\documentclass[12pt,a4paper]{report}
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\begin{document}
\title{Domains and domain walls in thin permalloy films}
\author{\\ \\ \textsc{Stage verslag}\\ \\ \\Y. de Boer\\ Faculteit der Wiskunde en Natuurwetenschappen \\}
\date{Leiden, \today}
\maketitle

The work described in this document has been done in the Magnetic
and Superconducting Materials group in Leiden university from
March 2005 until August 2005 under supervision of Prof Dr. J.
Aarts, Ing. R. W. A. Hendrikx and Ing. M. B. S. Hesselberth

\pagenumbering{roman}
\tableofcontents
\cleardoublepage
\pagenumbering{arabic}

\chapter{Introduction}

Permalloy is an alloy with the special property that it requires a
very weak magnetic field to let it switch the direction of
magnetization compared to other ferromagnetic materials, combined
with a high $M_{s}$ (saturation magnetization) and low
magnetostriction. For this reason it several possible applications
are being researched, like the behaviour while in contact with a
superconductor, or the possible usage as RAM memory in computers.
Several groups have studied the magnetization of rectangle shaped
structures\cite{Gomez, Barthelmess, Mamin, Yun-Sok}, others study
circular dots\cite{Shinjo, Schneider}, switching behaviour of
single domain elements\cite{Xiaobin, Jong-Ching}, or even more
complex shapes\cite{Mei-Feng, Machida, Hirohata}. In this project
rectangle shaped microstructured permalloy elements have been
studied. In particular, Bloch walls have been observed and the
ends of 2 $\mu m$ $\times$ 20 $\mu m$ rectangles have been imaged.
This has been done to give a better insight in the results of
(future) experiments with superconductor/ferromagnetic (SF) hetero
structures.

\chapter{Theory}
All theory described in this chapter originates from reference
\cite{Handley}, unless otherwise specified. For a more detailed
explanation, the reader should refer to \cite{Handley}, or another
book about magnetism.

When an external magnetic field is applied to a material, the
magnetization inside the material may change. Some materials
become magnetized parallel to the field, which is called
paramagnetism, other materials show antiparallel magnetization,
called diamagnetism. Both of these two effect are in general
relatively weak and disappear once the external field is turned
off.

The work described here is done on another type of magnetism,
called ferromagnetism. Ferromagnetic materials do not only align
their internal magnetic moments in the direction of the applied
field, they also remain magnetized after the field is removed and
show spontaneous magnetization. On a microscopic scale a
ferromagnet consists of domains. Within a domain the magnetization
is in one direction, but neighbouring domains are magnetized in
other directions, resulting in a net zero magnetization on the
macroscopic scale. When a field is applied, the domains in the
same direction as the field either become larger at the expense of
their neighbours, or domains in other directions rotate towards
the direction of the field. Once the field is turned off, these
domains may remain in the same directions. The macroscopic
behaviour of a magnet is often described in a hysteresis loop. A
typical hysteresis loop is shown in figure \ref{hysteresisloop}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{hysteresisloop}
\end{center}
\caption{\emph{Typical hysteresis loop for a ferromagnetic
material. The material is demagnetized at zero field. Once a
magnetic field ($H$)is applied the magnetization will follow the
'virgin' curve until saturation ($M_{S}$) has been reached (point
a). When the field is turned down, the remaining magnetization at
zero field is called remanence (point b). The opposite field
required to demagnetize the sample is called the coercive field
(point c). If the field is increased further in the opposite
direction, the sample will saturate in that direction (point d).
}\label{hysteresisloop}}
\end{figure}

\section{Ferromagnetism}
Before the existence of domains can be explained, it is first
necessary to explain the fundamental physics behind ferromagnetism
and the energies that play a role in determining the behaviour of
a ferromagnet. In particular, exchange energy will get a thorough
explanation, since this is the energy that 'creates'
ferromagnetism.

\subsection{Origin of exchange energy}
\subsubsection{Two electrons}
When two electrons are in the same potential, the Pauli exclusion
principle prevents those two electrons from being in the same
state. This means that either their wave functions or their spin
functions have to be antisymmetric. For weak electron
interactions, the effect can be calculated using perturbation
theory.
\begin{equation}\label{perturbed}
\begin{split}
&E_{S}=E^{0}+C_{ij}+\mathcal{J}_{ij}
\\&E_{T}=E^{0}+C_{ij}-\mathcal{J}_{ij}
\end{split}
\end{equation}
Here $E_{S}$ and $E_{T}$ are the singlet and triplet state
energies, $E^{0}$ is the unperturbed energy, $C_{ij}$ is the
Coulomb energy and $\mathcal{J}_{ij}$ is the exchange energy of
the two electrons in states $i$ and $j$. $\mathcal{J}_{ij}$ is
given by:
\begin{equation}\label{exchangeintegral}
\mathcal{J}_{ij}=\int\phi_{i}^{*}(1)\phi_{j}^{*}(2)\frac{e^{2}}{4\pi\varepsilon_{0}r_{12}}\phi_{i}(2)\phi_{j}(1)dv
\end{equation}

$\phi_{i}$ and $\phi_{j}$ are the wave functions for corresponding
to the states $i$ and $j$. If the Coulomb interaction is strong,
$C_{ij}$ and $\mathcal{J}_{ij}$ can no longer be considered
perturbations on the energy $E_{0}$, the ground state becomes one
of either parallel or antiparallel spin, depending on the sign of
$\mathcal{J}_{ij}$.

