- Email: [email protected]
An interaction matrix for the energy analysis of an n-layered magnetic thin-film system
248
Journal
of Magnetism
and Magnetic
Materials
96 (1991) 24X-760 North-Holland
An interaction matrix for the energy analysis of an n-layered magnetic thin-film system Guo Mian and R.S. Indeck of Electrical
Department
Received
13 August
Engineering,
* Wushmgton
Uniwrsi(~~.
St. Lours, MO
63130.
USA
1990
Normalized interaction factors have been developed for the analysis of the interaction energy and the magnetlc as well a\ the electric properties of a multi-layered magnetic thin-film system. Mathematical expressions of these factors with respect to dimensions and separations of the films have been developed. Interaction factors and self-demagnetization energies of an tl-layered magnetic thin-film system can be expressed by a matrix representation. For a magnetic thin-film system with k-dimensional magnetization. the interaction energy of the system can be described by X (X < 3) symmetric interaction matrices. Diagonal elements of the interaction matrices are zero because they represent the \elf-energy of the system. The off-diagonal elements of the interaction matrix are nonzero for a system which includes interactions. By using theae interaction the direction of magnetization in each film can he numerically factors and finding a minimum energy configuration, determined. The numerical solutions to the electrical property of a coupled MR thin-film system using these interaction factors agree with previous experimental findings.
1. Introduction Multi-layered thin-film systems utilizing high-permeability magnetic materials have been applied widely in modern electronics applications. For example, magnetoresistive gradiometers [1,2] and multi-layered thin-film inductive sensors [3] are used for sensing magnetic fields. Multi-layered thin films may also be used in microwave technology because of its particular resonance character and dispersion relation [4]. The large magnetization in highly permeable materials may create a strong interaction between nearby films. This interaction will, in turn, affect the magnetization within the films. Many properties, such as the magnetoresistive output, depend upon the magnetization within the films [5]. and will therefore depend upon the interactions between the films. Determination of the interaction energy of the multi-layered system is usually difficult. The interaction depends upon the direction of magnetization in each film, the separations of the films, and the film dimensions. Dove [6] calculated an interaction energy between two thin films by assuming a periodic (sinusoidal) distribution of magnetization in each film. The result was one dimensional and might be useful for analyzing the print-through effect in magnetic recording tapes but this analysis may not be useful for analyzing a multi-layered magnetic thin-film sensor because of the completely different magnetization distributions in the sensor elements. Rhodes and Rowland [7] calculated the demagnetization energy of a uniformly magnetized rectangular block. This result is expressed as a series expansion which is useful for analyzing the self-energy of the block but not convenient for the energy analysis of a multi-layered thin-film system. If very small coupled thin-film devices are considered, effects due to the edges may be significant. These edge effects usually depend strongly on such physical properties as the direction of the anisotropy and the
* Supported
in part by NSF Grant
0304-8X53/91/$03.50
No. ECS-8957148
?) 1991 - Elsevier Science
(Presidential
Publishers
Young
Investigator
B.V. (North-Holland)
Award)
Guo Mian, R.S. Indeck / Interaction matrix
249
dimensions of the device. Some micromagnetic analyses have been done to analyze the magnetization of such nonconventional devices [8-lo]. Smith [8] found significant edge effects near the edges of the films for a very small device when he assumed that the magnetization and the anisotropy field of the films were including a strong in plane and a 90” wall existed at edges of the films. With different assumptions perpendicular anisotropy field, Pohm and Slonczewski [9,10] calculated tunnel effects of a coupled thin film. Both of these works investigated the interactions of very small two-dimensional coupled thin-film devices. Most treatments of coupled thin-film systems to date are related to a two-dimensional thin-film device, this may not provide the information of the interaction of the films with respect to the dimension of the system. We have proposed interaction factors and their simple three-dimensional mathematical expressions for a symmetrically coupled, uniformly magnetized conventional magnetic thin-film system to analyze the interaction effects in such a system. We have investigated the interaction between two magnetic films, as well as electric and magnetic properties of the system with respect to the dimensions and the separation between the films [ll]. In the current contribution we develop a matrix representation of the proposed interaction factors for an n-layered magnetic thin film system to extend our previous two-film results. Equilibrium states of the magnetization in each film are determined by minimizing the magnetostatic energy of the n-layered film system.
2. The interaction
factors
Building on our previous work [ll], we assume that the system is a symmetric n-layered magnetic thin film system (each film is coaxial and parallel with the others) with each element being uniformly magnetized. The total interaction energy of the system can be given by the superposition of the interaction where the i and j are related to the i th and the jth films, respectively. The coordinates of energy Einter rj’ the ith and the jth films are shown in fig. 1. Since we consider here only conventional multi-layered thin-films systems, film thicknesses are assumed to be much smaller than that of the length and the width. This constrains the magnetization to be in the film plane and allows us to neglect edge effects. Additionally, most thin film magnetoresistive sensors are processed by a strong in-plane magnetic field, creating a uniaxial magnetic anisotropy in the y-z plane, further constraining the magnetization to lie in the plane. This in-plane magnetization assumption is used by many authors [8,12-141. In accordance with this conventional assumption, demagnetization fields due to the magnetizations in the ith and the jth films can be expressed by
Fig. 1. The coordinates
of two films for the n-layered
thin-film
system. The magnetization
is assumed
to be in-plane.
250
Cue Mian,
R.S. Indeck
/ Interaction
matrrx
where N is the demagnetization constant of the film and M is the magnetization within the film. Since the films are aligned and parallel with one and another, the maximum interaction occurs when the distance between two films, d,,, is zero, Assuming that the magnetizations within the films are uniform, the interaction energy due to the ith and jth films at infinite and zero separation can be given by E ,nterI, (4,
-w)=o,
E m&,.,=0)
= -%‘4*Ki,+Wf4/,) = f(M,,,M,.,N,
+ M_,M:,N:,
+ M@,,N,.,
+ M:,M,,N:,).
(3)
We have assumed that the films are aligned and parallel with one another so that the interaction energy due to two perpendicular magnetization vectors is zero. The above results can be generalized to an arbitrary separation by defining a normalized interaction factor related to an arbitrary direction, fa(d,,). which is proportional to the interaction energy due to the magnetization vectors in this direction (a^ direction) and is a function of the separation and coordinates of the films:
fa( 4, >= Em,,,a( d,, h’Emter a(0)3 which satisfies
the following
(4)
conditions:
= 0.
