%
% MSM.tex
%
% author:  O.W.B. Benningshof
% date:    April 5, 2005
%




\documentclass[12pt,a4paper,openright,twoside]{report}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{indentfirst}


\begin{document}
\title{Imaging Vortices in Mesoscopic Channels
\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{pictures/kaft}
\end{center}
\end{figure}
}
\author{O.W.B. Benningshof}
\date{Leiden, \today}
\maketitle


\tableofcontents

\chapter{Introduction}

Since the discovery of superconductivity by Heike Kamerlingh-Onnes
\cite{Kamerlingh-Onnes} in Leiden in 1911, a lot of experimental
and theoretical physicists have contributed to understanding of
this phenomenon. A great breakthrough was achieved by: Ginzburg
and Landau \cite{GL} with their phenomenological theory (GL
theory) and later by Bardeen, Cooper and Schrieffer \cite{BCS}
\cite{BCS2} with their microscopic theory (BCS theory).

A very interesting phenomenon in superconductivity is the
existance of so-called vortices that are created in a type II
superconductor when a magnetic field is applied. The vortex
consits of one fluxon (quantized magnetic flux) with circulating
supercurrent around it. The earliest development of the field
which is known as 'vortex physics' was the paper of Abrikosov
\cite{Abrikosov} in the context of GL theory.

When an electrical current is sent through the superconductor, the
vortex can be moved due to the Lorentz force. The physics
associated with the vortex dynamics is studied in detail by many
labs around the world. One of the experimental groups in this
field of research is the Magnetic \& Superconducting Materials
(MSM) group in Leiden, supervised by P. H. Kes and J. Aarts. The
MSM group is well known in fabricating good quality thin film
superconductors. These thin films make possible the study of
vortex dynamics and by means of scanning tunnelling microscopy
(STM) the imaging  of the vortices.

The vortex dynamics has been recently studied in samples which
contain easy flow vortex channels. A channel sample consists of
two layers of superconductors: a top layer of strong pinning
material and a bottom layer of weak pinning material. The channels
are created by etching away the strong pinning material from the
top layer, producing a channel of weak pinning material with
channel edges (CE) of strong pinning material. A DC current can be
tuned to move the vortices in the weak pinning material (in the
channel) while keeping them static in the strong pinning material
effectively creating easy flow vortex channels. Experiments on
this channels provide an insight in the vortex dynamics and the
shear force density acting on vortices. In the weakly pinned
channels the vortices are expected to form an order hexagon
lattice. However the channel edges are formed of strongly pinned
material \cite{kokubo}, which will lead to a randomly distributed
lattice. Two cases can be distinguished for the vortex rows inside
the channels, the commensurate case (the width of the channel fits
an integer number of the lattice distance of the vortices) and the
incommensurate case . The dynamics of vortex matter confined to
mesoscopic channels had been recently investigated by so called
'mode locking' experiments \cite{kokubo2}. In these experiments
the vortices are coherently driven through the potential provided
by the static vortices pinned in the channel edges. The
interference between the washboard frequency moving vortex lattice
(due to the potential provided by the channel edge) and the
frequency of the superimposed rf-signal causes (Shapiro-like)
steps in the $I$-$V$ curves. The positions of the voltage steps
uniquely determine the number of moving rows in the channels.
Maxima in flow stress occur at mismatch conditions, being due to
the traffic-jam-like flow impedance caused by disorder in the
edges. Numerical simulations predict that the disorder in the
edges should be small \cite{Besseling}, but so far there is no
experimental imaging of the lattice deformations in a sample of
channels.

Visualizing the deformation due the CE is the main goal of my 5th
year stage. This has been done with the help of a Scanning
Tunnelling Microscope (STM). The STM is able to distinguish
between a normal conducting metal and a superconductor. Since the
vortex core consists of normal conducting metal and because the
rest of the sample is superconducting, images of the vortex
lattice can be made. So, by fabricating a sample with channels,
images of changing vortex lattices near the CE should be possible
to made.

This report is organized as follows. Chapter 2 gives a brief
theoretical introduction in type II superconductors. In Chapter 3
the growth of the thin film superconductors and their
characterization are presented. Chapter 4 deals with the
fabrication of channels into a superconductor. Chapter 5 starts
with the explanation of the basic principles of the STM and how
vortices can be imaged with tunnelling spectroscopy. After that it
presents the results obtained from the STM measurements and gives
a detailed discussion of how to interpret the data. The report end
with Chapter 6 which gives an overall conclusion about the
project.


\chapter{Theoretical aspects of Type II Superconductor}

In this chapter we will try to introduce the reader in the physics
of type II conventional superconductors. The first questions to
answer would be: what is a type II superconductor and why, when
applying a magnetic field, it contains so called vortices
(circulating super current). The next step would be to see what is
the arrangement of the vortices into the material with and without
impurities.

The largest part of the chapter is devolved to basic notions but
which are very important in understanding the measurements
presented in later chapters. Therefore we made used of well known
text books of Tinkham \cite{M Tinkham}, Rose-Innes and Rhodericks'
\cite{Rose-Innes and Rhodericks}, Kittel \cite{Kittel},  Kes
\cite{Kes} and the PHD. Thesis of Marchevsky \cite{Marchevsky}.

\section{Superconductivity and Meissner Effect}

In 1911 superconductivity was discovered in Leiden by H.
Kamerlingh Onnes \cite{Kamerlingh-Onnes}. He found that electrical
resistance of mercury disappears below $4.2$ $K$. In the following
years more materials with this properties where found, all
perfectly conductors below a temperature which is material
dependent. This temperature is called the critical temperature
$T_c$ and is a function of magnetic field. These materials are
more than perfect conductors as was shown by Meissner and
Ochsenfield \cite{Meissner}. They show that an external magnetic
field is expelled by the superconductor when it is in the
superconducting state, which is called perfect diamagnetism or
Meissner (and Ochsenfield) effect. This property doesn't exist for
a perfect conductor, which makes it an essential property of the
superconducting state.

\begin{figure}[!ht]
\begin{center}
(a)
\includegraphics[width=5cm]{pictures/plaatje6}
(b) \caption{Meissner effect: Superconductor expels the external
magnetic field when temperature is below $T_c$. (a) superconductor
above $T_c$. (b) superconductor below $T_c$ }
\end{center}
\end{figure}


\section{Ginzburg-Landau Theory}

The first theory of superconductivity was developed by London
brothers \cite{London}. This theory was able to describe certain
phenomena of superconductivity based on the two fluid model, but
it did not take the quantum effects into account. The first theory
who was able to deal with the quantum effects was the
Ginzburg-Landau theory. Ginzburg and Landau realized that:
\begin{itemize}
\item  The superconducting state is more ordered then the normal
state.
\item The superconducting transition in the absence of magnetic
field a second order phase transition.
\item One needs an effective wavefunction of the (superconducting)
electrons, $\Psi(\mathbf{r})$ to develop a quantum theory
\end{itemize}
To deal with al this aspects, they considerer $\Psi(\mathbf{r})$
as an order parameter. The Landau theory of second order phase
transition is based on an expansion of the free energy density in
powers of the order parameter, which is small near the transition
temperature. Strictly speaking this theory is only valid when the
temperature is close to the transition temperature. However, at
least qualitatively, it proves to be quite right also at $T$ $<<$
$T_c$.

If we deal with an inhomogeneous superconductor in a uniform
external magnetic field, the Gibbs free energy density can be
expanded in powers of $\Psi$ as:

\begin{equation}
\begin{split}
G_{sH} = G_n + \int dV & \biggl[ \alpha |\Psi|^2 +
\frac{\beta}{2}|\Psi|^4 + \frac{1}{4m} \left| -i\hbar \nabla \Psi
- \frac{2e}{c}\mathbf{A} \Psi \right |^2 \\&  + \frac{(\nabla
\times \mathbf{A})^2}{8 \pi} - \frac{(\nabla \times \mathbf{A})
\cdot \mathbf{H_0}}{4 \pi} \biggl] \label{*}
\end{split}
\end{equation}

where $G_{sH}$ is the free Gibbs energy density of the
superconductor, $G_n$ the free energy density of the normal state,
$m$ the mass of the electrons, $e$ the charge of the electrons and
$\alpha$ and $\beta$ are some coefficients which are material
dependent. The term $\frac{1}{4m} \left| -i\hbar \nabla \Psi -
\frac{2e}{c}\mathbf{A} \Psi \right |^2$ is due to the kinetic
energy density of the superconducting electrons. $\mathbf{H_0}$ is
the external magnetic applied field, and $\mathbf{H}$ is the exact
microscopic field at a given point in the superconductor. The
integration is carried out over the entire volume of the
superconductor. The free energy density is in equilibrium position
when the function is in an absolute minimum. This absolute minimum
is found by deriving the free energy with respect to $\Psi$ and
$\mathbf{A}$ and set them equal to zero. From those the two
Ginzburg-Landau equations are obtained:

\begin{equation}
 \alpha \Psi + \beta \Psi
|\Psi|^2  + \frac{1}{4m} \left( i\hbar \nabla  +
\frac{2e}{c}\mathbf{A} \right)^2 \Psi = 0
\end{equation}\label{2.2}

\begin{equation}
j_s = - \frac{i \hbar e}{2 m} \left( \Psi^* \nabla \Psi - \Psi
\nabla \Psi^* \right) - \frac{2e^2}{mc} \left| \Psi \right|^2
\mathbf{A}
\end{equation}

where $j_s$ denotes the electrical current. The first equation is
analogue to the time-independent Schr\"{o}dinger equation with a
non-linear term. The second equation is identical to the quantum
mechanical equation describing the current density.

Next to the fact that the Ginzburg-Landau equations produce many
interesting and valid results, they predict two characteristic
lengths of a superconductor.

\begin{equation}
\xi^2(T) = \frac{\hbar^2}{4m \left| \alpha \right|}
\end{equation}

\begin{equation}
\lambda^2(T) = \frac{m c^2 \beta}{8 \pi e^2 \left|\alpha \right|}
\end{equation}

$\xi$ is the characteristic scale over which variations of the
order parameter $\Psi$ occur. This length is called the coherence
length. $\lambda$ is the characteristic depth for the penetration
of an external magnetic field in the superconductor. This length
is called the penetration depth. For a pure superconductor
($\xi_0$ $<<$ $l_e$),

\begin{equation}
\xi (T) = 0.74 \xi_0 \left( \frac{T_c}{T_c - T}
\right)^{\frac{1}{2}}
\end{equation}

\begin{equation}
\lambda (T) = \frac{1}{\sqrt{2}}  \lambda_{L}(0) \left(
\frac{T_c}{T_c - T} \right)^{\frac{1}{2}}
\end{equation}

where $\xi_0$ ($ \approx \frac{\hbar v_{F}}{k_{B} T_c} $)
represents the smallest size of wave packets the superconducting
charge carriers can form, $l_e$ is the electron mean free path and
$\lambda_{L}(0) =  \frac{m}{\mu_{0} n e^2} $ is the London
penetration depth by 0 $K$. In the so-called "dirty limit" case
($\xi_0$ $>>$ $l_e$) the characteristic lengths behave as:

\begin{equation}
\xi_{d} (T) = 0.855 \sqrt{ \xi_0 l_e } \left( \frac{T_c}{T_c - T}
\right)^{\frac{1}{2}}
\end{equation}

\begin{equation}
\lambda_{d} (T) = \lambda_{L} (0) \left( \frac{\xi_0}{1.33 l_e}
\right) \left( \frac{T_c}{T_c - T} \right)^{\frac{1}{2}}
\end{equation}

The critical behavior of both the coherence length and penetration
depth is given by their proportionality to $(T_c - T)^{-
\frac{1}{2}}$. Therefore a temperature independent ratio of these
two lengths can be introduced:

\begin{equation}
\kappa =  \frac{\lambda (T)}{\xi (T)}
\end{equation}

which is the so called Ginzburg-Landau parameter. The value of
this parameter divides the superconductors into two classes. For
bulk superconductors if $\kappa < \frac{1}{\sqrt{2}}$ , the
material is a type-I superconductor and if $\kappa >
\frac{1}{\sqrt{2}}$ it is a type-II superconductor. Compared to
the type I superconductors, when applying an external magnetic
field, the type-II superconductors have an extra state which is
called the vortex or the mixed state.

\section{The mixed state.}

As we already mentioned the superconductors will not allow
magnetic flux into its interior (Meissner effect). However, the
magnetic flux penetrates the superconductor a length $\lambda$
before it is totally cancelled. To cancel the applied magnetic
flux screening currents (which create a magnetization in opposite
direction) will appear, the result being diamagnetism. The
electrons in the superconductor (including screening current) are
paired in so-called Cooper pairs \cite{Cooper}, the spin of these
electrons are opposite. Pair breaking can be occur by high enough
magnetic fields, its then energetically more favorable, for both
electron, to aline with the magnetic field. The property of
superconducting disappear (no screening currents can be created),
so the magnetic flux can not be cancelled anymore and the
superconductor falls back into the normal state. The field at
which this occurs is called the critical field. The typically
expelling of the magnetic field for a type II superconductors is
shown in figure (\ref{plaatje14}). The superconductor behaves as a
perfect diamagnet for $H$ $<$ $H_{c_1}$ and falls back in the
normal state when $H$ $>$ $H_{c_2}$. Between $H_{c_1}$ and
$H_{c_2}$ the superconductor is in the so-called mixed state (or
"vortex" state) meaning that some flux started to penetrate the
superconductor were it makes the specimen locally normal, which is
a very interesting property of the type II superconductor and will
be the focus of our study.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=12cm]{pictures/plaatje14}
\caption{Magnetization versus applied magnetic field for a bulk
type II superconductor. Flux start to penetrate the specimen at a
field of $H_{c_1}$ which is in vortex state between $H_{c_1}$ and
$H_{c_2}$. Above $H_{c_2}$ the specimen is a normal conductor
}\label{plaatje14}
\end{center}
\end{figure}

