BAR-ILAN UNIVERSITY
Perpendicular Anisotropic
Magnetoresistance in Co/Pd
Multilayers
Yaniv Kachlon
Advisor - Dr. Amos Sharoni
Submitted in partial fulfillment of the requirements for the Master's
Degree in the Department of Physics and Institute of Nanotechnology and Advanced Materials,
Bar-Ilan University
Ramat Gan, Israel
2011
Contents
Motivation ??? .................................................................................................................................... 4
Theoretical overview ........................................................................................................................... 5
PMA Materials ................................................................................................................................. 5
Magnetoresistance.......................................................................................................................... 9
Experimental Methods ...................................................................................................................... 16
Deposition- PVD- sputtering .............................................................Error! Bookmark not defined.
Fabrication and nano-fabrication.................................................................................................. 18
Measurements .............................................................................................................................. 20
Numerical analysis ........................................................................................................................ 23
Results ............................................................................................................................................... 24
Discussion .......................................................................................................................................... 32
References......................................................................................................................................... 37
Abstract
We studied the anisotropic magnetoresistance (AMR) properties of thin multilayered
Co/Pd electrodes as function of magnetic field magnitude and direction and electrode
width. The multilayered structure induces perpendicular magnetization anisotropy
(PMA), found to be considerably different than in other thin Ferromagnetic (FM) films.
The magnetoresistance for fields out-of-plane (𝜌𝑜𝑝 ) is significantly different than for inplane fields transverse to current direction (𝜌𝑖𝑝 ), although in both cases the current is
perpendicular to the magnetic field. Moreover, opposed to other very thin films where
𝜌𝑂𝑃 is smaller than 𝜌𝑖𝑃 , our films show an opposite effects, the origin of which is not
clear.
We are able to model the rich AMR properties of the electrodes by an expanded
Stoner-Wolfarth model, where we introduce an additional energy scale related to the
PMA. By minimizing the free energy of the system, through a numerical refinement
process, we extract anisotropic constants of the electrodes. This is done by
reconstructing the magnetoresistance behavior of the electrodes, using only the linear
terms in the dependence of resistivity on the magnetization orientation. Our
anisotropic constants coincide remarkably with the literature (measured by other
methods).
Thus, our refinement process is an excellent method to easily and accurately extract
anisotropic constants also in nano-scale magnetic systems, which cannot be accessed
otherwise.
Motivation
Magnetic materials with perpendicular magnetization anisotropy (PMA) are
extensively used in commercial Spintronics related applications. Mainly as writing
media for magnetic memory hard disks based on giant magneto-resistance (GMR) {[1],
[2]}. PMA may be important in applications based on lateral spin valves, where a
perpendicular easy axis is preferred {MRAM [3]}. A recent example is the Spin Hall
Effect. Geometric consideration calls for a magnetic spin sensor with out-of-plane
orientation. this has been achieved by a strong magnetic field which pulls the
magnetization out of plane[4]. A PMA can be measured also at zero magnetic fields,
such as in []. In the latter case the deposition conditions were difficult. Thus, it is
favorable to use an PMA material which is easier to deposit, such as the multilayered
Co/Pd, [5]
Theoretical overview
Magnetization Anisotropy
In a ferromagnetic material there may be a preferred orientation of the magnetic
moments. This is referred as anisotropy. The preferred axis is called the easy axis, since
it’s the natural direction of magnetization even in the absent of an external field, while
an axis where in the absence of external field the magnetism will change is called the
hard axis.
There are three main sources of anisotropy, shape, crystal structure and atomic or
nano-scale texture. Additionally, a magnetic sample can be in a multi-domains mode
which will be described later or in a single-domain mode.