Einter
1 j (4,)
=-%ter&,,)
+Emtr&,,)
= t(M,,,MJ%,
In order to determine consider the interaction the a^ direction:
+ MI.,MI.,Y.,)fi.(d,,)
+f;(d,,)E,ntAO)
JO)
+ :(M:,M,,N,
+ M+Vz,)L(d,,).
(6)
we in
a ( d, 1)
+/&Jx,.
Y,,
4fL,b,.
Y,,
z,)
dx,
d.t
dz,
I’,
[I
+
=f,.(d,.,)Ler
the interaction factor with respect to the magnetization in the a^ direction, energy due to two plane-parallel magnetic films with the uniform magnetization
fmter a ( d, 1) CXEinter
at
(5)
tJMa,(x,, / I’,
J,,
+%,,(x,~
I;,
2,)
dx,
d-v,
dz,
(7)
where u, and u, are the volumes of the i th and the jth films, respectively. H, a, and H, a, are the demagnetization fields due to magnetizations in the i th and the j th films, respectively, and computed for the a^ direction. For films of identical dimensions, a result of the symmetry of the system, the expressions of H, il, and H,, a, are exactly the same when the coordinates of the i th film are reflected to those of the jth film. So the fd(dr,) is proportional to either term in eq. (7) because of the uniform magnetization. Let us consider the interaction factor with respect to the jj direction first:
f&)a~M,.,(x,.
Y,,
z,)K.I.,b,~
Y,,
z,) dx, dy, dz,+f~ ,.,b,.
Y,‘,
zt) dx, dy, dz,.
(8)
“3
Since the magnetization Hd ,,, by an integration
within each film is uniform, the volume magnetic charge is zero. We can calculate of the surface magnetic charge a,(~,, y,, f h/2) for the symmetric system. The
Guo Mian, R.S. Indeck / Interaction matrix
specific
charge,
a,, on two opposing
Hdyj( x,, y;, z;) = 2 [
surfaces
has the same magnitude
251
but opposite
sign. This yields [15]
dz, (Y, - h/2) 1”’ dxj jL’2 -t/2 -L~~[(X;-x,+d;j+t)~+(Y,-h/2)~+(z,-zJ)~]3’2 dzJ (Y; + h/2) (9) [(x,-xJ+diJ+t)2+(Y,+h,2)*+(zj-zj)~]3’2
Substituting
the above equation
fy( d,J) a /::z2dxl
I
’
into eq. (8) we get:
/hlT2dY,
/flz2dz,
/f’2 dxJ -r/2
( Y, - h/2)
X
(~,-x,+d,,+t)~+(y~-h/2)~+(z~-z~)~]~’~ ( Y; + h/2)
-
Considering be reduced
the symmetry to
X
Integrating
of the system and reflecting
1 dzJ.
[(x, - xJ + d,, + r)* + ( y, + h,‘2)2 + (z, - zj)‘] 3’2
y, in the second term to -y,,
(10)
the above integral
dzj ( Yt - h/2)
L/2
J-L~2[(~;-xj+d,j+t)2+(y,-hh/2)2+(z,-zJ)2]3’2’
the above with respect
fy(dJJ> a /::z‘lxl
Jr::,(
to zj while considering
Y, - i)
can
(11) the problem
symmetry,
we get
dY; J_fL;zdr,
X ~~~~2{(‘~-‘/‘)dr,)([(x,-Xj+d,J+t)2+(Y~-h/2)2]
x [(x,
Integrating
- xi +
the above integral
d,, + t)2 + ( y, - h,‘2)2 + (z, - L,2)2] with respect
to z, and y, yields
“‘) -I.
(12)
252
Guo Mum,
X
t’2
/ -t/2
+
il’
In
(x,
-x,
(x,
-
R.S. Indeck
/ Interaction
matrix
- L)iii’(x,
-x,
+
d,, +
q2+ L’
+
d,,
t)2
(d
+
+
_ x, + d,, + $
I
L, h > t. The second
x
i’
\lL’+h’+(d
+ L
(x,-.u,+d,,+r)?+h’+L.L
+ hZ] I”’ -((x,-x,+d,,+t)
:
dx,.
can be solved using a series expansions
(13) by considering
[S’+(d,,+r)‘]
term in eq. (13) may be approximated
,,+I)’
-Lj
(4,-t+
x,)
[S’+(d,,+t)‘]
when
L’
j:;;;x,j:;;*{[ (XI
i, + 4x,-
+ h? + L? + L)
d-y-, x,
The first and third terms in the above integrations that the term
R=
+ d,, + ty
by
+Lj/j\jL’+h’+(d,,+t)‘-L&/L’+(d,,+t)’+L~]
because
A maximum error due to the above approximation takes place at d,, = 0. The error of this approximation has been evaluated [see appendix] to be about one percent for L, h z+ t. This error may also be reduced by normalizing the interaction factors. By neglecting higher order terms of the series expansion, integrating and normalizing, we get iL’+h’+(d,,+$
f,.(d,,)
=
L In
c
[’
+L
I[’
\l’L’+(d,,+~)‘-L
I
[/L’+h’+(d,,+t)‘-L][/L’+(d,,+r)‘+L]
+2[/~+/h2+(d,,+t)2-/L’+h”+(d,,+r)L
-(d,,+f)] i
(14)
Guo Mian, R.S. Indeck / Interaction matrix
253
0
923
1846
2769
E2310
lb)
(a) view of the interaction factor &(t. d) for the interacting contour map of r,(d) versus t and d.
Fig. 2. (a) Three-dimensional
2310
The normalized interaction reflecting L to h:
factor
related
to the magnetization
films (h = 20 pm,
in the i direction
L = 600 pm). (b) The
can be obtained
by
1
[JL'+h2+(d,,+f)2 +h][{W-h]
fi(d;,)= h In
[JL'+h2+(d,i+1)2-h][~~+h]
+2[/iq77+JL2+(d,j+-/L2+h2+(d;,+t)2
-p,,+q] 1
The above interaction factors satisfy eq. (5). A three-dimensional view and contour map of f,(d,j) versus film thickness and separation are shown in fig. 2. From the figure we see that the interaction factor decreases rapidly as the distance is increased and that the rate of decrease is faster for thinner films.
3.A matrix representation We have obtained mathematical expressions of the interaction factors used for analyzing the magnetointeraction between two thin films in a three-dimensional n-layered thin film system. The total interaction energy of the n-layered magnetic thin film system can be generalized by static
k
i#j
[ Myify(dij)NyjMy,
+
MziL(dij)N,jMzj].