\subsubsection{Exchange in transition metals}

In order to understand the physics of magnetism in the transition
(3d) metals three concepts are needed.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{bandsplitting}
\end{center}
\caption{\emph{(a)Evolution of atomic 4s and 3d states at large
interatomic spacing to bands at smaller spacing (r$_{0}$ occurs
when the net repulsive force $-\partial E/\partial r$ from 4s
electrons exactly balances the net attractive force from 3d
electrons); (b) density of states of 4s and 3d states split to
reflect exchange preference for spins of one direction. Magnetism
occurs if the Fermi energy $E_{F}$\cite{Olaf} lies within the $d$
band. Taken from \cite{Handley}}\label{bandsplitting}}
\end{figure}

The first is the broadening of atomic levels into bands in solids
when the atoms get closer together, see figure
\ref{bandsplitting}a. The $4s$ electrons have a lower energy than
the $3d$ electrons and are further away from the nucleus, they
bond more when the interatomic distance increases. The lower half
of a band contains mainly bonding states, the upper half contains
mainly antibonding states. Figure \ref{bandsplitting}b shows a
broad $s$ band and a narrow $d$ band.

The second concept is the internal field $H_{E}=\lambda M$ caused
by the Coulomb interaction (to be exact, the exchange integral
$\mathcal{J}_{ij}$) that occurs as a result of the Pauli exclusion
principle. This interaction shifts the spin up and spin down parts
of the $d$ band relative to each other, as in figure
\ref{bandsplitting}b. A simple way of understanding magnetic
exchange in metals is by comparing it to Hund's first rule in
atoms; in degenerate states the parallel spins are filled first to
have a minimal spatial overlap. In a band however, states are not
degenerate; it costs energy to put all electrons in the spin up
band. The energy cost is greater if the density of states $Z(E)$
is small, that is, they are spread out over a broad energy range.
This leads to the Stoner criterion for the occurrence of
magnetism, formula \ref{stonercrit}.

\begin{equation}\label{stonercrit}
\mathcal{J}(E_{F})Z(E_{F})>1
\end{equation}

The last concept is that bonding states favour paired,
antiparallel spins, antibonding favours parallel spins. The first
half of a band is bonding, that's why ferromagnetism does not
occur in the first half of the $3d$ transition metals (V, Cr, Mn),
but does occur in the second half of the $3d$ series (Fe, Co, Ni).


\subsection{Energies in ferromagnetism}
\subsubsection{Exchange energy}
The discrete microscopic and the macroscopic energy density is
given by formula \ref{exchange}, where $A$ is the exchange
constant, it represents the energy cost to change the direction of
the magnetization, $a$ is the lattice constant, $M_{S}$ is the
saturation magnetization, $\theta$ is the angle between two
neighbouring spins and $S$ is the spin. It is a short range
effect, it is limited to direct neighbours.
\begin{equation}\label{exchange}
f_{ex}=-\frac{2\mathcal{J}S^{2}}{a^{3}}\cos{\theta_{ij}}=A\left(\frac{\partial\theta}{\partial
x}\right)^{2}\rightarrow A\sum^{3}_{i=1}\left(\frac{\nabla
M_{i}}{M_{S}}\right)^{2}
\end{equation}
\subsubsection{Magnetostatic energy}
Magnetostatic energy originates from discontinuities in the normal
component of magnetization across an interface. It is given in
formula \ref{magnetostatic}. This effect is long range, it is
generally much smaller than exchange energy at an atomic scale,
but can become much larger in larger volumes at a longer range. At
the edges of a magnet, or for small structures it is the energy
that causes shape anisotropy.
\begin{equation}\label{magnetostatic}
f_{ms}=-\mu_{0}M_{S}\cdot
H_{i}=\frac{\mu_{0}}{2}M^{2}_{S}\cos^{2}{\theta}
\end{equation}
\subsubsection{Magnetocrystalline anisotropy}
Magnetocrystalline anisotropy means it costs less energy to
magnetize the sample in a particular direction compared to other
directions. It originates from the crystal structure; in certain
directions the distance between atoms is different compared to
other directions. This favours magnetism in the so called easy
directions. It is given in equation \ref{magnetocrystalline}, $K$
is the anisotropy constant, $\alpha$ is a direction cosines.
\begin{equation}\label{magnetocrystalline}
\begin{split}
&f_{a}=K_{2}\sin^{2}{\theta}+K_{4}\sin^{4}{\theta}+\ldots(uniaxial)
\\&f_{a}=K_{1}(\alpha_{1}^{2}\alpha_{2}^{2}+\alpha_{2}^{2}\alpha_{3}^{2}+\alpha_{3}^{2}\alpha_{1}^{2})+K_{2}\alpha_{1}^{2}\alpha_{2}^{2}\alpha_{3}^{2}+\ldots(cubic)
\end{split}
\end{equation}
\subsubsection{Magnetoelastic energy}
Magnetoelastic is a type of magnetocrystalline anisotropy that is
proportional strain. For cubic materials it given in equation
\ref{magnetoelasticcubic}, for isotropic materials it is given in
equation \ref{magnetoelasticisotropic}, $B_{i}$ is the
magnetoelastic constant, $e$ is the strain constant.
\begin{equation}\label{magnetoelasticcubic}
\begin{split}
f^{c}_{me}=&B_{1}[e_{11}(\alpha_{1}^{2}-\frac{1}{3})+e_{22}(\alpha_{2}^{2}-\frac{1}{3})+e_{33}(\alpha_{3}^{2}-\frac{1}{3})]\\+&B_{2}(e_{12}\alpha_{1}\alpha_{2}+e_{23}\alpha_{2}\alpha_{3}+e_{31}\alpha_{3}\alpha_{1})+\ldots
\end{split}
\end{equation}
\begin{equation}\label{magnetoelasticisotropic}
f^{iso}_{me}\approx
B_{1}e_{33}\sin^{2}{\theta}=\lambda^{2}_{S}E\cos^{2}{\theta}=\frac{3}{2}\lambda_{S}\sigma\cos^{2}{\theta}
\end{equation}
\subsubsection{Induced anisotropy}
When a magnetic field is present during the sputtering process
(see Appendix), it may induce anisotropy. This will favour
magnetization in the direction of the field.
\subsubsection{Zeeman energy}
Finally, there is the Zeeman energy, the energy of a magnetic
moment in a field, given for both a single moment and per unit
volume in equation \ref{Zeeman}.
\begin{equation}\label{Zeeman}
\begin{split}
&F=-\mu_{m}\cdot B\, (single\: electron) \\
&f_{Zeeman}=-\mu_{0}M\cdot H\, (energy\: per\: unit\: volume)
\end{split}
\end{equation}

