Dr Elisabeth Blanquet

Dr Gabriel Ferro

Prof Lyon

Wellmann Peter

Prof Assoc Jianwu

Prof Takeshi Yoshikawa

Lingling Xuan

Tao Wang

Vijaya Xuefeng Han

Shanti Paul

Fanshi Meng

Yunfan Shao

Dawei Peng

Sunrise Wang

Prof. Junhong Chen

Extent of Bunching EF Extent of Faceting EMF Electromagnetic Field FEM Finite Element Method HHCF Height-Height Correlation Function PSDF Power Spectral Density Function PVT Physical Vapor Transport RMS Root Mean Square SEM Scanning Electron Microscopy SF Stacking Fault TED Threading Edge Dislocation TEM Transmission Electron Microscopy TSD Threading Screw Dislocation TSSG Top

Keywords: Silicon carbide, surface reconstruction, vicinal surface, step bunching, faceting, dissolution Autocorrelation Function AFM

The accurate description, understanding and control of the high-temperature behavior of the interface between 4H-SiC single crystal and the liquid phase are essential for the future development of SiC solution growth. The various parameters that cause morphological instabilities, such as step-bunching and micro-faceting, remain unclear. This thesis aims to provide new information on the origin of instabilities at the SiC/liquid interface during growth. First, a specific sessile drop approach was designed, combining experiments and numerical simulation. An isothermal environment was realized, in which an electromagnetic field was used as an actuator for carbon transport and thus enabled the transition from transient surface dissolution to stationary dissolution. All technical implementation issues were critically discussed and optimized. Next, the reconstructions of vicinal and concave surfaces of 4H-SiC {0001} in pure liquid silicon were studied in depth, considering the effect of external parameters (time, temperature) and intrinsic properties (crystallographic orientation, polarity). In general, a higher surface free energy leads to more pronounced faceting on the Si side, while a higher Ehrlich-Schwoebel barrier leads to stronger step-bunching on the C side. By growing on such reconstructed surfaces, we have clearly demonstrated total reversibility of bunching-debunching of the macrosteps on the C face, while the transition to a faceted surface is irreversible for the Si face. Moreover, an original sandwich experiment showed that the difference in surface free energy between the two polarities could generate carbon transport. Finally, using the same sessile drop configuration, we provided a new determination of the solubility of carbon in liquid silicon over a wide temperature range (1500 °C~2200 °C), clarifying the wide discrepancy in the literature.

Résumé

La description précise, la compréhension et le contrôle du comportement à haute température de l'interface entre le monocristal de 4H-SiC et la phase liquide sont essentiels pour le développement futur de la croissance en solution du SiC. Les différents paramètres qui causent des instabilités morphologiques, telles que le step-bunching et le microfacettage, restent peu clairs. Cette thèse vise à fournir de nouvelles informations sur l'origine des instabilités à l'interface SiC/liquide pendant la croissance. Tout d'abord, une approche spécifique de goutte sessile a été conçue, combinant expériences et simulation numérique. Un environnement isotherme a été réalisé, dans lequel un champ électromagnétique a été utilisé comme actionneur pour le transport du carbone et a ainsi permis la transition d'une dissolution de surface transitoire à une dissolution stationnaire.

Tous les problèmes de mise en oeuvre technique ont été discutés de manière critique puis optimisés. Ensuite, les reconstructions de surfaces vicinales et de surfaces concaves de 4H-SiC {0001} dans du silicium liquide pur ont été étudiées en détails, en considérant l'effet des paramètres externes (temps, température) et des propriétés intrinsèques (orientation cristallographique, polarité). En général, une énergie libre de surface plus élevée conduit à un facettage plus prononcé sur la face Si, tandis qu'une barrière d'Ehrlich-Schwöbel plus élevée conduit à un step-bunching plus fort sur la face C. En reprenant la croissance sur de telles surfaces reconstruites, nous avons clairement démontré une réversibilité totale du bunching-debunching des macromarches sur la face C, alors que la transition vers une surface facettée est irréversible pour la face Si. De plus, une expérience originale en sandwich a montré que la différence d'énergie libre de surface entre les deux polarités pouvait générer le transport du carbone. Enfin, en utilisant la même configuration de goutte sessile, nous avons fourni une nouvelle détermination de la solubilité du carbone dans le silicium liquide sur une large gamme de température (1500 °C~2200 °C), clarifiant le désaccord existant dans la littérature.

Mots clés : Carbure de silicium, reconstruction de surface, surface vicinale, stepbunching, facettage, dissolution.

Introduction

Silicon carbide (SiC) is a wide bandgap semiconductor material with unique physical and chemical properties. The special crystal structure gives this material very high hardness, chemical inertness and high thermal conductivity. Combining its exceptional electronic properties such as high critical electric field strength and high saturated electron drift velocity, SiC is regarded as a very promising semiconductor for high power and high temperature electronics [START_REF] Kimoto | Fundamentals of Silicon Carbide Technology[END_REF]. This chapter will briefly introduce the crystal structure and material properties of SiC. As a critical aspect for both crystal growth and electronic devices processing, the SiC surface reconstruction under various environments will be overviewed. Then the relevant mechanisms especially for the motion of the surface steps will be summarized. Last, this chapter will end by giving the purpose of the thesis work.

1.1. SiC background 1.1.1. Crystal structure of SiC SiC is a binary compound semiconductor which generally owns a stoichiometry ratio of silicon and carbon atoms of 1 (Si: C = 1:1). Since both Si and C atoms have four valence electrons, they form a strong covalent bonding in a SiC crystal by sharing electron pairs in sp 3 hybrid orbitals. Each Si (C) atom is surrounded by four C (Si) atoms, forming a tetrahedral structure as shown in Fig. 1.1a. The Si-C bonding energy is as high as 4.6 eV, which gives SiC a lot of outstanding properties.

SiC is perhaps the best known example of polytypism. Polytypism, sometimes referred to one-dimensional polymorphism, is the phenomenon that a substance of definite composition crystallizes into a number of modifications that differ only in one unit cell dimension [START_REF] Baronnet | Some aspects of polytypism in crystals[END_REF]. The polytypic structures (known as polytypes) are built up of elementary layers of invariant structure which are stacked along the variable dimension following the certain stacking sequences. Along the c-axis of SiC crystal structure, there are three possible occupied sites of Si-C pairs in the hexagonal close-packed system, denoted as A, B, and C (Fig. 1.1b). The layer where the most closed Si-C pairs locate is defined as the Si-C bilayer (comprise one layer of Si atoms and one layer of C atoms), which has a height of 0.25 nm. Each bilayer can occupy only one site among "A", "B" and "C". For example, on the top of an "A" bilayer, the next bilayer must occupy either "B" or "C" sites. Similarly, if the second bilayer occupies the B sites, the third bilayer could sit on "A" or "C" sites.

More than 200 polytypes of SiC have been reported, some of them have several hundred and even more than one thousand stacking bilayers in the primitive unit cell [START_REF] Cheng | Silicon carbide polytypes are equilibrium structures[END_REF]. Generally there are no significant differences in the potential energies and chemical properties between the polytypes (only some physical properties are more or less different). In contrast to polymorphic modifications, no stability domain is expected for polytypes. This is illustrated as the frequent coexistence of different polytypes in grown crystals even at a given pressure and temperature, which is in fact one of the most challenging parts for SiC single crystal growth [START_REF] Knippenberg | Growth phenomena in silicon carbide[END_REF]. The crystals are projected on {110} or {11 2 0} planes to better show the stacking sequences along the c-axis. Here the Ramsdell's notation is used to describe the different SiC polytypes [START_REF] Ramsdell | Studies on silicon carbide[END_REF]. In this nomenclature, the number in front defines the number of bilayers in one unit cell while the latter capital letter represents the crystal system (C for cubic, H for hexagonal, and R for rhombohedral). As we can see, the stacking sequences of 3C, 4H and 6H are ABC (or ACB), ABAC (or ABCB), and ABCACB respectively. Note that 3C-SiC is often called β-SiC, and all the other polytypes are referred to as α-SiC. 3C-SiC has a zincblende crystal structure thus the hexagonality is zero, while 4H-SiC and 6H-SiC have the hexagonality of 50% and 33.3%, respectively. Nevertheless, ab initio calculations of the polytype energy indicated that the bulk energy differences among different SiC polytypes are extremely small, below 1 meV/atom [START_REF] Ito | Theoretical investigations for the polytypism in semiconductors[END_REF]. The stability and nucleation probability of SiC polytypes strongly depend on temperature [START_REF] Knippenberg | Growth phenomena in silicon carbide[END_REF]. For example, 3C-SiC is considered as a low temperature form, and prone to be transformed into hexagonal SiC polytypes such as 6H-SiC at high temperatures above 1900-2000 °C [START_REF] Yoo | Solid-state phase transformation in cubic silicon carbide[END_REF]. For the hexagonal 4H-and 6H-SiC crystals, there is a change in stacking sequence along the c-axis after half of the unit cell and thus two sets of bilayers with different orientations are demonstrated. In the (0001) planes, each set of bilayers is rotated 60° relative to another neighboring set of bilayers. Therefore, one can imagine that two kinds of steps edge can form on SiC (0001) surface in <11 00> direction, which own different numbers of dangling bonds at the step edges (Fig. 1.3). They are referred to as SN and SD steps according to the nomenclature of Pechmen [START_REF] Pechman | Interactions of Br with Si(111)-7×7: Chemisorption, step retreat, and terrace etching[END_REF]. The SN steps have one dangling bond for the edge atoms while the SD steps have two dangling bonds. It is considered that these two kinds of steps have different energies and hence different advancing (during growth) or receding (during sublimation or dissolution process) rates. Owing to the difference of electronegativity between silicon and carbon, a small polarization exists along the Si-C bonding and thus along the c-axis of the tetrahedron.