(16)
Guo Mian, R.S. Indeck
254
/ InteractIon matrm
This equation may be expressed by a matrix representation. For simplicity, we consider a two-layer problem first. The interaction energy of a coupled thin film system given by eq. (6) can be expressed by a matrix representation as follows:
E
where
is the y component
of the magnetization
of the system in this matrix
is the z component
of the magnetization
of the system in the matrix
is a demagnetization
matrix
of the system with respect
representation,
and
representation.
to the j direction,
and
is the demagnetization matrix of the system with respect to the z^ direction. Similarly, matrices of the system with respect to the magnetization in the j and z^directions are
the interaction
(22) and 0
LP,,)
fZ=i h(d,,) The above results
0
(23)
’
can be generalized
to an n-layered
thin film system.
Let
1M,l
M!Jz My
=
My3
(24)
Guo Mian, R.S. Indeck / Interaction matrix
255
and
M, =
be y and z components of magnetization of the system in the matrix representation, the interaction and demagnetization matrices with respect to the j direction in this matrix representation can be expressed by n X n matrices 0
fy&)
fy(41) fy =
0
f,@31)
f&43)
...
f,(d,J
fyP23)
. . .
fyMn)
* **
fy(d3n)
fJd32)
0
o
...
. . .
0 ...
. . . f,MJ
f-(4*)
fy(43)
(26)
... 0
and
NY=
/ NY, 0
NY2 0
NY3 0
NY, ...
0
0
0 0
..... .
...
(27)
... ,$I
. . . Ym
0
...
Nyn/
The interaction and demagnetization matrices with respect to the 2 direction in this matrix representation are similar expressions except reflecting y to z. that is
f&4,)
...
fztd,,)
stead 0
f, =
@3A
f-2643)
f,(d32)
o
o
... 0
. . .
.. .
fz@n,) fiiid
:(d,,)
fX4J f*(d*n)
...
...
fzt;(d,J
(28)
... 0
and
N, =
N,,
0
0
.+.
0
0
Nz2
0
...
0
0
0 ...
Nr3
... N,,
0
0
... 0
...
0
... Nrm
. . .
%I
(29)
256
Guo MKUI, R.S. Indeck
Since all N,k(i =y, z; k = 1. 2,. therefore can be expressed by N,=N,I.
(i=,~,
/ Interaction
. , n) are the same for the symmetric
system,
the demagnetization
i),
E lnler = : [ &+f,NyM
matrix
(30)
where I is an n-dimensional n X n unity matrix. The interaction system expressed by eq. (16) now can be expressed by
which has exactly
matrix
+ M:f,N,M]
energy of the n-layered
magnetic
3
thin film
(31)
the same form as eq. (17) in this matrix
representation.
4. Results and discussions From eq. (4) we know that for a k-dimensional magnetization system (k I 3) there are k interaction matrices for describing the interaction of the system. We have assumed that the magnetization is in-plane for each of the n-layered thin films, so there are only two interaction matrices with respect to magnetizations in j and i directions. From eqs. (26) and (28) we see that the off-diagonal elements in the interaction matrices are nonzero. The physical meaning of this is that these off-diagonal elements represent the interaction of magnetizations of the thin film system. In a system which includes its interaction, the off-diagonal elements of the matrices have to be nonzero. Since f,(d,,) equals f,,(d,,) (a =y, z). the interaction matrices are symmetric matrices in this matrix representation. The diagonal elements of the matrix represent the self-demagnetization energy of each identical film and have to be zero in the interaction matrix because the self-demagnetization energy of each film does not contribute to the interaction energy of the system. The self-demagnetization energy of the system can be written by: Edemag= : [ A$!lN~,M,. + M,+INIMZ], where I is the n x n unity matrix. be given by [16]
(32)
Magnetostatic
energy of the multi-layered
magnetic
thin-film
system can
‘total = Ek + Ebiar + E,,i,, + ‘demag + Eappl. The first term, E,, which is the summation of anisotropy energy of each film, is the total anisotropy energy of the system; the second term, Ebias, which is the summation of the energy due to bias field in each film, is the total bias energy of the system; the last term is the energy due to the applied field. This magnetostatic energy may be written by a matrix representation: Et,,,, = -M,’
( Hk_k+ Hbiaa 1’+ Harp1 I’> - M=+ ( Hk- + Hhia\ : + Happi 2 >
+ : [M>: (fJ + I)N.,fl,
+ M.‘(f,
+ I)N,M;]
,
(33)
where
I+,.=
Hs,,
,
S
=
K, bias, appl,
(34)
Guo Man, R.S. Indeck / Interaction matrix
257
a
b
k -d=40
0
5
10
15
APPLIED Fig. 3. (a) Angle between
FIELD
20
25
30
0
10
APPLIED
Hy( Oe)
1
H
d=350
20
FIELD
30
;i
40
Hy( Oe)
thin films versus a perpendicular applied field (Hbias = 0, H, = 2.5 in the coupled films versus a perpendicular field. The bias fields in two film are anti-parallel along y axis ( Hbias = 8 Oe).
magnetizations
in two coupled
magnetic
Oe, MS = 8000 G, t = 450 A, h = 20 pm, L = 450 pm). (b) Angle between magnetization applied
is the y component of the fields (S = anisotropy, bias, applied fields) of the system, and ‘H
Sri
’
H Sr2 H,,=
f&z3
,
S
=
K, bias, appl,
(35)
is the z component of the fields of the system in the matrix representation. In the above equations, Hsy, and Hszn are y and z components of the S field (K, bias or applied) in the n th film, respectively. By minimizing the magnetostatic energy of the system, one can determine an equilibrium state of the system. Fig. 3a shows the numerical solutions of the angle between magnetizations in two films versus applied field in the 9 direction using the interaction factors. From the figure we see that at the initial point H = 0, the magnetizations in the i th and the j th = (i f 1)th films are anti-parallel. However, as the applied field increases, the magnetization vector in each film is rotated toward each other to reduce the energy due to the applied field. From the figure we also see that the closer the two films become, the strength of the field necessary to rotate two magnetization vectors parallel with each other increases accordingly. Here we can see a significant difference of magnetization of the system due to different separations of the films. A strong interaction between two films brings about a strong anti-parallel effect of the system as expected. Similar results shown in fig. 3b are found when there are perpendicular anti-parallel bias and anisotropy fields in the system. The numerical solution of the angle between the magnetization of the outer films, films 1 and 3 for a three-layered magnetic thin-film system, is shown fig. 4. For d 12= d,,, the magnetization in films 1 and 3 rotates synchronously. From the figure we see a correspondence between larger film separations and synchronous rotations of the magnetizations in films 1 and 3. Since d,, is smaller than d,, for the asymmetric case, the magnetization in film 3 will rotate toward to its final angle (270 o ) faster than that in film 1 because the interaction between films 1 and 2 is stronger than that of films 2 and 3. The peaks in the
258
Guo Mian, R.S. Indeck
0 Fig. 4. Numerical magnetic
/ Interaction
matrix
APPl%D FIEl,~” Hy( 0e)
60
solution of the angle between the magnetization of films 1 and 3 versus a perpendicular applied field of a 3-layered thin-film system (MS = 15000 G, L = 600 pm, h = 20 pm, t = 450 A, H, = 2.5 Oe. Hh,_ = 0, d12 = 40 A).
