PHYSICAL REVIEW B 71, 094434 共2005兲
Three-dimensional distribution of spin-polarized current inside „Cu/ Co… pillar structures
J. Hamrle,1,2 T. Kimura,1,2 T. Yang,1,2 and Y. Otani1,2,3
1FRS,
The Institute of Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2CREST, Japan Science & Technology Corporation, Japan
3ISSP, University of Tokyo, Kashiwa-shi, Chiba 277-8581, Japan
共Received 10 September 2004; revised manuscript received 12 January 2005; published 31 March 2005兲
Within diffusive transport, we calculate three-dimensional current distribution inside a 共Cu/ Co兲 pillar structure, where the pillar is terminated either by an infinitely large Cu layer, or by a Cu wire with a cross-sectional
area identical to that of the pillar. We study how pillar terminations 共infinitely large or finite兲 influence the
magnitudes and inhomogeneities of the spin-polarized current and electrochemical potential. We found that
infinitely large Cu layers work as strong spin scatterers, increasing the magnitude of spin-polarized current
inside the pillar twice and reducing spin accumulation to nearly zero. The inhomogeneities of the electrochemical potential are found to be much smaller than those of the spin-polarized current.
DOI: 10.1103/PhysRevB.71.094434
PACS number共s兲: 75.75.⫹a, 85.70.Kh, 85.70.Ay
I. INTRODUCTION
Spin injection, transport, and detection are key factors in
the field of magnetoelectronics. In particular, magnetization
reversal using spin-polarized current is of great interest1–7
due to its potential technological applications such as magnetic random access memory 共MRAM兲,8 spin transistors,9 or
spin batteries.10
To understand and optimize spin-transport behavior in
such devices, it is important to know the current distribution
in them. Particularly for MRAM applications, we have to
know the spin-current magnitude and distribution to optimize
the current density necessary for spin-injection-induced magnetization reversal. A variety of formalisms calculating magnetoelectronic transport in one dimension 共1D兲 even for noncollinear magnetization has been proposed.11–16
However, up to now, the spatial 共3D兲 calculation of the
spin-polarized current has been missing. To obtain spatial
distribution of the spin-polarized current 共and spin accumulation兲 in a given structure, we have proposed a formalism
expressing such a structure as a 3D circuit of spin-dependent
resistor elements 共SDREs兲, wherein the propagation is regarded as a 1D problem.17
In this article, we study how termination 共infinitely large
or finite兲 of 共Cu/ Co兲2 and 共Cu/ Co兲3 pillar structures influences spin-polarized jsp = j↑ − j↓, charge jch = j↑ + j↓ current
density, and electrochemical potential ␮↑/↓. In the literature,
pillars terminated with infinitely large continuous layers are
commonly used.2–4,18,19
This article is organized as follows: Section II introduces
studied 共Cu/ Co兲 pillar structures. Sections III and IV discuss
jsp and jch 共␮↑/↓兲, respectively, within various pillar terminations, their inhomogeneities, and agreement with 1D calculations. Section V discusses the influence of pillar terminations on magnetoresistivity.
II. DESCRIPTION OF „Cu/ Co… PILLAR STRUCTURE
We study structures consisting of two and three Co layers,
called 共Cu/ Co兲2, and 共Cu/ Co兲3, respectively, with dimensions in nanometers of Cu1 / Co1共40兲 / Cu2共6兲 / Co2共2兲 / Cu3
and Cu1 / Co1共40兲 / Cu2共6兲 / Co2共2兲 / Cu3共6兲 / Co3共20兲 / Cu4,
respectively. The square-shaped pillar 100 nm in size begins
with the Co1 / Cu2 interface. The considered structure types
are defined in Fig. 1 as 共a兲 cross-sectional area S-constant,
共b兲 column, 共c兲 constriction, and 共d兲 constriction with an infinite Co1 layer. In the following discussion, “constriction”
corresponds to case 共c兲. The “infinite” homogeneous layers
were approximated as a square pillar of 800 nm in size. The
magnetization of the Co1 layer is always fixed as “up” 共↑兲,
whereas the magnetic orientations of the other Co layers are
varied. Note that in our diffusive transport calculations, the
magnetization orientation with respect to the structure 共e.g.,
in-plane or out-of-plane兲 does not play any role, but only
mutual magnetization orientation 共parallel or antiparallel兲
does play an important role. The charge current passing
through the structure is assumed to be Jch = 1 mA, equivalent
FIG. 1. Sketches of studied structure types. 共a兲 S-constant 共infinitely long nanowire with constant cross-sectional area S兲, 共b兲 column
共infinitely long pillar deposited on infinitely large Cu1 layers兲, 共c兲 constriction 共both Cu3 cover and Cu1 buffer layers are infinitely large兲,
and 共d兲 constriction with Co1 infinite layer.
