JOURNAL OF APPLIED PHYSICS
VOLUME 85, NUMBER 8
15 APRIL 1999
GMR: Fundamentals I
W. H. Butler, Chairman
Effect of biquadratic exchange coupling on magnetoresistance
and magnetization process in magnetic bilayer systems
C. C. Kuo, M.-T. Lin,a) and H. L. Huang
Department of Physics, National Taiwan University, Taipei 106, Taiwan
An improved model is proposed to deal with the magnetic bilayer systems taking into account the
contribution of the anisotropy energy and biquadratic exchange coupling to elaborate on the
evolution of the magnetoresistance ~MR! ratio and magnetization process. The results indicate that
the characteristic behavior of the MR ratio depends distinctly on both the biquadratic coupling
constant and the layer thickness. The profile of the MR ratio was found to vary from an inverted bell
shape to a concave pyramid with increasing biquadratic coupling strength, and decays sharply with
the layer thickness. This model calculation helps us to provide a venue for further understanding the
MR or giant magnetoresistance behavior of the magnetic multilayer system. © 1999 American
Institute of Physics. @S0021-8979~99!48108-3#
angles between the neighboring magnetic moments of the
sublayers. When the magnetic field increases, these magnetic
moments tilt further, depending on the field strength, and
become saturated under a strong field, as depicted in Fig. 1.
Here, the field is assumed to be applied parallel to the plane
of magnetic layer along the easy axis.
As shown in Fig. 1, each magnetic layer A and B is
subdivided, respectively, into N 1 and N 2 sublayers which are
numbered from the edge to the interface of the layer. c and f
are defined as the angles between the field direction and the
magnetic moments at the edge of the magnetic layers, while
c i and f j correspond to the angle between the ith and
i11th moments in layer A and the one between the jth and
j11th moments in layer B, respectively. Schematically, the
configuration of the magnetic moments in layers A and B in
Fig. 1 can be projected onto a plane parallel to the interface,
as shown in Fig. 2.
The total energy E total of the system can be approximated
as the sum of the exchange energy E ex , the Zeeman energy
E z , the uniaxial anisotropy energy E k , and the biquadratic
coupling energy E bi . They are expressed as
In recent years, the phenomenon of giant magnetoresistance ~GMR! and exchange coupling in magnetic multilayers
has been of utmost interest for many physicists and engineers
concerned and has been extensively investigated. In addition
to bilinear exchange coupling, the biquadratic coupling ~i.e.,
the second-order Heisenberg exchange coupling! has been
found to exist in many magnetic multilayer system.1 The
biquadratic exchange coupling has also been linked with the
spatial fluctuations in the interlayer thickness,2,3 suggesting
that this particular coupling may be present whenever the
interface is not so smooth.
In order to describe the magnetic and GMR properties of
the multilayer systems in a tractable manner a simple analytical model calculation taking both the Heisenberg exchange and biquadratic exchange coupling into consideration
has been reported.4 For a bilayer system, there is also a
model calculation which subdivides each magnetic layer into
several sublayers for calculating the magnetization processes
of the system.5–7 However, neither the anisotropy energy nor
the biquadratic coupling has been properly taken into consideration in the latter. Our objective in this paper is therefore to
address this problem by incorporating these two terms into
this model calculation to examine its consequential effect
and significance on the magnetization processes and the MR
ratio of the bilayer system.
Let us consider a composite system of two ferromagnetic
layers A and B which couple antiferromagnetically at the
interface. Accordingly, in the absence of the magnetic field
or in a weak field, the magnetic moments in layers A and B
are expected to be either parallel or antiparallel to the field
direction. By introducing the anisotropy energy to the magnetic layers and biquadratic coupling at the interface between
the two layers one expects a continuous variation of the
N 1 21
E ex52A1 M 21
S
F
F
E z 52HM 1 cos c 1
2HM 2 cos f 1
Author to whom correspondence should be addressed; electronic mail:
mtlin@phys.ntu.edu.tw
4430
(
j51
N 1 21
1AM 1 M 2 cos c 1
a!