figure correspond to the situation where the magnetization in film 3 attains its final angle (270 o ) parallel to the applied field. At zero applied field, the magnetization in the film 2 is antiparallel to both the magnetization in films 1 and 3 and all of the magnetizations rotate more toward the direction of the applied field as the field grows larger. The smaller d,, is, the harder it is for the magnetization in film 3 to rotate parallel to the applied field. Fig. 5 shows the numerical results of the normalized resistance of a coupled Permalloy film device verses a uniform perpendicular applied field. The normalized resistance of the system is simulated to be equal to (cos28, + cos2B2)/2 [5,12], where 8, and 8, are the angles between easy axes and magnetizations of films 1 and 2, respectively. Similar to the experimental work of Van Ooyen et al. [12], the anisotropy field is uniaxial and in the z^direction. Since the external field is uniformly applied and the thickness of the films is much smaller than the width and the length of the films the edge effect is ignored. From the figure we see that there is a parabolic character of the magnetoresistance for films with a large separation. Since the normalized resistance versus the perpendicular applied field for a single film can be characterized by a parabola [5,12] the above results indicate that large separation of the films brings about a single-film character because of the weak coupling effect. These numerical results agree with the experimental findings of Van Ooyen et al. The general expression of the magnetostatic energy of an n-layered magnetic thin-film system has been represented using a matrix notation. Since there are many similarities between ferromagnetic and ferroelectric materials, we expect that the results may also be applicable to an n-layered ferroelectric thin-film system.
5. Conclusions A matrix representation has been proposed for the energy analysis of an n-layered magnetic thin-film system. The interaction energy of the n-layered magnetic thin-film system can be described by the proposed interaction matrices. For a k-dimensional magnetization, there are k (k I 3) symmetric interaction matrices. The off-diagonal elements of the interaction matrix are nonzero for an interacting system. The interaction matrices are symmetric n X n matrices for the n-layered system. A general
Guo Mian, R.S. Indeck
/ Interaction
259
matrix
0.60 w
a
z 2 zw 0.40 p: Es N
Li 2
ik, :f :: j: : :: :Li
o.20
5 z
0.00
0.00
-40
-20
APPLIED
F!lELD i&k)
I,,,:,,,,,,,,,,,,,,,,,,/,,,,,,,,,,,, \,,, -40
40
-20
APPLIED
FOIELD If&e)
40
g 0.40 N 2 2 p:
d .=4000 0.20
g 0.00 -30 Fig. 5. Numerical solution of the normalized resistance A uniaxial anisotropy field is along the 2 direction (t
-10
APPLIED
F’IELD1’Hy( Oe)
versus a perperdicular
applied
30
field for a coupled
Permalloy
thin-film
system.
= 450 A, h = 20 pm, L = 600 pm, Hbias = 8 Oe, If, = 2.5 Oe, MS = 8000 G).
expression of magnetostatic energy of the n-layered thin-film system in the proposed matrix representation has been found. Numerical results of uniformly magnetized, twolayered and a three-layered Permalloy magnetic thin-film system using the interaction matrix are presented. A strong interaction between two magnetic thin films bring about a strong anti-parallel effect of the magnetization in the films; larger separations between the films results in a single film magnetoresistive behavior because of a weak coupling effect. These results agree with experimental findings.
Acknowledgement We would like to thank Professor M.W. Muller for his valuable discussions.
Guo Mtan, R.S. Indeck / Interaction
260
matrtx
Appendix At d,, = 0, since L, h z+ t, the second term integration of a constant except for the term
The above integration
is equal to - 8.21 when
above L, h and t into the approximation tion is about - 8.1.
(L/2)
of the integral
L = 600 pm, ln(dm
in eq. (13) can be considered
h = 20 pm and - L)t’
we obtain
to be an
t = 450 A. Substituting the approximate
References [l] [2] [3] [4] [5] [6] [7] [E] [9] [lo] [ll] 1121 [13] [14] [15] 1161
R.S. Indeck, J.H. Judy and S. Iwasaki, IEEE Trans. Magn. MAG-24 (1989) 2617. H.S. Gill, V.W. Hesterman, G.J. Tarnopolsky, L.T. Tran, P.D. Frank and H. Hamilton, J. Appl. Phys. 65 (1989) 402 C. Denis Mee and Eric. D. Daniel, Magnetic Recording Handbook (McGraw-Hill, New York, 1989). C. Vittoria, J. Appl. Phys. 57 (1985) 3712. D.A. Thompson, L.T. Romankiw and A.F. Mayadas, IEEE Trans. Magn. MAG-11 (1975) 1039. D.B. Dove, Bell Sys. Tech. J. (Sept. 1967 1527). P. Rhodes and G. Rowlands, Proc. Leeds Phil. Sot. 6 (1954) 191. N. Smith, IEEE Trans. Magn. MAG-23 (1987) 259. A.V. Pohm and C.S. Comstock, IEEE Trans. Magn. MAG-20 (1984) 863. J.C. Slonczewski, B. Petek and B.E. Argyle, IEEE Trans. Magn. MAG-24 (1988) 2045. Guo Mian and R.S. Indeck, IEEE Trans. Magn. MAG-26 (1990) 2415. J.A.C. Van Ooyen, W.F. Druyvesteyn and L. Postma, J. Appl. Phys. 53 (1982) 2596. F.J. Jeffers, IEEE Trans. Magn. MAG-15 (1979) 1628. H. Chang, IBM J. (October 1962) 419. Guo Mian and T. Yamaguchi, J. Magn. Magn. Mat. 62 (1986) 325. S. Chikazumi, Physics of Magnetism (Krieger, Malabar, FL. 1964).
the
integra-
Journal
of Magnetism
and Magnetic
Materials
96 (1991) 24X-760 North-Holland
An interaction matrix for the energy analysis of an n-layered magnetic thin-film system Guo Mian and R.S. Indeck of Electrical
Department
Received
13 August
Engineering,
* Wushmgton
Uniwrsi(~~.