When cutting the crystal perpendicular to the c-axis, the vertical Si-C single bond is favorably cleaved and then two faces are formed as shown in Fig. 1. [START_REF] Knippenberg | Growth phenomena in silicon carbide[END_REF]. The (0001) surface that terminates with Si atoms is referred to as the Si face, correspondingly the other side ((0001) surface) which terminates with C atoms is called the C face. Similarly for the cubic SiC crystal (3C-SiC), the [START_REF] Bales | Morphological instability of a terrace edge during step-flow growth[END_REF] Si face and the (1 1 1 ) C face exist with the same top surface atomic configurations as the hexagonal {0001} surfaces. The polarity of SiC is very important as the two polar surfaces usually exhibit different behaviors during the surface processing and reconstructions. As one important property describing the surface, the surface free energies of the two polar surfaces for the three main polytypes are listed in Table 1-1. Even if there are discrepancies about the energy values in the literature, qualitative comparisons can be realized. Regardless of the polytype, the surface free energy of Si face is much higher than that of C face. This energetic difference between the two polar surfaces affects the surface reactivity and more generally its behavior. For example, the "hill-and-valley" (or faceted) structure is more often observed on the Si face whereas the C face exhibits lower surface roughness in the CVD growth of SiC [START_REF] Kimoto | Step bunching mechanism in chemical vapor deposition of 6H-and 4H-SiC{0001}[END_REF]. The enhanced macrostep formation on the Si face might be ascribed to the total surface energy minimization requirement, through which the area of the surface with the lower surface energy tends to increase.

Surface free energy

Si face C face 3C-SiC [START_REF] Pearson | Computer modeling of Si and SiC surfaces and surface processes relevant to crystal growth from the vapor[END_REF] 2220 300 4H-SiC [START_REF] Syväjärvi | Interfacial properties in liquid phase growth of SiC[END_REF] 1800 750 6H-SiC [START_REF] Takai | Reconstruction and energetics for surfaces of silicon, diamond and β-SiC[END_REF] 1767 718

1.1.2

. Basic properties and applications of SiC

As an IV-IV compound semiconductor material, SiC has many unique and attractive physical and chemical properties. Due to the very strong Si-C bonding, it owns outstanding mechanical properties like very high hardness (Mohs hardness: 9.1) and high Young's modulus (380-700 GPa) [START_REF] Zorman | Micro-and nanomechanical structures for silicon carbide MEMS and NEMS[END_REF]. SiC material can be used in harsh conditions because it is highly resistant to chemical attack and nuclear radiations. In addition, it has a thermal conductivity as high as 4.9 W•cm -1 •K -1 , which is dependent on the density of impurities or doping [START_REF] Slack | Thermal Conductivity of Pure and Impure Silicon, Silicon Carbide, and Diamond[END_REF]. All above properties make it a promising structural ceramic material especially for applications operated at high temperatures and in severe environments. For instance, SiC fiber reinforced-SiC matrix ceramic composites (SiCf/SiC) are among the most promising candidate as structural materials for fusion power reactors, which show even superior safety characteristics compared to metallic materials [START_REF] Riccardi | Issues and advances in SiCf/SiC composites development for fusion reactors[END_REF].

Apart from being an excellent structural material, SiC is an old but still very promising semiconductor materials. It is especially suitable for advanced power electronic devices because of its superior properties comparing to Si [START_REF] Kimoto | Fundamentals of Silicon Carbide Technology[END_REF]. All the SiC polytypes have an indirect bandgap structure; the bandgap at room temperature is 2.36 eV for 3C-SiC, 3.26 eV for 4H-SiC and 3.02 eV for 6H-SiC. The bandgap decreases with the increasing temperature and largely depends on the doping concentration. The breakdown electric field of 4H-and 6H-SiC is about eight times higher than that of Si at the same doping level. This is the main reason why it is one of the most attractive materials for power devices. In fact, GaN is another potential candidate because it has almost the same bandgap and breakdown electric field. However, the GaN technology (growth and device fabrication for power electronics) is not as mature as the one of SiC. For instance, up to now the large size bulk growth of GaN single crystal remains very challenging. Last but not least, SiC also shows a wide doping range for both n-type and p-type doping and owns a quite high saturated electron drift velocity.

The major physical properties of SiC together with other common semiconductors (Si and GaN) are summarized and compared in Table 1-2. The excellent properties including the wide bandgap, the high breakdown field and the large thermal conductivity, make SiC a unique and indispensable material for power electronic devices and systems. Note that 4H-SiC is the most widely considered polytype because of its slightly superior properties than other polytypes for power electronics applications [START_REF] Kimoto | Fundamentals of Silicon Carbide Technology[END_REF]. Driven by such applications, 4H-SiC has been more intensively developed and is currently the most mature polytype in terms of technology (wafer size and quality, epilayers, device processing and so on).

Nevertheless, with the lowest bandgap but fast saturation electron velocity, 3C-SiC is also a prospective choice for middle voltage electronic applications as long as high-quality single crystals can be obtained in the future. As we can see from the Si-C binary phase diagram (Fig. 1.5), there is a peritectic transformation at about 2830 °C where SiC decomposes into carbon and a silicon rich solution (about 18 atom% carbon [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. I. SiC and SiC-SiO2 mixtures under neutral conditions[END_REF]). This means it is not possible to grow SiC from a stoichiometric melt under standard pressure by using the conventional techniques like that applies for silicon ingots growth. Actually, theoretical calculations have indicated that stoichiometric melting would only occur for pressures >10 5 atm and temperatures >3200 °C [START_REF] Glass | SiC seeded crystal growth[END_REF]. From the Si-C diagram, there are generally two ways to grow SiC bulk crystals: 1) Sublimation growth. The Si-and C-bearing vapor phase species coming from a SiC source material are transported onto a substrate maintained at lower temperature. 2) Solution growth. SiC can be grown from a Si-rich liquid which is formed by dissolving limited amount of carbon (18 mol % maximum) in pure silicon. In this case, other alloying elements (like Cr and Fe) are usually added in order to increase the carbon solubility in the liquid. The details of above two approaches will be introduced as follows.

Figure 1.5 Binary phase diagram of Si-C system at a pressure of 1 atm [START_REF] Chaussende | Silicon carbide[END_REF].

1) Sublimation growth

Since the Lely method proposed in 1955 and then the breakthrough made by Tairov and Tsvetkov in 1978, the sublimation method has been significantly developed in the past decades [20,[START_REF] Tairov | Investigation of growth processes of ingots of silicon carbide single crystals[END_REF]. Now the modified-Lely method, also often referred to as physical vapor transport (PVT) method, has become the most mature method for SiC bulk growth. The principle idea of PVT method is to sublimate SiC powder and then re-crystallize on a single crystal seed. The main gaseous species formed from the SiC source are Si, Si2C and SiC2.

The driving force of this mass transport is controlled by the different partial pressure of the gases between the source powder and the seed, which is directly related to the temperature gradient in the growth cell. The illustration of the sublimation growth reactor and the axial temperature profile are shown in Fig. 1.6. The growth is carried out in a quasi-closed low porosity graphite crucible. The typical temperature of the source is 2300-2400 °C while the seed temperature is about 100 °C lower. Since the 3C-SiC is only stable at relatively lower temperature, the growth of 3C-SiC crystal needs to be carried out at a lower temperature. Now it is still difficult to realize 3C-SiC bulk growth because growth rate is slower at lower temperature and it is especially hard to keep single 3C-SiC polytype during the growth.

With the strong improvement of the modelling and simulation tools, combined with the in situ X-ray imaging techniques [START_REF] Wellmann | In situ visualization of SiC physical vapor transport crystal growth[END_REF], the "black box" of growth process has been opened up step by step. During the last two decades, SiC crystals with high quality and large diameter have been obtained at an industrial level. Wafers with low dislocation density near 100 cm -2 have been reported by using the "repeated a face" method where the growth was carried out two or even three times on either {112 0} or {11 00} faces subsequently [START_REF] Nakamura | Ultrahigh-quality silicon carbide single crystals[END_REF].

So far high quality SiC wafers of 150 mm in diameter are the standard commercial procedure [START_REF] Manning | Large area 4H SiC products for power electronic devices[END_REF], and 200 mm diameter wafers first demonstrated more than 10 years ago is currently ramping up [START_REF] Powell | Bulk growth of large area SiC crystals[END_REF]. Left side shows the schematic temperature along the symmetry axis of the crucible [START_REF] Chaussende | Silicon carbide[END_REF].

Beside the standard PVT method, a number of other vapor growth methods have been proposed such as HTCVD, halide CVD, and CF-PVT [START_REF] Chaussende | Status of SiC bulk growth processes[END_REF]. The common point of these new seeded growth methods is that continuous gas source precursors of high purity, rather than only gases generated from powder, are supplied into the cell during the growth. The last method mentioned, named the continuous feed-physical vapor transport method(CF-PVT), which was developed in our group, is a process which comprises CVD for the feeding of the SiC source and PVT for the growth of the single crystal [START_REF] Chaussende | Control of the Supersaturation in the CF-PVT Process for the Growth of Silicon Carbide Crystals: Research and Applications[END_REF]. This method not only has the potential of producing long boule as a result of the continuous feeding of the high-purity polycrystalline SiC source, but also can realize a precise control of SiC polytypes. In this sense, the CF-PVT is regarded as a prospective method for the growth of 3C-SiC single crystal.

2) Solution growth

SiC is the only semiconductor material, produced at a mass scale, that grown from the vapor phase. Other important semiconductors like Si, GaAs and InP are mainly manufactured from the liquid phase using melt growth techniques like Czochralski. In spite that the SiC growth from melt is not possible due to the incongruent decomposition of SiC, the growth from solution is still feasible based on the liquidus on Si side. The relevant works on solution growth started from around 1960s. Besides the pure liquid Si, the solution growth was carried out using other binary and ternary (even quaternary in some works) liquid phases like Si: Cr, Ti, Fe [START_REF] Griffiths | Growth of α-SiC Single Crystals from Chromium Solution[END_REF][START_REF] Pellegrini | Liquid phase epitaxial growth of SiC from transitionmetal silicide solvents[END_REF] as the solvent. But the research activities on solution growth stopped in the late 1970s probably due to the appearance and development of the more successful modified PVT method [START_REF] Tairov | Investigation of growth processes of ingots of silicon carbide single crystals[END_REF]. In the late 1990s, SiC growth from the liquid came back in force with some promising results, such as the ability to close the micropipes [START_REF] Rendakova | High quality 6H-and 4H-SiC pn structures with stable electric breakdown grown by liquid phase epitaxy[END_REF][START_REF] Müller | Studies on SiC Liquid Phase Crystallization as Technique for SiC Bulk Growth[END_REF]. Almost at the same period, Hofmann and Müller reported the successful solution growth of bulk SiC for the first time [START_REF] Hofmann | Prospects of the use of liquid phase techniques for the growth of bulk silicon carbide crystals[END_REF]. The growth rate in their work was 0.1-1 mm•h -1 , which was two orders of magnitude higher than that predicted in Tiller's model [START_REF] Tiller | Theoretical analysis of requirements for crystal growth from solution[END_REF]. This result restored researchers' confidence in solution growth and some of them reconsider it as a possible or even better bulk growth method for SiC.