As already mentioned before $\kappa = \frac{\lambda}{\xi} <
\frac{1}{\sqrt{2}}$ for type II superconductors. The difference
between the ratio of $\lambda$ and $\xi$ have a big consequence
for the free energy density. To make clear what happens to the
free energy we will consider the following picture. Suppose we put
a type II superconductor next to a normal material in an applied
magnetic field, as illustrated in Fig. \ref{plaatje15}. The
illustration shows that the superconductor has a negative free
energy density, which means that the normal regions created in the
mixed state ($H_{c_1}$ $<$ $H$ $<$ $H_{c_2}$) want to have the
surface to volume ratio maximized (maximal energy profit). It
turns out that this favorable configuration are cylinders of
normal cores lying parallel to the applied magnetic field. This
cores are not sharply defined, they don't have a sharp boundary
from superconducting state to normal conducting state. The
transition region is roughly two times the coherence length (2
$\xi$) wide, so only in the center of the core is pure normal
state. Furthermore, the magnetic flux associated with each core
spreads into the surrounding material with roughly two times the
penetration depth (2 $\lambda$). To maximize the surface to volume
ratio the radius must be as small as possible. This has as
consequence that the total flux generated at each core is just one
fluxon. To generate this fluxon the core has to be surrounded by
circling supercurrents which generates a total flux of one fluxon.
For this reason it's convenient to call the core and his circling
supercurrent a vortex. The vortex current encircling a normal core
interacts with the magnetic field produced by a vortex current
encircling any other core, which results in a repulsion of these
cores. As a consequence the vortices will arrange themselves in a
lattice.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=8.5cm]{pictures/plaatje15}
\caption{Type II superconductor attached to a normal conductor
with an applied magnetic field. (a) Show how deep the magnetic
flux penetrates into the superconductor and how the order
parameter varies. (b) Shows the individual energy density due to
the magnetic contribution and electron ordering. (c) Shows the
total energy density due to the magnetic contribution and electron
ordering.}\label{plaatje15}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=7cm]{pictures/plaatje10}
\caption{Mixed state in applied magnetic field of strength greater
than $H_{c_1}$. (a) Lattice of cores and associated vortices. (b)
Variation with position of the order parameter. (c) Variation of
flux density.}
\end{center}
\end{figure}



\section{The Fluxoid}

As is shown in the BCS \cite{BCS} \cite{BCS2} theory the
resistance less current involves motion of pair electrons. When
considering the current, each pair may be threaded as a single
particle whose velocity is that of center of mass of the pairs.
The cooper pairs do not randomly scatter, so their waves remain
coherent over indefinite long distances. If we deal with a uniform
current density, all electron-pairs in the superconductor have the
same momentum and therefore the same wavelength. It is easy to
make a superposition of all the waves with the same wavelength,
and reduces into one single wave. With this wave we are able to
describe the motion of all electrons. The wavelength is related
with the momentum by the Broglie relation $\lambda =
\frac{h}{\mathbf{p}}$. If no current and or magnetic field is
involved by this superconductor, $\mathbf{p}$ will be zero and
$\lambda$ goes to infinity. The phase difference of two randomly
chosen points $X$ and $Y$ in the superconductor will then be zero.
However if we deal with a current and or magnetic field,
$\mathbf{p}$ will be finite (not zero) consequently $\lambda$ will
finite. This leads to a phase difference between the points $X$
and $Y$.

\begin{equation}
(\Delta \phi)_{XY}= \phi_X - \phi_Y= 2 \pi \int_X^Y
\frac{\mathbf{\hat{x}}}{\lambda} \cdot d\mathbf{l} \label{D1}
\end{equation}

where $\mathbf{\hat{x}}$ is a unit vector in the direction of the
wave propagation and $d\mathbf{l}$ is an element of $\lambda$
joining $X$ to $Y$. When we dealing with an applied magnetic field
and a current the momentum is given as $\mathbf{p} = 2m\mathbf{v}
+2 e \mathbf{A}$. Where the velocity $\mathbf{v}$ is related by
the supercurrent density as $\mathbf{J_s} = \frac{1}{2}n_s 2e
\mathbf{v}$. Using the Broglie relation and being aware that
$\mathbf{\hat{x}}$ is parallel to $\mathbf{J_s}$ and $\mathbf{A}$
equation ({\ref{D1}}) chances in:

\begin{equation}
(\Delta \phi)_{XY}= \frac{4 \pi m}{h n_s e} \int_X^Y \mathbf{J_s}
\cdot d\mathbf{l} + \frac{4 \pi e}{h} \int_X^Y \mathbf{A} \cdot
d\mathbf{l} \label{D2}
\end{equation}

where the first term is the phase difference due to the current,
and the second term due to the magnetic field. If we are not
dealing with a line fragment, but with a supercurrent circulating
around a closed path, see Fig. \ref{circel}. Where S is the
superconductor and N normal metal. Suppose that in N is a flux
density due supercurrents flowing around it. The integrals of
equation (\ref{D2}) will then be circle integrals. Realizing that
superelectrons to be represented by a wave, the phase change
around a close path must be $2 \pi n$, where $n$ is an integer. If
this was not the case, destructive interference would destroy the
wave. This phenomena also goes under the name "phase condition" or
"quantum condition". So with a little rewriting equation
(\ref{D2}) becomes:

\begin{equation}
\phi' = n \frac{h}{2e}= \frac{m}{n_s e^2} \oint \mathbf{J_s} \cdot
d\mathbf{l} + \oint \mathbf{A} \cdot d\mathbf{l} \label{D3}
\end{equation}

where $\phi'$ is named the fluxoid. In general the penetration
depth $\lambda$ is small, nearly all circulating current will in
fact be concentrated very close to the boundary of the normal
region. So taken a curve far enough of the normal boundary (so the
first term of equation (\ref{D3})) can be neglected), the fluxoid
is equal to the enclosed flux. We see therefore that the flux
contained within a super conductor should only exist as multiples
of a quantum, the fluxon, $\Phi_0$.

\begin{equation}
\Phi_0 = \frac{h}{2e} = 2.07 \cdot 10^{-15} Wb
\end{equation}

\begin{figure}[!t]
\begin{center}
\includegraphics[width=5cm]{pictures/plaatje11}
\caption{Superconductor enclosing a non-superconducting
region}\label{circel}
\end{center}
\end{figure}


\section{The Perfect Vortex Lattice}

Let's suppose we deal with a perfect superconductor, without
defects. To know how the vortex lattice arranges itself, we study
the positions of the vortex cores in the $XY$-plane with an
applied magnetic field in the $Z$-direction. To simplify things we
consider the approximation of linearized GL equations, which
neglects the term $\beta \Psi |\Psi|^2$ in equation (\ref{2.2}).
If we choose a convenient gauge choice, i.e. $A_y = H x$, we can
rewrite the GL equation as

\begin{equation}
\left[ - \nabla^2 + \frac{4 \pi i}{\Phi_0} H x
\frac{\partial}{\partial y} + \left(\frac{2 \pi H}{\Phi_0}
\right)^2 x^2 \right] \psi = \frac{1}{\xi^2}\psi
\end{equation}

To simplify even more we will discuss this equation, at $H=
H_{c2}$ where it has infinite number of solutions of the form:

\begin{equation}
\psi_k (x,y) = \exp(iky)\exp\left[-\frac{(x - x_k)^2}{2
\xi^2}\right]\label{E1}
\end{equation}

with

\begin{equation}
x_k = \frac{k \Phi_0}{2 \pi H}
\end{equation}

Qualitatively, we expect a lattice array of vortices instead of a
random distribution, because a lattice would have a lower energy
compared to a random distribution \cite{M Tinkham}. This allows us
to enforce the periodicity of the solution, which can easily be
achieved by resticting the values of $k$ to a discrete set.

\begin{equation}
k = k_n = n q
\end{equation}

where n is an integer. The period in $y$ will then be

\begin{equation}
\Delta y = \frac{2 \pi}{q}
\end{equation}

This restriction also induces a periodicity in the $x$-direction
through:

\begin{equation}
x_n = \frac{k_n \Phi_0}{2 \pi H} = \frac{n q \Phi_0}{2 \pi H}
\end{equation}

So the periodicity in $x$ is given by:

\begin{equation}
\Delta x = \frac{q \Phi_0}{2 \pi H} = \frac{\Phi_0}{h \Delta y}
\end{equation}

which leads to:

\begin{equation}
H \Delta x \Delta y = \Phi_0
\end{equation}

So each unit cell of the periodic array carries one quantum of
flux. Although we have shown this to be true for $H = H_{c2}$, it
holds for all the mixed state $H_{c1} < H < H_{c2}$ if $H$ is
replaced by $B$.

The wavefunction can then be written as:

\begin{equation}
\psi_L = \sum_n c_n \psi_n = \sum_n c_n \exp(inqy)\exp\left[ -
\frac{(x - x_n)^2}{2 \xi^2} \right]
\end{equation}

with $c_n = c_{n + \nu}$, for some $\nu$. If $\nu = 1 (c_n = c_{n
+ 1})$ the lattice is square and if $\nu = 2$ ($c_n = i c_{n +
1}$) the lattice is triangular. As shown by Abrikosov
\cite{Abrikosov}, the parameter determining the faverability of
possible solutions is:

\begin{equation}
\beta_A \equiv \frac{\langle
\psi_L^{4}\rangle}{\langle\psi_L^2\rangle^2}
\end{equation}

So the most energetically favorable lattice is to find the set of
$c_n$ for which $\beta_A$ is the smallest. Numerical calculations
show for a square lattice $\beta_A = 1.18$ and a triangular
(hexagonal) lattice $\beta_A = 1.16$. This implies that a
$\mathbf{triangular}$ $\mathbf{lattice}$ is the most energetically
favorable. It's interesting that the results agrees with that of a
simple argument based of the fact that triangular array is a
"closet packed". In this array the distance of the nearest
neighbors is:

\begin{equation}
a_{\triangle} = \left(\frac{4}{3}\right)^{\frac{1}{4}}
\sqrt{\frac{\Phi_0}{B}} = 1.075 \sqrt{\frac{\Phi_0}{B}}
\end{equation}

where as for the four neighbors in a square.

\begin{equation}
a_{\Box} = \sqrt{\frac{\Phi_0}{B}}
\end{equation}

Thus, for a given flux density $a_{\triangle} > a_{\Box}$. Taking
into account of the mutual repulsion of vortices, it is reasonable
that the structure with greatest separation of nearest neighbors
would be favored.

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=10cm]{pictures/plaatje3}
(b) \caption{(a) Square lattice. (b) Triangular or hexagonal
lattice.}
\end{center}
\end{figure}


\section{Flux Pinning and Vortex Motion}

If we have a superconductor in the mixed state and send a current
through it the Lorentz force is acting on the vortices.

\begin{equation}
\mathbf{F_L} = \mathbf{J_c} \times \mathbf{B}
\end{equation}

where $\mathbf{F_L}$ is the Lorentz force density, $\mathbf{J_c}$
the current density and $\mathbf{B}$ the magnetic flux density. In
a perfect superconductor, with no defects, the vortices will move
due to the Lorentz Force. But perfect superconductors do not
exist. All superconductors contain lattice defects like
impurities, precipitates, grain boundaries, dislocation loops,
point defects, etc. These defects will pin the vortices down on
their position. So to make vortices move, we have to put high
enough current to overcome the pinning barrier. The current
density needed to overcome this pinning force is what we call the
critical current density $\mathbf{J_c}$. The (maximum) pinning
force density is then given by:

\begin{equation}
|\mathbf{F_p}| = |\mathbf{J_c} \times \mathbf{B}|
\end{equation}

So when we apply a current with $\mathbf{J}$ higher than
$\mathbf{J_c}$ the vortices start to move. At a constant
$\mathbf{J}$ the vortices will move with a constant velocity. This
happens because the Lorentz force will be in equilibrium with a
friction force $\mathbf{F_r}$, due to the viscosity of the
material (Bardeen-Stephen model \cite{M Tinkham}).

\begin{equation}
|\mathbf{F_L}| = |\mathbf{F_r}|
\end{equation}

As it is known from textbooks of electromagnetism this motion (in
this case of the vortices) will produce an electrical field
$\mathbf{E}$,

\begin{equation}
\mathbf{E} = \mathbf{B} \times \mathbf{v}
\end{equation}

which is perpendicular to both $\mathbf{B}$ and $\mathbf{v}$, and
therefore in the same direction with $\mathbf{J}$. Therefore,
there will be dissipation in the sample due to the motion of
vortices. As a consequence, although the system is still
diamagnetic its electrical resistance $R$ will be non-zero !!
Quantitatively this flux-flow resistivity was derived in
Bardeen-Stephen model as:

\begin{equation}
\rho_f \approx \rho_n \frac{\mathbf{H}}{\mathbf{H}_{c_2}}
\end{equation}

where $\rho_f$ is the flux flow resistance density and $\rho_n$
the resistance density by $\mathbf{B}_{c_2}$. Experimentally the
flux flow resistivity can be obtained by measuring the slope of
$\mathbf{E}$ versus $\mathbf{J}$ curves. From Ohm's law the flux
flow resistance of a homogenous wire is:

\begin{equation}
\rho_f = \frac{|\mathbf{E}|}{|\mathbf{J}|}
\end{equation}

One remark should be made about the non-zero resistivity in the
superconductor. The fact that vortices are pinned in the
superconductor (below $\mathbf{J}_c$) makes it possible to have
non-zero current resistance, otherwise vortices will move by all
currents and so will be the dissipation. Therefore pinning is
actually a useful phenomenon for applications.


\section{Pinning Mechanism}

The defects in the superconductor can lead to different pinning
mechanism: the $\delta \kappa$ pinning and the $\delta T_c$
pinning. This means that locally the values of $\kappa$ and $T_c$
can differ compared to the average values of $\kappa$ and $T_c$ of
the superconductor. This results in a local change of the free
energy, which can make the pinning attractive or repulsive
depending if, locally, $\kappa + \delta \kappa$ and $T_c + \delta
T_c$ are bigger or smaller then the average value. Furthermore it
can be said that the pinning centers, in general, will be randomly
distributed and when we consider a stiff an very large vortex
lattice the total net force will be given as:

\begin{equation}
\mathbf{F_p} \sim \frac{1}{\sqrt{V}}
\end{equation}

where $V$ the volume. If we deal with a macroscopically large
system ($V \rightarrow \infty$), the total net force will be zero.
However, fluctuations are possible, which will be larger when the
volume over which the average is taken becomes smaller. With a pin
concentration of $n$ the total number of pins $N$ in a correlated
volume $V$ is.

\begin{equation}
N = V n
\end{equation}

The total pinning force due to fluctuations in a region is given
by:

\begin{equation}
\delta \mathbf{F} = (N \langle \mathbf{f}_i^2
\rangle)^{\frac{1}{2}}
\end{equation}

with $ \mathbf{f}_i $ the individual pinning interaction. This
'behavior' of the fluctuation force is determined from random walk
arguments. The pinning strength $W$ of our sample can be defined
as follows:

\begin{equation}
W\equiv n \langle \mathbf{f}_i^2 \rangle
\end{equation}

If we assume that all pins are identical, $\mathbf{f}_i^2 \approx
\frac{1}{2} \mathbf{f}^2_p$, then the pinning strength will be:

\begin{equation}
W \approx \frac{1}{2} n \mathbf{f}^2_p
\end{equation}

So $W$ depends on the concentration of pins $n$ and the elementary
interaction with FLL. When we are dealing with two different
pinning mechanisms in the superconductor the pinning force $W$ can
be written as:

\begin{equation}
\begin{split}
W &\approx n_1 \langle \mathbf{f}_1^2 \rangle + n_2 \langle
\mathbf{f}_2^2 \rangle + (n_1 + n_2) \langle \mathbf{f}_1 \cdot
\mathbf{f}_2 \rangle \\&\approx \frac{1}{2} n_1 \mathbf{f}^2_{p_1}
+ \frac{1}{2} n_2 \mathbf{f}^2_{p_2} + (n_1 + n_2) \langle
\mathbf{f}_1 \cdot \mathbf{f}_2 \rangle
\end{split}
\end{equation}

In the case the different pinning mechanisms are caused by
different uncorrelated pins the dot product $\langle \mathbf{f}_1
\cdot \mathbf{f}_2 \rangle$ can be left out.

