1. Imaging of domain walls in small shape anisotropy
dominated magnetic structures
Michael Beljaars
May 9, 2005
2. Summary
In the development of fast magnetic memories (MRAMs) the ordering of
magnetic materials in domains is of great importance. The switching of
the magnetization of a memory element by means of the displacement of a
domain wall is the most important motivation for the research. To prevent
the existence of multiple domain walls within the element it is important
to create a structure with a preference for a single domain configuration.
When applying a rotating magnetic field to such a structure, a domain wall
can be induced. The final step is to propagate this domain wall using con-
trolled current pulses. Because of the extensive dimension of this research
only the search for a single domain configuration is subject to this internship.
In this experiment Magnetic Force Microscopy is used to examine the do-
main configurations in magnetic structures. The results from MFM highly
depend on the characteristics of the device itself and the magnetic probe,
the tip, used to scan the sample. Therefore an extensive amount of scans
are made to study the behavior of the MFM. Two kind of tips are examined,
MFM tips bought at Nanoworld [1] and contact tips bought at NT-MDT
[2], which are coated with a magnetic layer by means of sputtering. With
the Nanoworld tips, the whole system (tip, cantilever, cantilever holder) is
coated with a magnetic layer, whereas with the sputtered tips only the tip
is partially provided with a magnetic layer. This difference leads to a higher
sensitivity for the sputtered tips as can be seen in the scans. Furthermore
an important conclusion that can be drawn from this research is that still
few is known about the tip and tip-sample interaction. More insight in this
system makes it possible to draw more conclusions out of MFM scans.
The first step towards such a memory element is to create a structure small
enough to display single domain formation. Therefore a sample with small
shape anisotropy dominated CoFe structures is created using e-beam litho-
graphy. The structures of the first sample are approximately 1500 nm wide.
MFM scans of these structures reveal a multi domain configuration. Non the
less these scans are valuable as they confirm the possibility of the accurately
mapping of domain structures.
1
3. A second sample is created in the same way as the first, containing smaller
versions of the same structures. Also some alternative structures with elec-
trical contacts are made which may be suitable for the next step of the
research: the actual manipulation of a domain wall. MFM scans of one of
the last mentioned structures indicates single domain formation in a part of
the structure that is less than 1200 nm wide.
2
5. Chapter 1
Introduction
In the development of new high density storage media the dynamics of mag-
netization play a great role. With the development of research techniques
as Magnetic Force Microscopy (MFM) new possibilities to explore the or-
dering of magnetic structures on microscopic level have become available.
Although MFM provides high resolution images of static magnetization con-
figurations, it can not image the dynamics itself. By means of a discreet field
sweep and making scans at the different fields one can acquire insight in the
dynamic processes of magnetization.
One of the key features of magnetic materials that can be useful for appli-
cations is the ordering in magnetic domains, in particular the domain walls
that separate the domains. Above a certain characteristic length scale of
the magnetic structure, typically 1 µm, the formation of domains is a en-
ergetically favorable situation. Structures below that limit favor a uniform
magnetization, unless the form of the structure is chosen in a smart way.
Because of shape anisotropy such a structure can develop two domains sep-
arated by a single domain wall if an external field is applied in the right
direction.
Domain walls can be manipulated by an external field, but also by applying
current pulses perpendicular to the domain wall. This effect is called the
spin torque effect. It is due to the transfer of spin torque from the electrons
to the domain wall. A controlled shifting of the domain wall is possible given
the right length and intensity of the current pulses. Combining domain wall
movement with a Tunneling Magneto Resistance (TMR), one creates a elec-
trically switchable Magnetic Tunnel Junction (MTJ). This structure has all
the features necessary to create the ideal memory (Magnetic Random Ac-
cess Memory or MRAM): fast access time, fast read / write time, persistent
4
6. without power and no wear out.
The first step towards the development of these MRAMs is to creature shape
anisotropy dominated structures that are small enough to naturally display
single domain formation. Therefore special shaped magnetic structures are
created with e-beam lithography combined with sputtering and subsequently
scanned with the MFM. Then, by applying an external field in the right
direction, a domain wall should be created. A scan with the MFM will
provide the proof for the presence of a domain wall.
In chapter two a overview of the theory on the AFM / MFM setup is given
followed by a short introduction on magnetic domains and the relevant mech-
anisms that play a role in the formation of these domains. Chapter three
describes the actual setup that is used and subsequently the possible artifacts
that must be accounted for. In chapter four the results of the experiment are
published, beginning with some example scans made to explore the setup. In
the following subsection the behavior and resolution of the used tips (both
spatial as magnetic) is examined. The chapter ends with the presentation
of the results on two samples, E-beam sample I and II, specially created to
investigate the single domain formation. The report is concluded in chapter
five with a discussion on the results and a overall conclusion.
5
7. Chapter 2
Theory
2.1 Scanning Force Microscopy
2.1.1 Overview
Scanning Force Microscopy (SFM) is undoubtedly the most important break-
through in the imaging of surfaces of the 20th century. Using the right
equipment and settings it is possible to even make scans with atomic reso-
lution.
The general idea of this technique is to scan a surface, using a probe. This
probe, normally referred to as the ‘tip’, is mounted on a holder, called the
‘cantilever’. The tip can interact on various ways with the surface. It can
be used for example to measure the interaction due to the Coulomb force
or the Van der Waals forces. This gives an image of the topography of the
sample. But also the imaging of local electric or magnetic forces is possible,
given the right probe. The relevancy of the different forces for scanning
probe microscopy is described in the subsection Related forces.
The interaction of the tip with the sample causes a change in the orientation
of the cantilever, which can be measured in several ways. These are discussed
in the subsection Measure cantilever deflection.
Also there are a few different manners to probe the sample. Depending on
which interaction is to be measured and what kind of tip is used, certain
modes are suitable, where others aren’t. In the subsection Different modes
a variety of modes and their appliances is discussed.
6
8. Because the extensive area of possibilities only the relevant part of SFM is
discussed. Mainly Atomic Force Microscopy (AFM) in semi-contact mode
and AC Magnetic Force Microscopy (AC MFM) are relevant for the exper-
iments and are therefore are dealt with more extensively. For a full outline
on SFM it is advised to read Scanning Tunneling Microscopy II [5] and the
information on SFM on the website of NT-MDT [2].
