\documentclass[11pt,a4paper]{report} \usepackage{epsfig} \usepackage{afterpage} \title{\begin{LARGE}\textbf{Superconducting properties of Py/Al structures}\end{LARGE}} \author{Christianne Beekman} \date{Doctoraalscriptie} \begin{document} \maketitle \begin{titlepage} \vspace*{4cm} \begin{center} \begin{LARGE} \textbf{Superconducting properties of Py/Al\\} \end{LARGE} \end{center} \begin{center} \begin{LARGE} \textbf{structures\\} \end{LARGE} \end{center} \begin{center} Doctoraalscriptie\\Christianne Beekman\\Natuurkunde\\Studentnr.: 9932194\\ \end{center} \vspace{2cm} \begin{center} Faculteit der wiskunde en natuurwetenschappen\\Huygens laboratorium\\ \end{center} \begin{center} Leiden, 2005 \end{center} \end{titlepage} \pagestyle{plain} \pagenumbering{roman} \setcounter{page}{1} \tableofcontents \listoffigures \listoftables \newpage \pagestyle{plain} \pagenumbering{arabic} \setcounter{page}{1} \chapter{Introduction} In 1911 superconductivity was discovered. Certain materials undergo a phase transition and exhibit a resistance change from normal resistance to zero resistance when cooled through the superconducting transition temperature, T$_c$. Macroscopically, the resistance of these materials disappears, becoming a perfect conductor. However, at the origin of this phase transition lies the quantum character of the superconducting state. Superconductivity is a result of the formation of Cooper pairs when the superconductor is cooled through the transition temperature. A microscopic theory describing the behavior of a superconductor was developed by Bardeen, Cooper and Schrieffer (BCS theory)\cite{bcs} and a phenomenological theory was proposed even before the BCS theory by Ginzburg and Landau\cite{landau} (GL theory). The BCS theory provides the tools to calculate the energy gap $\Delta$, if $\Delta$ is constant in space. The GL theory, which combines the quantum character of the superconductor with the theory for phase transitions, describes a superconductor with $\Delta$ varying in space for T close to T$_c$. The driving force behind conventional superconductivity is the phonon mediated electron-electron interaction, responsible for the formation of Cooper pairs. These Cooper pairs are large and "hazy" particles with a spatial extension of $\xi_0$, the BCS coherence length. This coherence length can vary, for example from 38 nm for niobium to about 2 $\mu$m for aluminum. When hybrids of S/X bilayers are fabricated, with X a normal metal or a ferromagnet, these layers exchange phase information, this is called the proximity effect. The "hazy" nature of the Cooper pair is responsible for the proximity effect. %This nature induces a process, called Andreev reflection, at the interface of hybrid S/X structures. One can imagine that a Cooper pair can extend over the interface, reducing the order parameter in S but inducing an order parameter in X. The existence of the energy gap, $\Delta$, in the superconductor induces a scattering process, Andreev reflection, at the S/X interface. When the material which is in contact with the superconductor is ferromagnetic in nature there is a pair-breaking effect on the Cooper pairs in the superconductor. This pair-breaking effect results in a suppression of T$_c$ of the superconductor. This effect can be used in spintronics devices, such as spin switches. A spin switch consists of a superconducting layer which is sandwiched between two ferromagnetic layers. The superconducting transition temperature of the superconducting layer will be more suppressed when the F layers have a mutual magnetization orientation which is parallel than in the antiparallel orientation. This is caused by the stronger pair-breaking effect in the parallel orientation. %What if an external perturbation is imposed on a superconductor? %In the GL theory gradients caused by fields or currents can be taken into account. However what if a superconductor is brought into contact with a non-superconducting layer? A deposited layer of normal metal or a ferromagnetic layer influence the behavior of the superconductor and vice versa, this interaction between the neighboring materials is called the proximity effect. This effect causes the suppression of T$_c$ when a ferromagnetic layer is in contact with a superconducting layer. This suppression will occur when the thickness of the superconductor is of the order of or smaller then two times the superconducting coherence length, $\xi_s$. The theory behind this effect is described in more detail in chapter 2. The proximity effect can be used in certain devices such as the spin switch. A spin switch can basically have two different geometries: \begin{itemize} \item{A "vertical" F/S/F geometry with one of the ferromagnetic layers pinned by an anti-ferromagnetic layer. This device was theoretically proposed by Tagirov\cite{Tagirov} and Buzdin\cite{buz}. The pinning layer is necessary to ensure different switching fields of the F banks (a wide interval were mutual orientation is antiparallel is needed). A first experiment was done by Gu and Bader \cite{gu}.}% but other explanations are possible to explain their results\cite{rus} (the multi domain structure of the ferromagnets). \item{A lateral F/S/F geometry. A pinning layer is not necessary because different aspect ratios \footnote{$Aspect\;ratio=\frac{length\; F\; strip}{width\; F\; strip}$} of the F banks can be used to ensure different switching fields. To see any effect in a lateral switch a superconductor with a long coherence length should be used, like aluminum.} \end{itemize} The ultimate goal of this graduation project was to explore the feasibility of producing and measuring a lateral spin switch. We want to produce and measure a Py/Al/Py lateral spin switch but, in contrast to Jedema\cite{jedema} (measured this device with Al in normal state), we want to measure the spin switch with the aluminum going through the superconducting transition until spin transport is inhibited by the formation of Cooper pairs. %Interesting processes occur when superconducting materials %are brought into contact with other systems like ferromagnets. %This thesis will be about these proximity systems, in this case %Aluminum (Al) and Permalloy (Py). In the next section a short %outline of the thesis will be given \subsubsection{Outline of report} An outline of the theory needed to interpret our experiments will be given in \textbf{chapter 2}. Among other subjects proximity effect and suppression of T$_c$ in S/F bilayers will be treated. Also spin switches will be briefly discussed. The experimental setup, the hardware and software interface implemented for transport measurements, the sample preparation: the sputtering process, the lift-off technique and the fabrication of the lateral spin switch is discussed in \textbf{chapter 3}. Furthermore, the calibration of the aluminum and permalloy is presented. The results of the experiments that we performed are shown in \textbf{chapter 4} and finally conclusions and suggestions for future research are given in \textbf{chapter 5}. \chapter{Theory} \section{Brief overview of superconductivity} In 1911 Kamerlingh Onnes discovered that in a small temperature range the electrical resistance of several metals (mercury, lead and tin) disappeared. The temperature were this phenomenon occurred differed for various metals indicating that this critical temperature $T_c$ is characteristic of the material. This phenomenon was named superconductivity. The concept of superconductivity has puzzled physicists for decennia. Today there exist several theories describing this phenomenon. These theories exist due to the effort of people like Bardeen, Cooper, Schrieffer, Ginzburg, Landau, Gorkov and many others. A superconductor cannot be just described as a perfect conductor. Meissner and Ochsenfeld \cite{meisner} discovered that an externally applied magnetic field is expelled from a superconductor when the temperature decreases through $T_c$. However a perfect conductor would tend to do exactly the opposite: trapping flux in. Furthermore, there will be a critical magnetic field $H_c$ were superconductivity will be destroyed. Thus, the characteristics of a superconductor are zero electrical resistance below a temperature $T_c$ and below $H_c$ expulsion of the applied magnetic field. Furthermore, the transition from the normal metal state to the superconducting state is a phase transition. The London brothers \cite{london} were the first to describe the behavior of superconductivity. They proposed the London equations. %\begin{equation} %\textbf{E} = \frac{\delta}{\delta t}(\Lambda \textbf{J}_s) %\end{equation} %\begin{equation} %\textbf{h} = -c \textrm{curl}(\Lambda \textbf{J}_s) %\end{equation} %\begin{equation} %\Lambda = \frac{m}{n_s}(e^2) %\end{equation} %with n$_s$ the number density of superconducting electrons. Later various physicists established the existence of an energy gap in the density of states (DOS) of the order of k$_B$T$_c$ \cite{daunt} \cite{bardeen}. After this the BCS theory was produced by Bardeen, Cooper and Schrieffer \cite{bcs}. Ginzburg and Landau introduced the Ginzburg-Landau (GL) theory \cite{landau} and thereby introduced the order parameter $\psi$, a pseudo wavefunction, which is related to the density of superconducting electrons. Both these theories will be presented in the following sections. \section{Formation of Cooper pairs} A metal can be described as the collective system of a positively charged lattice of ion cores surrounded by a negatively charged electron sea. Since the electrons move much faster than the ions the assumption could be made that the dynamics of the electrons is decoupled from the dynamics of the lattice. However, this does not hold for electrons with energies close to the Fermi energy and mixed dynamics have to be taken into account. These electrons will distort the lattice in their vicinity. This distortion can be seen as a polarization cloud and the electron together with the cloud can be indicated as a quasiparticle. This electron-lattice interaction causes the ions to move a little, this displacement creates a wave in the lattice, a phonon. In figure \ref{lat} the electron and the lattice distortion are shown. \begin{figure} \centering \includegraphics[width=8cm, angle=0]{/home/beekman/Documents/sctiptie/plaatjes/lattice.ps} \caption{The lattice distortion caused by a passing electron.}\label{lat} \end{figure} The distortion of the lattice locally causes an increase in charge density. Other electrons will be attracted by this distortion, in principle introducing an attractive force between the two quasiparticles. So indirectly these electrons exchange virtual phonons (see fig. \ref{eph}). This phonon exchange and thus the attractive interaction is responsible for the formation of Cooper pairs which is the driving force of superconductivity (dissipationless transport). \begin{figure} \centering \includegraphics[width=8cm, angle=0]{/home/beekman/Documents/sctiptie/plaatjes/phonon.ps} \caption{Exchange of a phonon with the lattice.}\label{eph} \end{figure} The frequency of the interacting phonon is a measure for the stiffness of the lattice. Electrons within an energy interval $\hbar\omega_D$ around the Fermi energy participate in the interaction, therefore $\omega_D$ is a cut-off frequency. The attractive interaction potential is zero outside this interval. %By definition, the Fermi sea contains electrons with unperturbed plane %wave energies. However, the e-ph interaction introduces a %perturbation on these plane waves. The wavefunction describing two %perturbed electrons added to the Fermi sea at T$=0$ is, %\begin{equation} %\psi(r_1,r_2)=\sum_{k}g_ke^{i k.r_1}e^{-i k.r_2} %\end{equation} %with eigenvalues E determined by, %\begin{equation} %(E-2\varepsilon_k)g_k=\sum_{k'>k_F}V_{kk'}g_k' %\end{equation} %with $\varepsilon_k$ the unperturbed plane wave energies. The %interaction potential $V_{kk'}$ scatters a pair of electrons from %state $(k',-k')$ to state $(k,-k)$. As already mentioned %above $V_{kk'}$ needs to be an attractive potential for Cooper %pairs to form. When T $=0$ the highest filled energy level is the %Fermi energy E$_F$. A bound pair state can only exist when %E $<2E_F$, the potential being repulsive when this condition is not %met.% The frequency of the interacting phonon is a measure for the %%stiffness of the lattice. Therefore, there will be some cutoff %%frequency above which the potential will become equal to zero %%(Debye frequency $\omega_D$). An (oversimplified) interaction potential is given by de %Gennes \cite{gennes} %\begin{equation} %V(q,w)=\frac{4{{\pi}e}^2}{\mathbf{q}^2+{k_s}^2}+\frac{4{\pi}e^2}{\mathbf{q}^2+{k_s}^2}\frac{w_q^2}{w^2+{w_q}^2} %\end{equation} %were $\textbf{q}=\textbf{k}-\textbf{k}'$, 1/$k$$_s$ is the screening length and $\omega_q$ the phonon frequency. This equation does show that the order of magnitude of the coulomb %repulsion is of the same order as the attractive potential. \section{The BCS theory} Bardeen, Cooper and Schrieffer \cite{bcs} were the founders of the so called BCS theory (weak coupling theory) which revolutionized our understanding of superconductivity. It is most convenient to write this theory in the language of second quantization, by describing everything in terms of creation (c$^*$$_{k\uparrow}$) and annihilation operators (c$_{k\uparrow}$). The ground state can be expressed as, \begin{equation} |\psi_G>=\prod_{k=k_1,\ldots,k_m}(u_k+v_k{c}_{k\uparrow}^{*}{c}_{-k\downarrow}^{*})|\Phi_0> \end{equation} with $|\Phi_0>$ the vacuum state (no particles present), $|u_{k}|^2$ the probability that a pair state is unoccupied and $|v_{k}|^2$ that it is occupied. The Hamiltonian that has to be diagonalized is \cite{zaanen}, \begin{equation}\label{BCS} H_{BCS}=\sum_{k,\sigma}(\varepsilon_k-\mu){c}_{k\sigma}^{*}{c}_{k\sigma}-V\sum_{k,k'}{c}_{k\uparrow}^{*}{c}_{-k\downarrow}^{*}{c}_{-k\downarrow}{c}_{k\uparrow} \end{equation} (see also \cite{tinkham}). Here $\mu$, the chemical potential, is more or less equal to the Fermi energy, $\varepsilon_k$ is the energy of an electron in state k with k$>$k$_F$ and $V$ is the pairing potential. % We define the %following quantities: %\begin{equation}\label{delta} %\Delta=V\sum_{k}<{c}_{-k\downarrow}{c}_{k\uparrow}> %\end{equation} %the superconducting order parameter, and %\begin{equation}\label{e} %E_k=({\Delta}_k^2+{\xi}_k^2)^{1/2}\qquad \textrm{with}\qquad %\xi_k=\varepsilon_k-\mu %\end{equation} %The ground state is a phase coherent superposition of many body states. Because of this coherence the expectation %value $<{c}_{-k\downarrow}{c}_{k\uparrow}>$ can be nonzero in contrast to a normal metal where it is always equal to zero. %For many particle systems the fluctuations around these expectation values are small, therefore terms which contain the terms %(${c}_{-k\downarrow}{c}_{k\uparrow}$-$<{c}_{-k\downarrow}{c}_{k\uparrow}>$)$^2$ or of higher order can be neglected. %Neglecting these terms and putting in the definitions mentioned in eqs. \ref{delta} and \ref{e} gives, %\begin{equation}\label{MF} %H_{BCS}=\sum_{k,\sigma}(\varepsilon_k-\mu)({c}_{k\uparrow}^{*}{c}_{k\uparrow}+{c}_{-k\downarrow}^{*}{c}_{k\downarrow}) %-\Delta({c}_{k\uparrow}^{*}{c}_{-k\downarrow}^{*}+{c}_{-k\downarrow}{c}_{k\uparrow})+\frac{\Delta^2}{V} %\end{equation} %When extra energy is introduced to the system, like increasing the %temperature or application of an external magnetic field, this %results in excitation of particles from the Cooper pair condensate. %Therefore magnetic fields and increase in temperature are %so called pair breakers. What is the best way to handle these %excitations and is there any advantage in using the earlier %defined creation (annihilation) operators? Because of the second %quantization formalism the Hamiltonian is written in terms of %products of two creation, two annihilation and a creation and an %annihilation operator. This form makes it possible to use the Bogoliubov transformation; %\begin{equation}\label{anni} %c_{k\uparrow}=u^*_{k}\gamma_{k0}+v_k\gamma^*_{k1} %\end{equation} %\begin{equation}\label{creae} %c^*_{-k\downarrow}=-v^*_{k}\gamma_{k0}+u_k\gamma^*_{k1} %\end{equation} %to diagonalize H$_{BCS}$(eq.\ref{MF}). %These operators (eq. \ref{gam}) in terms of electron creation operators create %quasiparticles from the ground state. %\begin{equation}\label{crea} %\gamma_{k0}|\psi_G>=\gamma_{k1}|\psi_G>=0 \\ %\end{equation} %\begin{equation}\label{crea2} %\gamma^*_{k0}|\psi_G>=c^*_{k\uparrow}\prod_{l\neq %k}(u_l+v_l{c^*}_{l\uparrow}{c^*}_{-l\downarrow})|\Phi_0> %\end{equation} %\begin{equation}\label{crea3} %\gamma^*_{k1}|\psi_G>=c^*_{-k\downarrow}\prod_{l\neq %k}(u_l+v_l{c^*}_{l\uparrow}{c^*}_{-l\downarrow})|\Phi_0> %\end{equation} %Putting eq.\ref{creae} and eq.\ref{anni} in eq.