The top seeded solution technique (TSSG), which is based on a Czochralski process configuration, is the most promising and studied method for SiC solution growth. The TSSG process is briefly illustrated in Fig. 1.7. A graphite crucible is filled with Si-based liquid metal, the liquid reacts with the crucible which also serves as the carbon source for the growth. The seed is dipped into the solution and then slightly pulled out, in this way a meniscus bridge is formed between the surface of the growing crystal and the liquid. By controlling the meniscus height during the growth, a remarkable enlargement of crystal size was achieved [START_REF] Daikoku | Top-seeded solution growth of 4H-SiC bulk crystal using Si-Cr based melt[END_REF]. The typical growth temperature range is 1750-2100 °C, where the temperature of the seed is slightly lower than the solution to provide the driving force for the growth. The accelerated crucible rotation technique (ACRT) is sometimes used to enhance the melt mixing, where the crucible and seed are usually rotated in opposite directions. Before the real growth, the seed is first dissolved slightly in the liquid (meltback), through this the polishing defects such as scratches on the seed surface can be removed. A maximum growth rate of 2 mm•h -1 has been realized when using Si-Cr based melt for the solution growth of 4H-SiC [START_REF] Kado | High-speed growth of 4H-SiC single crystal using Si-Cr based melt[END_REF].

Figure 1.7 Schematic illustration of the top-seeded solution growth method [START_REF] Daikoku | Mechanism of Replicating 4H-SiC Polytype during Solution Growth on Concave Surface[END_REF].

Compared to the standard PVT method, the temperature gradient near the growth front (solid/liquid interface) is much smaller, which provides a growth condition closer to the thermodynamics equilibrium. This is beneficial for achieving a lower defect density in the grown crystals. In addition, the solution growth method has shown significant effect in defect healing or conversion. The possibility to close the micropipes and to reduce the density of dislocations has been reported [START_REF] Yakimova | Current status and advances in the growth of SiC[END_REF] [START_REF] Kusunoki | Liquid Phase Epitaxy of 4H-SiC Layers on On-Axis PVT Grown Substrates[END_REF]. The synchrotron X-ray topography observation showed that threading screw dislocations (TSDs) in the off-axis Si-face seed crystal were efficiently converted to Frank-type basal plane stacking faults (SFs) [START_REF] Yamamoto | High-Efficiency Conversion of Threading Screw Dislocations in 4H-SiC by Solution Growth[END_REF]. This process was effectively assisted by step-flow growth on off-oriented (0001) seed crystals [START_REF] Ujihara | Conversion mechanism of threading screw dislocation during SiC solution growth[END_REF]. With the intensive studies on solution growth these years, the quality and size have been improved fast. The TSD density was significantly reduced from 2000 cm -2 to only 30 cm -2 and the TED density was also reduced to a large extent [START_REF] Harada | Dislocation conversion during sic solution growth for high-quality crystals[END_REF]. Very recently, 4-inch SiC boule of thickness 15 mm has been successfully obtained by the solution growth method using an Si-Cr based solvent [START_REF] Kusunoki | Solution Growth of 4-Inch Diameter SiC Single Crystal Using Si-Cr Based Solvent[END_REF]. However, at moment several challenges are still pending to compete with the more mature PVT method:

1) The carbon solubility in liquid.

The most ideal solvent for SiC solution growth is pure silicon. However, the growth rate of SiC is extremely slow because the carbon dissolved in pure Si melt is very low.

Numerous works have been done to determine the carbon solubility in pure liquid silicon by various methods [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF][START_REF] Olesinski | The C-Si (Carbon-Silicon) system[END_REF][START_REF] Durand | Carbon solubility in solid and liquid silicon -A review with reference to eutectic equilibrium[END_REF], generally it is as small as 1 at % or even less at temperatures below 2000 °C though it is still controversial in literature. In this thesis, we will also provide a new determination of the carbon solubility in liquid silicon equilibrated with SiC, by using a slightly modified sessile drop method. The details of our method and obtained results can be found in chapter 6.

To overcome the low solubility of carbon in pure silicon below 2000 °C, addition of other metals such as Cr, Ti, Fe and Tb in the silicon or even pure molten metals have been used [START_REF] Griffiths | Growth of α-SiC Single Crystals from Chromium Solution[END_REF][START_REF] Pellegrini | Liquid phase epitaxial growth of SiC from transitionmetal silicide solvents[END_REF][START_REF] Narumi | Thermodynamic evaluation of the C-Cr-Si, C-Ti-Si, and C-Fe-Si systems for rapid solution growth of SiC[END_REF][START_REF] Tairov | Liquid Phase Epitaxy of SiC in the System Si-Tb-SiC by Temperature Gradient Zone Melting (I). Investigation of Solubilities in the System Si-Tb-SiC[END_REF]. For example, iron-rich Fe-Si alloy can attain a SiC solubility 100 times higher than that of silicon solvents at the same temperature [START_REF] Yoshikawa | Solution Growth of Silicon Carbide Using Fe-Si Solvent[END_REF]. However other aspects other than the carbon solubility should also be considered. One possible problem is that the additives could cause unintentional doping into the grown crystal, which might be detrimental to the electronic properties of the semiconductor substrate. For example, Ga and Al elements are not suitable because they induce p-type conduction as shallow acceptor in SiC. In this sense, it would be still meaningful to consider and develop the solution growth by using pure silicon as solvent. Recently Fahlbusch et al developed a solution growth process combining a vertical Bridgeman approach to a vertical gradient freeze method, by which they achieved high quality SiC layers up to 200 μm in a pure liquid silicon [START_REF] Fahlbusch | High temperature solution growth of SiC by the vertical Bridgman method using a metal free Si-C-melt at 2300 °C[END_REF][START_REF] Fahlbusch | Solution Growth of Silicon Carbide Using the Vertical Bridgman Method[END_REF].

2) The stability of the growth front.

During crystal growth, the stability of the growth front (fluid/crystal interface) is of critical importance. Instability often takes place as surface roughening due to the interaction with the liquid. It causes various defects in the grown crystal. The detailed presentation and corresponding mechanisms will be exposed in section 1.2.3. In particular, huge macrosteps of several hundred of nanometres are often observed on the solution growth surface, as a result of strong step bunching. The propagation of macrosteps on the basal plane plays an important role in reducing the TSDs. However on the other hand, they can cause some undesired defects in the crystal (e.g., trenches, solvent inclusions) if the macrosteps are too large (giant macrosteps formed by accumulation of macrosteps) [START_REF] Zhu | Influence of solution flow on step bunching in solution growth of SiC crystals[END_REF].

Even some measures have been applied to eliminate the formation of giant macrosteps, the mechanisms of macrostep formation at the solid/liquid interface and their evolution during the solution growth is still not fully understood.

SiC/liquid interface properties

The chemical systems based on SiC and liquid metals (or semimetals) are involved not only in the field of crystal growth but also in many other aspects. For instance, the SiC/liquid interface plays an important role in the fabrication of SiC-metal composites, joining of SiC ceramic pieces and refining of liquid metal etc. Some common and critical properties that determine the performance of SiC-liquid interfaces are introduced in the following.

1) Surface wettability

Wettability is defined as the ability of liquids to form interfaces with solid surface, which is normally referred to three phases which meet at the triple line. The sessile drop technique is often used to implement the wetting process in order to evaluate the interfacial phenomena between the liquid and the solid substrate. A droplet wetting on a solid and flat surface is illustrated in Fig. 1.8. When a liquid phase (L) wets a solid substrate (S) in a vapor phase (V), the contact angle between the solid and the liquid is defined from the relationship between interfacial energies:

cos θ = (1-1)
Here 𝛾 and 𝛾 are the surface energies of solid and liquid phases, respectively, 𝛾 is the solid/liquid interfacial energy. In principle, contact angle (θ) can have any value between 0° and 180°. In materials science, it is generally used to distinguish two types of systems: ''good wetting'' is obtained when θ < 90° whereas ''bad wetting'' is obtained when θ >90°. In addition to the contact angle, the work of adhesion is often adopted to compare the wetting and bonding behavior at the solid/liquid interface. The work of adhesion is given by the equation:

𝑊 = (𝛾 + 𝛾 ) -𝛾 (1-2)
It can be explained as the work per unit area needed to separate a solid/liquid interface of energy 𝛾 into a solid/vapor interface of energy 𝛾 and a liquid/vapor interface of energy 𝛾 . By combining the Eqs. 1-1 and 1-2, one can obtain the expression of Wad as the function of contact angle:

𝑊 = 𝛾 (1 + cos 𝜃) (1-3)
A higher Wad value reflects a stronger interaction between the solid phase and liquid phase, and vice versa. This expression is very useful and practical because both values 𝛾 and θ can be measured experimentally.

For the SiC/liquid system studied in this thesis, the Si liquid can generally form strong interactions with SiC at high temperature resulting in the low value of θ and the high value of Wad. The contact angle of silicon is about 38° on α-SiC substrate and about 41.5° on β-SiC [START_REF] Nikolopoulos | Wettability and interfacial energies in SiC-liquid metal systems[END_REF]. The wetting angle on the C face (about 47°) is larger than on the Si face. However it was considered that the difference between the two polar surfaces is limited if one looks at the Wad values (which differ only in a few percent) [START_REF] Rado | Wetting and adhesion in metalsilicon carbide systems: The effect of surface polarity of SiC[END_REF]. In addition, the contact angle usually shows a decreasing trend when increasing the temperature as a result of the stronger interaction between the SiC and solvent (higher carbon solubility).