\subsection{Permalloy}
Permalloy, with a composition of ~78\% nickel and 22\% iron has
the property that both the magnetocrystalline and the
magnetostriction anisotropy pass through zero near this
composition. Since both of these effects are very weak it is
possible to change the magnetization direction in permalloy with a
much smaller field than other ferromagnets. Permalloy has an
uniaxial magnetocrystalline anisotropy ($K_{2}$ in formula
\ref{magnetocrystalline} is often callled $K_{u}$), however, due
to the sputtering process on a silicon substrate, the easy
direction varies throughout the sample. Induced anisotropy, on the
other hand, may play an important role.

\section{Domains and domain walls}
Now that we have the relevant energies, we are now able to explain
why domains are formed. While the exchange energy tries to align
all moments in the same direction (figure \ref{domainformation}a),
in particular the nearest neighbours. The magnetostatic energy,
which is long range, tries to prevent moments sticking out of an
interface; this is the main energy that forces the creation of
domains (figure \ref{domainformation}b). To reduce the
magnetostatic energy even further, closure domains are formed
(figure \ref{domainformation}c). At this point anisotropy (from
any source) begins to play a role. If there is a strong uniaxial
anisotropy, it may be able to prevent the formation of closure
domains, as this would force the magnetization into a hard
direction. The opposite can also happen, if the easy magnetization
direction is directed in the width of the rectangle, multiple
closure domains may form, as in figure \ref{domainformation}d.

Domain walls are often 180$^{\circ}$, but in cubic materials, or
low anisotropy uniaxial (like permalloy), domains can have
90$^{\circ}$ domain walls, although 71$^{\circ}$ or 109$^{\circ}$
are also possible in cubic materials if the easy axes are not in
the $<100>$ directions. In a wall itself it's always more
favourable to put some spins into the hard (high anisotropy
energy) direction, resulting in a gradual change of direction,
than an instant change the magnetization by of 180$^{\circ}$,
which costs more exchange energy, since this is the highest energy
at the atomic scale.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{domainformation}
\end{center}
\caption{\emph{A single domain structure can reduce it's energy by
forming domains. (a) Single domain structure, (b) Two domain
structure to decrease magnetostatic energy, at the expense of a
domain wall, (c) the formation of closure domains to reduce the
magnetostatic energy even further, (d) formation of smaller
domains, which may reduce the total energy even further in some
materials.}\label{domainformation}}
\end{figure}

Four types of 180$^{\circ}$ walls can be distinguished; Bloch,
N\'{e}el, cross-tie and C-shaped walls, they are schematically
displayed in figure \ref{walltypes}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{walltypes}
\end{center}
\caption{\emph{Four different types of walls. (a) A Bloch wall, in
this type of wall the magnetization rotates out of plane. Magnetic
'charge' (causing magnetostatic energy) is building up at the top
and bottom of the wall, indicated by + and - signs, (b) A N\'{e}el
wall, the magnetization rotates in the plane of the sample,
creating magnetic 'charge' at the sides of the wall, (c) A
cross-tie wall (top view). In order to reduce the magnetostatic
energy, the wall direction alternates, having both Bloch and
N\'{e}el like parts, (d) An asymmetric Bloch wall, also known as a
C-shape wall. At the center it is like a Bloch wall, but at the
top and bottom it behaves like a N\'{e}el wall. The magnetization
rotates even a bit further, creating another vertical component,
which reduces the magnetostatic energy in the z-direction.
}\label{walltypes}}
\end{figure}