1098-0121/2005/71共9兲/094434共7兲/$23.00
094434-1
©2005 The American Physical Society
PHYSICAL REVIEW B 71, 094434 共2005兲
J. HAMRLE, T. KIMURA, T. YANG, AND Y. OTANI
FIG. 2. jsp along the center axis of the 共a兲 共Co/ Cu兲2 ↑ ↑, ↑↓; 共b兲 共Co/ Cu兲3 ↑ ↑ ↑, ↑ ↓ ↑; and 共c兲 共Co/ Cu兲3 ↑ ↑ ↓, ↑ ↓ ↓ for S-constant, column,
and constriction structure types 共Fig. 1兲.
to the average charge current density in the pillar jch = 100
⫻ 109 A / m2. The electrical properties of materials are of
room temperature:16,20,21 electric conductivity ␴Cu = 48.1
⫻ 106 ⍀−1 m−1, ␴Co = 4.2⫻ 106 ⍀−1 m−1, spin-flip lengths
␭Cu = 350 nm, ␭Co = 60 nm, and Co spin bulk asymmetry ␥
= 0.35. We assume no interface resistance, and no interface
or surface scattering. The largest SDRE grid size is 10 nm.
III. CURRENT DENSITY IN THE PILLAR
STRUCTURE
Figure 2 shows the profile of jsp along the center axis of
the structures: 共a兲 共Cu/ Co兲2, 共b兲 共Cu/ Co兲3 with parallel Co1
and Co3 layers, and 共c兲 共Cu/ Co兲3 with antiparallel Co1 and
Co3 layers for S-constant, column, and constriction type
structures. In all cases, jsp for parallel Co1 and Co2 layers is
larger than for the antiparallel configuration. Furthermore, jsp
is enhanced at the position of the free Co2 layer for the
column and constriction types compared to the S-constant
type structure. For example, in the case of the constriction
type structure with 共Cu/ Co兲2 ↑ ↑ configuration, jsp is en-
hanced by a factor of 1.75, and in the case of 共Cu/ Co兲3 ↑ ↑ ↑
by 1.5. Figure 2共c兲 shows that for 共Cu/ Co兲3 ↑ ↑ ↓ and ↑ ↓ ↓,
jsp is significantly reduced for all types of structures.
The origin of the jsp enhancement is as follows: the pillar
is attached to an infinitely large Cu layer, which acts as a
reservoir with ⌬␮ = ␮↑ − ␮↓ = 0. In other words, the infinitely
large Cu layer provides a large volume for the spin current to
be scattered, thus acting as a strong spin-flip scatterer. Hence,
the infinitely large Cu layer works as a small shortcutting
resistance between up and down channels. Consequently,
shortcutting of up and down channels leads to an increase in
jsp. The increase of jsp is related to an increase of spinpolarization efficiency p = jsp共Jch / Spillar兲, as the charge current
flowing through the pillar 共Jch兲 is fixed in all our calculations.
Consequently, an increase of p may lead to a decrease of the
critical switching current Js,ch, which is necessary to reverse
the magnetization direction of the free layer.
