Monica Bengt

Miriam Derek

Keywords: 4, 4 Critical current fluctuations in SQUIDs

I thank the support of my brother Patrik and his new family Virág and Janka, and also my brothers-in-

Acknowledgement

It is not a secret that this journey was not easy. That this work is finished is not only my achievement. It would not have been possible without the support of my supervisors, colleagues, friends and family. I would like to express my gratitude to them in the following paragraphs.

I would like to start by thanking Dominique Mailly, Grégory Abadias, Alain Marty and François Lefloch for accepting to be part of my jury, and taking interest in my research.

I owe thanks to the Nanosciences Foundation, particularly Alain Fontaine, Marie-Anne Carré and Fériel Kouiten-Sahraoui for funding the project my PhD was part of. I am especially grateful for the one month (July 2017) I could spend in Grenoble to conclude the writing of this manuscript.

My sincere thanks go to my supervisors, Klaus Hasselbach and Bruno Gilles, for proposing a great project. The scope of it, encompassing two fields of research, is what attracted me to it in the first place. Admittedly, this made my work difficult at times, but now I know that it was good that I was constantly challenged, and never quite comfortable. After all, how boring life would be if there was nothing left to learn. I am also grateful for my supervisors' kindness, guidance, their patient discussions with me and the encouragement they provided over the years. I consider myself lucky for having had supervisors who were generous with their time, and were available when I asked them to discuss. Above all, I am incredibly thankful that they did not give up on me during my long absences in the past year. I owe thanks to Olivier Buisson and his group. A large part of my work was done with his group, and although I was formally not part of it, I felt like I belonged. I had several interesting discussions with Olivier, and he kindly advised me and guided me during my time at Institut Néel. I am grateful to Cécile Naud for introducing me to cryogenics and lithography fabrication. It is thanks to her dedication that we succeeded in fabricating rhenium SQUIDs.

I would like to thank the whole NanoFab team for the training they gave me, and for i ii their help and advice on the fabrication of rhenium. I thank Guillaume Beutier and Marc Verdier for introducing me to synchrotron experiments. I count our time in Diamond Light Source among the most exciting experiences during my PhD.

I would like to thank Stéphane Coindeau, Frédéric Charlot, Florence Robaut and Sabine Lay for contributing to my work with measurements, and for the help they gave me in the analysis of the data. I thank Pierre Rodière for his measurement on one of my samples, and for his input on my defence presentation. I thank Virginie Simonet for taking me on my first tour of Institut Néel, and introducing me to Klaus.

During my time in Grenoble I made many friends.

Eva and Rosen were not only my office mates, but were my neighbours, and we quickly became good friends. We shared dinners and games, sometimes several times a week. With Eva we learned to sew and perfected our knitting skills together. Eva, we have yet to make our big skirts and wear them on top of a windy hill.

I would like to thank again the hospitality of Andrew, Nat, Sarah and Amy. I spent a week with them before my defence, and had a really excellent time. I hope to visit them again soon. I thank Markus for helping me out, and for arranging my inscription for the last year of doctoral school. I had lots of beers and pizza at Brasserie du Carré with Kristijan, Daria, Marek, Hannu, Tilo, Ingo, Andrew and Markus. These were really great times, and I miss our dinners and their company a lot. I had way too much coffee with Rémy, Javier, Luca, Etienne, Jorge and Farshad during my time at Institut Néel. I really did not need to drink all that coffee; I just enjoyed spending time with them. I am also grateful to have met and spent time with John, Marian, Dipankar, Gyuri, Márti, Alex, Alexadre, Dibyendu, Ivy, Clément, Farida, Hanno, Benjamin, Hasan, Solène and Maxime.

I owe the greatest gratitude to my parents, Marianna and József Ratter. I only have the abilities which allowed me to pursue a PhD because they provided me with the opportunity to learn. They supported all my endeavours, and allowed me to make my own mistakes, for which I am grateful. Köszönet mindenért! My parents-in-law Klazien and Dennis W. T. Nilsen were already familiar with the process of getting a PhD. They told me that it was hard for Dennis and for Gøran as well. It gave me a lot of comfort knowing that I was not alone. I also thank them for all of their encouragement and support and for their belief in me.

Introduction

The foundations of today's computers were laid down by Alan Turing in 1936, who developed a model for a programable machine, now known as the Turing machine. The first electronic computers appeared shortly after. With the invention of the transistor in 1947, hardware development took off, and computer power has been growing exponentially since, changing the world at an unprecedented pace and scale [START_REF] Michael | Quantum Computation and Quantum Information[END_REF].

However, it is argued that conventional computers will not be able to keep up with this established trend much longer. Due to the decreasing size of the electronic components, quantum effects are beginning to interfere with their operation. Furthermore, the time required to solve a problem with a conventional algorithm grows exponentially with the number of operations. This puts constraints on the finesse in a simulation [START_REF] Michael | Quantum Computation and Quantum Information[END_REF].

One path proposed is to redefine computation as we know it, and use quantum computers. In a quantum computer, bits are replaced by quantum bits or qubits. Unlike a bit, which can either be '0' or '1', a qubit can have a state which is the superposition of '0' and '1':

|ψ = α |0 + β |1 ,
where α 2 + β 2 = 1. N bits in both a quantum and in a conventional computer can have a total of 2 N states. The qubits can occupy all these states simultaneously. Algorithms that can exploit the superposition of states already exist. An example of this is Shor's algorithm, which demonstrated that factorisation of large integers can be solved efficiently. It is believed that this problem has no efficient solution on a conventional computer [START_REF] Michael | Quantum Computation and Quantum Information[END_REF][START_REF] Dumur | A V-shape superconducting artificial atom for circuit quantum electrodynamics[END_REF].

Bits in a computer are expected to preserve their states for a period of time. This is not different for a quantum computer either. However, the loss of coherence in qubits makes developing a quantum computer challenging.

One of the candidates to realise a qubit is based on Josephson junctions. Josephson junctions consist of two superconductors separated by a thin (∼ 1 nm) insulator barrier.

INTRODUCTION

They are described in more detail in section 3.3.1. The time scales over which a qubit preserves information are called the coherence times. These coherence times are limited by noise, which is ascribed to fluctuating charges in the insulator barrier. A frequently used material for the barrier is aluminium oxide. It is prepared by the subsequent deposition and oxidation of aluminium. The result is an amorphous layer, which is noted with the chemical formula AlO x . It is suspected that the aluminium is not fully oxidised in this form, and that this is the origin of the two-level fluctuations that lead to decoherence [START_REF] Grigorij | Strain tuning of individual atomic tunneling systems detected by a superconducting qubit[END_REF].

Consequently, the path to the quantum computer goes via employing new or unconventional materials and exploring the parameter space of deposition and growth conditions, in order to obtain high quality superconductor-insulator-superconductor junctions, which prerequisite for qubits.

The aim of this project was to eliminate disorder by growing epitaxial films. For epitaxial growth, the lattice parameter match is an important criterion. The lattice of rhenium, which is superconducting below 1.7 K, has an excellent match with the lattice of Al 2 O 3 . Furthermore, crystalline rhenium is very stable, and does not oxidise.

Rhenium has been the subject of a few studies in the recent past. Oh et al. grew rhenium thin films onto Al 2 O 3 using DC and RF sputtering. They observed the growth of epitaxial islands with spiral structure [START_REF] Oh | Epitaxial growth of rhenium with sputtering[END_REF].

Welander grew rhenium films on niobium surfaces [START_REF] Paul | Structural evolution of re (0001) thin films grown on nb (110) surfaces by molecular beam epitaxy[END_REF]. Niobium was first grown onto an Al 2 O 3 substrate, and this growth has been shown to be epitaxial [START_REF] Wildes | The growth and structure of epitaxial niobium on sapphire[END_REF]. Rhenium films grown this way were smooth, and fully relaxed by 20 nm thickness.

The molecular beam epitaxy growth of rhenium onto Al 2 O 3 substrate was started by B. Delsol in SIMaP [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. These films were used to fabricate microwave resonators [START_REF] Dumur | A V-shape superconducting artificial atom for circuit quantum electrodynamics[END_REF][START_REF] Dumur | Epitaxial rhenium microwave resonators[END_REF], and to study the proximity effect with graphene [START_REF] Tonnoir | Induced superconductivity in graphene grown on rhenium[END_REF]. Rhenium, indeed, grows epitaxially onto single crystal Al 2 O 3 . In the next step , crystalline aluminium would be deposited onto the flat surface of epitaxial rhenium, and oxidised. Our aim was that this would result in a crystalline, fully oxidised barrier. However, as is shown throughout this work, it is not so easy to produce a flat rhenium surface. The surfaces of our epitaxial films are decorated with spirals and deep holes. Such a topography is not adequate for the deposition of a second layer. Consequently, we achieved an understanding of the processes driving the growth mechanism, as we successfully identified dewetting as the culprit for the presence of holes.

The epitaxial growth of rhenium thin films onto single crystal Al 2 O 3 substrates was realised using molecular beam epitaxy. Following the characterisation of the films, wires and SQUIDs (superconducting quantum interference device) were fabricated using laser and electron lithography. The transport properties of these devices were studied at low temperatures.

Chapter 1 starts with a short description of the materials used in this work, rhenium and the single crystal Al 2 O 3 substrate. Following that, the requirement of ultra-high vacuum in epitaxial growth is explained. Sections 1.3 and 1.4 describes the molecular beam epitaxy setup, and the characterisation techniques that were used to prepare and study the films presented in this work. The final section of this chapter deals with the theoretical background of crystal growth, spiral growth, and thermal grooving. These theories are referred to in chapter 2.

The first section of chapter 2 discusses several aspects of rhenium growth on Al 2 O 3 specifically. The preparation procedure of the substrate is described first, then the temperature of the evaporating rhenium is estimated from the observed deposition rate. The critical thickness, above which dislocations are expected to appear, is also given here.

The following section studies how the temperature of the sample during growth influences the surface topography and the crystallography of the film. Section 2.3 shows that rhenium undergoes dewetting during growth when approximately 10 nm thickness is reached. Finally, the thermal transfer model is described, which was developed to calculate the temperature of the growing film.

The first section of chapter 3 presents the phenomena of superconductivity, and explains the basic theories that were developed to describe it. In the following section superconducting devices, namely the Josephson junction and the SQUID, are described. The final section of this chapter gives a short description on the two refrigerators that were used to reach temperatures below 1 K, and measure the transport properties of our samples.

In the first two sections of chapter 4 the lithography process and the fabricated circuit designs are presented. Section 4.3 discusses the transport measurements on the wires. The shape and width of the superconducting transition with respect to the topography and crystallography is studied. The final section presents the critical current oscillations of two low-noise SQUIDs.

1

Molecular beam epitaxy

In this chapter, first rhenium and the substrate material are introduced, then the theoretical and experimental basis for molecular beam epitaxy and crystal growth is given. In the second section the motivation for the use of ultra-high vacuum environment is explained. After that, the molecular beam epitaxy setup, and the available characterisation techniques are described. The following section deals with the basic theories of crystal growth, and the roll of misfit strain on dislocation formation. Finally, Burton, Cabrera and Frank's theory on spiral growth, and Mullins' theory on thermal grooving is introduced. These theories will be referred to in chapter 2, where the experimental results are discussed.

Materials

Rhenium

History and occurrence

The existence of rhenium was predicted by Mendeleev. It is the last discovered element that has a stable isotope, and was first detected in platinum ore in 1925 by Walter Noddack, Ida Tacke, and Otto Berg in Germany [START_REF] Noddack | Die ekamangane[END_REF]. It was named after the river Rhine.

In 1928 Noddack et al. were able to extract 1 g of pure rhenium by processing 660 kg of molybdenite (MoS 2 ) [START_REF] Noddack | Die herstellung von einem gram rhenium[END_REF].

Rhenium is among the rarest elements in the Earth's crust. It has not yet been found in pure form, and the only known rhenium mineral, ReS 2 called rheniite, was described as recently as 1994. It was discovered condensing on the Kudriavy volcano, on the Iturup island, disputed between Japan and Russia [START_REF] Korzhinsky | Discovery of a pure rhenium mineral at kudriavy volcano[END_REF]. The volcano discharges 20-60 kg rhenium per year mostly in the form of rhenium disulfide.

The primary commercial source of rhenium is the mineral molybdenite (MoS 2 ) which contains about 3% Re. Chile has the world's largest rhenium reserves and was the leading producer as of 2005. The total world production of rhenium is between 40 and 50 tons/year, and the main producers are in Chile, the United States, Peru, and Poland.

Physical and chemical properties

Rhenium is a silvery-white heavy metal from the third row of the transition metal block, with atomic number 75. Its melting point (3186 • C) and boiling point (5630 • C) are among the highest among the elements.

It crystallises in the hexagonal close-packed structure, shown in figure 1.1, with lattice parameters a = 0.2761 nm and c = 0.4456 nm [START_REF] Liu | Effect of pressure and temperature on the lattice parameters of rhenium[END_REF]. The density of rhenium is also among the highest, with 21.02 g/cm 3 measured at room temperature.

Rhenium has one stable isotope, 185 Re, which is in minority. Naturally occurring rhenium is composed of 37.4% 185 Re, and 62.6% 187 Re. 187 Re is unstable but its half life is very long, about 10 10 years [START_REF] Audi | The nubase evaluation of nuclear and decay properties[END_REF].

MATERIALS

Electron configuration of rhenium is [Xe] 4f 14 5d 5 6s 2 . Its oxidation state is known to vary between -3 and +7 skipping over -2, the most common being +7, +6, +4, and +2. There are several rhenium oxides, the most common is Re 2 O 7 , is colourless and volatile. Other oxides include ReO 3 , Re 2 O 5 , ReO 2 , and Re 2 O 3 [START_REF] Greenwood | Chemistry of the Elements[END_REF].

Pure rhenium is a superconductor, and its first recorded transition temperature was 2.42 K [START_REF] Daunt | Superconductivity of rhenium[END_REF]. Rhenium alloys show higher transition temperatures: rhenium-molybdenum is superconductive under 10 K [START_REF] Lerner | Magnetic properties of superconducting mo-re alloys[END_REF], and tungsten-rhenium at around 4-8 K [START_REF] Neshpor | Superconductivity of some alloys of the tungsten-rhenium-carbon system[END_REF].

Application

As a refractory metal, rhenium shows extraordinary resistance against heat and wear. Most of its applications are centred around this property.

Nickel based alloys that contain up to 6% of rhenium are used in jet engine parts or in industrial gas turbine engines. 70% of the worldwide rhenium production is used in this field. Tungsten-rhenium alloys are used as X-ray sources and thermocouples for temperatures up to 2200 • C.

The low vapour pressure of rhenium makes it suitable to be used as filaments in mass spectrometers, gauges and photoflash lamps.

Alloyed with platinum, it is used as a catalysts in the production of lead-free, highoctane gasoline.

Al 2 O 3 substrate

Rhenium thin films were deposited onto single crystal α-Al 2 O 3 substrates. Several polymorphs of Al 2 O 3 exists, α-Al 2 O 3 is the most stable, and is the only phase that occurs naturally. α-Al 2 O 3 is called corundum. Corundum is a rock-forming mineral. It is transparent, and in its chemically pure form has a white hue. In nature, corundum is rarely pure, and can appear in many different colours depending on the impurities. Coloured corundum is frequently used as a gemstone, best known varieties of it are ruby and sapphire.

Synthetic Al 2 O 3 crystals are prepared with the Czochralski growth process. A precisely oriented seed crystal is introduced into the molten Al 2 O 3 , and slowly pulled. The melt crystallises onto the seed matching its orientation.

Al 2 O 3 crystallises in the trigonal crystal system, in the R3c space group. Its lattice parameters are a = 0.476 nm and c = 12.993 nm [20]. In the lattice, six oxygen ions form a slightly distorted octahedron around an aluminium ion. Two octahedra are shown in two different orientations in figure 1.2(a) and 1.2(b). In figure 1.2(c) the stacking of the octahedra is shown, as they form the lattice.

Along the c-axis the structure is an alternation of one oxygen and two aluminium layers, shown in figure 1.3(a). The two neighbouring Al layers are separated by approx- imately 0.06 nm and shifted laterally. The separation between two layers of oxygen is the sixth of the c lattice parameter, 0.22 nm. The double Al layers also have an average spacing of 0.22 nm.

In figure 1.3(b) the view of the c plane is shown, terminated by oxygen ions, that form triangles over the Al ions.

All our substrates were c-plane, they were cut perpendicular to the crystallographic c-axis, along the (001) plane.

The basics of molecular beam epitaxy

Epitaxy is a Greek composite word, epi meaning 'above', and taxis meaning 'an ordered manner'. It roughly translates 'arranging upon'. Epitaxy occurs when a metastable material nucleates onto a crystalline substrate in registry with its crystalline order, as shown on figure 1.4 [START_REF] Arthur | Molecular beam epitaxy[END_REF]. This process allows the preparation of single crystal thin films.

Depending on the phase of the metastable material the epitaxy can be solid phase, liquid phase or vapour phase epitaxy. In chemical vapour deposition volatile precursors decompose onto or react with the substate to produce the layer. In other vapour epitaxy techniques the source is sputtered or ablated. These techniques allow fast growth of thin films therefore they are reliably used in the semiconductor industry and in research. Molecular beam epitaxy utilises beams of atoms or molecules in an ultra-high vacuum environment (10 -10 Torr) that are incident upon a heated crystal whose surface is atomically flat and clean [START_REF] Arthur | Molecular beam epitaxy[END_REF]. Depositions rates are much lower than in the above mentioned techniques, around 1 monolayer/minute, allowing the growth of single crystals and sub-monolayer composition control. The ultra-high vacuum conditions makes it possible to incorporate characterisation techniques, such as electron diffraction, X-ray photoemission spectroscopy, and sample preparation techniques such as ion etching. All these make MBE the ideal research tool for developing new materials.

The development of MBE was driven by the decreasing dimensions of semiconductor devices [START_REF] Cho | 20 years of quantum cascade lasers (QCLs) anniversary workshop[END_REF], and by the interest in heterostructures made out of semiconductors with different energy gaps [START_REF] Mccray | MBE deserves a place in the history books[END_REF]. Several unsuccessful attempts were made to grow such structures [START_REF] Collins | Electrical and optical properties of GaSb films[END_REF][START_REF] Günther | Aufdampfschichten aus halbleitenden III-V-verbindungen[END_REF]. Breakthrough came from the field of surface sciences in 1968, when Arthur observed that growth rate is not only the function of vapour pressure, but is strongly influenced by vapour-surface interactions [START_REF] Arthur | Interaction of Ga and As 2 molecular beams with GaAs surfaces[END_REF][START_REF] Arthur | GaAs, GaP , and GaAs x P 1 -x epitaxial films grown by molecular beam deposition[END_REF]. His discovery paved the way for the stoichiometric growth of compounds where the components have very different vapour pressures. 1968 marks the birth of MBE.

The advance of the supporting techniques was essential to the rapid evolution of MBE. Quadruple mass spectrometry was used in the study of surface-vapour interactions by Arthur [START_REF] Arthur | Interaction of Ga and As 2 molecular beams with GaAs surfaces[END_REF], and it remains a key component of MBE chambers to ensure a clean UHV environment. In 1969 A. Y. Cho was the first to use reflection high energy electron diffraction setup (RHEED) in the MBE chamber to investigate the growth process in situ. He showed that MBE is capable to produce atomically flat, ordered layers [START_REF] Cho | Morphology of epitaxial growth of GaAs by a molecular beam method: The observation of surface structures[END_REF][START_REF] Cho | GaAs epitaxy by a molecular beam method: Observations of surface structure on the (001 ) face[END_REF]. During these years compact electron guns became available, which made it possible to routinely combine MBE with RHEED, allowing the study of wide range of materials. From then on MBE was an essential part of several important studies and discoveries: (b) Time required for the formation of a monolayer as the functions of pressure. In ultrahigh vacuum, the mean free path is so long that collisions can be neglected, and it takes several hours for a monolayer to form from the residual molecules.

fractional Hall effect [START_REF] Tsui | Two-dimensional magnetotransport in the extreme quantum limit[END_REF], band-gap engineering [START_REF] Capasso | Band-gap engineering via graded gap, superlattice, and periodic doping structures: Applications to novel photodetectors and other devices[END_REF], quantum cascade laser [START_REF] Faist | Quantum cascade laser[END_REF], zerodimensional structures [START_REF] Goldstein | Growth by molecular beam epitaxy and characterization of InAs/GaAs strained-layer superlattices[END_REF], quantum dot lasers [START_REF] Ledentsov | Optical properties of heterostructures with InGaAs -GaAs quantum clusters[END_REF], giant magnetoresistance [START_REF] Fert | The origin, development and future of spintronics[END_REF][START_REF] Grünberg | From spinwaves to giant magnetoresistance (GMR) and beyond[END_REF]. In the following sections the operational principles of MBE and the supporting techniques are detailed, starting with the importance of ultra-high vacuum conditions. MBE operates in ultra-high vacuum. To reach 10 -10 Torr from atmospheric pressure, the chamber has to be evacuated by running high performance pumps for several days. After, the chamber walls, and all the instruments and surfaces are heated to aid the evaporation of molecules that were absorbed from the air. At the end of this procedure, in the ultra-high vacuum regime, the residual gas mainly consists of hydrogen molecules and methane. To maintain the low pressure, continuous pumping is necessary using ion pumps.

The kinetic gas theory demonstrates the necessity of ultra-high vacuum. The residual molecules are moving rapidly around the chamber, occasionally colliding with the wall, instruments, samples, or with each other. From the kinetic gas theory, the mean free path of the particles (λ) and the rate of collisions with a surface (N coll ) at pressure P can be calculated:

λ = k B T √ 2πd 2 P , N coll = P √ 2πk b T m ,
where k B is the Boltzmann constant, T is the temperature inside the chamber in kelvin, d is the diameter, and m is the mass of a molecule [START_REF] Henini | Molecular beam epitaxy: from research to mass production[END_REF].

In figure 1.5(a) the mean free path and the rate of collisions are shown as the function of pressure. In the calculation a hydrogen molecule was considered. Values are in the same range for the residual molecules that are commonly found in ultra-high vacuum. At atmospheric pressure the mean free path is in the range of nanometers, but at pressures where MBE operates it is around 100 km. This means that in a chamber with dimensions of 1-2 meter the particles can move without collisions. The beam of molecules/atoms can reach the substrate without reacting with other species on the way. Another advantage is that ultra-high vacuum allows the use of electron beam at high or low energy, the beam will not be scattered even at long distances (∼1 meter).

To calculate the time it takes for a monolayer to form from the residual molecules, the following is considered: on a surface of area of 1 m 2 there are approximately 10 19 atoms. Using the collision rate and assuming the colliding molecule sticks to the surface, one can calculate how long it takes for a monolayer to form from the residual particles: τ [s] = 10 19 N coll . This time is plotted as the function of pressure in figure 1.5(b). In ultra-high vacuum τ can be measured in hours. In MBE deposition rates are low, therefore, deposition of a sample can take hours. Keeping the rate of collisions low by keeping the pressure in the ultra-high vacuum range ensures the purity of the sample.

MBE instrumentation

The MBE setup used for this work is shown in figure 1.6. It consists of four interconnected chambers.

The introduction chamber, noted by label 1 in figure 1.6, is the only chamber that is brought to atmospheric pressure regularly, as it is used for the introduction of the substrates. It is pumped to 10 -7 Torr before it is opened towards the other chambers with higher vacuum levels. Otherwise it is kept at static vacuum. Before opening it towards the atmosphere, it is flooded with nitrogen gas. The intermediate chamber, noted by label 2 in figure 1.6, connects the other three chambers together.

Deposition chamber

The deposition chamber, labeled as 'Dep. chamber' in figure 1.6, is where the thin films are deposited. It is equipped with a Leybold quadrupole mass spectrometer that is used to monitor the composition of the residual gas inside the chamber. The schematics of the spectrometer is shown in figure 1.7. A quadrupole mass spectrometer has three parts. The first part is an ioniser, that ionises the molecules passing through it by electron bombardment. The second part is a mass-to-charge ratio filter, and the third part is the detector. The mass-to-charge ratio filter consists of two pairs of cylindrical electrodes in quadrupolar arrangement, as shown in figure 1.7. A potential of ±(U + V sin(ωt)) is applied between them, where U is a DC voltage and Vsin(ωt) is an AC voltage. The trajectory of ions travelling between the the four rods will be affected by the field, so that only ions with the set mass-to-charge ratio will reach the detector (red path in figure 1.7). The others will be thrown off course (blue path in figure 1.7). A massto-charge ratio spectrum is obtained by changing the voltage applied to the electrodes. From the spectrum, the composition of the residual gas can be determined.

The deposition chamber is also equipped with two Riber evaporation systems which consist of an electron gun, bending magnet, metal charges, and controlling electronics. The schematics of the evaporation system is shown in figure 1.8. The metal charge is heated with a 10 kV electron beam extracted from a tungsten filament. The beam scans the charge to ensure uniform heating. To adjust the heating power, and thus the rate of deposition, the current of the beam can be adjusted. The metal charge used in the present studies was 99.95% rhenium supplied by Neyco. To achieve a deposition rate of 0.1 Å/s -0.2 Å/s of rhenium, the beam current was set to approximately 200 mA.

Figure 1.8: Schematics of the evaporation system: electrically heated tungsten wire biased by 10 kV ejects electrons that are directed onto a metal charge using a magnetic field.

The substrate is placed horizontally on a manipulator above the charge. At its position the flux of atoms arriving at the surface is homogeneous. The deposition can be turned on and off with the use of a shutter located below the substrate. The manipulator is equipped with a furnace that consists of a tungsten filament, shown in figure 1.9. The substrate can be heated in two ways using this furnace: either by thermal radiation, or by electron bombardment. Infrared radiation is emitted by the tungsten filament when it is heated by a current running through it (up to approximately 10 A). Increasing the current will increase the temperature of the substrate. We can reach around 900 • C this way. When applying a voltage (400 V -800 V) between the sample and the filament, electrons are emitted. The temperature of the substrate is adjusted by the emission current (up to approximately 100 mA). We can reach around 1000 • C by electron bombardment.

The manipulator head is shown in figure 1.9. The temperature of the substrate is measured by a thermocouple that is located in the middle of the manipulator head, and is pressed against the back side of the substrate. There is an uncertainty in the contact between the thermocouple and the substrate, thus the value measured this way is an approximate of the real surface temperature. Also, the thermal and optical properties of the sample can change during growth, which affects the surface temperature. This change cannot be detected with the thermocouple. Figure 1.9: Furnace and thermocouple in the manipulator: the substrate is placed on top of the tungsten filament, the thermocouple is pressed against its back side.

Another way of measuring the temperature in an ultra-high vacuum environment, is to use a pyrometer. The pyrometer is located outside of the chamber, looking at the sample through a viewport. It measures the thermal radiation emitted by the material. For this method to give a reliable result, the viewport has to be made out of a material whose transmission as the function of wavelength is well known (usually Al 2 O 3 ). Also, the sample surface has to be aligned parallel with the window of the pyrometer. In the deposition chamber we cannot fulfil these requirements due to geometric constraints. The only way to measure the temperature of the sample during growth is by the thermocouple.

Using molecular beam epitaxy, films with thicknesses ranging from a few Å to 100 nm are routinely deposited. To be able to prepare samples in this wide range of thickness, precise measurement of the deposition rate is necessary. A microbalance made out of a quartz single crystal is the most commonly used tool to monitor the deposited thickness. The quartz microbalance consists of a quartz crystal, cut along a specific crystallographic orientation, with an alternating voltage applied to it. Due to the piezoelectric effect this voltage generates a standing wave in the crystal at a well defined frequency (resonance frequency) in the MHz range. When the mass of the crystal increases, the resonance frequency decreases. From the frequency shift the deposited mass and the thickness can be calculated. There are two Leybold quartz balances in our deposition chamber, located close to the sample, shown in figure 1.6.

The lifetime of a quartz microbalance is limited due to the deposit building up on it. To lengthen its lifetime, we are able to turn the measurement on and off using a shutter placed in front of the quartz crystal.

Characterisation chamber

There is a second ultra-high vacuum chamber connected to the deposition chamber. This is the characterisation chamber, labeled as 'Char. chamber' in figure 1.6. It is equipped with instruments that allow the investigation and preparation of the sample before or after the deposition, without exposing it to air. Instruments available in the chamber are the following: X-ray photoelectron spectroscopy (XPS), argon ion gun, low energy electron diffraction, a furnace that can reach over 2000 • C, and a pyrometer to measure the temperature. In this work, only the XPS was used, only that technique is discussed in detail in the following section.

Thin films characterisation techniques

In situ characterisation techniques

Some investigative techniques are available without having to remove the sample from the vacuum chamber. XPS is used to check the chemical composition of the surface of the substrate or the deposited film before or after deposition. RHEED can be used before, after, or during deposition to monitor the crystallographic properties of the film.

X-ray photoelectron spectroscopy

XPS is used to study the chemical composition of the surface. The principle of the technique, shown in in figure 1.10(a), is the following: the sample is irradiated with a known energy X-ray beam, and the electrons (mostly photoelectrons) that escape the material are sorted by their kinetic energies, and counted.

The setup consists of an X-ray tube, shown by the upper arrow in figure 1.6, and a detector, shown by the lower arrow. The anode material in the X-ray tube is magnesium, and the radiation corresponding to its Kα line with an energy of 1253.6 eV is used. The detector has two parts: an energy analyser with an energy window that is scanned over a given voltage range, and an electron multiplier for amplifying the current of the electrons.

The binding energy of the electrons are the characteristics of an atom or a molecule. From the kinetic energy of an emitted electron its binding energy can be calculated as follows:

hν = E B + Φ + E k → E B + Φ = hν -E k , (1.1)
where h is the Planck constant, ν is the wavelength of the exciting X-ray beam, E B is the binding energy of an electron, Φ is the work function that depends on the material and the instrument, and E k is the kinetic energy of the electron. Precise value of Φ is not known but it is small [START_REF] Moulder | Handbook of X-ray photoelectron spectroscopy[END_REF].

A typical XPS spectra is shown on figure 1.11. Most of the peaks indeed correspond to photoelectrons that were excited from the core shells of the atoms. There is, however, an other process, called Auger effect, which can result in peaks: a photoelectron leaves a vacancy on an inner shell that is filled by an electron from a higher shell. Then a second electron, an Auger electron, is emitted, carrying off the excess energy, leaving behind a doubly-charged ion. the first electron, the photoelectron, originated from the K level, its place was taken by an electron from the valence level, and the Auger electron was also from the valence level [START_REF] Moulder | Handbook of X-ray photoelectron spectroscopy[END_REF].

A ghost peak is also noted in figure 1.11 in blue. This is the result of copper contamination in the anode. The energy of the X-ray photons emitted by the copper contamination are different, therefore, the kinetic energy of the electrons they excite from the same shells are different too. When calculating the binding energy, only the Kα line of magnesium is considered in equation 1.1. This gives small intensity peaks in the spectrum at a wrong binding energy [START_REF] Moulder | Handbook of X-ray photoelectron spectroscopy[END_REF].

Even though the penetrations depth of the X-rays are relatively large (1-10 µm), the mean free path of electrons at these energies is restricted to a few nanometers due to strong electron-electron scattering. Thus we only get information from the top few atomic layers. A significant number of electrons undergo inelastic scattering processes, losing some of their kinetic energy, and thus add to the background. This is the reason for the step-like structure of the graph, that can most clearly be observed between the peaks Re4d and Re4f.

From the intensity of the XPS peaks, the surface monolayer coverage can be calculated. We used this method in other projects, and the detailed derivation is given in appendix B.

Reflection high energy electron diffraction

The deposition chamber is equipped with a Staib RHEED setup. The technique has been widely used to monitor the surface structure of the films during growth since the '70s. The setup consists of an electron gun, shown in figure 1.6 by an arrow labeled as RHEED, and a phosphor screen on the opposite side. The electron gun produces an electron beam with an energy of 20 keV, that is directed onto the surface of the growing crystal at a grazing angle (1 • -3 • ). Geometry of the RHEED setup is shown in figure 1.12. From the diffraction pattern the physical state of the surface can be determined: in-plane lattice parameter, orientation, symmetry of reconstruction.

Figure 1.12: Geometry of RHEED: monochromatic electron beam is directed onto the growing crystal surface, the diffraction pattern is detected by a phosphor screen.

In an elastic scattering process the energy of the scattered particle is conserved:

E I = E F = 2 k 2 I 2m = 2 k 2 F 2m → k F = k I = k, (1.2)
where E I and E F are the energies of the incident and scattered electrons, is the reduced Planck constant, k I and k F are the magnitudes of the wave vectors of incident and scattered electrons, and m is the electron mass. Laue's condition of diffraction states that the wave vector in diffraction can only change by a vector that is a reciprocal vector (g hkl ) of a scattering crystal:

k F -k I = g hkl , (1.3)
where

g hkl = ha * 1 + ka * 2 + la * 3 and a * i = 2π a j × a k a i • (a j × a k )
.

(1.4) a i,j,k and a * i,j,k are real and reciprocal lattice vectors respectively, and h, k, and l are integers [START_REF] Sólyom | Fundamentals of the Physics of Solids, volume I. Structure and Dynamics[END_REF][START_REF] Ichimiya | Reflection high energy electron diffraction[END_REF]. A more detailed discussion on diffraction can be found in section 1.4.2.

The solutions of equations 1.2 and 1.3 can be obtained geometrically by the Ewald construction, where the vector k I is placed in the reciprocal lattice of the diffracting volume so that its tail end is on a reciprocal point. Then a sphere with radius k is drawn around the head of the vector k I . Diffraction occurs in all the directions, where the sphere intersects a reciprocal lattice point [START_REF] Sólyom | Fundamentals of the Physics of Solids, volume I. Structure and Dynamics[END_REF][START_REF] Ichimiya | Reflection high energy electron diffraction[END_REF]. The detecter is placed in the forward direction, as shown in figure 1.12, thus we can only observe waves, that are diffracted forward.

The radius of the Ewald sphere can be calculated from the de Broglie wavelength of the electrons:

λ = h p , (1.5)
where h is the Planck constant, and p is the momentum of the electrons.

In case of high energy electron beams (>50 keV), relativistic effects have to be taken into account. For a 20 keV electron beam the relativistic correction in the wavelength is only 1%, but for the sake of completeness the relativistic calculation is shown here [START_REF] Ichimiya | Reflection high energy electron diffraction[END_REF].

Energy (E) of a particle with rest mass of m e (electron mass) is

E = p 2 c 2 + m 2 e c 4 = T + m e c 2 , (1.6)
where c is the speed of light, and T is the kinetic energy. From equation 1.6 the momentum can be expressed as follows:

p 2 c c = T 2 + 2T m e c 2 .
(1.7) Kinetic energy of a particle with charge e (electron charge) accelerated by a voltage U is the following:

T = 1 2 m e v 2 = U e. (1.8)
Using equations 1.5, 1.7, and 1.8, choosing an accelerating voltage of 20 kV, the wavelength, and the magnitude of the wave vector is:

T = 1 2 m e v 2 = U e = 0.09Å → k = 2π λ = 73Å -1 .
(1.9) The advantage of grazing incidence is its sensitivity to the surface structure of the sample. Just by glancing at the diffraction pattern it can be determined whether the surface is flat or has grain structure. In the following, the construction of the diffraction patterns are discussed starting with the case of island growth.

In figure 1.13(a) diffraction from a surface, that is covered with islands, is shown, schematically in real space. The electron beam travels through these islands in transmission. The diffracting volume is extended in all three directions, which in reciprocal space corresponds to reciprocal lattice points. This reciprocal lattice is shown in figure 1.13(b) with the Ewald sphere. Constructive interference occurs in directions where the Ewald sphere intersects the reciprocal lattice points. In figure 1.13(c) a cross section of the Ewald sphere and the reciprocal lattice is shown. The points of intersections are clearly visible, they will define all the possible directions of the outgoing wave vector (k F ). The intersection of a reciprocal lattice point and the electron beam is projected onto the phosphor screen. Due to the finite crystallite size, the reciprocal lattice points have an finite width. This results in spherical diffraction spots with a finite diameter. In figure 1.14(a), diffraction from a smooth surface is shown schematically, in real space. Diffraction happens in reflection, and the penetration depth of the electron beam is restricted to a few atomic layers. The third dimension of the diffracting volume is reduced. The reciprocal lattice of a two dimensional periodic structure consists of rods, that are perpendicular to the surface. The distance between the reciprocal lattice rods corresponds to the inverse of the in-plane lattice constant.