\subsection{Bloch walls}
In a Bloch wall the magnetization rotates out of plane, creating
'free' poles at the top and the bottom of the sample. The surface
energy density is given by:

\begin{equation}\label{blochwall}
\sigma=\int_{-\infty}^{\infty}\left[f_{a}(\theta)+A\left(\frac{\partial\theta}{\partial
z}\right)^{2}\right]dz
\end{equation}

where $f_{a}$ is the total anisotropy energy density, $\theta$ is
the angle between the magnetization at $z$ and the magnetization
at $-\infty$ and $z$ is perpendicular to the wall in the plane,
which reduces to $\sigma_{dw}=4(AK_{u})^{1/2}$ for uniaxial
materials (like permalloy). For uniaxial materials the
magnetization direction, as function of $z$ is given by:

\begin{equation}\label{magnetizationdirection}
\theta(z)=\mathrm{arccot}\left[\sinh\left(\frac{\pi
z}{\delta_{b}}\right)\right]+\pi=\arctan\left[\sinh\left(\frac{\pi
z}{\delta_{b}}\right)\right]+\frac{\pi}{2}
\end{equation}

$\delta_{b}$ is the domain wall thickness and is given by:

\begin{equation}\label{blochwallthickness}
\delta_{b}=\pi\left(\frac{A}{K_{u}}\right)^{1/2}
\end{equation}

\subsection{90$^{\circ}$ walls}
90$^{\circ}$ walls (or in some materials 71$^{\circ}$ or
109$^{\circ}$) are very common near the corners of microstructured
materials with a cubic anisotropy and low uniaxial anisotropy,
where closure domains are formed. They are visible in figure
\ref{domainformation} c and d. 90$^{\circ}$ walls are always in
the plane of the film.

\subsection{N\'{e}el walls}
In thin films, the magnetostatic energy density increases rapidly
\cite{Trunk} since the charged area at the top and bottom of the
sample increase compared to the wall area. In order to reduce the
total energy, it is more favourable to rotate within the plane. At
thicknesses close to the boundary thickness this will increase the
magnetostatic energy, but will significantly lower the exchange
energy. In even thinner films both forms of energy are lower than
in a Bloch wall. For $t \ll \delta_{M}$ The energy density can be
approximated as:

\begin{equation}\label{neelwallenergy}
\sigma_{N}\approx\pi tM_{S}^{2}
\end{equation}

and the wall thickness can be approximated as:

\begin{equation}\label{neelwallthickness}
\delta_{N}\approx\pi\left(\frac{2A}{K}\right)^{1/2}
\end{equation}

\subsection{Cross-tie walls}
In order to reduce the magnetostatic energy in both N\'{e}el and
Bloch walls, the magnetization direction alternates in a cross-tie
wall, as illustrated in figure \ref{walltypes}c. This type of
walls are typically found in the intermediate thickness between a
N\'{e}el wall and an asymmetric Bloch wall. If it is assumed that
the magnetization does not change throughout the thickness of the
film, but only within the plane of the film itself, this wall has
less energy than a Bloch wall\cite{Metlov}.

\subsection{Asymmetric Bloch walls (C-shaped walls)}
A C-shaped wall is like a Bloch wall in the bulk, but like a
N\'{e}el wall at the surfaces. By creating N\'{e}el like walls at
the top and the bottom and rotating the magnetization even a bit
further in the opposite direction of the Bloch part of the wall,
the magnetostatic energy is minimized. This wall is found in
thicker films\cite{Trunk} in simulations and using MFM by
Barthelmess et al\cite{Barthelmess}. No analytical calculation of
its energy has been found in the literature so far, but
simulations do exist\cite{Trunk}.


\chapter{Magnetic Force Microscopy}
The measurements on the samples has been done using magnetic force
microscopy (MFM). This is a special type of atomic force
microscopy (AFM) which does not measure the topography but the
magnetic field perpendicular to the sample. First topography scans
have been made using tapping mode AFM, followed by an MFM
measurement where the tip was kept above the surface, instead of
touching the surface. The AFM used in these experiments was a
commercial Nanoscope from Digital Instruments, but instead of
using the default hardware and software, an 'SPM 100' controller
from RHK Technology Co. with 'SPM 32' software has been used.
\section{Atomic Force Microscopy}

An AFM scans the surface using a small cantilever with a sharp
tip. A laser beam reflects from the back of the cantilever towards
2, 4 or more photodiodes, depending on the model, which allows the
detection of the cantilever deflection, as illustrated in figure
\ref{afm}. The scanning is done line by line using piezoelectric
actuators (piezos), which allows sub-\AA\, movement in both the
lateral and vertical directions. There are several methods of
measuring a sample with an AFM, the most common are contact and
tapping mode. A feedback loop (usually PI, but sometimes PID
controlled) keeps the deflection (in contact mode), or the
oscillation amplitude (in tapping mode) constant by adjusting the
tip-sample distance with the z-piezo.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{afm}
\end{center}
\caption{\emph{Typical detection method of the cantilever
deflection in an AFM, a reflected laser beam gets detected by
photo diodes. Taken from \cite{detection}}\label{afm}}
\end{figure}

\subsection{Contact mode}

In contact mode the tip is brought into direct contact with the
sample, keeping the cantilever slightly bent. Depending on the
surface topography the cantilever bends up- or downwards, which
will move the laser spot on the photodiodes. The feedback will
then move the sample up or down with the z-piezo to keep a
constant deflection.