\ref{MF} gives %a Hamiltonian expressed in terms of $\gamma_{ki}$ and %$\gamma_{ki}^*$. %We choose u$_k$ and v$_k$ in such a way %that the Hamiltonian is diagonalized which means that all terms %containing a product of two creation or a product of two %annihilation operators have coefficients equal to zero. Bearing %this in mind the equation for v$_k$ becomes, %\begin{equation}\label{v} %v_{k}^2=1-u_{k}^2=\frac{1}{2}\Bigg(1-\frac{(\varepsilon_k-\mu)}{E_k}\Bigg) %\end{equation} %Because probabilities are always normalized, the equation for $u_k$ is %\begin{equation} %u_{k}^2+v_{k}^2=1 %\end{equation} The diagonalized Hamiltonian can be written as \begin{equation}\label{Hdia} H_{BCS}=\sum_{k}\Big((\varepsilon_k-\mu)-E_k\Big)+\frac{\Delta^2}{V}+\sum_{k}E_k(\gamma_{k0}^*\gamma_{k0}+\gamma_{k1}^*\gamma_{k1}) \end{equation} with \begin{equation}\label{e} \Delta_k=V\sum_k\textrm{,}\;E_k=({\Delta}_k^2+{\xi}_k^2)^{1/2}\; \textrm{with}\; \xi_k=\varepsilon_k-\mu \end{equation} and the Bogoliubov transformations \begin{eqnarray}\label{gam} \gamma^*_{k0}=u^*_{k}c^*_{k\uparrow}-v^*_{k}c_{-k\downarrow} \\ \gamma^*_{k1}=u^*_{k}c^*_{-k\downarrow}+v^*_{k}c_{k\uparrow} \end{eqnarray} The operators $\gamma^*_{k0}$ and $\gamma^*_{k1}$ excite a quasiparticle into one of the states of the pair state (k$\uparrow$,-k$\downarrow$) leaving the other one unoccupied. This process effectively takes away the possibility for this pair state to participate in the many body wavefunction of the condensate and therefore raises the energy of the system. Therefore the first two terms in the Hamiltonian (eq. \ref{BCS}) refer to the ground state and the third refers to excitations. It is immediately evident that the ground state energy is \begin{equation}\label{Eg} E_G=\Big((\varepsilon_k-\mu)-E_k\Big)+\frac{\Delta^2}{V} \end{equation} To determine $\Delta$ at T=0 the ground state energy has to be minimized. Differentiating and putting $\delta$E$_G$/$\delta{\Delta}=0$ gives, \begin{equation} \Delta=\frac{w_D}{sinh(\frac{1}{VN(0)})}\simeq2\omega_De^{\frac{-1}{VN(0)}} \end{equation} This approximation is valid because we are in the \emph{weak coupling limit} were $VN(0)$$\ll$1. Here $\omega_D$ is the Debye frequency and N(0) is the electronic density of states at the Fermi energy. Even if the attractive potential is infinitely small, there will still be a finite order parameter (energy gap). %Thisf %approximation can be made since N($\varepsilon$) hardly varies on %a scale of V. Apparently, What is the relationship between $\Delta$ and the superconducting transition temperature T$_c$? The gap is largest when T=0 and it decreases with increasing T. At the superconducting transition temperature, T$_c$ the gap will vanish (i.e. $\Delta$=0). Consequently, the system ceases to be superconducting and will go back to the normal state. By minimizing the energy (including the third term of eq. \ref{BCS}, which refers to the excitations) and taking $\Delta$$\rightarrow$0 as T approaches T$_c$ results in, \begin{equation} \beta_c^{-1}\simeq1.13\omega_De^{\frac{-1}{VN(0)}} \end{equation} with $\beta$$_c$$^{-1}$=$k_BT_c$. In the weak coupling case (i.e. weak coupling between electrons and phonons) the ratio of the gap to T$_c$ becomes a universal number, \begin{equation} \frac{2\Delta(0)}{k_BT_c}=3.53 \end{equation} This relation holds for superconductors like aluminum. \section{The Ginzburg Landau theory}\label{secGL} The BCS theory is suited in dealing with an order parameter which is constant in space. However, the microscopic theory becomes very difficult when $\Delta$ starts to be inhomogeneous in space (for example: in S/F systems). In this case the Ginzburg Landau theory\cite{landau} (GL), a macroscopic theory based on Landau's theory on second order phase transitions, is more convenient. The GL theory only holds for T close to T$_c$ and introduces the complex order parameter, $\psi$ \begin{equation} \psi=|\psi|e^{i\phi} \end{equation} Here $|\psi|$ is the amplitude of the order parameter and $\phi$ contains the phase information. When $\psi$ is varying slowly in space, the free energy density can be expanded in series of $\psi$ $\footnote{ Since the free energy is real and $\hat{\psi}$ is complex, the expansion is be carried out in powers of$ |\psi|^2$}$\cite{tinkhamgl}; \begin{equation}\label{f} f=f_{n0}+\alpha|\psi|^2+\frac{\beta}{2}|\psi|^4+\frac{1}{2m^*}\Bigg|\Bigg(\frac{\hbar}{i}\nabla-\frac{e^*}{c}\textbf{A}\Bigg)\psi\Bigg|^2+\frac{h^2}{8\pi} \end{equation} where $h$ is an externally applied field, $f_{n0}$+$\frac{h^2}{8\pi}$ is the free energy density of the normal state and $\textbf{A}$ is the vector potential. Here $e^*$ and $m^*$ are the effective charge and the effective mass of a Cooper pair and therefore, $e^*$=$2e$ and $m^*$=$2m$ with $e$ and $m$ the electronic charge and mass. The parameters $\alpha$ and $\beta$ are functions dependent on temperature and the fourth term deals with fields and gradients. %When fields and gradients %are put equal to zero close inspection of eq.\ref{f} reveals that %$\beta$ should be positive. Otherwise, the lowest free energy %would coincide with an arbitrary large value of $\psi$. With only the first two terms to worry about one %also immediately recognizes that these only behave well for %temperatures near T$_c$. What about $\alpha$? When it is positive %the minimum free energy occurs when $\psi$$\rightarrow$0 thus when %the system is in the normal state. However, when $\alpha$ becomes %negative $f$ will reach its minimum when $\psi$=$\psi$$_S$ has the %finite value -$\alpha$/$\beta$ (superconducting state). Obviously, %$\alpha$ changes its sign when T crosses the superconducting %critical temperature. %The third term of eq.\ref{f} can be written as, %\begin{equation} %\frac{1}{2m^*}\Bigg[\hbar^2(\nabla|\psi|)^2+\Bigg(\hbar\nabla\phi-\frac{e^*}{c}\bf{A}\Bigg)^2|\psi|^2\Bigg] %\end{equation} %The first term gives the gradient in the magnitude of $\psi$ and %the second term gives the kinetic energy caused by the %supercurrents. If there is no boundary condition imposed on %eq.\ref{f} then fields, currents etc. will be absent and %$\psi$=$\psi$$_S$ everywhere. If there are fields etc., %$\psi$($\bf{r}$)=$|$$\psi$($\bf{r}$)$|$e$^{i\phi(\bf{r})}$ will %adapt itself to minimize the free energy F. Minimizing $f$ is done by taking $\delta$f/$\delta$$\psi$=0, which results in the well known GL differential equations, \begin{equation}\label{Sv} \alpha\psi+\beta|\psi|^2\psi-\frac{\hbar^2}{4m}\Bigg(\nabla-\frac{2ie}{\hbar c}\textbf{A}\Bigg)^2\psi=0 \end{equation} \begin{equation} \textbf{j}=\frac{c}{4\pi}\textrm{curl}\textbf{h}=-\frac{ie\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*)-\frac{2e^{2}}{mc}\psi^*\psi\textbf{A} \end{equation} Equation \ref{Sv} is very similar to the Schr\textrm{$\ddot{o}$}dinger equation for particles with eigenvalues -$\alpha$. The second term in the equation can be seen as a repulsive potential. Since this potential of $\psi$ basically acts on itself, the wavefunction $\psi$($\textbf{r}$) tends to spread out throughout space as much as possible. %If we assume that the edges of the system do not %effect the order parameter a boundary condition can be imposed here such that, %\begin{equation}\label{GLb} %(\frac{\hbar}{i}\nabla-\frac{e^*}{c}\textbf{A})\psi\Bigg|_n=0 %\end{equation} %The assumption is that there is no current flowing out of the superconductor. %However, when a metal is %placed in contact with a superconductor, according to de %Gennes\cite{gennes} the boundary condition needs to be modified to, %\begin{equation}\label{gen} %(\frac{\hbar}{i}\nabla-\frac{e^*}{c}\textbf{A})\psi\Bigg|_n=\frac{i\hbar}{b}\psi %\end{equation} %with $b$ a real constant. This means that $\psi$ will not go %abruptly to zero at the interface but the order parameter extends %over a length $b$ into the adjacent material. %Now, if we put $\textbf{A}$=0 (no applied external field) and take $\psi$ real, the first GL %differential equation becomes, %\begin{equation} %\frac{\hbar^2}{2m^*|\alpha|}\frac{d^2f}{dx^2}+f-f^3=0 %\end{equation} If there are no external magnetic fields or gradients present ($\textbf{A}$ and $\nabla\psi$ are zero) the solution to eq. \ref{Sv} with the lowest free energy is given by, \begin{equation} |\psi|^2=-\frac{\alpha}{\beta} \end{equation} Close to T$_c$, $\beta$ is taken constant and $\alpha$=$a(T-T_c)$. Furthermore, close to T$_c$, the amplitude of the order parameter is small. Neglecting terms in eq. \ref{Sv} containing products of $|\psi|$ gives the linearized GL equation, \begin{equation}\label{Svlin} \alpha\psi-\frac{\hbar^2}{4m}\Bigg(\nabla-\frac{2ie}{\hbar c}\textbf{A}\Bigg)^2\psi=0 \end{equation} When there are no applied magnetic fields ($\textbf{A}$=0), the second term (eq. \ref{Svlin}) describes the variation of $\psi$ and therefore it is natural to define a characteristic length for variation of $\psi$ in space, namely the GL coherence length, $\xi(T)$, \begin{equation} \xi_{GL}^2(T)=\frac{\hbar^2}{4m|\alpha(T)|} \end{equation} Near T$_c$ the following equations for $\xi$ hold, \begin{equation}\label{xic} \xi_{GL}(T)=0.74\frac{\xi_0}{(1-\frac{T}{T_c})^{1/2}}\qquad clean:l\gg\xi_0 \end{equation} \begin{equation}\label{xid} \xi_{GL}(T)=0.855\frac{(\xi_0l_e)^{1/2}}{(1-\frac{T}{T_c})^{1/2}}\qquad dirty:l\ll\xi_0 \end{equation} with $l_e$ the mean free path and $\xi_0$=$\hbar$v$_F$/$\pi$$\Delta(0)$ the BCS coherence length, which basically is the spatial extension of the Cooper pair at T = 0. In our case the aluminum is a dirty superconductor and therefore eq. \ref{xid} must be used. \subsection{Type I and type II superconductors.} When a superconducting material is in the normal state a magnetic field can completely penetrate the material. However when the material is cooled through T$_c$ the magnetic field is expelled from the interior of the superconductor. Inside the superconductor the magnetic field decays exponentially to zero. The distance over which the magnetic field has decayed to $\frac{1}{e}B_{ext}$ (with $B_{ext}$ the magnetic field outside the superconductor) is called the penetration depth, $\lambda_L$. %It is known that a superconductor expels a magnetic field. %However, the magnetic field does penetrate the edges of the %superconductor. This penetration depth has to be a function of %temperature because at T=T$_c$ the field penetrates the system The penetration depth is given by \cite{tinkhaml}, \begin{equation}\label{L} \lambda_L(T)=\frac{\lambda_L(0)}{|2(1-\frac{T}{T_c})|^{1/2}} \end{equation} with $\lambda_L$(0) the London penetration depth, which is given by \begin{equation}\label{L} \lambda_L(0)=\big(\frac{mc^2}{4\pi n_se^2}\big)^{1/2} \end{equation} with $m$, $e$ the electronic mass, $c$ the speed of light and charge and $n_s$ the number density of superconducting electrons (electrons participating in Cooper pair formation). The two characteristic lengths $\xi(T)$ and $\lambda_L(T)$ have similar T-dependence and when T$\rightarrow$T$_c$ both approach infinity. The ratio of these quantities is \begin{equation} \kappa=\frac{\lambda_L(T)}{\xi(T)} \end{equation} Using equations \ref{L} and \ref{xid} gives \begin{equation} \kappa=0.715\frac{\lambda_L(0)}{l}\qquad dirty:l\ll\xi_0 \end{equation} Superconductors can be divided into two groups based upon their different response to magnetic fields: type I and type II superconductors. In type I superconductors superconductivity is destroyed at a certain critical field H$_c$ while type II superconductors go into an intermediate state at H$_{c1}$ where normal state channels (vortices) coexist with the superconducting state. This intermediate inhomogeneous superconducting state is destroyed at a second critical field H$_{c2}$ leaving the system in the normal state. When $\kappa$$<$$\frac{1}{\sqrt{2}}$ the superconductor is type I and when $\kappa$$>$$\frac{1}{\sqrt{2}}$ it is of type II. \section{Ferromagnetism} In contrast to non-magnetic materials, ferromagnets have a spontaneous magnetic moment (in zero field). The spin direction of the electrons is not randomized but ordered and this ordering induces an internal field, the \emph{exchange field}. However, thermal energy competes with this ordering trying to randomize the spin direction and at a certain temperature the thermal energy destroys the ordering completely. This destruction happens at the Curie temperature T$_C$ $\footnote{T$>T$$_C$: disordered paramagnetic phase.\\ T$<$T$_C$: ordered ferromagnetic phase.}$. When all spins are pointing in the same direction, i.e. there is uniform magnetization throughout the material, the ferromagnet is saturated. Usually this saturation can only be reached by application of an external magnetic field. In zero field the material will be divided in regions with different spin direction called domains. Inside the domains there is uniform magnetization. \subsubsection{Domain walls} %The energies playing a crucial role in the domain formation are, %\begin{itemize} %\item{Exchange energy: Energy cost of a change in direction of magnetization.} %\item{Magnetostatic energy: Introduced by a discontinuous normal component of magnetization across an interface.} %\item{Magnetocrystalline anisotropy: Certain crystallographic directions are directions of easy magnetization.} %\item{Magneto-elastic anisotropy: Magnetocrystalline energy proportional to strain.} %\item{Zeeman energy: Potential energy of magnetic moment in a field.} %\end{itemize} %The formation of domains is a process of competition between the energies mentioned above. During the magnetization process domains are formed which are separated by walls. Mainly, the walls are either Bloch walls or N\textrm{$\acute{e}$}el walls. \begin{itemize} \item{Bloch walls: Consider two adjacent domains with opposite direction of magnetization. The wall connects these domains and thus contains spins which rotate from one direction to the opposite direction. In the case of a Bloch wall the rotation is out of the plane of the magnetization direction of the domains.} \item{N\textrm{$\acute{e}$}el walls: Consider again these two domains (again opposite direction of magnetization) but this time separated by a N\textrm{$\acute{e}$}el wall. In this case the spins inside the wall rotate in plane instead of out of plane.} \end{itemize} Creating Bloch walls costs energy since inside the walls the spins are directed away from the direction of easy magnetization i.e. the anisotropy energy increases. However, their formation also lowers the magnetostatic energy (introduced by a discontinuous normal component of magnetization across an interface). If the ferromagnet is saturated i.e. single domain, the magnetostatic energy (MS energy) is large. Creating domain walls lowers the magnetostatic energy as shown in fig. \ref{dom}. However, formation of closure domains is necessary to reduce the MS energy completely to zero. \begin{figure}[!] \includegraphics[width=12cm,height=7.5cm,angle=180]{/home/beekman/Documents/sctiptie/plaatjes/MSen.ps} \centering\caption{Effect of domain formation on magnetostatic energy.}\label{dom} \end{figure} When the thickness of the ferromagnet is reduced the magnetostatic energy will increase. Because of the out of plane rotation in the Bloch wall the wall will, at some point, extend throughout the whole thickness of the ferromagnet, introducing two charged surfaces (see fig.\ref{bn}). Moving these surfaces closer together increases the magnetostatic energy. Below a certain thickness, in plane rotation inside the wall will be energetically more favorable than out of plane rotation. The Bloch walls will become N\textrm{$\acute{e}$}el walls. Now the charged surfaces are inside the film, however, the area of these charged surfaces decreases with decreasing thickness. \begin{figure}[!] \includegraphics[width=12cm,height=5cm,angle=180]{/home/beekman/Documents/sctiptie/plaatjes/blochneel.ps} \centering\caption{Bloch and N\textrm{$\acute{e}$}el domain wall extending throughout the whole thickness of the film.}\label{bn} \end{figure} %\begin{figure}[!] %\includegraphics[width=8cm,height=4.5cm,angle=180]{/home/beekman/Documents/sctiptie/plaatjes/enth.ps} %\centering\caption{Energy cost of domain wall vs. thickness of the ferromagnetic film.}\label{bn2} %\end{figure} \subsubsection{Shape anisotropy} A system always tends towards the state with lowest possible energy. In the case of a long and narrow strip, this means finding the balance between creating domain walls (which costs energy) and minimizing magnetostatic energy. In such a strip it is therefore energetically more favorable to have the magnetization parallel to the long axis of the strip, shape anisotropy. It only causes a magnetostatic field at the far edges of the strip which can be minimized by the formation of closure domains. When the aspect ratio of the strip is large, the strip is effectively single domain. However, there will be two closure domains at the far edges of the strip. % Since there is a magnetic field in the sputtering chamber, caused by the target, the axis of easy magnetization is along the long axis of the strip. \subsubsection{Coercivity} Application of an external magnetic field alters the direction of magnetization. \begin{itemize} \item{Weak field: Domains with their magnetization in the same direction as the applied field will grow (this is energetically more favorable). The other domains will shrink and ultimately vanish. This is accomplished by domain wall motion.} \item{Strong field: In this case the spins which are not aligned with the field direction will rotate until they are parallel to the applied field.