Tcomp and the

dashed lines are the transition temperature Tc.

D=O.O

0.5 1 kBT/J1 1.5

I MI vs. T curves in the ferrimagnetic (-/3 < 0) bilayer system (a) with a fixed value J2 /Ja = 0.05 and D = 0.0, when the three values of J3/Jl are selected; .I3/J1 = - 0 . 0 5 (curve a), -0.5 (curve b) and -1 .0 (curve c). See also the

solid and dashed lines labeled c in fig. 2.

criterion of D separating the two ground states as

~11 = t J1 q- " (31 )

Since J2/J1 = 0.05, the critical value is satisfied at I J3 I / J x = 0 . 1 for the solid curve b and at I J3 I/J1 = 0.3 for the solid curve a.

Now, let us investigate the temperature dependence of magnetization predicted from the re- suits of fig. 2, which can be obtained by solving the coupled eqs. (21) numerically. Fig. 3 shows the thermal variations of M defined by

M = ½(tr + m) , (32)

when the parameters (J1, J2, D) of the bilayer system are fixed at J2/J1 --- 0.05 and D = 0.0 and the parameter J3 is selected as J3/J1 = -0 .05 , - 0.5 or - 1.0. That is to say, they correspond to the case of curve c in fig. 2. As predicted by curve c of fig. 2, the I M I with I J3 I/J1 = 0.05 (curve a) does show the compensation point just at the solid line of I J3 I/J1 = 0.05, while the I M I with I J3 I 1 J1 = 0.5 (curve b) and 1.0 (curve c) does not

exhibit the compensation point (or the N-type behavior in the magnetization curve).

Next, we study the variation of T~omv versus I J31 in the bilayer system with a fixed value

T. Kaneyoshi, M. Ja~ur / Compensation temperatures of ferrimagnetic bilayer systems 23

kBT

Jl

/ / / /

J2 /J1 =0.5 / / / / / / / -

A / / / / / /~' ~ / c / / b / / a / 4 / .

/ / / / / / / / / / / " I...-" |

2 / /

// /// //~ / / / / / - / "

/ . , t / / 1 / . . ~ j , a

dl c/

o 2 i jal a

J1

Fig. 4. Phase diagram (Toamp and T c vs. I-/3 I) of the bilayer system (a) in fig. 1, when the value of J2/J1 is fixed at 0.5 and the value of D is changed; D / J l = -4 .0 (curve a), -3 .0 (curve b), -2 .0 (curve c) and -1 .5 (curve d). The solid and

dashed lines represent T,x,mp and T¢ respectively.

J2/J1 = 0.5. In the system, the compensation temperature cannot be obtained for a positive value of D. The results are depicted in fig. 4. As is seen from the figure, the Tcomp cu rves have some outstanding features different from those of fig. 2. (i) The T~omp plots may take the S-type form for a large negative value of D, such as curve a for D/JI = - 4 . 0 . It means that in a very narrow

region of the curve the magnetization curve M can show the two compensation points. (ii) The Tcomp plot gradually becomes vertical from the S-type form when increasing D from a large negative value. The Tcomp cannot be obtained when D is larger than D/J 1 = -0.9.

Here, the Tcomp in fig. 4 can be obtained only for a large negative value of D. It implies that the magnetization curve M takes a form different from that of fig. 3; the ground state of the B layer

1 must be in the spin ~ state (S,~ = ± ½). In fig. 5, therefore, the thermal variations of tr, m and M are plotted for the bilayer system with JE/J1 = 0.5, Ja/J1 = -2.75 and D/J 1 = -4 .0 (or see curve a in fig. 4). As plotted in fig. 5a, the layer magnetizations tr and m express the normal behavior in their thermal variations. But, notice that the satu-

1 ration magnetization of m is given by m = ~ since the B layers with D/J = -4 .0 are in the

Z _ _ 1 S,~ - _ ~ state at T = 0 K. As shown in fig. 5b, on the other hand, the temperature dependence of the I M I has the two compensation points in the region of 0 < T < T c which are equivalent to those predicted by curve a of fig. 4. Moreover, the saturation value of I MJ at T = 0 K is given

1 by zero because of the S z = ± ~ state in the B layers. Thus, the results obtained in figs..4 and 5 are completely new phenomena which have not been found in the long history of ferrimagnetism;

0.5

0 '

~ " ~ m (a) \

D=-4.0J 1 ",\ \

J 2 / J 1 =o.5 \ \

\

, , I I [ I

' 0J8 kBT/J1

-0.5

IMI

0.0008

0.0004

A D=_4.0J 1 ' (b)

/ / J 2 / J 1 : 0 . 5

1 kB T/J1 2

Fig. 5. (a) The thermal variations of layer magnetizations (m and ~) in the ferrimagnetic bilayer system (a) with S A = 1 and S B = 3, when the parameters (D, ./2, ./3) are fixed at (D = - 4.0Ji, J2 ffi 0.5J1, J3 ffi - 2.75J1). See curve a in fig. 4. (h) The I M [ vs. T curve

of (a).