2.1.2 Measure cantilever deflection
The deflection of the cantilever is due to the interaction of the tip with the
surface. The relation between the interacting force F and the deflection δz
is given by The Hooke’s Law
F = c · δz (2.1)
where c is the force constant of the cantilever. The magnitude of the force
constant depends on the dimensions of the cantilever as well as it’s material
and the temperature. The latter isn’t relevant because the measuring occurs
at constant (room) temperature.
To measure the deflection several techniques, like tunneling current method,
optical interference, capacitance method and laser beam deflection can be
used. The last mentioned is used in the experiments. The functioning of
this method is displayed schematically in figure 2.1.2.
Because of the deflection of the cantilever, the reflection of a laser beam on
the rear side of the cantilever changes in orientation. A position-sensitive
detector (PSD) senses the location of the reflected beam. Trough calibra-
tion the signal of the PSD can be used to ascribe a absolute value to the
deflection. In comparison to other techniques the Laser Beam Deflection
method is very basic and easy to apply. It’s influence on the cantilever is
negligible and gives a reasonably good resolution. Furthermore it does not
need a very clean (vacuum) environment like the tunneling current method.
7
9. Figure 2.1: Laser Beam Deflection
2.1.3 Related forces
Van der Waals forces
The attractive Van der Waals Forces are due to the interaction of electric
dipoles and exist between all atoms or molecules. The interaction of these
forces is used to make topographic scans of surface There are basically three
types of such an interaction:
• permanent dipoles interacting by a dipole-dipole interaction
• the induction of a dipole in a non-polar molecule by a permanent
dipole, therefore creating a interaction between the permanent and
the induced dipole
• spontaneous dipole creation due to fluctuations in charge distribution,
with the possibility to induce another dipole in a non-polar molecule,
which can interact with the permanent dipole.
The Van der Waals Forces are relevant for distances between a few and a few
hundred ˚Angstroms. As shown in figure 2.1.3 at short distances the overlap
of electron orbits causes a repulsive interaction, when at large distances the
dipole interaction causes a attractive force.
Capillary forces
Because of the small curvature of the tip a meniscus of water can be formed
around it. Typically a tip of 1000 ˚A or less is a nucleus of condensation,
8
10. Figure 2.2: Van der Waals potential curve
providing that the environment contains vapor. The formation of this wa-
ter layer causes an additional force between the surface and the tip. An
approximation of this force is given by the equation
F =
4πRγ cos Θ
(1 + s/(R(1 − cos φ))
(2.2)
where γ is the surface tension, R the radius of curvature, Θ the contact angle,
s the distance between tip and sample and φ the angle of the meniscus as
shown in figure 2.1.3.
Figure 2.3: Schematic drawing of the tip
Given this equation a value for the maximum force
Fmax = 4πRγ cos Θ (2.3)
can be derived. For a tip with a curvature of 1000 ˚A a maximum force
Fmax = 9.3 · 10−8N is found. This relatively large in comparison to the
typical operating forces, which are of the order of 10−7) to 10−8 N.
9
11. Magnetic forces
Providing the AFM with a magnetic tip it becomes possible to image mag-
netic structures on surfaces. Because of the long range of magnetic forces
scans can be performed at several hundreds of ˚Angstroms distance from the
surface. If the distance to the surface is maintained constant, the Van der
Waals forces do not change as function of the place above the sample. In
figure 2.1.3 and figure 2.1.3 these forces are visualized. Figure 2.1.3 shows
the distance dependance of the topographic forces. The lighter the color,
the weaker the force is. As appears in the figure, the force is constant for a
certain fixed distance to the surface.
Figure 2.4: Distance dependence of the Van der Waals forces
The magnetic forces however still vary as function of the place over the sam-
ple due to the difference in magnetization at the surface of the sample (see
figure 2.1.3). The colors assigned to the different magnetization direction are
arbitrarily chosen. Furthermore for simplification only four possible mag-
netization directions are used. The colors representing magnetization are
drawn lighter as the distance to the sample increases, indicating that also
the strength of magnetic forces diminishes with distance. The difference
in magnetization depending on the position over the sample nevertheless
remains.
Figure 2.5: Distance dependance of the magnetic forces
10
12. Electrostatic forces
Similar to the imaging of magnetic forces it is possible to scan a surface mea-
suring the electrostatic interactions between a charged tip and the sample.
This scanning method is mainly useful for measurement at insulators. Prac-
tice has proven it is possible to measure individual electrons or currents down
to 10−19A. In this experiment a possible influence of electrostatic forces is
not desirable, so the device is provided with an earth connection. This rules
out the possibility of images features caused by electrostatic forces.
2.1.4 Different modes
The possibilities and modes of Scanning Force Microscopy are almost infi-
nite, therefore only the techniques that are used, are described more exten-
sively. Depending on the used mode an other region of the surface potential
is used for scanning as shown in figure 2.1.3.
Contact techniques
The highest spatial resolution is achieved in contact mode. Depending on
the environment and the size of the tip a atomic resolution can be achieved.
As the name suggests in this mode the tip is in constant contact with the
surface of the sample, which means that the interaction takes place in the
repulsive region of the Van der Waals force. One important condition for
this mode to work properly is a relatively hard sample, otherwise the tip
just scratches the atoms or molecules of the substrate. Biochemical samples
for example are mostly to soft to be scanned in contact mode.
Non-Contact techniques
The non-contact mode is most suitable to scan soft samples. Because the
working region is the attractive part of the potential (figure 2.1.3), the tip
doesn’t touch the surface. In this way damaging of the surface is prevented.
Unfortunately the non-contact technique has a lower spatial resolution and
sensitivity compared to contact or semi-contact techniques.
11
13. Semi-contact techniques
Semi-contact mode or Tapping mode means briefly tapping the surface while
scanning. Briefly tapping means jumping from the attractive to the repulsive
region of the surface potential (see figure 2.1.3). In this way a number
of point measurements is combined to form a image of the surface. The
advantage of this mode in comparison to the contact mode is that non or
less damage is done to the sample and the spatial resolution is better than in
non-contact mode. However the optimal resolution is still below the atomic
resolution of the contact mode. Semi-contact techniques can be divided in
two methods, which are discussed in the next two paragraphs.
Semi-contact mode In semi-contact mode the cantilever is driven in its
resonance frequency. By means of a feedback signal the amplitude of the
cantilever is maintained constant. As a result the feedback signal contains
information about the topography of the sample.