In fact, the wetting of liquid on SiC surface plays an important role in the solution growth process. One application is the so called meniscus formation technique, which is used to control the shape of the growth front, is actually based on the wetting of the solution on the SiC growth surface [START_REF] Daikoku | Top-seeded solution growth of 4H-SiC bulk crystal using Si-Cr based melt[END_REF].

2) Surface roughness

Roughness is one of the most important characteristics of surface texture, which can impact the chemical and physical properties of the surface. For the substrates used as growth seed, the irregular structures such as polishing scratches directly impact the final density of defects of epi-layers and crystals [START_REF] Sanchez | Assessment of Polishing-Related Surface Damage in Silicon Carbide[END_REF]. There is a direct correlation between the growth mode (surface mechanisms) and the roughness. On the other hand, the performance of SiC electronic devices is dependent on the morphology of the surface or the junction interface. The rough junction interface induced by the so-called 'microroughness' could cause electric field crowding in high-power devices, resulting in the reduction of blocking voltages. It has been reported that the surface roughness could affect the channel mobility and the susceptibility of gate oxide to defect-related breakdown in metal-oxide-semiconductor field effect transistors (MOSFET) [START_REF] Chung | The Effects of Low-Angle Off-Axis Substrate Orientation on MOSFET Performance and Reliability[END_REF].

There are several different ways to quantify the surface roughness, including the roughness average (Ra), Root Mean Square (RMS) roughness and so on. The Ra represents the arithmetic average of deviation from the center plane while the RMS roughness gives the standard deviation of the Z values within a given area. They are expressed as following:

R =

(1-4)

𝑅𝑀𝑆 = ∑ ( ) (1-5)
Where the Zi is the current Z value, Zave is the average Z value within the given area, Zcp is the Z value of the center plane. As can be inferred from the formulas, the single peak or valley on the surface will influence the RMS value more than the Ra. In literature RMS value is more widely used to study the macro roughness on the SiC surface.

3) Total surface energy

From the energetic point of view, surface reconstruction processes are usually driven by the minimization of the total surface energy. One thing to note here is that the total surface energy is not the same as the interfacial (or surface) energy (γij) we discussed before. The γij is referred to the specific interfacial energy, i.e. free energy per unit area. If the surface with an area of A is isotropic, the total surface energy (Et) would be calculated as

𝐸 = ∫ 𝛾 dA (1-6)
By analogy, if the surface was reconstructed into two different areas B and C with different free interfacial energy γB and γC, the total surface energy Et should be:

𝐸 = ∫ 𝛾 𝑑𝐵 + ∫ 𝛾 𝑑𝐶 (1-7)
In the real cases, the surface energy is not a simple sum of the surface energy of different parts. Other issues such as elastic potential energy between the steps and effect of surface stress at step edge should also be considered. For example, when vicinal SiC (0001) surface is reconstructed into nano-faceted structure, consisting of (0001) and (112 n) nano-facets, the total surface free energy (E) includes three terms [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF]:

𝐸 = 𝐸 surf + 𝐸 edges + Δ𝐸 elastic (1-8)
where Esurf is the free energy of the (0001) terraces and the (112 n) facets, Eedges is the shortrange energy of the edges, and Eelastic is the elastic energy due to discontinuity of the surface stress at the crystal edges. These three terms are expressed as follows [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF][START_REF] Marchenko | Elastic properties of crystal surfaces[END_REF]

𝐸 = ( ) ( ) ( ) ( ) ( ) (1-9) 𝐸 = (1-10) 𝛥𝐸 = - 𝑙𝑛 𝑠𝑖𝑛 𝜋 (1-11)
Here, E(0001) and E(112̅ n) are the surface energies of (0001) and (112 n) planes, respectively, ε is the short-range energy of the convex and concave edge, σ is the Poisson's ratio, F is the surface stress, Y is the Young's modulus, and a is the lattice parameter.

4) Surface stiffness and step stiffness

The surface stiffness describes the relationship between the load acting on the surface and the elastic deformation of the surface. Generally, the stiffness depends on the material's mechanical properties (elastic modulus, shear modulus, and Poisson's ratio) and on the geometry of the bearing structure as well.

In this thesis, the step stiffness rather than the surface stiffness is introduced to study the stability of the step movement. It can be defined as the resistance to an applied force causing a step to deform. The larger step stiffness indicates a more stable step when it propagates on the surface. In another word, the steps with large step stiffness are not sensitive to the external disturbances. For the vicinal surfaces, the step stiffness is closely related to the atomic diffusion length as well as the step distance [START_REF] Uwaha | Fluctuations and instabilities of steps in the growth and sublimation of crystals[END_REF]. The interplay between step stiffness and diffusion anisotropy induces a meandering instability during vicinal Si (001) surface growth [START_REF] Frisch | Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface[END_REF]. For steps on SiC surface, it was reported that the adhesion of Al atoms on growing SiC surface could reduce the step stiffness and then could induce instability in the regularly arranged step trains [START_REF] Mitani | Effect of aluminum addition on the surface step morphology of 4H-SiC grown from Si-Cr-C solution[END_REF].

SiC surface reconstruction

Generally, the semiconductor crystal surface is not ideally flat without any patterns or textures, either for the as-grown surface or mechanically polished surface. In most cases, the surface structure is composed of regular or irregular step trains, separated by flat terraces. The steps can originate from surface defects such as screw dislocations or twodimension (2D) nuclei, as well as from artificial process like off-axis cutting (to produce vicinal surfaces). Under different circumstances, different kinds of step-and-terrace structures on SiC surface will be formed through different surface reconstruction processes. When the effect of external conditions is excluded, the surface properties would dominate the reconstruction behavior and in return, the reconstruction features can also reflect the materials properties. In this section, the general reconstruction features of SiC surfaces under various conditions are introduced, emphasizing on the evolution of surface steps and terraces.

SiC surface reconstructed by gas etching

The commercially available SiC wafers usually contain a large number of scratches arising from the polishing process. One effective approach for removing the defects and obtaining atomically flat terraces is H2 etching [START_REF] Hallin | In situ substrate preparation for high-quality SiC chemical vapour deposition[END_REF][START_REF] Hartman | Characterization of hydrogen etched 6H-SiC(0001) substrates and subsequently grown AlN films[END_REF][START_REF] Xue | Two-step preparation of 6H-SiC(0001) surface for epitaxial growth of GaN thin film[END_REF][START_REF] Ramachandran | Preparation of atomically flat surfaces on silicon carbide using hydrogen etching[END_REF][START_REF] Powell | Process-Induced Morphological Defects in Epitaxial CVD Silicon Carbide[END_REF] or HCl/H2 etching [START_REF] Nakamura | Formation of periodic steps with a unit-cell height on 6H-SiC (0001) surface by HCl etching[END_REF][START_REF] Nakagawa | Self-Ordering of Nanofacets on Vicinal SiC Surfaces[END_REF]. Along with that the scratches are sufficiently removed, the ordered step-terrace structures by step bunching are usually observed on the etched surface.

After gas etching, the steps on vicinal SiC surfaces are mostly bunched into unit-cell height steps on Si face, whereas half unit-cell high steps dominate on the C face. Sometimes on nominal (0001) surface, periodic arrays of atomically flat terraces separated by steps of unit-cell height are also observed, which was believed originating from inadvertent miscut angles [START_REF] Ramachandran | Preparation of atomically flat surfaces on silicon carbide using hydrogen etching[END_REF]. On the Si face miscut towards the <11 00> direction, the formation of surface ripples are observed and the development of small facets (nanofacets) are found for higher miscut angles. However, much less facets are observed on vicinal C surfaces. On the Si face the etching rate along <11 00> (normal to the <112 0> direction) is slower than along < 112 0 > (normal to the < 11 00 > direction). On the contrary, on the C face, steps perpendicular to <112 0> are found having comparable stability to those perpendicular to <11 00>, which is attributed to a much larger spatial anisotropy in step energy for the Si face than for the C face [START_REF] Nie | Step formation on hydrogen-etched 6H-SiC{0001} surfaces[END_REF].

The step height and shape depend significantly on cut off directions. On the surfaces tilted toward <11 00>, the steps are straight (aligned along the <112 0> direction), whereas the steps have a zigzag shape on surfaces tilted toward <112 0> direction. Nakajima et al.