A similar effect can be realized by inserting a layer with
small characteristic spin-flip resistance AR␭ = ␭ / ␴, such as
Pt, Ag, Au, or Ru, above the last Co layer or below the first
Co layer. Such cover layers have been used22,23 since the first
TABLE I. Values of jsp, ⌬␮ = ␮↑ − ␮↓ at the position of the free Co2 layer for 共Cu/ Co兲2 and 共Cu/ Co兲3
structures. MR is determined between the first and last Co/ Cu interfaces. All values are determined as
averages over the whole pillar cross-sectional area. In square brackets, we present values calculated from the
1D model. Symbols in parentheses denote structure notation in Fig. 10.
j sp 关109A / m2兴
共Cu/ Co兲 ↑ ↑
共Cu/ Co兲2 ↑ ↑
共Cu/ Co兲2 ↑ ↑
共Cu/ Co兲2 ↑ ↑
S-constant 共+兲
column 共䊐兲
constriction 共〫兲
constr., Co1 infinite
⌬␮ 关meV兴
MR 关%兴
关14.93兴
关19.8兴
关33.7兴
关23.3兴
−0.223 关−0.220兴
−0.267 关−0.276兴
−0.052 关−0.009兴
−0.050 关0.001兴
0.483
0.63
1.01
1.43
共Cu/ Co兲3 ↑ ↑ ↑ S-constant 共⫻兲
共Cu/ Co兲3 ↑ ↑ ↑ column 共䉭兲
共Cu/ Co兲3 ↑ ↑ ↑ constriction 共䉮兲
19.51 关19.67兴
23.6 关24.5兴
31.0 关33.9兴
−0.078 关−0.077兴
−0.171 关−0.184兴
−0.019 关−0.005兴
0.444 关0.448兴
0.55 关0.57兴
0.74 关0.81兴
共Cu/ Co兲3 ↑ ↑ ↓ S-constant 共*兲
共Cu/ Co兲3 ↑ ↑ ↓ column 共䉯兲
共Cu/ Co兲3 ↑ ↑ ↓ constriction 共䉰兲
5.94 关5.93兴
8.08 关8.49兴
9.7 关10.5兴
−0.442 关−0.438兴
−0.490 关−0.494兴
−0.458 关−0.456兴
0.116 关0.115兴
0.162 关0.171兴
0.192 关0.210兴
2
14.85
19.0
29.4
28.7
094434-2
关0.485兴
关0.66兴
关1.17兴
关2.48兴
PHYSICAL REVIEW B 71, 094434 共2005兲
THREE-DIMENSIONAL DISTRIBUTION OF SPIN-…
FIG. 3. jch along the center axis of the 共Co/ Cu兲2 structure for
S-constant, column, and constriction structure types 共Fig. 1兲.
pioneering experiment,3 but their contributions to the jsp enhancement
have
only
recently
been
observed
experimentally23–25 and predicted theoretically.26
Figure 3 shows profiles of charge current jch = j↑ + j↓ along
the center axis of the 共Cu/ Co兲3 structure for S-constant, column, and constriction type structures. Obviously, for the
S-constant structure, the jch is constant. In the case of infinite
Cu termination, jsp decreases approximately exponentially
over the characteristic length of 50 nm. The same decay of
jsp is presented in Fig. 2 for infinitely large Cu layers.
Table I summarizes average values of jsp in the position of
the Co2 layer in all the types of studied structures. These jsp
values may differ from those presented in Fig. 2 due to lateral inhomogeneity of jsp inside the pillar, discussed in the
next section. The largest average jsp are obtained for the
共Cu/ Co兲3 ↑ ↑ ↑ constriction structure 共31.0⫻ 109 A / m2兲 and
the 共Cu/ Co兲2 ↑ ↑ constriction structure 共29.4⫻ 109 A / m2兲,
providing jsp enhancement by a factor of 2 with respect to
the 共Co/ Cu兲2 S-constant structure 共14.85⫻ 109 A / m2兲. This
tendency has already been explained in the above paragraph.
A. 3D versus 1D model
The values in square brackets in Table I show values calculated by 1D formalism, where an infinitely large Cu termination is simulated by 100 nm of Cu 共with the same crosssectional area as the pillar兲, followed by the Cu reservoir,
where ⌬␮ = 0. We can see a very good agreement between
the 1D approach and averaged values provided by 3D calculations.