Phase Imaging mode Using a different method also other data of the
surface of the sample can be acquired. Due to the force working on the tip
there is a change in frequency, which causes a difference in phase between
the resonance frequency, in which the cantilever is driven, and the actual
frequency in which the cantilever oscillates. By looking at this phase shift
information about local forces on the surface can be obtained. Still keeping
the amplitude at a constant level, the surface is scanned. Depending on the
interaction of adhesive forces a different phase shift is to be found, giving
information about the homogeneity and composition of the surface.
Many-pass techniques
When scanning the long range interactions of a sample, like magnetic or
electric interactions, it is important to keep a certain constant distance be-
tween the tip and the surface of the sample. Maintaining a fixed distance
the Van der Waals force is constant. The only way of doing this properly is
exactly knowing the topography of the surface. Therefore in the first pass,
the topography is scanned and subsequently, in the second pass, the tip is
raised en maintained at a certain height, while scanning the same line again.
AC Magnetic Force Microscopy A sensitive way to measure magnetic
interactions is in vibrating, AC MFM, mode, using a magnetic coated tip.
12
14. In stead of detecting the interacting force, the gradient of the stray field is
mapped, which makes it possible to detect much smaller interactions than
in non vibrating DC MFM mode.
For the second pass the feedback is turned off leaving the cantilever freely
vibrating in the driven (resonance) frequency ω0. For small amplitudes
the cantilever can be approximated as a harmonic oscillator. In case of
no additional interacting force the force F on the end of the cantilever is
proportional to the amplitude z and is given by
F = −k0z (2.4)
where k0 is the spring constant. The resonance frequency ω0 of the system
in terms of the spring constant k0 and the mass m is then given by
ω0 =
k0
m
. (2.5)
When a magnetic force acts on the tip, the spring constant is altered in
accordance to
k = k0 +
dFm
dz
. (2.6)
Substituting this in the equation for the resonance frequency, the shift of
the resonance frequency is found.
ω = ω0 1 +
1
k0
dFm
dz
(2.7)
For small gradients of the force the square root can be approximated by
1 +
1
k0
dFm
dz
≈ 1 +
1
2k0
dFm
dz
(2.8)
resulting in a shift of the resonance frequency given by
∆ω = ω − ω0 =
1
2k0
dFm
dz
ω0. (2.9)
13
15. To determine this shift the new resonance frequency has to be redetermined
for every point of measurement. That is why this method is a time con-
suming way of scanning a magnetic surface. Therefore normally not the
frequency shift, but the shift in phase is used as a measure for the gradient,
which is given by
∆φ =
Q
k0
dFm
dz
(2.10)
The magnitude of the phase shift clearly depends on the quality factor Q,
which is a measure to what degree the system retains its energy. For a high
sensitivity a good quality factor is required. Figure 2.6 shows the phase shift
for a relatively high and a relatively low quality factor.
Figure 2.6:
The magnetic force is caused by the interaction of the stray field H of a
sample with the magnetization M of the tip and is given by the equation
F = (M · H). (2.11)
Approximating the tip as a magnetic dipole, there are two cases to con-
sider: (1) the perpendicular Mz and (2) the parallel alignment Mxy of the
magnetization of the tip to the sample surface. The first case relates the z
derivative of the force to the z component of the stray field by
14
16. dFm
dz
= Mz
d2Hz
dz2
(2.12)
whereas the second case gives the equation
dFm
dz
= cos γMxy
d2Hxy
z2
(2.13)
relating the z derivative of the force and the in plane component of the stray
field Hxy, where γ is the angle between the magnetization of the tip Mxy
and the stray field Hxy. In both cases the z derivative of the force is related
to the second derivative of the stray field.
Because in this mode the MFM is only sensitive to the gradient the best
contrast is seen where there’s a lot of divergence in the stray field. Because
the divergence of Hxy is generally small compared to Hz most contributions
of the signal will be due to the z component of the stray field. Therefore
best contrast may be expected near transitions in magnetization, where the
stray field comes out or goes into the sample.
2.2 Magnetic materials
Many books are written about the extensive subject of magnetism and the
origins of domains that it would be too much to discuss all theory here.
Therefore only a qualitative treatment of the subject is given here, which is
basically enough to provide a logical explanation of the results. For more
information it is recommended to read [4].
The magnetic properties of materials originate from the magnetic moment of
the electrons in the materials. This moment consists of a magnetic moment
due to the angular motion of the electron around the nucleus and of an
intrinsic magnetic moment which can be associated with the spin of the
electron.
Depending on the filling of the atomic shells and the density of states at
the Fermi level, a material can have a certain magnetic character like dia-
magnetic, paramagnetic or ferromagnetic. At room temperature the CoFe
used in the experiment belongs to the latter category. Above the so called
Curie temperature ferromagnetic materials become paramagnetic. For Co
15
17. and Fe this temperature is much higher than room temperature. Ferromag-
netic materials have a net magnetic moment and align parallel to an external
field.
The way in which the density of states is filled determines several charac-
teristics. One of these characteristics that is of importance to evaluate the
mutual influence of magnetic systems is coercivity. The coercivity is related
to the field necessary to reduce the materials natural magnetization to zero.
A material with large / small coercivity is often referred to as a hard / soft
magnetic material. Thus a soft magnetic material is easily influenced by a
external magnetic field, whereas a hard magnetic material is persistent in
its magnetic configuration.
As all system encountered in physics until today, a magnetic system evolves
towards a minimum energy given its conditions. For a magnetic material
energy contributions from several effects are important to consider.
The preference to align with structural axes of the sample is called anisotropy.
The energy associated with this effect depends on the direction of the mag-
netization relative to the specific axis and has its minimum if the magne-
tization aligns with this axis. The most important anisotropous effects are
crystal anisotropy (the preference to align with crystal axis), surface and
interface anisotropy and exchange anisotropy (the preference to align with
neighboring spins).
Ferromagnetic structures tend evolve to a equilibrium with a constant mag-
netization direction. In this case all magnetic moments are aligned. Any
deviation of this ideal case will cause a increase in so called exchange energy.
This energy is due to the interaction of neighboring spins.
The magnetic field energy of a magnetic sample can be divided in two parts.
The first is the applied field energy, also called the Zeeman energy, which is
caused by the interaction of the magnetization vector field with an external
field. The second part is the stray field energy. This is the field generated
by the magnetic structure itself. An effective way to minimize this field is
flux closure. This means that the magnetization aligns in such a way that
most of the magnetic flux is kept within the sample. This effect makes the
origination of domains a energetically favorable situation.