St. Lours, MO
63130.
USA
1990
Normalized interaction factors have been developed for the analysis of the interaction energy and the magnetlc as well a\ the electric properties of a multi-layered magnetic thin-film system. Mathematical expressions of these factors with respect to dimensions and separations of the films have been developed. Interaction factors and self-demagnetization energies of an tl-layered magnetic thin-film system can be expressed by a matrix representation. For a magnetic thin-film system with k-dimensional magnetization. the interaction energy of the system can be described by X (X < 3) symmetric interaction matrices. Diagonal elements of the interaction matrices are zero because they represent the \elf-energy of the system. The off-diagonal elements of the interaction matrix are nonzero for a system which includes interactions. By using theae interaction the direction of magnetization in each film can he numerically factors and finding a minimum energy configuration, determined. The numerical solutions to the electrical property of a coupled MR thin-film system using these interaction factors agree with previous experimental findings.
1. Introduction Multi-layered thin-film systems utilizing high-permeability magnetic materials have been applied widely in modern electronics applications. For example, magnetoresistive gradiometers [1,2] and multi-layered thin-film inductive sensors [3] are used for sensing magnetic fields. Multi-layered thin films may also be used in microwave technology because of its particular resonance character and dispersion relation [4]. The large magnetization in highly permeable materials may create a strong interaction between nearby films. This interaction will, in turn, affect the magnetization within the films. Many properties, such as the magnetoresistive output, depend upon the magnetization within the films [5]. and will therefore depend upon the interactions between the films. Determination of the interaction energy of the multi-layered system is usually difficult. The interaction depends upon the direction of magnetization in each film, the separations of the films, and the film dimensions. Dove [6] calculated an interaction energy between two thin films by assuming a periodic (sinusoidal) distribution of magnetization in each film. The result was one dimensional and might be useful for analyzing the print-through effect in magnetic recording tapes but this analysis may not be useful for analyzing a multi-layered magnetic thin-film sensor because of the completely different magnetization distributions in the sensor elements. Rhodes and Rowland [7] calculated the demagnetization energy of a uniformly magnetized rectangular block. This result is expressed as a series expansion which is useful for analyzing the self-energy of the block but not convenient for the energy analysis of a multi-layered thin-film system. If very small coupled thin-film devices are considered, effects due to the edges may be significant. These edge effects usually depend strongly on such physical properties as the direction of the anisotropy and the
* Supported
in part by NSF Grant
0304-8X53/91/$03.50
No. ECS-8957148
?) 1991 - Elsevier Science
(Presidential
Publishers
Young
Investigator
B.V. (North-Holland)
Award)
Guo Mian, R.S. Indeck / Interaction matrix
249
dimensions of the device. Some micromagnetic analyses have been done to analyze the magnetization of such nonconventional devices [8-lo]. Smith [8] found significant edge effects near the edges of the films for a very small device when he assumed that the magnetization and the anisotropy field of the films were including a strong in plane and a 90” wall existed at edges of the films. With different assumptions perpendicular anisotropy field, Pohm and Slonczewski [9,10] calculated tunnel effects of a coupled thin film. Both of these works investigated the interactions of very small two-dimensional coupled thin-film devices. Most treatments of coupled thin-film systems to date are related to a two-dimensional thin-film device, this may not provide the information of the interaction of the films with respect to the dimension of the system. We have proposed interaction factors and their simple three-dimensional mathematical expressions for a symmetrically coupled, uniformly magnetized conventional magnetic thin-film system to analyze the interaction effects in such a system. We have investigated the interaction between two magnetic films, as well as electric and magnetic properties of the system with respect to the dimensions and the separation between the films [ll]. In the current contribution we develop a matrix representation of the proposed interaction factors for an n-layered magnetic thin film system to extend our previous two-film results. Equilibrium states of the magnetization in each film are determined by minimizing the magnetostatic energy of the n-layered film system.
2. The interaction
factors
Building on our previous work [ll], we assume that the system is a symmetric n-layered magnetic thin film system (each film is coaxial and parallel with the others) with each element being uniformly magnetized. The total interaction energy of the system can be given by the superposition of the interaction where the i and j are related to the i th and the jth films, respectively. The coordinates of energy Einter rj’ the ith and the jth films are shown in fig. 1. Since we consider here only conventional multi-layered thin-films systems, film thicknesses are assumed to be much smaller than that of the length and the width. This constrains the magnetization to be in the film plane and allows us to neglect edge effects. Additionally, most thin film magnetoresistive sensors are processed by a strong in-plane magnetic field, creating a uniaxial magnetic anisotropy in the y-z plane, further constraining the magnetization to lie in the plane. This in-plane magnetization assumption is used by many authors [8,12-141. In accordance with this conventional assumption, demagnetization fields due to the magnetizations in the ith and the jth films can be expressed by
Fig. 1. The coordinates
of two films for the n-layered
thin-film
system. The magnetization
is assumed
to be in-plane.
250
Cue Mian,
R.S. Indeck
/ Interaction
matrrx
where N is the demagnetization constant of the film and M is the magnetization within the film. Since the films are aligned and parallel with one and another, the maximum interaction occurs when the distance between two films, d,,, is zero, Assuming that the magnetizations within the films are uniform, the interaction energy due to the ith and jth films at infinite and zero separation can be given by E ,nterI, (4,
-w)=o,
E m&,.,=0)
= -%‘4*Ki,+Wf4/,) = f(M,,,M,.,N,
+ M_,M:,N:,
+ M@,,N,.,
+ M:,M,,N:,).
(3)
We have assumed that the films are aligned and parallel with one another so that the interaction energy due to two perpendicular magnetization vectors is zero. The above results can be generalized to an arbitrary separation by defining a normalized interaction factor related to an arbitrary direction, fa(d,,). which is proportional to the interaction energy due to the magnetization vectors in this direction (a^ direction) and is a function of the separation and coordinates of the films:
fa( 4, >= Em,,,a( d,, h’Emter a(0)3 which satisfies
the following
(4)
conditions:
= 0.