In figure 1.14(b) the reciprocal lattice rods and the Ewald sphere are shown. Diffrac-tion is observed in directions where the Ewald sphere intersects the reciprocal lattice rods. In figure 1.14(c) a cross section of the reciprocal lattice and the Ewald sphere are shown. Dimensions in the figures 1.14(b) and 1.14(c) are not accurate, the radius of the Ewald sphere is much larger than the spacing between the reciprocal lattice rods. Therefore, the intersections between them are extended along the direction of the surface normal. This is illustrated in figure 1.14(d). This is the reason why in the RHEED pattern of a film with a smooth surface and good crystalline quality sharp streaks are observed.

Ex situ characterisation techniques

Surface topography and the crystallographic properties of the substrates and the thin films were investigated using several techniques outside of the vacuum chamber. Atomic force microscopy was used to measure the topographic features; X-ray diffraction was used to check the orientations and verify the thicknesses of the films. These two techniques are described below.

Atomic force microscopy

Topography of the films and the substrates were measured using a Veeco Dimension 3100 atomic force microscope. Atomic force microscopy (AFM) belongs to the family of scanning probe microscopes. The AFM probe is an atomically sharp silicone tip attached to a cantilever. They have resonance frequency around 300 kHz.

The AFM cantilever is very flexible, and small forces that act between the sample and the tip can bend it according to Hook's law:

F = kz, (1.10)
where k is the spring constant of the cantilever, and z is the displacement of the tip.

Forces can have different sources depending on the sample, mostly it is due to electrostatic interaction. What is important, that the magnitude of the force decreases with the distance. This allows imaging the topography, by keeping the interaction between the tip and the surface constant [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

In this work, the AFM was used in tapping mode. In tapping mode the cantilever is oscillated so that it lightly taps on the surface of the sample at the lowest point of its swing. The frequency of the oscillation is near to the resonance frequency of the cantilever, where the amplitude is most sensitive to changes. The sample surface is scanned with the oscillating tip, while a feedback loop maintains a constant amplitude, ie. constant CHAPTER 1. MOLECULAR BEAM EPITAXY surface-tip distance, by lowering or lifting the probe. The feedback signal on the vertical module is calibrated, so that it gives the vertical movement of the AFM tip. Plotting this over the scanned are gives the topographic image of the surface [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF]. The schematics of the AFM is shown in figure 1.15(a). The cantilever is connected to a tube made out of a piezoelectric ceramic. This tube is composed of two parts corresponding to the lateral (x, y), and the vertical (z) directions. The vertical module, shown in figure 1.15(b), consists of two cylindrical electrodes separated by the piezoelectric ceramic. The voltage applied to the piezoelectric ceramic is adjusted by the feedback loop, and causes the part to contract or to extend, lifting or lowering the tip, respectively [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

The lateral module, also shown in figure 1.15(b), has four pairs of electrodes arranged around the piezoelectric ceramic tube. The ones opposite to each other receive the same signal but with opposite sign, so while one side extends the other contracts, thus causing the tube to tilt. The shape of the signal applied to these electrodes to generate the scanning raster motion is shown in figure 1.15(b) [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

The movement of the tip can be monitored with the use of a laser light that is directed onto the backside of the cantilever, as shown in figure 1.15(a). It is reflected towards a split photodiode detector that has two separate parts: A and B. The output of the detector is I A -I B I A +I B , where I A and I B are the signals on each diode. From this value the vertical position of the tip can be reconstructed [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

Three types of image can be obtained from an AFM scan: height, amplitude, and phase image. The height image is the one mentioned above, when the vertical position of the oscillating tip is adjusted to keep a constant amplitude. The vertical movement of the tip is plotted as the function of the coordinates of the scanned area, which directly corresponds to the topography of the surface [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF]. The change in the signal on the photodiode detector can also be plotted, this corresponds to the changes in the amplitude of the oscillation, so it is referred to as amplitude image. The feedback loop should keep this value constant, but rapid changes in the topography will show in the amplitude image.

The third value that can be used to create an image is the phase difference between the driving AC signal, and the oscillation of the cantilever. This can show changes in the interaction between the tip and the sample. This is called phase image. Determining what causes the changes in the phase is a science in itself [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

In figures 1.16 and 1.17 the differences between the height and amplitude image, and Images were analysed using the software Gwyddion [42]. Noise was reduced on all of them with in-built algorithms. Also, an algorithm called planefit was used on all of them, unless otherwise stated. Planefit is used to remove the slope across an image that could be caused by uneven mounting of the sample. In the case of stepped surfaces, which have a slope by nature, planefit has an effect that is illustrated in figures 1.18.

In figure 1.18(a), a simulated stepped surface is shown. The intensity of the colour is proportional to the surface height. A cross section of the surface is shown in figure 1.18(c) with the blue line. The plane fit algorithm determines the average slope of the surface, shown in green in figure 1.18(c), and subtracts it from the raw height data. The result is shown in figure 1.18(b), and its cross section in figure 1.18(c) in red.

After the slope is subtracted, the surface appears to be jagged. Nonetheless, it is preferred to use the planefit algorithm even on stepped surfaces, because it reduces the range of the vertical scale, making the features easier to observe.

Lateral resolutions of the AFM is ideally around 1 nm, but it strongly depends on the quality of the tip. The tip degrades over time, because it keeps touching the surface again and again. Vertical resolution is approximately 0.1 nm [START_REF]Scanning probe microscopy training notebook. Digital Instruments[END_REF].

X-ray diffraction

X-ray diffraction (XRD) was used to determine the crystal structures of the deposited thin films. Diffraction of high energy electrons was discussed briefly already in section 1.4.1 using Laue's condition of diffraction, and the Ewald sphere. In this section a short summary is given on the hexagonal crystal structure, then the diffraction phenomena and XRD is described in more detail.

Notes on the hexagonal crystal system. In a crystalline material the atoms are arranged periodically in all three directions of space, forming a crystal lattice. The smallest volume that have the overall symmetry of the crystal is called the unit cell. The length of the vectors (lattice vectors: a, b, c) that define it are the lattice parameters. Rhenium crystallises in the hexagonal system. The a and b hexagonal lattice vectors make an angle of 120 • , and the c vector is perpendicular to the ab plane. a and b are equal in length (a = 0.2761 nm), but c is longer (c = 0.4456 nm) [START_REF] Liu | Effect of pressure and temperature on the lattice parameters of rhenium[END_REF].

Atoms on a crystal lattice form a series of crystal planes. Infinite number of such planes can be defined. A crystal plane intercepts the lattice vectors at points a h , b k , c l . (hkl) are the Miller indexes and they define the orientation of a plane with respect to the coordinate system of the unit cell. Parallel planes are noted using the same Miller indices, and are spaced at equal distances (d hkl ). d hkl for hexagonal crystals can be obtained using the following equation:

1 d 2 hkl = 4 3
h 2 + hk + k 2 a 2 + l 2 c 2 . (1.11)
It is sufficient to use the three Miller indices to identify a plane or a direction in the hexagonal system, however, it does not have the same convenience, as it has in an orthogonal system. In an orthogonal system, indices of equivalent planes and directions can be generated by the permutation of the three Miller indices. This does not work with the Miller indices of a hexagonal crystal. However, permutation does work with the Bravais-Miller indices.

In the Bravais-Miller coordinate system a fourth, redundant axis is introduced in the ab plane, with 120 • apart from a and b. Crystal planes and directions are noted with the four Bravais-Miller indices, (hkil). Equivalent directions and planes can, in this notation, be obtained by the permutation of the first three indices. For example, a hexagonal prism is shown in figure 1.19(a). A plane parallel to the c axis is highlighted. All six of such planes around the prism are equivalent. Figure 1.19(b) shows the in-plane axes of the Miller coordinate system in blue (a and b axes), and the Bravais-Miller coordinate system in red (a 1 , a 2 , and a 3 axes). The intersections of the planes with the axes give the indices in the two system. The Miller indices are: [START_REF] Matthias | Superconductivity and electron concentration[END_REF], [START_REF] Cooper | Bound electron pairs in a degenerate fermi gas[END_REF], and (010). The Bravais-Miller indices are: (1100), (1010), and (0110). This demonstrates that indices in the four axis notation can be obtained by the permutation of the first three indices. In this work, mostly the Miller indices are used.

Indices can be transformed from one notation to the other. In case of a plane, the fourth index (i) is obtained as follows:

i = -(h + k).
(1.12)

A direction [U V T W ] can be converted to the three indices [uvw] as follows:

u = 2U + V, v = 2V + U, w = W. (1.13)
X-ray diffraction. In an XRD experiment, the sample is subjected to an X-ray plane wave (e -ik i r ) with a known wave vector (k i ), therefore known energy and propagation direction. We used two laboratory XRD setups: the Huber 4-cycle diffractometer and the Rigaku SmartLab high-resolution diffractometer. Both instruments use the Kα line of copper. The intensity of this emission is split in two: 2/3 Kα 1 with wavelength 1.540562 Å, and 1/3 Kα 2 with wavelength 1.544398 Å. In the Huber 4-cycle diffractometer both wavelengths were used, in the Rigaku SmartLab high-resolution diffractometer the Kα 2 line is removed. The X-rays are scattered by the electrons in the sample. The scattering is assumed to be elastic, only momentum transfer occurs. This means, the outgoing wave vector has the same length as the incoming wave vector (|k i | = |k f |), and their vectorial difference is called the scattering vector:

q = k f -k i .
(1.14) The scattered amplitude is the sum of the scattering from each atom in the illuminated volume, which, because X-rays are scattered by electrons, can be expressed as the Fourier transform of the electron density:

A(q) = V f (r)e iqr dr, (1.15)
where the integral is taken across the illuminated volume [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF].

The periodic arrangement of the atoms in a crystal lattice results in constructive (in specific cases destructive) interference whenever the scattering vector coincides with a reciprocal lattice vector (g hkl ) [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF]. This gives Laue's condition of diffraction:

g hkl = q.
(1.16)

Laue's condition of diffraction is illustrated in figure 1.20. k i , k f and g hkl form an isosceles triangle, where the angle enclosed by the two equal sides is 2θ, thus the following relation holds:

|g hkl | = 2|k i | sin θ.
(1.17)

Using the properties of the reciprocal lattice, it can be shown that g hkl is perpendicular to the plane series with indices (hkl), and its length is related to the spacing d hkl :

|g hkl | = m 2π d hkl , (1.18)
where m is an integer, which refers to the order of the reflection [START_REF] Sólyom | Fundamentals of the Physics of Solids, volume I. Structure and Dynamics[END_REF]. Substituting 1.18 in equation 1.17, and using relation |k| = 2π/λ we obtain Bragg's condition for diffraction: Bragg's condition of diffraction is also illustrated in figure 1.20.

Bragg's law shows a simple relationship between wavelength, angle of reflection, and lattice spacing. During an elastic diffraction experiment the angular distribution of the scattered intensity is measured. From the angles, the lattice spacings can be determined. Different lattice spacings correspond to different orientations, thus the texture of the film can be determined from a few measurements.

The schematics of a 4-cycle diffractometer is shown in figure 1.21. All circles are aligned so that their centres coincide with the centre of the sample. The detector and the source can move along the red circle. The angle between the incident beam and the surface of the sample is θ. In the symmetric, θ-2θ measurement, the angle between the incident beam and the detector is 2θ, and the source and the detector are moved symmetrically, as shown in figure 1. 22(a). During this measurement the direction of the scattering vector remains perpendicular to the surface, and its length changes. The sample is scanned for all d hkl values of planes that are parallel to the surface. Grains with different orientations are detected this way. Symmetric reflections are often called specular reflections. In the rocking curve measurement the detector and the source are fixed at a θ and a 2θ value where a specular peak was found. The sample is 'rocked' along the red circle in small steps. This is shown in figure 1.22(b). In this case, the length of the scattering vector is fixed, and its direction changes. Small rotations of grains with the same orientation can be detected this way. This is called mosaicity.

It was mentioned above that infinite number of planes can be defined in a crystal lattice. This means that, besides the specular reflections, several asymmetric reflections can be found. This concept in shown in figure 1.23(a). The scattering vector shown in pink was found by a θ-2θ scan. α is the angle between the specular (pink), and the asymmetric (blue) reflections. For a hexagonal structure α can be computed using the following expression:

cos α = 4 2a 2 h 1 h 2 + k 1 k 2 + 1 2 (h 1 k 2 + h 2 k 1 ) + 3a 2 4c 2 l 1 l 2 d h 1 k 1 l 1 d h 2 k 2 l 2 .
(1.20)

Using the angle α, the source and the detector can be moved on the asymmetric reflection.

To verify the crystallinity of the sample in-plane, a Φ scan is conducted, which is shown in figure 1. 23(b). The sample is rotated around the scattering vector of a specular reflection (|| surface normal), along the blue (Φ) circle in figure 1.21, while the detector is set on an asymmetric reflection. If the sample is crystalline, the number of reflections seen in a full rotation reflects the symmetry of the rotation axis. For example, rhenium grows epitaxially on Al 2 O 3 with orientation (002). The (002) axis has hexagonal symmetry, so when we set the source and the detector on the (103) asymmetric reflection, and rotate the sample around the (002) direction we expect to observe 6 bright signals coming from the planes equivalent to [START_REF] Maxwell | Isotope effect in the superconductivity of mercury[END_REF]. As an illustration of the technique, the above example is shown in figure 1.24. Here, Φ was scanned in a 180 • interval, and in addition a χ scan was performed. A 2D projection of the diffraction peaks can be observed. Indeed, within half a circle, 3 diffraction peaks appear. This shows that the rhenium film has a single in-plane orientation. This technique was used to determine the in-plane relationship between the substrate and the film which is presented in section 2.1.3. The scattered amplitude (A N ) is the sum of amplitudes from each plane:

A N (q) ∝ N -1 n=0 e -iqnd = with k=e -iqd 1 + k + k 2 + ... + k N -1 = 1 -k N 1 -k . (1.21)
Equation 1.21 can be arranged in the following form:

A N (q) ∝ sin qN d 2 sin qd 2 • e -iq(N -1)d , (1.22)
From this, the equation that describes the intensity is:

I N (q) ∝ sin 2 qN d 2 sin 2 qd 2 .
(1.23) Equation 1.23 is called the interference function, and was used to fit high-resolution X-ray data presented in chapter 2.

MBE growth

Adsorption and growth modes

During molecular beam epitaxy growth, a charge is heated to temperatures where it slowly evaporates. The deposition chamber contains the vapour phase of the material to-be-deposited and also a heated substrate in the solid phase. Crystal growth happens at the interface of the two phases [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]. Atoms of the vapour phase arrive on the surface of the solid phase. Growth will take place when the arriving atoms of the vapour phase attach to the solid phase at a higher rate than they reevaporate, which implies a departure from equilibrium conditions [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF].

Atoms arriving at the substrate have a temperature distribution corresponding to the source (T source ). Upon arrival they either reach thermal equilibrium with the substrate at the substrate temperature (T substrate ), or reevaporate at a temperature T reevap . This process is quantitatively described by the accommodation coefficient [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]:

α = T source -T reevap T source -T substrate . (1.24)
Thus equation 1.24 expresses the extent to which the arriving atoms reach thermal equilibrium with the surface. α equals zero when T reevap = T source , which means that the atoms reevaporate immediately, before they had time to loose from their energy and lower their temperature. The other limit is when T reevap = T substrate , and α = 1. In this case thermalisation is perfect, the arriving atoms cool to the temperature of the substrate. Atoms that have reached the equilibrium do not necessarily remain on the surface permanently. It is still possible for them to reevaporate at the temperature of the substrate. Sticking or condensation coefficient gives the probability that an atom will adhere to the surface [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]. It is defined as the number of adhered atoms (N adh ) over the total number of arriving atoms (N tot ):

s = N adh N tot , (1.25)
where for the accommodation coefficient only the temperatures are considered, in the sticking coefficient the nature of the physical or chemical bond is also included.

Absorption of an atom can be chemical, when ionic or covalent bonds are formed between the adsorbate and the adsorbent: electrons are transferred. It can be physical, when there is no electron transfer, van der Waals bond connects the two parts. Usually in MBE growth both of them are present subsequently [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]. Once atoms are adsorbed on the surface three things can happen: they can be incorporated into the crystal where they are, they can diffuse to find an energetically more favourable location, or they can reevaporate. This is shown in figure 1. 26(a). What an energetically favourable location for an adsorbate is, depends on the surface tensions between the interfaces, and the amount of material that has already been deposited. The relation developed by Young, which explains the shapes of liquid droplets on solid surfaces, is valid for solid adsorbate too. It demands that the forces acting on the surfaces are in balance:

γ SV -γ SA -γ AV cos θ = 0 → cos θ = γ SV -γ SA γ AV , (1.26)
where γ SV and γ SA are the surface tensions between substrate and vapour, and substrate and adsorbate, respectively. γ AV cos θ is the projection of the surface tension between the adsorbate and vapour to the plane of the substrate surface. θ is the angle between the surface of the substrate and the adsorbate. Geometry is shown in figure 1.26(b).

When γ SV < γ AV + γ SA , θ has a finite value, and it is energetically favourable to keep the area of the substrate-vapour interface at maximum, which will force the adsorbate to form islands. This is called Vollmer-Weber island growth mode, and is depicted in figure 1. 27(b). This growth mode is often observed when metal is grown on an insulator [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]. When the relation is reversed, γ SV > γ AV + γ SA , θ angle cannot be defined. It is now favourable to reduce the substrate-vapour interface by the formation of an adsorbate layer. This is called Frank-van der Merve layer-by layer growth mode, shown in figure 1.27(a). This growth mode is observed in the case of adsorbed gases on metals, semiconductors grown on semiconductors, or in metal-metal systems [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF].

There is a third growth mode, which is called Stranski-Krastanov layer-plus-island growth mode, and is shown in figure 1.27(c). In this case the growth starts layer by layer. After a few monolayer was deposited the growth mode changes into island growth. The change in the growth mode is triggered by the changing of the surface tension with increasing thickness. Surface tension is affected by many factors including strain or surface reconstruction [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF].

Dislocations and misfit

Dislocations

Dislocations are two dimensional defects in crystals. Depending on the orientation of the Burgers vector with respect to the dislocation line, edge and screw dislocations can be defined. A closed loop (M N OP Q) is drawn in the crystal that encloses the dislocation. (b) Burgers circuit is copied into a perfect crystal, where it is not closed. Burgers vector connects the starting point (M ) and the final point (Q) of the Burgers circuit [START_REF] Hull | Introduction to dislocations[END_REF].

In case of an edge dislocation, shown in figure 1. 28(a), an extra half plane is present in the crystal. A screw dislocation, shown in figure 1.29(a), can be imagined by cutting the crystal in half but not all the way, and displacing one half of the crystal by one lattice spacing relative to the other half. If a screw dislocation reaches the surface of the crystal, a step appears [START_REF] Hull | Introduction to dislocations[END_REF].

Dislocations can be characterised by their Burgers vectors. A Burgers circuit is any atom-to-atom path which forms a closed path. Burgers circuits are shown in figures 1.28(a) and 1.29(a). In figure 1.28(a), the circuit M N OP Q encloses an edge dislocation, in figure 1.29(b), a screw dislocation. If the same path is taken in a dislocation free crystal, as shown by the arrows in figure 1.28(b) and 1.29(b), and the path does not close, it must contain at least one dislocation. The vector required to close the loop, is called the Burgers vector. In figures 1.28(b) and 1.29(b), vectors pointing from points Q to M are the Burgers vectors. It can be observed, that the Burgers vector of a pure edge dislocation is perpendicular to the line of the dislocation. In case of a pure screw dislocation, it is parallel [START_REF] Hull | Introduction to dislocations[END_REF].

Dislocations in real materials are neither pure edge nor pure screw type. They are a mixture of both. A closed loop (M N OP Q) is drawn in the crystal that encloses the dislocation. (b) Burgers circuit is copied into a perfect crystal, where it is not closed. Burgers vector connects the starting point (M ) and the final point (Q) of the Burgers circuit [START_REF] Hull | Introduction to dislocations[END_REF].

Dislocations distort the crystal lattice, they induce elastic stress in the material. The stress around a dislocation scales with 1 r , where r is the distance from the dislocation. When r = 0, the stress in infinite which is not possible. This divergence is caused by the break down of the elastic theory at the vicinity of the dislocation. The elastic theory neglects the atoms, and treats the material as a continuum. To avoid infinite stress, an arbitrary cutoff radius, core radius (r 0 ) is defined, and calculations are stopped there. Reasonable values for the core radius are in the range of 1 nm [START_REF] Hull | Introduction to dislocations[END_REF].

Misfit

Heteroepitaxy refers to the growth of a layer onto a chemically different material. Due to the chemical difference they favour different interatomic distances, i.e. their bulk lattice parameters are different. This is shown in figure 1. 30(a). The difference can be expressed by the misfit:

i = a si -a li a li , i = x, y (1.27)
where a s and a l are the lattice parameters in the two directions (x, y) perpendicular to the growth direction. This means that if the layer grows epitaxially, its structure matches perfectly with the substrate. Thus, the layer experiences a homogeneous strain.

Strain energy scales with the volume, it increases with thickness. Above a critical thickness it becomes energetically favourable to release part of the strain by the spontaneous formation of dislocations, shown in figure 1.30(b).

Critical thickness

The existence of the critical thickness, where misfit dislocations appear, was first predicted by Frank and van der Merwe. It was treated theoretically by several authors and confirmed experimentally.

Formula for the critical thickness can be derived by comparing the work that is required to form a dislocation (W d ), and the work that can be gained from the stress field when a dislocation is formed (W m ). The thickness where the work gained equals to the work required, defines the critical thickness.

The geometry of a misfit dislocation is shown in figure 1.31(a). The coordinate system is taken so that the y axis is perpendicular to the surface. The dislocation line lies along the z axis, at the interface between the film and the substrate. Along this axis the strain is uniform. The grey plane is the plane where the dislocation can glide, and it divides the crystal in two, signed as (+) and (-) [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF].

To calculate the work that is required for the formation of a dislocation, the following is considered: a stress free crystal is cut along the glide plane from the surface to the dislocation, and material with radius r 0 is removed, centred on the eventual dislocation line. The path of the cut is shown in figure 1. created by the cut are displaced by on offset defined by the Burgers vector (b) of the dislocation. The energy per unit length that is stored in the material as the result of these operations is the following:

W d (η) = Γ 1 2 T i u i dl, (1.28)
where index i denotes the x, y, z coordinates of the corresponding vectors, Γ is the boundary of the region created by the cuts in the material, l is the arc length along this boundary, and T is the traction required to maintain the imposed displacement. Traction is related to the stress tensor (σ ij ) and the surface normal (n j ): T i = σ ij n j . Evaluation of the integral is lengthy and beyond the scope of this work. More details can be found in reference [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF]. The following expression is a good approximation of the final result:

W d (η) = µ[b 2 x + b 2 y + (1 -ν)b 2 z ] 4π(1 + ν) ln 2η r 0 , (1.29)
where b x , b y , and b z are the components of the Burgers vector, η is the distance of the dislocation from the surface of the crystal. µ is the elastic sheer modulus, it is the property of the material defined as the ratio of sheer stress to the sheer strain. Finally, ν is the Poisson ratio, it is also the property of the material, defined as the transversal strain over the axial strain. Poisson ratio is the measure of the Poisson effect: when a material is compressed in one direction, it tends to expand in the two other perpendicular directions. Most often Poisson ratio is negative [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF].

Next, the work done by the background stress field in forming a dislocation, where the stress is caused by the the misfit, is calculated. The same thought process, shown in figure 1.31(b), is followed, except this time the crystal is strained by misfit, and the stress field is considered to be unaffected by the formation of the dislocation. The work done by the field can be calculated by the same formula as before:

W m (η) = Γ 1 2
T i u i dl.

(1.30)

The difference here is that the stress is the misfit stress, not the stress caused by the dislocation. In this case, the formula for the stress is simply σ m = µ m , where m is the misfit strain from equation 1.27. The result of the integration is the following:

W m (η) = -b x σ m η. (1.31)
Dislocations spontaneously appear in the film when W d (h c r) + W m (h c r) = 0. The full expression for the critical thickness can be found in reference [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF]. An approximation, which is valid when the critical thickness is larger than the magnitude of the Burgers vector is the following:

b 2 x + b 2 y + (1 -ν)b 2 z 8π(1 + ν)b x h cr ln 2h cr r 0 = m h cr b. (1.32)
Positive critical thickness can only be defined, when the misfit and b x have the same sign, which means that only dislocations that relive the strain are allowed [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF]. Equation 1.32 was used to derive the critical thickness of rhenium on Al 2 O 3 . This is presented in section 2.1.3

Growth on a stepped surface and spiral growth

The best way to grow a good quality film is to use a single crystal substrate which was cut along a low energy crystal plane. It is impossible to cut a substrates from a bulk precisely along a certain direction, there will always be a miscut angle, which is usually in the range of 0.1 • . This small deviation from the low energy configuration is going to drive the atoms in the surface region to rearrange themselves and form steps with one atomic height, as shown in figure 1.32. The widths of the steps are equal, and are defined by the angle of the miscut [START_REF] Pimpinelli | Physics of crystal growth[END_REF].

The step edges provide efficient nucleation sites for the adatoms. They allow a so called step flow growth, shown schematically in figure 1.33. This was first described by Burton, Cabrera and Frank before MBE existed. In this growth mode, growth only happens at the step edges, and the terraces move or flow as more and more atoms are deposited [START_REF] Pimpinelli | Physics of crystal growth[END_REF]. Without the presence of steps, the adatoms diffuse on the surface until enough of them meet, and a critical nucleus is formed. The critical nucleus contains the minimum number of adatoms that can be stable on the substrate surface. They can capture further atoms and initiate the growth of islands. The size of the critical nucleus depends on the temperature. At low temperature a single adatom can be stable, at higher temperature two or three or more atoms are needed. These islands grow according to the mode defined by the surface free energies, and when they are big enough they coalesce. Along the line of coalescence defects can easily occur, such as grain boundaries or holes.

Step flow growth overcomes these issues as arbitrary lines of coalescence have no time to form [START_REF] Pimpinelli | Physics of crystal growth[END_REF].

The presence of steps is not enough for step flow growth to occur. If adatoms have no time to reach a step edge before forming a critical nucleus, islands grow on the terraces. Diffusion length defines the length an adatom can travel, before meeting an other adatom. Assuming a dimer is stable, it can be said that the requirement for step flow growth to occur is to have larger diffusion length than step width [START_REF] Pimpinelli | Physics of crystal growth[END_REF].

When discussing growth on a stepped surface, an important effect has to be mentioned: the Schwoebel barrier. Schwoebel barrier is the energy potential an adatom has to overcome when diffusing over a step. An edge dislocation produces a slanting step on the surface, which will act as a nucleation site for the arriving atoms. Nucleating adatoms keep creating steps, as a result the surface will grow in a spiral manner [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF].

Schwoebel barrier is explained in figure 1.34. The adatom is show in grey. The potential felt by the adatom is shown schematically in this figure. The potential has a maximum at the step edge. When an adatom reaches the end of a terrace, to step down to the terrace below, it has to pass through a position where it does not have many neighbours. This is what creates the potential barrier [START_REF] Pimpinelli | Physics of crystal growth[END_REF].

The Schwoebel barrier is felt by an adatom diffusing from a lower to a higher terrace as well. In this case the adatom is in a potential well when it reaches the step edge, as the coordination number is the highest there.

A special case of step flow growth was described by Burton, Cabrera and Frank in 1951 [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF]. It occurs when a screw dislocation reaches the surface, and creates a step. This step will act as a nucleation site, and the adatoms arrange themselves along it. Because the step created by the screw dislocation slopes, and disappears into the crystal, the adatoms perpetually create steps as they nucleate. This is shown in figure 1. [START_REF] Fert | The origin, development and future of spintronics[END_REF]. Growth around such step creates a spiral structure. Topology of the spirals depends on the sign of the screw dislocation that initiated the growth. A single spiral is created by a single dislocation, shown in figure 1.36(a). If there are two dislocations present with opposite signs, separated by a distance larger than 2πρ c , where ρ c is the critical radius, they start to grow independently, and form a double spiral when they overlap. This is shown in figure 1.36(b). If the distance between them is smaller, no spiral growth occurs [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF]. Dislocation pairs of like signs separated by larger than 2πρ c exhibit similar growth to opposite sign pairs. They turn separately until they meet. Following that, they grow as one spiral, as it is shown in figure 1.37 [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF].

When the dislocations of like sign are closer, growth still occurs. These spirals have no intersection point except in the origin, which means that they will grow separately. Turns of both spirals reach the whole area. This statement is true for any number of dislocations within a 2πρ c distance. This case is illustrated in figure 1.38 [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF].

The shape of the spirals are determined by the dependence of the growth rate on crystallographic orientation. In the case when it is independent, the shape is circular, when it is dependent, spirals are deformed into polygons [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF].

Thermal grooving

In polycrystalline thin films grooves can spontaneously develop along grain boundaries at elevated temperatures. This process is called thermal grooving, or dewetting, and was theoretically investigated by W. W. Mullins [START_REF] Mullins | Theory of thermal grooving[END_REF]. He derived the time dependent profile of a surface around a grain boundary during thermal grooving. Two cases were considered: one where the transport of the matter was driven by evaporation-condensation, and one where it was driven by surface diffusion. For both cases partial differential equations are derived, solved and the results are compared.

In the case of evaporation-condensation, it is shown that evaporation is proportional to the surface area. Therefore, the flux of atoms emitted by a curved surface is higher than by a flat surface. On the other hand, the mean free path of the metal atoms in the vapour phase is large, thus the density of metal vapour is equal all across the surface. The result is that less atoms condensate on the curved surface than evaporate, the grain boundary walls shift away from their original position, and the groove deepens [START_REF] Mullins | Theory of thermal grooving[END_REF].

The partial differential equation is derived from the approximation of the Gibbs-Thompson formula, which gives the equilibrium vapour pressure (p = ∆p + p 0 ) of a surface segment with curvature K:

∆p p 0 = K γΩ k B T , (1.33)
where p 0 is the equilibrium vapour pressure of a plane surface, γ is the surface free energy, Ω is the molecular volume, k B is the Boltzmann constant, and T is the temperature. This approximation is valid when p/p 0 is close to 1 [START_REF] Mullins | Theory of thermal grooving[END_REF].

The number of atoms emitted by the surface can be calculated from the equilibrium pressure. It is required that the densities of the vapour over the flat and the curved surfaces are equal. The net loss of atoms of the curved surface equals the difference between the number of atoms leaving the curved surface and the number of atoms leaving the flat surface. From this, the rate of advance of a profile element can be obtained. Using the definition of curvature, the differential equation for the time evolution of the surface profile is the following:

∂y ∂t = Ay , where A = p 0 γΩ 2 (2πM ) 1 2 (k B T ) 3 2 , y = ∂ 2 y ∂x 2 , (1.34)
and M is the molecular mass. This partial differential equation is solved with boundary conditions y(x, 0) = 0, and y (0, t) = tan β = m, where β is the angle of the surface, shown is figure 1.39 [START_REF] Mullins | Theory of thermal grooving[END_REF].

The solution is the following:

y ec (x, t) = -2m(At) 1 2 ierfc x 2(At) 1 2
, where (1.35)

ierfc (t) = ∞ t erfc (u)du = 2 √ π ∞ t ∞ u e -z 2 dz.
The time evolution of the surface profile is plotted in figure 1.40. Parameter A was arbitrarily chosen as 5000, β was 5 degrees. The units along the x, and y axis are a measure of length, the numbers in the legend are a measure of time. When surface diffusion drives the transfer of matter, the partial differential equation is derived from the dependence of the chemical potential (µ) on the surface curvature:

µ(K) = KγΩ.
(1.36)

Nernst-Einstein relation gives the average velocity of the surface atoms in the presence of a chemical potential gradient:

v = - D s k B T ∂µ ∂s = - D s γΩ k B T ∂K ∂s , (1.37)
where D s is the surface diffusion coefficient, and s is the arc length along the profile [START_REF] Mullins | Theory of thermal grooving[END_REF]. Equation 1.37 multiplied by the number of atoms per unit area (ν) gives the surface current. Divergence of the surface current gives the increase of the number of atoms per surface area. From this, the partial differential equation for the surface profile is the following:

∂y ∂t = -B ∂ ∂x 1 (1 -y 2 ) ∂ ∂x y (1 + y 2 ) 3 2
, where

B = D s γΩ 2 ν k B T . (1.38)
An approximation of equation 1.38 was solved. The approximation is valid in cases when m is small, and is referred to as small slope approximation: The boundary conditions were: y(x, 0) = 0, y (0, t) = tan β = m, and y (0, t) = 0. The solution of equation 1.39 is the following function:

∂y ∂t = -By . (1.39)
y sd (x, t) = m(Bt) 1 4 Z x (Bt) 1 4
, where

Z(u) = inf n=0 a n u n . (1.40)
The a n coefficients are:

a 0 = - 1 2 1 2 Γ 5 4 = -0.7801, a 1 = 1, a 2 = - 1 2 3 2 Γ 3 4 = -0.2885,
a 3 = 0, a n+4 = a n • n -1 4(n + 1)(n + 2)(n + 3)(n + 4) (1.41)
The time evolution of the surface profile is plotted in figure 1.41. Parameter B was arbitrarily chosen as 10 10 , and β was 5 degree. The units along the x, and y axis are a measure of length, the numbers in the legend are a measure of time.

For both cases the groove (x = 0) deepens as time passes. The overall shape of the surface profile does not change with time. The most important difference is that while evaporation-condensation profile increases monotonously along the profile line, surface profile that was shaped by surface diffusion shows a local maxima close to the groove, after which it flattens [START_REF] Mullins | Theory of thermal grooving[END_REF]. These results are used to describe our films in section 2.3.

Growth and characterisation of rhenium thin films

In this chapter the growth of rhenium onto single crystal Al 2 O 3 is presented. First the preparation of the substrate, then the evaporation of rhenium is described. From the frequently observed deposition rates, the temperature of the evaporating rhenium is estimated. Next, the epitaxial relationship between the rhenium and the substrate is presented, and the critical thickness of the rhenium is calculated from the misfit strain. In the following section, the effect of temperature on the properties of the film is studied on samples with 3 different thicknesses, then it is shown that rhenium undergoes dewetting, when its thickness reaches approximately 10 nm. Lastly, a model to calculate the temperature of the growing film is presented.

Growth procedure

Preparation of the substrate

Single crystal α-Al 2 O 3 substrates were purchased from Neyco. They were all 0.5 mm thick, and measured either 15 mm x 15 mm or 13 mm x 13 mm in the plane.

Al 2 O 3 is a frequently used substrate material, as many preparation procedures as users can be found in literature [START_REF] Yoshimoto | Atomic scale formation of ultrasmooth surfaces on sapphire substrates for high quality thin film fabrication[END_REF][START_REF] Heffelfinger | Steps and the structure of the (0001 ) α-alumina surface[END_REF][START_REF] Heffelfinger | Mechanisms of surface faceting and coarsening[END_REF][START_REF] Van | Evolution of steps on vicinal (0001 ) surfaces of α-alumina[END_REF][START_REF] Ribič | Behavior of the (0001 ) surface of sapphire upon high-temperature annealing[END_REF][START_REF] Cuccureddu | Surface morphology of c-plane sapphire (α-alumina) produced by high temperature anneal[END_REF][START_REF] Wang | Effect of the environment on α -Al 2 O 3 (0001 ) surface structures[END_REF][START_REF] Scheu | Manipulating bonding at a Cu/(0001 )α -Al 2 O 3 interface by different substrate cleaning processes[END_REF][START_REF] Eng | Structure of the hydrated α -Al 2 O 3 (0001 ) surface[END_REF][START_REF] Oh | Control of bonding and epitaxy at copper/sapphire interface[END_REF][START_REF] Neretina | The role of substrate surface termination in the deposition of (111 ) CdTe on (0001 ) sapphire[END_REF][START_REF] Walters | The surface structure of α -Al 2 O 3 determined by low-energy electron diffraction: aluminum termination and evidence for anomolously large thermal vibrations[END_REF][START_REF] Toofan | The termination of the α -Al 2 O 3 (0001) surface: a LEED crystallography determination[END_REF]. Based on these examples, we have also developed our own predeposition treatment. AFM height image taken of a substrate as received is shown in figure 2.1(a). The surface is covered with particles of various sizes. From analysing the profile of the surface, the height of the larger particles was found to be approximately 10 nm. Number density measured on a 6 µm x 6 µm AFM image was 130 per µm 2 . These small islands can influence the growth of rhenium by acting as a nucleation site. They have to be removed.