Magnetic domains are regions of uniform magnetization and appear even
in unstructured magnetic samples. At the border of the domains, so called
domain walls are formed. In these domain walls, which are typically several
hundreds of lattice constant in width, the magnetization gradually changes
direction. Although it takes some energy to form domain walls, in many
cases the formation of domains (and thus domain walls) reduces the stray
16
18. field energy more than the system gains on domain wall energy. The figures
2.7 to 2.9 show some examples of possible domains.
Figure 2.7: Magnetic structures of permalloy (left: topography, right: mag-
netic image)
Figure 2.8: Magnetic structures of permalloy (zoom) (left: topography,
right: magnetic image)
Figure 2.9: Magnetic domains of garnet film
Below a certain spatial limit the formation of domains is not energetically
17
19. favorable any more. A structures with one or more dimensions below this
limit will have a uniform magnetization direction. However domains can be
forced into these structures if their shape anisotropy is large enough to allows
for this. Therefore if the shape and dimensions of a structure are smartly
chosen, an external field applied in the right direction should induce a single
domain wall in the structure.
18
20. Chapter 3
Experimental setup
3.1 AFM Setup
For the experiments the Solver P47H Pro of the Russian company NT-MDT
is used. To minimize distortions due to floor vibrations the device is placed
on a actively dimmed table which itself is placed on a heavy stone table.
(a) (b)
Figure 3.1: a)Overview of the setup and b) a zoom in on the actual AFM
setup.
The base of the Solver P47H Pro contains a sample holder driven by a step
engine, which automatically brings the sample to the tip until a certain value
of the feedback signal, the setpoint, is reached. On top of the base, above
the sample holder, the actual scan head of the AFM (see figure 3.2) is found.
19
21. Figure 3.2: The AFM head
The setup contains a camera which images the sample trough a mirror in
the AFM head. The image of the camera is displayed on a monitor. This
image is very useful for placing the laser spot correctly on the cantilever and
to search the right area of the sample.
The sample holder can be equipped with a electromagnet. This electromag-
net consists of a pair of poles, creating a field parallel to the surface. The
maximum field that can be applied is approximately 500 Gauss.
A Hall probe is used to determine the strength and direction of the applied
field.The AFM is electronically connected to a computer to collect, store
and analyze the data.
3.2 Magnetic tips
Two different types of magnetic tips are used in the experiment. One type
of the magnetic tips is bought at Nanoworld [1]. These tips are coated with
CoFe, which is a relatively hard magnetic material. The silicon cantilever
of these tips is also covered with a CoFe layer.
The other type is created from normal non-magnetic tips from NT-MDT [2]
are provided with a magnetic layer. For deposition of the magnetic layer a
sputter mask is used. The mask can contain 4 cantilevers with tips. The
cantilevers are mounted vertical in the mask in such a way that the front
side of the tip is aligned towards the opening in the mask (see figure 3.3).
The mask with cantilevers is placed in a sputter machine. By aiming high
energy particles on a target sample material is sputtered. Mostly an Argon
20
22. plasm is used for this purpose, because Argon is an inert gas. Therefore it
doesn’t attach to the sample surface. Part of this material damp evaporated
by the Argon floats trough the mask and condenses on the tip, forming a
layer of the desired material.
Figure 3.3: Deposition of a (magnetic) layer on a tip by sputtering.
3.3 Artifacts
It’s important to realize that the images created with AFM / MFM are
not always representing real structures or magnetic properties. To draw
conclusions from these pictures a critical look is therefore necessary. There
are a lot of different artifacts that can occur. These are discussed in the
following sections.
3.3.1 Probe artifacts
The shape of the tip has a large impact on the final image that is obtained.
Depending on the characteristics of the tip, a surface object can look bigger,
less deep, deformed or appear multiple times.
Figure 3.4 gives an example of how a structure looks bigger than it actually
is. When the curvature of the tip is of the order of the size of the structure,
the image of the structure is magnified. If the curvature is one or more
21
23. order of magnitude smaller than the structure being imaged, this effect can
be neglected.
Figure 3.4: Depending on the tip curvature a object can look bigger.
The width of the tip is of importance when it comes to measuring features
that are below the surface (see figure 3.5). In this case the tip gets stuck
because of it’s width before it reaches the bottom of the pit.
Figure 3.5: Tip artifact affecting the measured depth.
Furthermore performing a scan correlates the tip with the surface, creating
strange fake features that originate from the shape of the tip as shown in
figure 3.6.
Figure 3.6: Tip artifact affecting the shape of features.
Small features that are scanned width a relatively large tip reflect the geom-
etry of the tip, rather than their own. In figure 3.7 a example of a so called
double tip is shown. This defect causes an object to appear double in the
scan image, first scanned by the front tip , second by the tip on the back.
The basic rule for artifacts caused by the geometry of the tip is when the
tip curvature is small compared to the structures that are to be imaged, the
influence of these artifacts is small.
22
24. Figure 3.7: Double tip causing multiple appearances of the same object.
3.3.2 Scanner artifacts
When evaluating the results of a AFM / MFM setup one should consider the
whole system because also the characteristics of the scanner will be visible
in the images.
One of the most common scanner artifacts is due to a too high scanning
speed. In this case the system is to slow to keep track of the surface. De-
pending on the scan direction, structures in the image will appear sharp on
one side but blurred on the other side because of a artificial slope in the
signal as shown in figure 3.8. The extra bump in the signal is called an edge
overshoot and is a very common AFM artifact.
Figure 3.8: With the scan made in the positive x direction, the slope on the
right side is much steeper than on the left side of the structure.
An other kind of scanner artifact can occur when the angle between the
cantilever and the sample is too large. The side of the tip hits the structure,
instead of the point of the tip, which leads to a distorted images as shown
in figure 3.9.
Figure 3.9: Because of a too large angle, the tip hits the structure at the
beginning with its side.
To avoid deformed images (see figure 3.10) it is necessary to be able to count
23
25. on the linearity of the movements of the scanner. Although one can correct
for possible non-linearities in the piezoelectric elements, other artifacts may
arise from the non-linearity that can not be accounted for. Therefore it
is preferable to work with a (near) linear system. To be able to relate
dimensions to scanned structures the movements of the X and Y piezoelectric
elements have to be properly calibrated.