For 1D calculations, the case of constriction structure
with Co1 infinitely large layer is simulated as a Co1 layer
with the same cross-sectional area as the pillar, attached to
the Cu reservoir. The disagreement between the 3D and 1D
models is only with respect to the magnetoresistance ratio
共MR兲. In this case, ⌬R = R↑ − R↓ 共R is a resistance between
the first and last Cu/ Co interfaces兲 is the same for both
cases, but it is the ohmic resistance between the first and last
Co/ Cu interfaces that is different between the 1D and 3D
calculations.
Although the current distribution can be rather inhomogeneous, there is good agreement between the 1D and 3D diffusive models. We also observed such behavior in the case of
a Py/ Cu lateral spin-valve structure,27 where we found a
very inhomogeneous current distribution, but “external”
measurable values, such as nonlocal voltage, were to some
extent in agreement with the diffusive 1D models.
B. Current inhomogeneity inside pillar
Figure 4 presents a map of the spin-polarized current density jsp inside the 共Cu/ Co兲2 constriction type structure for
parallel 关共a兲 and 共b兲兴 and antiparallel magnetization configurations 共c兲. Case 共a兲 is a structure without an infinitely large
Co1 layer, whereas 共b兲 and 共c兲 have it. A map of jch is not
presented here as it looks similar to jsp with ↑↑. The jsp
inside the Co2 layers is more homogeneous and flows well
perpendicular to the interfaces, although the to-pillar-center
component of jsp can be rather large. The jsp tends to be more
homogeneous when passing the Co layer, causing the jsp in
the adjacent Cu layers to have rather large in-plane components. This is remarkable in case 共c兲. A very similar tendency
is found for jch.
Figure 5 shows cross-sectional parallel-to-interface 共in the
x direction兲 profiles of 共a兲 jsp and 共b兲 jch through the pillar in
the middle of the Co2 layer for the 共Cu/ Co兲2 structure. Both
jsp and jch are inhomogeneous, having minima at the structure center. This is due to inhomogeneous current injection
into the pillar from infinitely large layers. There is no jsp and
jch inhomogeneity for the S-constant and nearly no inhomogeneity in case of the column type structure as the jsp and jch
are homogenized by a Co1 layer. For the constriction type
FIG. 4. Spin-polarized current
in 共Cu/ Co兲2 constriction type
structure 共a兲 without infinitely
large Co1 layer, ↑↑ and 共b兲 with
infinitely large Co1 layer ↑↑. 共c兲
Same as 共b兲 but ↑↓.
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J. HAMRLE, T. KIMURA, T. YANG, AND Y. OTANI
FIG. 5. 共a兲 jsp and 共b兲 jch in the
middle of the Co2 layer inside the
共Cu/ Co兲2 structures.
structure, the jsp inhomogeneity is 12% for ↑↑ and 5% for
↑↓. Inhomogeneity is defined as 共jside − jcenter兲 / 共jside + jcenter兲,
where the side and center values are taken from the crosssectional current distribution in Fig. 5. The inhomogeneity is
reduced for ↑↓ due to different conductivities of up and
down channel conductivities, leading to other current homogenization in the Cu2 spacer layer. If the Co1 layer is
infinitely large, the inhomogeneities are increased to 29%
and 10% for ↑↑ and ↑↓, respectively. The jch inhomogeneities are 8% and 20% for the constriction type structure with
and without an infinitely large Co1 layer, respectively.
Figure 6 shows the dependence of the inhomogeneity of
the perpendicular-to-plane 共z兲 component of jsp inside the
Co2 layer in the constriction type structure with Co1 infinitely large, when varying ␴ and the spin-flip resistance
AR␭ = ␭ / ␴ of both Co layers. Note that AR␭,Co = 14.3 f⍀ m2,
AR␭,Cu = 7.3 f⍀ m2; i.e., spin flip is more likely to occur inside Cu than inside Co. For ␴ Ⰶ ␴Co, the inhomogeneity is
determined solely by the value of ␴ 关Fig. 6共a兲兴. When the
inhomogeneity is expressed as a function of AR␭ 关Fig. 6共b兲兴,
it saturates for ␴ Ⰷ ␴Co, increasing with the reduction of AR␭
and being constant for large AR␭. Figure 6共b兲 also shows that
jsp inhomogeneity for ␴ = ␴Co, R␭ = R␭,Co is determined by the
interplay between ␴Co and ␴Cu, and not by the value of R␭.
