Materials Transactions, Vol. 47, No. 1 (2006) pp. 11 to 14
Special Issue on Present Status of the Research on Ordered Alloys and the Application to Magnetic Devices
#2006 The Japan Institute of Metals
Uncompensated Spin Elements in Ferromagnetic and Antiferromagnetic Bilayer
with Non-Collinear Spin Structure
Chiharu Mitsumata1 , Akimasa Sakuma2 , Kazuaki Fukamichi3 and Masakiyo Tsunoda4
1
Advanced Electronics Research Laboratory, Hitachi Metals Ltd., Kumagaya 360-0843, Japan
Deptartment of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
3
Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan
4
Department of Electronic Engineering, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
2
The exchange bias in a ferromagnetic (FM) and antiferromagnetic (AFM) bilayer with a -phase disordered structure has been investigated
within a framework of Heisenberg model. The magnetic atoms in the AFM layer realize the triple Q structure due to geometrical spin
frustrations. This non-collinear spin structure can bring about an exchange bias in the FM/AFM bilayer system. Under the influence of the
exchange bias, the uncompensated spin element appears in the AFM layer, accompanied by a shifted loop in magnetization curves, in accord
with the measured loop by X-ray magnetic circular dichroism spectroscopy for an AFM layer.
(Received August 5, 2005; Accepted September 8, 2005; Published January 15, 2006)
Keywords: antiferromagnetic, exchange bias, exchange coupling, spin frustration, spin structure, non-collinear spin, Heisenberg model
1.
Introduction
The exchange bias in a ferromagnetic (FM) and antiferromagnetic (AFM) bilayer system plays a key role in enabling
high density in magnetic storages. FM/AFM bilayer systems
are attractive in spin valve heads and perpendicular recording
media to control magnetic domains. For such FM/AFM
bilayer systems, a variety of AFM materials have been
investigated, and many models for the exchange bias have
been proposed.1,2) Several kinds of AFM materials have a
non-collinear spin structure.3) It should be noted, however,
that the proposed conventional models discuss the collinear
spin structure only. Therefore, we have focused the noncollinear spin structure in the AFM alloy, and shed light on
the mechanism of its exchange bias.3–5)
Recently, the domain observation by X-ray magnetic
circular dichroism (XMCD) spectroscopy for AFM materials
becomes very important to investigate the exchange bias in
FM/AFM systems. In general, the summation of AFM spin
vectors shows zero, and that cancellation of spins is called
‘‘the compensated state’’. On the other hand, as a result of the
summation of spin vectors, the uncompensation stands for the
generation of finite spin elements. The uncompensated spin
in the domain state model was predicted by Nowak et al.6)
Such uncompensated spin elements in AFM layers were
actually found in a Mn–Ir/Co system by Ohldag et al.7) It was
considered that the interfacial roughness forces an unbalance
in the number of spins on AFM sub-lattice. The collinear spin
structure in AFM layers has been formed the base of the
physical aspect of the domain state model. However, it
should be emphasized that the spin structure for several kinds
of -phase disordered alloys is not collinear.3) In the present
paper, we discuss the appearance of the non-collinear spin
structure in the -phase disordered AFM alloy, and investigate the uncompensated spin state at the interface of FM/
AFM bilayer system.
2.
Calculation Method of the Present Model
The present model works within the framework of the
classical Heisenberg model. In order to define the magnetic
structure in the AFM/FM bilayer, the following Hamiltonian
described in eq. (1) is evaluated in the numerical calculations.
X
X
J1;i; j hSi S j i J2;i;k hSi Sk i
H¼
hi; ji
X
i
hi;ki
AðSi Þ gB
X
hSi H app i
ð1Þ
i
where Si is the unit vector of the spin at the ith atom, and the
summation is carried out over all possible spin pairs by using
the exchange constants, J1 and J2 . The indices j and k denote
the first and second nearest neighboring spins, respectively.
The third and forth terms in eq. (1) describe the magnetic
anisotropy energy and the Zeeman energy, respectively.
Here, the function AðSi Þ stands for the magnetic anisotropy
energy, and the vector H app represents the applied field.
When spin Si belongs in the FM layer, the anisotropy energy
can be expressed AðSi Þ ¼ Du hSi ni2 , where Du and the unit
vector n respectively stand for the anisotropy constant and
the easy axis direction. In the present case, the vector n points
out ½01 1 direction. On the other hand, when spin Si belongs
in the AFM layer, the magnetic anisotropy function can be
AðSi Þ ¼ D1 ð2b 2c þ 2c 2a þ 2a 2b Þ þ D2 2a 2b 2c , where D1
and D2 denote the second and fourth orders of magnetic
anisotropy constants in the AFM layer, respectively. Also,
a , b , and c designate the elements of the spin vector Si in
the directions of the a-, b-, and c-axes in the -phase alloy,
respectively. In the AFM layer, the face centered sites in the
fcc lattice are randomly occupied by magnetic atoms and
non-magnetic atoms, composing a -phase disordered alloy.
