Solid State Communications,Vol. 106. No. 2, pp. 107t IO, 1998 0 1998 Elsevin Science Ltd Printed in Gut Britain. All ri$tts resewed 0038lO9&!98 f19.Olb.00
Pergamon
PII: soo381098(97)102~5
GIANT MAGNETORESISTANCE IN MAGNETIC GRANULAR SOLIDS Rong Yang,” Wei Zhangb and W.J. Song” “Department of Physics, Suzhou, University, Suzhou, 215006, China qnstitute of Theoretical Physics, Academia Sinica, Beijing, 100080, China (Received 23 September 1997; accepted 24 November 1997 by R.C. Dynes)
Based on a simple model of concentrically coated magnetic granules embedded in a nonmagnetic metallic matrix, the transport properties in systems of cylindrical or spherical magnetic granules are investigated analytically. It is shown that GMR as well as the resistivities of these systems depend strongly on the size of the granules. 0 1998 Elsevier Science Ltd
1. INTRODUCTION Giant magnetoresistance (GMR) has been observed in magnetic multilayers [I] and then, recently, it has been found that giant magnetoresistance can also be achieved in magnetic granular films, such as CoCu, CoAg, FeAg, NiFeAg, FeAu, NiAg [281. Granular films thus become another candidate for magnetic sensor or memory cells technology. The physical interpretation of the GMR effect is usually based on the spindependent interface scattering as well as on the spindependent bulk scattering. For granular films, as the spatial distributions of the field and the currents are more complicated, the theories of the MR in granular systems have not been quite as well developed. Recently, Rubinstein [9] made an attempt to extend the theory of Valet and Fert [lo] which deals with the CPPMR of multilayers to the granular systems, but had great difficulty in accounting for experimental results of GMR. Gu et al. [l l] developed a macroscopic theory of GMR in granular magnetic materials improving in that of Rubinstein. Now, we present a simple model of concentrically coated granules embedded in a nonmagnetic metallic matrix to investigate the transport properties in inhomogeneous ferromagnetic granular films based on the MaxwellGarnett approximation, Many authors show their experiments with spherical granule samples. And very recently, Teixeira et al. [12] also found the GMR effect in systems of (CoFe),AgI_, granular alloys with columnar structure. This paper we focus our attention on
systems of cylindrical magnetic granules. We find that both the GMR and the resistivities depend strongly on the grain sizes and that there is always an optimum grain size for the GMR. 2. FORMALISM 2.1. The general expression for conductivity in the MaxwellGamett approximation We first consider a simple model of a heterogeneous medium. Let the medium consists of cylindrical granules with conductivity d in a matrix with conductivity a”. The average field and current density of this system are
69 =_E + (1 _f)Eo =f
to=fdE, =fd
gy0 +(1 f)Eo,
(1)
+ (1 f)$Eo
2un + (1 fWEo, Eo a”+$
then we get the expression for the effective conductivity in the MaxwellGarnett approximation (MGA) of a simple cylindrical granular system uefl
_

o”
(d + 6) +f(d  6) = fdQ + a”( 1 f) (J+oYf(Ao”)
fQ+
’ (3)
with Q = 2646
107
+ o”).
108
GIANT MAGNETORESISTANCE IN MAGNETIC GRANULAR SOLIDS
2.2. The MaxwellGamett expression for the effective
with
conductivity of the coated granular system
Let us see the system of coated cylindrical granules (the core’s conductivity &and the shell’s conductivity u”) in a matrix with conductivity a”. According to Xinyu Liu and Zhenya Li [13], they consider that the coated particles in the whole system can be regarded as equivalent solid particles. Using their method, we can get that the expression for a concentrically coated cylindrical particle is u’ = J. Here
cc
Y(l +d+w
=
’
with y = S/d, A = (a/a + t)2 and a and t are the radius of the core and the thickness of the shell, respectively. Also using the method of Xinyu Liu and Zhenya Li, we get the effective conductivity of the MaxwellGamett approximation of a coated cylindrical granular system uefl
=
(u’ + 0”) + F(o’  6) =
un
(a’ + a”)  F(u’  8)
Fu'Q
+ (1  F)u”
FQ+(lF)
with F = f/X, Q = 2ti”lu’ + 6.
clef/@,$) = sin(e)
coated granular system
B,r + a2C,lr
(a+t>r>a)
i &r + &(a + t)2/r
(r > a + t).