Substrates were first washed in an RBS detergent solution purchased from Chemical Products then rinsed with deionised water. Afterwards, they were cleaned with acetone in ultrasonic bath. Finally, they were put in ethanol and dried in nitrogen flow. An AFM height image taken after the cleaning procedure is shown in figure 2.1(b). A few larger particles are still visible but their density is reduced ten fold, to only 10 per µm 2 .

After the cleaning, substrates were placed in a clean quartz tube to be annealed in a muffle furnace in air atmosphere. Quartz at this temperature can get soft, and deform due to creeping. For this reason, we designed a special tube: the inner quartz tube is supported by an outer alumina tube. A drawing of our design is shown in figure 2.2.

The temperature was raised linearly to 1100 • C from room temperature in 7 hours. Substrates were annealed at this temperature for an hour, then the furnace was switched off and let to cool. It took 4-5 hours to reach room temperature.

AFM height image taken after the heat treatment is shown in figure 2.3. As a result For the sample shown in figure 2.5(a), the annealing time was reduced to 30 minutes. Monoatomic steps began to form on the substrate, but small islands can be observed along the edges. The time was not long enough to complete the development of the steps. In case of the example shown in figure 2.5(b), the temperature of the annealing was reduced to 1000 • C. The step edges appear to be sharp, well defined but they are decorated with kinks, and large islands can be observed in-between. The temperature does not appear to be high enough to straighten the steps, and atoms do not have enough energy to reach the edges, so they form islands. If islands grow large enough they coalesce with the step edge, and form a structure similar to a peninsula. On a nominally flat (001) Al 2 O 3 surfaces only steps with single atomic height develop when they are annealed under 1200 • C. The coalescence of steps occurs at higher temperatures [START_REF] Yoshimoto | Atomic scale formation of ultrasmooth surfaces on sapphire substrates for high quality thin film fabrication[END_REF][START_REF] Heffelfinger | Steps and the structure of the (0001 ) α-alumina surface[END_REF][START_REF] Ribič | Behavior of the (0001 ) surface of sapphire upon high-temperature annealing[END_REF][START_REF] Cuccureddu | Surface morphology of c-plane sapphire (α-alumina) produced by high temperature anneal[END_REF]. Our setup was limited to 1100 • C. states that Al is the most stable termination even in O 2 atmosphere at high pressures, and oxygen can only be stable if hydrogen is present at the surface, in which case the termination is hydroxide [START_REF] Wang | Effect of the environment on α -Al 2 O 3 (0001 ) surface structures[END_REF]. We followed the same substrate preparation procedure for all our substrates and the resulting properties of the substrates and the deposited thin films were reproducible. We do not know what the terminations of the substrates are but we expect them to be consistent.

As was described before, in crystalline Al

Evaporation of rhenium

Rhenium has a low vapour pressure, and is therefore difficult to evaporate. As such, it needs to be heated to high temperatures (∼3000 • C) to achieve a reasonable deposition rate.

The first investigation of liquid evaporation of mercury into vacuum was conducted by Hertz in 1882 [63]. He concluded that the evaporation rate of a liquid cannot exceed a maximum value at a certain temperature, and this theoretical maximum is obtained only when as many atoms or molecules leave as would be required to exert the equilibrium vapour pressure (P v ) on the surface, and none of them return. This means that the number of atoms/molecules (dN source ) evaporating from a surface area A source during time dt has to be equal to the impingement rate on the surface corresponding to the pressure inside the chamber (P ):

dN source dt = A source (P v -P ) N A 2πM k B T , (2.1)
where N A is the Avogadro number, M is the molar weight, k B is the Boltzmann constant and T is the temperature measured in K [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF].

The observed evaporation rates are generally below the theoretical maximum. Based on this Knudsen argued that a certain fraction of the molecules contribute to the vapour pressure but not to the evaporation rate. The theoretical evaporation rate should be multiplied by the thus defined evaporation coefficient, α. This form of equation 2.1 is known as Hertz-Knudsen equation. α is measured experimentally, and here is considered to be 1. Later Langmuir showed that the Hertz-Knudsen equation applies to the evaporation from the surface of a solid as well [START_REF] Herman | Molecular beam epitaxy: Fundamentals and current status[END_REF]. [START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF], [START_REF] Carl | Handbook of Vapor Pressure, volume 4: Inorganic compounds and elements[END_REF], and [START_REF] Plante | Vapor pressure and heat of sublimation of rhenium[END_REF].

The vapour pressure can be calculated using the empirical equations 2.2 and 2.3. The parameters and their range of validity for rhenium is listed below.

Between 298 K -2500 K [START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF]:

log P v (Pa) = 5.006 + A + BT -1 + C log T,
where A = 11.543, B = -40726, C = -1.1629.

(2.2)

Between 2480 K -5915 K [START_REF] Carl | Handbook of Vapor Pressure, volume 4: Inorganic compounds and elements[END_REF]:

log P v (Pa) = (A + BT -1 + C log T + DT + E 2 ) • 133.322, where A = -31.5392, B = -3.2254e4, C = 12.215, D = -1.2695e-3, E = 3.7363e-8. (2.
3)

The vapour pressure of rhenium was measured by Plante et al. in a narrow temperature range that overlaps with the range of validity of equations 2.2 and 2.3 [START_REF] Plante | Vapor pressure and heat of sublimation of rhenium[END_REF]. This experimental data was used to check the values given by the equations. The results of equations 2.2 and 2.3, and the values found in reference [START_REF] Plante | Vapor pressure and heat of sublimation of rhenium[END_REF] are shown in figure 2.6. The calculated data from both equations and the experimentally measured data match well. The deposition rate is given by the evaporated particles that reach and stick to the substrate. For the sake of simplicity, the assumption is that the sticking coefficient is 1. Every particle reaching the substrate sticks to it.

The number of particles that reach the surface of the substrate depends on the geometry of the setup. To calculate the arrival rate, the following equation can be used for the Knudsen cell, which is an evaporation cell, where the source material is enclosed, and its vapour can escape through a small hole:

dN sub A sub dt = dN source dt 1 πr 2 cos θ cos (θ + φ), (2.4)
where r is the distance between the source and the substrate, and the angles are shown in figure 2.7 [START_REF] Henini | Molecular beam epitaxy: from research to mass production[END_REF].

In the MBE setup used, an open crucible was employed rather than a Knudsen cell, but equation 2.4 can be used to provide an estimate. The distance between the the substrate and the source is about 40 cm. The angles are small, and therefore considered to be 0 for this calculation.

The deposition rate was calculated using equation 2. Several streaks are visible, caused by the white-hot rhenium particles leaving the charge at high speed.

Several of these droplets were found when the chamber was opened. They are all perfectly round and have a smooth surface. A scanning electron microscope (SEM) image taken on one of them is shown in figure 2.9b. The analysis of the emitted characteristic X-rays induced by the electron beam in the SEM confirmed that the metal ball is indeed rhenium. The shape and the surface of the ball suggests that part of the rhenium is molten during deposition, and that the ejected droplets most likely to come from the liquid phase.

Rhenium on Al 2 O 3

The aim of this section is to study the epitaxial relationship between rhenium and Al 2 O 3 , and to determine the thickness above which dislocations are expected to appear in the film.

The atomic arrangement in the first layers of the growing film mimics the lattice of the substrate, and thus the film is under a strain, induced by the substrate. This strain is called misfit, and its consequences in the relationship between Re and Al 2 O 3 are discussed below.

In figures 2.10(a) and 2.10(b) the lattices of rhenium and Al 2 O 3 are depicted with the epitaxial orientations. positioned on top of the Al atoms, and are thus neighboured by 3 oxygen atoms at each surface site. The number density of the rhenium atoms is higher in the lattice than the aluminium density in Al 2 O 3 , where only 2/3 of the octahedral positions are filled. This is why rhenium can be observed to be positioned in the empty hexagonal spaces as well. These sites have the same oxygen coordination as the Re atoms on top of the Al 2 O 3 octahedra. We note that 3 out of 6 oxygens that are visible around the Re atoms belong to the lower plane of oxygens, and do not coordinate the rhenium atoms.

In figure 2.10(b) the view of the two lattices are shown viewed along the a axis of the substrate. The spacing between two rhenium planes is 0.22 nm.

The rhenium and the Al 2 O 3 lattices match very well. There is an epitaxial relationship between the lattices, which with the Bravais-Miller indices is (0001)Al 2 O 3 //(0001)Re and <2110>Al 2 O 3 //<0110>Re. The two lattices are rotated by 30 • in-plane with respect to each other, which can be observed in figure 2.10(a). The angle of rotation was confirmed using XRD. The Φ scans measured on a film and its substrate are shown in figure 2.11. The (102) reflection of rhenium and the (104) reflection of the substrate were located, and the sample was rotated around the specular (001) direction. (001) axis of the substrate has trigonal symmetry, and indeed the equivalent reflections appear 120 • apart. For rhenium this axis has sixfold symmetry, and its equivalent reflections are 60 • apart. The angular separation between the reflections of the two materials is consistent with the 30 • the rotation.

The 30 • rotation is taken into account when calculating the misfit. Furthermore, the a lattice parameter of the the Al 2 O 3 lattice was divided by 2. The misfit strain at room temperature is the following:

a (RT) = a s /2 -a Re cos 30 • a Re cos 30 • = a s - √ 3a Re √ 3a Re = -0.0043. (2.5)
This value corresponds to a very small misfit strain. As a comparison, ZnSe grown on GaAs has a lattice mismatch of 0.27%, BeTe on GaAs has -0.48%, and both are said to be nearly lattice-matched. The lattice mismatch and the elastic properties of the rhenium defines how thick it can grow in registry with the substrate, without defects [START_REF] Henini | Molecular beam epitaxy: from research to mass production[END_REF].

The negative sign means that the rhenium lattice in bulk has a larger a lattice parameter than the substrate, and therefore when grown pseudomorphically on Al 2 O 3 , is compressed in-plane. As a result of the compression along both in-plane directions, the lattice extends out-of-plane. The strains are connected via the Poisson's ratio ν, which is 0.2894 for rhenium [START_REF] Murli | Ultrasonic equation of state of rhenium[END_REF]:

c (RT) = - 2ν 1 -ν a = 0.0035. (2.6)
The sample is heated during deposition, which means both lattices are expanded. The value of misfit is therefore different at the deposition temperature than it is at room temperature. The high temperature lattice parameters can be obtained from the thermal expansions of the two materials along the a axis, which are the following:

Rhenium from 293 K to 1900 K [START_REF] Touloukian | Thermal Expansion: Metallic Elements and Alloys[END_REF]: The misfit was calculated for a wide temperature range, and it is shown in figure 2.12 by the green curve. Figure 2.12 also shows the thermal expansion coefficients of the two materials along the crystal axis a. The sample temperature is between 700 • C -1000 • C during deposition. The misfit changes one tenth of a percent between room temperature and 1000 • C.

∆L L (%) = -0.195 + 6.513e-4 • T + 5.412e-8 • T 2 -1.652e-11 • T 3 . (2.
The strain caused by the misfit can only be accommodated by the growing film up to the critical thickness. When the critical thickness is reached, dislocations spontaneously appear to relieve the strain. The derivation to obtain the critical thickness was given in section 1.5.2. The obtained formula is an approximation, which is valid until the critical thickness is larger than the magnitude of the Burgers vector. It is the following:

b 2 x + b 2 y + (1 -ν)b 2 z 8π(1 + ν)b x h cr ln 2h cr r 0 = m h cr b, (2.9)
where b x , b y , and b z are the components of the Burgers vector, ν is the Poisson ratio, r 0 is the dislocation core radius, h cr is the critical thickness, and m is the misfit strain.

In equation 2.9 the Burgers vectors of the dislocations are given by their Cartesian coordinates. How to obtain the Cartesian coordinates from Miller or Bravais-Miller indices of directions is explained in appendix C.

Six different Burgers vectors can exist in a hexagonal close-packed system. Four of them have the correct direction to relieve in-plane strain: 1/3 < 1120 >, 1/3 < 1123 >, 1/3 < 1100 >, and 1/6 < 2203 > [START_REF] Hull | Introduction to dislocations[END_REF]. Each of these four Burgers vectors include six equivalent directions, which can be obtained by the permutation of the first three indices. Of these, the ones where the second index is negative, can relieve compressive strain.

The misfit strain as the function of the critical thickness calculated using equation 2.9 is shown in figure 2.13 for the four Burgers vectors. The core radius of the dislocation was chosen to be half the magnitude of each Burgers vector [START_REF] Freund | Thin film materials: stress, defect formation and surface evolution[END_REF]. The horizontal line shows the misfit strain calculated for the temperature during deposition. According to the calculation, rhenium grows pseudomophically up to approximately 10 nm thickness onto the Al 2 O 3 substrate. The critical thickness obtained with Burgers vectors 1/3[1100] and 1/3 [1210] are very close to each other, 10 nm and 13 nm, respectively. These dislocations are expected to be present above the critical thickness.

We observe spirals on films that are thicker than 20 nm. Screw dislocations can cause spirals to grow by creating a step on the surface. To create a step, the Burgers vector needs to have a nonzero final (Bravais-)Miller index. Burgers vector 1/6[0223] gave a critical thickness slightly below 25 nm. These dislocation can be present in the relaxed film and can be responsible for spiral growth.

The fourth Burgers vector 1/3[1213] has too high energy cost to be expected in the films.

Thin film growth

The instruments mentioned below are described in detail in section 1.3.1.

During the deposition of rhenium the substrate has to be heated to provide kinetic energy for the adatoms. There are two ways to heat the substrate, either by infrared radiation or by electron bombardment. Al 2 O 3 is an insulator. If we applied electron bombardment, the sample would become charged because the excess electrons cannot be removed. The substrate cannot be heated with infrared radiation either, because it is transparent in that wavelength range. To overcome these issues 300 nm of tungsten is deposited onto the back side of the substrates.

After, the substrate is mounted on a sample holder with a hole in the middle. This way thermocouple is in contact with the back side, and also, the sample is heated directly, not through the sample holder. After the substrate is mounted, it is transferred to the deposition chamber, where it is degassed for a few hours at approximately 350 • C. Figure 2.14 was taken through a view port of the MBE setup. The sample holder with the sample is mounted on the manipulator inside the chamber, and it is in position for deposition.

Before starting the deposition, the temperature of the substrate is set using the furnace in the manipulator head.

A 10 kV electron gun is used for the evaporation of rhenium. To achieve a deposition rates between 0.1 Å/s and 0.2 Å/s, the electron emission current of the gun is slowly increased to about 150 mA -200 mA. When the deposition rate is stable, the shutter covering the substrate is opened, and the deposition begins.

The deposition rate is monitored, and kept constant by manually adjusting the emission current of the electron gun. The time required to deposit the desired thickness is calculated and measured with a stopwatch. When the thickness is reached, the shutter is closed, and the electron gun is turned off. The temperature of the sample is slowly decreased, the heating electronics are turned off, and the sample is left to cool to room temperature before removing it from the vacuum.

Influence of the growth temperature

The influence of the substrate temperature on the surface topography and crystallographic properties of the thin film was investigated. 7 samples were deposited: two with 25 nm, three with 50 nm, and two with 100 nm thickness. Temperatures during the deposition of all the samples from each thickness group were different. Thicknesses and temperatures are summarized in table 2.1.

At the lowest temperature the current running through the heating filament behind the sample was set to 7.5 A. The thermocouple that touched the back side of the sample measured 800 • C. Based on the model described in [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF], the actual surface temperature is estimated to be 700 • C. At the second temperature the current was set to 8.5 A, the thermocouple measured 900 • C, and according to the model the surface temperature was 770 • C. For the highest deposition temperature, which was only used for sample E, the sample was heated by electron bombardment. 600 V was applied between the heating filament and the sample, and the emission current was set to 50 mA. The temperature of the sample was approximately 1000 • C. Temperature ( • C) 800 900 800 900 1000 800 900

Table 2.1: Thicknesses and deposition temperatures of the samples discussed in this section.

Surfaces of these samples were investigated with AFM, and their crystallography was studied with X-ray diffraction. In this section, the results of these measurements are organized and discussed according to the thickness of the films.

Sample A and sample G was used to fabricate microwave resonators by Dumur et al. From the resonance frequency at a low temperature, the London penetration depth was determined [START_REF] Dumur | Epitaxial rhenium microwave resonators[END_REF]. The two surfaces are similar, both are covered with grains that have two distinct geometries: small ones, with approximately spherical shape, and larger ones with elongated, polygonial shape. The diameter of these grains are very similar also. Measuring 10 of both types, and averaging, it was found that the diameter of the larger ones is (96 ± 28) nm, and the smaller ones is (45 ± 12) nm on sample A. On sample B, the larger ones have diameter (73 ± 13) nm, and the smaller ones have (26 ± 4) nm. Grain sizes are more uniform, their standard deviations are smaller, on the sample which was deposited at higher temperature. This is visible in figure 2.15(b) in case of the small grains, which are almost identical, and appear to form a continuous, smooth layer.

25 nm thick films

Both samples are relatively flat, and aside from a few holes, there are no large deviations in height. However, the surface of sample A is jagged, it is rougher than sample B. The average roughness (R a ) can be measured by the arithmetical mean deviation. The Higher deposition temperature results in a smoother surface, with more uniform grain size. However, on the sample deposited at 900 • C there are still two distinct types of grains. They most likely have different orientations.

XRD θ-2θ measurements θ-2θ scans of both samples are shown in figure 2.16. Both curves were normalised with respect to the (002) peak of rhenium, so that the differences can be read more easily. Both graphs are dominated by the (001), epitaxial orientation, which is signalled by the large, higher order (002) and (004) peaks. There are 5 much lower intensity rhenium peaks, corresponding to 3 different orientations. Peak ( 110) is only present in the lower temperature sample. Orientation ( 101) is featured in both samples. Peaks from this orientation, ( 101) and ( 202), have a slightly lower intensity on the sample B. Finally, the intensity of ( 100) and ( 200) are higher on the sample B. It is confirmed that rhenium grown on the (001) plane of Al 2 O 3 prefers to grow along the (001) direction. Two orientations remain present in the sample grown at 900 • C: (100) and ( 101). This could either mean both these orientations are stable as well, or it could be an anomaly. It could have been caused by contamination on the substrate, which decreased the mean free path of rhenium adatoms, and caused nucleation and growth along these direction. Conclusion can only be drawn after looking at more samples.

The shape of the (002) diffraction peaks appear similar on both samples. To verify this, they were fitted, and the fitting procedure is described below.

A close up of the (002) peak of rhenium measured on sample A is shown in figure 2.17.

The sample was probed with the copper Kα radiation. The intensity of the incoming X-ray beam is composed of two parts Kα1 radiation with wavelength 1.540562 Å, and one part Kα2 with wavelength 1.544398 Å. The substrate peaks are double, which can be observed on the (00 12) reflection in figure 2. [START_REF] Daunt | Superconductivity of rhenium[END_REF]. A slight asymmetry can be observed in case of the rhenium peak as well. For this reason the sum of 2 functions was used to fit the data shown in figure 2.17: one corresponding to the copper Kα 1 radiation, the other to the Kα 2 . As mentioned, the intensity ratio of the two components of the incoming beam is Kα1:Kα2 = 2:1. Thus, the integrated intensity ratio of the respective diffraction peaks (I Kα 1 /I Kα 2 ) has to be 2:1. The integrated intensity of a peak depends on its amplitude and its full width half maximum (FWHM), both of which depend on the resolution of the diffractometer. The resolution of the diffractometer depends on the 2θ angle. The two peaks, corresponding to Kα 1 and Kα 2 , are so close to each other, that resolution can be considered constant in that range. Their full width half maxima are expected to be equal, and the ratio of their amplitudes is expected to be 2:1. In the fitting procedure the ratio of amplitudes were fixed at the expected value, the full with half maxima were set to be equal, and the position of the Kα 2 peak was calculated from the Kα 1 peak using Bragg's law, equation 1.19.

The angular separation between the two peaks increases with the Bragg angle. In the interval shown in figure 2.17 its value is around 0.1 • . A few degrees of misalignment in the experimental setup can cause variations in the second digit after the decimal point of the separation. A parameter ( ) was allowed to correct the position of the second peak in the fitting procedure.

Two model functions are used for the fit of diffraction peaks. These two functions are the Gauss function (G(x)) and the Cauchy or, as it is also known, the Lorentz function (L(x)). The copper Kα emission line of the X-ray tube have a Lorentzian shape. Broadening due to small crystal size is also associated with a Lorentzian shape. However, broadening due microstrains is described by a Gauss function, because microstrain fields often exhibit a normal distribution of lattice spacing values around an average d 0 value. Therefore, diffraction peaks are usually well described by a mixture of these two functions [START_REF] Birkholz | Thin film analysis by X-ray Scattering[END_REF].

The (002) rhenium peaks were fitted with the sum of two Voigt functions. The Voigt function is the convolution of a Gauss (G) and a Lorentz (L) function:

V (x) = ∞ -∞ G(x )L(x -x )dx where (2.10) G(x) = 1 σ √ 2π e -(x-x 0 ) 2 2σ 2 , F W HM G = 2 √ 2 ln 2σ L(x) = 1 πγ 1 + (x-x 0 ) 2 γ 2 , F W HM L = 2γ
The fit of the (002) rhenium peak of sample A is shown in figure 2.17. The individual Voigt curves corresponding to the two wavelengths are also shown. The results of the fits with their standard deviations are summarised in table 2.3.

XRD line profile analysis of epitaxial thin films is not straightforward. The film has a preferred orientation determined by the substrate that will reduce the number of diffraction peaks. A Williamson-Hall plot would allow us to separate size and strain contribution to peak broadening [START_REF] Williamson | X-ray line broadening from filed aluminium and wolfram[END_REF]. In this approach the parameters of each diffraction peak are plotted in a coordinate system with axis FWHM• cos θ and 2 sin θ/λ, and a straight line is fitted across the points. In our case, the film has a single dominant orientation, that results in two peaks which can reliably be fitted. This is not sufficient for the Williamson-Hall plot, this method is not suitable for epitaxial films.

The Warren-Averbach method makes use of the Fourier coefficients of at least two harmonic reflections [START_REF] Warren | The separation of coldâĂŘwork distortion and particle size broadening in xâĂŘray patterns[END_REF]. For this, first, the 2θ scattering angle has to be transformed to 25 nm (002) Voigt fit 800 the magnitude of the scattering vector (q) using the following expression: q = 4π sin θ/λ. The presence of a secondary wavelength makes this transformation uncertain. The analysis that relies on a single line takes advantage of the observation that was mentioned above: size broadening has Lorentzian shape, strain broadening has Gaussian shape. Based on this, the Lorentz and Gauss fractions in the fitted Voigt functions are interpreted to signal effects of size and strain, respectively [START_REF] Th | Use of the voigt function in a single-line method for the analysis of x-ray diffraction line broadening[END_REF].

• C (A) 900 • C (B) Peaks @ 40.462 • ± 0.001 • 40.474 • ± 0.004 • 40.654 • ± 0.007 • 40.64 • ± 0.02 • F W HM G 0.372 • ± 0.006 • 0.31 • ± 0.01 • F W HM L 0.122 • ± 0.004 • 0.127 • ± 0.009 •
Particle size can be determined from the FWHM using the Scherrer equation [START_REF] Scherrer | Bestimmung der größe und der inneren struktur von kolloidteilchen mittels röntgenstrahlen[END_REF]. This equation can be derived from the interference function, given in equation 1.23, and gives a lower limit to the size of cubic shaped crystallites in the direction perpendicular to the reflecting planes. In our case this size is the thickness (t). The Scherrer formula is the following:

t = Kλ FWHM • cos θ , (2.11)
where K is a geometrical factor approximately unity, and FWHM is taken in radians.

Thicknesses calculated from the Lorentzian width of the (002) peaks are 72 nm for sample A and 70 nm for samples B. This cannot be correct, because thickness was measured during deposition, and and is known to be approximately 25 nm.

The Gaussian contribution is thought to carry the strain broadening. Strain is a dimensionless quantity, that describes the variations of interplanar spacings in the crystal relative to the undistorted lattice parameter, d 0 : = ∆d/d 0 . The relationship between line broadening and strain can be obtained by differentiating Bragg's law:

∆d ∆(2θ) = - d 0 2 cot θ → 2 1/2 = ∆(2θ) 2 cot θ.
(2.12)

∆(2θ) is identified as the integral breadth of the Gaussian part of the Voigt function, which can be converted to the FWHM: FWHM G = ∆(2θ)/ √ 2π [START_REF] Birkholz | Thin film analysis by X-ray Scattering[END_REF]. Thus the root mean square of the strains calculated from the (002) reflections are:

2 A 1/2 = 0.004, 2 B 1/2 = 0.003.
(2.13)

The obtained strains are in the range where misfit strain is expected to be.

What we can safely conclude based on the θ-2θ scans presented above is that the shape of the main (002) diffraction peak, ie. arrangement of the lattice planes parallel to the surface, is not greatly affected by temperature during deposition at these temperature values. We can see a slight decrease in the Gaussian width with the temperature increasing (table 2.3), which can mean there are less defects present in sample B. The nature of these defects cannot be established based on these measurement.

The Lorentzian widths are equal within the error bar, which is good sign considering that this width is expected to carry the size component of the broadening, and the two films have the same thickness. However, the calculated thickness is not what we know it is. The Scherrer equation was derived for cubic materials and cubic shaped grains. It is possible that rhenium thin films fall outside of its limits.

Discrepancies could arise when one tries to deconvolve the effect of size and strain on X-ray diffraction peaks. Soleimanian et al. used several methods to extract the crystallite size and strain from the same set of lines. The values they obtained from different methods differed by a factor of 2 or 3, but were of the same order. They did not obtain the same strain value for harmonic reflections either. Voigt profile fitting, the one used above, gave them the largest values [START_REF] Soleimanian | Comparison methods of variance and line profile analysis for the evaluation of microstructures of materials[END_REF].

XRD rocking curve measurements

The rocking curves of the (002) rhenium peaks of sample A and B were measured. Rocking curve measurement probes the angle distribution of the reflecting lattice planes around the lattice normal. The lateral coherence length, ie. lateral grain size also contributes to the broadening.

To compare the widths of the rocking curves of the two samples, the data was fitted using a function which is also a mixture of a Gauss and a Lorentz function, but easier to compute than the Voigt function. This is the Pearson VII function (P (x)), or as also known, the modified Lorentz function, and given by the following equation:

P (x) = 1 1 + ( m √ 2 -1) (x-x 0 ) 2 w 2 m , F W HM P = 2w. (2.14)
This function is a Lorentz function in the m = 1 limit, and a Gauss function in the m → ∞ limit [START_REF] Birkholz | Thin film analysis by X-ray Scattering[END_REF]. The lower part of the rocking curve of the (002) rhenium reflection measured on sample B is shown in figure 2.18(a). The single Pearson VII function that was first fitted to the data is shown in blue, and it does not describe the data well. The tails of the experimental data exceed the tails of the model function. To account for the tail, a Gaussian contribution was added to the Pearson VII. This is shown in red in figure 2. 18(a). This curve describes the data well, however, it is still not perfect. The measured peak is slightly asymmetric due to the two wavelengths. To perfectly describe the rocking curve, the number of functions would need to be doubled. Since the parameters are only used to compare between samples, to describe growth qualitatively, and not to extract quantitative parameters, the blue and the red curves in figure 2.18(a) are both deemed satisfactory.

The rocking curves of both samples A and B are shown in figure 2 In figure 2.18(b) the G + P fit is shown. For both samples one of the functions has a smaller width and a larger amplitude contributing mainly to the peak (peak contribution), and the other function has a larger width and a smaller amplitude contributing to the tail (tail contribution). These functions are not the same for the two samples. Where one is a Pearson VII, the other is a Gaussian. This was taken into account when comparing the parameters. The widths of the tail contribution has decreased from 0.731 to 0.52, the widths of the peak-function decreased over two folds, from 0.384 to 0.178, when the deposition temperature was increased.

The integral of the two components were calculated for both samples, and the peakto-tail contribution ratios are listed in table 2.4. The ratios have a similar value for both samples. As it will be shown in the following sections, fitting of rocking curves measured on thicker samples does not require the addition of a second function. This means that the pronounced tails we see in case of these samples are due to the small thickness of the films. It is possible that it is caused by an intermediate layer of rhenium on the Al 2 O 3 . The volume fraction of this layer is reduced as the thickness grows, thus its effect cannot be observed on the rocking curves of thicker samples.

Conclusion on the 25 nm thick films

AFM study of the surfaces of the two samples revealed that higher deposition temperature results in a smoother surface with more uniform grains. XRD data shows that the dominant orientation is the epitaxial (001). There are three other orientations present in the films, intensities of two decrease with higher deposition temperature, rhenium favours the epitaxial orientation. The width of the rocking curve decreased by half on the film which was deposited at higher temperature. This means that the out-of-plane orientation of the grains are more uniform on sample B, it has lower mosaicity. The surface of sample D is covered with large, even spirals, that are connected to each other by ridges. There are deep holes in between them. The profile of the holes cannot be determined using AFM, as they are too steep. Only the shape of the probe is measured.

50 nm thick films

A double spiral is shown in figure 2.22(b). The profile was measured along its slope, shown with the white line, and is plotted in figure 2. 22(a). It shows several regular steps, and flat terraces. The average step height between consecutive turns extracted from this profile is 0.24 nm, which corresponds to the atomic spacing in the rhenium lattice along the c axis. The spirals on this sample are larger than on sample C, they measure up to 500 nm -600 nm across.

The surface of the third sample, sample E, is shown in figure 2.23, and is very different than the others discussed before. The sample has partially dewetted during the deposition. The film is not continuous, but composed of large islands. The discontinuous nature of the film has been confirmed by transport measurement [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF].

The surface of sample E appears to be very smooth, but in fact it is decorated by steps and terraces. An AFM image is shown in figure 2. 23(b), where the colour scale was set to highlight the topography of the topmost surface. It shows large flat terraces and shape of the AFM tip. The depth, however, can be measured, and it is 40 nm. This suggests that the whole thickness of the film took part in the dewetting process, and formed large islands. On all three samples there are steps with heights corresponding to the spacing of planes in the lattice of rhenium along the c axis, which suggests that these objects have (001) orientation. Higher deposition temperature resulted in larger spirals on sample D, and in the reduction of the uneven grains that were present on sample C. Also, deep holes appeared in between the spirals when deposition temperature was increased. Sample E, where the temperature was further increased partially dewetted. Islands formed that are not connected to each other. There are no signs of spirals, which suggests that dewetting eliminated the dislocations which, we assume, are responsible for spiral growth.

XRD θ -2θ measurements

Standard resolution data. The crystallographic properties of the films were studied with X-ray diffraction. θ-2θ scan of all three samples are shown in figure 2.25. Graphs were normalized to the (002) peak of rhenium to help comparison. The (001) orientation is the dominant in this case as well. Besides, there is a small peak corresponding to the (101) orientation on sample C which is not present in the spectrum of sample D or E. The (100) reflection, and its higher order (200) reflection are only featured on sample D. Finally, the (110) peak gradually decreases as the deposition temperature is increased. Samples E does not show any other orientation but the (001).

The (002) peak of rhenium was was fitted with the sum of two Voigt functions, The Lorentzian widths are increasing with the deposition temperature. The reason behind this increase is unknown. The Scherrer equation (eq. 2.11) used with the Lorentzian width does not give correct results for the film thickness.

The Gaussian widths decrease significantly with the increasing deposition temperature. This is expected, especially in case of the dewetted sample, sample E.

The root mean square of the strains can be calculated from the Gaussian widths using equation 2.12. The following strain values were obtained:

2 C 1/2 = 0.004, 2 D 1/2 = 0.003, 2 E
1/2 = 0.002.

(2.15)

All three of them are in the range where misfit strain is expected.

High-resolution data. Samples C and D were also measured using the Rigaku Smart-Lab high-resolution diffractometer.

In figure 2.26 the θ-2θ scan on the (002) rhenium peak of sample D is shown. The sharp, lower intensity peak at the higher angle side in figure 2.26 is the (006) reflection of the substrate.

On both sides of the largest, central peak oscillations can be observed. The frequency of these oscillations is inversely proportional to the number of lattice planes scattering in phase. The presence of these clear fringes indicates that the layer is highly crystalline, with a well-defined lattice spacing throughout the thickness.

The experimental data was first fitted with the interference function (I(q)), given in equation 1.23. This function describes the scattering by N number of parallel lattice planes with d spacing. To use equation 1.23, 2θ angles had to be converted to scattering vector q using the following formula:

q = 4π sin θ λ . (2.16)
The (006) peak of the substrate was included in the fit, a Lorenz function was used to describe it.

The fit of the interference function is shown with the blue line in figure 2.26. It looks almost perfect at the scale of the plot: the periodicity matches the data, the intensity of each peak looks correct, and also the shoulder that appear on the side of the substrate peak is well described. However, upon magnification, the shortcomings of this model appear. This is shown in figure 2.27. A definite broadening can be observed on the main peak, and on the fringes too. This is the reason that while the positions of the minima align well between the measurement and the blue curve, the maxima are slightly shifted. The intensities at the minima are at larger values on the measured data, while the model function minima go to zero.

To account for the broadening, disorder was introduced in the model lattice in the form of a Gaussian distribution of crystal plane spacings. This concept is shown in figure 2.28. The lattice is composed of N lattice planes, all with slightly different spacings. To achieve the Gaussian distribution of lattice parameters, a constant ∆d multiplied by a random number was added to the average d 0 . This random number was chosen from a Gaussian distribution centred on 0, with standard deviation 1.

The structure factor was then calculated by adding up the scattered plane wave from each lattice planes with a phase factor, which was calculated from the distance the radiation travels in the crystal. The scattered intensity was obtained by taking the absolute square of the structure factor. The formula describing this is the following:

I mod (q) = 2000 i=1 1 + e -iqd 1 + e -iq(d 1 +d 2 ) + • • • + e -iq(d 1 +d 2 +d 3 +•••+d N -1 ) 2 ,
(2.17)

where d i = d 0 + ∆d• rand(0, 1). Due to the random number generator, the resulting curves are not consistent. To improve this, a summation running from 1 to 2000 was introduced. This can have a physical interpretation as well: it can account for inhomogeneities that are inevitable in the sample. This aspect is not investigated further here, the number of sampling was increased until the resulting curves were consistent. Function 2.17 was fitted by hand, because it is a demanding calculation. Parameters were adjusted to the last digit until improvement could be observed on the fit and in the value of χ 2 . The errors listed in table 2.6 were taken as 1 on the last digit. Results and standard deviations of the parameters obtained from both fits are listed in table 2.6. 50 nm (002) fit by equation 2.17

800 • C (C) 900 • C (D) N 215 ± 0 204 ± 0 d 0 (0.22302 ± 1e-6) nm (0.22303 ± 1e-6) nm F W HM d (0.0093 ± 1e-4) nm (0.0065 ± 1e-4) nm Table 2
.6: Parameters of the fit of the high-resolution (002) rhenium peaks with the modified interference function, equation 2.17.

The fit is shown by the red curve in figure 2.26 and 2.27. It perfectly describes the intensity variations of the fringes, and the broadening as well.

In figure 2.29 the (002) peaks of both samples are shown. The fringes are denser on sample C, more lattice planes take part in the scattering process. Also, the intensity difference between the minima and the maxima is smaller on this sample, which is a sign of a more disordered film.