The magnetostatic energy of a ferromagnetic ellipsoid with magnetization 𝑀𝑠 is:
𝜀𝑚 =
1
𝜇 𝑉𝒩𝑀𝑠2
2 0
In a uniformly magnetized sample having the form of an ellipsoid the demagnetizing
field 𝐻𝑑 is also uniform. The relation between 𝐻𝑑 and 𝑀 is:
𝐻𝑑𝑖 = −𝒩𝑖𝑗 𝑀𝑗
𝑖, 𝑗 = 𝑥, 𝑦, 𝑧
where 𝒩𝑖𝑗 is the demagnetizing tensor, which is generally represented by a symmetric
3 × 3 matrix. A sum over the repeated index is implied. Along the principal axes of the
ellipsoid, 𝐻𝑑 and 𝑀 are collinear and the principal components of 𝒩 in diagonal form
(𝒩𝑥 , 𝒩𝑦 , 𝒩𝑧 ) are known as demagnetizing factors. Only two of the three are
independent because the demagnetizing tensor has unit trace:
𝒩𝑥 + 𝒩𝑦 + 𝒩𝑧 = 1
The general ellipsoid has major axes (𝑎, 𝑏, 𝑐). Defining 𝜏𝑎 = 𝑎/𝑐, 𝜏𝑏 = 𝑏/𝑐
a general expression for 𝒩𝑐 is:
1 ∞
1
(𝜏
)
𝒩𝑐 𝑎 , 𝜏𝑏 = ∫
𝑑𝑢
2 0 (1 + 𝑢)3⁄2 (1 + 𝑢𝜏𝑎2 )1⁄2 (1 + 𝑢𝜏𝑏2 )1⁄2
The other principal components are obtained by rotation:
𝒩𝑎 = 𝒩𝑐 (1⁄𝜏𝑎 , 𝜏𝑏 ⁄𝜏𝑎 ), 𝒩𝑏 = 𝒩𝑐 (𝜏𝑎 ⁄𝜏𝑏 , 1⁄𝜏𝑏 )
The energy of a sample in its demagnetizing field 𝐻𝑑 gives a contribution to the selfenergy, which depends on the direction of magnetization in the sample. This cannot be
an intrinsic property of the material, as it depends on the sample shape.
Other shapes can be approximated to ellipsoids. The anisotropy energy is related to the
difference in energy ∆𝜀 when the ellipsoid is magnetized along its hard and easy directions.
For example; for the magnetostatic energy of a ferromagnetic ellipsoid:
𝜀𝑚 =
1
𝜇 𝑉𝒩𝑀𝑠2
2 0
1
𝒩𝑐 is the demagnetizing factor for the easy axis and 𝒩𝑎𝑏 = 2 (1 − 𝒩𝑐 ) is the
1
1
demagnetizing factor for the hard axes. Therefore: ∆𝜀𝑚 = 2 𝜇0 𝑉𝑀𝑠2 [2 (1 − 𝒩𝑐 ) − 𝒩𝑐 ],
which gives for a prolate ellipsoid:
1
𝐾𝑠ℎ = 𝜇0 𝑀𝑠2 (1 − 3𝒩𝑐 ).
4
Shape anisotropy is only fully effective in samples which are small enough not to break
into Domains.
Generally, the tendency for magnetization to lie along an easy axis is represented by
the energy density term:
𝐸𝑎 = 𝐾𝑢 sin2 𝜃
where 𝜃 is the angle between 𝑀 and the anisotropy axis and 𝐾𝑢 is the anisotropy
constant.
Perpendicular Magnetic Anisotropy
In uniformly magnetized thin films, magnetization usually lies in the plane of the film
for magnetostatic reason. However, perpendicular anisotropy can arise when an
oriented or epitaxial film of hard magnetic material is grown with its easy axis
perpendicular to the film plane. In very thin films, a few nanometers thick, surface
anisotropy can sometimes lead to perpendicular magnetization.
Multilayer stacks with perpendicular anisotropy can be built up of alternating thin
ferromagnetic and nonmagnetic layers. If 𝜗 is the angle between the magnetization
and the film normal, and there is some perpendicular anisotropy 𝐾𝑢 , the energy per
unit volume, has a minimum at 𝜗 = 0.
1
𝐸𝑡𝑜𝑡 = 𝐾𝑢 sin2 𝜗 + 𝜇0 𝑀𝑠2 cos 2 𝜗
2
The condition for perpendicular anisotropy is:
1
𝐾𝑢 > 𝜇0 𝑀𝑠2
2
In small magnetic field, ferromagnetic materials tend to break into magnetic domains
in order to reduce the…. . As mention earlier, when a sample breaks up into Domains,
the shape anisotropy is not fully effective.
Magnetoresistance
The FM electrodes in lateral Spintronics devices have width on the order of 100 nm.
This elongated shape might hinder the preferred PMA of the thin films due to shape
and size effects.[6] It is essential to compare the shape anisotropy energy and the PMA
energy, to verify the electrodes maintain their perpendicular magnetization easy axis.