24 T. Kaneyoshi, M. Jag~ur / Compensation temperatures of ferrimagnetic bilayer systems

kBTk , , I , ~ I I / 7 / " / / / _

J1 J2/"1 :o.os . ." . . " ..-"

i" r'~ ~_~_,~/i ~ / " ~ ""~ --~ 0.8 ~b/''"

/ ,(. 0.4 0.8 ij31 1.2

J1 Fig. 6. Phase diagrams (T~,mp and T c vs. I-I31) of the two bilayer systems (a) and (b) in fig. 1, when the value of ,I2/J1 is fixed at 0.05 and the value of D is changed; D / J 1 = -0 .4 (curves a and a'), 0.0 (curves b and b') and 1.0 (curves c and c'). Solid and dashed lines represent T~m p and T c, respectively in the two bilayer systems. The curves a', b' and c' are the results of the bilayer system (b), while the curves a, b and c are the same as the corresponding ones (a, c and d) in fig. 2.

The two compensation points may be possible in the magnetization curve I MI when the bilayer system with an appropriate condition shown in fig. 4 has a large negative crystal-field constant (or the spontaneous magnetization vector of B layers is oriented in the plane of the multilayer film).

4.2, Bilayer system of fig. lb

For the bilayer system of fig. lb with J2/J1 = 0.05, let us at first examine the variations of T~or~ p and T~ versus 1131 by solving the coupled eqs. (22) under the condition (29) numerically. Fig. 6 shows the plots of T~omp (solid line) and T~ (dashed line) when selecting the same values of D as those in fig. 2. In order to compare the values of T~omp in the bilayer system of fig. lb with those in the bilayer system of fig. la, the two values are depicted in fig. 6.

Here, we should notice that the dashed lines (or T~) in the bilayer system of fig. lb reduce to the same value

4kBTc/J ~ = 4.081 (33)

when -/3 = 0.0. The transition temperature Tc in eq. (33) is also equivalent to that of the spin bilayer thin film obtained within the framework of the Zernike approximation [13], since the bilayer system then decomposes into two independent layers and the transition temperature is given by that of the A layers (or the two spin-1 layers) because of J2/J1 = 0.05. On the other hand, the critical condition separating the two ground states

3 (S z = + ~ and + ~) of the B layers in the bilayer system of fig. lb with a negative value of D is given by, upon comparing the ground state energies,

o - - = - ~ + , ( 3 4 )

J1 ~ J1

instead of eq. (31) for the bilayer system of fig. la. In fact, for the solid curve a' of fig. 6 with D / J t = - 0.4, the critical value I ./3 I / J t at which the T~omp rapidly reduces to zero is given by I J3 I/J1 = 0.55, as shown in the figure.

As depicted in fig. 6, the Tcomp for the bilayer system of fig. lb changes almost linearly in the region below T c, when it has a value of D equal to or larger than D / J 1 = 0.0. At this place, we can find some outstanding features in the figure. (i) for the bilayer systems with D/J1 > 0 the T~omp curve of the bilayer system (b) crosses the T~omp for the bilayer system (a) at a very small value of ] J3 [/J1 which may be termed as ( [ J3 [/J1)crit-

(ii) In the region of 0 < [ J3 [l J1 < ( I .]3 I/J1)~rit, the T~omp of the bilayer system (a) is larger than that of the bilayer system (b). On the other hand, in the region of ( [ .]3 [/J,)¢~it < [ J3 l~ J1, Tcomp of the bilayer system (b) is smaller than that of the bilayer system (a). In other words, the result indicates that, in the bilayer systems with D > 0 and (1J3 [//J1)crit ( 1J3 I /J1, the compensation temperature of the systems with equal thickness of the RE and TM layers must decrease with increasing thickness of the single layers. Experi- mentally, the Teomp of multilayer T b / C o films with equal thickness of the Tb and Co layers exhibits such a decrease when increasing the thickness of the single layers [14]. Furthermore, the results of fig. 6 also indicate that, in the bilayer systems with D > 0 and 0 < I J3 I/J1 <

T. Kaneyoshi, M. Ja~ur / Compensation temperatures o f ferrimagnetic bilayer systems 25

kBT

J1

i n n J2/J1 :o.5 2 i II

/ I / /

///" t

a

b

I

0

/r ,// ~ u

// / /

/ / / / / 1 / / /

/ . / / / "

lal Jl

Fig. 7. Phase diagrams (Tcomp and T¢ vs. I J31) of the two bilayer systems (a) and (b) in fig. 1, when the value of , /2 /J1 is fixed at 0.5 and the value of D is changed; D / J I = - 4 . 0 (curves a and a ' ) and - 3 . 0 (curves b and b ') . The curves a' and b ' are the results of the bilayer system (b), corresponding

to the results a and b of the bilayer system (a) in fig. 4.