It is visible on figure 2.29 that the disorder introduced in the system does not destroy the fringes on the diffraction pattern. The agreement between the thus modified interference function and the experimental data is improved with the disorder. The widths and positions of the peaks and fringes, and the vertical positions of the minima and maxima are all well matched. The root mean square strains can be calculated from the full width half maxima using the following formula:

2 1/2 = FWHM d /d 0 • 1/(2 √ 2 ln 2).
The obtained strain values are the following: The strain values calculated here are a magnitude larger than the ones obtained from the low-resolution measurement, shown in equation 2.15. Thickness values confirm the thickness expected from the quartz balance measurement during deposition.

Simulation of the standard resolution data. To verify the validity of the modified interference function, the standard resolution data was simulated from the fitted modified interference function curves.

The substrate is a high quality single crystal, and its (006) peak is very close to the (002) peak of the rhenium. The (006) reflection of the Al 2 O 3 should only show the instrumental broadening. As an approximation, its Lorenzian fit shifted to the rhenium (002) peak position (θ 1 ) was used as the resolution function of the instrument.

The modified interference function fitted to the high-resolution (002) peak of sample D (I film(002) mod (θ, θ 1 )), and the Lorentz function fitted to the Cu Kα 1 (006) peak on the standard resolution data (L sub(006) (θ, θ 1 )) were convolved. The result of the convolution is what would have been measured with the standard resolution instrument, using only the Cu Kα 1 radiation (S λ 1 (θ, θ 1 )). The Cu Kα 2 contribution (S λ 2 (θ, θ 2 )) needs to be added. The result of the convolution was divided by two, as dictated by the intensity ratio of the two wavelengths, and then it was shifted by the angular difference corresponding to the secondary wavelength (∆θ). This is the Kα 2 contribution. The two parts were then added.

The following formulation attempts to summarise the procedure described above:

S(θ) = S λ 1 (θ, θ 1 ) + S λ 2 (θ, θ 2 )
, where (2.20)

S λ 1 (θ, θ 1 ) = ∞ -∞ I film(002) mod (θ, θ 1 )L sub(006) (θ -θ , θ 1 )dθ , and
S λ 2 (θ, θ 2 ) = 0.5 S λ 1 (θ, θ 1 + ∆θ).
All the functions above have two arguments, the first argument (θ) is a running parameter, the second argument refers to the position of the maximum. The thus computed curve (S(θ)) should look similar to the (002) reflection measured by the low-resolution instrument.

The result of this simulation is shown in figure 2.31. The fringes are still visible, but significantly damped. The shape of the simulated curve matches well the shape of the measured data. The width of the simulated (002) is narrower than the measured, but the difference is small. The shape of the low-resolution data is reproducible from the high-resolution data.

We see the signature of lattice distortions on the high-resolution data. These measurements do not tell what the source of the lattice distortion are, or where in the lattice they may be. The lattice mismatch can only account for some of the strain, but it only affects the bottom part of the film, in proximity of the substrate. Spirals cover the surface of the sample D, and some can be observed on sample C too. This suggests, according to the theory of spiral growth [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF], the presence of screw dislocations. The strain field of a screw dislocation is proportional to b/(4πr) with a cosine or sine coefficient depending on the component of the strain. Here b is the Burgers vector of the dislocation, which is in the range of the lattice parameter, and r is the distance measured from the core of the dislocation. The strain a few nanometers away from a dislocations can be in the range calculated in 2.18, thus dislocations could account for the at least some of the strain.

XRD rocking curve measurements

The rocking curves of samples C, D, and E were measured. All three of them are shown in figure 2.32. They were fitted with Pearson VII function, equation 2.14. The fits are also shown in figure 2.32.

A single Pearson VII function describes these rocking curves well, the addition of a second function was not necessary this time. The parameters of the fits are listed in table 2.7.

The rocking curves are becoming significantly narrower with the increasing deposition temperature. This is consistent with the larger objects observed on the AFM images of sample D and E. This result also suggests that the mosaicity of the film is decreased.

Conclusion on the 50 nm thick films

AFM study of the surfaces showed spirals on both C and D films. On the lowest temperature sample several grains could be observed also. These grains disappeared from sample D. The size of the spirals grew 5 fold when 900 • C temperature was applied, compared to 800 • C. The third sample, which was deposited at he highest temperature dewetted, and formed atomically flat islands, which are not connected to each other. XRD data shows that the dominant orientation is the epitaxial (001). The three other orientations, which were observed on the 25 nm thick samples also, had low intensities, which decreased with higher deposition temperature. The only exception was the (100) peak, which was present only on sample D. The high-resolution (002) curves measured on samples C and D evidenced less disorder in the higher temperature sample. The width of the rocking curves decreased with increasing deposition temperature.

100 nm thick films

AFM study of the surfaces The structure of the surfaces look very similar: they are both covered with spirals that have atomic step heights. Spirals are connected to each other by ridges, and there are deep holes around them. On sample F there are a few grains in between the spirals, and the spirals are slightly smaller. However, the difference in size is not as pronounced as it was on the 50 nm samples. While spirals on sample G measure about 200 nm -400 nm, on sample F they measure about 100 nm -300 nm.

XRD θ-2θ measurements

The θ-2θ scan of both samples are shown in figure 2.34. The graphs were normalized to the (002) peak of rhenium. The dominant orientation on both films is (001). Aside from that, there is a low intensity (101) peak on sample E that is not present on sample F. There is also a low intensity [START_REF] Matthias | Superconductivity and electron concentration[END_REF] peak on both samples. The broad unindexed peak on sample F between 60 • and 70 • does not belong to rhenium nor to the substrate. It is possible that that it is a reflection from ReO 2 .

In figure 2.35(a) a close up of the (002) Re and (006) Al 2 O 3 peaks of both samples are shown. The peaks look almost identical. Neither reflection can be described by the sum of two Voigt functions (equation 2.10) well because of the clear asymmetry of the peaks. Asymmetry is expected as they were measured using two wavelengths. However, on these samples, the angles of the higher and lower intensity contributions are reversed. It appears that they have a lower intensity contribution towards the lower angles. The unusual shape could be due to the relaxation of the film. As the film grows the lattice spacing gradually becomes what it is in bulk rhenium. Adjusting the function to describe such a situation would require the addition of several extra parameters, which would reduce the reliability of the fit.

Samples F and G were also measured using the Rigaku SmartLab high-resolution diffractometer. The measured curves are shown in figure 2. 35(b). The interference function, equation 1.23, was used to fit the data. Parameters are listed in table 2.8. This model has the same shortcomings as was seen before. The peaks are broad on both of these samples, and the minima are shallow. However, the number of scattering planes and average lattice parameter can accurately be determined.

The thickness of the films calculated from the fit parameters are:

t F = 99 nm, t G = 86 nm. (2.21)
The thicknesses are in agreement with what was measured during deposition. An attempt was made to fit the high-resolution data with the modified interference function. In this case, the model fails to accurately describe the intensity variations of the experimental data and the broadening of the main peak. The reason why this simple model fails could be that there are defects in the film that cannot be described by a single Gaussian distribution of lattice planes.

Even without the parameters to compare, it is visible in figure 2.35(b) that the fringes are more damped on sample F than on sample G. This suggests that the film which was deposited at lower temperature (sample F) is more disordered.

XRD rocking curve measurements

The rocking curves of the 100 nm samples were fitted with the Pearson VII function, equation 2.14. The parameters of the fit are summarised in table 2.8.

The rocking curve of the sample which was deposited at higher temperature is narrower. This is consistent with our previous observations, and with the presence of larger objects on the surface.

Conclusion on the 100 nm thick films

The trend observed in case of the 25 nm thick films and the 50 nm thick films continues with 100 nm thick films. AFM revealed that spirals decorate the surface of both films. The spirals grew in size with higher deposition temperature, however, the difference is not as significant as for the 50 nm thick film. XRD data shows that the dominant orientation is the epitaxial (001). Two additional orientations appear with low intensities. Only one of them persists on the higher temperature sample. On the lower temperature sample there is a broad, unidentified peak. It is possible that it comes from an oxide of rhenium. No difference can be observed between the standard resolution (002) peaks, however, they have a distinct asymmetric shape, which might be due to the relaxation of the films. Fringes of the high-resolution data are more pronounced on the higher temperature sample, which indicates less disorder. We could not confirm this with the modified interference function. The width of the rocking curves decreased with increasing deposition temperature for these films as well.

Conclusions on the effects of the temperature

Higher deposition temperature resulted in more uniform grain sizes, smoother surface on the 25 nm thick films, and larger spirals on the 50 nm and 100 nm thick films. Overall, surfaces deposited at 900 • C appear more homogeneous. However, with the spirals, holes appeared also. The presence of holes is explained in the following section.

The 50 nm sample that was deposited at 1000 • C dewetted, which resulted in a surface covered with large, atomically flat islands, comparable to a mesa landscape. Every sample had a single dominant orientation, (001), in accordance with the substrate lattice. Several other orientations appeared on the graphs, but for almost every thickness, their intensity decreased or vanished with increasing deposition temperature.

The detailed study of the (002) reflections revealed that for every thickness the higher temperature deposition resulted in less disorder in the lattices. Less disorder is indicated by the decreasing Gaussian FWHMs of the Voigt functions. These values are shown as the function of temperature in figure 2.36(a). Disorder was quantified for two of 50 nm thick films (samples C and D), by the introduction of a distribution of lattice parameters. As was shown in figure 2.30, the distribution was narrower for the higher temperature film.

For every thickness the rocking curves of the higher temperature samples were significantly narrower. The Pearson VII FWHM values obtained from the fits are shown as the function of temperature in figure 2.36(b). The improvement of rocking curves is consistent with the larger objects observed on the AFM images, and they also indicate that the mosaicity of the epitaxial grains is reduced by higher temperature deposition.

Thermal grooving of the surface

During the deposition of the rhenium thin films, the crystallography of the surface is monitored by reflection high energy electron diffraction. The technique is described in chapter 1.4.1. Prior to evaporation, Kikuchi lines corresponding to the lattice of the Al 2 O 3 can be observed on the screen. As the rhenium deposition starts, the Kikuchi pattern gradually changes to broad rings on a diffuse background, indicating the growth of crystalline islands with different orientations. Then spots appear, indicating 3D growth. When the thickness reaches approximately 10 nm, the RHEED pattern changes again, suddenly rods appear. This is a sign that electrons are diffracted by a volume with single orientation and flat surface. The depth and the true shape of the holes that interrupt the ridges cannot be determined. Due to the restrictions of AFM imaging, these holes reflect the shape of the probe. They may reach all the way to the substrate.

Many of the ridges show a slight increase in height around the holes, as shown by two examples in figure 2.39(b). Height profiles were extracted along the paths that are drawn on the insets of figure 2.39(a) for this plot.

The patterning that can be observed in figure 2.37, and is highlighted in figure 2.39 resembles curves of thermal grooving, where the transport of the matter was driven by surface diffusion. Mullins' theory of thermal grooving was described in chapter 1.5.4. The theoretical profile, that develops along a grain boundary in case of surface diffusion, also shows a maxima. The equation of the surface profile (y sd (x, t)) was given in equation 1.40, and was used to fit the height graphs extracted from the AFM data.

An example of a fit is shown in figure 2.40 with the parameters. The theoretical curve describes the shape of the measured profile well, our observation is consistent with the theory of thermal grooving. However, the atomic steps, that can be observed on the tail of the measured data cannot be reproduced. Mullins' model is a continuous model, and cannot account for discontinuities, such as steps, that develop on a low energy crystal surface at high temperatures.

One of the parameters of the fit was (Bt)

1 4 , which equals to the following:

(Bt)

1 4 = D s γΩ 2 νt k B T 1 4 , (2.22)
where D s is the surface diffusion coefficient, γ is the surface free energy, Ω is the molecular volume, ν is the number of atoms per unit area, t is the time, k B is the Boltzmann constant, and T is the temperature.

From the (Bt) 1 4 parameter, the surface diffusion coefficient of rhenium can be determined, and compared to the value found in reference [START_REF] Goldstein | Atom and cluster diffusion on re[END_REF]. t was taken as the time it took to deposit the sample, approximately 1000 seconds. The growth temperature was 1150 K. The molecular volume is the volume of a rhenium atom, which can be calculated from the atomic radius: Ω = 3/4 πr 3 = 3/4 π(0.137 nm) 3 = 6.059 • 10 -3 nm 3 . The number of atoms per unit area was calculated from the rhenium hexagonal closed packed unit cell. Considering (001) orientation, the surface is covered with hexagons, and rhenium atoms are placed in the corners, and in the middle of each hexagon. The area of one such unit can be obtained using the lattice parameter a:

A hexa = 3 √ 3/2 a 2 = 3 √ 3/2 (0.276 nm) 2 = 0.198 nm 2 .
All rhenium atoms on the corners are shared by three hexagons, so there are 6 • 1/3 + 1 = 3 atoms on the area calculated above. The number of atoms per unit area can be calculated by a division:

ν = 3/A hexa = 15.147 nm -2 .
The value of the surface free energy can be found in references [START_REF] Tyson | Surface free energies of solid metals: Estimation from liquid surface tension measurements[END_REF], [START_REF] De Boer | Cohesion in Metals[END_REF], and [START_REF] Vitos | The surface energy of metals[END_REF]. Tyson and Miller calculated the surface free energy from liquid surface tension measurement data. They obtained a value of 3.626 Jm -2 [START_REF] Tyson | Surface free energies of solid metals: Estimation from liquid surface tension measurements[END_REF]. Surface free energy found in reference [START_REF] De Boer | Cohesion in Metals[END_REF] is in good agreement with Tyson and Miller, 3.600 Jm -2 . In a more recent article, Vitos et al. determined the surface free energy of low-index surfaces of 60 metals, including rhenium, using density functional theory [START_REF] Vitos | The surface energy of metals[END_REF]. For the (001) surface of rhenium they found a value of 4.214 Jm -2 .

The value reported by Vitos et al. was used to determine the surface diffusion coefficient from the averaged (Bt) 1 4 parameters. We obtained the following value:

D s = 4.06 • 10 -12 cm 2 /s.
(2.23)

It can be determined from the surface diffusion coefficient how far an atom can travel in one second (λ = √ D s t). λ is approximately 20 nm/s. This is in good agreement with the width of the atomically flat terraces, which measure a few tens on nanometers across.

Temperature dependence of the surface diffusion coefficient of the Re(001) surface was measured by Goldstein and Ehrich in the 210 K -235 K temperature range [START_REF] Goldstein | Atom and cluster diffusion on re[END_REF]. Temperature dependence of the diffusion coefficient follows the Arrhenius law:

D s (T ) = D 0 e -Ea k B T , (2.24)
where E a is the activation energy. The parameters reported in reference [START_REF] Goldstein | Atom and cluster diffusion on re[END_REF] are the following: E a = 11.11 ± 0.43 kcal/mol and D 0 = 6.13(•2.6 ± 1) • 10 -6 cm 2 /s.

(2.25)

Using these values, the surface diffusion coefficient was calculated at the temperature of the deposition. It is plotted in the relevant temperature range in figure 2.41. According to this, surface diffusion coefficient should be in the order of 10 -8 cm 2 /s at 1000 K. This value corresponds to a λ of 1µm/s. 1µm is much larger than the size of the terraces on the film, and the reported surface diffusion coefficient is four orders of magnitude higher, than what we obtained.

The authors of reference [START_REF] Goldstein | Atom and cluster diffusion on re[END_REF] conducted their experiments at much lower temperatures, at a relatively small temperature range: 210 K -235 K. Extrapolated values in the region of 1000 K should be taken with caution.

In our experiment, the sample was an extremely thin film, 15 nm, and the full thickness dewetted. Holes that developed as a result probably reach the substrate. Their profile cannot be determined but it can be assumed, that they have a similar stepped structure as observed on the ridges. Steps provide an energy barrier known as the Schwoebel barrier against the atoms diffusing through them. The effect was described in section 1.5.3. This can lower the diffusion coefficient we obtain. Lastly, the theoretical curve cannot account for the steps observed on the surface, but describe the overall shape well. The results should be viewed as qualitative due to the restrictions of Mullins' theory. He based his paper on the observations in copper samples that were polycrystalline and bulk. Our rhenium samples, on the other hand, had thickness about 15 nm. One of his assumptions was that the properties of the interface are independent of the orientations of the crystals. The orientations of the rhenium grains are very close to a low energy surface (001), thus this assumption cannot be valid in our case. Recrystallisation of the whole sample competes with thermal grooving. Recrystallisation to a low-index orientation flattens the surface. On the low-index surface steps develop, which provide a diffusion barrier, and stop the process of thermal grooving. The step structure can be observed on the ridges, highlighted in figure 2 That the dewetting process is accompanied by recrystallisation is confirmed by the transformation of the RHEED pattern during growth. The initial concentric rings correspond to grains with random orientation, and the regularly spaced rods that appear after, to a single orientation and a flat surface. X-ray diffraction data acquired on the sample further confirms that the layer has a single orientation, which is the epitaxial (001) orientation. The high-resolution θ-2θ scan around the Re(002) peak is shown in figure 2.42. The sharp, unresolved peak between 41 and 42 degrees corresponds to the Al 2 O 3 (006) orientation, the broader peak between 40 and 41 degrees is the rhenium (002) peak. A few fringes can be observed on both sides of the rhenium peak which is a sign that the X-ray beam was diffracted by well-arranged, parallel lattice planes. The interference function, equation 1.23 was fitted, to determine the number of lattice planes (N ) and their spacing (d). N is related to the periodicity of the fringes, d is to the angular position of the central peak. The fit is not shown in figure 2.42, because it fails to describe the intensity ratio of fringes and the main peak. The film is very thin, so the fringes are damped. However, this does not affect the position of the peak, and the periodicity. Parameters obtained from the fit are also shown in figure 2.42. By multiplying N and d, the thickness of the layer can be determined. We obtain (15.5 ± 0.5) nm.

Conclusion thermal grooving

The surface which develops as the result of the dewetting can be described by Mullins' theory of thermal grooving, where the matter was driven by surface diffusion. Dimensions observed on the topography are consistent with the surface diffusion coefficient we obtain from Mullins' model. Changes in the RHEED pattern indicate the coalesce of initial islands and the full recrystallisation of the film. This was confirmed by high-resolution X-ray diffraction.

We believe that dewettting and recrystallisation happens on most rhenium samples when they reach approximately 10 nm -15 nm thickness. This leaves behind a flat surface with a single orientation, and deep holes and ridges. Initially holes observed on the samples with spirals were thought to be the result of impurities on the substrate [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. We now believe that the holes are the result of dewetting during the early stages of growth. Spiral will grow onto the terraced ridges, that can be observed in figure 2.37.

Thermal transfer during crystal growth

Measurement of temperatures in a vacuum chamber is not a trivial task. Its difficulties were discussed in chapter 1.3.1.

To estimate the temperature of the surface of the growing rhenium, a model was developed by Delsol [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. His model is outlined here, and modifications are introduced, that can explain the thermal grooving that occurs when thickness of 10 nm -15 nm is reached by an increase in temperature.

Theory outlined here is described in more detail in reference [START_REF] John | A Heat Transfer Textbook[END_REF].

Elements of the model

Definitions

Thermal radiation is modelled by the ideal radiator, the black body, which absorbs all and reflects none of the radiation arriving at its surface. Planck predicted the emitted power flux (monochromatic emissive power) black body at temperature T at wavelength λ:

M λ,bb (λ, T ) = 2πhc 2 λ 5 1 exp hc k B T λ -1 , (2.26)
where h is the Planck constant, c is the speed of light in vacuum, and k B is the Boltzmann constant.

The total emissive power is obtained by an integration over the wavelengths, and its temperature dependence is given by the Stephan-Boltzmann law:

M bb (T ) = ∞ 0 M λ,bb (λ, T )dλ = σT 4 , (2.27)
where σ is the Stephan-Boltzmann constant. Thermal radiation of a real body is given by a comparison to the black body. Emittance ( ) is the ratio of the emitted radiation by the real and the black body:

λ (T ) = M λ (λ, T ) M λ,bb (λ, T ) and (T ) = M (T ) M bb (T ) . ( 2

.28)

Thus the Stephan-Boltzmann law for a real body is modified as follows:

M (T ) = σT 4 . (2.29)
In general, emittance is the function of the wavelength. A body whose emittance is independent of the wavelength is a grey body ( = λ ).

Besides the radiations that is emitted by a body, it is also important to discuss how it interacts with radiation that arrives at its surface. This is shown in figure 2.43. To simplify the problem, let us consider the incoming radiation to be 1. The fraction of the radiation that is absorbed is called absorbance (α). ρ is reflectance, and it measures the fraction that is reflected, and τ is the transmittance, gives the portion that is transmitted. In equilibrium, the following condition is fulfilled:

1 = α + ρ + τ → α = 1 -τ -ρ.
(2.30)

Kirchhoff's law connects the absorbance and the emittance. It states that a body in equilibrium absorbs as much energy as it emits in every direction and at each wavelength:

λ (T, θ, φ) = α λ (T, θ, φ), (2.31)
where θ and φ are angular coordinates. When the surface is diffuse, emittance and absorbance does not depend on the direction. Furthermore, if the body is grey, wavelength dependence can be neglected as well. Kirchhoff's law then simplifies to the following:

(T ) = α(T ).
(2.32)

Oppenheim's electrical analogy

An analogy to electric circuits was developed to study heat exchange between grey diffuse bodies by Oppenheim. Two new quantities need to be defined. Irradiance (H) is the flux of energy that irradiates the surface, and radiosity (B) is the flux leaving a surface. The flux of energy leaving a surface is the sum of reflected irradiance and the emitted flux:

B = ρH + σT 4 .
(2.33)

The net flux leaving a surface can be expressed as

Q = B -H = ρ σT 4 - 1 -ρ ρ B. (2.34)
If the body is opaque (τ = 0) and grey, using equations 2.30 and 2.32, equation 2.34 takes the shape of Ohm's law:

Q = σT 4 -B 1- , (2.35)
where Q takes the place of the current, (σT 4 -B) acts as the potential difference, and 1-is the resistance. This analogy makes heat transfer problems easier to handle. For example, heat transfer between two planes can be described as two resistors connected in series.

Heat conduction

The above analogy is restricted to planes with diffuse, opaque, grey surface. Our system consists of 4 parts: furnace, tungsten, substrate-rhenium, and the chamber wall. The approximation can be valid to all but one part: the substrate. Al 2 O 3 is transparent not opaque, and thick relative to the tungsten and rhenium. Heat conduction through the substrate has to be considered.

Heat transfer between two surfaces with temperatures T 1 and T 2 can be expressed as follows:

Q C = T 1 -T 2 t/k , (2.36)
where t is the distance between the two surfaces (thickness), and k is the thermal conductivity of the material.

Heating of the sample in UHV

The sample is heated with a tungsten filament that is located behind it. The setup is shown in figure 1.9. It can be operated in two modes: either emits infrared radiation as a result of Joule heating, or a voltage is applied between the filament and the sample, and electrons are emitted and bombard the backside of the substrate.

When the filament is heated by a current, the dissipated power is the product of the resistance of the wire and the square of the current:

P = R(T )I 2 .
(2.37) Some of this heat is lost through the hooks that keep the wire in place. It is estimated that 30% of the power heat the sample.

When electron bombardment is applied, it is assumed the all the power carried by the electrons heat the sample, thus the power is the product of the voltage applied between the filament and the sample and the electron current that is extracted from the filament (I e ):

P = U I e .
(2.38)

Heat transfer during growth

Delsol used the elements discussed above to build a model to calculate the temperature of the rhenium surface during growth. The model is shown in figure 2.44. All the parts of the system was assumed to be an infinite plane. The plane noted with F is the furnace. Besides radiosity, which is the result of the hot filament, Q E has to be included in the equations for experiments when electron bombardment is applied. W refers to the tungsten backing on the substrate. SRe is the substrate and rhenium, which is considered as one unit, and conduction of heat through the substrate (Q C ) is included in the model. Finally, B denotes the wall of the vacuum chamber ('bâtiment'), which is at room temperature.

The problem has three unknowns: the temperature of the tungsten (T W ), the temperature of the substrate-rhenium (T SRe ), and the heat flux between the surfaces, which in equilibrium have to be equal (Q). Three equations can be be written down to define Q using the radiosities of the surfaces:

Q = B f -B f W + Q E , (2.39)
Q = B SRe W -B W SRe + Q C , (2.40)
Q = B b SRe -B b .
(2.41)

The radiosities have to be expressed as the function of only the three unknown. To do this further equations have to defined. The heat exchange on each surface also have to be equal to Q. Writing these down as the difference between radiosity and irradiance gives further 6 equations.

Radiosity of a surface is the sum of the thermal radiation due to its temperature ( σT 4 ), the reflected irradiance (ρH), and transmitted irradiance (τ H). All except the CHAPTER 2. GROWTH AND CHARACTERISATION substrate-rhenium is considered opaque, which means τ = 0. The radiosities can then be expressed as follows:

B W SRe = ρ SRe H W SRe + SRe σT 4 SRe + τ SRe H b SRe , (2.42)
B b SRe = ρ SRe H b SRe + SRe σT 4 SRe + τ SRe H W SRe , (2.43)
B f = ρ f H f + f σT 4 f , (2.44)
B b = ρ b H b + b σT 4 b , (2.45)
B f W = ρ W H f W + W σT 4 W , (2.46)
B SRe W = ρ W H SRe W + W σT 4 W .
(2.47)

Using the 6 heat exchange equations and equations (2.42) -(2.47) the equation system (2.39), (2.40), (2.41) can be expressed as the function of only the three unknowns. The derivation is long but not complicated. It is shown in detail in appendix D. The final form of the equation system is the following:

Q = σT 4 f -σT 4 W 1 + R f + R W + Q E , (2.48)
Q = σT 4 W -σT 4 SRe 1 + R W + R SRe + r SRe 1 + R W + R SRe + 1 Q C , (2.49)
Q = σT 4 SRe -σT 4 b 1 + R B + R SRe - r SRe 1 + R B + R SRe Q C , (2.50)
where the following notations were used:

R f = ρ f f , R b = ρ b b , R W = ρ W W , R SRe = ρ SRe -τ SRe SRe + 2τ SRe , r SRe = τ SRe SRe SRe + 2τ SRe .
Substrate and rhenium was is treated as a single object. The common transmittance and emittance was calculated as follows:

τ SRe = τ S τ Re , SRe = (1 -τ Re ) Re + τ Re S .
(2.51)

Thermal and optical properties of the materials

Thermal conductivity of the substrate Thermal conductivity of Al 2 O 3 is listed in reference [START_REF] Touloukian | Thermal Conductivity, Nonmetallic Solids[END_REF]. It was fitted with a 5th degree polynomial. The fit is given by the equation below:

k Al 2 O 3 (T ) = 97.155 -0.32723 • T + 5.3582e-4 • T 2 -4.8283e-7 • T 3 + +2.2971e-10 • T 4 -4.4808e-14 • T 5 ,
where the temperature is measured in Kelvin.

Optical properties of the substrate-rhenium plane and tungsten

Transmittance be calculated from the complex refraction index (κ), which can be found tabulated for rhenium in reference [START_REF] Edward | Handbook of Optical Constants of Solids[END_REF]. In the calculation the 0.1 eV -2 eV energy range was used which corresponds to a wavelength range of 0.62 µm -12.4 µm. This is the lower (in wavelength) end of the infrared range.

If we assume that the total emissive power (equation 2.27) falls on the surface of a material with complex refraction index κ, the intensity that persists down to thickness z is obtained as follows:

M λ,t (z) = M λ,bb e -4πκz λ , (2.52)
Transmittance of material with thickness d can then be expressed as the ratio of the total transmitted intensity through depth d and the total emissive power (equation 2.27):

τ = M λ,t (z = d) dλ M bb . (2.53)
Emittance of rhenium was measured by Marple [START_REF] Marple | Spectral emissivity of rhenium[END_REF], but the temperatures they worked at (>1500 • ) are much higher than what we can achieve in the vacuum chamber. Their range of wavelength was 0.4 µm -3 µm. In the model by Delsol the following expression was used to calculate the emittance of rhenium [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]:

Re (T ) = -0.18906 + 4.9151e-4 • T -1.5979e-7 • T 2 + 1.8357e-11 • T 3 , (2.54)
where the temperature is measured in Kelvin. The emittance curve is shown in figure 2.45 with the solid line. The three red dots are the emittance values calculated from the curves found in reference [START_REF] Marple | Spectral emissivity of rhenium[END_REF]. The two datasets are in agreement.

Emittance is related to the full surface area of a body. Our thin films are smooth on a nanometer scale, thus their total surface area is smaller than the sand blasted rhenium sheet which was used for emissivity measurements by Marple [START_REF] Marple | Spectral emissivity of rhenium[END_REF]. Thus, we expect that emission from our rhenium films is smaller than what was presented in reference [START_REF] Marple | Spectral emissivity of rhenium[END_REF]. For this reason, the Re (T ) curve was reduced by 25%.

Emittance and transmittance of sapphire were taken to be 0.077 and 0.93, based on references [START_REF] Kisler | Direct emissivity measurements of ir materials[END_REF][START_REF] Wittenberg | Total hemispherical emissivity of sapphire[END_REF][START_REF] Dobrovinskaya | Sapphire Material, Manufacturing, Applications[END_REF].

The common transmittance, emittance and reflectance can now be computed using equations 2.51. The results are shown in figure 2.46 as the function of the thickness of the rhenium film. The temperature was fixed at 800 • C.

Emittance of tungsten is given by the following expression:

W = -2.6875e-2 + 1.819696e-4 • T -2.1946163e-8 • T 2 , (2.55)
where the temperature is measured in Kelvin [START_REF] Lassner | Tungsten -Properties[END_REF]. This equation is valid between 400 K (127 • C) and 3600 K (3327 • C).

Results and discussion

The temperature of the rhenium was calculated for two cases: with or without tungsten backing on substrate. The current through the furnace filament was set to 8.5 A, and no electron bombardment was applied. This is a setting we used most frequently.

The temperature of the rhenium is shown in figure 2.47 as the function of its thickness. The effect of the tungsten is clear. The substrate is almost completely transparent for infrared radiation, thus it cannot be heated effectively. Rhenium heats as the thickness builds up, and its transmittance decreases. Compared that to the case when tungsten backing is applied to the substrate, the temperature is relatively stable.

An increase in temperature, that starts around 10 nm, can be observed in on the red curve. This is consistent with the dewetting described in section 2.3. The temperature of the rhenium surface is not constant during growth, but increases significantly, which induces changes in the growth process. To eliminate the holes, the dewetting process needs to be avoided. To achieve this, the tendency for the temperature increase has to be compensated by manually lowering the power of heating. This calculation allows us to deduce how much we have to reduce the thermal power to achieve a constant temperature during the growth.

Superconductivity

This chapter starts with the short history of the discovery of superconductivity, following which the basic theories that describe this phenomena are introduced. In the third section superconducting devices, namely Josephson junction and the superconducting quantum interference device are explained. Finally, a description is given on the two refrigerators that were used to measure our superconducting circuits.

History of superconductivity

In 1908 Heike Kamerlingh Onnes succeeded in liquefying a cup of helium for the first time, which opened the door for low temperature physics [START_REF] Van Delft | Little cup of helium, big science[END_REF]. It was also he, who, in order to study the conduction of metals at low temperatures, measured the abruptly vanishing resistivity of mercury at 4.2 K in 1911 [89].

The concept of phonons did not exist at the time, but it was accepted that electrons are responsible for electrical conductance and that scattering by ions causes resistivity. What was not known is how the electron-ion scattering amplitude and the mobility of electrons change upon approaching absolute zero. (The lowest temperature achieved before 1908 was 14 K using liquid hydrogen.) It was also observed that impurities have an effect on resistivity. The Leiden laboratory, where Onnes worked, had a lot of experience in 111 purification of mercury by distillation, therefore it was a convenient choice of a pure sample to measure. Onnes' cryostat was made out of glass, and was able to cool below 2 K. Interestingly, on the day when the first superconducting transition was measured, 8th of April 1911, in his notebook Onnes described the superfluid transition of helium as well, without realising that what he saw was also something brand new and equally as baffling as superconductivity [START_REF] Van Delft | The discovery of superconductivity[END_REF].

After the experiments on mercury, Onnes' team discovered that tin and lead are also superconductors [91], and that magnetic field destroys superconductivity [92].

Zero resistivity is the first, most obvious hallmark of superconductivity. The second, which is more important for today's applications, was discovered over 20 yeas later [START_REF] Meissner | Ein neuer effekt bei eintritt der supraleitfähigkeit[END_REF]. The magnetic induction is zero inside a superconductor as long as the magnetic field is below a certain critical field, regardless the order of the following procedures: cooling below T c , turning on magnetic field. This is called Meissner-Ochsenfeld effect. The expulsion of an applied field is what distinguishes the superconductor from a perfect conductor.

A few years after the discovery of the Meissner-Ochsenfeld effect, the London brothers described the electrodynamic properties of superconductors by introducing modifications in the Maxwell equations [START_REF] London | The electromagnetic equations of the supraconductor[END_REF]. They assumed the existence of superconducting electrons, that can move in the lattice without resistance and their density increases from zero as temperature decreases below T c . A "simple, but unsound derivation" [START_REF] Tinkham | Introduction to superconductivity[END_REF] of the London equations is presented in section 3.2.1.

In another 15 years phenomenological description of superconductors was developed based on Landau's theory of second order phase transitions [START_REF] Ginzburg | [END_REF]. This allowed the investigation of spatial variations in the superconducting electron density, among others, that led to the discovery of flux quantisation [97].

The understanding the underlying microscopic physics came in the '50s in the form of the Bardeen-Cooper-Schreiffer (BCS) theory [START_REF] Bardeen | Microscopic theory of superconductivity[END_REF][START_REF] Bardeen | Theory of superconductivity[END_REF]. Cooper showed that even a small attractive interaction between the electrons causes the Fermi sea to become unstable against the formation of two-electron bound states, Cooper pairs, which are responsible for superconductivity [START_REF] Cooper | Bound electron pairs in a degenerate fermi gas[END_REF].

Theories of superconductivity

London equations

The London brothers assumed that in superconductors, besides the normal electrons, there are also superconducting electrons with charge e * , mass m * , and density n * s [START_REF] London | The electromagnetic equations of the supraconductor[END_REF].

These superconducting electrons, unlike the normal electrons, are not scattered by the ions of the metal, accelerate freely in the electric field. Their equation of motion is the following:

m * dv dt = -e * E. (3.1)
Using the expression for current density, j s = -e * n * s v s , the first London equation is:

dj s dt = n * s e * 2 m * E, (3.2)
which shows infinite conductance.

The second London equation can be obtained by combining the first, 3.2, with the Maxwell equation ∇ × E = -∂B/∂t:

d dt ∇ × j s + n * s e * 2 m * B = 0, (3.3)
and assuming the expression in the bracket is not only independent on time but zero, we obtain

∇ × j s = - n * s e * 2 m * B. (3.4)
This is the second London equation. When the second London equation is combined with another one of the Maxwell equations,

1 µ 0 ∇ × B = j s , (3.5)
we get the following differential equation:

∆ × B = - 1 λ 2 L B, (3.6)
where λ L is called London penetration depth. When solving equation 3.6 in one dimension, where x = 0 in the boundary of a superconductor, and x > 0 is the inside of a superconductor, we get an exponentially decreasing function: B(x) = B 0 e -x λ L . This equation describes the Meissner-Ochsenfeld effect: magnetic field inside a superconductor exponentially decreases. Characteristic length of this screening is the London penetration depth. Knowing that the superconducting electrons are electron pairs with charge 2e, mass 2e and density n s /2, the London penetration depth is:

λ 2 L = m * n * s e * 2µ 0 = m n s e 2 µ 0 (3.7)

Ginzburg-Landau theory

In 1950 Ginzburg and Landau used Landau's previously developed theory for second order phase transitions [101] to describe the superconducting phase transition [START_REF] Ginzburg | [END_REF]. The theory assumes the existence of an order parameter, Ψ(r) = ψ 0 (r)e -iθ , that is zero in the normal state and increases to a finite value in the superconducting state. This order parameter describes the superconducting electrons, and their local density, n s (r) = |Ψ(r)| 2 . A further assumption is that, in the vicinity of the transition, free energy can be defined, and it can be expressed as the series expansion of the order parameter, as follows:

f s = f n + α(T )|Ψ(r)| 2 + 1 2 β(T )|Ψ(r)| 4 + + 1 2m * 2 i ∇ + e * A(r) Ψ(r) 2 + 1 2µ 0 B(r) 2 , (3.8)
where f n and f s are the free energy density in the normal and the superconducting state, respectively, α(T ) and β(T ) are coefficients of the expansion. The second to last term is the kinetic energy of superconducting electrons (with m * mass and e * charge) in magnetic field (A is the vector potential), and the last term is the energy of the magnetic field (B). It is a requirement that the energy minimum in the normal state be at Ψ(r) = 0, and below the transition temperature the energy has to reduce. Form this the coefficients are chosen as follows: α(T ) ∼ (T -T c ) and β(T ) = β > 0. The energy minimum is found by taking the variational derivatives with respect to Ψ * (r) and A(r), which results in two differential equations, respectively:

1 2m * i ∇ + e * A(r) 2 Ψ(r) + αΨ(r) + β|Ψ(r)| 2 Ψ(r) = 0, (3.9) j(r) = 1 µ 0 ∇ × B(r) = - e * 2m * Ψ * (r) i ∇ + e * A(r) Ψ(r) + c. c. (3.10)
Equation 3.9 is the non-linear Schrödinger equation of the superconducting electrons, where the non-linear term can be interpreted as a repulsive potential. 3.10 is the quantum mechanical current carried by the superconducting electrons.