\section{The Theory of Collective Pinning}

As we have shown in the section about the perfect lattice (section
2.5), the FLL would be, in absence of the pinning barriers, a
perfectly hexagonal (triangular) lattice. However in practice we
are always dealing with pinning centers which deform the lattice
structure. A theory of collective pinning has been developed for
an elastically deformed FLL by Larkin and Ovchinnikov \cite{A. I.
Larkin and Yu. N. Ovchinnikov}. In this theory, which is best for
weak pinning, it is assumed that independent domains are formed
which act collectively. To determine such a region, a displacement
correlation function is defined:

\begin{equation}
g(\mathbf{r}) \equiv \langle \left[\mathbf{u}(0) -
\mathbf{u}(\mathbf{r}) \right]^2 \rangle
\end{equation}

where $\mathbf{u}(\mathbf{r})$ is the 2D displacement vector (due
to shear and/or tilt stress) in the $xy$-plane to be found for
every lattice point $\mathbf{r}_i$ and the average is taken over
random point disorder. Correlated regions are elastically
independent of each other and their dimensions are defined by the
conditions \cite{Marchevsky}:

\begin{equation}
\langle \mathbf{u}^2(R_c,0)\rangle=r_p^2 ; \ \ \ \ \langle
\mathbf{u}^2(0, L_c)\rangle =r_p^2
\end{equation}

where $r_p$ is the elementary pinning interaction (also called the
pinning force range). $r_p$ $\approx$ $\xi$ for $B$ $<$ 0.2
$B_{c_2}$ and $r_p$ $\approx$ $\frac{a_0}{2}$ for $B$ $>$ 0.2
$B_{c_2}$. So, the correlated volume $V_c$ is given by:

\begin{equation}
V_c = R_c^2 L_c
\end{equation}

with $R_c$ the correlation length perpendicular to $\mathbf{B}$
and $L_c$ the correlation length parallel to $\mathbf{B}$. The net
pinning force acting on the vortices within this correlated $V_c$
depends of the collective action of the individual pinning center.
The pinning force density is given by:

\begin{equation}
|\mathbf{F_p}| = \frac{ | \delta \mathbf{F} |}{V_c} =
\left(\frac{W}{V_c}\right)^{\frac{1}{2}}= \left(\frac{W}{R_c^2
L_c}\right)^{\frac{1}{2}}\label{E5}
\end{equation}

While this is true for elastically deformed FLL, it is shown by
Mullock and Evetts \cite{S.J. Mullock and J.E. Evetts} that it is
also true for plastically deformed FLL.

\subsection{Elastic Tensor of the FLL}

The elasticity of the Fll is described by

\begin{equation}
\sigma_i =c_{ij} \varepsilon_j
\end{equation}

$\sigma$ is the stress, $\varepsilon$ the strain and $c$ the
elasticity modules, $\sigma$ is a force per surface unit.

\begin{equation}
\sigma_{ij} = \frac{df_i}{dO_j}
\end{equation}

$O_j$ is a surface with normal in the $j$ direction, $\varepsilon$
is displacement per unit length. $c_{ij}$ is the elastic tensor,
the unit of coefficients is $\frac{N}{m^2}$. For a hexagonal
lattice:

\begin{equation}
\left(
\begin{array}{c}
\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{yz} \\ \sigma_{xz} \\
\sigma_{xy}
\end{array}
 \right)=
 \left(
 \begin{array}{ccccc}
c_{11} & c_{12} &        &    \O  &  \\
c_{12} & c_{11} &        &        &  \\
       &        & c_{44} &        &  \\
       &        &        & c_{44} &  \\
       &   \O   &        &        & c_{66} \\
 \end{array}
 \right)
\left(
\begin{array}{c}
\varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{yz} \\ \varepsilon_{xz} \\
\varepsilon_{xy}
\end{array}
 \right) \label{E2}
\end{equation}

$c_{66} = \frac{1}{2}(c_{11} - c_{12} )$, so there are only three
independent coefficients in the elastic tensor. Where $c_{11}$ is
the compression modulus, $c_{44}$ the tilt modulus and $c_{66}$
the shear modulus.

\subsection{3 Dimensional Case}

What we already have seen above, is that, the correlated region
$V_c$ is given by $V_c = R_c^2 L_c$ with a corresponding pinning
force density $F_p$ of $(\frac{W}{R_c^2 L_c})^{\frac{1}{2}}$. To
have a better understanding of $R_c$ and $L_c$ we should look at
the energy terms, which concerns the elastic energy and potential
energy due pinning. The elastic energy of the vortex lattice is,
in analogy to free energy density of a deformed crystal.

\begin{equation}
F= \frac{1}{2} \sum_{i,k,l,m} c_{iklm} \varepsilon_{ik}
\varepsilon_{lm}\label{E3}
\end{equation}

where $c_{iklm}$ is the elastic modulus tensor of rank four. From
thermodynamics we know , that for infinitesimal small steps, $F$
can be written as:

\begin{equation}
dF = -SdT + \sum_{i,k} \sigma_{ik}d\varepsilon_{ik}
\end{equation}

So, by constant temperature.

\begin{equation}
\left(\frac{\partial F}{\partial \varepsilon_{ik}}\right) =
\sigma_{ik} = c_{iklm} \varepsilon_{lm}
\end{equation}

Substituting this back into equation (\ref{E3}) gives,

\begin{equation}
F = \frac{1}{2} \sigma_{ik} \varepsilon_{ik}
\end{equation}

Because the shear and tilt deformation of the lattice exceeds the
compression displacement we will only consider the energy density
due to the shear and tilting. With the use of the the elastic
tensor (\ref{E2}), $F$ becomes:

\begin{equation}
F = \frac{1}{2} c_{44} \varepsilon_{yz}^2 + \frac{1}{2} c_{44}
\varepsilon_{xz}^2 + \frac{1}{2} c_{66} \varepsilon_{xy}^2 \approx
\frac{1}{2} c_{66} \left(\frac{r_p}{R_c} \right)^2 + c_{44} \left(
\frac{r_p}{L_c} \right)^2
\end{equation}

where the elastic displacement caused by pinning centers gives
$\varepsilon_{yz}\approx \varepsilon_{xz} \approx \frac{r_p}{L_c}$
and $\varepsilon_{xy} \approx \frac{r_p}{R_c}$ is taken into
account. The work (potential energy) done by the pinning centers
per unit volume is given by:

\begin{equation}
Work = - \mathbf{F_p} \cdot r_p = - \left(\frac{W}{R_c^2 L_c}
\right)^{\frac{1}{2}} r_p
\end{equation}

The total energy density is given by:

\begin{equation}
U_{tot} \approx \frac{1}{2} c_{66} \left(\frac{r_p}{R_c} \right)^2
+ c_{44} \left( \frac{r_p}{L_c} \right)^2 - \left(\frac{W}{R_c^2
L_c} \right)^{\frac{1}{2}} r_p
\end{equation}

The correlation lengths can be determined to minimize the total
energy with respect to $R_c$ and $L_c$, $ \frac{dU_{tot}}{dR_c} =
\frac{U_{tot}}{dL_c}= 0 $ So,

\begin{equation}
R_c \approx \frac{2 \left(c_{44} c_{66}^3 \right)^{\frac{1}{2}}
r_p^2 }{W}
\end{equation}

\begin{equation}
L_c \approx 2 \left(\frac{c_{44}}{c_{66}} \right)^{\frac{1}{2}}
R_c \approx \frac{4 c_{44} c_{66} r_p^2 }{W}\label{Lc}
\end{equation}

By strong disorder the Larkin Domain will be small. This could be
due to strong pinning ($W$ is large) or to a soft lattice
($c_{44}$, $c_{66}$ are small). In the other cases the Larkin
correlation lengths will be much larger than the FLL parameter
$a_0$.

\subsection{2 Dimensional Case}

The thin films amorphous superconductors usually have $d$ $<<$
$L_c$. In that case (due to the weak pinning) the vortex lattice
remains ordered along the field direction and the flux lines are
considered to be straight. In that case the Larkin domain $V_c$
and pinning force $\mathbf{F_p}$ will change in:

\begin{equation}
V_c = R_c^2 d
\end{equation}

\begin{equation}
|\mathbf{F_p}| = \left(\frac{W}{R_c^2 d} \right)^{\frac{1}{2}}
\end{equation}

and the total energy density will change in:

\begin{equation}
U_{tot} \approx \frac{1}{2} c_{66} \left(\frac{r_p}{R_c} \right)^2
+ c_{44} \left( \frac{r_p}{d} \right)^2 - \left(\frac{W}{R_c^2 d}
\right)^{\frac{1}{2}} r_p
\end{equation}

Because we have such a small sample, we assume that the vortices
will be straight and we can neglect the tilt deformation. To find
$R_c$ we should minimize the total energy density with respect to
$R_c$, $\frac{dU_{tot}}{dR_c}=0$.

\begin{equation}
R_c \approx \frac{c_{66} r_p \sqrt{d}}{\sqrt{W}}
\end{equation}

More accurate calculations, done by Kes and Tsuei (see \cite{P.H.
Kes and C.C. Tsuei}) obtained for $R_c$ :

\begin{equation}
R_c = \frac{r_p c_{66} \sqrt(d) }{\sqrt{W}} \left[\frac{2
\pi}{\ln{\frac{w}{R_c}}} \right]^{\frac{1}{2}}
\end{equation}

where $w$ is the width of the sample. If $b \left(=
\frac{B}{B_{c_{2}}} \right) < 0.2$, $r_p$ can be taken equal to
$\xi$. For $b > 0.2$ $r_p$ is equal to $\frac{a_0}{2}$. So in this
two dimensional case the pinning force is proportional to:

\begin{equation}
\mathbf{F_p} \sim \frac{W}{r_p c_{66} d } \sim \frac{1}{d}
\label{E4}
\end{equation}

Furthermore we can say that if $R_c \leq a_0 $ we are dealing with
strong pinning. The FLL will the be as randomly distributed as the
distribution of the pinning centers. When we deal with weak
pinning, the FLL within the Larking domain can be very well
described with a hexagonal lattice.

When we look at a typical force density versus field for weak
pinning materials (see Fig. \ref{peak}) we can distinguished three
regimes. For low field the system is in a 2 dimensional collective
pinning (2DCP) regime. The pinning force can be described with
equation (\ref{E4}). The relation between $c_{66}$ and field is
depending on the type of pinning. At the onset of the peak, there
is a dimension cross over (DCO). The system will go from a $2D$ to
a $3D$ regime and is thus a 3 dimensional collective pinning
regime. This occurs when $L_c = \frac{d}{2}$ \cite{R. Wordenweber
and P.H. Kes} and the pinning force will be described by equation
(\ref{E5}). At the top of the peak $R_c \sim a_0$. There is no
longer collective pinning and the only field field dependence is
of pinning strength $W$. At the highest field the pinning force
decreases as $(1 - b)$ with increasing field. If we dealing with
thin films we are in the 2DCP regime all the time. The behavior of
the graph will still be the same, only the onset of the peak can
be interpret as a crossover of the FLL from elastic to plastic
deformation.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=10cm]{pictures/piek}
\caption{Normalized bulk pinning force versus $B/B_{c_2}$ for thin
a-NbGe film ($d$ = 1.24 $\mu m$, $T_c$ = 3.8 $K$) at various
values of $t$ $=$ $T/T_c$.}\label{peak}
\end{center}
\end{figure}


\section{Tunnelling}

If we have two identical normal conducting metals close to each
other, with vacuum between them (or a nonconducting material), it
can happen that an electron crosses this barrier from one metal to
the other. This is called tunnelling. In this process an electron
can be represented by a wave function, whose amplitude (outside
the conductor) drops with $
\sim\exp{\left(\frac{-x}{\zeta}\right)}$. $\zeta$ is of the order
$10^{-8}$ $cm$. To give the electron a significant chance to
tunnel between the metals, they must be very close ($<$ $10^{-9}$
$m$) to each other. But for tunnelling two more conditions (next
to small separation) have to be met. Firstly, the energy of the
system must be conserved, so before and after tunnelling the
energy of the whole system must be equal. Secondly, electrons can
only tunnel to states which are empty, otherwise the process is
forbidden by the Pauli principle. To make a difference in the
occupied states a voltage difference between the metals should
exist. The energy levels of the electrons in the metals will be
shifted with respect to each other, so tunnelling can occur. The
net constant tunnelling current can be written as:

\begin{equation}
I = A |T|^2 \int_{-\infty}^{\infty} N_1(E)N_2(E + eV)\left[f(E) -
f(E + eV)\right] dE \label{F1}
\end{equation}

where $A$ is a constant depending on the materials, $T$ is a
constant tunnelling matrix element, $N_1(E)$ and $N_2(E)$ are the
density of states of material 1 and 2, respectively.
\begin{equation}
f(E) = \frac{1}{\exp\left[(\varepsilon - \mu)/k_B T \right] + 1}
\end{equation}
is the Fermi-Dirac distribution function. $eV$ is the resulting
difference in the chemical potential, due to the applied voltage
$V$. The factors $N_1(E)f(E)$ and $N_2(E + eV)f(E + eV)$ give the
number of occupied initial states for material 1 and 2,
respectively. The difference between these two factors gives the
amount of electrons which can participate in tunnelling. When we
deal with two normal metals, equation (\ref{F1}) becomes like:

\begin{equation}
\begin{split} I_{nn} &= A|T|^2 N_{1}(0) N_2(0) \int_{-\infty}^{\infty}
[f(E) - f( E + eV )] dE  \\&= A|T|^2 N_1(0) N_2(0) eV \equiv
G_{nn} V
\end{split}
\end{equation}

where $G_{nn}$ is a well defined conductance independent of $V$.
The two density states could be removed from the integral, because
for normal metals $N(E) \approx constant$. The tunnel current for
normal to normal metals is proportional with $V$ and is
independent of temperature.

In the case of tunnelling between a superconductor and normal
metal the situation is a bit more complicated. As shown in the BCS
theory \cite{BCS} \cite{BCS2}, the energy of a superconductor is
lowered due to the phonon-electron interaction. The electrons form
a pair (cooper pair) and together they condensate into ground
state of the BCS hamiltonian, at $T=0$ all the pairs will condense
into the this ground state. So to excite the electrons in the
ground state, first this lowering energy must be over won. This
has as consequence that there is an energy gap $\Delta(0)$ in the
energy distribution. The rest of the energy distribution can be
approximated to be continuous. For this reason the density state
will differ compare to a normal metal, see Fig (\ref{density2}).