} \end{itemize} When the field direction is perpendicular to the direction of easy magnetization the spins will rotate towards the field direction quasi statically when the field is increased. If the field is parallel to the easy axis domain wall motion governs the change of the magnetization direction. When the material is in a single domain state the magnetization tends to remain in the same direction even when the field changes sign. At a certain field the energy becomes so large that the magnetization switches abruptly to align with the field direction. Therefore, there is hysteresis in the M-H loops of ferromagnetic materials. A hysteresis loop is shown in fig. \ref{hystloop} and the coercivity field is indicated. The coercivity field is defined as the field that has to be applied, to reduce the magnetization M of a sample back to zero, after it has been completely magnetized (M$_s$). \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/hystloop.ps} \centering\caption{Hysteresis loop with the coercivity field H$_c$ indicated.}\label{hystloop} \end{figure} \section{Proximity effect} What happens when a superconductor is in contact with another material? The exchange of phase information between the superconductor and the other material is called the \emph{proximity effect}. \subsection{Superconductor/Normal metal interface.} \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/proxgsn.ps} \centering\caption{Proximity effect at a S/N interface.}\label{SN} \end{figure} The proximity effect in a superconductor-normal metal interface is shown in figure \ref{SN}. %The induced order parameter decreases exponentially into the normal metal because of de-phasing of the electron and hole of the Cooper pair. %This exponential decay can be explained by looking at the boundary condition of the GL equations. Ginzburg and Landau postulated a boundary condition (eq.\ref{GLb}) for the behavior of the order parameter at the edge of the superconductor. But at a SN interface the boundary condition of de Gennes (eq.\ref{gen}) holds. The spatial extension of the Cooper pairs induces a finite order parameter in the normal metal. The Cooper pairs can extend over the S/N interface and the part in the normal metal will still feel the pairing force of the superconductor, which not only induces a finite order parameter in the normal metal but also reduces the order parameter in the superconductor (see fig. \ref{SN}). The reduction of the order parameter in the superconductor extends over a length scale of the order of the superconducting coherence length, $\xi_s$. The jump of the order parameter at the interface (j) is caused by a finite interface resistance. Interface resistance is a result of scattering processes occurring at the interface, such as scattering on impurities. When there are no external fields applied the following equation holds at the S/N interface, \begin{equation} -i\hbar\nabla\Psi=i\frac{\hbar}{\xi_N}\Psi \end{equation} So the order parameter will become, \begin{equation} \Psi(r)=\Psi_0 e^{-\xi_N r} \end{equation} Thus, the order parameter decays exponentially with increasing distance into the normal metal over a distance $\xi_N$ (fig. \ref{SN}). %In the next section a scattering process, called Andreev reflection, also occurring at the interface is discussed. \subsection{Andreev reflections} The \emph{proximity effect} is caused by phase coherence between electrons in the superconductor and those in the adjacent material for instance a normal metal. This exchange of phase information is driven by scattering of electrons at the interface: Andreev reflection\cite{andreev}. When the system is in equilibrium one only has to take into account energies which lie below the energy gap, $\Delta$. There are in principle two processes which can occur at the interface (for $\epsilon<\Delta$), the so called \begin{itemize} \item{Specular reflection: the incoming electron (coming from the normal metal) is completely reflected back into the normal metal as an electron.} %\item{Transmission without branch crossing: the electron is transmitted into the superconductor and is still on the same side of the Fermi sphere (positive k).} %\item{Transmission with branch crossing: the electron is transmitted into the superconductor but going to the other side of the Fermi sphere during the process (negative k).} \item{\emph{Andreev reflection}: The incoming electron has an energy below $\Delta$ (energy gap in DOS of superconductor) and therefore there are no available energy levels in the superconductor. The electron collides with the interface and is scattered back as a hole. The hole has opposite spin direction compared to the incoming electron and therefore spin is conserved during Andreev reflection. Energy is also conserved since the electron has an energy $\epsilon$ above the Fermi surface while the hole has an energy $\epsilon$ below the Fermi surface. However, during this process charge is not conserved in the normal metal. A Cooper pair is formed at the interface (the incoming electron takes an electron from below the Fermi surface to form a Cooper pair and is transmitted into the superconductor, leaving a hole in the normal metal). When the hole propagates back along the same path as the incoming electron their phases will be correlated inducing a finite pair amplitude in the normal metal. The induced order parameter decreases exponentially into the normal metal because of dephasing of the electron and hole of the Cooper pair. When the excitation energy of the incoming electron is $\epsilon$ above the Fermi energy the hole will travel on a slightly different path than the incoming electron but they will still be phase correlated to a certain extend.} \end{itemize} \subsection{Superconductor/Ferromagnet interface.} When a superconductor is brought into contact with a ferromagnet the proximity effect is even more evident. A ferromagnet is a so called pair breaker because superconductivity and magnetism are two competing phenomena. Andreev reflection also occurs at the interface in a S/F system but the exchange field has a large influence on the amount of Andreev reflection. When the exchange field is large, Andreev reflection will be suppressed. This can be understood when looking at the rotation symmetry of the spins. In a normal metal all spin directions are equivalent; however, in a ferromagnet there is a majority and a minority spin band (the spin bands are separated by the exchange energy). A spin up electron is Andreev reflected into a spin down hole so the spin has to move from the majority spin band to the minority spin band. Depending on the density of states in both bands a certain amount of electrons (N$_{\downarrow}$/N$_{\uparrow}$ with N the electronic density of states) will be Andreev reflected. When the Cooper pairs diffuse into the ferromagnet there is an exchange field with a pair-breaking effect on the Cooper pairs. As the Cooper pair moves into the ferromagnet the spins will be subjected to different forces. The spin up electron lowers its potential energy by $h$ (=exchange field energy) and therefore has an increase of kinetic energy. Similarly, the spin down electron raises its potential energy and lowers its kinetic energy$\footnote{The change in potential energy must be compensated by change in the kinetic energy because energy must be conserved.}$. Because of this energy change the momentum of the Cooper pair changes upon entry in the ferromagnet. The center of mass momentum will change with, \begin{equation} Q=\frac{2h}{v_F} \end{equation} \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/proxgsf.ps} \centering\caption{Proximity effect at a S/F interface.}\label{SF} \end{figure} As a consequence, the order parameter will still decrease exponentially but there is an oscillation superimposed on this decay (see fig. \ref{SF}). However, in this thesis we are mainly interested in what happens in the superconductor. The superconducting transition temperature is suppressed by the exchange field of the ferromagnet, because not only do the Cooper pairs leak into the ferromagnet, but their extension over a coherence length also leads to a lowering of the order parameter in the superconducting layer. The suppression of T$_c$ can be approximated by using the GL theory, which will be explained in the next section. \section{Suppression of T$_c$.}\label{secsup} The behavior of a bulk superconductor is somewhat different from that of a superconducting thin film. When the thickness of the superconductor is decreased to around $\xi$ boundary effects become more important.%Reducing the thickness, $d$, of a superconductor results in a change in T$_c$ boundary effects become more important). These boundary conditions for a thin superconducting film are, \begin{equation}\label{bounds} \frac{\delta\psi(z)}{\delta z}\Bigg|_{z=\pm\frac{d}{2}}=0 \end{equation} meaning that there is no current flowing out of the superconductor. The linearized GL equation \ref{Svlin} can be solved using the boundary conditions (eq. \ref{bounds}) and a trial solution of the form, \begin{equation}\label{trial} \hat{\psi}=\psi(z)e^{(ik_xx+ik_yy)} \end{equation} \begin{figure}[!] \includegraphics[width=7cm,height=7cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/Sd.ps} \centering\caption{Superconducting thin film.}\label{sd} \end{figure} with $\psi(z)$ the amplitude of the complex order parameter varying with $z$ (see fig. \ref{sd}).%$\footnote{$z$=0 is the center of the superconductor and $z$=$\frac{-d}{2}$ is the edge of the superconductor.}$ Imagine a superconducting thin film sandwiched between to ferromagnetic layers. In this configuration the linearized GL equation can be used with the trial solution \ref{trial} as in the case of the superconducting thin film. However, the boundary conditions are different, \begin{equation} \psi(z)\Bigg|_{z=\pm\frac{d}{2}}=0 \end{equation} When no external magnetic field is present ($\textbf{A}$=0) and the trial solution mentioned above is used, the linearized GL equation becomes, \begin{equation} \frac{-\hbar^2}{4m}\frac{\delta^2\psi}{\delta z^2}+\alpha\psi=0 \end{equation} This equation is readily solved and gives, \begin{equation} \psi(z)=\psi_0cos(kz)\quad \textrm{with}\quad k^2=-\frac{\alpha4m}{\hbar^2} \end{equation} with $kd=\pi$ for the ground state. For the ground state, \begin{equation} -\alpha=\frac{\hbar^2\pi^2}{4md^2} \end{equation} with $d$ the thickness of the superconducting layer. Using the temperature dependence of $\alpha$ (see section \ref{secGL}) the suppression of T$_c$ of the superconducting film (due to the presence of the ferromagnetic banks) can be written as, \begin{equation}\label{sup} T_c=T_{c0}-\frac{\hbar^2\pi^2}{4mad^2}\quad\textrm{with}\quad \alpha=a(T_c-T) \end{equation} or \begin{equation}\label{sup2} T_c=T_{c0}-\frac{\pi^2\xi_{GL}^2}{d^2} \end{equation} Where T$_{c0}$ is the bulk superconducting transition temperature. The transition temperature of the F/S/F trilayer is inversely dependent on $d^2$. However, when the superconducting layer is thinner than $d_{cr}$, the critical thickness, superconductivity will be destroyed completely by the exchange fields of the ferromagnets. %\subsection{Green's function and the quasi classical approximation.} %the concept of the Green's function (i.e. propagator) will be introduced. The Green's function is necessary to formulate the Gorkov equations \cite{gorkov} and from there to form the Usadel equation \cite{usadel} for a dirty superconductor. The definition of a single particle Green's function is$\footnote{* = complex conjugate}$, %\begin{equation} %i\hat{G}(\mathbf{x}t,\mathbf{x'}t')=\frac{<\Psi_0|T[\Psi_{H\alpha}(\mathbf{x}t)\Psi_{H\alpha}^{*}(\mathbf{x'}t')]|\Psi_0>}{<\Psi_0|\Psi_0>} %%\end{equation} %\begin{itemize} %\item{$|\Psi_0>$: the Heisenberg ground state of the interacting system\\ ($<\Psi_0|\Psi_0>$=1 because of normalization).} %\item{$\Psi$$_{H\alpha}$(\textbf{x}t)=e$^{iHt/\hbar}$$\Psi$$_{\alpha}$(\textbf{x})e$^{-iHt/\hbar}$ is a Heisenberg operator.} %\item{H: the Hamiltonian of the system.} %\item{$\alpha$ and $\beta$ label the components of spin.} %\item{T: the time operator, to ensure that processes occur in the correct (chronological) order.} %\end{itemize} %$\Psi$$_{H\alpha}$(\textbf{x}t) and $\Psi$$_{H\alpha}$$^{*}$(\textbf{x'}t') are field operators and the Green's function is the expectation value of their product. To evaluate how such field operators evolve with time the Heisenberg equation of motion can be used to obtain the field equation. The equation of motion of the Heisenberg operator is, %\begin{equation} %i\hbar\frac{\delta}{\delta t}\Psi_{H\alpha}(\mathbf{x}t)=e^{iHt/\hbar}[\Psi_{\alpha}(\mathbf{x}),H]e^{-iHt/\hbar} %\end{equation} %$\Psi$$_{\alpha}$(\textbf{x}),H] is the commutator. One can write down the equations of motion for the field operators ($\Psi$$_{H\uparrow}$(\textbf{x}$\tau$) and $\Psi$$_{H\downarrow}^*$(\textbf{x}$\tau$)) for arbitrary parameter $\tau$ using the Heisenberg equation of motion. Doing this and using the definition of the single particle Green's function lead to the Gorkov equations \cite{gorkov}. %\begin{eqnarray} %\Bigg[-\hbar\frac{\delta}{\delta \tau}-\frac{1}{2m}\Bigg(-i\hbar\nabla+\frac{e\mathbf{A}}{c}\Bigg)^2+\mu\Bigg]G(\mathbf{x}\tau,\mathbf{x'}\tau')+\Delta(\mathbf{x})F^*(\mathbf{x}\tau,\mathbf{x'}\tau')\\ \nonumber %=\hbar\delta(\mathbf{x}-\mathbf{x'})\delta(\tau-\tau') %\end{eqnarray} %\begin{eqnarray} %\Bigg[\hbar\frac{\delta}{\delta \tau}-\frac{1}{2m}\Bigg(i\hbar\nabla+\frac{e\mathbf{A}}{c}\Bigg)^2+\mu\Bigg]F^*(\mathbf{x}\tau,\mathbf{x'}\tau')-\Delta(\mathbf{x})^*G(\mathbf{x}\tau,\mathbf{x'}\tau')=0 %\end{eqnarray} %here \textbf{A} is the vector potential and %\begin{equation} %G(\mathbf{x}\tau,\mathbf{x'}\tau')=- %\end{equation} %\begin{equation} %F(\mathbf{x}\tau,\mathbf{x'}\tau')=- %\end{equation} %The matrix version of the Gorkov equations is, %\begin{equation} %\Bigg[i\omega\tau_3+\frac{\hbar^2}{2m}\frac{\partial^2}{\partial\mathbf{r}_1^2}+\mu+\hat{\Delta}(\mathbf{r}_1)\Bigg]\hat{G}(\mathbf{r}_1,\mathbf{r}_2)=\delta(\mathbf{r}_1-\mathbf{r}_2) %\end{equation} %with $\tau_3$ $\footnote{Pauli matrices:$\tau_1=\left(\begin{array}{cc}0&1\\ %1&0 \end{array}\right)$ $\tau_2=\left(\begin{array}{cc}0&-i\\ %i&0 \end{array}\right)$ $\tau_3=\left(\begin{array}{cc}1&0\\ %0&1 \end{array}\right)$}$ %the third Pauli matrix. %Here it was assumed that there is no magnetic field present (\textbf{A}=0) also since the Hamiltonian is generally independent of time a Fourier transform introduces the term with $\omega$ (instead of the partial derivative). The following definitions for $\hat{\Delta}$ and $\hat{G}$ hold, %\begin{equation} %\hat{\Delta}=\left(\begin{array}{cc} %0&\Delta\\ %-\Delta^*&0 %\end{array}\right) %\end{equation} %\begin{equation} %\hat{G}=\left(\begin{array}{cc} %G&F\\ %F^*&-G %\end{array}\right) %\end{equation} %And the self consistency relation of $\Delta$ is %\begin{equation} %\Delta(\mathbf{r})=-\frac{\lambda}{2}T\sum_{\omega}Tr(\tau_1+i\tau_2)\hat{G}(\mathbf{r},\mathbf{r}) %\end{equation} %with $\omega$ the Matsubara frequency %\begin{equation} %\omega=\pi T(2n+1) %\end{equation} %What is the physical meaning of these functions? $G$(normal Green's function) describes the creation and annihilation of quasiparticles and is only nonzero outside the energy gap. $F$(anomolous Green's function) gives information about the creation of Cooper pairs and is closely linked to the pair amplitude (is nonzero inside the gap and reduces to zero as T$\rightarrow$T$_c$. %Before solving the Gorkov equations they need to be simplified. This simplification is called the \emph{Quasi classical approximation}. %First a coordinate transformation is done, the Wigner transformation. The Green's function (both normal and anomalous) are functions of the coordinates of two particles (members of Cooper pair) and therefore a transformation to a center of mass coordinate of the particles is logical. So \textbf{R}=(\textbf{r1}+\textbf{r2})/2 and r=\textbf{r1}-\textbf{r2} (x=r1 and x'=r2). Second, the Fourier transform of the Green's function makes the function frequency dependent, the spectrum of the Green's function shows two peaks corresponding a fast oscillating part (short range interactions, oscillations on the scale of the Fermi wavelength) and a slow oscillating part (long range interactions, on the scale of $\xi_0$). Finally, integrating over energy and impurities eliminates the fast oscillations of the Green's function. In principle, by doing this approximation, we discard the information about the excitation spectrum ((\textbf{r1}-\textbf{r2}) dependence), however there is enough information contained in the macroscopic field (Cooper pair wave function)((\textbf{r1}+\textbf{r2})/2 dependence). Leaving a simplified version of the Gorkov equations, the Eilenberger equations\cite{eil}. %\begin{equation} %\nu_F\nabla_r\hat{g}+\Bigg[\omega\tau_3+\hat{\Delta}-\frac{1}{\tau}<\hat{g}>,\hat{g}\Bigg]=0 %\end{equation} %here $\tau$ is the elastic scattering time. The self consistency equation becomes, %\begin{equation} %\Delta ln\frac{T}{T_c}-\pi T\sum_{\omega}\Bigg(\frac{\Delta}{\omega}-\Bigg)=0 %\end{equation} %Also, %\begin{equation} %\hat{\Delta}=\left(\begin{array}{cc} %0&\Delta\\ %\Delta^*&0 %\end{array}\right)\quad %\hat{g}=\left(\begin{array}{cc} %g&f\\ %f^*&-g %\end{array}\right)\quad \textrm{and}\quad %\hat{g}^2=1 %\end{equation} %$g$ and $f$ are the quasi classical (or Eilenberger) Green's functions. %\subsection{Suppression of T$_c$.