Einter
1 j (4,)
=-%ter&,,)
+Emtr&,,)
= t(M,,,MJ%,
In order to determine consider the interaction the a^ direction:
+ MI.,MI.,Y.,)fi.(d,,)
+f;(d,,)E,ntAO)
JO)
+ :(M:,M,,N,
+ M+Vz,)L(d,,).
(6)
we in
a ( d, 1)
+/&Jx,.
Y,,
4fL,b,.
Y,,
z,)
dx,
d.t
dz,
I’,
[I
+
=f,.(d,.,)Ler
the interaction factor with respect to the magnetization in the a^ direction, energy due to two plane-parallel magnetic films with the uniform magnetization
fmter a ( d, 1) CXEinter
at
(5)
tJMa,(x,, / I’,
J,,
+%,,(x,~
I;,
2,)
dx,
d-v,
dz,
(7)
where u, and u, are the volumes of the i th and the jth films, respectively. H, a, and H, a, are the demagnetization fields due to magnetizations in the i th and the j th films, respectively, and computed for the a^ direction. For films of identical dimensions, a result of the symmetry of the system, the expressions of H, il, and H,, a, are exactly the same when the coordinates of the i th film are reflected to those of the jth film. So the fd(dr,) is proportional to either term in eq. (7) because of the uniform magnetization. Let us consider the interaction factor with respect to the jj direction first:
f&)a~M,.,(x,.
Y,,
z,)K.I.,b,~
Y,,
z,) dx, dy, dz,+f~ ,.,b,.
Y,‘,
zt) dx, dy, dz,.
(8)
“3
Since the magnetization Hd ,,, by an integration
within each film is uniform, the volume magnetic charge is zero. We can calculate of the surface magnetic charge a,(~,, y,, f h/2) for the symmetric system. The
Guo Mian, R.S. Indeck / Interaction matrix
specific
charge,
a,, on two opposing
Hdyj( x,, y;, z;) = 2 [
surfaces
has the same magnitude
251
but opposite
sign. This yields [15]
dz, (Y, - h/2) 1”’ dxj jL’2 -t/2 -L~~[(X;-x,+d;j+t)~+(Y,-h/2)~+(z,-zJ)~]3’2 dzJ (Y; + h/2) (9) [(x,-xJ+diJ+t)2+(Y,+h,2)*+(zj-zj)~]3’2
Substituting
the above equation
fy( d,J) a /::z2dxl
I
’
into eq. (8) we get:
/hlT2dY,
/flz2dz,
/f’2 dxJ -r/2
( Y, - h/2)
X
(~,-x,+d,,+t)~+(y~-h/2)~+(z~-z~)~]~’~ ( Y; + h/2)
-
Considering be reduced
the symmetry to
X
Integrating
of the system and reflecting
1 dzJ.
[(x, - xJ + d,, + r)* + ( y, + h,‘2)2 + (z, - zj)‘] 3’2
y, in the second term to -y,,
(10)
the above integral
dzj ( Yt - h/2)
L/2
J-L~2[(~;-xj+d,j+t)2+(y,-hh/2)2+(z,-zJ)2]3’2’
the above with respect
fy(dJJ> a /::z‘lxl
Jr::,(
to zj while considering
Y, - i)
can
(11) the problem
symmetry,
we get
dY; J_fL;zdr,
X ~~~~2{(‘~-‘/‘)dr,)([(x,-Xj+d,J+t)2+(Y~-h/2)2]
x [(x,
Integrating
- xi +
the above integral
d,, + t)2 + ( y, - h,‘2)2 + (z, - L,2)2] with respect
to z, and y, yields
“‘) -I.
(12)
252
Guo Mum,
X
t’2
/ -t/2
+
il’
In
(x,
-x,
(x,
-
R.S. Indeck
/ Interaction
matrix
- L)iii’(x,
-x,
+
d,, +
q2+ L’
+
d,,
t)2
(d
+
+
_ x, + d,, + $
I
L, h > t. The second
x
i’
\lL’+h’+(d
+ L
(x,-.u,+d,,+r)?+h’+L.L
+ hZ] I”’ -((x,-x,+d,,+t)
:
dx,.
can be solved using a series expansions
(13) by considering
[S’+(d,,+r)‘]
term in eq. (13) may be approximated
,,+I)’
-Lj
(4,-t+
x,)
[S’+(d,,+t)‘]
when
L’
j:;;;x,j:;;*{[ (XI
i, + 4x,-
+ h? + L? + L)
d-y-, x,
The first and third terms in the above integrations that the term
R=
+ d,, + ty
by
+Lj/j\jL’+h’+(d,,+t)‘-L&/L’+(d,,+t)’+L~]
because
A maximum error due to the above approximation takes place at d,, = 0. The error of this approximation has been evaluated [see appendix] to be about one percent for L, h z+ t. This error may also be reduced by normalizing the interaction factors. By neglecting higher order terms of the series expansion, integrating and normalizing, we get iL’+h’+(d,,+$
f,.(d,,)
=
L In
c
[’
+L
I[’
\l’L’+(d,,+~)‘-L
I
[/L’+h’+(d,,+t)‘-L][/L’+(d,,+r)‘+L]
+2[/~+/h2+(d,,+t)2-/L’+h”+(d,,+r)L
-(d,,+f)] i
(14)
Guo Mian, R.S. Indeck / Interaction matrix
253
0
923
1846
2769
E2310
lb)
(a) view of the interaction factor &(t. d) for the interacting contour map of r,(d) versus t and d.
Fig. 2. (a) Three-dimensional
2310
The normalized interaction reflecting L to h:
factor
related
to the magnetization
films (h = 20 pm,
in the i direction
L = 600 pm). (b) The
can be obtained
by
1
[JL'+h2+(d,,+f)2 +h][{W-h]
fi(d;,)= h In
[JL'+h2+(d,i+1)2-h][~~+h]
+2[/iq77+JL2+(d,j+-/L2+h2+(d;,+t)2
-p,,+q] 1
The above interaction factors satisfy eq. (5). A three-dimensional view and contour map of f,(d,j) versus film thickness and separation are shown in fig. 2. From the figure we see that the interaction factor decreases rapidly as the distance is increased and that the rate of decrease is faster for thinner films.
3.A matrix representation We have obtained mathematical expressions of the interaction factors used for analyzing the magnetointeraction between two thin films in a three-dimensional n-layered thin film system. The total interaction energy of the n-layered magnetic thin film system can be generalized by static
k
i#j
[ Myify(dij)NyjMy,
+
MziL(dij)N,jMzj].