An excellent method to measure magnetization orientation in thin films and submicron electrodes is via the anisotropic magneto-resistance.[7]
Ordinary Magneto-resistance
The electrical resistivity of all metals can be changed through the presence of a
magnetic field. The magnitude of the Ordinary Magneto-resistance (OMR) is usually
small; this effect is due to the Lorentz force causes an acceleration of electrons
perpendicular to the velocity when moving across a magnetic field and produce
circular motion.
The ordinary magnetoresistance effect has been described by Kohler’s rule. Kohler
expressed the OMR magnitude-field dependency by:
∆𝜌
𝐻 2
∝[ ]
𝜌
𝜌
Anisotropic Magneto-resistance
The AMR is the dependence of the magnetic film resistivity on the relative angle
between the current direction and the magnetization direction, and is given by:
𝜌(𝜃) = 𝜌0 + (𝜌∥ − 𝜌⊥ ) cos 2 𝜃
where 𝜃 is angle between the current and the magnetization, 𝜌0 is the resistivity at
zero field and 𝜌∥ − 𝜌⊥ is the difference between the parallel and the perpendicular
configurations.
The resistivity when magnetization is parallel to the current direction, is usually larger
than when the magnetization is perpendicular to the current direction.[8] In some
cases, thin films show an additional dependence: resistivity of magnetization
perpendicular to the current in the film plane 𝝆𝒊𝑷 , is larger than for magnetization out
of plane 𝝆𝑶𝑷 , which is also perpendicular to the current direction.[9] In order to be
able to infer from the AMR signal the magnetic state of the sample, one has to fit the
measurements to a model. These have been done to some extent in the past for single
element ferromagnets,[10] usually fitting only a limited part of the resistivity vs. field
data-set.
Giant Magnetoresistance
A further large magnetoresistance effect is present in nano scale multilayer and nano
scale granular metallic systems which combine ferromagnetic and normal metals. This
mechanism is termed Giant Magnetoresistance (GMR). In GMR, the resistance state
depends upon the relative orientations of the direction of magnetization in the
different layers or granules.
The GMR can be analyzed by the Valet-Fert equations [11], using a general 2 fluid
model, where each spin current is considered separately. The main assumption is that
spin flipping processes are slower than other electron related scattering times.
In FM/NM interfaces, spin polarized electric current is injected from the FM metal to
the NM metal, the current I is flowing perpendicular to the FM/NM interface. As the
conductivities for the spin-up and spin-down electrons in a ferromagnetic metal are
unequal, the usual charge current (I↑ + I↓) in FM is accompanied by a spin current
(I↑− I↓) transporting magnetization in (or opposite) the direction of charge current.
The scattering rate of electrons depends on its spin and the magnetization orientation
of the layer it is passing through.
a
c
b
d
Figure 1: For the same magnetic orientation the resistance is small and for opposite it is large. In the tri-layered spin
valve this leads to lower resistance of the parallel state, relative to the anti-parallel one.
Domain walls
Domain walls (DW) in ferromagnetic metals are known to be another source of
resistance.
Magnetic domains are regions of uniform magnetization in macroscopic samples,
which are separated by planar regions- the Domain Walls - where the magnetization
rotates from one easy direction to another. An applied field changes the net
magnetization of the sample, either by causing the walls to move or by making the
magnetization in the domains rotate towards the applied field direction. [12]
There are two common types of domain walls; the commonest is the 180° Bloch wall,
where the magnetization rotates in the plane of the wall. The Bloch wall creates no
divergence of the magnetization. Since ∇ ∙ 𝑀 = 0, there is no magnetic charge and no
source of demagnetizing field in the wall. The second one is the Néel wall, where the
magnetization rotates within the plan of the domain magnetization, and is normally
higher in energy than the Bloch wall.
Anomalous Hall Effect
The Ordinary Hall Effect (HE) is a phenomenon where an electric field, EH is produced
perpendicular to the current density, J ,in the presence of a perpendicular magnetic field.
EH  RH ( J  0 H )
1
𝑅𝐻 = 𝑛𝑒𝑑 is the Hall coefficient, 𝑛 is the charge carrier density, 𝑒 is the electron charge
and 𝑑 is the thickness of the layer.