} %To calculate the suppression of T$_{c}$ of a SF bilayer compared to the T$_c$ of a unperturbed superconductor, the Usadel\cite{usadel} equations are needed. These equations can be derived by considering another simplification: the dirty limit, which is applicable to the case of Aluminum. The dirty limit is characterized by l$\ll$$\xi$, with l the mean free path of the system and $\xi$ the coherence length. The process dominating the dynamics of the system on coherence length scale is diffusion. Because the motion is diffusive the Green's function does not depend strongly on angular direction (denoted by the vector \textbf{n}) and can be considered isotropic. Because of this weak dependence on \textbf{n}, the quasi classical Green's function can be expanded in spherical harmonics, %\begin{equation} %g(\mathbf{n},\mathbf{r})=g_0(\mathbf{r}) + \mathbf{n}.g_1(\mathbf{r})+ \ldots %\end{equation} %The equations for the isotropic part of the quasi classical Green's function are (Usadel equations\cite{usadel}), %\begin{equation}\label{usa1} %i\hbar\nabla[\hat{g}(\mathbf{r})\nabla\hat{g}(\mathbf{r})]=[\omega\tau_3+\hat{\Delta}(\mathbf{r}),\hat{g}(\mathbf{r})] %\xi_S^2\pi T_{cS}\frac{d^2F_S}{dx^2}-|\omega|F_S+\Delta=0\quad \textrm{in the superconductor} %\end{equation} %\begin{equation}\label{usa2} %\xi_F^2\pi T_{cS}\frac{d^2F_F}{dx^2}-(|\omega|+ih\textrm{sgn}\omega)F_F=0\quad %\textrm{in the ferromagnet} %\end{equation} %and the self consistency equation, %\begin{equation}\label{del} %\Delta \textrm{ln} \frac{T_c}{T}=\pi T\sum_{\omega}\Big(\frac{\Delta}{\omega}-F_S\Big) %\end{equation} %Boundary conditions have to be imposed on the equations \ref{usa1}-\ref{del}. %At the outer surfaces of the bilayer, %\begin{equation} %\frac{dF_S(d_s)}{dx}=\frac{dF_F(-d_F)}{dx}=0 %\end{equation} %and at the interface between the two materials, %\begin{equation} %\xi_S\frac{dF_S(0)}{dx}=\gamma\xi_F\frac{dF_F(0)}{dx} \quad \gamma=\frac{\rho_S\xi_S}{\rho_F\xi_F} %\end{equation} %\begin{equation} %\xi_F\gamma_B\frac{dF_F(0)}{dx}=F_S(0)-F_F(0) \quad \gamma_B=\frac{R_BA}{\rho_F\xi_F} %\end{equation} %Here A is the area of the interface, $\rho_{F,S}$ are the normal state resistivities, $R_B$ is the interface resistance and the coherence lengths %%\begin{equation} %\xi_F=\sqrt{\frac{D_F}{2\pi T_{cS}}}\quad\xi_S=\sqrt{\frac{D_S}{2\pi T_{cS}}} %\end{equation} %with $D=vl/3$ the diffusion constant (these equations and the coming equation in this section are taken from the thesis of Fominov \cite{fominov}). %These are in principle SN boundary conditions but they can be applied if the following constraint holds, $h/E_F\ll 1$. This means that if the exchange field is relatively weak , smaller than the Fermi energy, these boundary conditions are also applicable to the SF systems. The thickness of the interface has atomic scale proportions, therefore when the exchange energy is considerably smaller than the Fermi energy the characteristic length of the magnetic properties is much larger then the atomic scale. As a consequence the boundary conditions are determined by the properties of the interface and therefore the contacting materials have no influence on the boundary conditions ensuring their validity in the SF systems. %However, Py is a ferromagnet of intermediate strength and these boundary conditions will be an approximation. %The boundary conditions of the Usadel equations are complex, however we are interested in the real part only. To obtain the real part F is written as, %\begin{equation} %F^{\pm}=F(\omega)\pm F(-\omega) %\end{equation} %The symmetry $F^*(\omega)=F(-\omega)$ implies that $F^+$ ($F^-$) is a real (imaginary) function. The solution of the Usadel equation in the superconductor using only $F^+$ has the form, %\begin{equation} %F_S^+(x,\omega)=f(\omega)cos\Bigg(\Omega\frac{x-d_S}{\xi_S}\Bigg) %\end{equation} %and, %\begin{equation} %\Delta(x)=\delta\Bigg(\Omega\frac{x-d_S}{\xi_S}\Bigg) %\end{equation} %Satisfying the boundary conditions yields, %\begin{equation} %f(\omega)=\frac{2\delta}{\omega+\Omega^2\pi T_{cS}} %\end{equation} %Now, the self consistency equation takes the form, %\begin{equation}\label{tc} %ln\frac{T_{cS}}{T_c}=\psi\Bigg(\frac{1}{2}+\frac{\Omega^2}{2}\frac{T_{cS}}{T_c}\Bigg)-\psi\Bigg(\frac{1}{2}\Bigg) %\end{equation} %where $\psi$ is the digamma function $\footnote{The digamma function is the logarithmic derivative of the gamma function: $\psi(z)=\frac{1}{\Gamma(z)}\frac{d\Gamma(z)}{dz} \quad (\Gamma(n+1)=n!)$}$. %With, %\begin{eqnarray} %\Omega\textrm{tan}\Bigg(\Omega\frac{d_S}{\xi_S}\Bigg)=W(\omega) \\ \nonumber %W(\omega)=\gamma\frac{A_S(\gamma_B+\textrm{Re}B_h)+\gamma}{A_S|\gamma_B+B_h|^2+\gamma(\gamma_B+\textrm{Re}B_h)}\\ \nonumber % A_S=k_S\xi_S\textrm{tanh}(k_Sd_S)\quad k_S=\frac{1}{\xi_S}\sqrt{\frac{\omega}{\pi T_{cS}}}\\ \nonumber % B_h=\Bigg[k_h\xi_F\textrm{tanh}(k_hd_F)\Bigg]^{-1}\quad k_h=\frac{1}{\xi_F}\sqrt{\frac{|\omega|+ih\textrm{sgn}\omega}{\pi T_{cS}}} %\end{eqnarray} %With equation \ref{tc} the suppression of the transition temperature of a superconductor in contact with a ferromagnetic layer can be calculated. \section{Superconducting spin switches} \subsection{Buzdin, Tagirov, Gu and Bader.} The suppression of the transition temperature caused by the proximity of a ferromagnet can be used in spintronic devices, such as a superconducting spin switch. The spin switch was theoretically proposed by Tagirov\cite{Tagirov} and Buzdin\cite{buz}. \begin{figure}[!] \centering \includegraphics[width=5cm, height=5cm, angle=0]{/home/beekman/Documents/sctiptie/plaatjes/tagspin.ps} \caption{The device, proposed by Tagirov.}\label{spintag} \end{figure} Tagirov proposed a structure (fig. \ref{spintag}) where the superconducting layer is sandwiched between two ferromagnetic layers. The magnetization direction of one of the F layers is pinned by the presence of an antiferromagnetic layer while the direction of the second F layer remains free to rotate. The T$_c$ of such a device can be controlled by the exchange fields of the F layers, when the thickness of the superconductor (d$_s$) is of the order of the superconducting coherence length $\xi_s$. The Cooper pairs are to probe the spin directions of both F banks at the same time. In this case the exchange field can be used to control T$_c$. The exchange field can be altered by changing the mutual orientation of the magnetization directions of the F layers. This can be seen by looking at the electrons which leak into the superconductor. These electrons are spin polarized and will reside in states near the Fermi energy. However, the attractive interaction, leading to the formation of Cooper pairs, takes place in these states.% resulting in the fact that when electrons all with the same spin orientation leak in$\footnote{The F banks have a parallel mutual magnetization direction.}$, no Cooper pairs will form and there will be less available states for the formation process. Imagine electrons with the same spin direction entering the superconductor$\footnote{Mutual orientation of the exchange fields of the F banks is parallel.}$. They do not participate in the formation of Cooper pairs. However, since these electrons occupy states near the Fermi energy, these states will become unavailable for the Cooper pair formation process and less Cooper pairs are formed. When electrons with opposite spin direction leak in$\footnote{Mutual orientation of the exchange fields of the F banks is antiparallel.}$, Cooper pairs can form. As a result the order parameter (and thus T$_c$) is suppressed more in the Parallel (P) state and much less in the Antiparallel (AP) state. Ideally when switching form P to AP state one can switch from normal state resistance to zero resistance. Baladi\textrm{$\acute{e}$} et al\cite{buz2} calculated the suppression of T$_c$ for the superconducting spin switch which is shown in fig. \ref{Rpair}. Here $\phi$ is the angle between the exchange field and the z-axis. \begin{figure}[h] \includegraphics[width=12cm,height=17cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/balbuz.ps} \centering\caption{Top: superconducting spin switch (F/S/F), the exchange field is given by large arrows and $\Phi$ is the angle between the exchange field and the $z$-axis. Bottom: the suppression of T$_c$ as function of $d^*/d_S$.}\label{Rpair} \end{figure} \afterpage{\clearpage} In the figure, T$_c^*$/T$_c$ is given as function of $d^*/d_S$, where T$_c$ is the transition temperature of a single superconducting layer, T$_c^*$ is the suppressed transition temperature of a F/S/F trilayer, $2d_S$ is the tickness of the superconducting layer and $d^*$ is the effective length, \begin{equation} d^*=\gamma\sqrt{\frac{I}{D_F}}\frac{D_S}{4\pi T_c} \end{equation} with $I$ the exchange field, $\gamma$ the interface transparency$\footnote{$\gamma=0$: interface not transparent.\\ $\gamma=1$: interface fully transparent.}$ and $D_F$ and $D_S$ are diffusion coefficients in the ferromagnet and the superconductor. %\begin{equation} %R^{''}=\Bigg(\frac{N_Fd_F}{N_Sd_S}\Bigg)\Bigg(\frac{d_S}{\xi_S}\Bigg)^2\Bigg(\frac{I}{2\pi T_{c0}}\Bigg) %\end{equation} The effective length $d^*$ is proportional to the exchange field $I$ and therefore becomes larger for increasing exchange fields. From the figure it becomes clear that for a certain range of exchange fields the superconductivity is completely destroyed in the P case but only weakly suppressed in the AP case. When $d_{cr}^{AP}>d_S>d_{cr}^{P}$ (with $2d_{cr}$ critical thickness of S), it is possible to go from normal state resistance of S to zero resistance by changing the mutual orientation of the exchange fields of the F banks from parallel to antiparallel orientation. This effect has been the subject of the experimental research of Gu et al \cite{gu}. They measured the effect in the structure shown in figure \ref{gustruct} \begin{figure}[h] \includegraphics[width=7cm,height=6cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/gustruct.ps} \centering\caption{The device which was measured by Gu and Bader.}\label{gustruct} \end{figure} The dimensions of the device which they measured are: \begin{itemize} \item{FeMn: thickness = 6 nm} \item{Py (Ni$_{82}$Fe$_{18}$): thickness = 4 nm} \item{Cu$_{0.47}$Ni$_{0.53}$: thickness = 5 nm} \item{Nb: thickness = 18 nm} \item{Cu$_{0.47}$Ni$_{0.53}$: thickness = 5 nm} \item{Py (Ni$_{82}$Fe$_{18}$): thickness = 4 nm} \end{itemize} The layers were deposited on a silicon substrate. The critical thickness, d$_{cr}$, of Nb in a Cu$_{0.48}$Ni$_{0.52}$/Nb/Cu$_{0.48}$Ni$_{0.52}$ trilayer is 14 nm \cite{rus2}. The composition of the CuNi of Gu and Bader is comparable to the composition used in \cite{rus2} and it is stated that the critical thickness of Nb is approximately the same for every measured concentration of Ni. \begin{figure}[h] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/guRH.ps} \centering\caption{Resistance vs. applied magnetic field H. Blue symbols: T=5 K($>$T$_c$). Red symbols: T=2.81 K($\sim$T$_c$).}\label{guRH} \end{figure} The results of their measurement are depicted in figures \ref{guRH} and \ref{guRT}. Figure \ref{guRH} shows no change in resistance when switching from P to AP state for a temperature above the superconducting transition temperature. However, near T$_c$ the figure shows only a $\sim$25$\%$ decrease in resistance going from the P to the AP state. The suppression of T$_c$ for the P state compared to the AP state is shown in figure \ref{guRT}. \begin{figure}[ht] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/RTgu.ps} \centering\caption{Resistance vs. temperature. Inset: difference in resistance between P and AP state vs. temperature.}\label{guRT} \end{figure} Figure \ref{guRT} shows that the shift in T$_c$ is only 6 mK and this shift is much smaller than the width of the transition. This explains the fact that the resistance in figure \ref{guRH} does not drop completely to zero. However, other explanations are possible. For example, the suppression of T$_c$ due to effects caused by the multi domain structure of the ferromagnet\cite{rus}. In this case the Cooper pairs also probe multiple directions of magnetization (different directions on either side of the domain wall) which causes the same effect as in the spin switch. \subsection{Lateral spin switch.} We want to measure a lateral spin switch. The structure of the device is shown in figure \ref{latt}. \begin{figure}[h] \includegraphics[width=5cm,height=3cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/lafsf.ps} \centering\caption{The lateral spin device.}\label{latt} \end{figure} Gu et al used an S layer of Niobium in their spin switch which has the following characteristic length scales, \begin{equation}\nonumber \xi_0= 38\;\mathrm{nm} \quad \textrm{and} \quad \xi_{GL}(0)= 12\; \mathrm{nm} \end{equation} As mentioned before the BCS coherence length, $\xi_0$ is a measure for the spatial extension of Cooper pair and $\xi_{GL}$ is the Ginzburg-Landau coherence length. Therefore Nb is excellent for usage in a spin switch such as the device proposed by Tagirov. However, for a lateral spin switch a much longer coherence length is needed. The spacing between the F banks in our lateral switch has to be more than an order of magnitude larger than the thickness of the S layer used in the device of Gu et al (18 nm); because spacings smaller than 500 nm become very difficult to fabricate. Consequently we chose aluminum as superconductor in the lateral device, since Al has the following characteristic values, \begin{equation}\nonumber \xi_0\approx 1-2\; \mathrm{\mu m} \quad \textrm{and} \quad \xi_{GL}(0)\approx 200\; \mathrm{nm} \end{equation} and a T$_c$ around 1.2 K. We expect that Al has a coherence length which is long enough for a lateral device, which is tested by measuring the Al/Py bilayers. Because of the low T$_c$ of aluminum the devices are measured in a He$^3$ cryostat. Jedema\cite{jedema} performed spin injection measurements on a Py/Al/Py lateral device with Al in the normal state. The ferromagnets are used to inject a current of spin polarized quasiparticles into the adjacent material (Al). In a normal metal quasiparticles with different spin directions have identical conductivities. However, in the ferromagnet spin-up and spin-down quasiparticles have different conductivities. Because of the conductivity difference in F and the conductivity "mismatch" between F and N, spin accumulates near the F/N interface. Because of the diffusive behavior of the quasiparticles in N and F the accumulation (spin polarization) decays with distance from the interface. The characteristic length scales over which this decay occurs is the relaxation length, $\lambda_N$ and $\lambda_F$ for normal metal and ferromagnet respectively. When the device in consideration has a electrode spacing L, which is much larger than $\lambda_N$, the decay is exponential with L. When $\lambda_F>L>\lambda_N$ %is much smaller than L and $\lambda_N$ is much larger the decay of the spin accumulation has a 1/L dependence. Jedema measured the resistance difference between parallel and antiparallel configuration of the F electrodes. The results of their measurements are shown in figure \ref{jedema}. \begin{figure}[h] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/jedema.ps} \centering\caption{The difference in resistance between the P and AP state of a lateral Py/Al/Py lateral spin valve. The spin relaxation lengths, obtained from the fits, are also given.}\label{jedema} \end{figure} In the graph the resistance difference, $\Delta R$ between P and AP configuration of the electrodes is shown versus the electrode spacing, L. The squares are the results of a measurement at T=4.2 K and the circles are the result of a measurement at T= 293 K. The squares at L=250 nm and L=500 nm deviate considerably from the remainder of the curve. This deviation was attributed to the granular character of the aluminum (grains with sizes comparable to the width of the Al strip). We used this measurement to determine which dimensions our device should have and still have a reasonable resistance difference. However, we want to measure the device going through the superconducting transition of Al. The amount of spin polarized quasiparticles in the Al will decrease with decreasing temperature until spin transport is inhibited %and the $\Delta R$ will decrease until the resistance difference between the P and the AP orientation completely disappeares when the Al becomes superconducting. \chapter{Experimental setup} \section{$^3$He cryostat}\label{he3} To characterize the Al films and the Al/Py bilayers a $^3$He cryostat (in our case an Oxford Instruments Helios) is needed because the transition temperature of aluminum is around 1.2 K. A sketch of the insert is shown in figure \ref{cryo}. The insert is used in a $^4$He storage dewer. \begin{figure}[h] \includegraphics[width=5cm,height=10cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/cryostat.eps} \centering\caption{Schematic drawing of the $^3$He insert.