(J3 I/J1)crit, the Tcomp may increase when increasing the thickness of each layers. (iii) When D < 0, on the other hand, the T~omp of the bilayer system (b) may have values in the region different from those of the bilayer system (a), as shown in the

curves a and a' for D / J 1 = -0.4 . So, one cannot plot the Tcomp as a function of thickness of each layer when the parameters (J1, J2, .]3 and D) in the systems are fixed. 0

Corresponding to fig. 4 with a fixed value J2/J1 = 0.5, the variations of Tcomp are plotted in fig. 7 for the bilayer system (b) with J2/J 1 = 0.5 by selecting the same values of D as those in fig. 4. The Tcomp of the bilayer system (b) may also show the S-type behavior. The result also indicates that the two compensation points may be possible in the magnetization curve I MI when the bilayer system (b) satisfies an appropriate condition in the S-type curve (or see the curve a' for D / J 1 --- - 4.0).

In order to understand some predictions in figs. 6 and 7 more clearly, let us study the thermal variation of I MI by solving the coupled eqs. (22) numerically. Fig. 8a shows the temperature dependence of the two bilayer systems with fixed values (J3/J1 = -0.2 , D = 0.0 and J2/J1 = 0.05). The solid and dashed lines represent I M I of the bilayer system (b) and the bilayer system (a) respectively. As is seen from the figure, Tcomp of the bilayer system (b) is significantly lower than that of the bilayer system (a). It means that the Tcomp

may decrease with increasing thickness of the single layers.

0.5

IMI

0.25

0 . 0 0 0 9

IMI

0 . 0 0 0 6

--'• D=O.O

~ J2/'11 =o.o5 \\ Js/J =-o.2 \',,

k B T / J 1

/ \ J2/J1 :o . s

\ D/J 1 =-4.0

Fig. 8. (a) The [ M I vs. T curves in the two ferrimagnetic bilayer systems (a) and (b) when the parameters (3"2, J3, D) are fixed at • I2/ .I1 = 0.05, . /3 /J1 = - 0 . 2 and D / J 1 = 0.0. The solid line is obtained for the bilayer system (b) and the dashed line is equivalent to the [M[ of the bilayer system (a) in fig. 3. (b) The [M[ vs. T curves in the two ferrimagnetic bilayer systems when the parameters (-/2, D) are fixed at . /2 / J1 = 0.5 and D / J t = -4 .0 . The solid line is obtained for the bilayer system with J3 /J1 = - 5 . 2

and the dashed line is equivalent to the I M [ of the bilayer system (a) in fig. 5b.

o ooo / / %

K ? ,b, },iV'I' ! 1 kB T/J1 0 0.3 0.6 0.9 0 ½

26 T. Kaneyoshi, M. Ja~ur / Compensation temperatures of ferrimagnetic bilayer systems

F , u / / / C ' / / ZY

4g j< //

I , I I ~

0 5 I J31 10 i i

J1 Fig. 9. The variation of T e in the two bilayer systems (a) and (b) of fig. 1 as a function of J3. The solid curves a and b are obtained for the bilayer systems (b) with J2/J1 = 0.05, when the value of D is changed; D / J 1 = 0.0 (curve a) and - 0 . 4 (curve b). The solid curves c and d are obtained for the bilayer system (b )wi th JE/Jl = 0.5, selecting D / J 1 = 0.0 (curve c) and - 0 . 4 (curve d). The dashed lines a' and b ' represent T e of the bilayer system (a) when the values of D and J2 are taken the same as those of a and b. The dashed lines c ' and d ' are T c of the bilayer system (a) when the values of D and ,/2

are taken the same as those of c and d.

On the other hand, fig. 8b shows the thermal variations of I MI in the two bilayer systems when the parameters J2 and D are fixed at J2/J1 = 0.5 and D / J 1 = - 4 . 0 . The solid line is obtained for the bilayer system (b) with J3/J1 = -5 .2 and the dashed line is the same as that of fig. 5b for the bilayer system (a). As is seen from the figure, two compensation points are obtained in I M I of the bilayer system (b), which is equivalent to the prediction of fig. 7.