As discussed above with respect to Fig. 4, the to-pillarcenter component of jsp inside Co2 nearly vanishes, although
the to-pillar-center component of jsp in surrounding Cu layers exists. To explain this feature, we have calculated the
to-pillar-center component of jsp inside the Co2 layer for
various ␴ and AR␭ of Co layers 共see Fig. 7兲. When ␴ 艋 ␴Co
and AR␭ 艌 AR␭,Co, the to-pillar-center jsp vanish. When ␴
艋 ␴Co and AR␭ ⬍ AR␭Co, the to-pillar-center components of
jsp become positive, because in this case the Co2 layer effectively blocks the flow of jsp from the bottom Cu2 layer, and
then the to-pillar-center jsp equalize the current inhomogeneity incoming from the Cu2 layer. When AR␭ is large and ␴
Ⰷ ␴Co, the to-pillar-center jsp saturates to negative 共i.e., out
from the pillar center兲 value, mirroring the flow of the topillar-center component of jch. In conclusion, the vanishing
of the to-pillar-center current inside the Co2 layer is due to
␴Cu Ⰷ ␴Co and to a large enough value of R␭,Co.
There remains a question as to whether such jsp inhomogenities make current magnetization reversal easier or
not. An advantage of the jsp inhomogeneity may be that it
locally enhances jsp near the pillar edge, whereas it decreases
jsp at the pillar center. As shown in Table I, the mean value of
jsp is about the same for the 共Cu/ Co兲2 constriction structure
with and without an infinitely large Co1 layer. However, in
the second case, jsp is much more inhomogeneous. This is
not well presented in Fig. 5共a兲: we do not see the largest jsp
flowing in the vicinity of the corners of the square pillars. A
disadvantage of jsp inhomogeneity may be that a different
magnitude of torque is exerted on magnetic spins of the free
Co2 layer.
IV. ELECTROCHEMICAL POTENTIAL INSIDE
STRUCTURE
˜ 共hereafter,
Figure 8 presents profiles of ␮↑, ␮↓, and ␮
␮-profiles兲 along the center axis of the 共a兲 共Cu/ Co兲2 ↑ ↑, ↑↓;
共b兲 共Cu/ Co兲3 ↑ ↑ ↑, ↑ ↓ ↑; and 共c兲 共Cu/ Co兲3 ↑ ↑ ↓, ↑ ↓ ↓. The
S-constant structure is presented for both magnetization directions of the Co2 layers, although column and constriction
types are presented only for ↑ magnetization of the Co2
FIG. 6. Dependence of jsp inhomogeneity inside the Co2 layer
in the 共Cu/ Co兲2 constriction type
structure on 共a兲 conductivity of the
Co layer ␴ and 共b兲 spin-flip resistance AR␭ of the Co layer.
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THREE-DIMENSIONAL DISTRIBUTION OF SPIN-…
Hence, there is effectively no spin accumulation ⌬␮ inside the Co2 layer in the case of a commonly used constriction type structure with two FM layers. It may be one of the
reasons that this contribution to magnetization reversal 共predicted in Ref. 29兲 was not observed in the Co/ Cu structure.18
To obtain nonzero ⌬␮, it is necessary to use either a three
FM layer system with antiparallel configuration of the first
and last FM layers, or to ensure that the structure above the
free layer will not contain spin scatterers. It can be reached
when the pillar structure above the free layer 共i兲 does not
contain any strong spin-scatterer layers 共as Au, Ag, Pt, Ru兲
and 共ii兲 the cover layer does not contain a large volume of
metal. It means that pillar current drain should be realized by
a long pillar or by a thin cover layer.