}\label{cryo} \end{figure} This cryostat is capable of reaching a base temperature of 300 mK by lowering the vapor pressure of a bath of liquid $^3$He ($^3$He-pot) by means of a cryogenic sorption pump (sorb pump). The $^3$He is condensed$\footnote{The $^3$He condenses at temperatures below $\sim$2.7K.}$ in the 1K-pot shown in figure \ref{cryo}. The system is implemented such that the temperature can be varied between 300 mK and 2.5 K. Temperature stability within 1 mK is achieved for several hours. It is possible to perform AC and DC transport measurements. \section{Electronics}\label{meas} In this section several devices used for measuring the samples and controlling the temperature will be explained. Four point measurements are necessary in order to avoid measuring the resistance of the wiring inside and outside the cryostat and the contact resistance. \subsection{DC measurement} In performing the DC measurements the following devices were used. A Keithley 220 Programmable current source was used to send a DC current through the sample. A Keithley 181 nanovoltmeter was used to measure the voltage at the sample. To measure these I-V characteristics of single Al layers typically currents between 1 nA up to 1 $\mu$A were used. This (DC) setup can measure signals down to just below 1 $\mu$V. %Also IV characteristics were measured using a Keithley 181 nanovoltmeter and a Keithley 220 Programmable current source. Software was written in LabView to control these devices and to do a IV characteristic automatically. \subsection{AC measurement} A LR-700 resistance bridge$\footnote{The LR-700 resistance bridge is used to do a four point measurement of the sample, excluding the wiring and contact resistance from the measurement. The bridge sends a fixed AC excitation current through the sample. The current depends on the resistance range (2 m$\Omega$-2 M$\Omega$) and full scale excitation voltage (20 $\mu$V-20 mV), which can be set. The excitation current frequency is 15.9 Hz. The settings of the bridge were controlled by software written in LabView.}$ was used to measure the temperature dependence of the resistance of single layers of aluminum. Also an SR830 lock-in amplifier and a home built AC+DC current source ("Delftse kast") were added to the setup for the measurement of the lateral spin switches. %The bridge is quite slow in measuring the sample which is a disadvantage because the measurement time is in the order of seconds. %The bridge puts an excitation voltage over the sample which induces a current. %When set to the correct resistance range the resistance is measured and displayed. \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/AC.eps} \centering\caption{A DC voltage is given by the lock-in amplifier together with an AC-excitation. V$_{AC}$= 10.58 mV(rms), $V_{DC}$=200 mV and f$_{AC}$=131.31 Hz.}\label{AC} \end{figure} To perform AC-transport measurements with standard lock-in technique$\footnote{DC: a signal constant in time.\\ AC: A signal with a certain frequency and amplitude which is compared to a reference signal (Lock-in technique) to filter out any noise. Usually, a AC-excitation is used together with a DC signal to measure not only an I-V characteristic but also get information about dI/dV.}$ we used a Delftse kast and a lock-in amplifier. The lock-in amplifier has a signal generator giving out a sine wave with certain amplitude and frequency and a DAC which outputs a DC voltage (see fig. \ref{AC}) (for a voltage sweep) (V$_{AC}$= 10.58 mV(rms), $V_{DC}$=200 mV and f$_{AC}$=131.31 Hz). The DAC is controlled to give a DC voltage and the sine from the Lock-in amplifier is the excitation voltage. The DC voltage from the DAC and the sine wave are the input$\footnote{V$_{AC}$ is connected to a 1/100 input in the Delftse kast and V$_{DC}$ is connected to a 1/5 input in the Delftse kast.}$ for the Delftse kast where both are mixed and converted into current. This current is sent through the sample. The voltage drop over the sample is amplified by the Delftse kast before being measured by the lock-in amplifier. The DC voltage is directly measured by the nanovoltmeter. This DC voltage represents a point on the IV characteristic of the sample and the AC excitation current, going through the sample, induces an oscillation around this point giving information about the derivative $\frac{dI}{dV}$ of the IV characteristic. The excitation voltage which is read is compared to a reference signal, the sine wave. By multiplying the measured AC voltage with the reference wave, the $\frac{dI}{dV}$ is obtained while the noise with random frequency components is filtered out by a low band pass filter. In this way the Lock-in amplifier is capable to measure signals in a noisy background. The Lock-in amplifier settings are controlled by LabView as well as the data acquisition.%, the Delftse kast has no interface and therefore has to be set manually. %The noise on the lock-in amplifier is seen to be about $\pm$4nV. \subsection{Temperature control and external magnetic field} %During the R-T measurements with the LR-bridge the temperature was controlled using the sorb pump (see fig \ref{cryo}. However, during a IV sweep the temperature must be much more stable and therefore a 1k$\Omega$ heater was mounted near the sample. During the resistance vs. temperature measurements, the temperature was controlled with two PID regulation systems: \begin{itemize} \item{by controlling the temperature of the sorption pump (using a Oxford Instruments PID regulator (ITC 503)). This is done for the coarse regulation of the sample temperature.} \item{by controlling the sample temperature with a heater (1 k$\Omega$) built close to the sample. This is done for fine temperature control and improved temperature stability.} \end{itemize} The sorb pump temperature is set to a value, such that the $^3$He-pot temperature is just below the setpoint T$_{set}$, while the heater is used to approach and maintain the setpoint. The heater output is controlled by a software PID regulator which is written in LabView. As input for the PID controller the resistance of the 2 k$\Omega$ RuO$_2$ thermometer, which was mounted on the sample holder, is measured continously by the LR-bridge. For calibration of the thermometer see Appendix \ref{a1}. The temperature could be held stable within 1 mK for several hours. Also the application of an external magnetic field is possible. A home made superconducting coil (see Appendix \ref{a2} for calibration) is used to sweep a magnetic field between $\pm$600 Gauss. This coil can produce the fields necessary for the switching of the Py electrodes. The direction of the field (parallel to the axis of the coil) is fixed, but the sample can be mounted perpendicular or parallel to the field direction. \section{Sample preparation} In this paragraph the fabrication process of the single Al films, the Al/Py bilayers and the lateral spin switch is explained. Almost all films were sputtered on Si substrates, except the ones which were fabricated for the X-ray measurements (here SrTiO$_3$ (STO) was used). \subsection{UHV sputtering system} A UHV (Ultra High Vacuum) system was used to sputter Al and Py layers. The sputtering chamber is constantly evacuated by a turbo pump maintaining a background pressure of about 3$\cdot$10$^{-9}$ mbar. The substrates are put into a load lock separated from the chamber, which is pumped down by a second turbo pump. When the pressure is low enough the substrates are transported into the UHV chamber. Inside, layers are deposited by magnetron sputtering. \subsubsection{The sputtering process.} Before sputtering the films argon gas is let into the system. A voltage is applied to the sputtering target, which spontaneously starts to ionize the Ar atoms. %Due to natural cosmic radiation part of the Argon gas atoms are ionized. Because of the applied voltage the Ar${^+}$ ions are accelerated towards the target. The ions which collide with the target set atoms of target material free. Besides the atoms secondary electrons are formed, ionizing even more Ar atoms and setting more target material free. Dependent on the argon pressure, at some point a stable glow discharge ignites. The sputtered particles are deposited all over the sputtering chamber and therefore also onto a substrate positioned carefully inside the chamber. The ionization efficiency can be enhanced by applying a magnetic field (the magnet is positioned just behind the target). The applied magnetic field influences the trajectories of the secondary electrons by trapping them in cycloids. In the region near the target the ionization efficiency will increase because of the confinement of the glow discharge. The enhanced ionization has the advantage that the discharge can ignite at lower argon pressures. The mean free path of the atoms is larger at lower pressures, so the atoms collide with the substrate with higher kinetic energy. There are several parameters which influence the properties of the film which is deposited on the substrate. The most important parameters are: \begin{itemize} \item{sputter current: determines the deposition rate of the film. Therefore this parameter determines the time a sputtered particle has to participate in surface diffusion and agglomeration. For the Al layers a sputter current of 220 mA is used and for the Py layer 165 mA is used.} \item{Ar pressure: as mentioned above the pressure in the chamber determines the mean free path of the target particles approaching the substrate. The distance of the target to the substrate and the pressure determine the number of collisions the target particles encounter on their way to the substrate. This can affect the crystallinity of the film. For Al a pressure of 6.0$\cdot$10$^{-3}$ mbar and for Py layers a pressure of 4.0$\cdot$10$^{-3}$ mbar was used.} \end{itemize} During sputtering the substrate is positioned in the center under the targets. The sample holder can be rotated from an angle of 45 $^\circ$ (angle between normal to target surface and sample holder plane) and the configuration where the sample holder plane is perpendicular to the normal of the target. For the Py lift-off procedure to succeed the Py layer needs to be sputtered in the perpendicular configuration. This decreases the resputtering against the walls of the bottom layer resist (see section \ref{ebeam}). During sputtering the thickness of the film is monitored by a crystal monitor. The crystal is situated inside the chamber and has a certain resonance frequency (excited by an oscillator). When material is deposited onto the crystal the resonance frequency is shifted. The shift in the resonance frequency is a measure for the amount of material on the crystal. Beforehand parameters like density have to be specified and the crystal monitor needs to be zeroed. When the growth rate of a target is calibrated the thickness of the sample can be determined with the crystal. Growth rates are calibrated with RBS (Rutherford Backscattering Spectroscopy) measurements and X-ray measurements. \subsection{Optical lithography} For the transport measurements of single Al and Al/Py layers, where micro-sized structures were needed, we used optical lithography. An optical resist called HPR 204 was spincoated on top of the Al layer. After baking a few minutes at 90 $^\circ$C the resist was exposed, with the use of a optical mask, to UV light for 15 seconds. After exposure the sample is put in a developer (AZ 312 MIF) which takes away the exposed resist (positive resist). Normally, after the development an etching step follows, in order to remove the Al which was under the exposed resist. This can be done by wet etching or ion beam etching. In this case we planned to use wet etching with a solution of the following composition 2$\%$ HNO$_3$, 75$\%$ H$_3$PO$_4$ and 23$\%$ H$_2$O. Surprisingly, this last step was not necessary because the developer already etched away the Al which was underneath the exposed resist. The fabricated structures were strips with the following dimensions: length = 1.4 mm, width = 200 $\mu$m and varying thicknesses between 21 nm and 475 nm. Included in the structure are the current and voltage leads and the contact pads for a 4-point measurement. \subsection{E-beam lithography}\label{ebeam} \subsubsection{Aluminum with gold contacts} \begin{figure}[h] \includegraphics[width=15cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/liftofftotg.ps} \centering\caption{The lift-off process.}\label{lift of} \end{figure} At first also the Al/Py bilayers were structured with optical lithography. However, the developer (AZ 312 MIF) reacted agressively with the Al. This process caused the destruction of some of the bilayers, making this a highly irreproducible structuring process. So the decision was made to start using e-beam lithography instead. Furthermore, e-beam lithography is required for fabrication of submicron-sized structures. Since the aim of this work is to study the possibilities of lateral spin switch devices, where sizes between 300 to 1500 nm have to be achieved, we also tested the lift-off fabrication procedure for a single submicron Al-strip. There are two possible structuring processes besides optical lithography which are applicable to our case. The first involving dry etching and the second the lift-off process. Dry etching has the disadvantage that during etching the material which is already etched away is piled up against the resist walls resulting in high edges, "ears". These "ears" are difficult to remove (a possible solution is to etch under different angles or rotate the sample while etching). With the lift-off technique these ears can also form. However, they can be avoided by optimizing the following three parameters of the fabrication process, \begin{itemize} \item{Undercut. In the lift-off technique a bilayer of resist is used. When the top layer of resist has a large overhanging profile compared to the bottom layer (undercut), the resputtering of target particles against the bottom layer resist walls will decrease. Furthermore, a large undercut improves the resolution of the structures significantly.} \item{The thickness of the bottom layer of resist. When the bottom layer of resist is too thick, more target material is resputtered against the resist walls introducing the so called "ears". So the bottom layer should preferably be as thin as possible. However, when the bottom layer resist is too thin, the sputtered material (in the resist structure) will be attached to the material which is sputtered on top of the resist stack. The structure will be destroyed during the lift-off.} \item{Ar pressure. Since the Al atoms are quite light a low Ar pressure should be used during sputtering. With a high pressure the atoms are scattered in every direction (also in the undercut) and therefore, the material will pile up against the bottom layer resist walls. One should use the lowest Ar pressure possible (more "directional" sputtering), at which a stable glow discharge can still be ignited.} \end{itemize} We chose to apply the lift-off procedure which is depicted in figure \ref{lift of}. First a bilayer of resist was spincoated onto a Si substrate. The bottom layer is PMGI (low resolution resist) and the top one is PMMA (high resolution resist). Both are positive resists$\footnote{Positive resist: during development only the exposed resist is removed. Negative resist: during development only unexposed resist is removed. This definition can depend, for the same resists, on the dose and on the developer.}$. When a single layer of resist is used the achievable resolution is poor. Here sputtered material is piled up against the resist walls. If the resist is thin this pile up causes damage to the structure during lift-off. When a PMMA/PMGI bilayer is used, the (high resolution) PMMA will form a overhanging profile in combination with the undercut PMGI (low resolution) (see fig. \ref{lift of}). This will significantly improve the resolution and edge definition of the structure, but not necessarily the "ears". The e-beam writes a structure by exposing this bilayer of resist to an electron beam (typical current: 50 pA). This is followed by development$\footnote{PMGI: PMGI developer. \\PMMA: MIBK:IPA 1:3.}$ of the resist taking away the exposed parts. The sample is put into the UHV to sputter Al or Py on it by magnetron sputtering. Now the real lift-off can start, after sputtering the sample is put in the NMP$\footnote{NMP= 1-methyl-2-pyrrolidinone}$, which causes the PMGI layer to swell and dissolve. So the whole film will come off except for the part which is sputtered directly onto the substrate, see fig. \ref{lift of}. \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/SEM/nanog.ps} \centering\caption{Al strip with Au contacts.}\label{struc} \end{figure} A typical result of this procedure is the Al structure shown in fig. \ref{struc}. The dimensions are: length = 5 $\mu$m, width = 800 nm and thickness = 42 nm. A second e-beam step is required to make contact pads. Here a single layer of PMMA is sufficient because we do not require a very high resolution. After exposure the Au/MoGe contacts are sputtered in the Z-400. The MoGe layer is an adhesion layer. A final lift off is done with acetone. \subsection{Lateral spin switch: The four lithography steps.