4.3. Variation of T~

Finally, let us investigate the behavior of T¢ in the two bilayer systems (a) and (b) of fig. 1 by solving eqs. (25) and (27) numerically as a function of J3. The results are depicted in fig. 9. Here, the solid lines are obtained for the bilayer system (b) and the dashed lines are obtained for the bilayer system (a). As is seen from the figure, the T¢ of the bilayer system (a) (dashed line) may

increase with the increases of J3, while the variation of T¢ in the bilayer system (b) (solid line) is strongly effected by the value of J2 in the B layers. That is to say, when the interaction J2 in the B layers is taken as a very small value (J2/J~ = 0.05), the transition temperature T¢ (curves a and b) of the bilayer system (b) can show a broad maximum in the vicinity of J3/J1 = 2.5. Thus, the result indicates that the spin configuration, inside the B layers with a thickness thicker than the monolayer may be disturbed by a strong interlayer interaction J3, when the interlayer interaction J2 is very weak.

5. Conclusions

In this work, we have studied mainly the compensation temperatures of the bilayer systems consisting of two magnetic layers A (SA = ½) and B (S B = 3) coupled ferrimagnetically by the use of a new formulation. This formulation corresponds to the Zernike approximation superior to the standard mean-field theory. In particular, such a study is very important from the technological point of view as well as for academic research, since the ferrimagnetic multilayered systems are considered to be possibly useful materials for magnetooptical recording.

As shown in figs. 2-9, a lot of interesting phenomena have been found in this work. In order to compare the present results with the experimental data of multilayer R E / T M films, however, one must. notice some important facts. As was discussed in refs. [2,14,15], the interface between RE and TM layers is more complicated than in the present model calculations. In this work, we simply assume that the interlayer coupling Ja is negative and the crystal-field constant D is included in the B layers. On the other hand, owing to the special geometry of the multilayered structures, the effect of the interfaces may play an important role in the macroscopic magnetic behavior. That is to say, the strong random single-ion anisotropy in the interface seems to play a major role for the macroscopic magnetic features of the R E / T M thin films because of the mixing of RE and TM ions in the interface region. Even

T. Kaneyosh~ M. Ja~ur / Compensation temperatures of ferrimagnetic bilayer systems 27

if so in the real R E / T M films, we find in this work that the compensation point of the bilayer system with D > 0 and Ja/J1 > ( I Ja I/J1)c~it may decrease when increasing the thickness of single layers, as observed in the Tb /Co multilayer films [14]. As shown in figs. 5b and 8b, the bilayer system with an appropriate condition may show two compensation points in the total magnetization curve. The phenomenon has not been predicted in the N6el theory of ferrimagnetism. Thus, the research of the bilayer system consisting of the two ferromagnetic layers coupled ferrimagnetically may be very interesting from the mate- rial scientific point of view.

References

[1] See the colloquium digest of the 13th Int. Coll. on Magnetic Films and Surfaces, Glasgow (1991).

[2] T. Tejada, F. Badia, B. Martinez and J.M. Ruiz, J. Magn. Magn. Mater. 101 (1991) 181.

[3] L.L. Hinckey and D.L. Mills, J. Appl. Phys. 57 (1985) 3687.

[4] T. Kaneyoshi and H. Beyer, J. Phys. Soc. Jpn. 49 (1980) 1306.

[5] E.L. Albuquerque, E.F. Sarmento and D.R. Tilley, Solid State Commun. 58 (1986) 41.

[6] A.M. Ferrenberg and D.P. Landau, J. Appl. Phys. 70 (1991) 6215.

[7] T. Hai, Z.Y. Li, D.L. Lin and T.F. George, J. Magn. Magn. Mater. 79 (1991) 227.

[8] M.G. Cottam and D.R. Tilley, Introduction to Surface and Superlattice Excitation (Cambridge University, Press, Cambridge, 1989).

[9] H.B. Callen, Phys. Lett. 4 (1963) 161. [10] T. Kaneyoshi, J.W. Tucker and M. Jag~ur, Physica A 186

(1992) 495. [11] R. Honmura and T. Kaneyoshi, J. Phys. C 12 (1979) 3979. [12] F. Zernike, Physica 7 (1940) 565. [13] T. Balcerzak, J. Mielnicki, G. Wiatrowski and A. Urba-

niak-Kucharcki, J. Phys.: Condens. Matter 2 (1990) 3955. [14] L. Ertl, G. Endl and H. Hoffman, p. 214 in ref. [1]. [15] T. Kaneyoshi, J. Phys.: Condens. Matter 3 (1991) 4497.

compensation temperatures of ferrimagnetic bilayer systems

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