The atomic configuration of the present model is illustrated in
Fig. 1. The bilayer is stacked to the h111i orientation, and the
thickness of FM and AFM layers are 9 monolayers (MLs) and
100 MLs, respectively. The thickness of 100 MLs corresponds to 18 20 nm in representative Mn-based alloys.
12
C. Mitsumata, A. Sakuma, K. Fukamichi and M. Tsunoda
[111]
FM 9 ML
φ
θ = cos −1 2 / 3π
[011]
θ
[101]
AFM 10 − 100 ML
θ
J1
J2
φ
Magnetic Non-magnetic FM atom
φ = π/2
Binary AFM alloy
Fig. 1 The atomic configuration near the interface of the AFM/FM bilayer
model.
For the sake of simplicity, the values of J1 are assumed to
be 20 meV in the FM layer and 20 meV in the AFM layer,
respectively. The exchange constants J2 is defined by
J2 ¼ jJ1 j=2.8) J1 at the interface is assumed to be equivalent
to J1 in the AFM layer. These exchange constants give the
Néel temperature TN ¼ 695 K, which roughly agrees with TN
of the -phase disordered Mn-based alloys (Mn–Ir, Mn–Rh,
etc.).3) The anisotropy constant D is assumed to be Du =J1 ¼
3 106 in the FM layer and D1 =J1 ¼ D2 =J1 ¼ 3 103 in
the AFM layer, respectively.5) The magnetization process is
solved by using the following motion equation of spin within
the scheme of a forward difference method;
dS
¼ ðS H eff Þ fS ðS H eff Þg;
dt
ð2Þ
where and are the gyromagnetic and damping constants,
respectively. The effective field H eff is obtained by the
derivation of Hamiltonian (1).
3.
Results and Discussion
The spin structure of -phase alloy is illustrated in Fig. 2 in
which 75% of magnetic atoms and 25% non-magnetic atoms
are included in the unit cell. As seen from this figure, the
AFM alloy realizes a triple Q (3Q) spin structure where an
angle between AFM spins indicates cos1 ð1=3Þ, showing a
typical non-collinear spin configuration. The 3Q spin
structure can be generated by spin frustrations caused by
the geometry of magnetic atoms in the -phase disordered
structure. The calculated spin structure is provided by the
single spin flip Metoropolis scheme in the Monte Carlo
simulation. Also, the theoretical approach with a quantum
spin predicts the generation of the 3Q structure in AFM spins
on an fcc lattice.9)
For the AFM spins, four equivalent atoms are composed in
the unit cell, and the summation of those spins should
become zero in the ground state. For the h111i stacking of the
FM/AFM bilayer, the (111) plane of the AFM layer faces to
the FM layer. In the case of (111) surface of the -phase AFM
alloy, the spin configuration can be compensated where the
summation of AFM spins is expected to becomes Si ¼ 0.
On the other hand, in the case of interface in the FM/AFM
Fig. 2 The spin structure of the -phase disordered AFM alloy. 75% of
magnetic atoms and 25% of non-magnetic atoms are included.
bilayer, the uncompensated spin element of the AFM layer
needs to remain at the interface at least, because of the finite
coupling energy at the interface as following. The coupling
energy at the FM/AFM interface is given as
X
JI hSFM SAFM i:
ð3Þ
EI ¼ Since, all FM spins SFM are equivalent vectors, eq. (3) can be
rewritten as
D
E
X
EI ¼ JI SFM SAFM :
ð4Þ
At the interface preferring the h111i orientation, the coupling
energy becomes zero from
P eq. (4), if the summation of AFM
spins SAFM becomes
SAFM ¼ 0. However, the exchange
coupling bias is actually observed in thePh111i stacked
bilayer.10) Accordingly, the condition of
SAFM 6¼ 0 is
required for the (111) interface.
The exchange coupling between the FM and AFM layers
influences the relaxation of the direction of spins. The spin
configuration near the interface when the 3Q spin structure is
realized in the -phase disordered AFM layer is shown in
Fig. 3(a). The indices of the AFM spins represent the
equivalent atomic sites. The ground state of 3Q spins is
illustrated in Fig. 3(b), in which the angle between AFM
spins is cos1 ð1=3Þ. The direction of the AFM spin near the
interface is different from that of the ground state. The
relaxation of spins influences the deviation of angles between
the AFM spins. Consequently, the resultant spin element
becomes finite, forming the uncompensated spins in the AFM
layer. The appearance of uncompensated spins was reported
by Nowak et al.6) within the scheme of Ising spin model
based on an interfacial roughness to retain the uncompensated spin component. On the contrary, in the present model,
no roughness is also able to create the uncompensated spin
element. Consequently, the frustration of spins at the interface could realize a finite value of the magnetization in the
AFM layer. Such uncompensated spin element is quite small,
but can be observed by X-ray magnetic circular dichroism
(XMCD) spectroscopy.7)
The characteristics of magnetization in the AFM layer
shows a shifted loop both in the field and magnetization axes
of M–H loops in the XMCD measurements,7) when the
magnetization loop in the FM layer is shifted by the bias field.
Uncompensated Spin Elements in Ferromagnetic and Antiferromagnetic Bilayer with Non-Collinear Spin Structure
On the other hands, there is no loop shift in the AFM layer
when the magnetization loop in the FM layer is not shifted.