} The structures in the previous section were used to determine whether the use of the PMMA and PMGI would "pollute" the aluminum and thereby affecting its resistivity and superconducting transition temperature, T$_c$. The resistance measurements showed that the use of these resists does not influence the characteristic parameters of the Al such as the superconducting transition temperature T$_c$ and its mean free path. The final goal of this research was to measure a lateral spin switch with the Al in the superconducting transition. The creation of such a device takes four separate writing steps with the e-beam. In figure \ref{ebstruc} the complete design of the device is shown. \begin{figure} \includegraphics[width=12cm,height=12cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/SEM/structure.ps} \centering\caption{Structure designed with e-beam software. Blue structure: Py, red structure: Al and yellow structure: Au.}\label{ebstruc} \end{figure} The blue squares are alignment markers to ensure that future writing steps overlap with the previous ones. Such accurate alignments can be done thanks to a computer motorized stage, a beam blanker and software which allows accurate post alignment and stitching. \subsubsection{Py electrodes} The first writing step is to write the Py electrodes. In order to achieve different switching fields for the two Py strips the following dimensions were chosen (see chapter 4, fig. \ref{pyH}): %To decide on the measures of the Py electrodes simulations using the OOMMF code were performed (for results see chapter 7). Based on these simulations the following dimensions were chosen, \begin{itemize} \item{top electrode :10 $\mu$m long and 1.5 $\mu$m wide.} \item{bottom electrode: 10 $\mu$m long and 0.3 $\mu$m wide.} \item{thickness: 40 nm} \item{electrode spacing: 750 nm.} \end{itemize} \begin{figure}[h] \includegraphics[width=5cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/SEM/Py2.023-6-2.ps} \centering\caption{Permalloy electrodes.}\label{pyel} \end{figure} Figure \ref{pyel} shows the structure after lift-off, the parameters used in the writing and the development are given in Appendix \ref{a3}. From the figure the real sizes can be determined, they are, \begin{itemize} \item{top electrode:10 $\mu$m long and 2 $\mu$m wide.} \item{bottom electrode: 10 $\mu$m long and 0.75 $\mu$m wide.} \item{thickness: 40 nm} \item{electrode spacing: 300 nm.} \end{itemize} The structures are significantly larger than the sizes aimed for. This is caused by the fact that the Ar pressure was too high during sputtering. When the pressure is high the sputtered atoms are scattered more and they will scatter into the undercut and reduce the resolution. This process also causes the formation of "ears" (AFM studies showed that the "ears" were several hundreds of nanometers high). The argon pressure during sputtering should be decreased to a value even lower than the used 4 $\mu$bar. During sputtering the long axis of the electrodes were aligned with the external magnetic field, caused by the total magnetic field from the magnetron system. Thus, the easy axes of magnetization of the electrodes is directed along the long axis of the electrodes. \subsubsection{Al strip.} The second writing step includes the aluminum strip (red structure) in figure \ref{ebstruc}. The aimed-for size of the strip is (for characteristic values of Al see chapter 4, table \ref{tal}), \begin{itemize} \item{width: 500 nm} \item{length: 8 $\mu$m} \item{thickness: 80 nm} \end{itemize} In figure \ref{Al1} the structure after lift-off is shown, the parameters used in the writing and the development are given in Appendix \ref{a3}. However, the measures in the structure after lift-off are larger than the aimed-for sizes, \begin{itemize} \item{width: 2 $\mu$m} \item{length: 8 $\mu$m} \item{thickness: 80 nm} \end{itemize} Also here the pressure during sputtering was too high (6 $\mu$ bar), which has decreased the resolution dramatically. For aluminum the effect is larger because the Al atom is lighter than the Ni and Fe atoms. \begin{figure}[h] \includegraphics[width=6cm,height=6cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/SEM/Christianne_struct_20040830/Alstrip.ps} \centering\caption{Aluminum strip on top of the Py electrodes.}\label{Al1} \end{figure} Also one can clearly see some ears at the edges of the Al but also on the edges of the Py (damaging the Al because of decreased film thickness in the "shadow" of the "ear"). The bottom layer resist was probably also too thick. %This was also caused by the fact that the pressure was too high and maybe the bottom layer resist was too thick. However this sample is still measurable. \subsubsection{Au contacts.} Finally two writing steps have to be done to put contacts on the sample for the transport measurements. Both the intermediate and large gold contacts are shown in figure \ref{ebstruc} as the yellow structures. The intermediate contacts after lift-off are shown in fig. \ref{Au}. The parameters used in the writing and the development are given in Appendix \ref{a3}. \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/SEM/ALPyAu.ps} \centering\caption{Structure with intermediate Au contacts.}\label{Au} \end{figure} The lift-off of the intermediate gold contacts failed, on the left side the top contact was short cutting the bottom contact and at the right side the bottom contact is not overlapping with the Py electrode. This can be explained by the fact that only a single PMMA resist layer was used instead of a bilayer for the fabrication of the Au-intermediate contacts. The shortcut is there because resolution of the resist is higher in a bilayer (PMMA/PMGI) which is apparently necessary for such small structures. (Before a single (PMMA) layer was enough when the structures had larger dimensions.) The damage on the right side was probably caused by underexposure of the smallest structures. The alignment of the gold with the Py is quite good, but it would have been easier and less prone to errors, if the Py electrodes would have had different lengths. Due to such problems, the final measurement could not be performed. \section{Calibration of the growth rate of aluminum and permalloy} \subsection{RBS-measurements} To calibrate the growth rate of Al and Py films Rutherford Backscattering Spectroscopy (RBS) and X-ray measurements were performed. The RBS measurements were performed at AMOLF $\footnote{AMOLF (Instituut voor Atoom en Molecuul Fysica) is one of the research institutes of F.O.M. (a government institution for Fundamental Research on Matter).}$. This method can be used to measure densities and consequently thicknesses of thin films. Figure \ref{rbssetup} shows the mechanism of RBS measurements. \begin{figure}[h] \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/rutherford.ps} \centering\caption{Setup for RBS measurement.}\label{rbssetup} \end{figure} The thin film is bombarded with light nuclei like $\alpha$-particles with an energy of 2 MeV. When these particles hit the sample they will be reflected into a detector. The energy of the particles that enter the detector are characteristic of the atom they collided with and the position of this atom in the sample. For example when the $\alpha$ particle collides with a surface atom the energy of the reflected particle depends on the recoil (weight) of the atom. If the atom is heavier the ion will loose less energy during collision. If the ion enters the sample it will slow down because of the charges that are present. After collision with a sample atom it also has to travel back through the sample to be detected. Consequently it has a lower energy when detected compared to the $\alpha$-particles which are scattered by the surface atoms. The energy difference between the surface and the bottom of the layer is a measure for the thickness of the sample. \begin{figure}[h] \includegraphics[width=12cm,height=7cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/rbsspec.ps} \centering\caption{spectrum of a RBS measurement.}\label{rbsspectrum} \end{figure} Figure \ref{rbsspectrum} shows the RBS spectrum of a Al/Py bilayer. The peak at lower energies is characteristic for the Al layer and the peak at 1.5 keV is the one characteristic for the Py layer. The latter, however, is shifted to lower energy in comparison with a single Py layer. This is caused by the presence of the Al layer, slowing the $\alpha$ particles down. Therefore, the energy over which the Py peak has shifted is also a measure for the thickness of the Al layer. \subsection{X-ray measurements} \begin{figure} \includegraphics[width=10cm,height=5cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/xray2.ps} \centering\caption{Setup for X-ray reflectometry.}\label{xray} \end{figure} To confirm the RBS measurements also some X-ray measurements have been performed. However, these films were sputtered on SrTiO$_3$ (STO) substrates because Si substrates are almost equal in electron density as aluminum and therefore unsuitable for X-ray measurements. In figure \ref{xray} a simple representation of the X-ray measurement setup is given. The detector measures the intensity of the reflected photons. When the angle of incidence $\alpha$ increases the intensity of the reflected photons decreases rapidly over 5-6 orders of magnitude. The photons will be partially reflected on the Al surface but some photons will go through the Al layer and reflect upon the substrate. The different paths taken by the photons have different lengths causing constructive or destructive interference. This causes the so called Kiessig fringes. The spacing between subsequent fringes is dependent on the thickness of the film. To prevent fringes due to spreading of the X-ray beam a knife is placed, so only rays which are reflected directly below the edge of the knife will end up in the detector. However, there where no fringes visible for the measurements on Al making it very difficult to determine the thickness. This is caused by the roughness of the Al films ($\sim$10 nm) (see section \ref{gran}). %Sputtering the Al films in the configuration that the normal of the sample holder is parallel to the normal of the target shows some improvement. In the next section the results of the thickness calibration of the aluminum and permalloy target are given. \subsection{Conclusion: RBS and X-ray measurements} When films are sputtered the thicknesses of the films need to be verified accurately. The results of the RBS data together with the results of the X-ray measurements are given in tables \ref{t1} and \ref{t2}. \begin{table}[h] \begin{tabular}{|c|c|} \hline d$_{Xtal}$ (1 nm$\big|_{Xtal}$=0.25 nm ) & d (RBS and X-ray data) \\ \hline 125 $\dot{A}$ & 21 nm\\ \hline 250 $\dot{A}$& 42 nm\\ \hline 375 $\dot{A}$& 62 nm\\ \hline 500 $\dot{A}$& 83 nm\\ \hline \end{tabular} \centering\caption{Calibration of the aluminum target}\label{t1} \end{table} From this we get the new calibration, 1 nm$\big|_{Xtal}$ =0.63 ($\pm$ 0.015) nm. Which means that when the Xtal crystal in the UHV system measures 0.63 ($\pm$ 0.015) nm, the film on the substrate has a thickness of 1 nm. \begin{table}[h] \begin{tabular}{|c|c|c|} \hline sputter time & d$_{Xtal}$&d (X-ray data) \\ \hline 3 min& 92 $\dot{A}$& 34.5 nm\\ \hline 6 min& 181 $\dot{A}$& 69.6 nm\\ \hline \end{tabular} \centering\caption{Calibration of the permalloy target}\label{t2} \end{table} The calibration of the new Py target at an argon pressure of 4 $\mu$bar is 1 nm = 2.54 $\dot{A}$. For Py it was much easier to determine the thickness from the X-ray measurements because there were clear Kiessig fringes in the data. \chapter{Results} \section{Al films: dependence of T$_c$ on thickness of the films.} First single Al layers were measured (4-point measurement) with the LR bridge (see section \ref{meas}). The results are presented in fig \ref{Al}. The graph shows the resistance dependence on temperature of Al thin films of different thicknesses. All films undergo a phase transition from the normal state to the superconducting state. The temperature at which this phase transition occurs, the superconducting transition temperature T$_c$, decreases when the thickness of the film increases. %Consequently, T$_c$ increases with increasing disorder. \begin{figure}[h]\label{Al} \includegraphics[width=13cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/alal.ps} \centering\caption{Resistance vs. temperature for Al films of various thicknesses.}\label{Al} \end{figure} From these results one can extract not only the T$_c$, but also the residual resistivity, $\rho$$_{0}$, the residual resistance ratio, RRR, the electronic mean free path, l$_e$ and the BCS and GL coherence lengths $\xi_0$ and $\xi_{GL}$. The characteristic values for our Al films are given in table \ref{tal}. The first three columns present the dimensions (d: thickness, w: width, L: length) of the measured films and the third contains the superconducting transition temperature T$_c$, which can be directly obtained from fig. \ref{Al}. The residual resistivity, $\rho$$_{0}$, can be calculated by extrapolating the normal state resistivity to T=0. The residual resistance ratio is the ratio, $\frac{R(300 K)}{R(4.2K)}$, of resistance at room temperature and resistance at 4.2 K. The electronic mean free path can be calculated with eq. \ref{no} using N(0) the density of states (DOS) at the Fermi energy, N(0)=2.2$\cdot$10$^{47}$ J$^{-1}$m$^{-3}$. \begin{equation}\label{no} l_e=\frac{3D}{v_F}\quad\textrm{with}\quad D=\frac{1}{N(0)e^2\rho_0} \end{equation} with D the diffusion coefficient and $e$ the electronic charge. The BCS coherence length is calculated from, \begin{equation} \xi_0=\frac{\hbar v_F}{\pi\Delta(0)}\quad \textrm{with}\quad \Delta(0)=1.764k_BT_c \end{equation} with $v_F$=1.3$\cdot$10$^6$ m/s the Fermi velocity and $k_b$ the Boltzmann constant. The GL coherence length is \begin{equation} \xi_{GL}=0.855\Bigg(\frac{\xi_0 l_e}{1-t}\Bigg)^{1/2} \end{equation} with t=T/T$_c$. \begin{table}[h] \centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \cline{1-9} d(nm)& L(mm) & w($\mu$m) & T$_c$(K)&$\rho$$_{0}$($10^\textrm{-8}$$\Omega$m) & RRR & l$_e$(nm) & $\xi$$_{0}$($\mu$m)& $\xi$$_{GL}$(nm) \\ \cline{1-9} 21 & 1.4 & 200 & 1.353 & 3.24 & 2.12 & 12.6 & 1.32 & 110 \\ 42& 1.0 & 5.5& 1.266 & 1.40 & 3.06 & 29.1 & 1.42 & 171 \\ 62 & 1.0 &5.5 & 1.242 &1.26 &3.29 & 32.5 & 1.44 & 185 \\ 83 & 1.4 & 200& 1.217 &1.12 & 4.18 & 36.6 & 1.47 & 196 \\ 380 & 1.4 & 200& --&0.86 & 5.29 & 47.4 & 1.45& 223 \\ 475 & 1.4 & 200& 1.237 &2.53 & 2.62 & 16.2 & 1.45 & 130 \\ \hline \end{tabular} \caption{Measured in $^3$He.}\label{tal} \vspace{0.5cm} \end{table} \begin{figure}[h] \includegraphics[width=8cm,height=12cm,angle=270]{/home/beekman/le.ps} \centering\caption{Mean free path, l$_e$, vs. thickness, d, of the Al films .}\label{ld} \end{figure} In fig. \ref{ld} the mean free path vs. thickness of the measured Al layers is shown. The mean free path increases for increasing film thickness. The figure clearly shows that the mean free path saturates at a value around 45 nm. The figure also shows that when the thickness is decreased to below the coherence length the interfaces of the film limit the mean free path. Furthermore, l$_e$ is a measure of disorder. When the disorder of the material is decreased, the electrons scatter less, resulting in a larger mean free path. The mean free path increases with decreasing disorder and therefore the superconducting transition temperature, T$_c$, increases with increasing disorder. \section{T$_c$ dependence on disorder.} Increasing the amount of disorder in thin superconducting film can induce a change in the transition temperature T$_c$. The induced change in T$_c$ depends on the structure of the density of states (DOS) near the Fermi energy, since the phonon mediated electron-electron interaction takes place around E$_F$. There are two different mechanisms which can explain the behavior of T$_c$ with disorder. The change in T$_c$ can be induced by either lattice imperfections or grains. \subsection{Imperfections}\label{Tdis} Lattice imperfections can induce degradation as well as enhancement of the energy gap (T$_c$) \cite{belitz}. Typically, T$_c$ degradation is explained as a density of states effect. The metals which exhibit degradation generally have sharp structures in their DOS and have a large N(0) (the DOS at the Fermi energy). Due to scattering these sharp structures around E$_F$ become smeared, reducing N(0) and thereby T$_c$.