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{pictures/density2}
\caption{Density of states in superconducting compare to normal
state. All $\mathbf{k}$ states whose energies fall in the gap in
the normal metal are raised in energy above the gap in the
superconducting state. }\label{density2}
\end{center}
\end{figure}

The tunnelling current of equation (\ref{F1}) becomes:

\begin{equation}
I_{ns} = A |T|^2 N_{2}(0) \int_{-\infty}^{\infty} N_1(E) [f(E) -
f(E + eV) ] dE
\end{equation}

The density of states of the superconductor is no longer constant,
for $E$ $<$ $\Delta$,

\begin{equation}
N_1(E) = 0
\end{equation}

and for $E$ $>$ $\Delta$,

\begin{equation}
N_1(E) \sim \frac{E}{\left(E^2 - \Delta^2 \right)^{\frac{1}{2}}}
\end{equation}

therefore it can not be removed from the integral.  In general
this integral has to be solved numerically, and is also depending
on temperature, this because the gap $\Delta(T)$ is depending on
temperature. Plotting both $I_{nn}$ and $I_{ns}$  vs $V$ curves we
obtain Fig. \ref{plaatje2}. It's clear the $I_{nn}$ vs $V$ goes
linear, but for $I_{ns}$ there is no tunnelling current until $eV
\geq \Delta$ (at least for $T$ = 0). For finite temperatures the
gap will effectively becomes smaller, and behave like: $\Delta(T)
\sim \Delta(0) \left(\frac{T_c - T}{T_c} \right)^{\frac{1}{2}} $


\begin{figure}[!t]
\begin{center}
\includegraphics[width=8cm]{pictures/plaatje2}
\caption{I-V curves of tunnelling. $I_{nn}$ is the curve
corresponding by tunnelling between two normal conductors.
$I_{ns}$ is the curve corresponding by tunnelling between normal
and superconductor.}\label{plaatje2}
\end{center}
\end{figure}


\chapter{Growth and Characterization of a-MoGe and NbN thin films}

\section{Introduction}

As mentioned already in the previous chapter, to do our final
experiment we need a weakly pinned and a strongly pinned
superconductor. Both superconductors should have their $T_c$ above
$4.2$ $K$, because the STM measurements will be done in liquid
Helium ($^4$He) at $4.2$ $K$. Two good candidates will be the thin
films of amorphous Molybdenum Germanium (a-MoGe) and
poly-crystalline Niobium Nitrite (NbN).

The fact that NbN is poly-crystalline means that, instead of being
a single piece of crystal, it actually consists of more pieces of
crystal which can have different orientations with respect to each
other. The sudden change of different orientations in the crystal
leads to a lot of mismatches in the crystal lattice. The regions
of mismatches are called grain boundaries and form a strong
potential barrier for the vortices, in other words: they pin the
vortices very strongly.

a-MoGe is an amorphous material, which means that its lattice has
no long range order. For this reason big defects, like ground
boundaries, do not occur. Only short scale fluctuations like
density and chemical compositions may act as pinning centers. So,
abrupt changes do not occur, which makes the potential barrier
weaker than in the case of grain boundaries. This makes a-MoGe a
weak pinning material.

The NbN and the a-MoGe samples are thin film samples. Thin film
properties can deviate from their bulk properties, so before using
them it is important to test their properties. Resistivity
measurements should be done to determine the $T_c$ of the samples.
The Quantum Design physical properties measurement system (PPMS)
is used for this purpose. This system can measure the resistivity
of the sample as function of temperature (between $2$-$300$ $K$)
and applied magnetic field (up to 9 $T$). Rutherford Back
Scattering (RBS) is done on the a-MoGe samples to determine the
chemical composition and the ratio between molybdenum and
germanium and X-ray diffraction (XRD) to determine the
amorphousness or crystalline structure of the samples.

\section{Fabrication of thin films of a-MoGe and NbN samples}

The resistivity measurements of a-MoGe and NbN, as functions of
temperature and magnetic field, (in the PPMS) are four points
measurements. This requires that the samples should be fabricated
using the design shown in \ref{design}. The structure has 4
contact pads attached to a long strip (0.6 $mm$ long and 100 $\mu
m$ wide). The contacts and strip are made from the same material.
Copper wires are mounted by pressing them with Indium on the
contacts. The current will flow through the strip via the
$I$-contacts and the voltage can be measured with the other two
contacts.

To create such a structure, the lift off procedure is used, which
consists of 5 main steps. In this process two electron sensitive
positive resists are used, namely polymethyl methacrylate (PMMA)
and polydimethylglutarimide (PMGI) \cite{chem}. Electron beam
lithography is used to expose the resist and a sputtering machine,
to deposit the materials on the Si substrates.

\begin{figure}[hb]
\begin{center}
(a)
\includegraphics[width=6cm]{pictures/structuur}
\includegraphics[width=6cm]{pictures/zoom}
(b)
 \caption{(a) Designed structure. (b) Zoom in on the strip.}\label{design}
\end{center}
\end{figure}

\subsection{The Lift-off Procedure}

The detailed parameters for this procedure can be found in
appendix A.

\paragraph{Step 1.}

On a clean silicon (Si) substrate the PMGI resist is spined. After
baking, to get rid of the solvent, PMMA is spined on top of it.
Again the solvent is baked away. (Fig \ref{step1})

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{pictures/step1}
\caption{Step 1.}\label{step1}
\end{center}
\end{figure}

\paragraph{Step 2.}

PMMA and PMGI are both sensitive to electrons. The resists can be
exposed with the electron beam. We used a JEOL 820 scanning
electron microscope (SEM) to expose the pattern of the designed
structure on the sample. (Fig \ref{step2})

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{pictures/step2}
\caption{Step 2.}\label{step2}
\end{center}
\end{figure}

\paragraph{Step 3.}

Both resists are positive. This means that the exposed area will
be removed after developing. First the PMMA will be developed with
a PMMA developer. This PMMA developer hardly develops any PMGI and
develops the exposed PMMA much faster than the unexposed PMMA.
After that, the PMGI layer under it will be developed with PMGI
developer. This developer does not develop the PMMA but develops
the exposed PMGI faster then the unexposed PMGI. However, if all
exposed PMGI is developed a certain amount of unexposed PMGI will
also be developed. This creates a undercut, which will be useful
in the last step. (Fig \ref{step3})

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{pictures/step3}
\caption{Step 3.}\label{step3}
\end{center}
\end{figure}

\paragraph{Step 4.}

The shape of the designed structure is now created on top of the
substrate. The desired material (MoGe or NbN) can be sputtered on
the sample. The sputtering is done by the Leybold Z-400, which is
an RF diode sputtering system. In the MSM group exists a great
deal of expertise and experience on how to make good amorphous
thin superconducting films with the Leybold Z-400 set-up. Taking
advantage of all this prior knowledge, the sputtering was done
with a deposition pressure of $3 \cdot 10^{-2}$ $mbar$ and a base
pressure of the order of $ \sim 2-3 \cdot  10^{-6}$ $mbar$. The
system is equipped with 3 targets (MoGe, Nb, Au) and 3-channel gas
blending (Ar, N$_2$ and O$_2$). An Ar-plasma is created to sputter
the material from the target on the substrate. The cooling is done
with circulating water.  After sputtering the situation looks like
in figure \ref{step4}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{pictures/step4}
\caption{Step 4.}\label{step4}
\end{center}
\end{figure}

\paragraph{Step 5.}

Of course not only the developed areas contain the sputtered
material, but also the top of the remaining resists. However, the
resists can be removed by 1-methyl-2-pyrrolidinone. By removing
the resist, we automatically removed the sputtered material on top
of it as well. This is why the whole procedure is called lift-off.
The undercut took care that the remaining sputtered material did
not touch the PMGI layer. This prevented that the structure would
be damaged during the lift off.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{pictures/step5}
\caption{Step 5.}\label{step5}
\end{center}
\end{figure}

This 5 step procedure, just described above, is a reliable
procedure to create well structured samples necessary to do four
point measurements with current applied in a controlled way.

\section{Sample characterization}

The properties of four different samples were tested. One of them
was a $42$ $nm$ thick NbN film. Because its  growth procedure was
quite well known \cite{private} this sample was only tested in the
PPMS for final resistivity measurements. The three other samples
were a-MoGe films with thicknesses around $500$ $nm$. The
differences between the a-MoGe samples were in the different
sputtering procedures. The reason for using different procedures
is that during sputtering the samples could heat up and the
increased temperature can promote the appearance of crystalline
grains (the crystallization temperature of MoGe is not that big).
To suppress the possible heating, sputtering can be done with
interruptions as follows: after one minute of sputtering, the
target will be moved from the substrate for a minute and then
moved back. This will be repeated for the whole sputtering
procedure. Another alternative to suppress the heat is to glue the
sample with silver paint on the holder which improves the heat
link with the circulating cooling water and therefore cools down
the sample In order to study the effect of interruptions and the
use of silver paint, three different sputtering procedures were
used. Sample a-MoGe-1 is sputtered with interruptions and silver
paint, sample a-MoGe-2 is sputtered with silver paint only and
sample a-MoGe-3 is sputtered continuously without Ag paint.

\begin{table}[!hb]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline
 & Base   & $Ar$ &
 $N_2$ & $S_t$  & Breaks & Ag  & $T_{c}$ & $\mu_{0} H_{c2}$  \\
 & Pressure      &  Rate & Rate  &
 & & Paint & & \\
 & ($mbar$) & ($ \% $) & ($ \% $) & ($min$) & &
 & ($K$) & ($T$) \\
\hline \hline NbN & $1.9 \cdot 10^{-6}$
& 18 & 10 & 16 & no & no & $10.6 $ & $>9.0$ \\
\hline MoGe-1 & $3.2 \cdot 10^{-6}$
& 25 & 0 & 91 & yes & yes & 7.3 &$7.2$ \\
\hline MoGe-2 & $2.4 \cdot 10^{-6}$
& 25 & 0 & 91 & no & yes & 7.2 & 7.2\\
\hline MoGe-3 & $1.5 \cdot 10^{-6}$
& 25 & 0 & 91 & no & no & 7.4 & 7.4 \\
\hline
\end{tabular}
\caption{Parameters of the sputtering conditions, where $S_t$
stands for the sputter time. All samples are sputtered by a
$V_{DC}$ of $1.0$ $kV$ and measured by 4.2 $K$. The NbN sample is
42 $nm$ and the a-MoGe samples are around $500$ $nm$}
\end{centering}
\end{table}

On samples a-MoGe-2 and a-MoGe-3 RBS measurements were performed
(by M. Hesselberth in AMOLF, Amsterdam) in order to determine the
chemical composition and the ratio between Mo and Ge in the the
samples. The results are plotted in Fig. \ref{RBS} and Fig.
\ref{RBS2} with the best possible fit function. From these results
it is clear that only the expected chemical elements Mo and Ge
were sputtered and that the chemical composition for a-MoGe-2 is
a-Mo$_{76}$Ge$_{24}$ and for a-MoGe-3 Mo$_{78}$Ge$_{22}$. We can
conclude that there are no significant chemical composition
differences between the samples used in this study.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=13cm]{pictures/RBSMoGe-2}
\caption{RBS measurements on a-MoGe-2 sample.}\label{RBS}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=13cm]{pictures/RBSMoGe-3}
\caption{RBS measurements on a-MoGe-3 sample.}\label{RBS2}
\end{center}
\end{figure}

\newpage

XRD measurements were performed by R. Hendrikx on all three the
a-MoGe samples. In order to determine how amorphous or crystalline
the samples are. The results are plotted in the Fig. \ref{XRD} In
general, crystals, due to their LRO lattice structure, have a
clear preferred diffraction direction and show sharp peaks in
intensity specific to the chemical elements and the type of
lattice. Amorphous samples do not have LRO and therefore they
diffract more randomly, which will give a broader peak. So the
width of the peak is a measure of the amorphousness of the
samples.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=6.5cm]{pictures/XRDMoGe-1}
\includegraphics[width=6.5cm]{pictures/XRDMoGe-2}
\includegraphics[width=6.5cm]{pictures/XRDMoGe-3}
\caption{X-ray intensities vs the incident angle.}\label{XRD}
\end{center}
\end{figure}

Comparing the three a-MoGe samples, it is clear that a-MoGe-1 is
the most and a-MoGe-3 is the least amorphous. However, since the
peak amplitude of a-MoGe-3 is still small when compared to
crystalline sample one can speculate that instead of being
complete amorphous the sample contain also some small crystalline
regimes

We can conclude, on the basis of the XRD results, that there is a
difference due to the different sputter procedure used. It is
clear that better cooled samples show more amorphousness, but we
cannot really discern between the interval sputtering or Ag paint
glued sputtering.


\section{Transport measurements}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=13.5cm]{pictures/RvsT_NbN}
\includegraphics[width=13.5cm]{pictures/RvsT_MoGe}
\caption{Resistance vs temperature for NbN and a-MoGe
samples.}\label{RvsT}
\end{center}
\end{figure}

In Fig \ref{RvsT} resistance versus temperature measurements (in
the absence of magnetic field) for NbN and a-MoGe samples are
plotted. The superconducting transitions are very clear: NbN has a
$T_c$ around $10.6$ $K$ and the a-MoGe samples around $7.3$ $K$.
They satisfy the condition to have a $T_c$ above $4.2$ $K$, which
makes it possible to use them for the STM experiments.

While the $T_c$ of the a-MoGe samples lie very close to each
other, the normal state resistivity of a-MoGe-2 differ compared to
the other the a-MoGe samples. Since no detailed investigation was
made we can only speculate that the different sputtering procedure
could be responsible.

More important for our purpose is the magnitude of the critical
current density $J_c$ of the samples, because it is a direct
measure of the pinning strength. In practice the critical current
$I_c$ is measured, from which the critical current density $J_c$
can be derived. As explained in the section 2.6 of chapter 2 the
$I_c$ is the minimum current needed to overcome the pinning force.
A current higher than $I_c$ starts vortex motion and due to the
produced electric field there will be dissipation. Experimentally
the $I_c$ can be obtained from $V$-$I$ measurements which are done
by sweeping the currents and recording the voltage change across
the sample at a constant $T$ and $H$. The $I_c$ is magnetic field
dependent, so $V$-$I$ measurements at different applied magnetic
fields (between 0 and 7 $T$) at 4.15 $K$ were done for all
samples. A typical $V$-$I$ result is shown in Fig. \ref{VI} for a
field of 2 $T$.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=13.5cm]{pictures/IV}
\caption{Typically $V$-$I$ graph.}\label{VI}
\end{center}
\end{figure}

At small currents ($I$ $<$ $77$ $\mu A$) the votices are still
pinned and don't move. When the applied current is increased above
77 $\mu A$ the force exerted on them is large enough to make them
move. This creates dissipation and an electrical field, and
therefore a measurable voltage appears across the sample.
Remarkably this voltage is linear with $I$ which is in agreement
with previous measurements done in literature and can be explained
as flux flow resistivity (see section 2.6 of chapter 2).