0021-8979/99/85(8)/4430/3/$15.00
(
i51
N 2 21
cos c i 2A2 M 22
N 1 21
(
n51
N 2 21
(
n51
(
S
S
D
N 2 21
c i1 f 1
i51
cos f j
cos c 1
n
(
j51
( ci
i51
cos f 1
n
(
j51
DG
DG
fj
fj ,
~1!
,
~2!
© 1999 American Institute of Physics
J. Appl. Phys., Vol. 85, No. 8, 15 April 1999
Kuo, Lin, and Huang
4431
FIG. 1. Configuration of the bilayer system under various strength of the
magnetic field.
F
N 1 21
E k 5K 1 sin c 1
2
F
(
sin
n51
2
S
c1 ( ci
N 2 21
1K 2 sin2 f 1
(
n
i51
S
n
sin2 f 1
n51
DG
(
j51
fj
DG
and the biquadratic coupling energy
S
N 1 21
E bi5BM 21 M 22 cos2 c 1
(
i51
N 2 21
c i1 f 1
(
j51
~3!
,
D
fj ,
~4!
where M 1 and M 2 are the moments of the sublayers in A and
B, A1 and A2 are the intra-molecular ~exchange! field coefficients in the ferromagnetic layers, A is the antiferromagnetic intermolecular exchange field coefficient at the interface between layers A and B, and H is the applied field. K 1
and K 2 are the uniaxial anisotropy constants in layers A and
B, respectively. B is the biquadratic field constant between
the layers across the interface between layers A and B. By
introducing the dimensionless constants8 the total energy « is
expressed in terms of c, c i , f, and f j . After some algebraic
manipulations, we obtain a recursion relation for c 1 and c n
as follows:
c 1 5arcsin
S
F
D
h
k1
sin c 1 sin 2 c ,
a1
a1
S
DG
~5!
n21
h
c n 5arcsin sin c n21 1 sin c 1
ci
a1
i51
S
n21
k1
1 sin 2 c 1
ci
a1
i51
(
(
.
D
FIG. 3. MR ratio ~a! and corresponding magnetization profile ~b! of the
bilayer system as a function of the magnetic field with different biquadratic
coupling strength b between the two layers for N 1 5N 2 53. The inset in ~a!
is that of the MR ratio for b50.
From Eqs. ~5! and ~6!, all the angles c n and f n can be
uniquely expressed in terms of the angles c and f. Thus, all
the undetermined variables are reducible to the two angles c
and f which can be determined by seeking the extremum
conditions of «(5E total /AM 2 ) for c and f.
It is of particular interest to examine the characteristic
behavior of the bilayer system such as the magnetoresistance
~MR! ratio and magnetization process under the magnetic
field. The MR ratio and magnetization process can be expressed as
MRratio5
~6!
By the same procedure, the corresponding relations for f 1
and f n can be obtained easily by simply replacing c i , a 1 ,
k 1 by f i , a 2 m, k 2 /m, respectively.
F
21
Dr1
r t,0
N 1 21
i51
S
and
F
N 1 21
F
(
n51
1M 2 cos f 1
(
j51
N 1 21
1D r 12 cos c 1
M 5M 1 cos c 1
FIG. 2. Projection of the magnetic moments shown in Fig. 1 on the plane
parallel to the interface of the magnetic layers.
(
N 2 21
cos c i 1D r 2
(
i51
S
N 2 21
(
n51
N 2 21
c i1 f 1
n
cos c 1
S
cos f j
( ci
i51
cos f 1
DG
n
(
j51
fj
(
j51
DG
,
fj
DG
, ~7!
~8!
where r t,0[(N 1 21) r 1,01(N 2 21) r 2,01 r 12,0 , r i( j)0 and
D r i( j) are the resistivity of the sublayer ~interface! at zero
field and change of the resistivity of the sublayer ~interface!,
respectively. Upon substituting the calculated values of c,
c i , f, and f j , it is thus possible to numerically evaluate the
MR ratio and magnetization process in a straightforward
manner. For easy manipulation, all the pertinent physical
4432
J. Appl. Phys., Vol. 85, No. 8, 15 April 1999
FIG. 4. MR ratio ~a! and corresponding magnetization ~b! of the bilayer
system as a function of the magnetic field with different number of sublayers for b50.3. The inset in ~a! is the comparison between the MR ratio
~solid line! and the 2(M 2M 0 ) 2 profile ~dotted line! for N 1 5N 2 53.
quantities such as the exchange constants and the anisotropy
constants, etc., are expressed in terms of the dimensionless
constants a 1 , a 2 , k 1 , k 2 , m and b, as given in Ref. 8.