The calculated corresponding magnetization loops are shown
in Fig. 4. The magnetization m is obtained by the average of
all the AFM or FM spins. The average of the AFM spins
includes all equivalent atomic sites shown in Fig. 3. The
thickness of the AFM layer is adopted for 10 and 100 MLs in
Figs. 4(a) and (b), and in Figs. 4(c) and (d), respectively. The
thickness of the FM layer is 9 MLs in both the cases.
[111]
[011]
[101]
(a)
(b)
IV
III
I
IV
II
III
II
I
Unit cell
FM spin
AFM spin
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Magnetization (m/ms)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
(a)
-6
-4 -2
0
2
4
Applied field (H/D)
6
(c)
-6
0
2
4
-4 -2
Applied field (H/D)
For the thin AFM layer (10 MLs), the magnetization loop
in the FM layer exhibits a symmetric switching field, giving
no exchange bias as seen from Fig. 4(a). The resultant
magnetization in the AFM layer shows a finite value. Such
uncompensated spin element forms the hysteresis loop
corresponding to the magnetization loop in the FM layer.
The switching field in the AFM layer is equivalent to that in
the FM layer. On the other hand, in the case of thick AFM
layer (100 MLs), the FM spins show a shifted magnetization
loop under the influence of the exchange bias in Fig. 4(c). In
the AFM layer, at the same time, the resultant spin vector
becomes finite, forming an uncompensated spin element, as
well as spins in the thin AFM layer. The uncompensated spin
element in the AFM layer shows a hysteresis loop in a similar
manner as the FM layer, and it also indicates the biasing shift
in the direction of the abscissa (field axis) in Fig. 4(d). The
biasing field in the AFM layer is equivalent to the bias in the
FM layer. Consequently, the exchange bias influences the
loop shift in both the FM and AFM layers, and the agreement
between such bias values of the FM and AFM layers could
reproduce the measured results in XMCD spectroscopy.7)
The sign of resultant spin in the AFM layer is dominated by
the exchange constant J1 at the interface. In the present
model, J1 < 0 is assumed, although the positive J1 at the
interface reverses the sign of resultant spin in Fig. 4(d),
because the uncompensated spin element is sensitive to the
exchange interaction at the interface. For example, the depth
profile of uncompensated spin elements was not uniform in
the case for the L12 -type ordered AFM layer.11) The depth
profile of uncompensated spins in -phase AFM layer will be
discussed in a separate paper.
Magnetization (m/ms)
Magnetization (m/ms)
Magnetization (m/ms)
Fig. 3 The spin configuration of the AFM/FM bilayer. (a) The spin
configuration near the interface, (b) the ground state of the spin structure in
the magnetic unit cell.
6
13
0.03
0.02
0.01
0.00
-0.01
-0.02
-0.03
-6
(b)
-4 -2
0
2
4
Applied field (H/D)
6
0.003
0.002
0.001
0.000
-0.001
-0.002
-0.003
-6
(d)
-4 -2
0
2
4
Applied field (H/D)
6
Fig. 4 The magnetization loops in the AFM/FM bilayer system. (a) The magnetization in the FM layer with the thin AFM layer, (b) the
resultant magnetization in the thin AFM layer with the thickness of 10 MLs, (c) the magnetization in the FM layer with the thick AFM
layer, (d) the resultant magnetization in the thick AFM layer with the thickness of 100 MLs. The resultant magnetization is averaged all
over the AFM layer.
14
C. Mitsumata, A. Sakuma, K. Fukamichi and M. Tsunoda
In Fig. 4, the asymmetry between the positive and
negative value of resultant magnetization is determined by
the following eq. (5),
¼ ðmpos þ mneg Þ=ðmpos mneg Þ:
ð5Þ
where mpos and mneg are the resultant magnetizations in the
remanent state of AFM layers. The calculated value of
asymmetry in the AFM layer becomes 10 ML ¼ 0 in
Fig. 4(b). Thus, in the case of no exchange bias, a loop shift
in the ordinate (magnetization axis) is not obtained. Meanwhile, in Fig. 4(d), the value of asymmetry shows 100 ML ¼
0:19. Namely, the asymmetry 100 ML is significantly larger
than 10 ML in the calculation of magnetization process. The
finite value of asymmetry gives a loop shift downward in
the ordinate. Consequently, in both the thin and thick cases,
the calculated loops would reproduce the XMCD measurements.
4.
Conclusion
The magnetic structure and the magnetization process of
AFM/FM bilayer are calculated within the framework of
classical spin model. For the -phase disordered alloy in the
AFM layer, the uncompensated spin element is generated due
to the frustration of spins, which appears to be independent of
the AFM thickness.
Such uncompensated spin results in a hysteresis loop
corresponding to the magnetization loop of the FM layer.
For the thick AFM layer, the loop of uncompensated spin
element is shifted in both directions of the applied field and
the magnetization. On the other hand, in the case of thin AFM
layer, the resultant magnetization becomes finite, but the
magnetization loop shows no shift. These calculated results
are consistent with the magnetization loops by XMCD
spectroscopy.
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