To find the $I_c$ we used the so-called "$1.0$ $\mu V$ criterium".
This means that if $V$ $<$ $1$ $\mu V$ the vortices are considered
static and the value of the current corresponding to overcoming
the 1 $\mu V$ threshold is defined as $I_c$. We mentioned here
that the same qualitative results are obtained also if we use "100
$nV$ criterium" or a constant velocity criterium. The $J_c$ is
derived by dividing the $I_c$ by the dimensions of the sample.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=13.5cm]{pictures/JvsH_NbN}
\caption{Critical current density vs applied magnetic field for
NbN sample.}\label{JvsH}
\end{center}
\end{figure}

In Fig. \ref{JvsH} and Fig. \ref{JvsH2} $J_c$ is plotted as
function of applied magnetic field at temperature of $4.15$ $K$
for the NbN and a-MoGe samples, respectively. The shape of the
curve for the a-MoGe samples agrees with the theoretical
predictions. At low fields (individual pinning regime) the pinning
is effectively stronger than by high fields (collective pinning 2D
regime), this explains the divergent behavior of the $J_c$ for
small fields. However, the peak regime is not observed because no
measurements were done by significantly high enough fields. The
behavior of the three a-MoGe samples is qualitatively the same,
indicating that the different sputtering procedures do not
influence much the critical current density. When camping with
NbN, it turns out that the critical current density of NbN is at
least two orders of magnitude higher than for the a-MoGe samples.
Therefore the pin strengths significantly differ from each other
indicating that the vortices in NbN are strongly pinned and in
a-MoGe weakly pinned.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=13.5cm]{pictures/JvsH_MoGe}
\caption{Critical current density vs applied magnetic field for
a-MoGe samples.}\label{JvsH2}
\end{center}
\end{figure}

In order to complete the picture, we have checked whether the
a-MoGe sample show the "peak effect" by measuring resistance
versus the applied magnetic field. The results are plotted in Fig.
\ref{piek}. From this graphs $\mu_{0} H_{c2}$ can be obtained
\cite{Geers}, which is the field for which the material falls back
in his normal state. The a-MoGe samples have a $\mu_{0} H_{c2}$ of
$7.2$ $T$, $7.2$ $T$ and $7.4$ $T$, respectively. I should mention
that we also measured NbN, which $\mu_{0} H_{c2}$ is still not
reached at $9$ $T$ (which is the limit of our set-up). This accord
with the literature value for NbN, which has an $\mu_{0} H_{c2}$
above $12.5$ $T$ at 4.2 $K$.

The magnetic field of the STM can reach a value of $2.2$ $T$.
Measuring in high fields gives a better representation of the
vortex lattice, because more of them can be imaged at once. All
measured $\mu_{0} H_{c2}$ of the samples are bigger then $2.2$
$T$, which enable us to use the STM magnet in full range.

The sudden drop in the $R$ versus $\mu_{0} H$ graph (Fig.
\ref{piek}) of the a-MoGe sample indicates the "peak effect". This
effect is interpretable as a crossover of the FLL from elastic to
plastic deformation. This is an important feature of the
collective pinning theory and again shows us to good quality of
our a-MoGe samples.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=13.5cm]{pictures/RvsH_MoGe}
\caption{Resistance vs applied magnetic field for the a-MoGe
samples.}\label{piek}
\end{center}
\end{figure}

\section{Conclusion}

The NbN satisfies the conditions for strong pinning
superconductor. The $T_c$ and $\mu_{0} H_{c2}$ are both high
enough, and is a useful candidate for the final experiment.

In principle all a-MoGe samples satisfy the conditions for weak
pinning superconductor. Also all $T_c$'s and $B_{c2}$'s are high
enough. Despite of the apparently crystalline part in the a-MoGe-3
sample the behavior is qualitatively the same as the other two
a-MoGe samples. The advantages of using the third sputtering
procedure are of practical nature: firstly, if no interruptions
are made during the sputtering process the speed of the procedure
is increased and the risk of non-uniform sputtering are reduced.
Secondly, and most importantly, by avoiding the use of Ag paint
one reduces the risks of dirt deposited on top of the sample,
which is a big problem for the STM. However, for a future
experiment it is desirable to use a sample that shows no
crystalline peak.

\chapter{Fabrication of Channels}

\section{Fabrication of Channels in the Sample}

Now that we know how to fabricate good quality thin films, we can
proceed with the following step. The idea is to fabricate a sample
with channels using electron beam lithography. The method used
with success in this group \cite{Besseling} \cite{Pruijmboom}
\cite{Benningshof} to create easy flow vortex channels requires a
double layer of superconductors, for instance, a strong pinning
superconductor as the top layer and a weak pinning superconductor
as the bottom layer. Channels of few hundred of nanometers wide
are created by etching away the strong pinning superconductor. In
this case sharp channel edges can be achieved, which is an
essential ingredient for a controlled study of vortex dynamics.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8cm]{pictures/NbGe}
\caption{Cross-section of sample used for the study of easy flow
vortex channels.}
\end{center}
\end{figure}

However, if one wants to do STM measurements on such a sample it
will have problems with the sharp edges. The STM tip is very sharp
at its end but usually to a pyramid or cone that is quite broad at
its base. Therefore when scanning across the channel it cannot
follow sharp edges and may crash An alternative design is to
change the position of layers. The strong pinning layer with
channels as the bottom layer (having sharp edges) with the weak
pinning layer deposited on it. Although this gives a more smooth
top layer, which can be followed by the STM the pinning felt by
vortices in the top layer will still have sharp step. The
fabrication of such a sample has been done using a $7$ step
procedure, which will be described below.

\subsection{The 7 Step Procedure}

\paragraph{NbN sputtering.}On a clean silicon substrate $50$ $nm$ of NbN
is RF sputtered in Z-400 (see the previous chapter and appendix D
for details ). PMMA is spined on top and then baked to remove the
solvent (Figure \ref{step6}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step6}
\caption{Step 1.} \label{step6}
\end{center}
\end{figure}

\paragraph{E-beam exposure.}The channels are written with the E-beam
in the PMMA by directing the beam perpendicularly on the substrate
at the designed locations. The cross-section of the structure
looks like in figure \ref{step7}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step7}
\caption{Step 2.}\label{step7}
\end{center}
\end{figure}

\paragraph{Developing.}The exposed area is removed with a PMMA
developer, creating channels in the resist. Since the PMMA is a
very high resolution material the channels will have quite sharp
edges.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step8}
\caption{Step 3.}
\end{center}
\end{figure}

\newpage

\paragraph{Ion Beam etching.} In this step channels will be created in
the NbN layer. The method consist of etching using an ion beam
etcher (Kauffmann type). The etching is done by an Ar pressure of
$4.0 \cdot 10^4$ $mbar$, beam voltage and current of $350$ $V$ and
$10$ $mA$, respectively. This set-up accelerates Ar-ions to a high
speed and then the Ar-ions collapse on the sample. Consequently
the NbN film that is not protected by PMMA is removed from the
sample. The remaining PMMA on the sample serves as a mask and
protects the parts of NbN which should not be etched. In this way
we etch through the NbN layer. To be sure that all NbN is etched
through, a small amount of the silicon will be etched as well.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step9}
\caption{Step 4.}
\end{center}
\end{figure}

During this process, however, a part of the etched NbN can be
redeposited on the side of the PMMA. The redeposition is usually a
problem: it is hard to remove and will remain after removing the
PMMA. This problem is known technically as 'ears' and is
successfully solved in our group by van Rijn \cite{Rijn} using a
rotating etching holder.

\paragraph{PMMA removal.}The remaining PMMA was removed with
1-methyl-2-pyrrolidinone. To be sure that no PMMA remained on the
top, the sample was etched in a (home-built) RF oxygen-etcher. The
operation principle of the etcher is the following: By a pressure
of $1.0 \cdot 10^{-1} $ $mbar$ and a RF current of $100$ $W$
oxygen plasma is created. The plasma consists of oxygen radicals,
which react very strongly with the remaining PMMA (and other dirt)
cleaning the sample surface very effectively. To get rid of any
oxidant layer, created during the oxygen etching, the sample was
etched again for 30 seconds. The results, as shown in figure
\ref{step10}, is a clean NbN sample with $\pm$ $50$ $nm$ deep
straight channels.

\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{pictures/step10}
\caption{Step 5.}\label{step10}
\end{center}
\end{figure}

\paragraph{a-MoGe sputtering.} About $500$ $nm$
a-MoGe is sputtered on the sample (see Fig. \ref{step11}.) at
Z-400 continuously and without silver paint glued on the substrate
(see the previous chapter).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step11}
\caption{Step 6.}\label{step11}
\end{center}
\end{figure}

\paragraph{Au sputtering.} Immediately after a-MoGe sputtering and
just before the STM measurements is performed a thin layers of
gold (few $nm$) is sputtered on top. This method was developed in
this group \cite{Au} and has proved to be essential for STM
measurements. Without this protection layer the sample would
oxidize. Oxidation (even in small amounts) is detrimental to map
the superconducting gap, which is essential to image the vortices.
The gold layer will protect the sample against oxidation and still
allow to perform gap spectroscopy due to the proximity effect. By
proximity effect it is meant that the the wavefunction can
overcome the gold later, which make it still possible to to
tunnelling spectroscopy.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{pictures/step12}
\caption{Step 7.}\label{step12}
\end{center}
\end{figure}

\newpage

\section{Problems with Exposure of 'Small' Structures.}

An important remark should be made about step 2 (e-beam exposure),
where PMMA was exposed to create the desired shape of the channel.
In order to make the PMMA sensitive to develop, it should be
exposed using a high enough dose (intensity) of the electron-beam.
The dose is defined as:

\begin{equation}
Dose = \frac{I_B \times t_D}{(\Delta A)^2}
\end{equation}

where $I_B$ is the beam current, $t_D$ the area dwell time and
$\Delta A$ the area step size. For 'big' ($ > 1 \mu m$ )
structures the PMMA is usually exposed with a dose of $161$
$\frac{\mu A s}{cm^2}$. For 'small' structures ($ < 1 \mu m$ )
three problems arise: The resolution of the set-up (Jeol 820 SEM),
the spot size of the electron-beam and the proximity effect.

Firstly, the Jeol 820 SEM has an accuracy of $\approx$ $50$ $nm$.
Structures with this dimension or smaller are impossible to made.
Secondly, the spot size of the electron beam depends on the
magnitude of the current (see Fig. \ref{beamcurrent}) obviously
being larger at higher currents. The limitation imposed by the
spot size is usually not a problem for structures of few microns.
However, for structures of a few hundreds of nanometers, as it is
the case with our channels, this might play a role: if a high
current is used one can expose more area than designed. Last, but
not least the "proximity effect" limits the resolution of electron
beam lithography as well. Although it has the same name as the one
discussed is section 4.2 this proximity effect refers to a
completely different issue. It is, basically due to scattering of
electrons in substrate and resist. During the exposure the
electrons will not only expose the resist at the position they are
applied, but they can also be scattered in the neighborhood. Again
more area could be exposed than designed. All these problems mean
that the design for 'small' structures (channels) is not perfectly
reproduced by the e-beam if the exact dimensions are to be
introduced as parameters in the program.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=13.5cm]{pictures/beamcurrent}
\caption{E-beam spot vs current}\label{beamcurrent}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12.5cm]{pictures/channelsall}
\caption{Top view SEM pictures taken on channels written with
different dose but all designed with a width of $400$ $nm$. (a)
Dose is $161$ $\frac{\mu A s}{cm^2}$ and corresponding channel
width is 360 $nm$. (b) Dose is $322$ $\frac{\mu A s}{cm^2}$ and
corresponding channel width is 480 $nm$. (c) Dose is $483$
$\frac{\mu A s}{cm^2}$ and corresponding channel width is 732
$nm$. (d) Dose is $644$ $\frac{\mu A s}{cm^2}$ and corresponding
channel width $>$ 1 $\mu m$ }\label{Dose1}.
\end{center}
\end{figure}

To fabricate channels with the desired channel width one should
tune the parameters, such as the dose, current, developing time of
the PMMA and channel width. In our case we kept the current (0.5
$nA$), developing time (1.0 $min$) the same and introduced as
parameter for channel width the same value of 400 $nm$. We only
changed the dose by varying the dwell time and the area step size.
As a result channels with the doses of $161$ $\frac{\mu A
s}{cm^2}$, $322$ $\frac{\mu A s}{cm^2}$, $483$ $\frac{\mu A
s}{cm^2}$ and $644$ $\frac{\mu A s}{cm^2}$ are written, as shown
in figure \ref{Dose1}. It is quite clear that increasing the dose
increases the channel width well above the aimed value of $400$
$nm$ (except channel in Fig. \ref{Dose1} (a)). By tuning the dose
the desired width can be achieved. However, the channel (Fig.
\ref{Dose1} (a)) written by a dose of $161$ $\frac{\mu A s}{cm^2}$
is smaller then $400$ $nm$ (namely, 360 $nm$). It can be explained
as that the PMMA is under exposed by a dose of $161$ $\frac{\mu A
s}{cm^2}$, but because of the proximity effect and beam diameter
certain parts (middle of the channel) still receives enough dose
for developing. The channel width of 360 $nm$ qualifies to the
experimental purposes and was therefore preferred.


\section{Cross-linking PMMA.}

Another important remark should be made about step 4 (ion beam
etching). It is a known problem \cite{PMMA2} that PMMA can form
'bubbles' when etched with a ion beam etcher. The 'bubbles' are in
fact decross-linked polymers of the PMMA, which causes
inhomogeneity in the PMMA mask. The feature prevents us to etch
homogenously in the NbN, which results in channel edges that are
not sharp as shown in \ref{Dose2} (a). As mentioned already in the
introduction the sharpness of the edges is an essential feature of
easy flow vortex channels and, as shown by R. Besseling
\cite{Besseling} it can lead to a completely different dynamics
behavior.

A solution to this problem \cite{PMMA} is to cross-link the PMMA
polymers back to their initial state before being etched. We
succeeded in this by directly rewriting all PMMA (so the whole
sample) after developing (step 3) with a 'very' high dose, which
will cross-link the PMMA polymers again. The used dose is $20000$
$\frac{\mu A s}{cm^2}$, which is 124 times more then the dose used
when writing channels. As a result the PMMA mask becomes
homogenous again, which makes it possible to etch homogenously
through NbN. The result of etching is compared in Fig.
\ref{Dose2}. (b) to the case where this procedure was not used
\ref{Dose2}. (a). The result is quite clear. The channels look
homogenous with sharp edges. As a side effect, however, we have
discovered that the cross-linking has also increased the channel
width by $30$ $nm$ at around 390 $nm$, which for our samples is,
in fact, no problem since our aimed channel width was 400 $nm$.