In ferromagnetic materials there is an additional contribution to the Hall resistivity due
to spin-dependent scattering of the charge carriers, this is the Anomalous Hall Effect
(AHE), which varies with the magnitude of the magnetization 𝑀.[13]
H 
EH
  0 ( R0 H  R A M )
J
The first part of the Hall resistivity expression is from the ordinary HE and the second
part is from the anomalous HE, the anomalous Hall coefficient 𝑅𝐴 depend on the
measured material and temperature.
There are two scatter mechanisms which produce the AHE phenomenon, the first,
named Skew scattering, change the direction of the electron’s velocity and defined by
θ
θs
x
s
Skew scattering
l
Side jump
𝜃𝑠 from its origin course. The second mechanism is a quantum effect which cause
widthwise displacement of the electron’s location in ∆𝑥, this effect termed “Side
Jump”. The angle of the digression due to side jump is depend with inverse relation to
𝑙, the mean free path, where 𝜃𝑠 ≈
∆𝑥
𝑙
and 𝑅𝐸 ∝ 𝜌2 .
The two mechanisms are usually combined in the AHE and superposition of both will
give dependency of: 𝑅𝐸 = 𝑎𝜌 + 𝑏𝜌2
Planar Hall Effect
Another phenomenon which we briefly discuss is the Planar Hall Effect which depends
on the angle between the current density J and the magnetization M (AMR).
The AMR yields a transverse electric field when J in not parallel or perpendicular to M:
𝐸𝑦 = 𝑗𝑥 𝜌∆ sin 𝜃 cos 𝜃.
Y
𝑗𝑥
X
Here 𝑗𝑥 is the current density along the X axis, 𝜌∆ is the resistivity difference between
the parallel and the perpendicular configurations and 𝜃 is the angle between the
current and the magnetization.
Experimental Methods
Deposition- Physical Vapor Deposition- Sputtering
All necessary parts of the devices, magnetic and non-magnetic, were deposited in our
AJA high vacuum magnetron sputtering system. The chamber has a base pressure of
1x10-8 Torr, 6 targets with pneumatically controlled shutters and the ability to deposit
materials from 3 targets simultaneously. All chamber components (power supplies,
shutters, and gas flow controllers) are computer controlled (for more details see Fig.
1). In the magnetron sputter process Argon plasma is confined using magnets to the
area of the target and accelerated toward the target, which is made of a desired
composition. This will cause continues flux of the specific material to accumulate on a
substrate, which is positioned above the target. Changing deposition conditions, such
as argon pressure, plasma power and substrate temperature modifies the deposited
layer’s properties.
Main chamber
Load Lock
Power Supplies
Sample Holder
Targets
Computer Control
Figure 2: Our AJA high vacuum magnetron sputtering system. Samples held by the sample holder and insert
through the load lock into the main chamber, where the sputtering process is controlled by the computer.
Co 99.98% and Pd 99.95% targets were sputtered sequentially at room temperature
and argon pressure of 3mTorr, with typical deposition rate of 0.3A/s and 0.6A/s for Co
and Pd respectively (Figure 2). Films with different Co/Pd thickness ratios and
repetitions were prepared and
Pd
characterized in order to achieve films with
Co
high PMA.
Figure 3 : Co/Pd multilayers
Micro-Fabrication and nano-fabrication
From the multilayered films we fabricate measurement bridges or electrodes for
transport measurements that enable both magnetoresistance (MR) and Hall
measurements (see cartoon of bridge).
Figure 4: A device sketch from our photolithography mask of multiple bridges with different length scale
Nano scale electrodes were prepared in an E-beam lithography system- Crestec Inc.
(40nm line writing resolution). There are two ways of preparing the electrodes; the
first one is to deposit the Co/Pd multilayers through an appropriately designed e-beam
mask, followed by a lift-off process depicted in figure 5 (see figure caption for details).
A second method consists of first depositing the Co/Pd superlattice on the entire
substrate. The electrodes are defined using inverse e-beam lithography, [14], and are
then etched from the film using high energy argon ion source. This process is shown in
figure 6.
e-beam exposure
development
PMMA
deposition
lift-off
Co/Pd superlattice
Substrate
Figure 5: Schematic description of the lift-off process. An electron sensitive resist is exposed to a high
energy electron beam (left). The exposed areas are dissolved in a suitable developer and material is
deposited on top of the remaining resist structure (two center images). As a last step the sample is put in a
strong solvent that removes all resist. Material deposited on top of the resist is hence removed (right).