(a) (b)
Figure 3.10: Providing a test grid with squares, a linear scanner (a) and a
scanner with a non-linearity in the X direction (b).
In the same way the Z direction should be calibrated and act linear to
guarantee non-deformed images representing real heights.
Because of the curved way the piezoelectric elements move the cantilever
over the surface during a scan, there is a certain background bow to be seen
when scanning a large area (see figure 3.12). This effect is unavoidable but
can be compensated by subtracting the background from the image. Non
the less, for the most reliable results it is better to keep the scan area limited,
thereby keeping the scan movements almost linear.
Figure 3.11: Curved motion of the cantilever, creating a background bow.
A second possible cause for an edge overshoot is hysteresis in the piezo-
electric element that controls the Z position of the cantilever. Edges of
structured samples appear higher on one side and lower on the other side
(see figure 3.12). This effect makes the image look better, because the edges
appear sharper. Non the less it is not a real feature of the sample.
Logically hysteresis can also occur in the X and Y piezoelectric elements.
This artifact also often happens at the beginning of a new scan, when the
24
26. Figure 3.12: Edge overshoot / undershoot because of hysteresis.
new area is just set. The tip has to be moved into the right position and it
takes some time to stabilize (see figure 3.13).
Figure 3.13: Hysteresis in the X and Y elements cause drift at the begin of
a new scan.
Due to changes in temperature or thermal drift in the piezoelectric elements
the scan area can be shifted over time. If the temperature during a scan
can be supposed constant, this effect is mostly seen between two successive
scans.
3.3.3 Vibrations
Because of the scale of measurement a AFM is very sensitive to both floor
and acoustic vibrations. To avoid influence from floor vibrations the AFM
setup is placed on an actively dimmed table, which itself is placed on a
heavy stone table. Non the less one should not underestimate the effect
of acoustic noise. Something little as a cough can dramatically distort the
signal. Therefore it is advisable to cover the AFM with a protective cap
while measuring.
3.3.4 Interference
When using laser beam deflection it is inevitable that some of the laser light
reflects on the surface. Unfortunately it can happen that the light is reflected
25
27. on the photodiode and interferes with the reflected light of the cantilever.
This interference creates a image of a seemingly ordered structure on the
surface, but is nothing real. To solve this problem it mostly is enough to
change the angle between the sample and the AFM head, thereby directing
the reflection away from the photodiode.
Figure 3.14: Interference from the reflection of the surface with the reflected
light of the cantilever.
3.3.5 MFM artifacts
Compared to AFM artifacts many MFM artifacts are difficult to recognize.
The first difficulty is to determine wether or not the signal that is detected is
really magnetic. Scanning high above the surface(> 3000˚A) makes it likely
to only detect magnetic signal, but mostly gives a poor signal. Scanning close
to the surface however results in a clearer signal, but makes it questionable
if the signal is purely magnetic. At these distances it is possible that the
tip slightly touches the surface. In this way not a pure magnetic signal is
imaged, but the image will contain some topographic features. Sometimes
hitting the surface gives such a distortion on the phase signal that it is
easy recognizable as an artifact. However most of the time it is not easy to
say if the magnetic scan is indeed purely magnetic or partially (or entirely)
topographic.
In explaining the behavior of the tip-sample magnetic system two proper-
ties are relevant, which are the coercivity and strength of the stray field.
The meaning of coercivity is already explained in de chapter on Magnetic
Materials. The strength of the stray field is the external field induced by a
magnetic material. It is because of these stray fields that a technique like
MFM works.
26
28. Measuring a magnetic sample with a magnetized tip, one must take in ac-
count the possible mutual influence of the tip and the sample. As follows
out of the theory on coercivity and strength of the stray field, the most de-
sirable situation is to work with a magnetically hard tip with a small stray
field. Magnetically hard means in this case that the tip has a coercive field
higher than the stray field of the scanned sample. The stray field may be
called small as long as it is lower than the coercive field of the sample. With
such a tip it is possible to scan relatively soft magnetic samples with a large
stray field without significantly influencing the tip or the sample.
In practice too little is know about the effective stray field of both tip
and sample, even as their effective coercivity. Different experiments can
be thought of to gain more insight in these properties, however this is not
the goal of this internship. Therefore, until there are methods to character-
ize the tip, the only way to determine influences of the tip on the sample
or the other way around is by looking at the scan image. Certain influences
are indeed easily to recognize out of a scan image. A common influence
artifact when scanning with a relatively soft magnetic tip is the switching of
the magnetization of the tip. It inverts the image from the scan line where
the switching took place. It’s very ease to recognize as an artifact because
of the inversion.
27
29. Chapter 4
Measure results
The AFM scans are made in semi-contact mode, while the MFM scans were
performed in AC MFM.
4.1 Exploring the setup
To explore the AFM and it’s possibilities and restrictions, at first a series of
scans on test samples is performed. The same is done for MFM with pieces
of harddisk. In the figures 4.1 and 4.2 some examples of scans are showed.
(a) 3.5 x 3.5 µm topographic image (b) 8 x 8 µm topographic image
Figure 4.1: The step structure of Aluminum Oxide (a) and Aluminum Oxide
with gold particles (b).
Figure 4.1a shows how Aluminum Oxide (AlOx) grows in a order manner
28
30. depending on the crystal lattice. Furthermore the image shows a strange
developed canyon within the AlOx, probably due to a contamination during
the grow of the sample. In figure 4.1b a scan of a AlOx sample covered with
gold particles is showed. The gold particles appear sharp at the left side and
rather blurred at the right side. This is not a real feature as discussed in the
Artifacts part of the Experimental Setup, but is caused by the apparently
too high scanning speed.
(a) 28 x 28 µm topographic image (b) 28 x 28 µm magnetic image
Figure 4.2: MFM scan of a piece of hard disk.
Because of the movement of the hard disk head on the disk scratches appear
on the surface as is to see in figure 4.2a. The bit pattern aligns with these
scratches (figure 4.2b).
4.2 Tips
The tips used for AFM / MFM form a very important part of the setup,
because the characteristics of the tip are reflected in the scan image as
discussed in the chapter on probe artifacts. To develop more understanding
in the behavior of the magnetic tips both the tips bought at Nanoworld as
the sputtered tips are tested.
4.2.1 Nanoworld tips
Figure 4.2 shows a scan of a piece of harddisk. This scan is made using a
Nanoworld tip. Evaluating the width of the smallest bit the spatial resolu-
tion is of approximately 200 nm.