Ginzburg-Landau theory was derived phenomenologically, 7 years before the microscopic origins of superconductivity were understood. In 1959 Gorkov derived the the Ginzburg-Landau equations from the microscopic theory [102], and showed, what is well known today, that e * and m * is the charge and mass of two electrons.

Bardeen-Cooper-Schrieffer theory

In 1950 it was experimentally shown by two groups that the critical temperature and field of mercury is sensitive to the isotope mass, which is today known as the isotope effect [START_REF] Maxwell | Isotope effect in the superconductivity of mercury[END_REF][START_REF] Reynolds | Superconductivity of isotopes of mercury[END_REF]. Shortly after, Fröhlich submitted his paper, where he proposed that superconductivity is the result of the interaction of ions and electrons [START_REF] Fröhlich | Theory of the superconducting state. i. the ground state at the absolute zero of temperature[END_REF], predicting the isotope effect, and claiming he came to this conclusion independently, without the knowledge of the experimental confirmation [START_REF] Fröhlich | Isotope effect in superconductivity[END_REF]. A stance that is today generally accepted [START_REF] Hirsch | Did herbert fröhlich predict or postdict the isotope effect in superconductors?[END_REF]. The idea that ions could be responsible for superconductivity, and the discovery of the isotope effect were the foundation stones of the first successful theory, the BCS theory.

The first step towards BCS theory was the realisation that the normal ground state of the electron gas, the Fermi sea, becomes unstable when an attractive interaction, however small it may be, acts between the electrons, and two-electron bound states appear [START_REF] Cooper | Bound electron pairs in a degenerate fermi gas[END_REF]. In the derivation of the Cooper instability the Schrödinger equation with an attractive potential is solved for two electrons that are added to the Fermi sea. When evaluating the two-electron wave function, an important observations can be made. The lowest energy state is expected to have zero momentum, which means the electrons have equal and opposite momenta. To ensure that the wave function is antisymmetric, the electron spins must be opposite. This is an s-wave state, and indeed all conventional superconductors were found to have s-wave Cooper-pairs. It might sound surprising to assume an attractive interaction between electrons at first. Coulomb interaction is repulsive, and even when considering screening that occurs in metals, the potential remains repulsive. Motions of the ions have to be considered to get an effective attractive interaction. An intuitive image could be the following: the first electron that passes through the lattice, polarises it by attracting the positive ions, then these positive ions attract the second electron. If this attraction can override the repulsion between the two electrons, an effective attractive potential occurs [START_REF] Tinkham | Introduction to superconductivity[END_REF].

Characteristic vibrational frequency of the ions, phonon frequency, is the Debye fre-quency, ω D . It is assumed, that only the electrons whose energies are in the 2 ω D interval around the Fermi level (E F ), experience the attractive potential.

The energy eigenvalue of the two-electron Schrödinger equations is the following:

E ≈ 2E F -2 ω D e -2 N (0)V , (3.11)
where N (0) is the density of states at the Fermi level, and -V is the attractive potential. The energy is negative with respect to the Fermi level, no matter how small V is. This means that there is a bound state of two electrons with lower energy than the ground state in the normal phase. This is called Cooper instability. The only conclusion that can be drawn from the calculation outlined above is that the Fermi sea is not a stable state anymore. To find what the new ground state of electrons is, the Schrödinger equation of all electrons in the material must be solved, and that is what the BCS theory does.

The most important results are the prediction of the energy gap (∆) that opens in the electron spectrum around the Fermi energy, and linking that to the transition temperature (T c ). Both depend on Debye frequency, which signals the isotope effect:

k B T c = 1.14 ω D e -1 N (0)V , (3.12)
∆(0K) ≈ 2 ω D e -1 N (0)V . (3.13)
From equations 3.12 and 3.13, the relation between T c and ∆(0):

∆(0) = 1.754k B T c . (3.14)
BCS theory was a ground breaking theory, since it was the first microscopic theory that described superconductivity. Nonetheless, there are several experimental situations that cannot be explained by it, and thus it needs to be generalised. Processes other than electron-phonon scattering need to be considered. The reason for this are the several assumptions that are made to simplify the already complicated derivation. A few of these are: Fermi surface is assumed to be spherical, electron-phonon interaction is constant for all energies around the Fermi energy, and only singlet Cooper pairs are considered. Superconductors that can be described by the BCS theory are called conventional superconductors. Rhenium, alongside other pure metal superconductors, belongs to this group.

Characteristic lengths

One of the important characteristic lengths of a superconductor has already been defined in equation 3.7, it is the London penetration depth, which gives how deep the magnetic field can penetrate into a superconductor. Due to the limitations of the theory behind the London equations, λ L is a theoretical limit of the effective penetration depth at T → 0. The effective penetration depth is always larger, and diverges close to T c .

A coherence length can be defined based on the uncertainty principle arguing that only electrons within ∼ k B T c interval of the Fermi energy can play a role. The momentum of these electrons are p ≈ k B T c /v F , where v F is the Fermi velocity. For this the uncertainty of the location can be expressed as

∆x ∆p ≈ v F k b T , (3.15)
which can be identified as the coherence length, and the numerical factor (α) can be obtained from the BCS theory:

ξ 0 = v F π∆(0) = α v F k b T , (3.16)
where α is about 0.18. Another characteristic length can be defined based on the Ginzburg-Landau theory [START_REF] Tinkham | Introduction to superconductivity[END_REF]. From equation 3.9 the equilibrium value of the order parameter (Ψ 0 ) can be determined:

0 + αΨ 0 + β|Ψ 0 | 2 Ψ 0 = 0 → |Ψ 0 | 2 = - α β (3.17)
Using the normalised wave function f = Ψ/Ψ 0 , equation 3.9 (A = 0) can be rewritten in the following form:

- 2 2m * α ∇ 2 f + f -|f | 2 f = 0. (3.18)
The coefficient of the gradient term is defined as the characteristic length. This is referred to as Ginzburg-Landau coherence length:

ξ 2 GL (T ) = - 2 2m * α(T ) ∝ 1 T c -T (3.19)
ξ GL is different from ξ 0 , however for pure materials, far below T c they are equal.

The Ginzburg-Landau theory introduces the order parameter (Ψ), that was said to be related to the density of superconducting electrons (n s = |Ψ| 2 ). It was shown in equation 3.17 that the equilibrium value of the order parameter is -α/β. Writing this in the expression of the London penetration depth (equation 3.7), we can see that the penetration depth has the same temperature dependence as the Ginzburg-Landau coherence length, and they both diverge upon approaching T c [START_REF] Tinkham | Introduction to superconductivity[END_REF].

λ 2 L (T ) = m * 4µ 0 e * 2 |Ψ 0 | 2 = - βm * 4µ 0 e * 2 α(T ) ∝ 1 T c -T . (3.20)
The dimensionless and temperature independent Ginzburg-Landau number is introduced, it shows the relation between the penetration depth and the coherence length [START_REF] Tinkham | Introduction to superconductivity[END_REF].

κ = λ L (T ) ξ GL (T ) . (3.21)
Its significance will be discussed in chapter 3.2.7.

Dirty and clean superconductors

The term 'dirty superconductor' was coined by P. W. Anderson in his 1959 paper, where he "sketched" a BCS type theory for "very dirty superconductors" [START_REF] Anderson | Theory of dirty superconductors[END_REF]. It has been observed that superconductivity is often insensitive to the amount of impurities present in the material. These impurities include crystal defects and non-magnetic chemical impurities: beryllium was shown to display superconductivity in amorphous state with T c twenty times higher than in crystalline state [109,[START_REF] Matthias | Superconductivity and electron concentration[END_REF]. He divided superconductivity in two regions: clean superconductors that are sensitive to the introduction of additional impurities; and dirty superconductor that are insensitive.

Impurities cause conduction electrons to scatter. This scattering quantitatively is described by the mean free path (l). In the clean limit the electrons are rarely scattered, the mean free path is longer than the superconducting coherence length (l ξ). In the dirty limit, however, electron scattering is strong, electron mean free path is shorter than the coherence length (l ξ). The effective coherence length in dirty superconductors is reduced. Its value near the transition temperature can be obtained from the BCS theory [START_REF] Tinkham | Introduction to superconductivity[END_REF]:

ξ clean = 0.74 ξ 0 1 -T Tc 1 2 , ξ dirty = 0.855 ξ 0 l 1 -T Tc 1 2
.

(3.22)

Fluxquatization

It has been observed that inside a superconductor ring the magnetic field cannot take an arbitrary value, but it must be an integer multiple of the so called flux quantum, Φ 0 = h/2e [START_REF] Bascom | Experimental evidence for quantized flux in superconducting cylinders[END_REF].

Flux passing through a surface area (S with normal n) can be calculated by integrating the magnetic induction vector (B) over that surface [START_REF] Sólyom | Fundamentals of the Physics of Solids[END_REF]. Using Stokes theorem the surface integral becomes a line integral running around the boundary of the area (l), and the induction vector is replaced by the vector potential (A):

Φ = B • n dS = A • dl.
(3.23)

The vector potential in the superconducting regime can be expressed from one of the Ginzburg-Landau differential equations, equation 3.10. Current is 0 deep inside the superconductor, where the integration is considered. The order parameter is assumed to have the form Ψ(r) = ψ 0 (r)e -iθ , where ψ 0 is thought to have the equilibrium value, and does not change along the integral. Only the phase can change. Using these, the vector potential, and the flux inside the superconducting ring, respectively, is:

A = - e * ∇θ, Φ = - e * ∇θ dl. (3.24)
Phase must be a single valued function of space, θ(r), so, along a closed loop it can only change by integer multiples of 2π. The solution of the integral is then

Φ = - e * n2π = n h 2e = nΦ 0 , where Φ 0 = h 2e = 2.067 • 10 -7 Gauss • cm 2 . (3.25)
Φ 0 is called the flux quantum.

Two types of superconductors

In chapter 3.2.4 the dimensionless Ginzburg-Landau number was introduced, and its importance was promised to be explained. To recap, κ is defined as the ratio of the penetration depth (λ) and the coherence length (ξ). The penetration depth describes the disappearance of the magnetic field at the boundary of the superconductor, and the coherence length is related to the decay of the order parameter. Therefore, their relative value describes the properties at the superconducting-normal interface. In figure 3.1 the magnetic field curve (h) and the order parameter (Ψ) are shown on the boundary of a superconductor for the two extreme cases of κ. When κ is smaller than 1, the coherence length is longer than the penetration depth, and there is a region where the magnetic field and the order parameter are both small. Expelling the magnetic field costs energy, and this energy is not compensated by the condensation to the superconducting phase, thus this interface has a positive energy. However, in the case when κ is larger than 1, following the same argument, interfacial energy can become negative, meaning that walls of this type can spontaneously appear. The crossover between positive and negative interfacial energy is at κ = 1/ √ 2. This value for κ was already mentioned in Ginzburg and Landau's original paper [START_REF] Ginzburg | [END_REF], but it was Abrikosov who predicted how the magnetic field behaves in such superconductors [97], which will be briefly discussed below [START_REF] Tinkham | Introduction to superconductivity[END_REF].

Since the penetration depth and the coherence length are material properties, we can talk about two types of superconductors. Type I are the ones with κ < 1/ √ 2. Superconductivity is completely destroyed, and magnetic field enters the material at a certain critical filed H c (T ).

Type II superconductors have κ > 1/ √ 2. These materials have two critical fields, H c1 and H c2 . Their phase diagram is shown in figure 3.2. The Meissner phase, is the homogeneous superconducting phase, where the magnetic field is fully expelled. When the magnetic field is increased exceeding H c1 , type II superconductors transit to a so called vortex phase, where magnetic field and superconductivity mix. Above H c2 , the superconductor transits to the normal phase [START_REF] Sólyom | Fundamentals of the Physics of Solids[END_REF]. In the vortex phase, that is unique to type II superconductors, magnetic flux is present in the sample in the form of tubes, vortices, around which the superconducting phase persists. The profile of the magnetic induction (B) and the order parameter around a core of a vortex (r=0) is shown in figure 3.3. The order parameter drops in the core of the vortex, while the magnetic induction reaches its maximum. The supercurrent is circulating around the core of the vortex. One vortex can only contain integer multiples of the flux quanta (φ 0 = h/2e), where the integer is energetically favoured to be 1. When the magnetic field is increased, the density of the vortices increases. Due to the repulsion that acts between the vortices, they arrange themselves in a regular array, which is has trigonal symmetry in most cases.

The vortex phase was predicted by Abrikosov in 1957 [97], and it was first directly observed in 1967 by electron microscope using, magnetic particles for contrast [START_REF] Essmann | The direct observation of individual flux lines in type ii superconductors[END_REF].

Superconducting devices

Josephson junction

In 1962 Josephson pointed out that Cooper pairs can tunnel between two superconductors separated by a thin (L < ξ) barrier [START_REF] Josephson | Possible new effects in superconductive tunnelling[END_REF]. This construction is called a Josephson junction. The thin barrier can be an insulating layer, as it was originally imagined by Josephson, or a thin metallic constriction with dimensions below the coherence length. The tunnelling current can be derived from the Ginzburg-Landau equations, and it is outlined below.

Recalling equation 3.18 in one dimension:

ξ 2 d 2 f dx 2 + f -f 3 = 0, (3.26)
where f = Ψ/Ψ 0 . The superconductors on the two sides of the bridge are assumed to be in equilibrium, giving |f | = 1, but the phase of the superconducting wave function can still differ. An absolute phase cannot be defined without loss of generality, it can be chosen to be 0 at one end of the bridge and ∆θ at the other, which then gives the boundary conditions: f (x = 0) = 1; f (x = L) = e i∆θ . When L ξ, equation 3.26 is dominated by the first term, in which case the problem is reduced to Laplace's equation:

ξ 2 d 2 f dx 2 = 0. (3.27)
The most general solution in one dimension is f = a + bx, which after satisfying the boundary conditions gives:

f = 1 - x L + x L e i∆θ . (3.28)
The current running through the bridge can be obtained by inserting 3.28 into the Ginzburg-Landau expression of current, equation 3.10:

I j = I c sin ∆θ, where I c = e * Ψ 2 0 m * A cs L . (3.29)
A cs is the cross-sectional area of the bridge. This means that the tunnelling superconducting current is the function of the relative phase of the wave function in the two superconducting regions, and is limited by the critical current, which is dependent on the bridge dimensions [START_REF] Tinkham | Introduction to superconductivity[END_REF].

Superconducting quantum interference device

A superconducting quantum interference device (SQUID) is superconducting ring that is interrupted by at least one Josephson junction, shown in figure 3.4. Their currentmagnetic flux characteristics are very sensitive to small changes of the magnetic field, therefore they are used as magnetometers. In scanning SQUID microscopy, SQUIDs are the scanning probes, and they map the magnetic field across a magnetic or superconducting sample. In most SQUIDs the superconducting ring is interrupted by two junctions, as shown in figure 3.4. The ring is put in magnetic field such that the flux through it is Φ e . As a result of the magnetic field, a current will circulate in the ring. From equation 3.29, the current through the junctions is defined by the phase drop across, θ 1 and θ 2 :

I 1 = I c1 sin θ 1 I 2 = -I c2 sin θ 2 , (3.30)
where I c1 and I c2 are the critical currents.

CHAPTER 3. SUPERCONDUCTIVITY

The total current through the ring is the sum I 2 and I 1 :

I total = I 1 + I 2 = I c1 sin θ 1 -I c2 sin θ 2 . (3.31)
The sum of the phase differences across the two junctions is the integral of the vector potential along the ring:

θ 1 + θ 2 = 2e Adl = 2π Φ e Φ 0 . (3.32)
Using this in equation 3.31, the total current in the ring is:

I total = I c1 sin θ 1 + I c2 sin θ 1 -2π Φ e Φ 0 . (3.33)
To find the maximum current through the ring, equation 3.33 has to be minimised with respect to θ 1 : Summing the squares of equations 3.33 and 3.34:

dI total dθ 1 = I c1 cos θ 1 + I c2 cos θ 1 -2π Φ e Φ 0 = 0 (3.34)
I 2 total = I 2 c1 + I 2 c2 + 2I c1 I c2 cos 2π Φ e Φ 0 . (3.35)
Using the identity cos δ = 2 cos 2 δ 2 -1, equation 3.35 can be rearranged as follows:

I total = (I c1 -I c2 ) 2 + 4I c1 I c2 cos 2 π Φ e Φ 0 . (3.36)
If the junctions are assumed to be identical with equal critical currents (I c1 = I c2 = I c ), the critical current takes the following form:

I total = 2I c cos π Φ e Φ 0 . (3.37)
The flux dependence of the current is shown in figure 3.5. The current changes periodically as the flux increases or decreases. These devices can be used to detect small magnetic fields [START_REF] Hasselbach | Microsquid magnetometry and magnetic imaging[END_REF].

The above calculation is only valid if the inductance of the SQUID (L SQUID ) is zero, which is not true in general. When L SQUID is not zero, the flux inside the ring is modified by the current running through it: Φ L = Φ e + IL SQUID . The critical current can be determined numerically. The inductance of the SQUID depends on its size.

To measure the critical current of a SQUID, one end is connected to a current source, the other to the ground, and the current is increased until it reaches the critical current.

In our setup, the SQUID is biased by a current which is ramped up from 0 with a given slope. At the start of the current ramp, a 40 MHz quartz clock is starts, simultaneously. When the critical current is reached, voltage appears across the SQUID, and the ∂V /∂t pulse is detected, the clock is stopped and the current is set back to zero. The time laps measured by the clock is registered, and the ramp starts again. The biasing current versus time is shown in figure 3.6. One critical current data point is plotted after averaging 30 critical current measurements.

SQUIDs have a hysteric V(I) characteristic due to Joule heating in the normal state. When the critical current is reached, the voltage pulse heats the SQUID, and after the bias current is switched off, it takes time to cool down, and return to the superconducting state. This limits the frequency at which the current ramp can be repeated [START_REF] Hasselbach | Microsquid magnetometry and magnetic imaging[END_REF].

Refrigerators

Temperature and magnetic field dependence of transport properties of complete films and nanostructures were measured. For the low temperature measurements two different refrigerators were used: a table top Helium-3 cryostat where only the temperature can be adjusted, and a dilution refrigerator where magnetic field can be applied as well.

Table-top helium-3 cryostat

Low temperature resistivity measurement on the rhenium wires were performed in a table-top helium-3 cryostat, which was designed and built at Institut Néel [117]. This cryostat, as its name suggests, is a compact, easy-to-use refrigerator, is able to cool down to 300 mK in about 3 hours. A simplified schematics, and a photograph of the inside is shown in figure 3.7, and in figure 3.8, respectively. It consist of a vacuum chamber that houses two interlocked helium circuits: one open circuit for helium-4 (blue line in figure 3.7), that runs through the cryostat from the reservoir to the recovery exhaust; and a closed circuit for helium-3 (orange line in figure 3 The fridge can be cooled down to approximately 1.2 K by circulating only helium-4, and pumping on it in the 1 K pot, shown in figures 3.7 and 3.8. At this temperature the vapour pressure of 4 He is too small, the rate of evaporation is too low to reduce the temperature further. Below 3.19 K helium-3 starts to condense in its reservoir. The condensation process can be monitored by measuring the pressure in the helium-3 circuit. When the condensation is complete, the helium-3 liquid is pumped by an internal sorption system, which is activated charcoal placed in the neighbouring container, labeled as charcoal in figures 3.7 and 3.8. This way the temperature of the reservoir can be further reduced to 300 mK, where we meet the same limitation: the vapour pressure is too low for evaporation to cool down further. This is the minimum temperature that can be achieved using pure helium-3. Once all the helium-3 is adsorbed, the charcoal can be heated to 40 K to release the gas, and the condensation-adsorption cycle can be repeated.

The temperature of the sample can be adjusted by ohmic heating, running current through a resistor placed on the sample stage.

Inverted dilution refrigerator

A dilution refrigerator contains several circuits, one of which is a closed circuit, containing a mixture of helium-4 and helium-3. Using the other, only helium-4 circuits, the fridge is able to reach the 1.2 K mentioned above, by strong pumping on the liquid 4 He. At this temperature the mixture has condensed, and by pumping on it, the temperature is decreased further. Upon reaching 800 mK, the mixture spontaneously undergoes a phase separation, as shown in the phase diagram in figure 3.9: the pink region is forbidden, the liquid separates into two phases. The one marked with yellow is still a mixture of 4 He and 3 He, but contains only a small amount of 3 He. This is the diluted phase. The other phase marked with green is also a mixture of the two isotopes, but it is rich in 3 He, this is the concentrated phase.

The volume of the mixture and the concentration of 3 He is set so that the phase boundary between the diluted phase and the concentrated phase occurs in the mixing chamber, and the liquid surface of the concentrated phase is in the still. A simplified mixture circuit is shown in figure 3.10, noting the mixing chamber and the still. Diluted phase is marked with blue, concentrated phase with orange. The lighter shade of orange corresponds to the gaseous phase, the darker to the liquid. After the mixture separates to a diluted (blue) and a concentrated (orange) phase, 3 He is extracted from the still and resupplied to the mixing chamber. Cooling power is provided by the diffusion of 3 He from the concentrated to the diluted phase in the mixing chamber.

In the mixing chamber the concentrated phase floats on top of the denser diluted phase. The vapour in the still contains higher concentration of the component that has a lower boiling point. This is the helium-3, and is continuously pumped. The concentration of 3 He in the diluted phase drops as a result. To compensate the decreasing concentration, 3 He diffuses from the concentrated phase into the diluted phase in the mixing chamber. Dilution of helium-3 in the superfluid helium-4 requires heat absorption, superfluid helium-4 acts like vacuum. Dilution can be compared to evaporation, since the they are further apart like in a gas. As a result, the mixing chamber, and everything thermally connected to it (sample) cools down.

The diagram in figure 3.10 is grossly simplified. The mixture circuit is interlocked with the helium-4 circuit. On the left the vapour leaving the still is warmed up by running through the incoming liquid helium-4, reaches room temperature by the time it gets to the pump. After the compressor the mixture runs through a liquid nitrogen bath, that removes the contamination that might be present, then in enters the fridge through the outgoing helium-4 pipes, which pre-cools it. On its way to the mixing chamber it runs in close contact with the vapour that is leaving the still, that aids further cooling. To liquify it, its pressure is increased by driving it through narrow pipe sections called impedance.

Dilution fridges are limited by the increasing viscosity and thermal conductivity of the circulating fluids as the temperature is lowered. For this reason the lower the temperature the larger the diameter of the tubes must be which increases the cost. The lowest temperature where they are still practical to use is 2 mK. In conventional dilution fridges, the coldest part, the mixing chamber is at the bottom. We used an inverted dilution fridge, SIONLUDI, where the mixing chamber is at the top. Its schematic is shown in figure 3.11, it was designed and built in Institut Néel. The sample is easy to load, it is placed under concentric bells which are, from the outside: vacuum seal, and temperature shields for 300 K, 80K, 20 K, and 4 K. Helium-4 is fed from the bottom, and the fridge is placed on a vibration free table. The magnetic field can be adjusted by changing the current in the copper coil, which is outside of the vacuum bell.

"The world's wealth would be won by the man who, out of the Rhinegold, fashioned the ring which measureless might would bestow." Richard Wagner, Das Rheingold, Der Ring des Nibelungen 4

Transport properties of rhenium wires and SQUIDs

Rhenium thin films have shown long superconducting coherence lengths (up to 170 nm) and electron mean free paths (up to 200 nm) [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. This makes them suitable to study the effect of the decreasing size on the superconducting properties by reducing the dimensions of the samples to the vicinity of those two length scales. To explore this, and to study the effect of lithography on the superconducting properties of rhenium thin films, wires with width from 100 nm to 3 µm were fabricated. We have also successfully fabricated SQUIDs.

In this chapter it is shown that superconductivity is unaffected by the lithography process. Sensitivity of T c to lattice imperfections and orientation is studied. Finally, low noise SQUIDs are presented. The third design is shown in figure 4.2. This pattern was also fabricated using electron beam lithography. Widths of the wires were reduced below 1 µm. The pattern included 2 SQUIDs as well, with different bridge widths: 50 nm and 20 nm. In this design each wire and SQUID is connected to two pads on the bottom, and there is a common electrode on the top with four pads. Two versions of the design were fabricated alternating on the film. The first version had wire widths 100 nm, 200 nm, and 400 nm; the second had 500 nm, 700 nm, and 900 nm.

Fabrication

Drawings shown in figures 4.1(a), 4.1(b), and 4.2 are not to scale.

Lithography

Steps of the lithography process are shown in figure 4

.3.
First, the sample is spincoated with the resist. For laser beam lithography, S1818 resist was used, which forms a layer with thickness in the range of 1 µm on the surface of the sample. For electron beam lithography, PMMA (polymethyl methacrylate) resist was used, with a thickness in the range of 100 nm. After spincoating, the sample is baked to evaporate the solvent from the resist.

In the next step, the surface was exposed to a laser or electron beam along the pattern lines. Both of the resists are so called positive resists. They consist of long polymer chains, which break up to smaller fragments when exposed to light/electron beam. The smaller fragments can be dissolved easily in the developer, leaving behind a positive imprint of the pattern in the resist [START_REF] Rai-Choudhury | Handbook of Microlithography, Micromachining, and Microfabrication[END_REF].

Laser beam

Electron beam In the case of electron beam lithography, the designs had to be patterned without moving the sample stage. This was done in two steps. In the first step, a 1 mm 2 writing field was chosen, and the pads and the electrodes were written using a beam current (I) of a few mA. In the second step, the writing field was reduced to 50 µm 2 , and the wires and the SQUIDs were patterned using a beam current of 12-14 pA. The dose (D), charge received by unit area, can be obtained from the beam current using the formula D = It/A, where t is the exposure time, and A is the area. For the bridges in the SQUIDs, the dose was increased, and they were scanned only once.

Resist

After the development, the resist was removed from the surface of the rhenium where it was exposed. In the next step, 20 nm of aluminium was deposited using a Plassys MEB 550S evaporator. During lift-off, the sample was placed in acetone, which dissolved the remaining resist. The aluminium on top of the resist was also removed. To protect the thin bridges of the SQUIDs, we did not use ultrasonic bath at this step, only a pipette to stir the liquid a few times. Lift-off took approximately 2 hours. After the lift-off, aluminium only covered the rhenium film along the pattern lines. In the penultimate step the rhenium was removed everywhere by reactive ion etching (SF 6 ) using an RIE Plassys, except where it was protected by the aluminium. Finally, the aluminium covering the structures was removed by first immersing the sample in a developer called MF-319 developer (app. 3 % solution of N(CH 3 ) + 4 OH -in water), then rinsing it in distilled water and drying under nitrogen flow.

Description of the samples

Three rhenium films with different thicknesses, surfaces and crystallographic structures were patterned as described above, and transport measurements were conducted at low temperature. Before patterning the samples were studied with AFM and XRD techniques. The details of the measurements and analysis are described in chapter 2.2. A summary of the results and the patterns fabricated are listed in table 4

Measured wires 100 nm -400 nm 3 µm 3 µm SQUID × × Table 4.2: Crystallography and surface information on the samples that were used for the fabrication.

50 nm thick sample

Ten copies of design II, shown in figure 4.1(b), was patterned onto the 50 nm thick sample using electron beam lithography, as described in section 4.1. An image taken with an optical microscope of a completed pattern unit is shown in figure 4.4. The pattern is not perfect, as some rhenium remained on the sides of some of the electrodes, and in some corners. This is due to wrong electron dose used during patterning. Electrons are scattered in the resist, are backscattered, and excite secondary electrons from the film. All these processes affect the size and shape of the patterned volume, this is called proximity effect. In the third step shown in figure 4.3, the ideal shape of the imprint in the resist is shown. It is a trapezoid. If the shape of the imprint is correct, the deposited aluminium is not continuous, and is removed with the resist during lift-off. However, if the dose is wrong, and the imprint does not have the correct shape, the aluminium layer can be continuous, thus harder or impossible to remove with the resist during lift-off. Where aluminium is left, rhenium cannot be etched. The remaining rhenium created shortcuts and thicker wires on some of the patterns. These were excluded from the measurement. The correct dose can only be found by experimentation.

To verify the crystallography of the patterned area, EBSD measurements were carried out in SiMaP on the circuits after the lithography process using a FEG Zeiss scanning electron microscope. Here the colour is uniformly blue.

The intensity of the colour red and blue in these figures scales with the IQ, which measures the quality of the Kikuchi pattern recorded at a position [START_REF]Electron Backscatter Diffraction in Materials Science[END_REF]. In the areas where the intensity of red/blue is lower, points with the non-uniform colour can also be found. The low IQ of this area suggests that the orientations of these points were wrongly identified by the software due to weak contrast or blur in the Kikuchi patterns.

The surface was also investigated with AFM after the patterning. A SQUID is shown in figure 4.6. On the surface of the SQUID, a few nanometers of height variation can be observed. It corresponds to the spirals. They were left intact during the lithography process.

Unfortunately, the thin bridges that are the weak links between the two forks of the SQUID are not present on this sample. The dose during electron beam exposure was too low. The resist at the nanowire level was removed during development. For later samples the dose was adjusted.

25 nm thick sample

The two versions of design III were fabricated onto the sample using electron beam lithography. Version one had wires widths 100 nm, 200 nm and 400 nm, version two had 500 nm, 700 nm and 900 nm. The two version were alternating, and were repeated 18 times all over the sample. As with the previous sample, the electron dose was not perfectly set, and it caused short cuts in some places. Because the pattern was repeated many times all over the sample, there were a sufficient number to choose from, where the left-over rhenium did not cause a short circuit.

100 nm thick sample

Design I was patterned onto the 100 nm thick film, using the laser lithography technique. An optical image of the fabricated pattern is shown in figure 4.9

Transport measurements of the wires

The experimental setup

The resistances of the wires were measured using four-terminals. The principle of the technique is shown in figure 4.10.

Current is supplied to terminals 1 and 4, shown in blue, that are connected to the ends of the wire. The voltage is measured between terminals 2 and 3, shown in green, placed between the current electrodes. The separation of the voltage and current terminals means that only the resistance of the part of the wire that falls between the voltage electrodes is measured. The resistances of the electrodes, and contacts are excluded. This allows accurate measurement of low resistance values. The sample is glued on a sample holder with copper leads deposited onto it, as shown in figure 4.11. Connections were made between the copper electrodes of the sample holder and the large pads of the sample using a West Bond ultrasonic wire bonder with 25 µm diameter aluminium wire. There are 4 sets of 4 connectors inside the fridge, one of which is visible in figure 3.8, labeled as 'electrical connections'. They lead to the current source and the voltage probe outside the fridge, which was a TRMC2 controller.

The resistances of all three films before patterning was measured by B. Delsol, and was presented in detail in reference [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. Some of his results, along with the results of the The resistance of the thinnest (3 µm) wires was measured on the 50 nm and the 100 nm thick films. An optical microscope image of one of the connections made on the 50 nm thick sample is shown in figure 4.12. The voltage probe was connected to the two ends of the 3 µm part of the pattern, while the current source and the ground to points further to the sides. The resistivity was only characterised along the green line in figure 4.12.

On the 25 nm samples the 400 nm, 200 nm and the 100 nm wires were measured. The connections made for one of these experiments is shown in figure 4.13. The current source and ground were connected to the pads marked by the blue circles, and the voltage was measured between the two green circles. This allows the measurement of the 200 nm wide wire.

Calculation of the resistivity

The setup detailed above measures the resistance (R). Resistance depends on the geometry: the thickness of the film (d), the width of the wire (w), and the length, where the path of the current overlaps with the path of the voltage probe (l). If the geometry is known, resistivity (ρ), which depends only on material, can be calculated, as follows:

ρ = R wd l . (4.1)
Calculating the resistivity of the 3 µm wire is easy. It is shown in figure 4.12 that only the resistance of the wire is probed, and all its dimensions are known.

Resistivities in case of design II (figure 4.13) are more difficult to obtain. Magnified images of the pattern in figure 4.8 show that the thin wires are connected to thicker wires. Resistances of these thick parts are included in the measurement:

R output = R wire1 + R wire2 = ρ d l wire1 w wire1 + l wire2 w wire2 . (4.2)
Thus the resistivity can computed as follows:

ρ = dR output l wire1 w wire1 + l wire2 w wire2 . (4.3)
There is a significant uncertainty (>10%) in the obtained resistivity values of the 100 nm, 200 nm and 400 nm wires, because their precise thicknesses were not verified with SEM. Due to the proximity effect, they can be thicker then designed.

Transport characteristics of rhenium wires

Normal state properties

Resistivity is the result of the scattering of the conduction electrons. According to Matthiessen's empirical rule, resistivities corresponding to independent sources of scatter-ing add up. Dominant sources in metals are scattering on impurities and electron-phonon scattering, thus at room temperature:

ρ RT = ρ el-ph + ρ imp . (4.4)
Resistivity due to electron-phonon scattering is ∝ T at temperatures above the Debye temperature, and is ∝ T 5 below [START_REF] Sólyom | Fundamentals of the Physics of Solids[END_REF]. At low temperatures, electron-phonon scattering becomes negligible, and impurity scattering dominates, which is independent of the temperature [START_REF] Sólyom | Fundamentals of the Physics of Solids[END_REF]:

ρ res = ρ imp . (4.5)
Resistivity settles at a constant value, the residual resistivity.

The residual resistivity ratio (RRR) is defined as the ratio of the room temperature and the residual resistivity (RRR = ρ RT /ρ res ). As equations 4.4 and 4.5 show, it strongly depends on the magnitude of impurity scattering. It is used as the measure of sample quality. Higher RRR indicates, less impurity. In the case of rhenium, when lowering the temperature further, below the critical temperature (T c ) the resistivity vanishes. This corresponds to the superconducting transition. The resistivity of the wires measured at temperatures ranging from 250 K down to 300 mK is plotted in figure 4.14.

Film

As the Debye temperature of rhenium is relatively high, 413 K [START_REF] Purushotham | Mean square amplitudes of vibration and associated debye temperatures of rhenium, osmium and thallium[END_REF], the graphs shown in figure 4.14 should display a T 5 temperature dependence down to about 30 K. This could not be verified, because the thermocouples in the fridge are not calibrated for this temperature range.

The curves of the 200 nm and the 100 nm wires are not smooth around 50 K. Those are artefacts of the measurement, not real effects.

Below approximately 30 K, all the curves settle on a constant resistivity value, but these values are different for each wire. Between 1 K and 2 K the wires become superconducting.

The critical temperature, the width of the transition, the resistivity, and other calculated values are summarised in tables 4.3 and 4.4 for all the samples. Room temperature was considered 300K, and the residual resistivity was measured at 2.4 K. The critical temperature is taken where the resistivity decreases to half of the residual resistivity. The width of the transition was defined as the temperature interval between 90% and 10% of the residual resistivity.