(16)
Guo Mian, R.S. Indeck
254
/ InteractIon matrm
This equation may be expressed by a matrix representation. For simplicity, we consider a two-layer problem first. The interaction energy of a coupled thin film system given by eq. (6) can be expressed by a matrix representation as follows:
E
where
is the y component
of the magnetization
of the system in this matrix
is the z component
of the magnetization
of the system in the matrix
is a demagnetization
matrix
of the system with respect
representation,
and
representation.
to the j direction,
and
is the demagnetization matrix of the system with respect to the z^ direction. Similarly, matrices of the system with respect to the magnetization in the j and z^directions are
the interaction
(22) and 0
LP,,)
fZ=i h(d,,) The above results
0
(23)
’
can be generalized
to an n-layered
thin film system.
Let
1M,l
M!Jz My
=
My3
(24)
Guo Mian, R.S. Indeck / Interaction matrix
255
and
M, =
be y and z components of magnetization of the system in the matrix representation, the interaction and demagnetization matrices with respect to the j direction in this matrix representation can be expressed by n X n matrices 0
fy&)
fy(41) fy =
0
f,@31)
f&43)
...
f,(d,J
fyP23)
. . .
fyMn)
* **
fy(d3n)
fJd32)
0
o
...
. . .
0 ...
. . . f,MJ
f-(4*)
fy(43)
(26)
... 0
and
NY=
/ NY, 0
NY2 0
NY3 0
NY, ...
0
0
0 0
..... .
...
(27)
... ,$I
. . . Ym
0
...
Nyn/
The interaction and demagnetization matrices with respect to the 2 direction in this matrix representation are similar expressions except reflecting y to z. that is
f&4,)
...
fztd,,)
stead 0
f, =
@3A
f-2643)
f,(d32)
o
o
... 0
. . .
.. .
fz@n,) fiiid
:(d,,)
fX4J f*(d*n)
...
...
fzt;(d,J
(28)
... 0
and
N, =
N,,
0
0
.+.
0
0
Nz2
0
...
0
0
0 ...
Nr3
... N,,
0
0
... 0
...
0
... Nrm
. . .
%I
(29)
256
Guo MKUI, R.S. Indeck
Since all N,k(i =y, z; k = 1. 2,. therefore can be expressed by N,=N,I.
(i=,~,
/ Interaction
. , n) are the same for the symmetric
system,
the demagnetization
i),
E lnler = : [ &+f,NyM
matrix
(30)
where I is an n-dimensional n X n unity matrix. The interaction system expressed by eq. (16) now can be expressed by
which has exactly
matrix
+ M:f,N,M]
energy of the n-layered
magnetic
3
thin film
(31)
the same form as eq. (17) in this matrix
representation.
4. Results and discussions From eq. (4) we know that for a k-dimensional magnetization system (k I 3) there are k interaction matrices for describing the interaction of the system. We have assumed that the magnetization is in-plane for each of the n-layered thin films, so there are only two interaction matrices with respect to magnetizations in j and i directions. From eqs. (26) and (28) we see that the off-diagonal elements in the interaction matrices are nonzero. The physical meaning of this is that these off-diagonal elements represent the interaction of magnetizations of the thin film system. In a system which includes its interaction, the off-diagonal elements of the matrices have to be nonzero. Since f,(d,,) equals f,,(d,,) (a =y, z). the interaction matrices are symmetric matrices in this matrix representation. The diagonal elements of the matrix represent the self-demagnetization energy of each identical film and have to be zero in the interaction matrix because the self-demagnetization energy of each film does not contribute to the interaction energy of the system. The self-demagnetization energy of the system can be written by: Edemag= : [ A$!lN~,M,. + M,+INIMZ], where I is the n x n unity matrix. be given by [16]
(32)
Magnetostatic
energy of the multi-layered
magnetic
thin-film
system can
‘total = Ek + Ebiar + E,,i,, + ‘demag + Eappl. The first term, E,, which is the summation of anisotropy energy of each film, is the total anisotropy energy of the system; the second term, Ebias, which is the summation of the energy due to bias field in each film, is the total bias energy of the system; the last term is the energy due to the applied field. This magnetostatic energy may be written by a matrix representation: Et,,,, = -M,’
( Hk_k+ Hbiaa 1’+ Harp1 I’> - M=+ ( Hk- + Hhia\ : + Happi 2 >
+ : [M>: (fJ + I)N.,fl,
+ M.‘(f,
+ I)N,M;]
,
(33)
where
I+,.=
Hs,,
,
S
=
K, bias, appl,
(34)
Guo Man, R.S. Indeck / Interaction matrix
257
a
b
k -d=40
0
5
10
15
APPLIED Fig. 3. (a) Angle between
FIELD
20
25
30
0
10
APPLIED
Hy( Oe)
1
H
d=350
20
FIELD
30
;i
40
Hy( Oe)
thin films versus a perpendicular applied field (Hbias = 0, H, = 2.5 in the coupled films versus a perpendicular field. The bias fields in two film are anti-parallel along y axis ( Hbias = 8 Oe).
magnetizations
in two coupled
magnetic
Oe, MS = 8000 G, t = 450 A, h = 20 pm, L = 450 pm). (b) Angle between magnetization applied
is the y component of the fields (S = anisotropy, bias, applied fields) of the system, and ‘H
Sri
’
H Sr2 H,,=
f&z3
,
S
=
K, bias, appl,
(35)
is the z component of the fields of the system in the matrix representation. In the above equations, Hsy, and Hszn are y and z components of the S field (K, bias or applied) in the n th film, respectively. By minimizing the magnetostatic energy of the system, one can determine an equilibrium state of the system. Fig. 3a shows the numerical solutions of the angle between magnetizations in two films versus applied field in the 9 direction using the interaction factors. From the figure we see that at the initial point H = 0, the magnetizations in the i th and the j th = (i f 1)th films are anti-parallel. However, as the applied field increases, the magnetization vector in each film is rotated toward each other to reduce the energy due to the applied field. From the figure we also see that the closer the two films become, the strength of the field necessary to rotate two magnetization vectors parallel with each other increases accordingly. Here we can see a significant difference of magnetization of the system due to different separations of the films. A strong interaction between two films brings about a strong anti-parallel effect of the system as expected. Similar results shown in fig. 3b are found when there are perpendicular anti-parallel bias and anisotropy fields in the system. The numerical solution of the angle between the magnetization of the outer films, films 1 and 3 for a three-layered magnetic thin-film system, is shown fig. 4. For d 12= d,,, the magnetization in films 1 and 3 rotates synchronously. From the figure we see a correspondence between larger film separations and synchronous rotations of the magnetizations in films 1 and 3. Since d,, is smaller than d,, for the asymmetric case, the magnetization in film 3 will rotate toward to its final angle (270 o ) faster than that in film 1 because the interaction between films 1 and 2 is stronger than that of films 2 and 3. The peaks in the
258
Guo Mian, R.S. Indeck
0 Fig. 4. Numerical magnetic
/ Interaction
matrix
APPl%D FIEl,~” Hy( 0e)
60
solution of the angle between the magnetization of films 1 and 3 versus a perpendicular applied field of a 3-layered thin-film system (MS = 15000 G, L = 600 pm, h = 20 pm, t = 450 A, H, = 2.5 Oe. Hh,_ = 0, d12 = 40 A).