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=6cm]{pictures/extradoseslecht}
\includegraphics[width=6cm]{pictures/extradosegoed}
(b)
\end{center}
\caption{Top view SEM pictures taken after etching the channels
with Ar ion beam. Channels etched without (a) and with (b)
cross-linking procedure.}\label{Dose2}
\end{figure}


\section{Smoothing the Surface}

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=6cm]{pictures/AFMNbN}
\includegraphics[width=6cm]{pictures/AFMMoGe}
(b) (c)
\includegraphics[width=6cm]{pictures/AFMNbN2}
\includegraphics[width=6cm]{pictures/AFMMoGe2}
(d) \caption{Top view AFM pictures taken on a sample with channels
etched in NbN without (a) or with (b) an a-MoGe layer on top. (c)
and (d) show the corresponding line profile as indicated in (a)
and (b).} \label{AFM}
\end{center}
\end{figure}

As explained above, in step 6 a-MoGe is sputtered on the sample.
As a result we expected that the surface probed by STM will be
smoothed. In order to check that we have performed AFM
measurements. In figure \ref{AFM} AFM pictures with corresponding
profiles are shown. Fig. \ref{AFM} (a) shows an AFM picture of the
sample with channels in the etched NbN without a-MoGe on top, the
corresponding profile is plotted in \ref{AFM} (c). The AFM tip is
expected to have more or less the same size as the channel width,
which can lead to a convolution of the AFM in the profile, as
illustrated in figure \ref{AFM2}. This mat explains the small
roundings in the profile. Apart from that, the profile show
straight channel banks and sharp drops due to the channels. The
periodicity is around $1$ $\mu m$ and the etching depth is
$\approx$ $65$ $nm$. All these parameters correspond to our
expectation, which indicates that sharp channel edges are probaly
well fabricated.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=5cm]{pictures/AFM}
\caption{AFM tip scanning channel edge }\label{AFM2}
\end{center}
\end{figure}


Fig. \ref{AFM} (b) shows an AFM picture of the sample with
channels in the etched NbN with $\approx$ 500 $nm$ a-MoGe on top
for smoothness of the surface. After AFM measurements profile
\ref{AFM} (d) is obtained. By comparing the profiles of \ref{AFM}
(c) and (d) a clear smoothing of the surface is observed. Of
course the periodicity (of $1$ $\mu m$) is conserved, but more
important this surface is operable for STM measurements.


\chapter{Results and Discussion}

\section{Introduction}

In the previous chapter we described the fabrication of sample for
STM measurements. We have these STM measurement performed with the
help of F. Galli and G. van Baarle. Now is the moment to give a
short introduction in the STM technique and the set-up used. The
STM is home built, constructed by the group members of the MSM
group \cite{Nature}. The measurements are done in liquid helium
(He$^4$) at $1$ bar and at a temperature of $4.2$ $K$ see figure
\ref{vessel}. The Set-up is equipped with a $2.4$ $T$
superconducting magnet used in persistent mode. The STM tip used
in measurement is made of platinum-iridium, which is simply cut
from a thin wire by a scissor. The scissor tears the material
apart, forming sharp tips. The scanning range in the $XY$-plane is
approximately $1.5 \times 1.5$ $\mu m$ and is not equipped with a
$X$-$Y$ table. The lack of such a table prevents us from scanning
over the whole sample and forced us to fabricate a channel array
with a periodicity smaller than $1.5$ $\mu m$. For the same reason
the tip must be aimed at room temperature, hoping that during the
cooldown procedure it does not shift too much. To increase the
chance to stay aimed at the channels after cooldown, $900$ writing
fields (figure \ref{Channels}) were stitched toghter other. In
this way a sample of $3 \times 3$ $mm^2$ array of channels was
fabricated.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=4cm]{pictures/STMsetup}
\caption{Schematic drawing of the cryogenic system: (a) STM unit,
(b) 2.4 T magnet, (c) location of bias voltage divider, (d)
sliding seal, (e) flushing valve.}\label{vessel}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
(A)
\includegraphics[width=5cm]{pictures/channels}
\includegraphics[width=3.6cm]{pictures/zoom2}
(B) \caption{Designed structure of channels. (A) Writing field of
$100 \times 100$ $\mu m^2$, consisting of $100$ channels (blue)
with a length of $100$ $\mu m$, a width of $400$ $nm$ and a
periodicity of $1$ $\mu m$. More writing fields can be stitched
among each other. In this way a sample of $3 \times 3$ $mm^2$
array of channels can be made. (B) Zoom of the writing field.}
\label{Channels}
\end{center}
\end{figure}

In general it is not trivial to perform STM measurements, because
the sample surface should be smooth and free of oxidation. For the
first requirement the variation of height should be small,
otherwise the tip can crash. However in the MSM group there is a
large expertise in STM measurements on superconducting thin films.
The roughness of the surfaces of such thin film is of a few
angstr\"{o}m, which makes them excellent to use for STM scans. The
importance of having flat surfaces shows the difficulty of our
experiment. Although the surface is smoothen, because of the
channel banks, it still has high level differences. This makes the
experiment far from trivial.

Another well known problem in vortex imaging is that the surface
of the material has to be clean and not oxidized. This is
detrimental to map the superconducting gap, which is essential to
image the vortices. This problem can be solved as discussed in the
precious chapter by depositing a thin gold layer. It turns out
that vortices can still be imaged through a thin layer of gold, by
using the proximity effect. The gold layer should be homogenous,
which is not so difficult to achieve for amorphous
superconductors.

\subsection{STM used for Topography}

\begin{figure}[!t]
\begin{center}
\includegraphics[width=6.5cm]{pictures/STM}
\caption{STM scanning with constant current mode.}\label{STM}
\end{center}
\end{figure}

The STM topography measurements were done in a constant current
mode as follows: A bias voltage is applied between the tip and the
sample. Tunnelling occurs when the tip is approached to a few
angstr\"{o}m from the sample. In constant current mode the current
is kept constant by the feedback circuit. Vertical displacements
of the scanner (feedback signal) reflect, then,  surface
topography, see figure \ref{STM} \cite{STM}. The constant tunnel
current (few $nA$) is kept far above the gap value, see figure
\ref{IV}. The $I$-$V$ curves, as explained is section 2.6, for
normal conductors and superconductors overlap in this regime. So
the topographical information is obtained independently of the
state (superconducting or normal) of the scanned part.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=6.5cm]{pictures/IV-curve}
\caption{$I$-$V$ curves of tunnelling spectroscopy.}\label{IV}
\end{center}
\end{figure}

\subsection{Scanning Tunnelling Spectroscopy}

Scanning tunnelling spectroscopy  was used to determinate the gap
of various superconductors (probing locally the density of states)
\cite{1. G. Binnig and H. Rohrer: Surf. Sci. 126 (1983)} . The
same technique can be used to image the vortices in the material.
The difference in density of states between the superconductor and
the normal conductor (vortex core) makes it possible to
distinguish them. As shown in figure \ref{IV} the $I$-$V$ curves
differ for both conductors. Tunnelling spectroscopy consists in
measuring the tunnelling current I(V) as a function of the bias
voltage, see figure \ref{STM2} \cite{STM} at a fixed point above
the film.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=6.5cm]{pictures/STM2}
\caption{Tunnelling spectroscopy.}\label{STM2}
\end{center}
\end{figure}

All measurements of spectroscopy are done for our sample keeping
the same distance between tip and sample (determined by constant
current mode). $I$-$V$ curves as shown in Fig. \ref{IV} are
obtained. To make an image out of the $I$-$V$ curves the values of
derivative of the curve $I = f(V)$ for small $V$ (near 0) and big
$V$ (far above the gap) are divided. Normal state tunnelling
should give $1$, and superconducting state values between $0$ and
$1$. These numbers are then used for contrast difference, which
can be used to make a vortex image like Fig. \ref{vortex}
\cite{Au}. Fig. \ref{vortex}. (a)  gives an example of weak
pinning material (a-Moge) and (b) of strong pinning material
(NbN), where the white spots are normal conducting vortex cores
and dark structures are superconducting.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=5.5cm]{pictures/vortex2}
\caption{Vortex configurations as measured in a-MoGe and NbN at
4.2 $K$ and magnetic fields of 0.5 and 0.8 $T$, respectively. Scan
ranges are 704 $\times$ 704 $nm^2$ and 510 $\times$ 680 $nm^2$,
respectively. On the On the right, the corresponding
triangulations are shown. Black circles correspond to vortices
with coordination number $z$ $=$ 6, open circles have $z$ $<$ 6,
while the open squares mark vortices with $z$ $>$
6.}\label{vortex}
\end{center}
\end{figure}

\section{Topography Measurements}

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=5.5cm]{pictures/toposample}
\newline
(b)
\includegraphics[width=6cm]{pictures/toporaw}
\includegraphics[width=6cm]{pictures/toposmooth}
(c) \caption{Typically STM scan of 1.5 $\mu m$ $\times$ 1.5 $\mu
m$ region on top of the channel sample. (a) Topography image of
the sample. The brightness of the color indicates the relative
level height (bright is high, and dark is low). (b) Height
variation along the selected line (raw data). (c) The same height
variation but smoothed.}\label{topo}
\end{center}
\end{figure}

A typical STM image of the channel sample, is shown in Fig.
\ref{topo}.(a). The scan range is 1.5 $\mu m$ $\times$ 1.5 $\mu m$
and the measurements are done at a temperature of 4.2 K with no
applied magnetic field. Next to the topography image the height
variation along a selected line is also shown: in Fig \ref{topo}.
(b) the raw results, and smoothed  ones in Fig \ref{topo}. (c).
The images indicate that we have scanned across the channels and
the banks. The dark (red) colors represent the channels (small z)
and the bright (yellow) regions represent the banks (high z). The
periodicity is, as clearly results from Fig \ref{topo} (b) and
(c), around 1 $\mu m$ which agrees with the designed value.
Moreover the direction of the channels agrees with the direction
chosen when mounting the sample at room temperature. There are,
however, few discrepancies related to these images. First, the
figures show a level difference of maximum $11$ $nm$, but as
described in Chapter 4 and measured with the AFM (see Fig
\ref{AFM}. (b)), the channels should have a depth of approximately
$65$ $nm$. The piezo element which controls the $Z$-component of
the STM can have an error of few $nm$ (due to calibration
problems), but certainly not an error as big as $50$ $nm$.
Secondly, the plots in Figs \ref{topo}. (b) and (c) do not show a
smooth surface, as seen with AFM (see Fig \ref{AFM}. (b)) , but
rather jumps and plateaus of approximately 0.1 to 0.15 $\mu m$
wide.

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=5cm]{pictures/tipA}

(b)
\includegraphics[width=5cm]{pictures/tipB}

(c)
\includegraphics[width=5cm]{pictures/tipC}

(d)
\includegraphics[width=5cm]{pictures/tipD}
\caption{Illustration of tunnelling between a sample and a tip for
different geometries of tip and sample (a). Scan with a symmetric
tip on a flat sample. (b) Scan with symmetric tip on a sample with
a step. At the edge of the step, the shortest distance between
sample and tip shifts and tunnelling occurs on one side of the
tip. (c) Scan with a tip that is not oriented perpendicular to the
sample with a step. At the edge of the step the shortest distance
shifts and the tunnelling takes place on the side. (d) If the
scanning tip is asymmetric at the edge of the step the tunnelling
site in the tip can change position. All these 3 situation can
lead to a distortion of the topography image!}\label{tip}
\end{center}
\end{figure}

In order to explain this discrepancies we should start with some
observations regarding the STM tip. To get the right topography of
the sample the tunnelling current between tip and sample should
always enter in the same point of the tip. By moving the tip in
constant current mode (see Fig. \ref{STM}) the surface of the
sample can be mapped. For thin film measurements the size, the
asymmetry or the tilting of the tip normally do not distort the
topography image much, because unstructured thin films prepared
with the recipe discussed in the previous chapter hardly contain
any roughness. The situation is, however, different for our
structured samples. In order to illustrate some possible problems
we plotted in Fig. \ref{tip} three different situations. Since
tunnelling is very sensitive to the distance between tip and
sample, it always occurs between the point of the tip and the
position of the sample that are at the shortest distance. As it is
illustrated in Fig. \ref{tip} (b - d) due to the tip size,
asymmetry or tilting this shortest distance can shifts from the
end of the tip to a different position leading to a distorted STM
image. As a consequence, the topography of the tip can be
reflected in the final image, in which case we say that the tip
and the sample images are convoluted.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=6cm]{pictures/doubbletunnelling2}
\caption{Convoluted topography between tip and sample: an example
of repeated patterns is marked. Scanning direction is from top to
bottom, starting from the top right and ending at the bottom left.
}\label{convolution}
\end{center}
\end{figure}

If we return to Fig. \ref{topo}. (a) and study it in more detail,
we see indeed that the topography between sample and tip is
convoluted in the form of repeating patterns. To illustrate this
effect better we have plotted the scanned region in Fig.
\ref{convolution} and marked an area where we believe such
convolution occurs. We see continuing repeating regions (plateaus)
until the next channel bank is reached. The regions are 0.1 to
0.15 $\mu m$ wide (perpendicular to the channel direction) and
repeat themselves 7 times. Apparently a clear scan is made only
for the top of the channel banks, the rest seems to be a
convolution of the banks and the tip. The repeat plateaus indicate
an asymmetry and a multi-tip structure of the tip. It indicates as
well, that the tip size is too big for the channels. This prevents
us to properly image deep inside the channels, as already was
indicated by the level plots from Fig. \ref{topo} (b) and (c).

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=6cm]{pictures/STMtip1}
\includegraphics[width=6cm]{pictures/STMtip2}
(b) (c)
\includegraphics[width=6cm]{pictures/STMtip3}
\includegraphics[width=6cm]{pictures/STMtip4}
(d) \caption{Typical SEM image of a STM tip cut from a $200$ $\mu
m$ wide platinum-iridium wire. (a) Scan field of 360 $\times$ 330
$\mu m$, magnification 400 $\times$. (b) Scan field of 48 $\times$
19.5 $\mu m$, magnification 3000 $\times$. (c) Scan field of 900
$\times$ 825 $nm$, magnification 160,000 $\times$. (d) Scan field
of 240 $\times$ 220 $nm$, magnification 600,000 $
\times$.}\label{stmtip}
\end{center}
\end{figure}

Recently (after the STM measurements are performed), a new SEM
became available in our group, namely a FEI NanoSEM
\cite{nanosem}, which is a Schottky field emitter SEM that has
very high resolution. This SEM can reach an accuracy of
approximately $5$ $nm$. To gain more insight of the tip
configuration SEM scans of several tips were made. A typical image
is shown in Fig. \ref{stmtip}. As can be observed from the images
in Fig. \ref{stmtip} the tip is indeed 'big' compared to the
channel width and if we look at Fig. \ref{stmtip} (c) the tip
hardly fits into the channel. At the maximum magnification Fig.
\ref{stmtip} (d) multiple-tips can be recognized. The typical
distance between those tips is around $100$ $nm$, which
corresponds to the width of the plateaus. The asymmetry seen in
topography indicates that apart from multiple-tips the tip is
probably tilted as well.