Here A,, B,, C,, D, are constants to be determined from the continuity of the scalar potential and the current at r=a and r=a+t. Here a=+() for majority (minority) spin direction. The conductivities of the core, shell and matrix are u,f, 4 and a”. So the effective conductivity in the MaxwellGamett approximation of this system (CT; + CT”)+ F(ub,  a”) (ub,+o”)+(&a”)=
PL,=
Ilad,
(10)
Ydl + Ya)+ hy&  rol) (l+Ycr)X(lyu)
(11)
’
dt
(12)
7.
MR is defined as the difference of resistivity between the completely magnetized state and the completely demagnetized state. For the magnetized state, ail the granules have the same magnetization direction (pointed up), so the effective conductivity FQp;
$n
+ (1  F)u”
FQ++(lF)
FQ_u’_
+
+(l
 F)cJ“
FQ_ +(1F)
.
(13)
For the demagnetized state, there are equal numbers of granules with magnetization pointed up and down. So the effective conductivity of the demagnetized state is 2
O.WQ+u;
Q
+

u’_ ) +
(1  F)#
’
(14)
and then the MR of the system is obtained by MR = (8
 8)luD.
When t  0, tlg = r,,equations(lO)and(ll)change into ab, = cL&J,f= a ;!u!. a
(15)
a I& =
(16)
a + rauJ
and putting equation (15) into equation (8), Qa =
(6)
ueff= a=
’ = UC2
0.5F(Q+ + Q  ) + (1  F)
Now we consider a cylindrical system composed of parallel ferromagnetic coated granules in a nonmagnetic matrix. All the ferromagnetic granules are assumed to have only two magnetization directions: up (1) and down (I), with the quantization axis along the direction of the applied magnetic field. The electric field is assumed to be perpendicular to the cylinders. The equation of continuity for the current becomes Poisson’s equation in all regions. In this case, the effective potential in the xy plane has the form (a > r)
(8)
F = ;,
0~ =
2.3. The MaxwellGamett expression for the effective
&
a:+&
=
(5)
conductivity of theferromagnetic
26
Q,=
Ya =
7)
(1 +r>X(1y)
Vol. 106, No. 2
2u” =
pdJ(t+ a”
2uTa +
ad + aa”
r&J + r,a,fa”’
(17)
and putting equations (15) and (17) into equations (13) and (14), we can get the same values of g”, gD in [I 11. The above calculations can be extended straightforwardly to the case of spherical granules. We can find that equations (13) and (14) are still valid if the parameters are represented as Qa =
3u”
ub, + 2ti’ 3
(18) (19)
Fa’,Q, + (1  F)u”
FQ,+(lF)
’ (7)
l&Y=
Ya(l + 2y,) + 2X7,( 1  ya) (1+2%)X(1 vu) *
(20)
Vol. 106, No. 2
GIANT MAGNETORESISTANCE IN MAGNETIC GRANULAR SOLIDS
_._ 0.50 0.5

a: N30 b: NdO c: N10
c
0.1 0.3 ’
0
1
0.5
I
I
1
1
1.5
2
I
25
9
3.5
(v_&a
Fig. 1. The relative resistivity pDlpf as a function of (r_ of/a) for several spinasymmetric factors N. (N = o$at = 6,/u’!!, f = 0.3). Here r_ = t/d”, a and t are the radius of the core and the thickness of the shell; u!., o’l are the core’s conductivity and the shell’s conductivity. 3. DISCUSSION In Fig. 1, we plot the resistivity as a function of the inverse of radius, which approximately in linear proportion. It is shown that the resistivity is inversely proportional to the particle size, which is in good agreement with the experimental observations in [6]. Figure 2 shows the MR as a function of radius in systems of granules. There is always an optimum radius for the GMR. As seen from the experimental observations [6], there is always an optimum annealing temperature in preparing the magnetic granular sample for MR. In view of that the sizes of the particles are always increased with increasing the annealing
I
a:N=30
10
b:N30