Resistivity values measured at room temperature and at low temperature are in a similar range between the wires and the films they were fabricated on. Values of RRR are also very close for film and wire of the 50 nm and 25 nm thick sample. There is, however, a roughly 40% drop in the RRR value measured on the 100 nm film and the wire. The reason for this drop is unclear. One possible explanation is that the wire was fabricated on spot where impurity concentration was high. Another possibility is that the wire was damaged by the fabrication process. To determine the cause of the drop in the RRR value requires more experiments.

For the majority of samples, however, the fabrication did not alter their transport properties.

Properties of the superconducting transition

In this section the critical temperatures and the widths of the superconducting transitions of the films and the wires are compared to each other. The films were measured in a different cryostat by Delsol [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. Figure 4.15(a) shows the superconducting transitions of the 3 µm wide wire on the 50 nm film in green. In red, the transition of the full film is shown. The superconducting to normal transitions start at about the same temperature in both cases, however, the transition of the film is over ten times broader than that of the wire. In the case of the wire, the transport measurement probes only an area 3 µm wide and 120 µm long. The sharp transition of the wire indicates that the film is homogeneous in that small region. In comparison, in case of the film, the area probed is several millimetres long and wide. The transition reflects the lowest resistivity along a filament. The observed broadening The 3 µm wide wire that was fabricated on the 100 nm thick sample was measured and its transition is shown in figure 4.15(b) in red, along with the previously discussed 3 µm wide wire on the 50 nm thick film in green. To be able to compare the two curves, the resistivities were normalised with respect to their residual resistivities. The temperature scales are different for the two curves. The transition of the 100 nm thick film is 10 times broader than that of the 3 µm wide wire fabricated on it (from table 4.3). The transition of the wire made of the 100 nm thick film is also significantly narrower and smoother than the wire made of the 50 nm thick film. This can be observed in figure 4. 15(b).

The critical temperature of the wire fabricated onto the 100 nm film is the closest to what is expected for a bulk sample.

On the third, 25 nm thick sample, the 400 nm, 200 nm, and 100 nm wide wires were measured, their transitions, and the transition of the film is shown in figure 4. [START_REF] Daunt | Superconductivity of rhenium[END_REF]. The transition curves were normalised with respect to their residual resistivity. The temperature axis was left unchanged. About half way through the transition curve of the film a step can be observed. This suggests that the volume measured has two distinct transition temperatures. Similar steps can be observed in the same temperature region on the 100 nm wide wire. This curve has a third step also at a lower temperature. The 200 nm wide wire appears smoother compared to the film, and to the 100 nm wide wire, however, it is broad and perfectly envelopes both curves. The transition of the 400 nm wide wire is the sharpest of the four and relatively featureless.

Conclusions on the ρ(T ) curves

In the literature, previously reported critical temperatures of rhenium vary in a wide range. They can be found anywhere between 0.9 K and 5.5 K. After characterising several samples prepared with different methods, Hull and Goodman concluded that the correct critical temperature of a strain free rhenium single crystal is 1.699 K. They reported an increase of 1 K in the critical temperature after surface grinding [START_REF] Hulm | Superconducting properties of rhenium, ruthenium, and osmium[END_REF].

Alekseevskii et al. reported an increase in the critical temperature, and the κ parameter (defined in equation 3.21) of deformed, bulk rhenium samples. Increase of the κ parameter signals that bulk rhenium changes from a type-I to a type-II superconductor upon deformation [START_REF] Alekseevskii | The superconducting properties of rhenium[END_REF]. Kopetskii et al. suggested that dislocations increase the critical temperature, but do not affect the residual resistivity after studying deformed, annealed and quenched rhenium single crystals. Point defects, however, do affect both T c and the residual resistivity. They stated that planar defects, such as twin boundaries have an insignificant effect on both parameters [START_REF] Kopetskii | Critical temperature of superconducting transition in plastically deformed rhenium single crystals[END_REF]. Their last statement is in contrast with observations of localised superconductivity along twin boundaries, with critical temperatures higher than that of bulk [START_REF] Khlyustikov | Localized superconductivity of twin metal crystals[END_REF].

Haq et al. conducted annealing experiments on rhenium thin films, and, consistently with the previous authors, concluded that vacancies and dislocations are responsible for the increased residual resistivity and transition temperature, and that grain boundaries do not contribute to this effect [START_REF] Haq | Electrical and superconducting properties of rhenium thin films[END_REF]. Chu et al. studied the effects of pressure on the critical temperature of rhenium. They observed a non-linear dependence: the critical temperature decreases steeply initially, passes through a minimum, then levels off, as shown in figure 4.17 [START_REF] Chu | Study of fermi-surface topology changes in rhenium and dilute re solid solutions from T c measurements at high pressure[END_REF]. Here, the pressure directly translates to strain, and thus the expected transition temperature can be calculated from the strain measured in our films.

In figure 4.17, instead of strain the relative volume change is given. Relative change of the hexagonal unit cell volume can be calculated from the strain as follows:

∆V V 0 = 3 √ 3 2 (a 0 + ∆a) 2 (c + ∆c) -a 2 0 c 3 √ 3 2 a 2 0 c = 2 a + 2 a c + 2 a c + 2 a + c , where a = ∆a a 0 , c = ∆c c 0 = - 2ν 1 -ν a .
The misfit strain results in a relative change in volume of 0.5%. The strain obtained from the high resolution X-ray scans give a much larger value, ∼ 1%. As can be read from figure 4.17, the critical temperature only changes 0.1 K before it levels off on a value close to the bulk. If only strain were present in our samples, we would measure critical temperatures closer to the bulk value. Thus, we can conclude, in agreement with previous authors and our previous measurements, that dislocations (and possibly vacancies, though this cannot be confirmed at this point) are present in our films, which cause an increase in the residual resistivity and the critical temperature.

In figure 4.18 the residual resistivity ratios of the wires are shown as the function of their critical temperatures. The two are inversely proportional which further confirms, that the increase of the critical temperature is caused by crystallographic defects in the sample. The widths of the transition in case of the 100 nm and 50 nm thick samples are broader on the film. This is a sign that the films are composed of domains, that transit to the superconducting phase at slightly different temperatures. The samples are not homogeneous on the mm 2 scale, which is the size of the region probed by the transport measurement on the full film. The areas of the wires are 360 µm 2 , however. The narrower transition suggests that the domains are larger than the size of the wires. Inhomogeneity was reduced with the reduction of the size.

The transition curves of the 25 nm thick sample (film and wires) display a very clear structure. It was shown by the AFM images and by the XRD measurements that this film has a grainy structure, with two distinct grains that likely have two different orientations. Figure 4.17(a) and figure 4.17(b) shows that while the critical temperature of two polycrystalline samples behave slightly differently under applied pressure, the T c of two single crystal samples can be described by a single curve. This could indicate that crystal properties, such as orientation, could affect the critical temperature. The possible cause of the steps in figure 4.16 is thus that the two types of grains have slightly different transition temperatures. When fabricating the wires, their volume ratio changes, which affects the 'height' of the step in the transition curve. The sizes of the grains are much smaller than the width of the narrowest wire (see table 4.2), therefore the measured volume remains inhomogeneous. The widths of all the transitions measured on the 25 nm thick sample, listed in table 4.4, are in close agreement.

The critical temperature of a superconductor calculated from the BCS theory is given in equation 3.12, the important parameters are the density of states at the Fermi level and the pairing potential. The band structure of rhenium was calculated for the first time by Mattheiss [START_REF] Mattheiss | Band structure and fermi surface for rhenium[END_REF]. The introduction of crystal defects can lead to an increased unit cell volume and thus to increased density of states at the Fermi level in Re [START_REF] Mito | Large enhancement of superconducting transition temperature in single-element superconducting rhenium by shear strain[END_REF], as well as a change in the pairing potential which affects the critical temperature. Mito and collaborators observed T c as high as 3.2 K for Re polycrystals submitted to shear strain leading to a volume expansion of 0.7%.

The critical temperature of the 3 µm wire on the 50 nm film stands out. It is over 0.2 K lower than the bulk critical temperature, 1.7 K, which is tantalising. This film is decorated by large spirals and deep holes. We can only speculate how these affect T c : is there a Re wetting layer underneath, is the stress the same over the thickness of the film and in the spirals? These questions need further investigation.

Mean free path and coherence length

The electron mean free path and the superconducting coherence length can be obtained from the measured residual resistivity and critical temperature values.

According to the Drude model, the expression to calculate resistivity is the following:

ρ = m e n e e 2 τ = m e v F n e e 2 l , (4.6)
where n e is the density, e is the charge, m e is the weight of electrons. τ is the time between collisions, which is equal to l/v F , where v F is the Fermi velocity and l is the mean free path of the electrons [START_REF] Sólyom | Fundamentals of the Physics of Solids, volume I. Structure and Dynamics[END_REF]. Multiplying both sides with l shows that the product of the resistivity and the mean free path is nominally a constant:

ρl = m e v F n e e 2 . (4.7)
This product was measured by several authors for rhenium, yielding the following values: 4.5e-5 µΩcm 2 [START_REF] Haq | Electrical and superconducting properties of rhenium thin films[END_REF], 2.16e-5 µΩcm 2 [START_REF] Tulina | Point-contact spectroscopy of renium single crystal[END_REF], and an average of 2.01e-5 µΩcm 2 [START_REF] Tulina | Influence of purity on the magnetic properties of rhenium in the superconducting state[END_REF].

The last two values reported by Tulina et al. are in good agreement, but the first is very different. In the expression 4.7 the density of electrons is the only parameter that can change between samples. In reference [START_REF] Tulina | Influence of purity on the magnetic properties of rhenium in the superconducting state[END_REF], the resistivities and mean free paths of several rhenium samples were measured separately. Their RRRs are also listed. From reference [START_REF] Tulina | Influence of purity on the magnetic properties of rhenium in the superconducting state[END_REF], samples with the closest RRR values to our samples were chosen, and their ρl product was used to calculate the mean free paths of our samples. These values are listed in tables 4.3 and 4.4.

To calculate coherence length, the superconducting energy gap (∆(0)) needs to be obtained first using equation 3.14. Values for all the samples are listed in tables 4.3 and 4.4.

From the energy gap, the coherence length of the superconducting electrons (ξ 0 ) can be calculated using equation 3.16. A Fermi velocity was obtained by averaging the values published in reference [START_REF] Tulina | Influence of purity on the magnetic properties of rhenium in the superconducting state[END_REF], which gave 2. • 10 5 m/s. The coherence lengths are listed in tables 4.3 and 4.4.

The coherence length can be compared to the mean free path of the electrons to determine whether the sample is in the clean or in the dirty limit. In the clean limit, the electrons can travel the characteristic distance of superconductivity without scattering.

The 50 nm and the 100 nm thick film were in the clean limit, but after fabrication, the 3 µm wires are in the dirty limit. Whether this is due to the fabrication process or to the ageing of the sample is not known. However, the three characteristic lengths of both wires and both films are all larger or equal to 100 nm, except one.

The 25 nm thick film and its wires are in the dirty limit.

The effective coherence length is obtained from equations 3.22, which takes into account the effect of the electron mean free path. It is shorter than the coherence length.

In our study, we reached the clean limit with the 100 nm and the 50 nm thick film. In addition, the mean free path is larger than the thickness, conditioning the entrance in the ballistic regime along the thickness direction. Unfortunately, we did not pattern the narrowest wires on these films. We could not yet reach the ballistic regime in the width direction of the wire.

Critical current fluctuations in SQUIDs

Nanobridge SQUIDs were successfully fabricated on the 25 nm thick rhenium film using electron beam lithography. An example of a SQUID is shown in figure 4.19. The two forks of the SQUID are connected by two narrow bridges, with widths smaller than the superconducting coherence length (ξ 0 ). The fabricated pattern corresponds to figure 4.2 and included 2 SQUIDs. The bridges were designed to be 20 and 50 nm wide. Their dimensions were measured by SEM after fabrication: the narrower bridges were about 40 nm, the wider ones were about 70 nm. The SQUID shown in figure 4.19 has the wider bridges.

Considering that in the superconducting state the current flows in the middle of the the SQUID arms, which measure 200 nm across, the effective area of the SQUID loop is 1.2 µm 2 . A single electrode is used to bias the SQUID and to detect the ∂V /∂t pulse. One of the connections is schematically indicated in figure 4.20. The measurements were performed using a two-terminal method. In the case of this sample, the ground was connected to the pad marked on the top, and the current bias was connected to pad on the left. In this figure, an SEM image taken of the SQUID is also shown. The period of the oscillation is defined by the area (S) enclosed by the SQUID:

I c = 2i c cos πΦ Φ 0 = 2i c cos πBS Φ 0 , (4.8)
where i c is the critical current through one junction of the SQUID, and Φ = BS is the applied flux, given by the dot product of the magnetic induction vector and the vector loop area of the surface. Since these two vectors are parallel, the dot product is simply the product of the magnitudes in equation 4.8.

From the period, the area enclosed by the SQIUD is obtained as follows:

S = Φ 0 ∆B . (4.9)
The periodicity of the fast component corresponds to an area of 40.5 µm 2 , which is much larger than the 1.2 µm 2 size of the SQUID. However, the slow oscillation gives an area of 1.4 µm 2 , which is in good agreement with the SQUID loop. We are measuring two SQUIDs with different areas. This is shown in figure 4.22. The small SQUID (S 1 , Φ 1 ) is the one that was intentionally fabricated and wired. It is connected to either a Josephson junction or a second SQUID, and together they form a larger SQUID loop (S 2 , Φ 2 ).

On design III, there are two SQUIDs next to each other. They share the ground node, shown in figure 4.20. Their current pads and wires are very close together. It is possible, that there is an electrical connection between either the pads or the wires, that the optical microscope did not reveal. This way, the supplied current biases both SQUIDs, leading to this double SQUID. A second SQUID (SQUID2) was also measured. The critical current is shown in figure 4.23 as the function of the applied magnetic field. The maximum critical current modulation is approximately 20 µA, which is in agreement with SQUID1. The graph shows no obvious periodicity, we suspect a beating of several frequencies. There is a local maximum centred at zero applied field, as expected, and there are maxima on both sides of this peak. Some of these maxima have a clear period, and some do not. To have a clearer picture whether this data shown in this graph have the periodicity corresponding to the size of the SQUID, Fourier transform of the critical current was computed, and is shown in figure 4.24.

The most dominant peak is at 0.385 mT -1 , which corresponds to a period of 2.6 mT. The area enclosed by the loop calculated from the period is 0.8 µm 2 . This value is consistent with the 1.2 µm 2 SQUID loop area.

4.25(b

). An increasing DC current is injected into the SQUID. As the value of the current approaches the critical current, the probability that the SQUID switches to the normal state increases. The number of switching events exponentially increases with the current until I c is reached, where the probability of switching is almost 1. As a result, for currents close to the critical current the number of events rapidly decreases [START_REF] Rabaud | Courants permanents dans des anneaux mésoscopiques connectés[END_REF]. Switching current histograms are asymmetric, exhibit a tail towards the lower currents. In our case the number of switching events measured were low, so the tails are not well defined.

The mean critical current ( I c ) and its standard deviation (σ Ic ) were extracted from the histograms, and they are shown in figures 4.25(a) and 4.25(b).

From σ Ic the flux noise (∆Φ) of the SQUID can be determined using the following expression:

∆Φ = ∆I c ( ∆t Ic dI dt ) 1/2 1 dIc dΦ , (4.10)
where ∆t is the measurement time interval, dI/dt is the slope of the current ramp used in the measurement [START_REF] Rabaud | Courants permanents dans des anneaux mésoscopiques connectés[END_REF][START_REF] Hazra | Nano-superconducting quantum interference devices with suspended junctions[END_REF]. dIc dΦ was determined from the slopes of the I c (B) curves. For SQUID1 the high frequency oscillations were used. The slope is 1.03 ± 5 mA/mT, which corresponds to (using the area obtained in equation 4.9) 51 µA/Φ 0 . For SQUID2, on the positive side of the 0 T peak the slope is 36 µA/mT, and on the negative it is 50µA/mT. Using the obtained loop area, these correspond to 90 µA/Φ 0 and 124 µA/Φ 0 , respectively.

The obtained flux noise values are 2.6e-5 Φ 0 /Hz 1/2 for SQUID1 and 2.0e-4 Φ 0 /Hz 1/2 for the SQUID2.

Theoretically, the highest current a superconductor can carry without dissipation is defined by the depairing mechanism. Superconductivity vanishes when the kinetic energy associated with the supercurrent exceeds the condensation energy (binding energy of the Cooper pairs) [START_REF] Tinkham | Introduction to superconductivity[END_REF].

The depairing current density is given by the following expression:

j dp = e * Ψ 2 0 2 3 2 3 |α(T )| m * ,
where e * = 2e and m * = 2m are the charge and mass of the superconducting electron pairs, Ψ 0 is the equilibrium value of the superconducting order parameter, and α(T ) is a coefficient from the Ginzburg-Landau theory (see equation 3.8) [START_REF] Tinkham | Introduction to superconductivity[END_REF].

Using equations 3.19 and 3.20, the depairing current can be expressed as follows:

j dp = 1 3 √ 3 1 π Φ 0 µ 0 λ 2 L (T )ξ GL (T ) , (4.11)
where µ 0 = 4π • 10 -7 Wb/(Am) is the vacuum permeability.

The sample is in the dirty limit, so in equation 4.11, instead of the London penetration depth the effective penetration depth was used, which is obtained by the following equation:

λ ef f = λ L ξ 0 l . (4.12)
The London penetration depth of rhenium thin film was measured by Hykel [START_REF] Hykel | Microscopie à micro-SQUID: étude de la coexistence de la supraconductivité et du ferromagnétisme dans le composé UCoGe[END_REF] and Wang [START_REF] Wang | Superconducting properties of iron-based Ba-122 by transport measurements and scanning nano-SQUID microscopy[END_REF] by studying vortices, and by Dumur et al. [START_REF] Dumur | Epitaxial rhenium microwave resonators[END_REF] by studying rhenium microwave resonators. They obtained values of 79 nm, 103 nm, and 85 nm for λ L , respectively. Here, λ L was assumed to be 90 nm.

Instead of ξ GL , ξ ef f was used (dirty limit sample). The coherence lengths, effective coherence lengths and mean free paths of the wires fabricated from this sample was calculated. We took the average values of l, ξ 0 and ξ ef f to estimate depairing current density.

A theoretical maximum current density of 4e10 A/m 2 was obtained.

The maximum critical current we measured was about 72 µA. The current flows through the two arms of the SQUID. The cross sectional area of the SQUID arm can be calculated by multiplying the sample thickness (25 nm) with the width of the bridge (70 nm). The critical current density (j c ) is we measured is then

j c = 72 µA 2 • 25 nm • 70 nm = 2e10 A/m 2 . (4.13)
The measured current density is half of what an ideal rhenium wire could theoretically carry. This is not surprising, as achieving the theoretical critical current in superconductors is subject of active research. Attempts have been made to reach the depairing current either by reducing the dimensions of the superconductor below the characteristic lengths or by introducing artificial pinning sites to stop the motion of vortices [START_REF] Dinner | Depairing critical current achieved in superconducting thin films with through-thickness arrays of artificial pinning centers[END_REF], [START_REF] Nawaz | Approaching the theoretical depairing current in yba2cu3o7âĹŠx nanowires[END_REF], [START_REF] Xu | Achieving the theoretical depairing current limit in superconducting nanomesh films[END_REF].

Conclusion on the SQUID measurements

We have successfully fabricated SQUIDs from rhenium thin films. The critical current oscillations measured on two SQUIDs were imperfect, probably due to contaminations on the surface (left-over rhenium from fabrication or other superconductive particle). The width of the critical current histogram could reach values which make rhenium a promising candidate for low noise µSQUIDs. Rhenium has been shown to have long coherence length and electron mean free path, therefore it would be promising to continue the study of rhenium SQUIDs on samples that have better crystallographic properties, as the presented sample was a dose test to develop Re SQUID fabrication. Future patterns should include only SQUIDs, with all other superconducting structures as far away as possible. Before these preliminary experiments, the measurement of the wires reduced the number of SQUIDs available, and the sample was handled many times before we could undertake the SQUID experiments.

Conclusion and outlook

Conclusion

In this work, the epitaxial growth of rhenium thin films onto single crystal Al 2 O 3 using molecular beam epitaxy was realized and is discussed. An epitaxial relationship was found with orientations (0001)Al 2 O 3 //(0001)Re and <2110>Al 2 O 3 //<0110>Re. This was confirmed using X-ray diffraction. The misfit strain between the lattices is -0.43% at room temperature, which gives a critical thickness of about 15 nm. The substrates were heated during growth using either a Joule-heated tungsten filament located behind the sample or electron bombardment. An AFM study comparing films grown at temperatures of either 800 • C or 900 • C revealed that the higher deposition temperature results in a more homogeneous surface. On samples with thicknesses 50 nm and 100 nm, spirals are frequently observed. The diameter of these spirals grew over two fold when the higher deposition temperature was used. An XRD study of the sample films showed that they are all dominated by the epitaxial (0001) orientation. The few secondary orientations have low intensities which in almost all cases decrease with increasing deposition temperature. Deposition at a temperature of 1000 • C leads to dewetting of the 50 nm thick sample, and islands with atomically flat surfaces are formed.

The spirals that are often observed on thicker films are most likely the result of steps on the surface caused by screw dislocations. Among the spirals there are deep holes, whose origin is suspected to be partial dewetting and recrystallisation of the film. It was shown by a theoretical model that the temperature of the film starts to increase when the thickness of approximately 10 nm is reached, as the film becomes more opaque. Around this thickness a transformation of the RHEED pattern indicates a crystallographic change, and the observed surface shows signs of dewetting. The surface profile was modeled using Mullins' theory of dewetting, which allowed the determination of the surface diffusion coefficient, 4 × 10 -12 cm 2 /s. Wires with widths between 100 nm and 3 µm, and SQUIDs were fabricated on the films using the lithography process. Low temperature transport measurements showed that the fabrication did not affect the superconducting properties. The critical tempera-163 ture of the wires was found to vary in a wide range, between 1.43 K and 1.96 K. We found that this correlates with the crystallography and topography of the films. The mean free paths and the superconducting coherence lengths were determined. Two films were in the clean limit, but the wires fabricated on them were in the dirty limit. The mean free paths and the coherence lengths were larger than the thickness of the films for almost all the films and wires conditioning the ballistic regime in the thickness direction. The ballistic regime was not yet obtained in the width direction.

Critical current oscillations of two SQUIDs were measured using a dilution refrigerator. The lowest flux noise value obtained was 2.6 × 10 -5 Φ 0 /Hz 1/2 .

Outlook

The initial intention of this project was to grow epitaxial Re-Al 2 O 3 -Re junctions. Rhenium is a promising candidate for such junctions as it is known to resist oxidation. To manufacture a junction, a rhenium film with flat surface needs to be deposited, followed by the deposition of an aluminium layer which is subsequently oxidised. However, the epitaxial rhenium films are found experimentally not to have a flat surface. They are covered with spirals and deep holes. This topography is not adequate for the deposition of a second layer.

Welander grew rhenium films onto thick epitaxial niobium layers [START_REF] Paul | Structural evolution of re (0001) thin films grown on nb (110) surfaces by molecular beam epitaxy[END_REF]. His films were relatively flat and smooth, however, displayed several in-plane orientations, and rhenium mixed with niobium at the interface. Our films were flat in the case of grainy structure. The growth of grainy rhenium films is one possible way to avoid holes, and achieve a flat surface. Of course, in this case the aim of fully epitaxial, single crystal junction has to be sacrificed. A second line of investigation could be the use of a seed layer, that prevents the dewetting of rhenium, thus the formation of holes.

We obtained larger mean free path and coherence length values than the thickness of the films, and the width of the thinnest wire, 100 nm. Unfortunately, the 100 nm wide wires were fabricated on a sample which was in the dirty limit, and displayed small λ and ξ values. Wires that are in the ballistic regime in both the thickness and width directions are feasible on a clean limit film, which we can routinely deposit.

The SQUIDs fabricated on a thin film of rhenium showed a flux noise in the range of 10 -5 Φ 0 /Hz 1/2 and 10 -4 Φ 0 /Hz 1/2 . Flux noise values reported in literature for low noise µ-SQUIDs and nano-SQUIDs range between 10 -4 Φ 0 /Hz 1/2 and 10 -6 Φ 0 /Hz 1/2 [START_REF] Wang | Superconducting properties of iron-based Ba-122 by transport measurements and scanning nano-SQUID microscopy[END_REF][START_REF] Hasselbach | Microsquid magnetometry and magnetic imaging[END_REF][START_REF] Hazra | Nano-superconducting quantum interference devices with suspended junctions[END_REF][START_REF] Ketchen | Low noise dc squids fabricated in nb-al/sub 2/o/sub 3/-nb trilayer technology[END_REF][START_REF] Finkler | Self-aligned nanoscale squid on a tip[END_REF][START_REF] Hao | Measurement and noise performance of nano-superconducting-quantum-interference devices fabricated by focused ion beam[END_REF][START_REF] Hazra | Nano-superconducting quantum interference devices with continuous read out at millikelvin temperatures[END_REF][START_REF] Hykel | Microsquid force microscopy in a dilution refrigerator[END_REF][START_REF] Hazra | Retrapping current in bridge-type nano-squids[END_REF]. It is very encouraging that our preliminary results fall in this range. By using clean limit films, and refining the lithography parameters, state of art rhenium SQUIDs should be achievable.

A Ptychography

A.1 Phase problem in crystallography

In a scattering experiment, a sample is subjected to a parallel monochromatic beam with a known wave vector (k i ), therefore known energy and propagation direction. The angle (elastic scattering) and/or the energy distribution (inelastic scattering) of the scattered wave is then studied to draw conclusions regarding the crystallographic (or magnetic or dynamic) properties of the sample.

When the scattering is dominantly elastic, the energy of the incoming wave does not change during the interaction with the sample. This means, the outgoing wave vector has the same length as the incoming wave vector (|k i | = |k f |). Momentum transfer does occur, however, resulting in a directional change. The difference is called the scattering vector:

q = k f -k i . (A.1)
The amplitude of scattered X-ray waves is given by the Fourier transform of the electron density (f (r)), where the integral is taken across the illuminated volume (V ):

165 ξ L = λ ∆λ -1 λ 2 ≈ λ 2 2∆λ (A.4)
Equation A.4 tells us, that higher degree of monochromaticity results in a longer coherence length. Similarly, longer wavelength, therefore lower energy gives longer coherence length. The I13-1 beamline in Diamond Light source uses X-rays in the energy range of 6-20 keV, which corresponds to a wavelength in the range of 2-0.6 Å. λ = 1 Å midrange. Monochromatic beam is achieved by diffraction through a series of perfect single crystals. A double pass Si [START_REF] Bascom | Experimental evidence for quantized flux in superconducting cylinders[END_REF] monochromator can achieve a bandwidth of ∆λ/λ = 10 -4 , resulting in a longitudinal coherence length of 0.5 µm [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF][START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF]. A, until A and B have opposite phases. Within distance 2ξ T they have the same phase again. Considering the above, the transverse coherence length is ξ T = 1/2 λ/ tan ∆θ.

Beam divergence can be caused by the finite size of the source. In figure A.2, D denotes the size of the source, wavefronts A and B are emitted at either ends. R denotes the distance from the source. Using this, tan ∆θ can be expressed as D/2R, and substituting to the previously obtained expression, the transverse coherence length is the following:

ξ T = λR D (A.5)
The transverse coherence length increases with wavelength, lower energy X-rays are favoured. It also increases with distance from the source. This is why the I13-1 experimental hall in Diamond Light Source is located in a separate building about 130 m away from the main building. Lastly, ξ T is inversely proportional to the size of the source. This motivates reducing the spread of electron bunches in the storage ring, that produce the probing X-ray beam. Alternatively, a slit can be placed close to the source to create a virtual source reducing the size [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF][START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF][START_REF] Robinson | Optimisation of coherent x-ray diffraction imaging at ultrabright synchrotron sources[END_REF]. This is a major motivation for synchrotron facilities to upgrade their ring lattice, such as MAX-IV in Lund and ESRF in Grenoble.

A synchrotron X-ray source measures about 100 µm vertically and 10 µm horizontally. If the wavelength is approximately 1 Å, and the experiment is carried out 100 m away from from the source, the transverse coherence length is 100 µm horizontally, and 1 mm vertically. Specifically for the I13-1 beamline, coherence lengths of 200 µm (horizontal) and 350 µm (vertical) were demonstrated [START_REF] Wagner | Characterising the large coherence length at diamond's beamline i13l[END_REF].

Synchrotron radiation is only partially coherent. Individual electrons emit coherently, but the batch does not. Coherent beam is produced by inserting a slit of the size of the transverse coherence length in the beam. This only allows the coherent portion through, part of the flux has to be sacrificed [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF][START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF]. In case of the I13-1 beamline, the initial flux of 7 × 10 14 photon per second per 0.1% bandwidth (Ph/s/0.1%BW) is reduced to a coherent flux of about 10 10 Ph/s/0.1%BW [START_REF] Rau | The imaging and coherence beamline i13 at diamond[END_REF].

The diffraction pattern produced by a coherent beam differs from one created by an incoherent beam. A diffraction peak obtained using an incoherent beam is the incoherent sum of the scattering by different domains (n) in the sample (I(q) = n |F n (q)| 2 ). This results in a diffuse pattern. When using a coherent beam, the peak is the coherent sum of the scattering by different domains (I(q) = | n F n (q)| 2 ). The pattern then shows sharp intensity fluctuations, known as speckles [START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF][START_REF] Sutton | Observation of speckle by diffraction with coherent x-rays[END_REF]. If the scattering object is smaller than the footprint of the beam, fringes related to the shape of the object also appear.

A.3 Coherent diffraction imaging and ptychography

Coherent diffraction imaging (CDI) and ptychography are lensless imaging techniques that allow the reconstruction of the phase information. They are used with electrons and with X-rays as well. In both techniques a coherent X-ray beam is scattered by an object, and the scattered intensity is collected by a 2D detector in the far-field. In case of ptychography the complex amplitude of both the probing and the scattered wavefront can be reconstructed with iterative algorithms. In case of CDI one is assumed to be known (usually the probing wavefront) and the other (usually the scattered wavefront) is recovered.

CDI is used on samples smaller than the footprint of the coherent beam. This way the whole volume of the sample takes part in the scattering process.

Ptychography is the scanning version of CDI. The concept of ptychography was first put forward by Hoppe [START_REF] Hoppe | Beugung im inhomogenen Primärstrahlwellenfeld. I. Prinzip einer Phasenmessung von Elektronenbeungungsinterferenzen[END_REF][START_REF] Hoppe | Beugung in inhomogenen Primärstrahlenwellenfeld. II. Lichtoptische Analogieversuche zur Phasenmessung von Gitterinterferenzen[END_REF] to be used with scanning transmission microscopy (STEM), and the proof of the concept was demonstrated by Hoppe and Strube [START_REF] Hoppe | Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten[END_REF] using visible light. It was not developed for STEM at the time, because the instrumentation was not sufficiently developed [START_REF] Rodenburg | Ptychography and related diffractive imaging methods[END_REF]. Thanks to the advances that were made since in X-ray optics and computation, the advantages of ptychography are being discovered.

During the course of a ptychography measurement a large sample is scanned along a predefined path by a coherent beam. The diffraction patterns are collected at each point of the path. The complex amplitude of the scattered and the probing beam is reconstructed.

Redundancy is introduced in the data by partially overlapping the footprint of the beam between the steps in the scan. This overlap is then used as a constraint in the reconstruction algorithm. The ideal degree of overlap was determined to be 60% by Bunk et al. [START_REF] Bunk | Influence of the overlap parameter on the convergence of the ptychographical iterative engine[END_REF], with overlap (o) defined as o = 2r -a, where r is the radius of the footprint and a is the centre-to-centre distance.

A.3.1 Oversampling criterion

Retrieval of the phase relies on the concept of oversampling the diffraction pattern.

Nyquist-Shannon theorem states that a continuous function can be completely determined with a sampling frequency twice the highest frequency component of the signal [START_REF] Shannon | Communication in the presence of noise[END_REF]. This frequency is called the Nyquist sampling frequency. It is important to note that we are talking about the probing periodicity of the diffraction plane, which, considering a 2D detector, translates to spatial frequency. This minimum spatial sampling frequency determines the number of detector pixels required per fringe or speckle. When sampling at the Nyquist frequency, the amplitude can be recovered, but half the information, the phase, cannot. Sayre showed that if the diffraction pattern is sampled at at least twice the Nyquist sampling frequency, the phase of the scattered wave can also be recovered [START_REF] Sayre | Some implications of a theorem due to Shannon[END_REF].

A.3.2 Phase retrieval methods

The phase problem is solved by applying inverse Fourier transform to the diffraction patterns to recover the complex scattered wavefront ('object') via the use of the convolution theorem.

The first algorithm for ptychography was put forward by Rodenburg and was termed the ptychographic iterative engine (PIE) [START_REF] Rodenburg | A phase retrieval algorithm for shifting illumination[END_REF]. It was successful in solving the phase problem, however, it required an accurate knowledge of the incident probing wavefront ('probe') and the stage positions during the scan. The necessity of the accurate knowledge of the probe was removed by the development of the extended ptychographic iterative engine (ePIE), which can recover the probe from a rough estimate as well as the object [START_REF] Maiden | An improved ptychographical phase retrieval algorithm for diffractive imaging[END_REF]. Independently of Rodenburg and his coworkers, around the same time Thibault et al. developed their own algorithm based on the difference map algorithm (DM), which is also capable of reconstructing both the probe and the object [START_REF] Thibault | Probe retrieval in ptychographic coherent diffractive imaging[END_REF]. Later, it was demonstrated (using ePIE) that errors in samples positions can be corrected for in the algorithm [START_REF] Zhang | Translation position determination in ptychographic coherent diffraction imaging[END_REF]. Furthermore, the DM method was extended by Thibault et al. to take into account partial coherence in both the longitudinal and the transverse directions [START_REF] Thibault | Reconstructing state mixtures from diffraction measurements[END_REF]. This is useful, as it removes the strict restriction on coherence, which limits the flux [START_REF] Parsons | Coherent Diffraction Imaging using a High Harmonic Source at 40 eV[END_REF].

Below the two most commonly used iterative reconstruction algorithms, Rodenburg's ePIE and Thibault's DM method, are described.

Both algorithms rely on two assumptions: the interaction between the object function (O(r)) and the probe function (P (r)) can be modelled by a complex multiplication, and the scattered wavefront can be modelled by the Fourier transform (F).

The extended ptychographic iterative engine

Below the ePIE method is introduced following references [START_REF] Maiden | An improved ptychographical phase retrieval algorithm for diffractive imaging[END_REF]164,[START_REF] Stephan | High-Resolution 3D Ptychography[END_REF].

Based on the assumptions above the exit wave is given by the following:

ψ(r) = O(r)P (r -R), (A.6)
where R refers to the position of the beam on the sample along the path. The exit wave observed in the far field is:

where |P j (r -R j )| 2 max refers to the maximum value of |P j (r -R j )| 2 , P * j (r -R j ) is the complex conjugate, and the same stands for O j (r). α and β are constants that adjusts the step size of the update.

One iteration is complete when the algorithm ran through all J number of diffraction patterns. The updated object and probe functions are the new guesses in the next iteration.

The convergence is monitored by the following metric:

E = j q | I j (q) -|ψ j (q)|| 2 j q I j (q) , (A.14)
The aim is to minimise E.

The difference map algorithm

The DM algorithm is detailed below. The discussion here adheres to references [START_REF] Thibault | Probe retrieval in ptychographic coherent diffractive imaging[END_REF]164,[START_REF] Johannes | Ptychographic X-ray Microscopy and Tomography[END_REF][START_REF] Thibault | Algorimethric methods in diffraction microscopy[END_REF].

The DM method also iterates between real and Fourier space using the object and probe functions, but addresses all J diffraction patterns in the same time. It is parallel rather than sequential. The DM algorithm solves the phase problem by searching the intersection point of two constraint sets, one defined in real space, the other in Fourier space. Both constraint sets are associated with a projection operator, that map the iterations onto the constraint sets.