%, consequently decreasing the amount of electrons which can participate in the phonon mediated attractive potential. Therefore, the energy gap and thus T$_c$ are reduced. However, Al displays enhancement of T$_c$ with increasing disorder so another explanation is needed. %Enhancement of T$_C$ is caused by an increase of electron-phonon coupling. Enhancement generally occurs in metals with a small value of N(0) and a DOS without sharp structures. Imperfections influence the shape of $\alpha^2 F(\omega)$, the effective phonon density of states with $\alpha$ the frequency dependent electron-phonon coupling strength. The enhancement is caused by the increased coupling of the electrons to the phonons. There are basically two types of phonons, transverse phonons and longitudinal phonons \cite{el}. In the presence of imperfections two effects occur. The coupling to transverse phonons is increased and the coupling to longitudinal phonons is decreased. However, the first effect is dominant. The coupling to transverse phonons is caused by the fact that the lattice defects allow partial violation of the conservation of momentum during the electron-phonon collisions. This increases $\alpha^2 F(\omega)$ in the low frequency regime, making more phonon states available for the phonon mediated electron-electron interaction. Increase of e-ph coupling results in an increase of the energy gap and consequently enhancement of T$_c$. The second effect can be explained as follows, the electrons fluctuate in phase with the longitudinal phonons creating regions where the electron gas is compressed (electronic mean energy increased) and regions with less electrons (lower mean energy). Electrons diffuse from the dense regions to the less dense regions to compensate the gradient in mean energy and the energy of the phonon dissipates. However, when imperfections are involved the diffusion is partially inhibited and energy cannot be carried away from the phonon therefore increasing the lifetime of the longitudinal phonon and thus decreasing the coupling of the electron to this phonon. The enhanced coupling to transverse phonons is therefore partially canceled by the decrease in coupling to the longitudinal phonons. The dependence of T$_c$ on the mean free path l$_e$ has been derived by Keck and Schmid\cite{el}, \begin{equation}\label{Tl} \frac{T_c-T^p_c}{T^p_c}=\frac{\Big(\frac{12}{\pi}\Big)\Big(\frac{c_L}{c_T}\Big)^2-\Big[\Big(\frac{8}{\pi}\Big)-\Big(\frac{\pi}{2}\Big)\Big]}{\Big(\frac{g^2_LN_0q^2_D}{4p^2_0}\Big)}\frac{1}{q_Dl_e} \end{equation} They found that the transition temperature increases linearly with $l_e^{-1}$. With T$_c^p$ the T$_c$ of a pure superconductor (for Al T$_c^p$=1.16 K \cite{belitz}), $p_0$ the momentum at the Fermi energy and $c_L$ and $c_T$ are the longitudinal and transverse sound velocities. Here $q_D=\frac{\Theta_L}{c_L}\frac{k_b}{\hbar}$ (a factor of $\frac{k_b}{\hbar}$ is added to make sure that the dimensions are correct) with $\Theta_L$ the Debye temperature. Also present in eq. \ref{Tl} is $g_L$ which is given by \begin{equation} g_L=\hbar\frac{p_0^2}{3m\rho^{1/2}c_L} \end{equation} the multiplication with $\hbar$ is needed to obtain the correct dimensions. Furthermore, $\lambda_p$ is a dimensionless coupling constant which is a measure for the coupling strength of the electrons to the phonons and is given by \begin{equation} \lambda_p=\Bigg(\frac{N_0q_D^2}{4p_0^2}\Bigg)g_L. \end{equation} The coupling constant can be estimated for our aluminum. To estimate $\lambda_p$ the following values were used, \begin{itemize} \item{$N_0$, the electronic density of states at the Fermi energy per spin, is 1.1$\cdot$10$^{47}$ J$^{-1}$ m$^{-1}$\cite{mermin}.} \item{q$_D$, the phonon wave vector, is 1.32$\cdot$10$^{10}$ m$^{-1}$\cite{mermin}.} \item{p$_0$=$\hbar$k$_F$ with k$_F$=1.75$\cdot$10$^{10}$ m$^{-1}$ the wave vector at E$_F$\cite{mermin}.} \item{m, the electronic mass, is 9.1$\cdot$10$^{-31}$ kg.} \item{$\rho$, the ionic mass density, is 2.7$\cdot$10$^{3}$ kg m$^{-3}$.} \item{$\Theta_L$, the Debye temperature, is 394 K.} \item{k$_B$, the Boltzmann constant, is 1.38$\cdot$10$^{-23}$ J K$^{-1}$.} \item{$\Bigg(\frac{c_L}{c_T}\Bigg)$=4.13.} \end{itemize} Using the values above results in a coupling constant of the order of 0.5-0.6. This coupling constant induces a change of T$_c$, compared to T$_c^p$ of pure aluminum, of the order of 70 mK (for the 62 nm film) which is in good agreement with our results (see table \ref{tal}). In figure \ref{Tcl} the T$_c$ dependence on the mean free path, l$_e$, of our aluminum films is presented. The graph shows a linear increase of T$_c$ with l$_e^{-1}$ for the films with thicknesses 42 nm-83 nm (the results for the 21 nm film seems to deviate). Another form of a lattice defect that influences the energy gap is a grain boundary which exists in polycrystalline Al. \begin{figure}[h] \includegraphics[width=8cm,height=12cm,angle=270]{/home/beekman/programs/oommf/Tcle.ps} \centering\caption{T$_c$ vs. l$_e^{-1}$ for the aluminum films (see fig. \ref{Al}).}\label{Tcl} \end{figure} %\subsection{T$_c$ dependence on disorder} %There exist many possible sources of disorder. One can have %lattice defects or deformations but also atoms from other materials %(imperfections). This amount of disorder induces change in the %properties of the superconductor for example enhancement or %degradation of the superconducting transition temperature T$_c$ %with increasing disorder. In this thesis the properties of %Aluminum films will be discussed. %As was shown in the previous section we have a T$_c$ enhancement %with increasing disorder. Generally, two effects are dominant in %inducing change in T$_c$: %\begin{itemize} %\item{Change in the Density of states (DOS).} %\item{Change in the electron-phonon coupling ($\lambda$).} %\end{itemize} %The conventional explanation for degradation of T$_c$ with %disorder is the smearing of the DOS peak which is near the Fermi %energy. Since we have T$_c$ enhancement this explanation is not %applicable in our case. However, the second effect is a much more %satisfactory explanation. The electron-phonon coupling $\lambda$ %can be calculated using the Eliashberg equation\cite{el} but %neglecting the coulomb interaction and the DOS effects. They found %the correct order of magnitude for the increase in the transition %temperature (for weak coupling superconductors). %%Belitz\cite{belitz} did include these effects and could extend this theory %into the strong coupling regime. The Eliashbergfunction %represented by $\alpha^2$F($\omega$) is in principle the effective %phonon density of states were $\alpha^2(\omega)$ is the %electron-phonon coupling strength. According to B. Keck, A. %Schmid\cite{el} the most likely explanation for %T$_c$ enhancement is the following, the change in e-ph coupling is %induced by the fact that the momentum conservation law can be %violated when scattering on imperfections enters the picture. The %treatment of a disordered lattice is done by means of the standard %impurity technique\cite{abrikosov}. In this technique %Green's functions are used. In general a Green's function %represents a response to a delta function source (for example an %impurity). Therefore, this function gives us information about the %trajectory of a certain electron through the material thus it also %represents the probability that an electron which is in state %(p,s) scatters into another state (p',s'). The one electron %Green's function for the clean metal can be compared to the %impurity averaged one electron Green's function which turns out to %be broadened and has a lower peak at the Fermi energy than the one %for the clean metal. Obviously this originates from the fact that %adding imperfections will increase the probability to scatter into %more states in a larger band around the Fermi energy. Now also the %self energy which is weakly dependent on the momentum suddenly has %to be taken into account ( A noninteracting system has no self %energy but an interacting system has, making it a measure for the %amount of interaction in the system.). Finally B. Keck, A. Schmid %come to the following equations, %See Appendix for relevant characteristic values concerning the Al %films ( with N($0$)=$2.2 10$$^{47}$ %J$^{-1}$m$^{-1}$\cite{boogaard}). \subsection{Single crystalline or granular films?}\label{gran} The growth process of the films determines whether they will become single crystalline or polycrystalline (i.e. granular). For single crystalline films Fortuin \cite{fortuin} found a T$_c$ between $1.10$ K and $1.15$ K for varying line widths (0.04 and 0.14 $\mu$m and thicknesses of 20 and 100 nm). However, our films have a T$_c$ around $1.26$ K ($21$ nm) and the lowest T$_c$ was $1.217$ K for the $83$ nm film. Obviously our films are polycrystalline,% This %basically means that our films are divided into small crystals %(i.e. grains). which was confirmed by AFM measurements shown in figures \ref{grain} and \ref{grain2}. The boundaries of the grains (surface grains: 50-100 nm) influence the superconducting properties such as T$_c$. This change in T$_c$ is caused by the following two competing effects: \begin{itemize} \item{The grain boundaries induce an increase in the average phonon amplitude of the ions. The formation of these boundaries reduces the symmetry of the ions positioned near a grain boundary. Because of this symmetry loss the ions near the boundaries are held in place by weaker ionic forces then in the bulk of the crystal. Therefore, they can vibrate with larger amplitude and lower frequency then the bulk ions. The average phonon amplitude will therefore be enhanced and as a result T$_c$ will increase when the film becomes polycrystalline\cite{gar}.} \\ \item{The introduction of grain boundaries (i.e. disorder) makes more vibrational states available through scattering. This has a broadening effect on the phonon density of states. This effect will suppress T$_c$ instead of enhancing it. } \end{itemize} Since in our case we have T$_c$ enhancement compared to the single crystalline Al films of Fortuin, apparently the first effect is more important than the second. The amount of T$_c$ enhancement depends on the size of the grains. \subsubsection{T$_c$ dependence on grain size.} \begin{figure}[h] \includegraphics[width=6cm,height=6cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/Al_Si_top01.ps} \centering\caption{Topology of the surface of the Al layer on a Si substrate. Thickness of film: 62 nm. Roughness: $\sim$10 nm. Grain size (surface grains): 50-100 nm.}\label{grain} \end{figure} \begin{figure}[h] \includegraphics[width=6cm,height=6cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/AlPy_Si_3D.ps} \centering\caption{Topology of the surface of the Al/Py bilayer on a Si substrate.}\label{grain2} \end{figure} The transition temperature in bulk superconductors lies considerably lower than the T$_c$ of superconducting particle (i.e. grain). When an electron is trapped in a box (grain) its energy spectrum will become discrete. These discrete energy levels are raised when the dimensions of the box are decreased. Therefore, a polycrystalline film has elevated energy levels compared to the single crystalline films. This explains the increase in T$_c$ when superconducting film is more granular. In subsection \ref{Tdis} a mechanism was presented to explain the enhancement of T$_c$ with disorder. However, one might imagine a second mechanism to be responsible for the change in T$_c$ with disorder. Namely, a decrease in the size of the grain when the strain on the grain is increased. Due to lattice mismatch between the Si-substrate and the Al film, the Al film is under strain. This strain relaxes with increasing thickness of the film. Therefore one can imagine that the size of the grain will increase with film thickness, which could explain the induced decrease in T$_c$ with increasing film thickness. However, the T$_c$ of the single crystalline films of Fortuin is also enhanced with disorder. Therefore the grains are not responsible for the T$_c$ enhancement with decreasing film thickness, only for the enhancement compared to single crystalline films. %This can be seen easily when looking %at the analogy of an electron in a box. %Taking a 1-dimensional box and assuming that the walls are impenetrable %the schr$\ddot{o}$dinger equation is, %\begin{equation} %\frac{d^2\psi}{dx^2}+2E\psi=0 %\end{equation} %and the solution is, %\begin{equation} %\psi=Acos(\sqrt{2E}x)+Bsin(\sqrt{2E}x) %\end{equation} %The following boundary conditions must hold, %\begin{equation} %\psi(0)=\psi(a)=0 %\end{equation} %with a the size of the box. %These boundary conditions are responsible for the quantization of the energy$%\footnote{When $a$ increases the importance% %of the boundaries decrease. For large a the energy spectrum is continuous instead of discrete.}$. %The energy spectrum is given by$\footnote{In three dimensions this equation becomes E=$\frac{(n_x^2+n_y^2+n_z^2)\pi^2}{2a^2}$.}$, %\begin{equation}\label{Ebox} %E=\frac{n^2\pi^2}{2a^2}\qquad n=1,2,3... %\end{equation} %Each energy has an eigenfunction which fits exactly in the box, a standing wave. %As can be seen from eq.\ref{Ebox} the energy spectrum is shifted upwards when a is decreased. %A superconducting particle is a box with several electrons in it, also the walls are not infinitely high %barriers. Therefore the eigenfunctions will slightly extend beyond the particle walls, because tunneling between %neighboring particles is possible. As mentioned before the size effect transforms a continuous energy spectrum into a discrete one$\footnote{ %The inelastic mean free path should be large compared to the grain size otherwise through scattering the energy spectrum %will be broadened and the spectrum will appear almost continuous.}$. %To describe these grains the BCS theory can be used when the integrals are replaced %by summations. To understand the behavior of the energy gap, the free energy has to be evaluated. In this theory %sums can be evaluated with the Poisson sum formula: %\begin{equation} %\Sigma_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-î(\mathbf{\nu}.\mathbf{k})a}f(\mathbf{k})dt=2\pi\Sigma_{k} f(2\pi k) %\end{equation} %This can be seen as a transformation of the sum over direct lattice points \textbf{$\nu$}$a$ to the sum over the reciprocal %lattice points \textbf{$k$}. %The free energy of a system is given by, %\begin{equation} %F=U-TS=W_{EK}+W_I-TS %\end{equation} %with W$_{EK}$ the kinetic energy, W$_U$ the interaction energy and TS the entropy of the system. %To evaluate the change in energy gap and T$_c$ due to the size effect we need to take the expectation %value of the following Hamiltonian with respect to the wavefunction for an excited state. %\begin{equation} %H_{exc}=\Sigma_{k>k_F,\sigma}\epsilon_kn_{k\sigma}+\Sigma_{kk_F}\epsilon_k[f_k+h_k(1-2f_k)]-{} \nonumber\\ % {} -\Sigma_{kk'}V_{kk'}[h_k(1-h_k)h_{k'}(1-h_{k'})]^1/2(1-2f_k)(1-2f_{k'})+{} \nonumber\\ % {} +2k_BT\Sigma_k[f_klnf_k+(1-f_k)ln(1-f_k)] %\end{eqnarray} %The first sum is the kinetic energy with $\epsilon_k$ the Bloch energy relative to the %Fermi energy, with f$_k$ (1-f$_k$) the overall probability that state $k$ is occupied (not occupied) and h$_k$ (1-h$_k$) is the probability %that a pair state is occupied (not occupied). By using the Poisson sum formula we can see that the kinetic energy %term and the entropy term (third term in eq.\ref{free}) stay unchanged when the grain size is changed. However, the %interaction term (second term in eq.\ref{free}) does depend the grain size because of the product of k sums. %This gives an expression for the interaction term which shows that this energy oscillates rapidly with grain size. %Averaging over these oscillations the following equation for $V_{kk'}$\cite{parmenter} is obtained, %\begin{equation} %V_{kk'}^{eff}\equiv V\Bigg[1+\Sigma_{\nu}^{'}\frac{1}{(k_F\nu a)^2}cos\Bigg(\frac{\epsilon_{k}\nu a}{\hbar v_F}\Bigg)cos\Bigg(\frac{\epsilon_{k'}\nu a}{\hbar v_F}\Bigg)\Bigg] %\end{equation} %which is basically a correction of the interaction potential compared to the BCS case. Now the free energy has to be minimized with respect to $h_k$. %Together with the demand that $a$ is considerably smaller then the Pippard coherence length\cite{parmenter} %\begin{equation} %\xi_0=0.15(\hbar v_F/\pi T_c) %\end{equation} %and working through all the integrals gives the following equation for the enhancement of $T_c$, Parmenter\cite{parmenter} derived the following equation for T$_c$ enhancement with grainsize. \begin{equation}\label{gr} \frac{T_c}{T_{c\infty}}ln\frac{T_c}{T_{c\infty}}=\frac{1}{2}\pi(L/a)^3(C/4) \end{equation} with T$_{c\infty}$ the bulk superconducting transition temperature, $L\equiv(\lambda_F^2\xi_0)^{1/3}$ where $\lambda_f$ is the Fermi wavelength, $a$ is the grain size, $C\equiv\Delta(0)/k_BT_c=3.528$ and $\xi_0=0.15(\hbar v_F/k_b T_c)$ the Pippard coherence length. The T$_c$ of our polycrystalline film is 1.217 K (d=83 nm) and the T$_c$ of bulk aluminum is 1.16 K \cite{belitz}. When this is put in equation \ref{gr} one can calculate the size of the grains in our films. For Al L$\sim$6.2 nm and thus the grainsize which induces a T$_c$ enhancement of 0.05 K is $a\sim$20 nm. The surface grains are 50-100 nm. Maybe, the size of the grains inside the sample can be much smaller than the surface grains and are of the order of $20$ nm.%, so from figure below it can be seen that $T_c$ is slightly %elevated above the bulk value which coincides with our findings. \section{Al: type I or type II superconductor?