\subsection{Tapping mode}

In tapping mode the cantilever is driven close to mechanical
resonance, generally slightly below it's resonance frequency, at
the frequency where $\partial A/\partial f$ is the largest. When
the tip gets close to the sample it will hit the sample, which has
a dampening effect on the oscillation. The amplitude during
'contact' changes as a result of the topography. Just like in
contact mode, the feedback will move the sample up or down to keep
this amplitude at a constant level, typically 50\% of the
amplitude in free space.

\section{Magnetic Force Microscopy}

When measuring the magnetic field, the tip is brought into
resonance, but unlike tapping mode, it does not hit the surface.
The tip of the cantilever, coated with a magnetic material (during
most of the measurements this was Co), is often considered a point
dipole for simplicity. The force on the dipole $\mathbf{m_1}$ on
the cantilever as results of a dipole $\mathbf{m_2}$ in the sample
is given by:
\begin{equation}\label{forceoncantilever}
F=\nabla\left(\frac{3(\mathbf{m}_{1}\cdot\mathbf{\hat{r}})(\mathbf{m}_{2}\cdot\mathbf{\hat{r}})-\mathbf{m}_{1}\cdot
\mathbf{m} _{2}}{r^{3}}\right)
\end{equation}
However, in resonance, it is not the force, but the force gradient
which is being detected. A force gradient changes the effective
spring constant:
\begin{equation}\label{effspringconst}
c_{eff}=c-F'
\end{equation}
which in turn changes the resonance frequency:
\begin{equation}\label{resonancefrequencychange}
\omega'_{0}=(c_{eff}/m)^{1/2}\approx\omega_{0}\left(1-\frac{F'}{2c}\right)
\end{equation}
The approximation is valid if F' is small compared to c. Figure
\ref{tipnearsample} gives a illustration what happens to the
oscillation as a results of a force with a non zero spatial
derivative. The shift of resonance frequency changes the amplitude
and phase of the oscillation, which can be detected by a lock-in
amplifier. Measuring the phase gives slightly better (a factor of
$\sqrt{2}$) better signal to noise than the amplitude.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{tipnearsample}
\end{center}
\caption{\emph{A spring in a van der Waals potential. In the
attractive regime the parabola becomes wider as a results of the
van der Waals potential, thus changing the effective spring
constant. This does not apply to just van der Waals forces, but to
any force that has a non zero derivative.}\label{tipnearsample}}
\end{figure}

\chapter{Results}
There are many unknown factors during MFM measurements. The most
important uncertainty is the exact location of the magnetic moment
on the tip, and how much each part of the tip contributes to the
total signal, the average tip sample distance, and the exact
oscillation amplitude. For this reason, the strength of the field
is not calculated from the images. The images in this section only
give information about the domain structure within the sample, not
about the strength of the field (or rather, it's second
derivative). To calculate the actual field it is more convenient
to use a computer simulation to calculate the magnetization on a
similarly magnetized sample and calculate the magnetic field using
that data, similar to what has been done by Barthelmess et al
\cite{Barthelmess}.

For all figures in this section the top left (a) is the
topography, the top right (b) is the corresponding MFM measurement
and the bottom image (c) is a schematical  representation of the
domains within the sample, unless otherwise specified. Due to non-
linear effects in the x- and y-piezos the distances given in the
figures in this chapter are not correct. All lateral distances
given are how they appear in the image. In reality they could be
as much as 20\% smaller.

\section{50 nm permalloy}

Figure \ref{image08} and \ref{image09} are results of a
measurement of 50 nm thick permalloy. These are some early
results, the quality of the images is not as good yet as those
that will appear later in this chapter.

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo08}
\includegraphics[width=6cm]{imagemfm08}
b
\\c
\includegraphics[width=6cm]{domain08}
\end{center}
\caption{Topographic and MFM image of a $2 \mu m$ $\times$ $2.5
\mu m$ element. A short Bloch wall can be seen, which can be
recognized because it's straight, with a clear
contrast.}\label{image08}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo09}
\includegraphics[width=6cm]{imagemfm09}
b
\\c
\includegraphics[width=6cm]{domain09}
\end{center}
\caption{Topographic and MFM image of a $2 \mu m$ $\times$ $2.8
\mu m$ element. A cross-tie wall is visible, because a cross-tie
wall extends into the domains, therefore, it is not a straight
line. }\label{image09}
\end{figure}
\clearpage
\section{100 nm permalloy}

Figure \ref{image15}, \ref{image16} and \ref{image19} are elements
on a 100 nm thick permalloy sample. All observed $180^{\circ}$
domain walls are in the y-direction (some images are slightly
rotated). This could be the result of induced anisotropy caused by
the magnetic field inside the UHV, but it can also be coincidence.
It is unknown in which direction the sample was orientated during
the sputtering process.