Figure 3 shows the tendency of MR ratios and corresponding magnetization processes for different biquadratic
coupling strength b when the number of sublayer N 1 5N 2
53. Typically, the MR ratios are shown to decrease with the
increase in the magnetic field h as expected. One particular
noteworthy feature of these figures is that in the absence of
the biquadratic coupling constant b, the MR profile exhibits a
convex curvature, having an inverted bell shape. Then, with
an increasing biquadratic coupling strength b, the curvature
of the MR profile turns from convexness to concaveness, i.e.,
the inverted bell shape turns into a concave pyramid. Clearly,
the presence of an inverted bell shape of the MR ratio profiles is contingent upon the absence of the biquadratic coupling. Our result is consistent with the previous data9,10 indicating the existence of the biquadratic coupling in some of
Kuo, Lin, and Huang
the bilayer systems. Note that the magnetization processes
seem to be insensitive to changes in the biquadratic coupling
strength, as shown in Fig. 3~b!.
Figure 4 shows the trend of the behaviors of the MR
ratios and magnetization processes by altering the number of
sublayers from one to three for the case of biquadratic coupling constant b50.3. The MR ratio profiles exhibit a
pyramid-like shape rather than an inverted bell shape before
becoming saturated. Furthermore one sees from the figures
that the larger the number of sublayers the more sharply the
MR ratio profile declines with the applied field. Also, the
magnetization increases steeply with the magnetic field when
both the biquadratic constant b and sublayer number N become larger. In fact, the (M 2M 0 ) 2 profile, as suggested in
Ref. 10, bears a very good resemblance to the MR ratio
profile, as depicted in the inset of Fig. 4. However, unlike the
behavior of the magnetization process versus b shown in Fig.
3~b!, presently we see that the magnetization becomes relatively harder to saturate, almost behaving like a hard axis
magnetization process especially when N is small.
In summary, our model calculation indicates that the MR
ratio profile versus magnetic field may vary from an inverted
bell shape to a concave pyramid as the biquadratic coupling
constant b increases. Thus, the present simple model may
help to shed light on the role of the biquadratic coupling in
relation to the MR ratios and the magnetization processes in
the magnetic bilayer systems. Our calculation also suggests
that the MR ratio profile decays more sharply with the external field when the number of sublayers becomes larger.
This research was supported by the National Science
Council through Contract No. NSC-87-2216-002-005.
1
M. Rührig, R. Schäfer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov,
and P. Grünberg, Phys. Status Solidi A 125, 635 ~1991!.
2
J. C. Slonczewski, J. Appl. Phys. 73, 5957 ~1993!.
3
S. Demokritov, E. Tsymbal, P. Grünberg, W. Zinn, and I. K. Schuller,
Phys. Rev. B 49, 720 ~1994!.
4
H. Fujiwara and M. R. Parker, J. Magn. Magn. Mater. 135, L23 ~1994!.
5
M. Motokawa and H. Dohnomae, J. Phys. Soc. Jpn. 60, 1355 ~1991!.
6
H. Dohnomae, T. Shinjo, and M. Motokawa, J. Magn. Magn. Mater.
90&91, 88 ~1990!.
7
M. Motokawa, Prog. Theor. Phys. Suppl. 101, 537 ~1990!.
8
The dimensionless constants are «5E total /AM 21 , a 1 5A1 /A, a 2
5A2 /A, m5M 2 /M 1 , k 1 5K 1 /AM 21 , k 2 5K 2 /AM 21 , h5H/AM 1 , and
b5BM 22 /A, respectively.
9
F. Ngugen Van Dau, A. Fert, and M. Baibich, J. Phys. ~France! 49, C81633 ~1988!.
10
S. S. P. Parkin, Phys. Rev. Lett. 72, 3718 ~1994!.