figure correspond to the situation where the magnetization in film 3 attains its final angle (270 o ) parallel to the applied field. At zero applied field, the magnetization in the film 2 is antiparallel to both the magnetization in films 1 and 3 and all of the magnetizations rotate more toward the direction of the applied field as the field grows larger. The smaller d,, is, the harder it is for the magnetization in film 3 to rotate parallel to the applied field. Fig. 5 shows the numerical results of the normalized resistance of a coupled Permalloy film device verses a uniform perpendicular applied field. The normalized resistance of the system is simulated to be equal to (cos28, + cos2B2)/2 [5,12], where 8, and 8, are the angles between easy axes and magnetizations of films 1 and 2, respectively. Similar to the experimental work of Van Ooyen et al. [12], the anisotropy field is uniaxial and in the z^direction. Since the external field is uniformly applied and the thickness of the films is much smaller than the width and the length of the films the edge effect is ignored. From the figure we see that there is a parabolic character of the magnetoresistance for films with a large separation. Since the normalized resistance versus the perpendicular applied field for a single film can be characterized by a parabola [5,12] the above results indicate that large separation of the films brings about a single-film character because of the weak coupling effect. These numerical results agree with the experimental findings of Van Ooyen et al. The general expression of the magnetostatic energy of an n-layered magnetic thin-film system has been represented using a matrix notation. Since there are many similarities between ferromagnetic and ferroelectric materials, we expect that the results may also be applicable to an n-layered ferroelectric thin-film system.
5. Conclusions A matrix representation has been proposed for the energy analysis of an n-layered magnetic thin-film system. The interaction energy of the n-layered magnetic thin-film system can be described by the proposed interaction matrices. For a k-dimensional magnetization, there are k (k I 3) symmetric interaction matrices. The off-diagonal elements of the interaction matrix are nonzero for an interacting system. The interaction matrices are symmetric n X n matrices for the n-layered system. A general
Guo Mian, R.S. Indeck
/ Interaction
259
matrix
0.60 w
a
z 2 zw 0.40 p: Es N
Li 2
ik, :f :: j: : :: :Li
o.20
5 z
0.00
0.00
-40
-20
APPLIED
F!lELD i&k)
I,,,:,,,,,,,,,,,,,,,,,,/,,,,,,,,,,,, \,,, -40
40
-20
APPLIED
FOIELD If&e)
40
g 0.40 N 2 2 p:
d .=4000 0.20
g 0.00 -30 Fig. 5. Numerical solution of the normalized resistance A uniaxial anisotropy field is along the 2 direction (t
-10
APPLIED
F’IELD1’Hy( Oe)
versus a perperdicular
applied
30
field for a coupled
Permalloy
thin-film
system.
= 450 A, h = 20 pm, L = 600 pm, Hbias = 8 Oe, If, = 2.5 Oe, MS = 8000 G).
expression of magnetostatic energy of the n-layered thin-film system in the proposed matrix representation has been found. Numerical results of uniformly magnetized, twolayered and a three-layered Permalloy magnetic thin-film system using the interaction matrix are presented. A strong interaction between two magnetic thin films bring about a strong anti-parallel effect of the magnetization in the films; larger separations between the films results in a single film magnetoresistive behavior because of a weak coupling effect. These results agree with experimental findings.
Acknowledgement We would like to thank Professor M.W. Muller for his valuable discussions.
Guo Mtan, R.S. Indeck / Interaction
260
matrtx
Appendix At d,, = 0, since L, h z+ t, the second term integration of a constant except for the term
The above integration
is equal to - 8.21 when
above L, h and t into the approximation tion is about - 8.1.
(L/2)
of the integral
L = 600 pm, ln(dm
in eq. (13) can be considered
h = 20 pm and - L)t’
we obtain
to be an
t = 450 A. Substituting the approximate
References [l] [2] [3] [4] [5] [6] [7] [E] [9] [lo] [ll] 1121 [13] [14] [15] 1161
R.S. Indeck, J.H. Judy and S. Iwasaki, IEEE Trans. Magn. MAG-24 (1989) 2617. H.S. Gill, V.W. Hesterman, G.J. Tarnopolsky, L.T. Tran, P.D. Frank and H. Hamilton, J. Appl. Phys. 65 (1989) 402 C. Denis Mee and Eric. D. Daniel, Magnetic Recording Handbook (McGraw-Hill, New York, 1989). C. Vittoria, J. Appl. Phys. 57 (1985) 3712. D.A. Thompson, L.T. Romankiw and A.F. Mayadas, IEEE Trans. Magn. MAG-11 (1975) 1039. D.B. Dove, Bell Sys. Tech. J. (Sept. 1967 1527). P. Rhodes and G. Rowlands, Proc. Leeds Phil. Sot. 6 (1954) 191. N. Smith, IEEE Trans. Magn. MAG-23 (1987) 259. A.V. Pohm and C.S. Comstock, IEEE Trans. Magn. MAG-20 (1984) 863. J.C. Slonczewski, B. Petek and B.E. Argyle, IEEE Trans. Magn. MAG-24 (1988) 2045. Guo Mian and R.S. Indeck, IEEE Trans. Magn. MAG-26 (1990) 2415. J.A.C. Van Ooyen, W.F. Druyvesteyn and L. Postma, J. Appl. Phys. 53 (1982) 2596. F.J. Jeffers, IEEE Trans. Magn. MAG-15 (1979) 1628. H. Chang, IBM J. (October 1962) 419. Guo Mian and T. Yamaguchi, J. Magn. Magn. Mat. 62 (1986) 325. S. Chikazumi, Physics of Magnetism (Krieger, Malabar, FL. 1964).
the
integra-