\section{Tunnelling Spectroscopy Measurements}

Spectroscopy measurements were done for applied magnetic fields of
$0.2$, $0.5$, $1.0$, $1.5$ and $1.8$ $T$. We will first discuss,
in great detail, the measurements done at $\mu_{0} H$ $=$ $0.5$
$T$. These are plotted in Fig. \ref{0.5T}

First, Fig. \ref{0.5T}. (a) shows the topography of the whole
sample ( scan range: 1.5 $\mu m$ $\times$ 1.5 $\mu m$). In the
blue marked area (scan range: 1.0 $\mu m$ $\times$ 1.0 $\mu m$)
the tunnelling spectroscopy measurements were performed. The scan
ranges are adjusted to obtain as good a resolution and contrast as
possible for imaging vortices. A higher field increases the amount
of vortices, which needs smaller scanning areas for the same
contrast. This area is displayed in Fig. \ref{0.5T}. (b) which
shows the image (scan range: 1.0 $\mu m$ $\times$ 1.0 $\mu m$)
obtained, as explained in the previous section, from tunnelling
spectroscopy. In these images the dark red color (dark) displays
the normal conducting areas and the yellow color (bright) for
superconducting areas. Vortices can be distinguished in the
superconductor as round dark red spots in the yellow areas.
However, next to the red dark spots, big red dark continuous areas
appear as well. The reason for their appearance is not well
understood. A possibility is that the Au layer doesn't have
uniform thickness and is too thick locally, so that the proximity
effect doesn't survive. Other reasons can be local oxidization or
dirt. In order to clearly illustrate the correspondence between
the vortices and the different height in the sample we plotted in
Fig. \ref{0.5T}. (c) a 3D image of the topography on which are
imposed the colors of the spectroscopy. The vortices arrange
themselves in a hexagonal lattice independent on the position in
the sample. This lattice has the axes parallel to the channel
direction. Subsequent measurements at different magnetic field
show that this happens independent of the the magnitude of
magnetic field. It should be noticed here, that in thin film
measurements the lattice direction usually is random after
applying the magnetic field \cite{Au}, because in a weak pinning
film there is no preferred direction. All vortices enter the
sample from the side and randomly form a lattice in the middle of
the sample. The fact that we see, independent of the magnitude of
the magnetic field, the same direction of the lattice strengthens
our belief that we image a channel structure, because this imposes
a preferential orientation. Finally, in Fig. \ref{0.5T}. (d), the
same image as in Fig. \ref{0.5T} (b) is shown but this time
smoothed in order to accurately determine the coordinates of the
vortex cores. The average distance between the vortex cores is
calculated. The vortices which lie exactly at the plateau jump are
excluded in the calculation, because their distance seems
artificially increased. For 0.5 $T$ the average distance between
the vortices was found to be $69.1$ $nm$, which agrees well with
the theoretical value of $69.2$ $nm$ for a triangular lattice.

\begin{figure}[!t]
\begin{center}
(a)
\includegraphics[width=6cm]{pictures/Overviewp5T.png}
\includegraphics[width=6cm]{pictures/Vorticesp5T}
(b)
(c)
\includegraphics[width=6cm]{pictures/Topop5T}
\includegraphics[width=6cm]{pictures/Vortexp5TS}
(d) \caption{Results of topography en spectroscopy measurements at
$0.5$ $T$ and $4.2$ $K$. (a) Topography image of the sample 1.5
$\mu m$ $\times$ 1.5 $\mu m$. Blue marked area 1.0 $\mu m$
$\times$ 1.0 $\mu m$ is the area where tunnelling spectroscopy
measurement were performed by 0.5 $T$. (b) Image of vortices
obtained from tunnelling spectroscopy. Scanned area 1.0 $\mu m$
$\times$ 1.0 $\mu m$ (blue marked area of (a)). Vortices arrange
themselves in hexagonal lattice. (c) 3D landscape image, where
obtained results from tunnelling spectroscopy are put on the
topography image. Hexagonal vortices lattices seems to arrange
themselves parallel to the channel. (d) Analyzed image of the
spectroscopy to determine lattice structure (hexagonal) and the
average vortex distance ($69.1$ $nm$).}\label{0.5T}
\end{center}
\end{figure}

Unfortunately, from the tunnelling spectroscopy a large part of
the sample appears as being not superconducting. This reduces the
total amount of vortices imaged, complicating the analysis of the
data. However there is still a sufficient amount of imaged
vortices to conclude that they seem to arrange themselves in a
hexagonal lattice. However, this limitation makes it impossible to
determine the dimension $R_c$ of the Larkin domain, but we see
however that $R_c$ is at least $6a_0$, which in itself is an
unexpected result. For amorphous thin films as a-MoGe we expect to
be in the 2D case of the theory of collective pinning, described
in section 2.8.3. In this regime the tilt modulus is neglected and
vortices which now can be approximated with stiff tubes will
arrange in a hexagonal lattice. On the other hand strong pinning
in the NbN gives a randomly distributed lattice. For the a-MoGe
film deposited above the NbN, since the vortices are stiff tubes,
they will be pinned by the underlaying NbN layer. Therefore we
expect a randomly distributed lattice. As we mentioned before, we
probably image only the channel banks, which should give, due to
the NbN, a randomly distributed lattice. But the data show a clear
hexagonal lattice in contradiction with the above mentioned
scenario. At this stage, we cannot give a definitive explanation
for this disagreement, but we can state that al least one of our
assumptions seems to be wrong. Let's discuss each of them in
detail. A recent calculation has shown that for a rather thick
a-MoGe layer the tilt modulus is relevant and vortices may relax.
In equation (\ref{Lc}) we derived the dimension $L_c$, which give
a length scale (along the field) for the vortices to be
correlated. In analogy with the length scale $L_c$ there is a
length scale $d_{max}$. This $d_{max}$ is a length scale for
vortices to relax from a randomly distributed lattice to a regular
lattice. If the thickness of the film is much larger than
$d_{max}$, the vortex lattice can even relax from a randomly
distributed lattice to a hexagonal lattice. Not unexpectedly,
$d_{max}$ looks like $L_c$, as derived in \cite{Baarle}.

\begin{equation}
d_{max} \approx  \sqrt{ \frac{\pi}{32}} \frac{ a_0^2
\sqrt{(1-b)}}{\lambda} \sqrt{\frac{c_{44}(0)}{c_{66}}} \approx
\frac{\pi}{\sqrt{2}} \sqrt{\frac{\Phi_0 }{B_{c2}}
\frac{1}{b(1-b)}}\label{gejatte_formule_gertjan}
\end{equation}

where $b = \frac{B}{B_{c2}}$ and $B_{c2}$ is $7.4$ $T$ for our
a-MoGe. Equation (\ref{gejatte_formule_gertjan}) has been plotted
in Fig. \ref{gejatte grafiek gertjan}. The STM measurements are
done for $b$ between $0.03$ - $0.24$. In this regime it follows
from Fig. \ref{gejatte grafiek gertjan} that $d_{max}$ lie between
$90$-$225$ $nm$, which is much less then the $500$ $nm$ thick
a-MoGe film. This may enable the vortices to relax to a less
random configuration than in NbN.

\begin{figure}[t]
\begin{center}
\includegraphics[width=13.5cm]{pictures/Bending}
\caption{Plot of $d_{max}$ vs b.}\label{gejatte grafiek gertjan}
\end{center}
\end{figure}

Although, before a-MoGe was sputtered on the NbN, the NbN was
etched carefully to get rid of any oxidized layer, another
explanation can still be that the NbN remained oxidized during the
fabrication process. The vortex lattice between the NbN and a-MoGe
layers will then be decoupled and will order itself, as in absence
of the NbN, i.e. in a hexagonal lattice.

Finally, for completeness, we mention that the convolution can be
seen in the tunnelling spectroscopy images as well. When we look
in more detail to Fig. \ref{0.5T} (a), we see again a repetition
of scanned patterns (see Fig. \ref{double}). These are the same
repeating regions which were multi scanned in Fig. \ref{topo} (a).
It means that no vortices are scanned in the channels, only those
in the channel banks. It is unfortunate, because this way we were
not able to study the change in vortex distribution from the
channel banks to the channel valley. Adjustments should be made in
the future to deal with this problem.

\begin{figure}[t]
\begin{center}
\includegraphics[width=6.7cm]{pictures/Vorticesp5Tdouble}
\caption{Image of vortices obtained from tunnelling spectroscopy.
Scanned area 1.0 $\mu m$ $\times$ 1.0 $\mu m$. Again repeated
regions (encircled) can be found due to the
convolution.}\label{double}
\end{center}
\end{figure}

\section{Conclusion and Future Ideas}

In this Chapter STM measurements carried out are presented done
for the first time on structured thin film sample. Although it was
an "one-shot-experiment" (due to the experimental limitations)
very interesting results are obtained. The topography images show
clearly that we probed channels and (channel) banks with the
periodicity as expected and orientation as mounted. The
spectroscopy measurements successfully imaged vortices, from which
we can conclude that the coating procedure between Au and a-MoGe
was successful. The vortices seem to arrange themselves in a
hexagonal lattice, which orients itself along the channel edges.
Furthermore, the vortex lattice constant corresponds to the value
$a_0$ as predicted for an Abrikosov lattice.

However, some of the experimental data could not be properly
understood in the original framework. The depth scans and
convolution of the topography images suggest that the STM tip is
far from ideal. Subsequent SEM pictures revealed a very irregular
shape (asymmetric and multiple-tip structures) and a too wide tip
end. Due to all these features it was impossible to follow the
landscape of the sample, instead convoluted images were observed.
The most likely scenario is that only the vortices from the
channels banks could be imaged. The fact that they show a
hexagonal lattice oriented along the channels is still puzzling.
As possible explanation we can mention that the relaxation of the
vortices could decrease the disorder in the channel banks to such
an extent that the resolution of our STM cannot distinguish it
from a perfect hexagonal lattice. Alternatively, it is possible
that there is an oxide layer at the interface between the NbN and
a-MoGe layers which decouples the two layers. Although, a great
deal of new and interesting information was already obtained, for
the future adjustments should be made.

To solve the problems due to the size and the irregularity of the
tip it should be etched before mounting. The etching procedure
usually takes a lot of time and the risk to oxidize the tip
increases, but is a necessary step because a sharp tip will better
follow the landscape of the sample. It is even better, if one
could attach a carbon nanotube to the end of the tip gaining
access to less than 10 $nm$ resolution. Secondly the STM should be
equipped with a $XY$-table. The $XY$-table can move the sample
under the tip, which will increase the scanning range
significantly. A $XY$-table is already under construction in our
group, and will be operational in the near future.

As an alternative the growth procedure can be adapted by first
creating channels directly in the Si substrate followed by
depositing NbN in the channels by means of a lift off procedure.
On top a-MoGe can be sputtered, which may give a smooth surface,
as shown in Fig. \ref{laststep2}.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=8cm]{pictures/Laststep}
\caption{Alternative fabrication of a structured sample. With a
lift- off procedure channels are directly etched in the  Si
substrate and filled with NbN. Later a-MoGe is sputtered on top.
This procedure may give a smooth surface.}\label{laststep2}
\end{center}
\end{figure}

Since $500$ $nm$ a-MoGe is quite a thick layer one should consider
to reduce the thickness to below the predicted healing length.

\chapter{Conclusions}

This report presents the first (ever) attempt to image with STM
mesoscopic vortex channels. Due to the experimental complexity of
the problem a lot of effort has been put in optimizing the sample
preparation. For instance, the fabrication of small channels with
the e-beam was not a known recipe. Time, effort and systematically
tuning parameters were needed to optimize in order to obtain the
right channels pattern. The high dose procedure was the last step
to give the channels a sharp edge. This procedure can even be used
for other projects within the MSM group.

Another important problem was that sharp channel edges are
impossible to follow with a STM tip, which forced us to develop a
new procedure to design channels with the top sample surface
smoothed. First the channel banks (consisting of NbN) were
fabricated. Next, the a-MoGe layer was sputtered on top. The
result was a more smooth surface, but still containing the
properties of the easy flow vortex channels.

There are strong evidences that we were able to probe channels and
(channel) banks. For instance, the periodicity of the channels
corresponds to the designed periodicity of 1 $\mu m$, and the
vortex lattice, independent of the magnitude of the magnetic
applied field, arranges itself parallel to the channels.
Furthermore, the scanned vortices form a hexagonal lattice with
the lattice constant equal to $a_0$ (Abrikosov lattice). However,
some experimental data could not yet be properly understood. This
is most probably because the tip is far from ideal. Subsequent SEM
pictures and detailed studies of the topography and spectroscopy
images revealed indeed that the tip is too wide (compared to the
channels) and is multiple-tip structured. This leads to a
convoluted image, which resulted in a repeated image of the
channel banks only. Unfortunately, this feature made it impossible
to image deep inside the channels and therefore all the vortices
that we imaged are in the channel banks. Contrary to our initial
expectation these vortices appear to form a hexagonal lattice
instead of having a random configuration. Two possible
explanations are put forward. Either, the interface between the
NbN and a-MoGe is oxidized, which decouples the vortices from NbN
and a-MoGe, or the a-MoGe is too thick (thicker than the healing
length) and the positional of the vortex lattice disorder at the
top of a-MoGe is much smaller than at the bottom in the NbN.

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\addcontentsline{toc}{chapter}{Appendix A}
\chapter*{Appendix A}  % use *-form to suppress numbering

\section*{Parameters for the thin films samples}

\paragraph{Spinning and baking PMGI and PMMA resists on Silicon substrate.}
Name of machine: Spin Coater, Model: P6700. The used resists is
the so called 950 PMMA A4 and PMGI SF 11.

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Spinning rate & 6000 $rpm$ \\
\hline Baking temperature of PMGI & 190 $^{\circ}C$ \\
\hline Baking time of PMGI & 30 $min$ \\
\hline Baking temperature of PMMMA & 140 $^{\circ}C$ \\
\hline Baking time of PMMA & 60 $min$ \\
\hline
\end{tabular}
\caption{Parameters which belongs to the spin coater and baker.}
\end{centering}
\end{table}

\paragraph{Writing contact pattern on the samples.}
Name of machine: JEOL 820 SEM.