} As discussed in chapter 2, superconductors can be divided into two groups. In the first superconductivity is destroyed at a certain critical field. The other contains superconductors in which regions of normal state nucleate at a certain critical field and superconductivity is destroyed at a second larger magnetic field. Superconductors are categorized by using their value for $\kappa$, \begin{equation} \kappa=\frac{\lambda_L(T)}{\xi(T)} \end{equation} The following equation was used to calculate the bulk penetration depth, \begin{equation} \lambda=\lambda_L(0)\sqrt{\frac{\xi_0}{l_e}} \end{equation} and when the film is much thinner than the bulk penetration depth, \begin{equation} \lambda_{\perp}=\frac{\lambda^2}{d} \end{equation} should be used. The values for $\xi$, l$_e$, $d$ and $\xi_0$ can be found in table \ref{tal}. %\begin{equation} %\kappa=0.715\frac{\lambda_L(0)}{l} %\end{equation} When $\kappa<\frac{1}{\sqrt{2}}$ the superconductor is of type I and when $\kappa>\frac{1}{\sqrt{2}}$ it is of type II. Apparently, the very thin films are of type II while the thick film is of type I. The aluminum which is to be used in the lateral spin switch has a thickness of 80 nm and will be a type I superconductor with only one critical magnetic field. The value for $\kappa$ is given in table \ref{tk}. %\begin{table}[h] %\centering\caption{}\label{tk} \vspace{0.5cm} %\begin{tabular}{|c|c|c|} %\hline %thickness (nm)) & $\kappa$)& type \\ %\hline %21 nm &0.908 & II\\ %\hline %42 nm &0.393 &I\\ %\hline %62 nm &0.352 &I\\ %\hline %83 nm & 0.313&I\\ %\hline %\end{tabular} %\end{table} \begin{table}[h] \centering \begin{tabular}{|c|c|c|c|c|} \hline d (nm)) &$\lambda$ (nm)&$\lambda_{\perp}$ ($\mu$m)& $\kappa$& type \\ \hline 21 nm &163.7 &1.27&11.6 & II\\ \hline 42 nm &111.7 &0.297 &1.74 &II\\ \hline 62 nm &106.5 & 0.183&0.99 &II\\ \hline 83 nm &101.4 & 0.124&0.63&I\\ \hline \end{tabular} \caption{Characteristic values of our Al for the bulk penetration depth, $\lambda_{\perp}$ and $\kappa$. }\label{tk} \vspace{0.5cm} \end{table} Here $\lambda_L(0)$=16 nm \cite{romijn} was used. \section{Py film: AMR measurement and simulations.} Besides the single layers of Al there was also an AMR measurement performed on a Py strip. The spin of an electron can affect the transport of the electron. The AMR (anisotropic magnetoresistance) is an example, here the resistance of a ferromagnet is dependent on the angle between the magnetization and the current direction. The AMR is therefore a measure for the anisotropy and domain structure of the ferromagnet. When the ferromagnet has a preferred magnetization direction in the demagnetized state, an applied field along that direction will not induce a resistance difference. When the field is applied perpendicular to the axis of easy magnetization there is a change in resistance when the direction of magnetization is changed. When the magnetization direction is perpendicular to the current direction the resistance of the ferromagnet is smaller than for the case were the magnetization is parallel to the current. In literature a difference in resistance of 3-4 percent is given for permalloy. In figure \ref{pyamr} an AMR measurement is shown of a Py strip$\footnote{The sample was fabricated by Marcel Hesselberth: argon pressure of 2 $\mu$bar.}$ with a width of 1.5 $\mu$m (see fig \ref{py}). As can be seen from figure \ref{py} the resolution is considerably improved by sputtering at a lower argon pressure (2 $\mu$bar) compared to the structure in fig. \ref{pyel} (sputtered at 4 $\mu$bar). A resistance difference of about 2.7 percent is measured which is in reasonable agreement with literature. \begin{figure}[h] \includegraphics[width=8cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/pystrip.ps} \centering\caption{Py strip with dimensions: w=1.5$\mu$m, l=20$\mu$m and t=40nm.}\label{py} \end{figure} %\afterpage{\clearpage} \begin{figure}[h] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/AMRg.ps} \centering\caption{AMR measurement of a Py layer.}\label{pyamr} \end{figure} To investigate the switching fields of different sizes of Py strips simulations were performed using the OOMMF code\cite{oommf}. When a magnetic field is applied over a ferromagnet a torque, \textbf{L}, is exerted on the moments of magnetization (spins) resulting in rotation of the spins. The torque tends to bring the system back to equilibrium which is disturbed by the applied field. In equilibrium the torque must be equal to zero. The equation of motion for the magnetization \textbf{M} is, \begin{equation} \frac{d\mathbf{M}}{dt}=\gamma_0\mathbf{L}\quad \textrm{and}\quad \mathbf{L}=\mathbf{M}\textrm{x}\mathbf{H} \end{equation} with $\gamma_0$ the gyromagnetic ratio and \textbf{H} an effective magnetic field. However, in the above equation dissipation has not been taken into account. With a dissipative term added the equation becomes, \begin{equation} \frac{d\mathbf{M}}{dt}=-\gamma_{0}^{'}\mathbf{M}\textrm{x}\mathbf{H}-\frac{\gamma_{0}^{'}\alpha}{M_s}\mathbf{M}\textrm{x}\mathbf{M}\textrm{x}\mathbf{H} \end{equation} with $\gamma_{0}^{'}$ the Landau-Lifshitz gyromagnetic ratio, $\alpha$ the damping coefficient and $M_s$ the saturation magnetization. The integration of this equation of motion results in a hysteresis loop. This loop was simulated for several widths of the ferromagnetic strip keeping the thickness (40 nm) and the length (10 $\mu$m) constant. For the damping coefficient we kept the default value of 0.5, the magnetocrystalline anisotropy, K$_1$ was taken to be equal to zero, for the exchange constant, A, the default value, 13$\cdot$10$^{-12}$ J/m was used and the saturation magnetization was taken to be 800 10$^3$ A/m. The coercive fields which are extracted from the hysteresis loops are plotted versus the width of the simulated strip (see fig. \ref{pyH}). Using fig. \ref{pyH} the width of the Py electrodes of the lateral spin switch can be chosen. The widths of the electrodes were chosen to be 1.5 $\mu$m and 300 nm. The graph in fig. \ref{pyH} shows a significant difference in switching fields for these strips. %The results in fig. \ref{pyH} are compared to the results given by Nitta et al \cite{nitta}. The values for the coercive fields of Nitta et al were obtained also by doing micro magnetic simulations (they used M$_s$=860$\cdot$10$^3$ A/m, K1=0 J/m$^3$ and A=13$\cdot$10$^{-12}$ J/m). Our simulations give values which are somewhat higher than the results given in \cite{nitta}. The fact that Nitta finds lower coercive fields might be attributed to the difference in the value for M$_s$. \begin{figure}[!] \includegraphics[width=10cm,height=12cm,angle=270]{/home/beekman/Hvsd.ps} \centering\caption{Coercive field H$_c$ vs. width of Py strips simulated with OOMMF.}\label{pyH} \end{figure} %\begin{figure}[!] %\includegraphics[width=12cm,height=10cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/nitta2.ps} %\centering\caption{Coercive field H$_c$ vs. width of Py strips simulated with a micro magnetic code (triangles) by Nitta et al\cite{nitta}. The circles are measurements done by Nitta on LHE devices (see \cite{nitta}).}\label{pyH} %\end{figure} \section{Al/Py bilayers: Suppression of T$_c$.} \begin{figure}[h] \includegraphics[width=13cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/pyall.ps} \centering\caption{Suppression of T$_c$ due to proximity of a Py layer.}\label{pyal} \end{figure} To test whether the coherence length of Al is long enough for use in a lateral spin switch, Al/Py bilayers where measured. The results are given in Figure \ref{pyal}. Suppression of T$_c$ occurs when the thickness of the superconducting layer is of the order of twice the superconducting coherence length, $\xi_s$. The figure shows a single Al layer (green curve) of 475 nm and two Al/Py layers (red curve 475 nm/5 nm and black curve 352 nm/5 nm). The T$_c$ of the single layer of Al is 1.237 K, the bilayer with the same thickness for Al layer but with 5nm of Py underneath has a T$_c$ of 1.13 K and thus shows some suppression. The curve of the 352 nm Al layer with also 5nm of Py underneath has a T$_c$ of around 1.0 K. This T$_c$ is definitely not reachable by increasing the thickness of the Al layer (bulk value: T$_c$=1.18 K). At a thickness 350 nm there already is considerable suppression which is in good agreement with the coherence length of our aluminum (see table \ref{tal}). This test shows that Al has a coherence length which is long enough for usage in lateral spin switches. %\begin{figure}[h] %\includegraphics[width=13cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/Td.ps} %\centering\caption{T$_c$ vs. thickness, d, of the Al in the Al/Py bilayers (dots) compared to eq. \ref{sup} (line).}\label{tcd} %\end{figure} %In fig.\ref{tcd} the transition temperatures of the bilayers (see fig. \ref{pyel}) are plotted versus the thickness of the Al layers. I In section \ref{secsup} the suppression of T$_c$ of a bilayer, as function of the thickness of the superconducting layer, was derived. \begin{equation}\label{supp} T_c=T_{c0}-\frac{\pi^2\xi_{GL}^2}{d^2} \end{equation} with d the thickness of the superconducting layer. With eq. \ref{supp} the Ginzburg Landau coherence length (of the unperturbed aluminum layer) can be calculated using the information obtained from figure \ref{pyal}. This calculation results in a coherence length of 155 nm for the 475 nm Al layer. This is in reasonable agreement with the coherence length in table \ref{tal}. %This equation is also plotted in fig. \ref{tcd} for comparison. The agreement between the equation and the results of the measurements shown in fig. \ref{pyel} is poor. However, this can be attributed to the roughness of the Al layers ($\sim$10 nm) which made the determination of the thickness of the Al layers prone to errors. For the bilayers the graph shows a step in the transition from normal to superconducting state. Apparently, there are regions in the Al layer where T$_c$ is more suppressed than in other parts of the layer. The explanation of this effect may come from the fact that the Py film has multiple domains. It has been shown by Rusanov\cite{rus} that S/F bilayers where F has multiple domains shows the same effect as a spin switch. In a spin switch (F/S/F geometry) Cooper pairs probe both F layers, hence, a Cooper pair can also probe both magnetization directions on either side of a domain wall causing the same effect. In some parts of the Al layer the Cooper pairs probe domains with nearly parallel magnetization direction while in other parts they probe nearly antiparallel configurations. Therefore in some regions in the superconductor T$_c$ is suppressed more than in other regions. This also explains the "sharpness" of the step. Thickness gradients could also cause suppression of T$_c$ in some thinner regions of the sample but this effect can be ruled out because this would cause a more gradual decrease in resistance. \chapter{Conclusions} In this project it was tested if fabrication and measurement of a lateral spin switch is possible. Single layers were measured to characterize the Al. The T$_c$ dependence on disorder seems to be caused by the enhancement of electron-phonon coupling due to lattice imperfections. From measuring the single Al layers and the Al/Py bilayers the conclusion can be drawn that aluminum is suitable for usage in a lateral spin switch. We find that the coherence length is long enough ($\xi_0$=1.47 $\mu$m, $\xi_{GL}$=196 nm). According to the simulations (see fig. \ref{pyH}) it is also possible to fabricate Py electrodes with different switching fields. Therefore we can conclude that it is possible to fabricate a lateral spin switch. However, the fabrication process requires accurate choice of parameters, therefore dose tests for different resist thicknesses were performed resulting in the parameters given in Appendix \ref{a3}. Also the argon pressure during sputtering is a critical parameter, for Py sputtering at 2 $\mu$bar (instead of 4 $\mu$bar) improves the resolution enormously. As mentioned in chapter 6 the roughness of the Al greatly impedes the ability to calibrate the thickness of the Al layers. The roughness can be decreased also by decreasing the argon pressure (shown by others in the group) but another solution would be to change the sputtering process into an evaporation process. This will also solve the resolution and definition problems caused by sputtering. However, the fabrication process is now known to us and we have confirmed that Al would be an excellent candidate for the superconducting structure in the lateral spin switch. In the future a F/S/F lateral spin switch could be measured. \newpage \appendix %\begin{large} \chapter{Calibration of 2 k$\Omega$ RuO$_2$ thermometer.}\label{a1} %\end{large} \begin{figure}[h] \includegraphics[width=13cm,height=16cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/calRuOp1.ps} \end{figure} \begin{figure}[h] \includegraphics[width=13cm,height=6cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/calRuOp2.ps} \end{figure} \begin{figure}[!] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/gertjancal+dilfridgecal.ps} \caption{Calibration of a 2 k$\Omega$ RuO$_2$ thermometer.}\label{calr} \end{figure} Figure \ref{calr} shows the calibration of RuO$_2$ thermometer. Red symbols: Temperature vs. Resistance measured in the $^3$He cryostat (section \ref{he3}) and the black symbols is a fit to the red curve. The green and blue curve where measured in the dilution fridge. The table contains the values of the dilution fridge measurement. The measurements of the dilution fridge are in excellent agreement with the measurement of the $^3$He cryostat (the resistor used here was not the same as the one used in the dilution fridge however, they did come from the same batch). \\ \newpage %\begin{large} \chapter{Calibration coil.}\label{a2} %\end{large} In the figure below the "home made" coil is shown. \begin{figure}[h] \includegraphics[width=5cm,height=10cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/coil.ps} \centering\caption{Home made coil.} \end{figure} The coil was made of superconducting wire (NbTi with Cu filaments). The wire thickness is 0.12 $\mu$m and the coil has 1886 windings (2 layers). At 1 Tesla the critical current is 6.7 A. In the graph (fig. \ref{coil}) below the calibration of the coil is shown. The calibration was done with a Hall probe. The Hall probe was calibrated in the PPMS and the results are shown in fig. \ref{hal}. \begin{figure}[!] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/CalibrationCoil1886.EPS} \centering\caption{Calibration of the coil: magnetic field vs. current.}\label{coil} \end{figure} \begin{figure}[!] \includegraphics[width=12cm,height=8cm,angle=0]{/home/beekman/Documents/sctiptie/plaatjes/Hallprobecal.EPS} \centering\caption{Calibration of Hall probe: resistance vs. applied magnetic field at 300 K(red) and 4.2K(black).}\label{hal} \end{figure} \newpage %\begin{large} \chapter{Process parameters for the fabrication of the lateral spin switch using the lift-off technique.}\label{a3} %\end{large} \section{Py:} Spincoat PMGI SF 3.6 4000 rpm \\ ramp1: 2 s \\ ramp2: 1 s \\ time: 50 sec \\ thickness: ~100 nm\\ Bake 30 min 190 $^\circ$C \\ \newline Spincoat PMMA A2 4000 rpm \\ ramp1: 2 s \\ ramp2: 4 s \\ time: 50 sec\\ thickness: 60 nm\\ Bake 60 min 140 $^\circ$C\\ \newline Exposure parameters:\\ I=50 pA\\ Area Dose= 160 $\mu$As cm$^{2}$\\ Dose factor=2.3\\ The small structures (small strip and alignment markers) were exposed with relative dose 1.2 while the large strip and large alignment markers were exposed with a relative dose of 1.0.\\ \newline Development:\\ PMMA 30 sec MIBK:IPA 1:3 and IPA 1 min rinse and dry\\ PMGI 2.5 min PMGI developer and H$_2$O rinse and dry\\ Post bake: 30 min 140 $^\circ$C \\ \newline Sputtering:\\ UHV\\ I= 165 mA\\ P= 4 $\mu$bar\\ thickness: 40 nm \section{Al:} Spincoat PMGI SF 5 2300 rpm\\ ramp1: 2 s \\ ramp2: 1 s \\ time: 50 sec\\ thickness: 200 nm\\ Bake 30 min 190 $^\circ$C \\ \newline Spincoat PMMA A2 4000 rpm\\ ramp1: 2 s \\ ramp2: 4 s \\ time: 50 sec\\ thickness: 60 nm\\ Bake 60 min 140 $^\circ$C\\ \newline Exposure parameters:\\ I=50 pA\\ Area Dose= 160 $\mu$As cm$^{2}$\\ Dose factor=2.3\\ %The small structures (small strip and alignment markers) were exposed with relative dose 1.2 while the large strip and large alignment markers were exposed with a relative dose of 1.0. \\ Development:\\ PMMA 30 sec MIBK:IPA 1:3 and IPA 1 min rinse and dry\\ PMGI 3 min PMGI developer and H$_2$O rinse and dry\\ Post bake: 30 min 140 $^\circ$C \\ \\ Sputtering:\\ UHV\\ I= 220 mA\\ P= 6 $\mu$bar\\ thickness: 80 nm\\ \\ \section{Au:} Spincoat PMMA A2 6000 rpm\\ ramp1: 2 s \\ ramp2: 1 s \\ time: 50 sec\\ thickness: 60 nm\\ Bake 60 min 140 $^\circ$C\\ \\ Exposure parameters:\\ I=50 pA\\ Area Dose= 140 $\mu$As cm$^{2}$\\ Dose factor=2.0\\ %The small structures (small strip and alignment markers) were exposed with relative dose 1.2 while the large strip and large alignment markers were exposed with a relative dose of 1.0. \\ Development:\\ PMMA 30 sec MIBK:IPA 1:3 and IPA 1 min rinse and dry\\ %PMGI 3 min PMGI developer & H$_2$O rinse and dry\\ Post bake: 30 min 140$^\circ$C \\ \\ Sputtering:\\ Z-400:\\ P=4.9 10$^{-6}$ mbar\\ MoGe 20 sec (adhesion layer)\\ Au 200 sec \\ \begin{thebibliography}{99} \bibitem{bcs} J. 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