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo15}
\includegraphics[width=6cm]{imagemfm15}
b
\\c
\includegraphics[width=6cm]{domain15}
\end{center}
\caption{This $2.8 \mu m$ $\times$ $2 \mu m$ elements has 7
domains. At the left side a short Bloch wall can be seen and at
the right side a cross where four 90$^{\circ}$ walls come
together.} \label{image15}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo16}
\includegraphics[width=6cm]{imagemfm16}
b
\\c
\includegraphics[width=6cm]{domain16}
\end{center}
\caption{$2.8 \mu m$ $\times$ $2 \mu m$ element. A long Bloch wall
at the left and a cross at the right side.}\label{image16}
\end{figure}
\clearpage

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo19}
\includegraphics[width=6cm]{imagemfm19}
b
\\c
\includegraphics[width=6cm]{domain19}
\end{center}
\caption{Bloch wall in a $2 \mu m$ $\times$ $3 \mu m$
element.}\label{image19}
\end{figure}

\clearpage

\section{100 nm permalloy with 20 nm Niobium}
To prepare for future experiments concerning the interaction of
domain walls with superconductors a sample with a $20 nm$ thick
layer of niobium on top of $100 nm$ permalloy has been made. This
sample contains both $2 \mu m$ $\times$ $20 \mu m$ elements and
elements with both sides approximately $2 \mu m$.

In these images the oscillation amplitude of the cantilever has
been reduced to $\approx 10 nm$ (see appendix). On several MFM
pictures some dark lines with a width of $\approx 20 nm$ have been
observed, often even away from the elements. It is unknown where
they come from. Possibilities could be a tip effect, but such
effect would be expected to be visible on the topographic image as
well, or the formation of iron whiskers, but then the lines should
be straight. Similar lines were also measured later on a sample
with permalloy only (not shown).

\subsection{Small structures}
The same small structures were made on 100 nm permalloy with 20 nm
niobium as on the previous permalloy sample. This was done to see
whether the same magnetization patterns could be seen. Even though
there was an applied magnetic field in the UHV, there was no
evidence in the images that there was any induced anisotropy,
since two elements which were rotated 90$\circ$ in respect to each
other showed the same magnetization pattern. Some results are
displayed in figures \ref{image22}, \ref{image32} and
\ref{image33}.

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo22}
\includegraphics[width=6cm]{imagemfm22}
b
\\c
\includegraphics[width=6cm]{domain22}
\end{center}
\caption{Bloch wall in a $3.5 \mu m$ $\times$ $2.5 \mu m$ element,
dark lines of unknown origin are visible at both left corners and
the right side.}\label{image22}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo32}
\includegraphics[width=6cm]{imagemfm32}
b
\\c
\includegraphics[width=6cm]{domain32}
\end{center}
\caption{Bloch wall in a $2.3 \mu m$ $\times$ $2.6 \mu m$, some
dark lines are visible at the top of the element.}\label{image32}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo33}
\includegraphics[width=6cm]{imagemfm33}
b \\c
\includegraphics[width=6cm]{domain33}
\includegraphics[width=6cm]{blochprofile}
d
\end{center}
\caption{Another Bloch wall on a 3.5 $\mu m$ $\times$ 2.0 $\mu m$.
Due to the absence of drift during this scan, it was possible to
scan at a distance of approximately 10-20 $nm$ from the sample, a
total distance of 30-40 $nm$ from the permalloy. The wall profile
taken shows a FWHM of 62 $nm$ at this distance for the main peak,
but also shows a small negative peak, indicating that the wall is
not a perfect Bloch wall (figure \ref{walltypes}b), but an
asymmetric (C-shaped) Bloch wall (figure
\ref{walltypes}d).}\label{image33}
\end{figure}
\clearpage

\subsection{2 $\mathbf{\mu m}$ $\times$ 20 $\mathbf{\mu m}$ structures}
Since many SF experiments in this lab are done on 2 $\mu m$
$\times$ 20 $\mu m$ magnetic structures, some MFM images have been
made on elements with that shape. They are displayed in figures
\ref{image25}, \ref{image26}, \ref{image27} and \ref{image28}.
Most images have several domains at the ends, where they may be
under the influence of the stray fields of neighbouring elements,
but only one large in the center, which is caused by shape
anisotropy. For these images, the full scan range of the AFM ($8.3
\mu m$) has been used.


\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo25}
\includegraphics[width=6cm]{imagemfm25}
b \\c
\includegraphics[width=6cm]{domain25}
\end{center}
\caption{Two ends of 2 by 20 $\mu m$ elements. The left element
has only domains at the end, while the right has domains at the
entire scanned region. Both elements show some evidence that the
magnetization has been changed during the scan, indicated by the
dotted lines. There is a difference in contrast between the
elements, this has been caused by a plane fit, automatically done
by the software (WSxM) which has been used to process the images,
not by the scanning itself. }\label{image25}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo26}
\includegraphics[width=6cm]{imagemfm26}
b \\c
\includegraphics[width=6cm]{domain26}
\end{center}
\caption{Two ends of 2 $\mu m$ $\times$ 20 $\mu m$ elements. The
magnetiztion of the left element has been altered twice during the
scan, indicated by the dotted lines.}\label{image26}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo27}
\includegraphics[width=6cm]{imagemfm27}
b \\c
\includegraphics[width=6cm]{domain27}
\end{center}
\caption{Two ends of 2 $\mu m$ $\times$ 20 $\mu m$ elements. None
of these two show any domain structure towards the
center.}\label{image27}
\end{figure}

\begin{figure}[!ht]
\begin{center}
a
\includegraphics[width=6cm]{imagetopo28}
\includegraphics[width=6cm]{imagemfm28}
b \\c
\includegraphics[width=6cm]{domain28}
\end{center}
\caption{Two ends of 20 $\mu m$ $\times$ 2 $\mu m$ elements. The
top element has a long Bloch wall. There are some dark lines, in
the lower element it's not entirely clear whether a domain wall,
or a dark line is visible.}\label{image28}
\end{figure}
\clearpage