29
31. Because of the pour results with one of the Nanoworld tips, an Yttrium Iron
Garnet (YIG) sample is used to test the tip. YIG formes very nice magnetic
patterns. As was to be expected no magnetic contrast can be seen in the
scan. Remagnetizing the tip does not affect the magnetic interaction with
the surface. Combining this with the fact that the tip has crashed into a
sample a few times leads to the conclusion that most likely a part of the tip
has been damaged. This can cause a dramatically reduction of the effective
magnetic moment of the tip, resulting in almost no magnetic interaction
with the sample.
With a new Nanoworld tip the YIG sample is scanned again, obtaining
the images shown in figure 4.3. Although there are clearly some artifacts
in the image, it is clear that image 4.3(b) shows the magnetic influence of
the sample on the tip. The apparently frayed magnetic structure could be
the result of the magnetic influence of the tip on the sample. But after
reducing the scanning speed, the artifact disappears. The effect is clearly a
consequence of a too high scanning speed. The system basically can’t keep
up with the changing stray field.
(a) 23 x 23 µm topographic image (b) 23 x 23 µm magnetic image
Figure 4.3: YIG sample scan with working Nanoworld tip.
Figure 4.4 shows how the magnetic signal of the defective tip is affected by
an applied field. This image is obtained by varying the applied external field
during a single scan of the YIG sample. At 0 Gauss there is only a weak
magnetic signal and when applying an increasing external field the signal
eventually entirely disappears. The pour signal at 0 Gauss implies a small
tip magnetization perpendicular to the sample surface. As the field parallel
to the sample surface is increased, the tip magnetization is canted more
parallel to the sample surface. Eventually this results in a zero perpendicular
component of the magnetization (see figure 4.5). The damaging of the tip
30
32. makes it likely for this process to happen because of breaking the symmetry
of the tip and changing its shape anisotropy.
Figure 4.4: 16 x 16 µm magnetic scan showing the field dependance of the
MFM signal. From left to right the vertical stripes belong to an applied
field of 0, 500, 0, 100, 50, 25, 10 and 0 Gauss.
The defective tip is measured in a SQUID to look for anomalies in the
magnetization curve which may explain the strange behavior. The result of
the measurement is shown in figure 4.6. The jumps that are visible in figure
4.6(a) are caused by the measurement sequence. The shifted parts (C1 to
C4) of the diagram are measured after the other parts. Probably the tip has
shifted a little bit between these measurements.
A zoom of the relevant part of the figure is displayed in figure 4.6(b). It
doesn’t show any certain jumps, in fact it is what was to be expected: the
magnetization curve of the total tip and cantilever. To get information about
the magnetic behavior of the tip alone a more local measure technique is
necessary. The Magneto Optical Kerr Effect (MOKE) could be used for
this, providing that it is possible to focus only on the tip.
4.2.2 Sputtered tips
Normal AFM tips are coated with a magnetic layer using a mask and the
sputter machine. The advantage of this technique is that only the tip will
be coated with a magnetic layer.
In a first attempt to create MFM tips non-contact tips are coated with a
layer of CoFe. For good adhesion first a thin layer of Magnesium is sputtered.
31
33. Figure 4.5: Rotation of the tip magnetization as consequence of the increas-
ing applied field.
To avoid oxidation of the CoFe the layer is inclosed in Aluminum. In table
4.1 an overview of the different sputtered layers is showed.
The tips are tested on a piece of harddisk to characterize their response to
magnetism. Unfortunately it turns out that it is practically impossible to
retain a reasonable magnetic signal with these tips. The noise is of the same
order as the actual signal. In a second attempt the same coating sequence
is used on contact tips, with more success as shows figure 4.7. This scan is
made with the first tip that was to be tested.
Both the topographic as the magnetic image show one strange feature,
namely a horizontal band. The probable cause for this artifact is a additional
particle that got stuck on the tip. This explains why the topographic image
shows a sudden change in height and why the magnetic image is blurred at
the location of the band. Because of the particle the distance of the tip to
the sample is raised with a certain amount. This increase in distance also
causes a lower spatial magnetic resolution (the blur).
32
34. (a) Total measurement range
(b) Enlargement of the rectangular area (A1 and
A2)
Figure 4.6: SQUID measurement on the defect Nanoworld tip.
After this successful scan, the e-beam sample I (see next section) is scanned
with the same tip, resulting in the image shown in figure 4.8. This image
also shows a resolution improvement in comparison to the NanoWorld tips.
A second scan sequence with the first tip astonishingly hardly gives a visible
magnetic signal, whereas the third try again results in a very clear image
with very good contrast. The other three tips appear to be useless after
several attempts to optimize the magnetic signal by adjusting the distance
to the sample and the amplitude. To understand why these tips behave so
unpredictable and why some of them don’t even work, more research has to
be done on the coating of these tips. Most probably some Scanning Electron
Microscope (SEM) scans and SQUID measurements will tell more about the
sputtered magnetic layer.
33
35. Table 4.1: Sputtered layers on the tips
Material Rate (˚A/s) Time(s) Layer thickness (˚A)
Mg 1,2 300 50,5
Al 0,5 300 21,0
CoFe 0,6 3600 300
Al 0,5 600 42,1
(a) 28 x 28 µm topographic image (b) 28 x 28 µm magnetic image
Figure 4.7: Harddisk scan with self sputtered tip.
4.3 Results
4.3.1 E-beam sample I
On a GaAs substrate 50 nm high structures are created using e-beam litho-
graphy (see figure 4.9). The design of the sample is shown in figure 4.10.
The structures [3] are created in eight sets of nine, numbered as shown in
figure 4.10. Each set consists of three structures of 10 µm width, three of 5
µm width and three of 1 µm width. The length of the legs of the structures
scales with the width, going from approximately 150 µm for the 10 µm
width to 15 µm for the 1 µm width.
The structures consist of a 90 ◦ curve with on one end a diamond shaped
pad and on the other end a sharp needle. The diamond at one side of the
34
36. (a) 4 x 8 µm topo-
graphic image
(b) 4 x 8 µm mag-
netic image
Figure 4.8: E-beam sample I scan with self sputtered tip.
structure reduces the field necessary to form a domain wall. The other side
is sharpened to make it easy for domain walls to leave the structure.
At first a topographic scan is made to check if the structures are really there
and if they have sharp borders as they should. Figure 4.11 shows this scan.