suggested that such a difference in step morphologies results from the different etching rate in <112 0> and <11 00> directions [START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. This kind of anisotropy is also observed for the CVD growth surfaces, where the fastest growth direction is the <112 0> direction and the slowest is the <11 00> direction [START_REF] Kimoto | Nucleation and step motion in chemical vapor deposition of SiC on 6H-SiC{0001} faces[END_REF]. This distinction may be ascribed to the density of kinks along different orientations, which is clearly depicted in Fig. 1.9a. The kink density increases gradually when the direction deviates from < 11 00 > direction, and becomes maximum in the <112 0> direction. The etching process can be regarded as a reverse process of growth, which proceeds by atoms desorption at the kink positions; the rate is thus proportional to the kink density. In this sense the etching rate is faster for the <112 0> direction than for the <11 00> ones. As a result, the straight steps are formed on the surface tilted to the <11 00> direction and on the contrary, zigzag shaped steps are formed on other off-axis surfaces (Fig. 1.9b). It is worth noting that on the <112 0> vicinal surface, the steps are split into two sets of steps perpendicular to the <11 00> direction, with an angle of 60° between them and an offset in height of 0.75 nm (half unit cell of 6H-SiC) [START_REF] Ramachandran | Preparation of atomically flat surfaces on silicon carbide using hydrogen etching[END_REF]. This may be caused by the instability of the steps perpendicular to the <112 0> direction [START_REF] Powell | Process-Induced Morphological Defects in Epitaxial CVD Silicon Carbide[END_REF][START_REF] Nie | Step formation on hydrogen-etched 6H-SiC{0001} surfaces[END_REF]. The HCl etching of 6H-SiC (0001) vicinal surfaces also showed the strong dependence of step bunching on the offcut direction [START_REF] Nakamura | Formation of periodic steps with a unit-cell height on 6H-SiC (0001) surface by HCl etching[END_REF]. One extraordinary point is that on some samples, the macrosteps perpendicular to <112 0> direction were decomposed into two kinds of steps: straight steps perpendicular to <112 0> with two bilayer height and zigzag steps perpendicular to two <11 00> directions with one bilayer height. This morphology was not reported for the H2 etching. The reason for this is not clear, which is probably due to the presence of Cl adatoms. As for the reason for the alternation of two-and onebilayer-height steps, one possibility is related to the difference of bond configuration, i.e., hexagonal and cubic site in different bilayers of 6H-SiC. Apart from the steps of one or less unit cell height, the much higher steps of several unit cell height (also referred to "nanofacets") are also frequently observed on vicinal surfaces [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF][START_REF] Nie | Step formation on hydrogen-etched 6H-SiC{0001} surfaces[END_REF][START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. To avoid any confusion, in this thesis work only the {11 0n} or {112 n} planes formed at the step edge will be called "facet" while the {0001} surfaces will be called terrace. This step faceting phenomenon usually occur on the vicinal Si face rather than the C face. For example, the 6H-SiC (0001) surfaces off toward <11 00> are usually separated into two kinds of structures by H2 etching, which are (0001) terrace and (11 0n) facets with n≈12, the later one makes an angle of about 25° with the (0001) plane [START_REF] Nie | Step formation on hydrogen-etched 6H-SiC{0001} surfaces[END_REF][START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. On the <112 0> off 4H-SiC (0001) surfaces, the typical angle between the (112 n) nanofacet and the (0001) terrace is 15-16° which corresponds to n=25-30 [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF][START_REF] Nakagawa | Self-Ordering of Nanofacets on Vicinal SiC Surfaces[END_REF]. The surface splitting was considered as a result of the minimization of the total surface free energy, where the nanofacet planes ((11 0n) or (112 n) with specific n values) were considered as the minimum energy point on the surface free energy curve of vicinal surface (Fig. 1. 11).

Evolution of SiC surface during dissolution

SiC is well known for its excellent chemical stability in severe corrosive environments. On the other hand, the surface wet etching is extensively investigated since it has a variety of applications in wide-bandgap semiconductor technology, including defect detection, polarity and polytype identification, and device fabrication that complements dry etching techniques [START_REF] Zhuang | Wet etching of GaN, AlN, and SiC: A review[END_REF]. As a defect characterization method, wet etching owns many advantages like low cost and simple procedures compared with the other methods such as TEM and X-ray topography. The KOH etching is the most widely-used defect-selective etching technique. Generally, two steps are involved in the KOH etching process: first the oxidation of the surface and second, the dissolution of the resulting oxide. This means that the dissolved oxygen from the environment is essential for this technique, which is actually proved as the rate-limiting step for the etching [START_REF] Katsuno | Mechanism of Molten KOH Etching of SiC Single Crystals: Comparative Study with Thermal Oxidation[END_REF]. Though this kind of etching is powerful to reveal the defects, it is not appropriate for studying the inherent properties of SiC surface because the etching reaction occurs mainly with the oxide layer instead of the SiC surface directly. Therefore, the SiC surface evolution during wet etching is not included in the discussion here. With the aim of studying the quasi-equilibrium interface, the works reviewed below are mostly focusing on etching of SiC by solvents which are potential candidates for solution growth, i.e. Si-based liquid alloys.

The dissolution morphologies of on-axis SiC {0001} surfaces with the pure Si melt and Si-Sc solution are presented in Fig. 1.12 [START_REF] Syväjärvi | Interfacial properties in liquid phase growth of SiC[END_REF]. The Si face of both 6H-and 4H-SiC dissolve with pure Si and triangular shaped features with some circular pits appear, while the C face is much smoother without showing any specific patterns. The dissolution morphology becomes rougher with temperature on the Si face, while almost no change with temperature is observed on the C face. As a rule, isotropic dissolution happens on a low surface energy while preferential dissolution (such as defect selective etching) occurs on a higher surface energy. According to this rule, selective etching is more expected on the Si face than on the C face, due to the higher surface energy of Si face. This is actually similar with the case when the two polar faces are etched by molten KOH [START_REF] Yakimova | Preferential etching of SiC crystals[END_REF]. The addition of Sc into Si causes much rougher morphology, which can be explained by the enhancement of the process due to the increased carbon solubility in Si-Sc alloys. Real-time observation is a powerful approach to study the evolution of the structures with temperature at the interface between SiC and solution [START_REF] Kawanishi | Real-time observation of high temperature interface between SiC substrate and solution during dissolution of SiC[END_REF]. This interface can be observed

Evolution of SiC surface during growth

Monitoring of the interface during the growth of SiC is very important, because it directly impacts the final quality of the grown crystal. A stable growth front is always desirable while it is hard to be controlled during the high temperature growth process. Given the different nature of vapor growth and solution growth, the interface evolution among two kinds of growth routes are introduced separately.

1) During vapor growth

Similar to the vapor etching process, the evolution of surface structures during vapor growth also strongly depends on the surface properties including polytype, polarity and crystalline orientation. In the following, some typical surface characteristics are summarized, mainly referring to the chemical vapor deposition (CVD) and the PVT growth processes.

Step bunching on off-oriented {0001} SiC surfaces was investigated in the chemical vapor deposition process [START_REF] Kimoto | Step bunching mechanism in chemical vapor deposition of 6H-and 4H-SiC{0001}[END_REF][START_REF] Kimoto | Step bunching in chemical vapor deposition of 6H-and 4H-SiC on vicinal SiC(0001) faces[END_REF]. On the Si face, three or six Si-C bilayer-height steps are formed on 6H-SiC surface while two or four bilayer-height steps are observed on 4H-SiC surface.

On the contrary, single bilayer-height steps are the most abundant steps on C face for both 6H-and 4H-SiC epilayers. It should be mentioned that Tanaka et al. reported that single bilayer steps are predominant on both 3.5° off and on axis 6H-SiC (0001) surfaces [START_REF] Tanaka | Vicinal and on-axis surfaces of 6H-SiC(0001) thin films observed by scanning tunneling microscopy[END_REF].

This discrepancy may be related to the different experimental procedures. In Tanaka's work, the surface graphitization might have preliminarily occurred before the CVD growth process.

In CVD growth process, step bunching is usually enhanced under C-rich conditions when a higher C/Si gas ratio is used [START_REF] Kimoto | Step bunching mechanism in chemical vapor deposition of 6H-and 4H-SiC{0001}[END_REF]. This has been also reported with molecular beam epitaxy (MBE), in which the step bunching is enhanced when the gas flow ratio of C2H4/Si2H6 is larger than one [START_REF] Tanaka | Effects of gas flow ratio on silicon carbide thin film growth mode and polytype formation during gas-source molecular beam epitaxy[END_REF]. It is considered that the different C/Si ratios cause different surface reconstructions ((1×1) or (3×3)) at the outset of the growth, which affect the diffusion length of the adatoms and then the step bunching.

The surface topography of the as grown {0001} plane of 6H-and 4H-SiC crystals grown by the PVT method was also studied [START_REF] Ohtani | Step bunching behaviour on the {0001} surface of hexagonal SiC[END_REF]. The surface morphologies largely depend on the polytype and on the surface polarity, the morphology of 6H-SiC are shown in Fig. 1.14.

On the 6H-SiC Si face, two kinds of surface morphologies are observed, either an extremely smooth surface or concentric circular giant steps, which is possibly related to the fluctuations of the growth conditions such as different growth rates. Besides, highly regular equidistant step trains of 1.5 nm (unit cell) height were observed on the smooth large terrace between giant steps. On the 6H-SiC C face, a shallow growth hillock forms around the spiral growth center, the same equidistant steps of unit-cell height are also observed but they are less stable as compared to the Si face. There is no result for 4H-SiC Si face since the 4H polytype is usually not grown on (0001) Si face. On the 4H-SiC C face, star-like morphology is observed, which is composed by petal-like giant macrosteps protruding towards the < 112 0 > directions. The large terraces between the giant macrosteps exhibit highly regular steps with unit cell height. In addition, it is believed that nitrogen adsorption enhances the asymmetry in atomic incorporation rate from opposite direction of a step. Therefore, the equidistant step trains on both nitrogen doped 4H-and 6H-SiC C face are transformed into meandering macrosteps of height greater than 10 nm.

2) During solution growth

In solution growth, the surface morphology of the growth front is not only dependent on the substrate properties, but also influenced by the strong solid-fluid interaction which is absent in vapor growth. Therefore larger steps with irregular shapes and pronounced anisotropic behavior would occur in liquid phase growth [START_REF] Syväjärvi | Step-bunching in SiC epitaxy: Anisotropy and influence of growth temperature[END_REF]. Even if high density of small steps is more desired for the step-flow growth, macrosteps of few hundred of nanometers are often observed. Sometimes, giant macrosteps of few or tens of micrometers are even formed by the further bunching of macrosteps. The giant macrosteps, combined with the 2D nucleation on wide terraces can lead to the generation of trenches and solvent inclusions in the grown crystals [START_REF] Mitani | Effect of aluminum addition on the surface step morphology of 4H-SiC grown from Si-Cr-C solution[END_REF][START_REF] Kusunoki | Solution Growth of SiC Crystal with High Growth Rate Using Accelerated Crucible Rotation Technique[END_REF][START_REF] Kamei | Solution growth of single crystalline 6H, 4H-SiC using Si-Ti-C melt[END_REF][START_REF] Mitani | Morphological stability of 4H-SiC crystals in solution growth on {0001} and {110m} surfaces[END_REF]. To increase the carbon solubility and thus the growth rate, Si-based metal alloys rather than pure Si melt are usually used as solvents. In fact, the additives do not only change the carbon solubility, but also have a strong influence on the surface step morphologies. For this reason, the characters of the surface evolution during solution growth will be categorized in terms of different solvent systems.

a) Pure Si

Notwithstanding the low carbon solubility in silicon, solution growth with pure silicon melt is still persistently studied since high purity crystals can be potentially obtained. Komatsu et al. studied the surface morphology of 4H-SiC single crystal grown from Si-C solution at growth temperatures from 1840 to 2140 ºC [START_REF] Komatsu | Growth rate and surface morphology of 4H-SiC single crystal grown under various supersaturations using Si-C solution[END_REF]. On the crystal grown on on-axis (0001 )

C face, the surface become rougher with the increasing supersaturation and the general surface morphology transforms from step flow patterns to 3D island patterns at the higher supersaturation. In addition, there exist some polytype inclusions (3C and 6H) in the grown layer at high temperature, which suggests the two dimensional nucleation and growth under high supersaturation conditions. The effect of temperature gradient on surface morphology is also dependent on the polarity [START_REF] Endo | Effects of crystalline polarity and temperature gradient on step bunching behavior of off-axis 4H-SiC solution growth[END_REF]. Not like the C face, the surface steps grown on the Si face show a stronger wandering and a reduced extent of step bunching with increasing temperature gradient. The constitutional supercooling mechanism was used to explain the abnormal bunching on the Si face when under the higher supersaturation.