Inverse e-beam
exposure
development
etching process
lift-off
PMMA
Co/Pd superlattice
Substrate
Figure 6: Schematic description of the etching process. An electron sensitive resist is exposed to a high energy
electron beam (left). The unexposed areas are dissolved in a suitable solvent. An ion beam is used for etching
the material. As a last step the sample is put in a strong solvent that removes all resist.
In order to connect the electrodes to our measurement devices, a photolithography
process is used with a mask especially designed to connect macroscopic pads to the
nano scale electrodes (see cartoon). The pads are then connected to the measurement
electronics via wire bonding.
Measurements
Superconducting Quantum Interference Device
The Superconducting Quantum Interference Device (SQUID) measures the
magnetization under influence of a magnetic field. The SQUID does not detect directly
the magnetic field from the sample, but function as an extremely sensitive current to
voltage convertor. The sample moves through a system of superconducting detection
coils (Figure 7) which are connected to the SQUID with superconducting wires, any
Figure 7: superconducting detection coil
change of magnetic flux in the detection coils produces a change in the persistent
current in the detection circuit. The variations in the current produce corresponding
variations in the SQUID output voltage which are proportional to the magnetic
moment of the sample. . The device can generate magnetic field up to 7 T, and has
temperature range between 2 K to 400 K. It acts as a magnetometer to detect
incredibly small magnetic fields as low as 10−12 T. by dividing the magnetization
measured with the volume of the material, we can assess the magnetization per unit
volume of the material and Ms (saturated magnetization of the sample).
Transport measurements
Magnetoresistance (MR) and Hall Effect (HE) measurements were performed. A four
probe measurement was employed to see how films properties depend on the
geometrical features of the electrode, such as width and length. By doing a 4 probe
measurement the contacts resistance could be eliminated. All MR and HE
measurements were performed in our cryostat, Quantum Design Physical Properties
Measurement System (PPMS model 6000) at 15K. The system includes a
superconducting magnet capable of generating a magnetic field up to 9 T. The angle of
the sample relative to the magnetic field was defined using the PPMS rotatable sample
holder (Figure 6). The measurements were done either by sweeping fields at constant
angle or rotating the sample at constant field. The samples were rotated in the XY, XZ
and YZ planes (see Figure 8b) with different magnitudes, between 0T to 8T.
(a)
(b)
HE
Z
Y
I
X
MR
Figure 8: (a) A photo of a 50um bridge sample. (b) A schematic illustration of the defined axes relative to
the field, the MR and the Hall Effect 4 probe measurement.
By bonding different bridges on the device, we could change the length of the
measured area, and determine more accurately the changes in the domains structure
on different width scale devices.
Additional Characterization methods
Magnetic Force Microscopy
Magnetic Force Microscopy (MFM) is an atomic force microscope based technique. In
principle, an oscillating magnetic tip change its resonance frequency when subject to a
magnetic field gradient in the direction of oscillations. By imaging the change in
resonance frequency or amplitude while scanning the surface of a magnetic sample,
the MFM maps the out-of-plane component of the magnetic field. Measurements
were performed in an attoCUBE MFM-I system, with magnetic lateral resolution of ~20
nm. X-ray diffraction
X-ray diffraction (XRD) measurement can provide information of the crystallographic structure
and properties of a sample by The intensity of an x-ray beam is measured after reflecting from
the samples surface at a well-defined angle. The intensity is a function of the interference
pattern the beam experiences, providing sub angstrom information of the (averaged) ordered
(crystal) atomic distances. By measuring low angle refraction one can characterize multilayered structure parameters, such as the individual layer thickness and the interface
roughness.
Numerical analysis

Introduce effective model

Refinement process

Error analysis
By modeling the measurement to single domain magnetism, we develop a refinement
process on the full magnetization orientation data-set and derive from the MR
measurements the anisotropy energy, and identify when the single domain model
breaks down. We also measure 200nm electrodes, proving that the electrodes
maintain their anisotropy energy also in small scale, and as expected it is much larger
than the shape anisotropy.
Results
We had two ways to prepare the electrodes as shown in Figures 5 & 6, the etching
process (the second process) was the one we used in this experiment. The first process
(direct deposition) was easier to perform, but it modified and degraded the magnetic
properties of the electrode, which not occur with the etching process.