\chapter{Conclusion}
Bloch walls and the magnetic domain structure have been observed
in 100 $nm$ thick microstructured permalloy. Although the
direction of these walls was not completely reproducible, it is
certain they do appear in rectangles with dimensions of 1.5 $\mu
m$ $\times$ 2 $\mu m$, 1.5 $\mu m$ $\times$ 2.5 $\mu m$ and 2 $\mu
m$ $\times$ 2.5 $\mu m$. The magnetic signal was also very clearly
visible with a 20 $nm$ thick niobium layer on top of the
permalloy. the ends of 2 $\mu m$ $\times$ 20 $\mu m$ have been
imaged, the magnetization varies greatly from element to element,
which may be related to strain or anisotropy, but also their close
proximity to each other may have affected the magnetization.

Strange lines, with a width of approximately 20 $nm$ have been
observed on some elements after making the measurement method more
sensitive. They were reproducible in repeating scans, but appeared
to be randomly placed and in some cases not even above the element
itself. No explanation for this effect has been found yet.

\appendix

\chapter{Sample preparation}
The creation of the sample has 3 main steps which are explained in
detail below. A schematic overview of the production process is
given by figure \ref{samplepreparation}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{samplepreparation}
\end{center}
\caption{\emph{Preparation of the sample step by step. (a) A clean
silicon wafer, 10-15 $mm$ $\times$ 10-15 $mm$, (b) A sputtered
layer of permalloy, possibly with a layer of niobium on top (not
displayed), (c) a layer of resist gets spin-coated on top of the
material, (d) after writing a structure with a SEM the developed
resist is removed, (e) the permalloy (and niobium) is etched using
an ion etcher, (f) the final result after solving the last resist
in acetone. }\label{samplepreparation}}
\end{figure}

\section{Sputtering}
The samples have been sputtered in a home-built UHV magnetron
sputter system, which has a base pressure that can be as low as
$1.0 \times 10^{-10} mbar$, but is typically $1.0 \times 10^{-10}
mbar$. During sputtering, highly energetic argon ions bombard a
target, causing particles to break off, which then redeposit on
the sample. The sputter conditions for permalloy are an argon
pressure of $6.0 \times 10^{-3} mbar$ and a current of $200 mA$.
The niobium has been sputtered with an argon pressure of $4.0
\times 10^{-3} mbar$ and a current of $220 mA$. To measure the
thickness a crystal monitor was used. By measuring the changing
resonance frequency of this crystal, the amount of sputtered
material on it can be determined. For permalloy $0.272 \pm 0.03
nm$ on the crystal corresponds with $1 nm$ on the sample. This has
been measured using X-ray diffraction; after sputtering $15.7 \pm
0.1 nm$ on the crystal the sample thickness was $57.8 \pm 0.1 nm$.
For niobium the sputter rate was approximately $9.3 \pm 0.1 nm$ on
the crystal for $19 \pm 1 nm$ on the sample.

\section{E-beam lithography}
After sputtering a resist layer was spin coated on the sample.
With an Scanning Electron Microscope the elements have been
written on the sample. During the development of the resist only
the places that have been exposed with the electron beam remain.
To make sure the last bits of resist were removed from the places
where it should not be, the sample was put into an oxygen etcher.
An oxygen plasma with a pressure of $~1.0*10^{-1} mbar$ 'burned'
the remaining unwanted resist away from the sample.

\section{Ion-beam etching}
After writing the pattern the sample was etched in an ion etcher.
The base pressure is typically $1 \times 10^{-5} mbar$, argon is
then added with a pressure of $2.5 \times 10^{-4} mbar$. In this
etcher the sample is bombarded with argon ions, which will remove
material except at the places where the resist is located, leaving
the desired structure on the sample. The typical etch time is $30
sec$ for $10 nm$ of permalloy.

\section{complications}
A complication during this process is the formation of 'ears',
displayed in figure \ref{ears}. These ears are formed when
permalloy redeposits on the resist. A rotating sample holder,
while etching under an angle should prevent this, but it did not.
Several methods proved to be useful in removing those ears. The
first (and best) method was reactive etching with bromine, similar
to what others have done with chlorine\cite{Vasile}. First a gas
of bromine with a pressure of $2.8 \times 10^{-4} mbar$ was
inserted into the chamber, followed by argon for a total pressure
of $5.0 \times 10^{-4} mbar$. The ears created by this process can
easily be washed away with water, leaving nice flat edges. This
method has been used on the $100 nm$ permalloy sample.