The same trend is obtained when the growth is conducted at a lower temperature (1650 ºC), which is shown in Fig. 1.15 [START_REF] Harada | Surface morphology and threading dislocation conversion behavior during solution growth of 4H-SiC using Si-Al solvent[END_REF]. The surface morphologies obviously indicate that the growth proceeds following the step-flow growth mode. The growth surface on the C face is much smoother than that on the Si face. With the addition of 5 mol% Al, the step bunching on Si face is enhanced and macroterraces of several micrometers in width are formed. On the C face, the surface also becomes smoother by the addition of Al but the effect is not as obvious as on the Si face. The addition of other elements like B, Sn, N, and V in to Si solvent can also decrease the average macrostep height and stabilize the growth front [START_REF] Komatsu | Change in surface morphology by addition of impurity elements in 4H-SiC solution growth with Si solvent[END_REF]. Thought the C face has many advantages such as good polytype stability and smooth surface maintaining during solution growth, off-axis (0001) Si face is often preferred because it shows a higher TSD conversion rate [START_REF] Xiao | Conversion Behavior of Threading Screw Dislocations on C Face with Different Surface Morphology During 4H-SiC Solution Growth[END_REF]. To control the formation and evolution of macrosteps on the Si face, the antiparallel flow method, where the fluid flow close to the crystal surface has an opposite direction with respect to the step flow, has been proposed. It allows suppressing the step bunching for solution growth in pure Si [START_REF] Zhu | Influence of solution flow on step bunching in solution growth of SiC crystals[END_REF].

b) Si-Cr system

The surface morphology of 4H-SiC crystals grown with Si1-xCrx and Si1-x-yCrxAly solvents were investigated [START_REF] Mitani | Growth rate and surface morphology of 4H-SiC crystals grown from Si-Cr-C and Si-Cr-Al-C solutions under various temperature gradient conditions[END_REF]. In Si1-xCrx solvent, the surface exhibits many trench-like defects, the number of which increases with the ΔT (the temperature difference between the seed and crucible bottom). Addition of Al in Si1-xCrx suppresses the generation of trench-like defects, resulting in a growth surface covered only by macrosteps. In addition to the decrease of the macrosteps height, many smaller bunched steps are found on the terraces by Al addition. The morphology of terraces between macrosteps has been checked [START_REF] Mitani | Effect of aluminum addition on the surface step morphology of 4H-SiC grown from Si-Cr-C solution[END_REF].

Half and unit cell size steps are observed without Al addition while bunched steps of few nanometers occur with Al addition (Fig. 1.16). This implies that the adsorption of Al atoms on the surface reduces the repulsive interactions between steps, resulting in the smaller bunched steps on the terrace. This stability of step caused by the Al addition might play a role in suppressing the giant macrosteps, though the specific mechanism is not clear.

The addition of Sn into Si-Cr solvent also lowers the macrosteps, though it is not as efficient as Al addition. Surprisingly, combined use of these two additives is effective to modify the surface morphology of 4H-SiC, which shows a more significant effect than when each elements are added independently [START_REF] Komatsu | Modification of the surface morphology of 4H-SiC by addition of Sn and Al in solution growth with SiCr solvents[END_REF]. In addition, solution growth of 4H-SiC on seeds with various crystallographic orientations were carried out, showing that the morphology is dominated by the crystallographic orientation of seeds rather than the solvent composition [START_REF] Mitani | Morphological stability of 4H-SiC crystals in solution growth on {0001} and {110m} surfaces[END_REF]. The {11 0m} (m=4, [START_REF] Pearson | Computer modeling of Si and SiC surfaces and surface processes relevant to crystal growth from the vapor[END_REF]20) planes are reconstructed into {112 0} and {0001} planes, suggesting the morphological stability of the {112 0} planes in 4H-SiC solution growth. The long-term growth on (11 0m) (m=1 and 2) seeds gives smooth surfaces without any trench-defect, which seems more suitable for controlling the surface roughness than {0001} seeds.

c) Si-Ti system K. Kamei et al carried out TSSG growth of on-axis and off-axis 6H-SiC (0001) Si face in Si-Ti-C ternary solution [82][91]. The surface morphology of on-axis grown crystal shows wide terrace and high density of bunched macrosteps, indicating 2D nucleation and lateral growth regime. The grown off-axis 4H-SiC crystals exhibit very rough morphology with meandering and bunched macrosteps (Fig. 1.17). The surface is very irregular with many overhangs which then trap the solvent and severely degrade the crystal. A smoother surface with no inclusion is obtained when a smaller temperature gradient is applied around the interface. Using Si-Ti solvents, a growth rate exceeding 200 μm/h was achievable by using ACRT [START_REF] Kusunoki | Solution Growth of SiC Crystal with High Growth Rate Using Accelerated Crucible Rotation Technique[END_REF]. The ACRT improves the mixing of the solution and then results in a homogeneous concentration of solute at the growth interface. Using ACRT in Si-Ti solution growth results in specular surfaces in contrast to the macrosteps formed on the crystals grown without ACRT. This technique can also suppress the occurrence of solvent inclusions which usually form by the protrusive part of the growth interface (macrosteps).

However, the reported results are still controversial since some other researchers reported the opposite trends, i.e ACRT roughens the surface giving rise to step bunching [START_REF] Kamei | Solution growth of single crystalline 6H, 4H-SiC using Si-Ti-C melt[END_REF].

Mechanism of surface reconstruction and step motion

Either during the growth or dissolution, the surface will go through a reconstruction process. At atomic level, the surface atoms are rearranged forming special configuration, which usually happens at the beginning under ultra-high vacuum (UHV) or high purity inert gas conditions. At mesoscopic scale, the surface structures like steps and terraces are formed and evolve during the surface reconstruction process. In any case, the surface morphology evolution is controlled by both the surface properties and surrounding environment, through either the energetics or dynamics/kinetics. Based on the characteristics of surface structures during various surface reconstructions (discussed in last section), the underlying mechanisms are introduced in this section.

Energetic barriers on the surface

To illustrate the concept of surface energy barriers, let us look at the following situation:

considering that a marble is put on a smooth table, it will roll freely on the smooth surface.

When it comes to the table edge, it will drop down without pausing to "think" that it might fall and break [START_REF] Lagally | Thin-film cliffhanger[END_REF]. However, if a person walks closing to a steep cliff, he will look down the cliff and stop moving. It seems there is an invisible "barrier" at the edge of the cliff, which prevents him from jumping down. For the atoms moving on the surface (or interface), do some similar barriers exist at special locations such as step edge?

The terrace-step-kink model, developed in the famous paper of Burton, Cabrera and Frank, has depicted the picture of the atomic movement on the crystal surface [START_REF] Burton | The growth of crystals and the equilibrium structure of their surfaces[END_REF]. When the atoms adsorb on the surface, they will diffuse on the terraces until they find the proper place to stay. The ultimate location of the diffusing atoms, normally on the kinks of lower terrace, is considered as the most stable state from energetic point of view. One point to note is that they postulated that the adatoms could be incorporated into a step from two sides with equal probability.

With the rapidly developed surface characterization techniques, a much clearer picture was obtained for the atomic diffusion on a crystal plane. It has been confirmed that the diffusing atoms also behave like human on a cliff, which is, feeling a barrier when moving over the step toward the lower terrace. The first experimental evidence of this "edge barrier" was observed by Ehrlich and Hudda in using field-ion microscopy [START_REF] Ehrlich | Atomic view of surface self-diffusion: Tungsten on tungsten[END_REF]. They found that the single atom putting on the terrace preferentially being reflected back at the step edge instead of rolling over the step. Almost at the same time, Schwoebel and Shipsey found the same phenomenon independently [START_REF] Schwoebel | Step Motion on Crystal Surfaces[END_REF]. The atomic illustration of the step edge and the energy discrepancy (the so-called Ehrlich-Schwoebel (ES) barrier) close to the step edge compared with the terrace area are shown in Fig 1 . 18. It is clear that the presence of such a barrier at the edge makes the adsorbed or diffusing species easier to be incorporated at the ascending step site (b to c) than the descending step site (a to c) .This ES barrier could create an uphill trend, which can make the growth unstable and roughen the growing surface [START_REF] Villain | Continuum models of crystal growth from atomic beams with and without desorption[END_REF]. ES barrier could also influence the nucleation process, that is, the nucleation takes place at the downward step edge with the presence of ES barrier whereas the islands nucleate at the terrace center with the absence of the ES barrier [START_REF] Kaufmann | Critical impact of Ehrlich-Schwöbel barrier on GaN surface morphology during homoepitaxial growth[END_REF].