The first constraint set is the Fourier constraint, which relates the observed intensities to the scattered waves via the Fourier transform:

I j (q) = |F[ψ j (r)]| 2 , ∀j. (A.15)
The second is the overlap constraints, which states that the each scattered wave in the ptychographic scan can be factorised as a probe and an object function:

ψ j (r) = P (r -R j )O(r), ∀j. (A.16)
The task of the algorithm is to find the series of O and P that satisfy these two constraints.

A state vector is defined as Ψ(r) = {ψ 1 (r), ψ 2 (r), ψ 3 (r), ..., ψ J (r)} using the initial guesses for the the probe and the object functions.

The modified atomic form factor in case of X-ray scattering is the complex electron density [START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF][START_REF] Takagi | A dynamical theory of diffraction for a distorted crystal[END_REF].

Non-symmetric Bragg peaks can be decomposed into the symmetric and asymmetric contributions. The symmetric part can be considered to come from the average electron density, and the antisymmetric part is associated with a phase that equals to the projection of the local displacement along the scattering vector g. This displacement can be imaged as a real-space map of phase values across the illuminated or scanned area [START_REF] Robinson | Coherent x-ray diffraction imaging of strain at the nanoscale[END_REF].

Over the course of the Bragg ptychography measurement we only recorded the symmetric (002) reflection of rhenium. This reflection only carries information on the displacements along the direction of the surface normal.

It is possible to recover the complex amplitude along the depth of the scattering volume of the scanned area. This technique is known as 3D ptychography, and gives 3 dimensional maps of the modulus and the phase. To achieve this, the ptychographic scans are repeated at and around the Bragg angle along the rocking curve. The technique was successfully demonstrated by Godard et al [START_REF] Godard | Three-dimensional high-resolution quantitative microscopy of extended crystals[END_REF]. This is demanding measurement as it requires precise alignment and stability of the setup over several hours.

Hruszkewycz et al. recently demonstrated that 3D reconstruction is possible form the collection of lateral scans at a single angle [START_REF] Hruszkewycz | High resolution three dimensional structural microscopy by single angle bragg ptychography[END_REF]. Rocking the sample around the Bragg angle is not necessary.

A.4 I13-1 beamline in Diamond Light Source

Ptychography experiments were carried out on the I13-1 beamline in Diamond Light Source on rhenium thin films.

An areal photograph of the Diamond Light Source is shown in figure A.5(a). The schematics of a beamline is shown in figure A.5(b). The dimensions are specific to the Diamond Light Source and the I13 beamline. The electrons are traveling in the storage ring at an energy of 3 GeV. The ring is not a true circle but a 48 sided polygon. Undulators, shown in figure A.5(c), are composed of a series of dipole magnets with alternating polarity. These are placed in the straight sections of the ring to make the electrons oscillate, and emit X-ray radiation. A specific energy of this radiation is chosen by monochromators, and focused onto the sample under investigation. Most beamlines are located inside the main building, around the storage ring. However, on the I13-1 beamline coherent X-ray radiations is used, which is achieved by placing the experiment far away from the source of the X-ray. The experimental hutch is located in a building separate from the synchrotron storage ring. The total length of the beamline measured [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF] (c) Undulators are composed of a series of dipole magnets, that make electrons oscillate to generate X-ray radiation [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF].

from the source is 250 m.

The experimental setup is shown in figure A.6. The sample stage has a 30 • pitch by default with respect to the beam, and can be further tilted by ±15 • around the two in-plane axes. Lateral and vertical movement is achieved by two sets XYZ motors, one allows the rough alignment of the sample, another is for fine movements (5 nm resolution) [START_REF] Z D Pešić | Experimental stations at i13 beamline at diamond light source[END_REF][START_REF] Rau | Imaging in real and reciprocal space at the diamond beamline i13[END_REF].

The detector is placed on an industrial robot arm, part of which is visible in figure A.6. Including this arm, the setup is an 3+2 circle diffractometer, and is able to cover a wide range of hkl positions. To achieve sufficient sampling of the diffraction peak, the distance between the sample and the detector can be adjusted. Depending on the Bragg angle, it can be increased up to 5 m, the only limit is the ceiling. Using 9.4 keV X-rays, the Bragg angle for the (002) reflection of rhenium is 17.2 • , which allowed a distance of 2 m. Penetration depth of the X-rays can be obtained as the reciprocal of the attenuation coefficient, which can be found in tables [175]. At this energy for rhenium the penetration Spirals decorate the surface of all the rhenium samples that feature a single orientation. An AFM image of a spiral is shown in figure 2.22. Their sizes vary, and are thought to be related to the temperature of deposition, as was shown in section 2.2. Burton et al. explained the growth of spirals by the presence of dislocations with an edge component. This creates a step on the surface, which provides nucleation sites, and allows the spiral to grow [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF]. Whether this theory is valid or not in case of our rhenium films could be verified by the technique of ptychography.

Ptychography was preformed on features that were patterned onto the rhenium films using lithography. These clear cut features would allow us to check the resolution and the validity of the measurement.

The features included a square with size 2 µm x 2 µm on the 50 nm sample, which was described in section 2.2.2 (sample D deposited at 900 • C). An AFM image taken on one of the squares is shown in figure A.7. A square was scanned with the X-ray beam in a spiral fashion. Spiral path is often used, as it provides good overlap between spots, and eliminates artefacts associated with raster scan. in a 2 µm region of the starting point. Here, the complete footprint of the beam is on the rhenium. As the beam gets further away from the centre, the volume of rhenium that takes part in the diffraction process is reduced, so is the intensity recorded on the detector.

The distance between the consecutive points in the spiral was 0.4 µm, which (with spot size 1µm) gives the ideal overlap of 60%. Two examples of the diffraction patterns recorded in two different point of the spiral path are shown in figure A.9. The speckle pattern changes throughout the scan. The centre peak also shows some structure, it appears to be a double peak with small angular separation (approximately 0.002 • ). The direction and the magnitude of the separation was observed to change along the scan.

We recored scans for 3D ptychography on a 3 µm thick line by repeating the spiral scan at angles around the Bragg angle along the rocking curve. This is a demanding measurement, as it can take several hours, and the sample is required to be stationary, bar the rocking and the scanning motions. To ensure nothing moves, the experimental hutch is kept at a constant temperature by air conditioning, and the robot arm keeps the detector stable. Furthermore, the beam, the centre of the rotation of the stage, and the sample has to be aligned. A camera placed over the sample was used for the alignment. We did not manage to correct all the movements. Figure A.10 shows the position of the sample for each angle. Two of the intensity maps are also shown as a demonstration.

The centre of the line was obtained from the centre-of-mass of the modified intensity map. To correctly detect the middle of the line its weight in the centre-of-mass calculation had to be increased, so intensities below a fixed value were set to zero. This might indicate that there is some noise in the data, which would prevent the ptychographic reconstruction.

The position of the sample changes consistently with increasing angles in both negative and positive directions. The measurement was repeated latter for the same line, and the points are in close agreement. This suggest that the alignment was the best we could achieve with the setup available.

Reconstruction of the datasets is not a trivial task. To our knowledge, two reconstruction packages are available, both developed for Python environment. Reconstruction was attempted using the ptypy package [184] with no success yet. The pynx package currently only works for the small angle geometry and is being developed for the Bragg geometry [185,[START_REF] Vincent Favre-Nicolin | Fast computation of scattering maps of nanostructures using graphical processing units[END_REF]. Analysis of this data is a work in progress.

B Determination of surface coverage from XPS data

The technique of XPS was introduced in detail in section 1.4.1. The principle of the technique is the following: the sample is irradiated with a known energy X-ray beam, and the electrons (mostly photoelectrons) that escape the material are sorted by their kinetic energies, and counted. From their spectrum the chemical composition of the surface can be determined.

The surface monolayer coverage (σ = covered surface uncovered surface ) of an element (contamination or deposit) can be calculated from the intensity of the corresponding XPS peak. The intensity of a peak arising from the substrate can be expressed as follows:

I substrate ∝ (1 -σ)I ∞ substrate + σI ∞ substrate e -1 λ , (B.1)
where I ∞ substrate is the intensity that would be detected form the pure material. The first part on the left side of the equation B.1 is reduced by a factor of (1-σ), which corresponds to the reduced surface area which is not covered by contamination or deposit. The second part is the intensity that is transmitted through the monolayer coverage, therefore, it is reduced by e -1 λ , where λ is the inelastic mean free path of the photoelectrons, and can be found in tables.

Intensity that would be detected from a full monolayer can be calculated by integrating the exponential shown below:

I 1ML ∝ I ∞ bulk 1 0 e -z λ dz ∝ I ∞ bulk (1 -e -1 λ ),
where I ∞ bulk is the signal that would arise from a pure bulk of the same material. If only a fraction of the surface is covered by the contamination or deposit, the expression above is simply multiplied by σ:

I coverage ∝ σI ∞ bulk (1 -e -1 λ ).
From the ratio of the two intensities (R) the surface coverage can be calculated:

R = I substrate I coverage = I ∞ substrate I ∞ bulk 1 -σ(1 -e -1 λ ) σ(1 -e -1 λ )
.

C Transformation of the Bravais-Miller indices to Cartesian coordinates

Rhenium is a hexagonal closed-packed material, and the coordinate system of the hexagonal crystal system, shown in figure C.1(a), is not orthogonal. Transformation of the Bravais-Miller indices of a direction to Cartesian coordinates is not trivial, involves trigonometry. Directions in a hexagonal system are most often given by their four Bravais-Miller indices. In the first step of this transformation, the four Bravais-Miller indices, [U V T W ], have to be converted to the three Miller indices, [uvw], using the following formula:

u = 2U + V, v = 2V + U, w = W. (C.1)
Now we can develop the formula to convert between the two coordinate systems. The orientation of the Cartesian coordinate system shown in figure C.1(b) was chosen according to the critical thickness calculation detailed in section 1.5.2. Axes x and z are the in-plane orthogonal directions, and axis y is perpendicular to the surface. In the hexagonal crystal system the axis c, of length c lattice parameter, has six fold symmetry. The rhenium films grow along this direction, and therefore the crystal axis c is parallel to axis y. To convert from c to y involves a multiplication by c lattice parameter.

The transformation is demonstrated below, using the indices of a Burgers vector, which is expected to occur in our rhenium films: Bravais-Miller → Miller → Cartesian Radiosity of a surface is the sum of the thermal radiation due to its temperature ( σT 4 ), the reflected irradiance (ρH), and transmitted irradiance (τ H). According to this the radiosities on the two sides of the substrate-rhenium plane are the following:

B W SRe =
= 1 -ρ -τ.
The chamber wall and the furnace are opaque, their transmittances are 0. The tungsten layer on the backside of the substrate is also thick enough to be considered opaque. Thus, the radiosity (and the irradiance) of the furnace, chamber wall and the two sides of the tungsten layer are given as follows:

B f = ρ f H f + f σT 4 f → H f = 1 ρ f B f - f ρ f σT 4 f , (D.
B f W = ρ W H f W + W σT 4 W → H f W = 1 ρ W B f W - W ρ W σT 4 W , (D.14) B SRe W = ρ W H SRe W + W σT 4 W → H SRe W = 1 ρ W B SRe W - W ρ W σT 4 W . (D.15)
When τ = 0, equation = 1 -ρ -τ is modified as follows:

= 1 -ρ.

As was shown with equation (2.35), a quantity analogue to electric resistance can be defined for opaque objects:

R = 1 -= ρ .
This definition of resistance is used for the furnace, the chamber wall and the tungsten layer in the derivation.

Equations (D.4) -(D.15) have to be manipulated to obtain expressions for the radiosities that depend only on the 3 unknowns (Q, T W , T SRe ). This can be easily done for the surfaces of the opaque planes.

Using equations (D.4) and (D.12):

Q = B f -H f + Q E = 1 - 1 ρ f ρ f -1 ρ f =-f ρ f =-1 R f B f + f ρ f σT 4 f + Q E . B f = σT 4 f + R f (Q E -Q) (D.16)
Using equations (D.9) and (D.13):

Q = H b -B b = 1 ρ b - 1
Q = H f W -B f W + Q E = 1 ρ W -1 1-ρ W ρ W = W ρ W = 1 R W B f W - 1 R W σT 4 W + Q E . B f W = σT 4 W + R W (Q -Q E ) (D.18)
All the radiosities are expressed as the function of only the unknown parameters in equations (D.16), (D.17), (D.18), (D.19), (D.23) and (D.24). The final form of the equations system (D.1), (D.2), (D.3) can be computed.

Rewriting equation (D.1) using (D. [START_REF] Daunt | Superconductivity of rhenium[END_REF]) and (D.18):

Q = B f -B f W + Q E = = σT 4 f + R f (Q E -Q) -σT 4 W -R W (Q -Q E ) + Q E . Q = σT 4 f -σT 4 W 1 + R f + R W + Q E (D.25)
Rewriting equation (D.2) using (D.19) and (D.23):

Q = B SRe W -B W SRe + Q C = = σT 4 W + R W (Q C -Q) -σT 4 SRe -R SRe (Q -Q C ) + r SRe Q C + Q C . Q = σT 4 W -σT 4 SRe 1 + R W + R SRe + r SRe 1 + R W + R SRe + 1 Q C (D.26)
Finally, rewriting equation (D.3) using (D.17 ] ) ] # a n g u l a r p o s i t i o n s b e f o r e t h e s c a n data [START_REF] Wildes | The growth and structure of epitaxial niobium on sapphire[END_REF] '#L Two Theta Theta H K L Epoch Seconds D e t e c t o r \n ' # columns o f t h e data data [START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF] a r r a y ( [ [ 3 . 2 0 0 0 0 0 0 0 e +01 , 1 . 6 0 0 0 0 0 0 0 e +01 , 8 . 5 0 8 1 7 0 0 0 e -06 , . . . , 3 . 4 6 7 0 0 0 0 0 e +03 , 3 . 0 0 0 0 0 0 0 0 e +00 , 7 . 0 0 0 0 0 0 0 0 e +00] , . . . # t h e data E.1.2 Functions used to fit X-ray data

Voigt function

The Voigt function was used to fit the θ-2θ diffraction peaks. It is the convolution of a Gauss function and a Lorentz function, and is given in equation 2.10. Both the Gauss and the Lorentz functions are centred on a peak. The discrete convolution only preserves the position of the peak, if that is in the middle of the two arrays. This was considered when defining the Voigt function, which is given below. import numpy a s np import s c i p y . s i g n a l a s s i g n a l import s c i p y . i n t e r p o l a t e a s i n t e r p # nm s i n t h e t a 2 = ( CuKalpha2/CuKalpha1 ) * np . s i n ( t w o t h e t a 1 /2 * np . p i / 1 8 0 . ) t w o t h e t a 2 = 2 * np . a r c s i n ( s i n t h e t a 2 ) * 180/ np . p i r e t u r n t w o t h e t a 2 d e f f u n c ( x ) :

" Returns t h e h i g h r e s o l u t i o n f i t o f t h e Re ( 0 0 2 ) and t h e Al2O3 ( 0 0 6 ) peaks . " f , d_dist = mod_interf ( x , 1 1 3 0 0 . , 2 0 4 , 2 . 2 3 0 2 6 , 2 . 8 e -2) f = f + l o r e n t z ( x , 8 . 9 5 2 7 7 4 9 5 , 4 1 . 6 9 1 0 3 2 9 , 1 . l i n e s = meta . r e a d l i n e s ( ) a l l = l i n e s [ n : n+8] s t e p _ s t a r t = i n t ( r e . s e a r c h ( r ' \ d + ' , l i n e s [ n + 1 ] ) . group ( ) ) ramp_start = i n t ( r e . s e a r c h ( r ' \ d+ ' , l i n e s [ n + 2 ] ) . group ( ) ) s t e p _ h e i g h t = i n t ( r e . s e a r c h ( r ' \ d+ ' , l i n e s [ n + 3 ] ) . group ( ) ) ramp_slope = i n t ( r e . s e a r c h ( r ' \ d + ' , l i n e s [ n + 4 ] ) . group ( ) ) t h r e s h o l d _ v o l t = i n t ( r e . s e a r c h ( r ' \ d + ' , l i n e s [ n + 5 ] ) . group ( ) ) r e s i s t a n c e = i n t ( r e . s e a r c h ( r ' \ d+ ' , l i n e s [ n + 6 ] ) . group ( ) ) g a i n = i n t ( r e . s e a r c h ( r ' \ d+ ' , l i n e s [ n + 7 ] ) . group ( ) ) meta . c l o s e r e t u r n [ a l l , s t e p _ s t a r t , ramp_start , s t e p _ h e i g h t , ramp_slope , r e s i s t a n c e ]

Example:

In

Figure 1 . 1 :

 11 Figure 1.1: Hexagonal closed-pack structure.

Figure 1 . 2 :

 12 Figure 1.2: Lattice of Al 2 O 3 . (a) and (b) Six oxygen ions form a slightly distorted octahedron around an aluminium ion. (c) Stacking of the octahedra. The size of the atoms on the figure corresponds to their atomic radii [19].

Figure 1 . 3 :

 13 Figure 1.3: Lattice of Al 2 O 3 . (a) Alteration of one oxygen and two aluminium layers along the c axis. (b) View of the c plane. The size of the atoms on the figure corresponds to their atomic radii [19].

Figure 1 . 4 :

 14 Figure 1.4: Epitaxy refers to the growth of a crystalline layer onto a crystalline substrate following its lattice.

Figure 1 . 5 :

 15 Figure 1.5: (a) Mean free path and number of collisions as the function of pressure.(b) Time required for the formation of a monolayer as the functions of pressure. In ultrahigh vacuum, the mean free path is so long that collisions can be neglected, and it takes several hours for a monolayer to form from the residual molecules.

Figure 1 . 6 :

 16 Figure 1.6: MBE setup in SIMaP. It consists of four chambers: 1 -introduction chamber, 2 -intermediate chamber, Dep. chamber -deposition chamber, Char. chambercharacterisation chamber.

Figure 1 . 7 :

 17 Figure 1.7: Schematics of the quadrupole mass spectrometer: four electrodes placed parallel, with voltage applied between them. Depending on the voltage, only the particles with the set mass-to-charge ratio will reach the detector.

  Figure 1.10: (a) Schematics of the XPS measurement: the sample is irradiated with a monochromatic X-ray beam. As a result electrons escape from the surface region, their energy is measured by the detector. (b) Auger effect: a vacancy left by a photoelectron is filled up by an other electron from a higher energy level, the excess energy is carried away by a second emitted electron, called Auger electron.

Figure 1 .

 1 Figure 1.11: A typical XPS spectra recorded on a rhenium thin film.

Figure 1 . 13 :

 113 Figure 1.13: Electron diffraction from an uneven surface. (a) Diffraction happens in transmission through an island, a three dimensional object. (b) Sections of the Ewald sphere are shown in the reciprocal lattice, which consists of points. Constructive interference occurs in directions where the Ewald sphere intersects a reciprocal lattice point. (c) Cross section of the Ewald sphere is shown with the reciprocal lattice points and the wave vectors of the incoming and the outgoing, forward scattered waves. (d) The intersection of a reciprocal lattice point and the electron beam is projected onto the phosphor screen, which results in a spherical spot.

Figure 1 . 14 :

 114 Figure 1.14: Electron diffraction from an even surface. (a) Diffraction happens in reflection, the third dimension is reduced. (b) Sections of the Ewald sphere are shown in the reciprocal lattice, which consists of rods. Constructive interference occurs where the Ewald sphere intersects the reciprocal lattice rods. (c) Cross section of the Ewald sphere is shown with the reciprocal lattice rods and the wave vectors of the incoming and the outgoing beams. (d) The intersection of a reciprocal lattice rod and the electron beam is projected onto the phosphor screen, which results in an elongated rod perpendicular to the surface.

Figure 1 . 15 :

 115 Figure 1.15: Schematics of Tapping Mode AFM and the feedback loop. (a) An oscillating cantilever is attached to a piezoelectric ceramic tube. Movement of the tip is detected by a split photodiode. The amplitude of the oscillation is kept constant by a feedback loop [41]. (b) Cross section of the modules of the piezoelectric tube, and the applied voltage. (c) The feedback signal is converted to height, phase or amplitude, and are plotted as the function of the coordinates of the scanned area.

Figure 1 .

 1 Figure 1.16: 3 µm x 3 µm height and amplitude image taken of the same area of a rhenium thin film: variation of height shows on the height image, but it is easier to observe the edges on the amplitude image.

Figure 1 .

 1 Figure1.17: 6 µm x 6 µm height and phase image taken of the same area of a sapphire substrate: the phase image shows a contrast that is thought to be due to different chemical composition on the surface. Contrast is not visible on the height image.

Figure 1 . 18 :

 118 Figure 1.18: Effect of the planefit algorithm illustrated by a simulated stepped surface: the average slope of the image is subtracted, stepped surface appears to be jagged.

Figure 1 .

 1 Figure 1.19: Miller indices and Bravais-Miller indices of the hexagonal system.

Figure 1 . 20 :

 120 Figure 1.20: Bragg's and Laue's conditions of diffraction.

Figure 1 . 21 :

 121 Figure 1.21: Circles and angels of a 4-cycle diffractometer.

Figure 1 .

 1 Figure 1.22: (a) Schematics of the θ-2θ scan. (b) Rocking curve measurement.

Figure 1 .

 1 Figure 1.23: (a) Asymmetric reflection. (b) Φ scan.

Figure 1 .

 1 Figure 1.24: Φ and χ scan on the (103) equivalent reflections of rhenium grains with (002) orientation.

Figure 1 . 25 :

 125 Figure 1.25: Scattering from N planes with equal distances.

Figure 1 .

 1 Figure 1.26: (a) Processes that can occur when an atom reaches the surface of the growing crystal or the substrate. (b) Surface tensions that act between the adsorbate island (A), substrate (S), and vapour (V).

Figure 1 . 27 :

 127 Figure 1.27: Three growth modes: (a) Frank-van der Merve layer-by layer growth mode, (b) Vollmer -Weber island growth mode, (c) Stranski-Krastanov layer-plus-island growth mode.

Figure 1 .

 1 Figure 1.28: (a) Schematic representation of an edge dislocation with the Burgers circuit. A closed loop (M N OP Q) is drawn in the crystal that encloses the dislocation. (b) Burgers circuit is copied into a perfect crystal, where it is not closed. Burgers vector connects the starting point (M ) and the final point (Q) of the Burgers circuit [45].

Figure 1 .

 1 Figure 1.29: (a) Schematic representation of a screw dislocation with the Burgers circuit. A closed loop (M N OP Q) is drawn in the crystal that encloses the dislocation. (b) Burgers circuit is copied into a perfect crystal, where it is not closed. Burgers vector connects the starting point (M ) and the final point (Q) of the Burgers circuit [45].

Figure 1 .

 1 Figure 1.30: (a) Misfit of two hexagonal crystal lattices. (b) Due to the misfit, dislocations spontaneously appear in the film [44].

Figure 1 .

 1 Figure 1.31: (a) A dislocation lies η distance away from the surface, along axis z. Its glide plane divides the crystal in two: (+) and (-). (b) Sideview of (a): surfaces Γ + , Γ -are on the two sides of the glide plane. A volume around the dislocation line, with radius r 0 is excluded from the calculation [46].

Figure 1 .

 1 Figure 1.32: A low energy crystal surface with miscut of α tends to rearrange itself into a stepped structure.

Figure 1 . 33 :

 133 Figure 1.33: Step flow growth mode: steps are providing nucleation sites for the adatom, and the growth happens only at the step edges.

Figure 1 . 34 :

 134 Figure 1.34: Schwoebel barrier: theoretical potential felt by an adatom (shown in grey) on a step edge [47].

Figure 1 .

 1 Figure 1.35:An edge dislocation produces a slanting step on the surface, which will act as a nucleation site for the arriving atoms. Nucleating adatoms keep creating steps, as a result the surface will grow in a spiral manner[START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF].

Figure 1 .

 1 Figure 1.36: (a) Single spiral grown around a single dislocation. (b) Double spiral that grew around a dislocation pair of opposite signs [48].

Figure 1 . 37 :

 137 Figure 1.37: Growth of spirals initiated by a pair of screw dislocations of like sign [48].

Figure 1 . 38 :

 138 Figure 1.38: Growth of spirals initiated by dislocations within a 2πρ c distance [48].

Figure 1 . 39 :

 139 Figure 1.39: Grain boundary groove with angle β [49].

Figure 1 . 40 :

 140 Figure 1.40: Surface profile when shaped by evaporation-condensation plotted for different time intervals. The units along the x, and y axis are a measure of length, the numbers in the legend are a measure of time.

Figure 1 . 41 :

 141 Figure 1.41: Surface profile when shaped by surface diffusion plotted for different time intervals.. The units along the x, and y axis are a measure of length, the numbers in the legend are a measure of time.

Figure 2 . 1 :

 21 Figure 2.1: (a) 3 µm x 3 µm AFM height image taken of the Al 2 O 3 substrate as received. The surface is covered with particles of various sizes. (b) 3 µm x 3 µm AFM height image taken of the Al 2 O 3 substrate after cleaning, before annealing. A few larger particles are still visible, but their density is significantly reduced.

Figure 2 . 2 :Figure 2 . 3 : 3

 22233 Figure 2.2: Schematics of the tube used for the heat treatment of the substrates: the quartz tube is supported by an outer alumina tube to prevent deformation.

Figure 2 . 4 :

 24 Figure 2.4: Surface profile extracted from the AFM height image showing the step structure.

2 O 3

 23 along the c axis two Al layers and an O layer alternate. This means three different surface terminations are possible: single Al, double Al, or O. Which dominates in single crystal substrates, has been the subject of

Figure 2 . 5 :

 25 Figure 2.5: (a) Surface after annealing at 1100 • C for 30 minutes. (b) Surface after annealing at 1000 • C for an hour. The planefit procedure was not applied to the data.

Figure 2 . 6 :

 26 Figure 2.6: Vapour pressure of rhenium found in references[START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF],[START_REF] Carl | Handbook of Vapor Pressure, volume 4: Inorganic compounds and elements[END_REF], and[START_REF] Plante | Vapor pressure and heat of sublimation of rhenium[END_REF].

Figure 2 . 7 :

 27 Figure 2.7: Geometry of the source and the substrate.

  4, and it is shown in figure 2.8. The deposition rate reaches the frequently observed values at around 3000 • C.

Figure 2 . 8 :

 28 Figure 2.8: Deposition rate of rhenium. The observed deposition rate is shown with the blue dashed line.

Figure 2 . 9 :

 29 Figure 2.9: (a) Rhenium droplets ejected from the charge. (b) SEM image of a droplet found in the chamber.

Figure 2 .

 2 10(a) shows a single layer of rhenium on top of a single layer of Al -O octahedra viewed along the c axis. The rhenium atoms can be

Figure 2 .

 2 Figure 2.10: (a) A single layer of rhenium on top of a single layer of Al -O octahedra viewed along the c axis. (b) The view of the Re and Al 2 O 3 lattices along the a axis of the substrate [19].

Figure 2 .

 2 Figure 2.11: XRD Φ scans on the (102) equivalent reflections of rhenium, and (104) equivalent reflections of Al 2 O 3 show the 30 • rotation between the two lattices.

7 )Figure 2 . 12 :

 7212 Figure 2.12: Thermal expansion coefficient of rhenium and Al 2 O 3 and the obtained misfit as the function of temperature.

Figure 2 . 13 :

 213 Figure 2.13: Misfit as the function of the critical thickness calculated using equation 2.9.

Figure 2 .

 2 Figure 2.14: Sample inside the chamber, shutter and the quartz balance are also shown.

  Two samples with 25 nm thickness were prepared: sample A at 800 • C and sample B at 900 • C. AFM study of the surfaces AFM images taken of sample A and sample B are shown in figures 2.15(a) and 2.15(b), respectively.

Figure 2 .

 2 Figure 2.15: (a) 3 µm x 3 µm AFM height image shows the surface of the 25 nm thick sample deposited at 800 • C, sample A. (b) 3 µm x 3 µm AFM height image shows the surface of the 25 nm thick sample deposited at 900 • C, sample B.

Figure 2 .

 2 Figure 2.16: θ-2θ of the two 25 nm thick samples.

Figure 2 .

 2 Figure 2.17: (002) reflection of sample A fitted with the sum of two Voigt functions.

Figure 2 .

 2 Figure 2.18: (a) Rocking curve of the (002) peak of rhenium on sample B fitted with a Pearson VII (blue) and Pearson VII plus Gaussian (red). The red curve describes the wide tails of the data better. (b) Rocking curves of the 25 nm samples fitted with the sum of a Pearson VII and a Gaussian.

  AFM study of the surfacesThree samples with thickness 50 nm were investigated. Sample C, was deposited at 800 • C, sample D at 900 • C, and sample E at 1000 • C.

Figure 2 .

 2 Figure 2.19: 1 µm x 1 µm AFM height image shows the surface of the 50 nm thick sample deposited at 800 • C (sample C).

Figure 2 .

 2 Figure 2.20: Sample C: 350 nm x 400 nm AFM height image showing an irregular spiral with the profile measured along the blue line.

Figure 2 .

 2 Figure 2.21: 1 µm x 1 µm AFM height image shows the surface of the 50 nm thick sample deposited at 900 • C (sample D).

Figure 2 .

 2 Figure 2.22: Sample D: 600 nm x 250 nm AFM height image showing a double spiral with the profile measured along the white line.

Figure 2 .

 2 Figure 2.23: (a) 12 µm x 6 µm AFM height image showing the surface of the 50 nm thick sample deposited at 900 • C (sample E). (b) 3 µm x 1.5 µm AFM height image. The colour scale was set to highlight the terrace structure of the topmost surface.

Figure 2 .

 2 Figure 2.24: (b) 1 µm x 0.5 µm AFM image, a magnification of the area marked by the blue square in figure 2.23(b). The colour scale was set to highlight the bottom of the channels. (a) Surface profiles were measured along the coloured lines, and are plotted in corresponding colours.

Figure 2 .

 2 Figure 2.25: θ-2θ of the three 50 nm thick samples.

Figure 2 .

 2 Figure 2.26: High-resolution X-ray scan of the (002) peak of rhenium with the (006) peak of the substrate (sample D). Only every fifth datapoint is shown.

Figure 2 .

 2 Figure 2.27: A magnification of figure 2.26, the (002) peak of rhenium on sample D. Only every third datapoint is shown.

Figure 2 . 28 :

 228 Figure 2.28: Scattering of X-ray wave from parallel planes that are slightly disordered.

  The distributions of d lattice parameters used in the calculation are shown in a histogram in figure2.30. The full width half maximum of the distribution used for sample C is indeed wider.

Figure 2 . 29 :

 229 Figure 2.29: The high-resolution (002) peaks of rhenium measured on the 50 nm samples and fitted with the modified interference function, equation 2.17. Only every sixth datapoint of both datasets is shown.

Figure 2 . 30 :

 230 Figure 2.30: The distributions of lattice plane spacings used for the fits shown in figure 2.29, to describe the intensity variation of the fringes, and the slight broadening of the main peak.

Figure 2 . 31 :

 231 Figure 2.31: Standard resolution data simulated from the high-resolution data of the (002) peak of rhenium (sample D). As an approximation, the substrate peak was used as the resolution function. The red curve is the result of the convolution of the fitted high resolution curve, and the approximate resolution function (equation 2.20).

Figure 2 . 32 :Table 2 . 7 :

 23227 Figure 2.32: Rocking curves measured on the (002) peaks of the 50 nm samples.

Figure 2 .

 2 Figure 2.33: (a) 3 µm x 3 µm AFM height image shows the surface of the 100 nm thick sample deposited at 800 • C (sample F). (b) 3 µm x 3 µm AFM height image shows the surface of the 50 nm thick sample deposited at 900 • C (sample G).

Figure 2 .

 2 Figure 2.34: θ-2θ scan of the 100 nm samples.

Figure 2 .

 2 Figure 2.35: (a) The (002) peak of rhenium measured on the 100 nm samples. The curves measured on the two films are asymmetric, and almost identical. (b) High-resolution scan of the same (002) peaks of both 100 nm samples. The fringes on the curve of sample G are more pronounced.

Figure 2 .

 2 Figure 2.36: (a) Gaussian widths of the θ-2θ peaks as the function of temperature. (b) Pearson VII widths of the rocking curves as the function of temperature.

Figure 2 .

 2 Figure 2.37: 2 µm x 2 µm AFM hight image of the rhenium thin film with approximately 15 nm thickness, showing signs of dewtting.

Figure 2 .

 2 Figure 2.38: (a) 2 µm x 2 µm AFM hight image taken on the same sample as shown in figure 2.37 with two insets. The colour scale of the insets was adjusted to show the step structure of the ridges. (b) Profiles extracted from the AFM hight image, showing singe and double steps. Minima of both curves were set to 0.

Figure 2 .

 2 Figure 2.39: (a) 2 µm x 2 µm AFM hight image taken on the same sample as shown in figure 2.37 with two insets. The colour scale of the insets was adjusted to show the slight maximum in height around the holes. (b) Profiles extracted from the AFM hight image. The slight bump that appears around the holes resembles curves of thermal grooving driven by surface diffusion.

Figure 2 . 40 :

 240 Figure 2.40: Extracted height profile fitted by Mullins' theoretical curve. Parameters of the fit are shown above the plot.

Figure 2 . 41 :

 241 Figure 2.41: Temperature dependence of the diffusion coefficient according to the measurements of Goldstein and Ehrich [76].

. 38 .

 38 It is also visible on the tail of the measured profile in figure 2.40. Recrystallisation also causes the maxima after the hole to flatten, most visible on the blue profile in figure 2.39(b).

Figure 2 .

 2 Figure 2.42: θ-2θ scan of the Re(002) peak measured on the 15 nm thick rhenium film. Several fringes on both sides of peak show that the X-ray beam was diffracted by regularly arranged lattice planes.

Figure 2 . 43 :

 243 Figure 2.43: Radiation arriving on a surface (1). Proportions of it are reflected (ρ), transmitted (τ ) or absorbed (α).

Figure 2 . 44 :

 244 Figure 2.44: The model consists of a series of planes: furnace (F ), tungsten (W ), substrate-rhenium (SRe), and chamber (B). Irradiance and radiosity of the planes is considered.

Figure 2 . 45 :

 245 Figure 2.45: Emittance of rhenium: values calculated from reference [83] (red points) compared to curve used by Delsol [7] (blue line).

Figure 2 . 46 :

 246 Figure 2.46: Optical coefficients of the substrate-rhenium plane as function of the rhenium thickness.

Figure 2 . 47 :

 247 Figure 2.47: Temperature of the rhenium thin film with and without tungsten backing as the function of its thickness calculated from the model.

Figure 3 . 1 :

 31 Figure 3.1: The superconducting order parameter and the magnetic field at a normal(left)superconducting(right) interphase. [95].

Figure 3 . 2 :

 32 Figure 3.2: H-T phase diagram of a type II superconductor. [112].

Figure 3 . 3 :

 33 Figure 3.3: Order parameter and magnetic induction across a vortex. [112].

Figure 3 . 4 :

 34 Figure 3.4: Schematic illustration of a SQUID containing two Josephson junctions. I 1 and I 2 currents running through the two junctions are modulated by the phase drops θ 1 and θ 2 [115].

Figure 3 . 5 :

 35 Figure 3.5: I(Φ e ) characteristic of a symmetric SQUID with zero inductance.

Figure 3 . 6 :

 36 Figure 3.6: Oscilloscope reading of the current ramp repetitions applied to a Nb SQUID [116].

  .7). The volume of the fridge is divided by two radiation shields: an 80 K one (red in figure 3.7), and a 4 K one (green in figure 3.7). The coldest point is at the top, this is where the sample is placed, shown by purple in figure 3.7, and by an arrow in figure 3.8.

Figure 3 . 7 :

 37 Figure 3.7: Schematic diagram of the table-top Helium-3 cryostat. It contains two circuits: 4 He shown in blue, 3 He shown in orange. Coldest temperature, 300 mK is achieved by internal pumping using charcoal on the 3 He reservoir.