Magnetization vs. magnetic field measurement was performed for films with different
Co/ Pd thickness ratio and repetitions. We found that the anisotropy properties of the
films changes with these conditions. The SQUID measurement of 4 different films at
100K, in fields parallel and perpendicular to the film’s plane are presented in figure 9.
The eminence of the PMA could be verified by measuring high remnant magnetization
and square hysteresis loop for perpendicular measurements, and the lack off, for in
plane measurements.
Normalized Magnetization
1.0
0.5
Co/Pd (nm)
0.33/1
0.41/1
0.26/1
0.33/1.2
1.0
0.5
Co/Pd (nm)
0.33/1
0.41/1
0.26/1
0.33/1.2
0.0
0.0
(a)
-0.5
-0.5
-1.0
-1.0
-1.0
-0.5
0.0
H(T)
0.5
1.0
-2.0
(b)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
H(T)
Figure 9: SQUID measurements perpendicular (a) and parallel (b) to the films plane for films with different
Co/ Pd thickness ratio and repetitions.
The best PMA sample was found in the SQUID measurement for film structure of:
Pd (4nm) / [Co (0.33nm)/ Pd (0.12nm)]17/ Pd (4nm).
When comparing the SQUID measurement and the AHE measurement on the 50u
Bridge (figure 10), the perpendicular easy axis is maintained as the AHE measurement
is reflecting the magnetic state of the multilayer device correspondingly to the SQUID
measurement of the film [15].
1.0
1.0
R()
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Magnetization
Vertical
Horizontal
PPMS Hall
1.2
H(T)
Figure 10: SQUID measurement and PPMS AHE measurement comparison of
[Pd (4nm) / [Co (0.33nm)/ Pd (0.12nm)]17/ Pd (4nm)]multilayer film and bridge respectively.
In figure 11 we present magnetoresistance measurements of a typical sample as a
function of magnetic field applied along the three main axes: parallel to the current,
H|| (x-axis); in the film plane and perpendicular to the current, HIP (y-axis); and normal
to both the current and the film plane HOP (z-axis).
5.58
 (cm)
5.55
(H||)
5.52
5.49
(HOP)
5.46
Z
Y
X
5.43
5.40
I
(HIP)
-4
-2
0
2
4
H (T)
Figure 11: Magnetoresistance as a function of external field applied along the x, y and z
axis (red circles, blue triangles and black squares respectively).
For all three curves exhibited in figure 11, at high enough fields the resistivity reduces
linearly with the external field due to spin-magnon scattering. [16, 17]. When the
magnetic field is applied normal to the plane, the magnetization which is
spontaneously oriented perpendicular to the plane, changes its direction around 0.15
Tesla where a small peak in resistivity occurs. This peak is attributed to magnetic
domains breaking during switching of the magnetization which leads to resistivity
increase as a result of domain walls scattering [18, 19]. The rest of the R vs. H curve
shows only spin-magnon scattering.
At Zero field, || (red squares) and IP (blue triangles) show the same value. This is a
manifestation of the PMA, where at H=0, the magnetization of the sample lies out of
plane, and the resistivity is similar to that of a demagnetized sample. When increasing
the field the resistivity of || (IP) increases (decreases) until reaching a maximum
(shoulder) at Hsaturation ~ 2 Tesla (~ 2.5 Tesla), The curve for || is well described by AMR,
in which the resistivity depends on the angle between the magnetization and
current.[16] Since the film has perpendicular magnetization at H=0, increasing the
external field tilts the magnetization toward the current direction, increasing the film
resistance. At higher fields (above 2 Tesla), the magnetization aligns along the external
field and the magnon negative slope is obtained. As described before, the channel
length is longer than its width. As a result, the channel has a shape anisotropy which
drags the magnetization along the current direction. Therefore, to reach saturation in
the IP direction a higher external field is needed than for the parallel direction.
The magnetoresistance measured for IP does not refer to the conventional AMR. In
this regime, despite the fact that the magnetization is always vertical to the current
the resistivity reduces by almost 2 percent, twice as large as the AMR effect presented
above. This AMR is different than that observed in other thin film structures [20, 21]
where OP is larger than IP.
In order to follow the resistivity behavior of the whole regime, we applied a constant
magnetic field of 8T at x, y, z axes and measured the resistivity as a function of 𝜃 or 𝜑
defined as the angle between the applied magnetic field, B, and the z or y axis
respectively, see the three cartoons in Fig. 12.