Step bunching

Step bunching is a phenomenon frequently observed during the surface reconstruction process. There are several possible sources of step bunching, including geometrical, energetic and kinetic origins and so on. The final surface morphology can be resulted from single or multiple factors among them. As we have discussed above, during the vapor reconstructions (either etching or growth), step bunching on vicinal SiC {0001} surfaces often results in bunched steps of half or one unit cell [START_REF] Kimoto | Step bunching mechanism in chemical vapor deposition of 6H-and 4H-SiC{0001}[END_REF][START_REF] Kimoto | Step bunching in chemical vapor deposition of 6H-and 4H-SiC on vicinal SiC(0001) faces[END_REF]. While in the liquid environment, the giant macrosteps are frequently formed on the surface due to intense reconstruction. The relevant mechanism for step bunching, mainly but not only pertaining to SiC surface reconstruction, are summarized as following. In another word, the ES barrier stabilizes the step trains making the steps equidistant during the growth process. However, it can cause an opposite result during the evaporation or dissolution: the ES barrier leads to the formation of bunches of more closely spaced steps separated by a wider terrace and thus destabilizes the surface [START_REF] Hartman | Crystal growth: an introduction[END_REF].

This mechanism is well known as the Ehrlich-Schwoebel effect. In general, there are two different types of ES effects: one is the direct ES barrier which is on top of the step, another one is called the inverse ES barrier that is present below the step. The inverse ES barrier has an opposite effect, which could prevent adatoms from attaching to the step from the lower terrace.

However, the symmetric step kinetic theory is a short-range step-step interaction and thus has some limitations. For example, it cannot explain why the steps height on the reconstructed SiC surface is corresponding to one or half unit of SiC crystals.

2) Energetics of different bilayers

Heine et al. firstly proposed a model that each SiC bilayer has different surface free energies with one another, owing to the peculiar stacking sequence [START_REF] Heine | The Preference of Silicon Carbide for Growth in the Metastable Cubic Form[END_REF]. This model has been used

to explain the heterogeneous growth of 3C-SiC on 6H-and 15R-SiC [START_REF] Chien | Terrace growth and polytype development in epitaxial β -SiC films on α -SiC (6H and 15R) substrates[END_REF]. The formation of three or two bilayers on the homo-epitaxial grown surface of 6H-and 4H-SiC were also predicted by such model. Since 4H-SiC substrates are mainly used in this thesis, the energy change of depositing or removing Si-C layers on 4H-SiC surface is introduced as an example.

First the extra energy ∆U for adding one new layer (σnew) on top of the previous layer (σs) can be written by

𝛥𝑈 = 𝐸 -𝐽 * 𝜎 𝜎 (1-12)
Where,

𝐽 * = 𝐽 + 𝐽 𝜎 𝜎 + 𝐽 𝜎 𝜎 + 𝐾𝜎 𝜎 (1-13)
Here, the E0 is the energy of the crystal without the interaction between layers. Jn is the energy of interaction with the nth nearest neighbor plane, and K the coefficient of the four-spin coupling term that can be interpreted as a modification of the third-neighbor interaction. As for the reverse process, that is removing (etching or dissolving) one bilayer from the terraces, the variation of energies are given as follows (in meV per Si-C pair atoms):

∆ 𝑈 = -∆ 𝑈 = -𝐸 + 6.59 (1-18) ∆ 𝑈 = -∆ 𝑈 = -𝐸 -2.34 (1-19)
whereby the step bunching of 4H-SiC during growth or dissolution is discussed based on these results. For the growth process, the atom deposition on 4H2 terraces is the most favorable, and that on 4H1 terraces is the most costly from a view point of total energy.

Therefore, the step velocity may be the fastest on 4H2 terraces due to the higher probability of incorporation at the step edges. In other words, the edge of 4H1 terrace is catching up the 4H2 step edge, finally forming two bilayer-height (half-unit-cell) steps with AB or CB stacking.

3) Step edge bonding

As shown in section 1.1.1, there exist two types of step edges, named SN and SD, having one or two dangling bonds at the step edge. The two different situations occur when atoms attach to SN and SD steps, which is shown in Fig. 1.21. For the growth process, the attached atom is only singly bonded to the SN step and thus easy to be detached. While the attached atom is doubly bonded to the SD step and thus less likely to be detached. In this way the growth rate of SD step is faster than the SN. On the contrary, for the etching process, SN step would be etched more slowly than the SD step, because the former has triply bonded outermost atoms while the later has only doubly bonded outermost atoms. The different arrangements of SN or SD steps along two miscut directions are accounted for the different scenarios of step bunching. On the surface with miscut toward the <112 0> direction, the step edges have a natural zigzag shape which consists of alternating SN or SD steps (left column of Fig. 1.22). After the growth or etching, the triangular protrusions are formed where the straight parts are SN or SD steps. In the ideal case with exact miscut towards <112 0> direction, the formed half unit-cell height steps propagate with the same speed. As a result, no full unit cell high steps will be formed. The unit-cell high step observed in some works are possibly due to the slight deviation from <112 0> direction.

As for the surface off towards <11 00> direction, the bunching process is also not a simple catching game between SN and SD steps. It was reported that the triangular protrusions were appeared at the initial stage of etching [START_REF] Ramachandran | Preparation of atomically flat surfaces on silicon carbide using hydrogen etching[END_REF]. This can be explained by the instability

4) Elastic interaction of steps

In many cases, the bunched steps are distributed on the surface with highly regular ordering.

The extremely uniform distribution of surface steps is possibly ascribed to certain repulsive force between the steps. One of the reasons causing steps repel to each other is the elastic step-step interaction. Marchenko and Parshin derived an analytical expression for the elastic potential between steps at the surface [START_REF] Marchenko | Elastic properties of crystal surfaces[END_REF]:

𝑈(𝑥) = (𝛽𝑎) (1-20)
Where σ and E are Poisson's ratio and Young's modulus, respectively, β is the surface tension at the step (step stiffness), a and x are the step height and the distance between steps respectively. It can be told from the expression that the step repulsive interaction is stronger when it comes to the higher and denser steps.

On the other hand, such an equidistant step train with a repulsive step interaction can become unstable by the asymmetric step kinetics and be transformed into a densitymodulated step train [START_REF] Sato | Morphological instability caused by asymmetry in step kinetics[END_REF]. In another word, the step morphology is a competing result between the repulsive step-step interaction and the asymmetric step kinetics. For this, Sato and Uwaha derived the dispersion for the perturbation in the step density δn(x, t) = δne k t+ikx under the driving force f0 [START_REF] Sato | Morphological instability caused by asymmetry in step kinetics[END_REF]:

𝑤 = -𝑘 - (1-21)
Where ξ(n) and n are the step energy (chemical potential) and step density, dξ/dn is the surface stiffness of the vicinal face, λ+ and λ-are the dimensionless resistances of step kinetics for the lower and upper terraces, respectively. The λ value can be regarded as the reciprocal of ES barrier, therefore a larger λ value implies a smaller incorporation probability (thus for most cases 𝜆 < 𝜆 ). The f0 is the driving force which is the sum of all external forces, which is positive in growth (supersaturation) and negative in sublimation or dissolution (undersaturation).

The first term in the brackets in Eq. 1-21 comes from the repulsive step interaction and always stabilizes the equidistant steps (step stiffness). The second term (with "-" in front), which represents the destabilizing factor due to the asymmetric step kinetics, is positive in growth and negative in sublimation or dissolution (assumed that 𝜆 < 𝜆 ).The morphological stability of the surface can be analyzed by using this expression. In general, when ωk < 0, the step trains are stable and the equidistant steps are preserved. On the contrary, when ωk > 0, the steps are unstable and step meandering or bunching tends to occur.

Step faceting

Step faceting, usually along with the step bunching, can cause profound change of surface morphology during the growth of semiconductor crystals. Real surfaces often lower their surface free energy by forming a new (reconstructed) structure much different from the bulk termination of the solid lattice [START_REF] Williams | Thermodynamics of surface morphology[END_REF]. For the vicinal SiC {0001} surfaces, they are often reconstructed into periodic hill-and-valley morphology consisting of {0001} terraces and {112 0n} (or {11 0n}) nanofacets. Normally the special facet planes with "magic" n value can be observed at the step edge, which is believed having the lowest surface energy.

For example, it was reported that there exists a magic facet angle (φ) between the (0001)

and the {112 n} planes, being in the range of 12°-16° [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF][START_REF] Nakagawa | Self-Ordering of Nanofacets on Vicinal SiC Surfaces[END_REF]. In addition, the height of the nanofacets are usually quantified as a half-integer of the unit-cell height, which is related to the atomic stacking sequence of bilayers near the surface [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF][START_REF] Nakagawa | Self-Ordering of Nanofacets on Vicinal SiC Surfaces[END_REF][START_REF] Nie | Step formation on hydrogen-etched 6H-SiC{0001} surfaces[END_REF][START_REF] Kimoto | Step bunching in chemical vapor deposition of 6H-and 4H-SiC on vicinal SiC(0001) faces[END_REF][START_REF] Arima | Atomic-scale flattening of SiC surfaces by electroless chemical etching in HF solution with Pt catalyst[END_REF].

For the highly self-ordering faceting, a linear elastic theory proposed by Marchenko and Parshin was used to study the origin of characteristic ordering distance [START_REF] Marchenko | Elastic properties of crystal surfaces[END_REF][START_REF] Marchenko | Theory of the equilibrium shape of crystals[END_REF]. When the total surface free energy E is minimal, by setting zero to the differentiation of Eq. (1-8)

with respect to L (dE/dL = 0), it was obtained that:

𝐿 = 𝜋𝑎𝑒𝑥𝑝 ( ) 𝑠𝑖𝑛 𝜋 (1-22)
This indicates that the ordering distance L is dependent on the substrate vicinal angle (θ)

as well as the angle between (112 n) and ( 0001) planes (φ). When the condition φ = 2θ is satisfied, the characteristic ordering distance L will be independent of the vicinal angle.

Step meandering

Apart from step bunching, step meandering is another instability commonly observed during the surface reconstruction process, especially on the vicinal surfaces.