I
109
temperature, our results are in good agreement with these observations. To understand this feature, we notice that there are two factors that determine the GMR. With decreasing size of the granules, the ratio of interface area to the granule volume always increases; On the other hand, the decrease in size can enhance the effect that the currents bypass the granules, leading to the decrease in the spindependent scattering of the electrons. The competition between these two factors leads to a maximum of the GMR. The spindependent scattering in the granules and in the mixing films is considered to come from the same mechanism [ 141,thus, for simplicity we here assume the spinasymmetric factors in the granules and in the mixing films are of the same value N (N = u$uf = a”,/$!). According to [15], NNico> 13, NcOre> 12, etc. In our model we select N = 10, 20, 30 to see the influence of the spinasymmetric factor on GMP. It is clear from Fig. 2 that the heterogeneity among the spinup and spindown electrons also give rise to GMR [16]. In summary, with a simple model of concentrically coated granules embedded in a metallic matrix, we investigated the transport properties in inhomogeneous ferromagnetic granular films based on the MaxwellGamett approximation. With a meanfield framework, we calculate the conductivity and the GMR in systems of cylindrical and spherical magnetic granules. Our results are in good agreement with the experimental observations. REFERENCES Baibich, M.N., Broto, J.M., Fert, A., Nguyen Van Dau, F., Petroff, F., Etienne, P., Creuzet, G., Friederich, A. and Chazelas, J., Phys. Rev. L&t., 61, 1988, 2472. 2. Berkowitz, A.E., Mitchell, J.R., Carey, M.J. and Young, A.P., Phys. Rev. Lett., 68, 1992, 3745. 3. Xiao, J.Q., Jiang, J.S. and Chien, C.L., Phys. Rev. 1.
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4. Xiao, J.Q., Jiang, J.S. and Chien, C.L., Phys. Rev., B46, 1992,9266. 5. Jiang, J.S., Xiao, J.Q. and Chien, C.L., A&. Phys. Lett., 61, 1992,2362.
6. Wang, J.Q. and Xiao, G., Phys. Rev., B49, 1994, 3982.
0
0.6
1
1.6
2
2s
3
dfdJ
Fig. 2. Percent of GMR (Au/aD) as a function of (a/r _ uf ). (N = u(luf = S,lu’!!, f = 0.3). Here r _ = t/u”_. a and t are the radius of the core and the thickness of the shell; u’!! are the core conductivity and the shell conductivity.
uf,
7. Parent, F., Tuaillon, J., Stem, L.B., Dupuis, V., Prevel, B., Perez, A., Melinon, P., Guiraud, G., Morel, R., Barthelemy, A. and Fert, A., Phys. Rev., B55, 1997, 3683. 8. Gerber, A., Milner, A., Groisman, B., Karpovsky, M., Gladkikh, A. and Sulpice, A., Phys. Rev., B55, 1997, 6446. 9. Rubinstein, M., Phys. Rev., B50, 1994, 3830. 10. Valet, T. and Fert, A., Phys. Rev., B48,1993,7099. 11. Gu, R.Y., Sheng, L., Xing, D.Y., Wang, Z.D. and Dong, J.M., Phys. Rev., B53, 1996, 11685.
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GIANT MAGNETORESISTANCE IN MAGNETIC GRANULAR SOLIDS
12. Teixeira, S.R., Dieny, B., Chaxnbemd, A., Cowache, C.. Auffret, S., Auric, P., Rouviere, J.L., Redon, 0. and Pierre, J., J. Phys., C6, 1994, 5545. 13. Xinyu Liu and Zhenya Li, Phys. L&t., A223, 1996,475. 14. Levy, P.M., Zhang, S. and Fen, A., Phys. Rev. L&t., 65,1990,
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15. Campbell, LA. and Fert, A., Transpon Properties of Ferromagnets, in Ferromagnetic Materials, Vol. 3 (Edited by E.P. Wohlfarth), p. 747. North Holland, 1982. 16. Chang, C.R. andLo, K.H., J. Appl. Phys., 80,1996, 6888.