Figure 12: Resistivity versus magnetic field direction scanned around a major axis. Black triangles (a): φ=0, z to –z around y, red circles
(b): φ=90, z to –z around x, blue squares (c): θ=90, y to –y around x. The solid lines are fits to Eq. 1.
In Figure 12 we show the resistivty as a function of external field angle for H=8T at 15K.
The red black curve, which shows the data for the rotation presented in Fig.12a,
resistivity reduces when magnetic field tilts from z to y axis and returns to its initial
value when 𝜃 = 180. By increasing 𝜃, starting from z axis toward x axis (curve 12b) the
film resistance increases as the magnetic field approaches the current in the x axis.
When the magnetic field oriented in the film plane (𝜃 = 90), B rotates around the x
axis from y to –y, increasing 𝜑 from 0 to 180. The resistance increases with 𝜑 till 𝜑 =
90, in which the magnetic field and current are parallel. The resistance reduces
symmetrically when 90 > 𝜑 > 180.
The resistance changes in curves 12b and 12c is well understood as being due to AMR
effect, consistently with the curves presented in figure 11. In the absence of an
external magnetic field the magnetization preference is perpendicular to the film (z
axis). A positive contribution in the resistance is expected when the magnetization
follows the magnetic field which is rotated toward the current direction. As said
before, the case presented in curve 12a is less conventional, considering the fact that
the angle between the magnetization and the current stays constant. Geometrical size
effects can cause different resistance value in the perpendicular and transverse
direction [9, 22, 23]. However, in this effect, exhibited in Co, Ni and Permalloy, the
transverse resistance is larger than the perpendicular. Recently, Kobs et. al. [24]
reported an opposite effect, similar to our results, in Pt/Co/Pt sandwiches. They
attribute the increase in resistivity when the magnetization rotate from perpendicular
to transverse direction to a scattering mechanism of electrons at the interface, which
they named anisotropic interface magnetoresistance (AIMR). The origin for this effect
is still not clear. Nevertheless, we can see that also in the case of multilayers, the AIMR
effect governs.
H
0.4
-H
HAverage
Y
Z
(R-R0)/R0 ()
0.3
0.2
1000nm
1000nm
0.1
0.0
Y
Z
-10
0
10
20
30
40
50
60
70
80
90
100
angle
After understanding the origin of the magnetoresistance behavior, we can get
magnetic properties, as anisotropic constants, by minimized the free energy of the
system. This way that will be following described, allows us to extract the magnetic
state of system by doing simple resistivity measurements.
Discussion
In order to characterize the resistivity behavior we write a first order,
phenomenological equation for the dependence of resistivity on the magnetic
orientation. We expand the usual in plane AMR to include the OP component, possible
because of the symetery breaking induced by the multilayer structure (this is similar to
a crystolographic MR term):
𝜌(𝜃 ∗ , 𝜑 ∗ ) = (𝜌𝑥 𝑠𝑖𝑛2 𝜑 ∗ + 𝜌𝑦 𝑐𝑜𝑠 2 𝜑 ∗ )𝑠𝑖𝑛2 𝜃 ∗ + 𝜌𝑧 𝑐𝑜𝑠 2 𝜃 ∗ ,
(1)
Where 𝜃 ∗ and 𝜑 ∗ are the angles between the magnetization, M, and the z or y axis
respectively, and 𝜌𝑖 is the resistivity along i (i=x, y or z) axis. As opposed to Kobs et al,
our results do not fit the 𝑐𝑜𝑠 2 form, indicating that there is a difference between the
magnetization and applied magnetic field, even for the high field applied. We can still
consider a coherent rotation of the magnetization; and use the Stoner–Wohlfarth (SW)
model [25]. For every rotation in the experimental setup (see cartoons in Fig. 12) we
express the free magnetic energy considering the perpendicular anisotropy and the
shape anisotropy arising from the narrow channel measured:
(2a)
𝐹𝑎 = −𝐾𝑢 𝑐𝑜𝑠 2 𝜃 ∗ − 𝑀𝑆 𝐵𝑠𝑖𝑛(𝜃 ∗ − 𝜃)
(2b)
𝐹𝑏 = −(𝐾𝑢 − 𝐾𝑠ℎ )𝑐𝑜𝑠 2 𝜃 ∗ − 𝑀𝑆 𝐵𝑠𝑖𝑛(𝜃 ∗ − 𝜃)
(2c)
𝐹𝑐 = 𝐾𝑠ℎ 𝑐𝑜𝑠 2 𝜑 ∗ − 𝑀𝑆 𝐵𝑠𝑖𝑛(𝜑 ∗ − 𝜑)
𝐾𝑢 , 𝐾𝑠ℎ are the perpendicular and shape anisotropic constants respectively and 𝑀𝑆 is