FIG. 7. The to-pillar-center component of jsp on the side of the
Co2 layer in the constriction type structure with Co1 infinitely large
as a function of spin-flip resistance AR␭. Note AR␭,Cu / AR␭,Co
= 0.51, ␴Cu / ␴Co = 11.5.
layer. In contrast to jsp, the inhomogeneity of spin accumulation ⌬␮ at the position of the free Co2 layer is very small,
below 2%. Table I presents mean values of ⌬␮ for all types
of the studied structures.
Figures 8共a兲–8共c兲 show that ␮-profiles depend slightly on
magnetization of the free Co2 layer because tCo2 Ⰶ ␭Co. Figure 8共a兲 exhibits suppression of ⌬␮ in the vicinity of an
infinitely large Cu layer. The reason is exactly the same as
discussed in Sec. III: the infinitely large Cu layer works as a
strong spin scatterer, causing a small spin-flip resistance
共large scattering兲 between up and down channels. Obviously,
such a shortcut reduces ⌬␮. When finishing this article, we
found that this has been recently pointed out also by
Berger.28
This is contradictory to Refs. 23–25, where it is argued
that the presence of a spin scatterer increases spin accumulation ⌬␮ inside the pillar. It should be emphasized that presence of spin scatterers increases jsp 共and also magnetoresistance兲 in the pillar, but reduces ⌬␮.
Figure 8共b兲 shows that ⌬␮ is reduced when Co1 and Co3
layers have parallel magnetization configurations. Figure
8共c兲 shows an increase in ⌬␮ when Co1 and Co3 layers are
antiparallel, enhancing ⌬␮ by a factor of 2 with respect to
共Cu/ Co兲2 S-constant structure.
⌬␮ inhomogeneity
Figure 9 presents the inhomogeneity of ⌬␮ in the center
of the Co2 layer presented as a function of 共a兲 ␴ / ␴Co and 共b兲
AR␭, / AR␭,Co of the Co layers. We can see very similar features as presented for jsp inhomogeneity: for ␴ 艋 ␴Co, the
inhomogeneity does not depend on AR␭ 关Fig. 9共a兲兴. As ␴
⬎ ␴Co, the inhomogeneity is determined mainly by the value
of AR␭ 关Fig. 9共b兲兴. When ␴ ⬎ ␴Co and AR␭ is small, ⌬␮
inhomogeneity is very small 共about 2%, i.e., much less than
jsp inhomogeneity兲 and increases with the reduction of AR␭.
V. MAGNETORESISTANCE
Finally, we discuss the influence of the type of structure
on the MR, presented in the last column in Table I. The value
˜ ↑ − ⌬␮
˜ ↓兲 / 共⌬␮
˜ ↑ + ⌬␮
˜ ↓兲,
of MR is determined as MR= 共⌬␮
where ↑, ↓ denote the magnetization of the free Co2 layer,
˜ ↑/↓ = ␮
˜ last,↑/↓ − ␮
˜ first,↑/↓, where ␮
˜ first,↑/↓ and ␮
˜ last,↑/↓ are deter⌬␮
mined on the Cu side of the first and last Cu/ Co interface,
respectively.
The calculated value of MR in the case of the 共Cu/ Co兲2
S-constant structure is 0.48%. However, in the case of the
共Cu/ Co兲2 constriction type, it reaches 1.01% 共enhancement
by a factor of 2兲, and in the case of the constriction with an
infinite Co1 layer, even 1.43% 共enhancement by a factor of
3兲. The last value may be misleading as this increase is due
FIG. 8. Electrochemical potential ␮↑/↓ along the center axis of the 共a兲 共Cu/ Co兲2 ↑ ↑, ↑↓; 共b兲 共Cu/ Co兲3 ↑ ↑ ↑, ↑ ↓ ↑; and 共c兲 共Cu/ Co兲3 ↑ ↑ ↓,
↑ ↓ ↓ for S-constant, column, and constriction structure types 共Fig. 1兲.