The image shows a nicely sharp edged structure without serious pollution.
As will appear in following images, due to the exposition to normal air, over
time dust particles are attracted to the sides of the structure.
Second a MFM scan is performed on one of the 1 µm structures. This scan
is shown in figure 4.12.
Clearly visible is the appearance of a multi domain structure in the corner
of the structure as well as in the diamond shape. Evidently even the 1
µm structures are not small enough to display single domain. The 5 µm
and 10 µm structures are therefore refrained from scanning. Accurately
remeasuring the width of the supposedly 1 µm structure reveals an actual
width of approximately 1 µm for the straight parts but more than 1500 nm
in the corner of the structure. Further the structures appear to be 80 nm
height in stead of the expected 50 nm.
To research the effect of an external field on the domain structure several
scans are made at different field strengths. Figure 4.13 gives an overview of
the results.
The first scan at 0 Gauss is a little bit shifted, but it is clearly visible that the
35
37. Figure 4.9: This figure explains in six steps how e-beam lithography works.
First (a) the GaAs substrate is equipped with two different layers of photo
resist, PMMA950 and PMMA450. Second (b) a focused electron beam
is used to write the desired structures in the photo resist. Because the
PMMA450 is more sensitive to the electron beam more material will be
affected as shown in the figure. The next step is to develop the photo
resist and etch the exposed material away, leaving the structure as shown in
(c). Then with sputtering the sample is provided with a layer of CoFe (d).
Because of the sputtering there will not only be CoFe on top of the sample,
but also in the etched holes. After this step the other photo resist is etched
away, leaving only the substrate with the CoFe structures and a separate
layer of CoFe, which can be removed (e). The final step (f) is to sputter a
protective coating of AlOx to prevent the CoFe from oxidation.
36
38. (a) (b)
Figure 4.10: Layout of the first with e-beam lithography created sample.
Figure 4.11: 17 x 17 µm topographic scan of one of the L structures.
structure contains domains. When in creasing the field the image doesn’t
change dramatically. The only effect visible is the shrinking of some of the
domains between -50 and -75 Gauss. When the field is raised from -150 to -
160 Gauss, suddenly al domains are gone and there is apparently one uniform
magnetization. It is true that at relatively high field, one would expect a
uniform magnetization and no domains, but this transition is expected to
be gradually by growing of some domains (the ones that are aligned with
the field) and shrinking of others. Eventually this process leads to a uniform
magnetization. That this is not the process happening here, becomes clear
when again a scan is performed at 0 Gauss. The domain structures present
in the first scan at 0 Gauss stay out. Assuming that the domains are still
present in the structure (which is very likely, at least at 0 Gauss), one
must conclude that apparently the tip is mall functioning. One possible
explanation is the partially reversal of the effective magnetization of the
37
39. (a) 25 x 25 µm topographic image (b) 25 x 25 µm magnetic image
Figure 4.12: First MFM scan of one of the smallest structures of the E-beam
sample I
tip, leading to a state in which the tip has (almost) zero net magnetization.
Another explanation for the sudden reduction of the magnetization of the
tip could be due to some damage to the tip. Because of the applied field,
the tip is more drawn to the surface, which in several cases has lead to a
crash of the tip in the sample, possibly causing damage to the tip.
As is visible in 4.13 E-beam sample I still shows multi domain formation.
This means that the dimensions of the structures are to large to favor a
single domain state. Therefore a sample with smaller structures is created
as will be discussed in the next section.
4.3.2 E-beam sample II
Even though the very beautiful domain structures that appeared in E-beam
sample II were a useful to test the spatial resolution of the MFM, it is not
a desirable situation for single domain wall manipulation. To be certain to
have a single domain wall in the structure, it has to be down scaled. In
earlier publications structures of typically 200 nm are created to observe
single domain walls. Therefore a second sample is created with e-beam
lithography. This sample contains 6 arrays of 5 by 4 L shaped structures
of different sizes and exposure times. Each array is split by a special shape
structure connected to two contacts. In figure 4.14(a) the layout of the
total sample is drawn. Figure 4.14(b) shows a close up of one of the arrays.
The square in the middle can contain one of the structures shown in figure
38
40. (a) 0 Gauss (b) -50 Gauss
(c) -75 Gauss (d) -100 Gauss
(e) -150 Gauss (f) -160 Gauss
(g) -170 Gauss (h) 0 Gauss
Figure 4.13: 5.5 x 5.5 µm magnetic image at different strengths of the
applied field.
39
41. 4.14(c), depending on the location on the sample as appears from figure
4.14(a).
(a) overview layout
(b) closeup array (c) special structures
Figure 4.14: Structure of the E-beam sample II
The widths of the L like structures are, from left to right, 0.5 µm, 0.1 µm,
0.2 µm and 1 µm. Because of the limited focusing of the used e-beam
lithography, writing structures smaller than 1 µm is very likely to fail. To
enlarge the change on success al the four sizes are created in five fold using
different intensities of the e-beam. In figure 4.14(b) this is indicated by
different shades of gray: the lighter the color, the longer the exposure.
The sample is viewed under a microscope with a maximal enlargement of
100 times. As was expected most of the L structures smaller than 1 µm
are completely gone. However some of the smaller structures had some
luck and are (partially) still in tact. Figure 4.15 shows some pictures taken
with the digital camera attached to the microscope. Because of the low
40
42. resolution of the image, it is difficult to accurately determine the width of
these structures. The best estimation that can be made ranges from 7 to 9
µm.
(a) 23 x 23 µm photograph (b) 27 x 27 µm photograph
Figure 4.15: Microscopic photographs of two of the remained structures.
Also of the special structures photos are taken. These are shown in figure
4.16. As the photos show, all structures are present and without artifacts
except structure 2. Somehow some relatively large pieces material are stuck
to this structure. Furthermore it becomes clear that the structure is dupli-
cated. This most clearly can be seen in the middle of the photo, where the
structure consists of two parts. Also some of the L structures are double
projected on the substrate. Probably this is due to some error during the
writing of the structures.
Because of the few time that was left, only a couple test scans were per-
formed. Figure 4.17 shows nicely the domain configuration of the ellipse
formed structure (5). Because of the high scanning speed and the use of an
old MFM tip the image is somewhat unsharpen and contains some speed
artifacts.
The ellipse structure itself displays multi domain formation, but the con-
necting stripe on the right side of the image appears to be single domain.