Figure 3 . 8 :

 38 Figure 3.8: Photograph of the inside of the table-top helium-3 cryostat.

Figure 3 . 9 :

 39 Figure 3.9: Phase diagram of the 3 He and 4 He mixture. Below 800 mK the mixture spontaneously separates into a diluted (yellow) and a concentrated in 3 He phase (green) Image is from reference [118].

Figure 3 . 10 :

 310 Figure 3.10: Simplified diagram of the mixture circuit of a dilution refrigerator. After the mixture separates to a diluted (blue) and a concentrated (orange) phase,3 He is extracted from the still and resupplied to the mixing chamber. Cooling power is provided by the diffusion of 3 He from the concentrated to the diluted phase in the mixing chamber.

Figure 3 . 11 :

 311 Figure 3.11: Diagram of the SIONLUDI inverted dilution fridge. The coldest part is on the top, the sample is loaded there, placed under concentric bells. Helium-4 is supplied from the bottom. Modified image from reference [115].

4. 1 . 1

 11 Circuit designs 3 different patterns were fabricated using laser and electron beam lithography. The first design is shown in figure 4.1(a). It features a long wire that has 7 parts with different widths, ranging from 50 µm to 3 µm. This is shown in blue. The electrodes 133 that allow 2-point or 4-point resistivity measurements are shown in pink. Electrodes are connected to the ends of all the 7 parts, thus they can be measured independently of each other. External electrodes can be connected to the large pads shown in purple. This design was fabricated with laser beam lithography. The smallest object it included was the 3 µm wire, due to the resolution limitations of laser light (∼ 1µm).

Figure 4 . 1 :

 41 Figure 4.1: (a)Lithography design I: long wire with parts that have different widths (blue). Drawing is not to scale. (b) Lithography design II: long wire with parts that have different widths (blue), and 3 SQUIDs (green). Drawing is not to scale.

Figure 4 . 2 :

 42 Figure 4.2: Lithography design III: 2 SQUIDs and 3 wires, each have two electrodes at the bottom, and one common electrode on top. Drawing is not to scale.

Figure 4 . 3 :

 43 Figure 4.3: Steps of the lithography process used to fabricate wires and SQUIDs on rhenium thin films.

Figure 4 . 4 :

 44 Figure 4.4: Optical microscopy image taken on the completed pattern fabricated from the 50 nm thick sample.

Figure 4 . 5 :

 45 Figure 4.5: (a), (c) Out-of-plane (OP) orientation maps show that the orientation of rhenium along the 3 µm line and the SQUID is (001). (b), (d) In-plane orientation maps taken on the same areas as (a) and (c) shows that the in-plane orientations are also uniform.

Figure 4 . 6 :

 46 Figure 4.6: AFM height image taken on a SQUID. The spirals were left intact, however the bridges connecting the two forks of the SQUID are missing.

Figures 4 .

 4 Figures 4.5(a) and 4.5(c) show the out-of-plane (OP) orientation maps of a SQUID and a 3 µm line. The colour red is uniform along the measured area except in a few points. The red colour corresponds to the orientation (001), which is consistent with the results of the XRD.

Figures 4 .

 4 Figures 4.5(b) and 4.5(d) show the in-plane (IP) orientation maps of the same areas.Here the colour is uniformly blue.

Figure 4 . 7 :

 47 Figure 4.7: Optical and electron microscopy image taken on the completed design patterned onto the 25 nm thick sample.

Figure 4 . 8 :

 48 Figure 4.8: Optical microscopy images taken on the central parts of the two versions of the completed design on the 25 nm thick film.

Figure 4 .

 4 Figure 4.7 shows an optical microscopy image of one of the full, completed patterns. In figure 4.8, the wires and the SQUIDs are shown.

Figure 4 . 9 :

 49 Figure 4.9: Optical microscopy image of the completed design patterned onto the 100 nm thick sample.

Figure 4 .

 4 Figure 4.10: Four-terminal resistivity measurement.

Figure 4 . 11 :

 411 Figure 4.11: Sample is glued on the sample holder, and is connected by 25 µm diameter aluminium wires.

Figure 4 . 12 :

 412 Figure 4.12: Connection for resistance measurement of the 3 µm wide rhenium wire on the 50 nm thick sample.

Figure 4 . 13 :

 413 Figure 4.13: Connection for resistivity measurement of the 200 nm wide rhenium wire on the 25 nm thick sample.

Figure 4 . 14 :

 414 Figure 4.14: Resistivity the wires measured with decreasing temperature.

Figure 4 .

 4 Figure 4.15: (a) Superconducting transitions of the 3 µm wide wire, and the 50 nm thick film it was fabricated on (Sample D). (b) Superconducting transitions of the 3 µm wide wires fabricated on the 50 nm (D) and the 100 nm (F) thick samples with normalised resistivities

Figure 4 . 16 :

 416 Figure 4.16: Superconducting transitions of the 100 nm, 200 nm, and the 400 nm wide wires, and the 25 nm thick film they were fabricated on (Sample B).

Figure 4 . 17 :

 417 Figure 4.17: Critical temperature as the function of pressure/relative change of volume measured by Chu et al. (a) for two polycrystalline samples and (b) for two single crystal samples [128].

Figure 4 . 18 :

 418 Figure 4.18: Residual resistivity ratio as the function of the critical temperature.

Figure 4 .

 4 Figure 4.19: SEM image of a SQUID fabricated on the 25 nm thick rhenium thin film.

  The critical current as a function of the magnetic field measured on one of the SQUIDs (SQUID1) are shown in figures 4.21. A low frequency oscillation envelopes a fast critical

Figure 4 . 20 :

 420 Figure 4.20: The schematics of the electrical connections made to SQUID1 shown on the optical image of the pattern.

Figure 4 . 21 :

 421 Figure 4.21: The critical current oscillations as a function of the applied field measured on SQUID1. A low frequency component (a) envelops a fast oscillation (b).

Figure 4 . 22 :

 422 Figure 4.22: Schematics consistent with the data shown in figure 4.21. The small SQUID is connected to either a junction or another SQUID, forming a large SQUID loop.

Figure 4 . 23 :

 423 Figure 4.23: The critical current oscillations as a function of the applied field measured on SQUID2.

  That the I c (B) graph of SQUID2 in figure 4.23 appears irregular could be explained by left-over rhenium on the surface, that provides alternate path for the current. This has been a problem on other samples. The switching histogram of the measured SQUIDs are shown in figures 4.25(a) and

Figure 4 .

 4 Figure 4.24: Fast Fourier Transform of the I c (B) curve of SQUID2.

Figure 4 . 25 :

 425 Figure 4.25: Critical current histogram of (a) SQUID1 (b) SQUID2

Figure A. 1 :

 1 Figure A.1: Longitudinal coherence length.

Figure A. 2 :

 2 Figure A.2: Transverse coherence length (modified figure from reference [43]).

Figure A. 3 :

 3 Figure A.3: Nyquist sampling frequency demonstrated on a square function (corresponds to a slit) (a). The square of the Fourier transform is seen by the detector. To fully determine it, it has to be sampled at least once per fringe, shown in red (b).

Figure A. 5 :

 5 Figure A.5: (a) An areal photograph of Diamond Light Source (source: Science and Technology Facilities Council). (b) Schematics of a synchrotron beamline (dimensions are specific to I13-1 in Diamond Light Source)[START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF] (c) Undulators are composed of a series of dipole magnets, that make electrons oscillate to generate X-ray radiation[START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF].

Figure A. 7 : 4

 74 Figure A.7: 4 µm x 4 µm AFM image taken on one of the 2 µm x 2 µm squares fabricated for the ptychography experiments on the a 50 nm sample (sample D).

  The spiral scan is shown in figure A.8. At each point of the path a single slice of the (002) reflection of rhenium was recorded on the 2D detector. The colour of the points shown in figure A.8 correspond to the total, summed intensity measured on the detector. The scan started in the middle of the square. Maximum intensity is indeed observed

Figure A. 8 :

 8 Figure A.8: Spiral ptychography scan on a 2 µm x 2 µm rhenium square: the colour of each point corresponds to the total, summed intensity recorded on the detector.

Figure A. 9 :

 9 Figure A.9: Two examples of the diffraction pattern recorded on the detector at different points of the scan.

Figure A. 10 :

 10 Figure A.10: Movement of the sample during the 3D ptychography scan.

D

  Derivation of the equation system for the heat transferTo estimate the temperature of the surface of the growing rhenium, a model was developed by Delsol[START_REF] Delsol | Elaboration et caractérisation de couches minces supraconductrices épitaxiées de rhénium sur saphir[END_REF]. The complete derivation to obtain the equation system given in chapter 2.4 is given below.The model is shown in figure D.1. All the parts of the system was assumed to be an infinite plane. The plane noted with F is the furnace. Besides radiosity which is the

Figure D. 1 :

 1 Figure D.1: The model consist of a series of planes: furnace (F ), tungsten (W ), substraterhenium (SRe), and chamber (B). Irradiance and radiosity of the planes is considered.

12 )B b = ρ b H b + b σT 4 b→

 124

  σT 4 b + R b Q (D.17)Using equations (D.5) and (D.14):

= σT 4 SRe -σT 4 b 1 +.

 441 ) and (D.24):Q = B b SRe -B b = = σT 4 SRe -R SRe Q -r SRe Q C -σT 4 b -R b Q. Q R B + R SRe -r SRe 1 + R B + R SRe Q C (D.27)Notations used throughout this derivation are summarised below:Substrate and rhenium was is treated as a single object. The common transmittance and emittance was calculated as follows:τ SRe = τ S τ Re , SRe = (1 -τ Re ) Re + τ Re S .f o r k i n r a n g e ( n u m o f i n t e r v a l s +1): d a t a s t r = l i n e s [ n u m o f l i n e+11+k ] d a t a a r r = np . a r r a y ( d a t a s t r . s p l i t ( ' ' ) ) data [ k , : ] = d a t a a r r . a s t y p e ( np . f l o a t ) break e l s e : c o n t i n u e r e t u r n [ command , n u m o f i n t e r v a l s , [ s t a r t _ s c a n n e d 1 , stop_scanned1 ] , [ s t a r t _ s c a n n e d 2 , stop_scanned2 ] , time , [ anglename , a n g l e s ] , names , data ] Example: In : f i l e = open ( ' EJM216 ' ) l i n e s = f i l e . r e a d l i n e s ( ) f i l e . c l o s e scannum = 10 data = s p e c . get_scan_data ( l i n e s , scannum ) Out : data [ 0 ] '#S 10 a 2 s c a

  d e f l o r e n t z ( x , c , x0 , FWHM) : " Returns t h e L o r e n t z f u n c t i o n c e n t r e d on x0 . " gamma = FWHM/2 l o r e n t z f u n c = np . d i v i d e ( 1 , np . p i * gamma * ( 1 + np . d i v i d e ( np . s q u a r e ( x -x0 ) , gamma * * 2 ) ) ) r e t u r n c * l o r e n t z f u n c d e f g a u s s ( x , c , x0 , FWHM) : " Returns t h e Gauss f u n c t i o n c e n t r e d on x0 . " sigma = np . d i v i d e (FWHM, 2 * np . s q r t ( 2 * np . l o g ( 2 ) ) ) g a u s s f u n c = np . d i v i d e ( 1 , sigma * np . s q r t ( 2 * np . p i ) ) * np . exp (-1 * np . d i v i d e ( np . s q u a r e ( x -x0 ) , 2 * sigma * * 2 ) ) r e t u r n c * g a u s s f u n c d e f v o i g t ( x , c , x0 , FWHM_G, FWHM_L) : " Returns t h e Voigt f u n c t i o n c e n t r e d on x0 . " l o w e r = -100 upper = 100 nop = 8 e3+1 x _ t o f i t = np . add ( np . l i n s p a c e ( lower , upper , nop ) , x0 ) y l = l o r e n t z ( x _ t o f i t , 1 , x0 , FWHM_L) yg = g a u s s ( x _ t o f i t , 1 , x0 , FWHM_G) vv = s i g n a l . f f t c o n v o l v e ( yl , yg , ' same ' ) yv = c * vv * ( upperl o w e r ) / ( nop ) f v = i n t e r p . i n t e r p 1 d ( x _ t o f i t , yv , bounds_error = ' F a l s e ' , f i l l _ v a l u e = 1 e7 ) yyv = f v ( x ) # r e t u r n [ x _ t o f i t , yyv , yg , yl , yv ] r e t u r n yyv CuKalpha2 = 0 . 1 5 4 4 3 9 8

1 5 3 1 6 1 1 e 1 #

 11 -02) r e t u r n [ f , d_dist ] d e f r e s o l u t i o n ( x , c , x0 , FWHM) : " Returns an a p p r o x i m a t i o n o f t h e r e s o l u t i o n o f t h e s t a n d a r d XRD s e t u p . " f = l o r e n t z ( x , c , x0 , FWHM) r e t u r n f # c a l c u l a t i n g t h e peak p o s i t i o n s from t h e l a t t i c e parameter t wo t h e t a = 2 * np . a r c s i n ( con . CuKalpha1 / ( 2 * 2 . 2 3 0 2 2 ) ) * 180/ np . p i tw o t h e t a 2 = secondpeak ( t w o t h e t a ) d t h e t a = twotheta2-t w o t h e t a # high-r e s o l u t i o n peak n o r m a l i s e d f_hr = f u n c ( xx ) [ 0 ] f_hr = f_hr /np . max( f_hr ) # s t a n d a r d r e s o l u t i o n f u n c t i o n n o r m a l i s e d f _ r e s = r e s o l u t i o n ( xx , 1 , twotheta , 0 . 0 4 ) f _ r e s = f _ r e s /np . max( f _ r e s ) # s t a n d a r d r e s o l u t i o n peak from CuKalpha1 l r _ f u n c 1 = s i g n a l . f f t c o n v o l v e ( f_hr , f_res , ' same ' ) * 0 . uu i s t h e argument o f t h e i n t e g r a l e r r o r f u n c t i o n uu = np . z e r o s ( [ num_of_x , np . shape ( At2 ) [ 0 ] ] ) f o r i i n r a n g e ( np . shape ( At2 ) [ 0 ] ) : uu [ : , i ] = np . d i v i d e ( datax , At2 [ i ] ) # C a l c u l a t i n g t h e i n t e g r a l e r r o r f u n c t i o n i e r r f = np . z e r o s ( [ num_of_x , np . shape ( At2 ) [ 0 ] ] ) f o r i i n r a n g e ( np . shape ( At2 ) [ 0 ] ) : f o r j i n r a n g e ( num_of_x ) : i e r r f [ j , i ] = i e r f c ( uu [ j , i ] ) # C a l c u l a t i n g t h e p r o f i l e f u n c t i o n yy = np . z e r o s ( [ num_of_x , np . shape ( At2 ) [ 0 ] ] ) f o r i i n r a n g e ( np . shape ( At2 ) [ 0 ] ) : yy [ : , i ] = np . m u l t i p l y (-1 * mm * At2 [ i ] , i e r r f [ : , i ] ) E.2.2 Surface diffusion import numpy a s np import math import s c i p y . s p e c i a l # D e f i n i n g t h e a_n c o e f f i c i e n t s o f t h e Z f u n c t i o n nn = 51 # number o f e l e m e n t s i n t h e a_n s e r i e s coef_an = np . z e r o s ( [ nn ] ) # c o n t a i n s t h e a_n c o e f f i c i e n t s coef_an [ 0 ] = -1/(np . s q r t ( 2 ) * math . gamma ( 5 . / 4 ) ) coef_an [ 1 ] = 1 coef_an [ 2 ] = -1/(np . s q r t ( 2 * * 3 ) * math . gamma ( 3 . / 4 ) ) coef_an [ 3 ] = 0 f o r i i n r a n g e ( nn -4): coef_an [ i +4] = coef_an [ i ] * ( i -1)/(4 * ( i +1) * ( i +2) * ( i +3) * ( i +4)) num_of_x = 50000 datax = np . l i n s p a c e ( 0 , 1 8 0 0 0 , num_of_x ) # p r o f i l e l i n e t t = [ 5 . , 3 0 . , 9 0 . , 1 2 0 . , 1 8 0 . ] * 60 # time b e t a = math . r a d i a n s ( 5 ) # b e t a i n r a d i a n s mm = np . tan ( b e t a ) BB = 1 e10 Bt = np . power ( np . m u l t i p l y (BB, t t ) , 0 . 2 5 ) # uu i s t h e argument o f t h e Z f u n c t i o n uu = np . z e r o s ( [ num_of_x , np . shape ( Bt ) [ 0 ] ] ) p r i n t np . shape ( uu ) f o r i i n r a n g e ( np . shape ( Bt ) [ 0 ] ) : uu [ : , i ] = np . d i v i d e ( datax , Bt [ i ] ) # C a l c u l a t i n g t h e Z f u n c t i o n ZZ = np . z e r o s ( [ num_of_x , np . shape ( Bt ) [ 0 ] ] ) f o r j i n r a n g e ( np . shape ( Bt ) [ 0 ] ) : f o r i i n r a n g e ( nn ) : aa_n = coef_an [ i ] uu_n = np . power ( uu [ : , j ] , i ) ZZ [ : , j ] = ZZ [ : , j ] + np . m u l t i p l y ( aa_n , uu_n) # C a l c u l a t i n g t h e p r o f i l e f u n c t i o n yy = np . z e r o s ( [ num_of_x , np . shape ( Bt ) [ 0 ] ] ) f o r i i n r a n g e ( np . shape ( Bt ) [ 0 ] ) : yy [ : , i ] = np . m u l t i p l y (mm * Bt [ i ] , ZZ [ : , i ] ) E.3 Preparation of the SQUID data E.3.1 SQUIDbox function The following function retrieves the parameters of the SQUID critical current measurements. These parameters are used to calculate the critical current. import r e d e f SQUIDbox ( meta_name , n ) : " Returns t h e p a r a m e t e r s o f t h e SQUID box . I n p u t s : f i l e name , nnumber o f t h e l i n e ' P e r i o d e du s q u i d (SQUID P e r i o d ) " meta = open ( meta_name )

  : name = '2016 -03 -18_15-18-38_250p00mK . IcH_cleaned ' meta_name = name [ 0 : l e n ( name) -11] + ' meta ' p a r a m e t e r s = SQUIDbox ( meta_name , 6 1 ) a l l t h e l i n e s c o r r e s p o n d i n g t o t h e SQUID box Out : p a r a m e t e r s [ 0 ] [ ' P e r i o d e du s q u i d (SQUID P e r i o d ) : 255\ r \n ' , 'T du d e p a r t p a l i e r ( s t e p s t a r t time ) : 35\ r \n ' , 'T du d e p a r t rampe ( ramp s t a r t time ) : 103\ r \n ' , ' h a u t e u r du p a l i e r ( s t e p h e i g h t ) : 8815\ r \n ' , ' p e n t e de l a rampe ( s l o p e o f t h e ramp ) : 2047\ r \n ' , ' t e n s i o n de s e u i l ( t h r e s h o l d v o l t a g e ) : 1538\ r \n ' , ' R e s i s t a n c e [Ohm ] : 50000\ r \n ' , ' g a i n du p r e ampli ( A m p l i f i e r g a i n ) : 1\ r \n ' ] # a l l t h e l i n e s c o r r e s p o n d i n g t o t h e SQUID box p a r a m e t e r s [ 1 ] 35 # T du d e p a r t p a l i e r ( s t e p s t a r t time ) p a r a m e t e r s [ 2 ] 103 # T du d e p a r t rampe ( ramp s t a r t time )

Table 2 .

 2

	25 nm
	800 • C (A)	900 • C (B)
	Diameter of larger grains (nm) 96 ± 28 (29 %) 73 ± 13 (18 %)
	Diameter smaller grains (nm) 45 ± 12 (27 %) 26 ± 4 (15 %)
	Average roughness (nm)	1.27	0.98

N j=1 |r j |) is calculated. This value for sample A is 1.27 nm, for sample B it is smaller, 0.98 nm. Measurements on figures 2.15(a) and 2.15(b) are summarised in table 2.2. 2: Surface features measured in figures 2.15(a) and 2.15(b).

Table 2 .

 2 3: Parameters of the Voigt functions fitted to the (002) rhenium peaks of the 25 nm samples

Table 2 .

 2 .18(b), with the sum of a Pearson VII and a Gaussian fitted to each. The data was normalised to help com-parison. Parameters of the fits (both single Pearson VII and Pearson VII plus Gaussian) are listed in table 2.4. 4: Parameters of the rocking curves measured on the 25 nm thick films.It is immediately apparent from figure2.18(b) that the rocking curve of sample B is significantly narrower. The single Pearson VII full width half maximum of this sample is less than half than that of sample A. This signals that the grains have lower mosaicity.

		25 nm (002) Rocking curves
		800 • C (A)	900 • C (B)
		P	P + G	P	P + G
	χ 2	0.03	0.006	0.06	0.03
	I ratio	-	I P /I G = 0.95	-	I G /I P = 0.75
	Peak @ ( • )	20.2797	20.2799	20.2340	20.2338
		± 5e-4	± 2e-4	± 4e-4	± 3e-4
	F W HM G ( • )	-	0.731 ± 0.004	-	0.178 ± 0.002
	F W HM P ( • ) 0.531 ± 0.002 0.384 ± 0.003 0.225 ± 0.002	0.52 ± 0.02
	m	1.84 ± 0.03	1.21 ± 0.02	1.20 ± 0.02	5 ± 1

Table 2 .

 2 equation 2.10. The same fitting procedure was applied, as described previously for the 25 nm thick films. Parameters of the fits are summarized in table 2.5. 5: Parameters of the Voigt fits of the (002) rhenium peaks measured on the 50 nm thick samples.

	50 nm (002) Voigt fit
	800 • C (C)	900 • C (D)	1000 • C (E)
	Peaks @ ( • ) 40.454 ± 0.001 40.478 ± 0.001 40.4185 ± 0.0004
	40.640 ± 0.007 40.625 ± 0.006	40.624 ± 0.002
	F W HM G ( • ) 0.386 ± 0.006	0.364 ± 0.004	0.246 ± 0.003
	F W HM L ( • ) 0.028 ± 0.004	0.043 ± 0.003	0.092 ± 0.003

Table 2 .

 2 8: Parameters of the fit of the high-resolution (002) rhenium peaks with the interference function, equation 1.23, and the parameters of the fit of the rocking curves with the Pearson VII function, equation 2.14.

		100 nm (002) High-resolution data
		800 • C (F)	900 • C (G)
	N	448 ± 0	387 ± 0
	d 0	(0.2230084 ± 6e-7) nm (0.222972 ± 7e-6) nm
		100 nm (002) Rocking curves
	Peaks @ ( • )	20.28330 ± 6e-5	20.2668 ± 3e-5
	F W HM ( • )	0.3384 ± 0.0002	0.2794 ± 0.0001
	m	3.15 ± 0.02	3.83 ± 0.02

Table 4 .

 4 1: Resists, instruments and developers used for lithography.

		S1818	PMMA
	Thickness	∼ 1 µm	∼ 100 nm
	Bake	115 • C	180 • C
	Instrument Heidelberg DWL66FS	Léo 1530
	Developer	MF-319	1:3 MIBK:IPA
			methyl isobutyl ketone:isopropyl alcohol

  .2.

		Sample B	Sample D	Sample F
	Film thickness	25 nm	50 nm	100 nm
	Deposition temp.	900 • C	900 • C	800 • C
	AFM image	figure 2.15(b)	figure 2.21 figure 2.33(a)
	Features	grains	spirals	spirals
	Diameter	< 100 nm	∼ 500 nm	∼ 200 nm
	XRD	figure 2.16	figure 2.25	figure 2.34
		red curve	red curve	blue curve
	Orientations

Table 4 .

 4 3: Values obtained for the 50 nm and the 100 nm films, and the 3 µm wide wires fabricated onto them.

	thickness	50 nm (D)	100 nm (F)
			film	wire: 3 µm	film	wire: 3 µm
	ρ RT (µΩcm)	15.0 [7]	24.1	19.75 [7]	25.68
	ρ res (µΩcm)	1.1	1.4	0.915 [7]	2.14
	RRR		15.0 [7]	17.2	21.6 [7]	12.0
	T c (K)		1.66	1.48	1.85 [7]	1.77
	∆T (K)	0.43	0.03	0.10 [7]	0.01
	∆(0) (meV)	0.25	0.22	0.28	0.27
	l (nm) [121]	172	142	217	88
	ξ 0 (nm)	167	187	150	157
			clean	dirty	clean	dirty
	ξ ef f (0) (nm)	124	139	111	100
	Film thickness			25 nm (B)
		film	wire: 400 nm	wire: 200 nm wire: 100 nm
	ρ RT (µΩcm)	15.0 [7] 27.98 (extrapol.)	30.99	41.76
	ρ res (µΩcm)	3.8		7.16	5.82	8.57
	RRR	4.0 [7]	3.9	5.3	4.9
	T c (K)	1.89		1.95	1.89	1.96
	∆T (K)	0.17		0.13	0.15	0.17
	∆(0) (meV)	0.29		0.29	0.29	0.3
	l (nm) [121]	47		25	31	21
	ξ 0 (nm)	147		142	147	141
		dirty		dirty	dirty	dirty
	ξ ef f (0) (nm)	71		51	58	47

Table 4 .

 4 4: Values obtained for the 25 nm film, and the wires fabricated on it.

  ρ SRe H W SRe + SRe σT 4 SRe + τ SRe H b SRe , (D.10) B b SRe = ρ SRe H b SRe + SRe σT 4 SRe + τ SRe H W SRe . (D.11) Furthermore, from equations (2.30) and (2.31) the following is true, and is used in the derivation:

He leaves a phase where the atoms are densely packed, like in a liquid, for a phase where

A(q) = V f (r)e iqr dr.

(A.

2)

The Fourier transform is reversible, the inverse Fourier transform of the amplitude could recover the electron density distribution. However, it is the intensity that is recorded by the detectors, which is the absolute modulus of the structure factor:

(A.3) Thus, the phase information of the structure factor, which is required for the inverse Fourier transform, is lost upon the measurement. Only the amplitude of the structure factor can be recovered. This is known as the phase problem in crystallography [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF].

The consequence of the phase problem is that the electron density, and thus the atomic positions cannot be directly retrieved from the diffraction data. Of course, this has not prevented scientist from trying and succeeding reconstructing structures of materials. Many of the approaches rely on a priori information regarding the chemistry of the material, rather than directly recovering the phase [START_REF] Rodenburg | Ptychography and related diffractive imaging methods[END_REF]. Recovery of both phase and amplitude information is possible by holography, where a reference wave is used to interfere with the scattered wave [START_REF] Gabor | A new microscopic principle[END_REF][START_REF] Gabor | Microscopy by reconstructed wave-fronts[END_REF].

The recovery of the phase of the scattered wave is also possible with iterative phaseretrial algorithms. For this, the probing beam needs to be coherent, otherwise the phase is not well-defined. This technique is outlined in this chapter.

A.2 Coherence of the probing beam

Coherence of waves means that there is a known phase relationship. An incoherent beam consists of many coherent waves, between which the phase relationship cannot be defined.

Coherence lengths can be defined in the framework of geometrical optics.

Longitudinal coherence length (ξ L ) is related to the monochromaticity of the beam, this concept is illustrated in figure A.1 Two waves with slightly different wavelengths (λ and (λ -∆λ)) are emitted from a point source. They have the same phase at the point of emission. Longitudinal coherence length is defined as the distance it takes for the the two waves to have opposite phases. At a distance twice the coherence length, they will be in phase again. From this criteria longitudinal coherence length can be computed:

(A.7)

The algorithm starts with the initial guesses for the probe (P 0 (r)) and the object (O 0 (r)) functions. Both guesses get updated through subsequent iterations that move between the real and the Fourier space.

The J number of diffraction patterns that were collected during the ptychography scan are addressed in a random sequence in the algorithm.

In the first step the guessed scattered wave is calculated from the (updated or initial) shifted probe and the object functions at iteration j:

Then Fourier transform is applied:

In the next step, the modulus of the scattered wave in the Fourier space (ψ j (q)) is replaced by the modulus obtained from the measured, corresponding diffraction pattern ( I j (q)):

The updated scattered wave is calculated with the inverse Fourier transform:

Finally the object and probe functions are updated by adding the weighted correction of the wavefront to the guess wavefront. This is expressed by the following two equations:

(A.12)

The Fourier constraint is applied where (just as in the ePIE), the modulus of the Fourier transformed wave (ψ j (q) = F[ψ j (r)]) is replaced with the modulus obtained from the measured diffraction patterns. The associated projection (Π F ) takes the following form: Π F (Ψ(q)) : ψ j (q) → ψ F j (q) = I j (q) ψ j (q) |ψ j (q)| , ∀j.

(A.17)

The overlap projection is determined from the minimisation of distance |Ψ(r) -Ψ O (r)| 2 , where Ψ O (r) = { P (r -R j ) Ô(r)}. Thus, the following equation needs to be minimised with respect to P and Ô:

which defines the overlap projection (Π O ):

Setting the derivative of equation A.18 to 0, the solution for the minimum is the following equation system:

(A.20)

O and P cannot be uncoupled analytically. When both the object and the probe are unknowns, the above equations (A.20) are applied in turn to update Ψ.

Using the projections defined in equations A.17 and A.19, the reconstruction is implemented using the the following update rule [START_REF] Veit Elser | Phase retrieval by iterated projections[END_REF]:

The convergence is monitored by the difference map error:

The aim is the minimise the difference map error.

A.3.3 Sensitivity of the phase to atomic displacement

Ptychography on rhenium was executed in the Bragg geometry, as shown in figure A.4. This technique is known as Bragg Projection Ptychography (BPP). Intensity distribution from a perfect crystal, where all the atoms are in their ideal positions (r 0 ), is a periodic function of the reciprocal space coordinates, with Bragg peaks at positions defined by the crystal. The intensity distributions are also symmetric and identical around each Bragg peak. Most crystals, however, are not perfect, and non-symmetric Bragg peaks are often observed.

In a strained crystal atoms are displaced from their ideal positions. A new position is given as r = r 0 + u(r 0 ), where u(r 0 ) is the displacement. Substituting r in the amplitude in equation A.2, the phase in the vicinity of a scattering vector g becomes the following:

For small displacements (q -g) • u(r 0 ) 2π and the third term can be neglected [START_REF] Dupraz | Coherent X-ray diffraction applied to metal physics[END_REF]. Thus the scattered amplitude is:

where the modified atomic form factor is

depth is approximately 4 µm, which, taking the angle of incidence into account, gives a probing depth of 1 µm. The detector consists of 3 modules arranged horizontally. Each module contains 2x8 chips, and each chip has 256x256 pixels. This results in a 1536x2048 image. The size of one pixel is 55x55 µm 2 [START_REF] Rau | Micro-and nanoimaging at the diamond beamline i13l-imaging and coherence[END_REF].

The X-ray beam was focused onto the sample using a Fresnel-Zone Plate. The spot size was approximately 1 µm horizontally and 0.5 µm vertically. The the horizontal size of the footprint of the beam on the sample is unchanged, the vertical size is increased to 0.5µm/ sin(17.2 • ) = 1.7µm.

A.5 Bragg ptychography on rhenium

Ptychography has been successfully used to study displacement fields in a wide range of materials. Dzhigaev et al. combined finite element method simulations with ptychography to obtain the 3D strain distribution in InGaN/GaN core-shell nanowires, and showed asymmetry in the strain relaxation [START_REF] Dzhigaev | X-ray bragg ptychography on a single InGaN/GaN core-shell nanowire[END_REF]. Using 3D ptychography Yau et al. observed grain boundary and dislocation dynamics in individual gold grains of a polycrystalline thin film while the sample was subject to heating [START_REF] Yau | Bragg coherent diffractive imaging of single-grain defect dynamics in polycrystalline films[END_REF]. Hruszkewycz

result of the hot filament, Q E has to be included in the equations for experiments when electron bombardment is applied. W refers to the tungsten backing on the substrate. SRe is the substrate and rhenium, which is considered as one unit, and conduction of heat through the substrate (Q C ) is included in the model. Finally, B denotes the wall of the vacuum chamber ('bâtiment'), which is at room temperature.

In equilibrium the heat exchange between the surfaces (Q) are equal, and can be expressed using the radiosities:

These three equation give the equation system that needs to be solved. The three unknowns are Q, the temperature of the tungsten layer (T W ), and the temperature of the substrate-rhenium (T SRe ). Radiosities need to be expressed as only the function of the unknowns and material parameters.

In equilibrium the heat exchange on the surfaces are also Q, and can be expressed using the irradiance and the radiosity:

Using equations (D.6) and (D.15):

Obtaining the radiosity of the substrate-rhenium is a little more difficult, because due to the non-zero transmittance, the irradiances of the two surfaces mix.

An equation that describes the relationship between the irradiances on the two sides can be obtained by subtracting (D.8) from (D.7) and substituting (D.10) and (D.11):

Now the irradiance of one side of the substrate-rhenium can be expressed with the radiosity of the same side.

Expressing H b SRe from (D.20) and substituting it in equation (D.10):

where, analogous to opaque objects, the notation R SRe = ρ SRe -τ SRe SRe +2τ SRe was used.

Next H W SRe is expressed from (D.20) and substituted in equation (D.11):

Using (D.7) and substituting (D.21) for H W SRe , the final expression for B W SRe is obtained:

E

Python scripts

In this appendix a selection of scripts and functions that were written in the Python environment to calculate, treat, or simulate the data detailed in this work are presented. The modified interference function

E.1 X-ray diffraction

The interference function, given in equation 1.23, was modified to account for the disorder in the film. A Gaussian distribution of lattice parameters was introduced. The mathematical formula of the thus modified interference function is given in equation 2.17, and is defined below. d_sum = 0 f o r n i n r a n g e ( i n t ( round (N, 0 ) ) ) : d_act = d0 + dw * np . random . normal ( 0 . 0 , 1 , 1 ) d_sum = d_sum + d_act d_dist . append ( d_act ) f = f + np . exp(-1 j * xq * d_sum) i n t e n s i t y = f * np . ma . c o n j u g a t e ( f ) i n t e n = c * i n t e n s i t y /np . max( i n t e n s i t y ) r e t u r n [ np . a r r a y ( np . r e a l ( i n t e n ) ) , d_dist ]

E.1.3 Simulation of the high resolution data

In section 2.2.2 the standard resolution data was simulated from the fit of the high resolution data. Equation 2.20 describes this operation mathematically, and the script is given below. import numpy a s np import s c i p y . s i g n a l a s s i g n a l

E.2 Functions for Mullins' thermal grooving

Below, the script that was used to describe the surface profile that develops during thermal grooving are presented. The theory was detailed in section 1.5.4, and it was applied to a rhenium film in section 2.3.

E.2.1 Evaporation-condenstation

import numpy a s np import math import s c i p y . s p e c i a l # D e f i n i n g t h e i n t e g r a l e r r o r f u n c t i o n : d e f i e r f c ( x ) : " Returns t h e i n t e g r a l e r r o r f u n c t i o n . " r = np . m u l t i p l y ( x , s c i p y . s p e c i a l . e r f ( x ) ) + np . m u l t i p l y ( 1 . / np . s q r t ( np . p i ) , np . exp (-1 * np . s q u a r e ( x ) ) ) -x r e t u r n r num_of_x = 50000 datax = np . l i n s p a c e ( 0 , 1 8 0 0 0 , num_of_x ) # p r o f i l e l i n e t t = [ 5 . , 3 0 . , 9 0 . , 1 2 0 . , 1 8 0 . ] * 60

# time b e t a = math . r a d i a n s [START_REF] Paul | Structural evolution of re (0001) thin films grown on nb (110) surfaces by molecular beam epitaxy[END_REF] # b e t a i n r a d i a n s mm = np . tan ( b e t a ) AA = 5 e3 At2 = 2 * np . s q r t ( np . m u l t i p l y (AA, t t ) ) p a r a m e t e r s [START_REF] Grigorij | Strain tuning of individual atomic tunneling systems detected by a superconducting qubit[END_REF] 8815 # h a u t e u r du p a l i e r ( s t e p h e i g h t )

p a r a m e t e r s [START_REF] Oh | Epitaxial growth of rhenium with sputtering[END_REF]