the saturated magnetization measured by SQUID and equal to 520 emu/cm3. We
extract the magnetization direction, 𝜃 ∗ and 𝜑 ∗ , for every external field orientation
(expressed by 𝜃 and 𝜑) and for a particular choice of the anisotropy constants, by
numerically minimizing the free energy equations. In the inset of figure 13, the
difference between applied magnetic field and magnetization direction, ∆𝜃 = 𝜃 − 𝜃 ∗ ,
is plotted versus 𝜃, for the rotation plotted in figure 12a. Untill 𝜃 = 52° the shift
between magnetic field and magnetization increases indicating that even in applied
field of 8T the perpendicular anisotropy is strong enough to attract the magnetization
toward the z axis. When the applied field rotates toward the film plane, the difference,
∆𝜃, decreases and becomes zero when the magnetic field is applied in the y axis. By
increasing 𝜃 above 90°, ∆𝜃 becomes negative which means that 𝜃 ∗ is bigger than 𝜃
because the perpendicular anisotropic toward the –z axis. We calculate the expected
magntoresistivity by inserting the extracted values of 𝜃 ∗ or 𝜑 ∗ in eq. (1).
Next, we perform a refinement process in which we search for the numerical results
that minimize the difference from the measured magnetoresistance. we can extract
magnetic parameters of the multilayers. The solid lines in figure 12 are fits to the
calculated resistivity versus magnetic field direction 𝜃 or 𝜑 where the anisotropic
constant, 𝐾𝑢 and 𝐾𝑠ℎ , are the only refinement parameters. The extracted shape
anisotropic constant is equal to 95 KJ/m3. We calculated the shape anisotropic
constant of ellipsoid having major axis identical to our channels size [26]. For channel
of 50 𝜇𝑚 width, the anisotropic constant obtained is equal to 139.4 KJ/m3 which is a
good approximation to our extracted value. The perpendicular anisotropic constant we
received by refinement is 𝐾𝑢 = 630 KJ/m3 which is also reported for Co/Pd multilayers
measured by different methods [20, 27].
5.58
5.56




5.52
(deg)
R ()
5.54
5.50
8
6
4
2
0
-2
-4
-6
-8
0
5.48
0
30
60
90
120
30
60
90 120 150 180
(deg)
150
180
angle (deg)
Figure 13: Resistivity versus magnetic field direction scanned around the y axis (identical to the 2a position) at
different values of magnetic field. The solid lines are fits to Eq. 1 after refinement. The inset shows the shift
between the applied magnetic field and magnetization direction, ∆𝜽 = 𝜽 − 𝜽∗ as a function of the magnetic field
direction, 𝜽 at magnetic field equal to 8T.
We point out that the calculation was done under the assumption that the
magnetization reverses coherently, which allows us to use the SW model. However,
this assumption seems to be broken when small magnetic field are applied. Figure 13
presents the resistivity versus 𝜃 under different magnetic fields rotated from z to -z
axis around the y axis, after subtracting the magnon slope. The channel measured in fig
13 is a "twin" sample of the channel shown in fig 12, both were taken from the same
wafer. Therefore, the solid lines in fig 12 and 13 are calculated by substitution the
same anisotropic constants, 𝐾𝑢 and 𝐾𝑠ℎ . It can be seen that the calculated curves
follow the measured curves until the applied magnetic field oriented close to the
plane. We suggest that this deviation from the SW model results from magnetic
domains breaking. As the applied field becomes close to the y axis the magnetic
moment prefers breaking to several magnetic domains [28]. At this state the resistance
is higher because of GMR effect results from scattering from domains wall. It can be
seen in fig 3 that the lower the field the deviation from calculation is bigger until
applied magnetic field of 1T in which SW model is not describe the system at all.
It can be seen that the perpendicular anisotropic is five times bigger than the shape
anisotropic explaining the fact that even in a magnetic field of 8T the magnetization
and the applied field are not in the same direction (𝜃 ≠ 𝜃 ∗ ).
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