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FIG. 9. Inhomogeneity of ⌬␮
inside the Co2 layer in the
共Cu/ Co兲2 constriction type structure on 共a兲 conductivity of the Co
layer ␴ and 共b兲 spin-flip resistance
AR␭ of the Co layer.
mainly to the resistivity reduction of the Co1 infinite layer.
Figure 10共a兲 shows the dependence of jsp position in the
Co2 layer on A⌬R, where ⌬R = R↑ − R↓, and ↑ and ↓ mean up
and down magnetizations of the Co2 layer, respectively. This
shows that ⌬R increases linearly with jsp, rather than with
⌬␮.
Figure 10共b兲 shows the dependence of ⌬␮ on ⌬R. We can
see that with increasing ⌬R, ⌬␮ is reduced. Two different
slopes correspond to two different “sources” of ⌬␮ acting
with different “hardnesses.” The source hardness is determined by a presence or absence of scatterers below the fixed
Co1 layer. The hard source is for the constriction and column
structure types; i.e., when Cu1 is infinitely large. The weak
one is for the S-constant structure type; i.e., when Cu1 is not
infinitely large. The explanation of this behavior has already
been provided in Sec. IV: when the spin-flip scatterer is presented below the fixed Co1 layer, ⌬␮ vanishes on the
Cu1 / Co1 interface and hence provides a harder source of
⌬␮. Figure 10 also shows that when changing the surroundings of FM 共fixed兲/spacer/FM 共free兲 layers, an increase in
⌬␮ at the position of the free layer is related to a decrease of
MR.
VI. CONCLUSION
We have calculated a three-dimensional current distribution within a diffusive regime inside 共Cu/ Co兲2 and 共Cu/ Co兲3
pillar structures, where the pillar cross-sectional areas of the
starting/terminating layers were assumed to be either infinitely large 共column type, constriction type兲 or they have the
same cross-sectional area the as pillar 共S-constant types兲. For
averaged quantities 共as averaged jsp or magnetoresistivity兲,
we have found a good agreement between the 1D and 3D
models for all studied structures.
Inside the 共Cu/ Co兲2 pillar surrounded by infinitely large
layers, the jsp and jch are found to be inhomogeneous. Maximal jsp inhomogeneities are found to be 29% and 20% in the
cases of constriction structures with and without an infinitely
large Co1 layer, respectively. On the other hand, the profile
of ⌬␮ is found to be more homogeneous, with a maximum
inhomogeneity of 2%. The to-pillar-center components of jsp
and jch inside the Co2 layer nearly vanished, when ␴Co
Ⰶ ␴Cu. If AR␭ were smaller, then the inhomogeneity inside
Co2 layer might be further enhanced.
When the pillar is terminated by infinitely large layers,
they serve as spin- scatterers, shortcutting up and down channels and hence reducing ⌬␮ and magnifying jsp inside the
Co2 layer. When such spin scatterers 共this is also valid for
different spin scatterers, such as layers of Au, Ag, Pt, Ru,
etc.兲 are introduced below the fixed Co1 layer, they make ⌬␮
source harder. When they are placed above the free Co2
layer, they shortcut up and down channels, reducing ⌬␮ to
nearly zero, hence enhancing jsp. As most of the experimentally studied 共Cu/ Co兲2 structures have infinitely large layers
at both ends, ⌬␮ inside them is nearly zero.
In agreement with experimental results,23–25 we found that
j sp is linearly proportional to ⌬R = R↓ − R↑ 共and hence MR is
increased when increasing jsp at the position of the free Co2
layer兲. Furthermore, dependence of ⌬␮共⌬R兲 is also linear,
but ⌬␮ is reduced when increasing ⌬R 共and thus increasing
MR兲.
FIG. 10. Dependence of 共a兲 jsp
and 共b兲 ⌬␮ on A⌬R for different
studied 共Cu/ Co兲 structures; symbol notation in Table I. Lines are
the best linear fits, dashed line and
bold symbols 共full line and normal
symbols兲 denote parallel 共antiparallel兲 Co1 and Co2 layers, respectively. Cu1 infinite means column
and constriction types, Cu1 finite
means S-constant type.
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THREE-DIMENSIONAL DISTRIBUTION OF SPIN-…
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