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Filament & 260 $\mu A$ \\
\hline Acceleration Voltage & 30 $V$\\
\hline Diaphragm & 3 \\
\hline Magnification & 15 \\
\hline Work distance & 4 $\times$ 4 $mm$ \\
\hline Current & 5.0 $nA$ \\
\hline Dose & 161 $\mu A s / cm^2$ \\
\hline
\end{tabular}
\caption{Parameters of the SEM for the contact pattern.}
\end{centering}
\end{table}

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Dose  & 161 $\mu A s / cm^2$ \\
\hline Current & 5.0 $nA$ \\
\hline Step size & 0.1221 $\mu m$ \\
\hline Dwell time & 0.004801 $ms$ \\
\hline Beam speed & 25.435 $mm/s$ \\
\hline
\end{tabular}
\caption{Parameters to obtain the desirable dose.}
\end{centering}
\end{table}

\paragraph{Developing PMMA and PMGI for the contact pattern.}
The PMMA developer is a mixture of MIBK and IPA, with a ratio 1:3.
The PMGI developer is called the PMGI 101 developer.

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Developing PMMA & 0.5 $min$ \\
\hline Rinse and dry & 1.0 $min$ \\
\hline Developing PMGI & until through \\
\hline Undercut PMGI & 2.5 $min$ \\
\hline Rinse and dry & 1.0 $min$ \\
\hline
\end{tabular}
\caption{Parameters of the developing times of PMMA and PMGI.}
\end{centering}
\end{table}

\paragraph{Sputtering NbN and a-MoGe with the Z-400.}
Used sputtering machine: The Leybold Z-400.

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
 & Base  & $Ar$ &
 $N_2$ & Sputter  & Breaks & Silver  & Thick-  \\
 & Pressure      & Rate & Rate  &
 Time & & Paint & ness  \\
 & ($mbar$) & ($ \% $) & ($ \% $) & ($min$) & &
 &($nm$) \\
\hline \hline NbN & $1.9 \cdot 10^{-6}$ & 18 & 10 & 16 & no & no & 42 \\
\hline MoGe-1 & $3.2 \cdot 10^{-6}$ & 25 & 0 & 91 & yes & yes & $ \pm$ $ 500 $ \\
\hline MoGe-2 & $2.4 \cdot 10^{-6}$ & 25 & 0 & 91 & no & yes & $ \pm$ $ 500 $ \\
\hline MoGe-3 & $1.5 \cdot 10^{-6}$ & 25 & 0 & 91 & no & no & $ \pm$ $ 500 $ \\
\hline
\end{tabular}
\caption{Parameters of the sputtering conditions all sample were
sputtered by a $V_{DC}$ of 1.0 $kV$ and a sputter pressure of
$\pm$   $ 3 \cdot 10^{-2} $ $mbar$.}
\end{centering}
\end{table}

\section*{Parameters for the structured samples}

\paragraph{Sputtering NbN on silicon substrate with Z-400.} Used
sputtering machine: The Leybold Z-400. For the construction of
channels we started with a silicon substrate with NbN on it.

\begin{table}[!ht]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Base pressure & $2.4 \cdot 10^{-6}$ $mbar$ \\
\hline Sputter pressure & $\pm$ $ 3 \cdot 10^{-2} $ $mbar$ \\
\hline $V_{DC}$ & 1.0 $kV$ \\
\hline $Ar$ rate & 18 $ \% $ \\
\hline $N_2$ rate & 10 $ \% $ \\
\hline Sputter time & 22.6 $min$ \\
\hline Thickness & 50 $nm$ \\
\hline
\end{tabular}
\caption{ Parameters of the sputtered NbN on silicon substrate.}
\end{centering}
\end{table}

\newpage

\paragraph{Spinning and baking PMMA resists on Silicon substrate.}
Name of machine: Spin Coater, Model: P6700. The used resists is
the so called 950 PMMA A4.

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Spinning rate & 6000 $rpm$ \\
\hline Baking temperature of PMMMA & 140 $^{\circ}C$ \\
\hline Baking time of PMMA & 60 $min$ \\
\hline
\end{tabular}
\caption{Parameters of the spined PMMA on the sample.}
\end{centering}
\end{table}

\paragraph{Writing channels into the PMMA of the sample.}
Name of machine: JEOL 820 SEM.

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Filament & 260 $\mu A$ \\
\hline Acceleration Voltage & 30 $V$\\
\hline Diaphragm & 4 \\
\hline Magnification & 600 \\
\hline Work distance & 0.1 $\times$ 0.1 $mm$ \\
\hline Current & 0.5 $nA$ \\
\hline Dose & 161 $\mu A s / cm^2$ \\
\hline
\end{tabular}
\caption{Parameters of the SEM for the channels.}
\end{centering}
\end{table}

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Dose  & 161 $\mu A s / cm^2$ \\
\hline Current & 0.5 $nA$ \\
\hline Step size & 0.0305 $\mu m$ \\
\hline Dwell time & 0.003005 $ms$ \\
\hline Beam speed & 10.150 $mm/s$ \\
\hline
\end{tabular}
\caption{Parameters to obtain the desirable dose.}
\end{centering}
\end{table}

\paragraph{Developing PMMA for the channels}
The PMMA developer is a mixture of MIBK and IPA, with a ratio 1:3.

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Developing PMMA & 1.0 $min$ \\
\hline Rinse and dry & 1.0 $min$ \\
\hline
\end{tabular}
\caption{Parameters of the developing times of PMMA.}
\end{centering}
\end{table}

\paragraph{Writing high dose over the channels in the PMMA of the sample.}
Name of machine: JEOL 820 SEM.

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Filament & 260 $\mu A$ \\
\hline Acceleration Voltage & 30 $V$\\
\hline Diaphragm & 2 \\
\hline Magnification & 15 \\
\hline Work distance & 4 $\times$ 4 $mm$ \\
\hline Current & 20 $nA$ \\
\hline Dose & 20000 $\mu A s / cm^2$ \\
\hline
\end{tabular}
\caption{Parameters of the SEM for the high dose.}
\end{centering}
\end{table}

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Dose  & 4000 $\mu A s / cm^2$ \\
\hline Current & 20 $nA$ \\
\hline Step size & 17.4561 $\mu m$ \\
\hline Dwell time & 609.431 $ms$ \\
\hline Beam speed & 0.029 $mm/s$ \\
\hline
\end{tabular}
\caption{To obtain a dose of 20000 $\mu A s / cm^2$ the structured
is written 5 times with a dose of 4000 $\mu A s / cm^2$.}
\end{centering}
\end{table}

\newpage

\paragraph{Etching of the channels.}Name of machine: Ion Beam Etcher (Kauffmann
type).

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Base pressure & $2.9 \cdot 10^{-6}$ $mbar$ \\
\hline $Ar$ pressure & $4.0 \cdot 10^{-4}$ $mbar$ \\
\hline Discharge & 40.0 $V$ \\
\hline Beam current & 10.0 $mA$ \\
\hline Beam voltage & 350 $V$ \\
\hline Neutralizer & 10.0 $mA$ \\
\hline Etching time & 5:52 $min$ \\
\hline Rotating & 1.0 $V$ \\
\hline $N_{2}$ cooling & yes \\
\hline
\end{tabular}
\caption{Parameters of channel etching}
\end{centering}
\end{table}

\paragraph{Oxygen Etching} Hand made machine:

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|l|c|}
\hline Base Pressure & $\pm$ $1 \cdot 10^{-2}$ $mbar$ \\
\hline $O_{2}$-Pressure & $1 \cdot 10^{-1}$ $mbar$ \\
\hline Power & 100 $W$ \\
\hline Etching time & 2.0 $min$ \\
\hline
\end{tabular}
\caption{Parameters of $O_{2}$-etching.}
\end{centering}
\end{table}

\paragraph{Sputtering a-MoGe and Au with the Z-400.}
Used sputtering machine: The Leybold Z-400.

\begin{table}[!h]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & $Ar$ & $O_2$ & Sputter  & Thickness  \\
 & Rate & Rate  & Time &  \\
 & ($ \% $) & ($ \% $) & ($min$) &($nm$) \\
\hline \hline a-MoGe sputtering & 25 & 0 & 91 & $\pm$ 500 \\
\hline Au (1) sputtering & 25 & 0 & 2/15 &  $ 2 $ \\
\hline Au (2) sputtering & 25 & 39 & 2/15 & $ 2 $ \\
\hline
\end{tabular}
\caption{Parameters of the sputtering conditions all sample were
sputtered by a $V_{DC}$ of 1.0 $kV$ and a base and sputter
pressure of $1.5 \cdot 10^{-6}$ $mbar$ and $\pm$ $ 3 \cdot 10^{-2}
$ $mbar$, respectively.}
\end{centering}
\end{table}

\addcontentsline{toc}{chapter}{Bibliography}
\begin{thebibliography}{}
\bibitem{Kamerlingh-Onnes}H. Kamerlingh Onnes, Leiden Comm.
$\mathbf{120b}$, $\mathbf{122b}$, $\mathbf{124c}$ (1911)
\bibitem{GL}V.L. Ginzburg and L.D. Landau, Zh. Eksperim. Theor.
Fiz. $\mathbf{20}$, 1064 (1950)
\bibitem{BCS}J. Bardeen, L. N. Cooper and J. R. Schrieffer, phys. Rev. $\mathbf{106}$, 162
(1957)
\bibitem{BCS2}J. Bardeen, L. N. Cooper and J. R. Schrieffer, phys.
Rev. $\mathbf{108}$, 1175 (1957)
\bibitem{Abrikosov} A.A. Abrikosov, Phys. Chem. Solids
$\mathbf{2}$, 199 (1953); A.A Abrikosov, Sov, Phys. JETP,
$\mathbf{5}$, 1174 (1957)
\bibitem{kokubo} N. Kokubo, R. Besseling, and P.H. Kes, Physical
Review B $\mathbf{69}$, 064504 (2004)
\bibitem{kokubo2} N. Kokubo, R. Besseling, V.M. Vinokur and P.H. Kes, Physical
Review Letters $\mathbf{88}$, 247004 (2002)
\bibitem{Besseling} R. Besseling, Ph.D. Thesis, Depinning and
Dynamics of the Vortex Lattice in Thin Films and Narrow Channels,
University of Leiden (2001)
\bibitem{M Tinkham}M. Tinkham, Introduction to Superconductivity,
Krieger Publishing Company, Malabar, Florida (1980)
\bibitem{Rose-Innes and Rhodericks}A.C. Rose-Innens and E. H.
Rhoderick, Introduction to Superconductivity, Pergamon Press plc,
Oxford, England (1978)
\bibitem{Kittel}C. Kittel, Introduction to Solid State Physics,
John Wiley and Sons, Inc., New York, USA (1967)
\bibitem{Kes}P.H. Kes, "Pinning and dynamics of magnetic vortices in the vortex state",
N. Bontemps et al (eds:), p159-174, Kluwer Academic Publishers,
The Netherlands, 1994.
\bibitem{Marchevsky}M.V. Marchevsky, Ph.D. Thesis, Magnetic
Decoration Study of the Vortex Lattice in Superconductors,
University of Leiden (1997)
\bibitem{Meissner} W. Meissner, R. Ochsenfield, Naturwiss. $\mathbf{21}$, 787 (1933)
\bibitem{London}F. and H. London, Proc. Roy. Soc. (London)
$\mathbf{A149}$, 71 (1935)
\bibitem{Cooper}L. N. Cooper, Phys. Rev. $\mathbf{104}$, 1189 (1956)
\bibitem{A. I. Larkin and Yu. N. Ovchinnikov}A. I. Larkin and Yu. N. Ovchinnikov,
Journal of Low Temperature Physics $\mathbf{34}$, 409 (1979)
\bibitem{S.J. Mullock and J.E. Evetts}S.J. Mullock and J.E.
Evetts, J. Appl. Phys. $\mathbf{57}$, 2588 (1985)
\bibitem{P.H. Kes and C.C. Tsuei}P.H. Kes and C.C. Tsuei, Phys.
Rev. B. $\mathbf{28}$, 5126 (1983)
\bibitem{R. Wordenweber and P.H. Kes}R. W\"{o}rdenweber and P.H.
Kes, Phys. Rev. B $\mathbf{34}$, 494 (1986)
\bibitem{chem} Microchem, homepage: http://www.microchem.com/
\bibitem{private} M. Hesselberth, G.J.C. van Baarle and T. Sorop
(private communication)
\bibitem{Geers} J.M.E. Geers, Ph.D. Thesis, Superconductivity in
Thin Films and Multilayers, University of Leiden (1999)
\bibitem{Pruijmboom} A. Pruijmboom, Ph.D. Thesis,
Critical-Current-Limiting Mechanisms In Type-II Superconductors,
University of Leiden (1988)
\bibitem{Benningshof} O.W.B.
Benningshof, Student Report (IEN), Fabrication of Vortex Flow
Channels in Superconducting Thin Films, Univesity of Leiden (2002)
\bibitem{Rijn} R.T. van Rijn , Student Report, Micro-structuring
thin film, Optimizing the etching process, Technische Hoge School
Rijswijk (2004)
\bibitem{Au} G. J. C. van Baarle, A. M. Troianovski, T. Nishizaki,
P. H. Kes and J. Aarts, Applied Physics Letters $\mathbf{82}$,
1081 (2003)
\bibitem{PMMA2}T. B. Borzenko, Y. I. Koval, and V. A. Kudryashov,
Microelectron. Eng. $(\mathbf{23})$, 337 (1993)
\bibitem{PMMA}Y. Koval, T.Borzenko and S.Dubonos, J. Vac. Sci. Technol. B
$(\mathbf{21(5)})$, 2217 (2003)
\bibitem{Nature} A. Troyanovskii, J. Aarts, and P. H. Kes, Nature
(London) $(\mathbf{399})$, 665 (1999)
\bibitem{STM} NT-MDT,homepage: http://www.ntmdt.ru/
\bibitem{1. G. Binnig and H. Rohrer: Surf. Sci. 126 (1983)}G. Binnig and
H. Rohrer, Surf. Sci. $\mathbf{126}$ (1983)
\bibitem{nanosem} http://www.feicompany.com
\bibitem{Baarle}G.J. v Baarle, Ph.D. Thesis, as yet untitled, as
yet unpublished
\end{thebibliography}


\addcontentsline{toc}{chapter}{Acknowledgements}
\chapter*{Acknowledgements}

I started working in the MSM-group in September 2004 worked in
this group for almost a year. I really enjoyed being a member of
this group and I want to thank every member of this group, from
which I will mention only a few of them.

First of all I want to thank Prof. dr. P. H. Kes for attracting me
to assigning me on this subject, being interested in the results
we obtained and for the discussions about them.

Of course I am also very grateful to dr. T. Sorop. We worked
together for almost two whole year and he really learned me a lot
about vortex dynamics.

For the STM measurements I am really grateful for the assistance
of dr. F. Galli and drs. G. J. v. Baarle. Without their effort the
measurements wouldn't be that fine.

ing. Marcel Hesselberth and ing. Ruud Hendrikx I want to thank for
the help of the sample preparation and characterization.

Of course I am also very grateful to Youri de Boer, Onno
Broekmans, Steven Habraken, Jorina van der Knaap, Bas Leerink, Bas
van Leeuwen and Rob van Rijn for sharing the student room (HL 628)
with me. We had a lot of fun but we also learned quite much from
each other.

\end{document}