This is the first indication of succeeding in creating a single domain pre-
ferred structure. The width of the connecting strip is approximately 1200
nm, whereas the smallest structures of E-beam sample I were larger than
1500 nm. Considering this result it is very likely to find single domains in the
structures of figure 4.15, because the width of the structures is substantially
less than 1200 nm.
41
43. (a) Special structure 1 (no photo taken;
same as structure6)
(b) Special structure 2; 56 x 56 µm
(c) Special structure 3; 65 x 65 µm (d) Special structure 4; 54 x 54 µm
(e) Special structure 5; 53 x 53 µm (f) Special structure 6; 47 x 47 µm
Figure 4.16: Photographs of the 6 special structures.
42
44. (a) 11 x 11 µm topographic image (b) 11 x 11 µm magnetic image
Figure 4.17: Test scan of the 5 structure.
43
45. Chapter 5
Conclusion and discussion
One can safely say that this report tells more about the properties of AFM
and especially MFM than it does about domains. Of course this is not
very surprising considering the magnitude of the project, especially when
beginning from scratch. Non the less some major progress has been made.
5.1 Interpreting MFM images
One of the most important conclusions to draw from this research is that still
a lot has to be done to gain more insight in the behavior of the MFM. A great
step forward would be to be able to characterize the tip. By determining
its effective stray field and switching field(s), more can be said about its
influence on the sample. A small conducting loop on a substrate could be
used to deduce all these properties. Provided that the diameter of the loop is
known, it is possible to accurately calculate its magnetic field that it induces
when a certain current is send trough it. By knowing the magnetic field of
the sample, its conceivable, not only to calibrate the tip, but also to examine
the tips behavior as function of an external field.
During the experiments it turned out that the achieved signal due to mag-
netic interaction depends greatly on the different parameters of the AFM.
Until now it is just a case of monitoring the signal while varying the pa-
rameters to optimize the signal. In this way the height is more or less
optimized for a certain amplitude, when in fact both should be optimized.
To gain more insight in the relations between the height, amplitude and
sample magnetization and the signal, also the current loop as mentioned
above could be used.
44
46. The characterization of tips and the determination of the relationships men-
tioned above makes the development of tips towards the ’ideal’ tip possible:
high coercivity, low stray field, single domain.
5.2 Coating tips
Coating tips to use them for MFM scans is a very delicate activity. Only
one of the in total eight sputter-coated tips appears to work more or less.
Unfortunately even that one seems not reliable, sometimes for no apparent
reason giving no magnetic signal at all. But apart from these difficulties
the scans show the advantage of these tips above the totally coated tips
from Nanoworld. Because a smaller region of interaction and a smaller
tip magnetization, the spatial as well as the magnetic resolution is higher
than with the Nanoworld tips. It is definitely worth to do more research
optimizing these tips. The coating of the tips has to be controlled with
higher precision. Now a mask together with sputtering is used to equip the
tips with a magnetic coating. This system is very sensitive to the correct
placing of the tip. When the tip is not correctly in front of the mask,
there might be no magnetic layer on the tip at all. And even if the tip
is somewhat coated with magnetic material, the cover layer that prevents
the magnetic material from oxidizing might not cover the whole layer. The
magnetic moment of this tip will over time disappear as a consequence of the
oxidation. However it pays to solve these difficulties because higher contrast
and spatial resolution can be achieved with these tips.
5.3 Magnetic Domains
The imaging of domains in the L shaped structures has been successful. The
images give a good estimation of the spatial resolution of the MFM. Also the
dependance of domain structures on an external field have been examined
by means of a discreet field sweep. This remains rather difficult because the
tip is influenced greatly by the external field. Hopefully this can be solved
by using the sputtered tips. The width of at least 1500 nm was still to large
for single domain formation.
Also the first breakthrough in the imaging of a single domain has been made.
The connecting stripes to the special structures of the E-beam sample II are
small enough (<1200nm) to display single domain formation. This makes it
plausible to also find single domain formation in the other small L structures.
Therefore continuation of the research is valuable.
45
47. Bibliography
[1] Nanoworld website. http://www.nanoworld.com.
[2] Nt-mdt website. http://www.ntmdt.com.
[3] T. Ono A. Yamaguchi and S. Nasu. Real-space observation of current-
driven domain wall motion in submicron magnetic wires. PRL,
92(7):077205, 2004.
[4] Rudolf Sch¨affer Alex Hubert. Magnetic Domains. Springer, 1998.
[5] R. Wiesend¨anger and H.J. Guntherodt. Scanning Tunneling Microscopy
II. 1992.
46
![Summary
In the development of fast magnetic memories (MRAMs) the ordering of
magnetic materials in domains is of great importance. The switching of
the magnetization of a memory element by means of the displacement of a
domain wall is the most important motivation for the research. To prevent
the existence of multiple domain walls within the element it is important
to create a structure with a preference for a single domain configuration.
When applying a rotating magnetic field to such a structure, a domain wall
can be induced. The final step is to propagate this domain wall using con-
trolled current pulses. Because of the extensive dimension of this research
only the search for a single domain configuration is subject to this internship.
In this experiment Magnetic Force Microscopy is used to examine the do-
main configurations in magnetic structures. The results from MFM highly
depend on the characteristics of the device itself and the magnetic probe,
the tip, used to scan the sample. Therefore an extensive amount of scans
are made to study the behavior of the MFM. Two kind of tips are examined,
MFM tips bought at Nanoworld [1] and contact tips bought at NT-MDT
[2], which are coated with a magnetic layer by means of sputtering. With
the Nanoworld tips, the whole system (tip, cantilever, cantilever holder) is
coated with a magnetic layer, whereas with the sputtered tips only the tip
is partially provided with a magnetic layer. This difference leads to a higher
sensitivity for the sputtered tips as can be seen in the scans. Furthermore
an important conclusion that can be drawn from this research is that still
few is known about the tip and tip-sample interaction. More insight in this
system makes it possible to draw more conclusions out of MFM scans.
The first step towards such a memory element is to create a structure small
enough to display single domain formation. Therefore a sample with small
shape anisotropy dominated CoFe structures is created using e-beam litho-
graphy. The structures of the first sample are approximately 1500 nm wide.
MFM scans of these structures reveal a multi domain configuration. Non the
less these scans are valuable as they confirm the possibility of the accurately
mapping of domain structures.
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