The second method is using acid with Fe$_{3}$Cl to etch the ears
away. The major disadvantage of this method is that it does not
only etch the ears, but it also rounds all corners. Besides that,
it can only be used if there is some protective layer on top of
the structure, which could be the resist or any material that is
unaffected by the acid. The acid used was concentrated HCl in
water ($1:330\pm30$) with $1.1\pm 0.2 gram/240 ml$ Fe$_{3}$Cl.
This method has been used on the $100nm$ permalloy + $20nm$
niobium sample. The total etch time took a few minutes in an
ultrasound. Both the exact concentration and time still needs to
be optimized.

Another complication, which happened while making the sample with
niobium, is that the resist remains between the ears. This has
been removed in the oxygen etcher, which etched a total of 50
minutes, before acid had been used to remove the ears. It is not
known how long it actually took to remove the resist.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{ears}
\end{center}
\caption{\emph{Preparation of the sample step by step. (a-d) Same
as in figure \ref{samplepreparation}, (e) the permalloy redeposits
on the resist during the etching process, (f) 'ears' on the sides
of the structure after the resist has been removed.}\label{ears}}
\end{figure}

\chapter{Quality of the MFM images}
In addition to measuring the domain structure, one of the purposes
of this project was to optimize the quality of the MFM
measurements. Especially during the beginning of the measurements
the quality of the images was rather poor, as shown in
\ref{imagemfm01}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{imagemfm01}
\end{center}
\caption{\emph{One of the first MFM measurements. This is the
amplitude signal, which had to be expanded 100 times (after
subtracting an offset signal) by the lock-in amplifier to get
enough contrast. The phase signal did not have enough contrast to
produce a clear image.}\label{imagemfm01}}
\end{figure}

The cause was too large a driving voltage, resulting in tip
oscillations with an amplitude with an order of magnitude of 1
$\mu m$. The total signal (coming from the photodiodes was too
large for the lock-in amplifier (max 1 $V$ RMS) and was reduced by
a factor 100, while the relevant signal (after subtracting an
offset) had to be expanded 100 times again. This large amplitude
was useful for topographic measurements, since the changes in
topography are much less than the oscillation amplitude. However,
for magnetic measurements a small oscillation gives a much larger
(relative to the total amplitude) change in amplitude and phase
because the average tip-sample distance can be significantly
decreased. This can easily understood by recalling figure
\ref{tipnearsample}, if the parabola in the graph moves to the
right (away from the surface), the tip spends more time at places
where the field is much weaker and thus reducing the signal
strength. During the last measurement the oscillation amplitude is
reduced to approximately 10 nm by reducing the driving voltage.
This resulted in a major improvement of the image quality,
especially the phase signal, see for example figure
\ref{imagemfm29}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{imagemfm29}
\end{center}
\caption{\emph{One of the last measurements at the ends of 2 2
$\times$ 20 $\mu m$ elements. This image represents the phase
signal, with a typical tip-sample distance of 30-50 $nm$ and an
oscillation amplitude of $\sim 10 nm$.}\label{imagemfm29}}
\end{figure}

Furthermore, the lock-in settings have a lot of influence on the
quality of the images, especially the 'time constant', this is the
time over which the output signal is averaged before it is
changed. A high time constant reduces noise (more time to average
the signal) , but results in horizontal lines in the image (if the
time to average over is more than the time for one pixel), which
can be seen very clearly in figure \ref{imagemfm01}. Increasing
the time constant and scanning slower could solve both of these
issues, but becomes unviable due to drift in the z-direction.
Drift can come from several sources, for example the sample can
become warm from the laser. Typical scan times for were 500 $ms$
per line (~1 $ms$ per pixel) with a time constant of 30 $\mu s$.
The total size of a picture is 512 $\times$ 512 pixels, which took
~8.5 minutes.

There have also be a few measurements using 'Lift Mode' on another
AFM. This was the same type of AFM, except this one had the
original hardware and software. During Lift Mode each line is
scanned four times, first the trace and retrace topology is
scanned in a single line, then the tip is lifted to a set height
and follows the exact same height curve along the same line. This
is then repeated for each line in the image. The main advantage of
this method is that it basically eliminates drift in the
z-direction, during each line the drift is compensated. The
disadvantage is that if there is a lot of non-magnetic material on
the surface, the tip will be lifted there as well. This returns in
the magnetic image, when the tip is lifted over a non-magnetic
feature, the magnetic signal of the underlying magnetic material
becomes weaker and you will clearly see those non-magnetic
features in the magnetic images, especially if a blunt tip is
used. A blunt tip will image itself on any topographic feature,
causing tip sample convolution, for example, see figure
\ref{imagemfm10}. Besides surface effects this technique has
another major disadvantage; during each scan line the topography
is measured using tapping mode, which means the tip has to be in
direct contact at the time it hits the surface. The stray fields
of the tip can easily change the magnetization of the sample
during the measurement\cite{Polushkin, Temiryazev}. Many groups
have used this technique because it's very easy to use, it's part
of the default software package that comes with this type of AFM
and there is no drift in the MFM image.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm]{imagemfm10}
\end{center}
\caption{\emph{An MFM measurement using lift mode. Surface effects
are clearly visible in the magnetic image.}\label{imagemfm10}}
\end{figure}

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