Step bunching and meandering can be coexisting on the surface. The step bunching may cause meandering and in reverse the strong meandering can make the steps to bunch [START_REF] Misbah | Crystal surfaces in and out of equilibrium: A modern view[END_REF]. The meandering during step flow growth was first studied in the linear regime by Bales and

Zangwill [START_REF] Bales | Morphological instability of a terrace edge during step-flow growth[END_REF]. They proposed that the meandering happens when the energy barriers for adatoms approaching a step from two opposite directions are not equal. For instance, the ES barrier, as one of the origin of local potential inequality, might play an important role in the step meandering or waviness during the reconstruction.

The mechanism of this is briefly introduced in However, we should realize that the above mechanism actually increases the total length of the step edge and thus the total line energy of the step. To reduce the energy cost, the further diffusion process of adatoms must happen to "smoothen" the step. One possible way is capillary-induced diffusion by which the atoms could move along the step edge to change the step shape. Another possible way is through the Gibbs-Thomson effect: the atoms detach from convex portions of the step, followed by surface diffusion on the terrace and then reattach onto the concave portions of the step.

The above discussion of step meandering is only a linear stability analysis for a single steps.

However, in order to study the long-time behavior of a growing instability or to determine the final stage of the dynamic process, the much more complicated nonlinear analysis is necessary [START_REF] Bena | Nonlinear evolution of a terrace edge during stepflow growth[END_REF]. In addition, step-step interaction should be considered to make the step dynamic closer to the reality, especially for the steps on vicinal surfaces [START_REF] Misbah | Crystal surfaces in and out of equilibrium: A modern view[END_REF][START_REF] Bena | Nonlinear evolution of a terrace edge during stepflow growth[END_REF].

Objective of this thesis

This thesis focuses on investigating the morphological evolution on vicinal SiC surfaces reconstructed under high temperature liquid environment. On the one hand, the SiC The thesis will end by giving out a general conclusion and prospective studies.

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Experimental

High temperature experiments are always very challenging, especially when the aim is to uncouple the complex mixture of phenomena. To avoid the unnecessary influencing factors, the surface reconstruction experiment requires a well-controlled solid/liquid interface environment in terms of process parameters, i.e. well-controlled heat and mass transfers around the samples. For this, the sessile drop method is utilized in this thesis as the fundamental method to investigate the SiC/liquid interface reconstruction, in which a specific processing apparatus and procedure were designed and performed. In this chapter, we will firstly introduce the configuration and features of the apparatus used in this work and then detail the procedure of the sessile drop method. At the same time, the numerical simulation will be introduced as a tool for designing both the thermal field and the electromagnetic field. During the practical experimental process, the effect of the two important factors, the gas filling and the electromagnetic field, will be fully discussed. At last the surface characterization methods used in this thesis will be listed and introduced.

Several further quantitative analyses based on the first-hand experimental data will also be presented.

Surface reconstruction setup 2.1.1. Overview of the apparatus

The experimental apparatus is a laboratory-scale high temperature induction furnace, named "BELENOS" located in SIMaP laboratory; its photo is shown in Fig. 2.1. In general, the whole setup is composed of three main parts:

(1) The heating unit. It is composed of power generator and the capacitor box with the connected induction coil on it. The latter is fixed on a large scissor lift which can move vertically for adjusting the relative position of the induction coil to the crucible. This can be used to adjust the temperature distribution inside the crucible, especially the axial temperature gradient.

(2) The reactor. It is composed of the reaction chamber and the indispensable accessories including the vacuum system (equipped primary pump, with a limit vacuum of 1×10 -3 mbar) and gas lines, gas gauges for pressure measuring, two pyrometers (top and bottom) for temperature monitoring, and the cooling system (circular water-cooled quartz jacket and two external fans).

(3) The monitor and controller. All the experimental parameters are centrally supervised and controlled by a locally-designed computer-based interface. The temperature (1000~3000 °C) and chamber pressure can be automatically and accurately controlled by this system.

Simulation-aided thermal design

A quasi-isothermal condition is the prerequisite for observing the formation of the "native structure" of the solid-liquid interface. This ensures that after the initial, transient dissolution of the SiC surface, there is no more temperature driven evolution of the SiC/liquid interface. Then, once the liquid is thermodynamically equilibrated with SiC, the activity of carbon in the liquid becomes uniform. All the carbon transport phenomena related to the temperature gradient can be ruled out. This is the case for i) Fick's diffusion as the carbon activity is uniform, ii) natural convection as the liquid density is uniform, iii)

Marangoni convection, as the surface tension of the liquid is uniform. This is very important for the latter contribution as Marangoni convection can be activated by a very small temperature gradient along a free liquid silicon surface, of about 2 °C•cm -1 [START_REF] Mercier | Coupled heat transfer and fluid dynamics modeling of high-temperature SiC solution growth[END_REF].

Therefore, the thermal condition of the set-up, especially of the local area where the reconstructions take place, needs to be carefully studied. Nevertheless, it is hard to evaluate the thermal state merely from experiments because: 1) the temperature are measured from two different measurement positions: one from the bottom of the graphite sample holder while another one is from SiC substrate; 2) it is impossible to directly measure the temperature of SiC/liquid interface. In this sense, numerical simulation is a good alternative to study the thermal state of the SiC/liquid interface.

The whole system involves various physical and chemical phenomena, which are controlled by partial differential equations that contain multivariable functions as well as their partial derivatives. The Finite Element Method (FEM) is a powerful approach to solve these continuous equations without too much oversimplification. Nowadays, this method has been widely used for treating interface problems such as solid mechanism and magnetohydrodynamics. Herein the FEM method is adopted to analyze the heat and mass transport of the system especially at the solid/liquid interface. As a relevant part, the influence of the magnetic field on the interface reconstruction will also be discussed based on the same FEM simulation pack.

In this thesis, the FEM simulations are carried out within the COMSOL Multiphysics ® software package. A two-dimensional axisymmetric model is built to replicate the geometry of the real reactor including the induction coil and with a droplet of fixed shape (Fig. 2.4).

Note the outmost magnetic insulation is a virtual boundary where the tangential component of the magnetic potential is set to zero. The furnace is heated by electromagnetic induction which could be described by Maxwell's equations. The induced eddy currents are generated according to Lenz's law and then the crucible is consequently heated by Joule's effect. According to the skin depth effect, mainly the susceptor part is directly heated while inner parts are heated by the susceptor. For heat transfer, three mechanisms are considered in the studied system: conduction, convection, and radiation.

Among them, the radiation is the main mechanism where the heat is transferred from the susceptor to the inner parts. Although the chamber is filled with Argon at high temperature (> 1450 ⁰C), the heat transfer by its convection is not considered since the influence of convection was reported much weaker when compared to the radiation [START_REF] Ariyawong | Process modeling for the growth of SiC using PVT and TSSG methods[END_REF]. A magnetic insulation boundary is applied around all the reaction chamber and coils.

To be simple and save the calculation time, a stationary study was carried out in this work.

The electromagnetic problem was solved in the frequency domain and coupled with the calculation of the temperature distribution in the system. The density of AC current circulating in the coil was adjusted to reach a desired temperature at the SiC/liquid interface.

Furthermore, effective electromagnetic force acting in the droplet was introduced in the Navier-Stokes equation. The Si melt was considered to be an incompressible Newtonian fluid and the fluid flow generated by the Lorentz force was assumed to be laminar. We did not consider buoyancy convection related to the solutal gradient because the carbon solubility is very small, typically less than 0.1 at% over all the studied temperature range.

The relative position between the crucible and the coil was chosen by a series of simulation to obtain minimum temperature gradients in and around the liquid droplet. One optimized case is exampled and shown in Fig. 2.5, where the target temperature of the interface is 1600 °C. As we can see, the temperature in the central part of crucible is quite homogeneous, with typical temperature gradients being 1 °C•cm -1 along 5 cm from the droplet center, both in radial and axial directions (Fig. 2.5a). The isothermal line is marked on the close-up droplet shown in Fig. 2.5b, indicating that the temperature variation inside the droplet, and particularly along the liquid-solid interface, was apparently less than 0.1 °C at a sample temperature of 1600 °C. This very low temperature gradient is believed not only dependent on the relative position of the coils, but also ascribed to the small size of the droplet (typically 4 to 5 mm in diameter) as compared to the large homogeneous hot zone.

Sessile drop method 2.2.1. Principle and general procedure

The sessile drop method is initially a commonly-used geometric method to evaluate the wettability of solid substrate by liquid. The high temperature contact angle can be directly determined by placing a drop of liquid metal on a smooth and homogeneous surface in a well-controlled environment. Besides the contact angle, the interface morphology can be observed by etching off the solidified liquid after cooling to room temperature. However, this obtained surface is not the real, high temperature interface between the substrate and liquid, which indeed has been inevitably altered by the cooling down period. This effect has been confirmed in our work, in which the morphology of the solid/liquid surfaces obtained depends largely on whether the substrate was cooled down with or without the droplet. This is primarily due to the recrystallization as a result of the increasing carbon supersaturation in the droplet, which will be detailed in the following chapters. Except for the in-situ observation of the interface at high temperature, one feasible solution is to "freeze" the reconstructing interface and suppress any further evolution. This idea can be realized by removing the liquid from the SiC substrate at high temperature, i.e. at the end of surface reconstruction. In this way the interface morphology can be well preserved (no more reconstruction without the liquid) to be examined ex situ.

In this work, the sessile drop method is the primary and the standard method for studying the SiC surface reconstruction, its general procedure is schematically illustrated in Fig 2 .6.

The basic principle is: after the melting of silicon, first a transient dissolution of the SiC surface occurs, subsequently the SiC surface is thermodynamically equilibrated with liquid, lastly the SiC surface is ex situ studied. In our work, only pure silicon was used as the liquid, which was supplied by a 2 mm-thick electronic grade silicon wafer. Small silicon squares of 2.0×2.0 mm² were cut from the thick wafer using a high precision dicing saw, ensuring a precise and reproducible amount of silicon. The used SiC pieces were primarily issued from commercial 4H-SiC substrates with a miscut angle of 0° (on axis), 2° and 4° towards the <112 0> direction. All the SiC surfaces have been chemical mechanically polished (CMP) to ensure a surface free of subsurface damage. After degreasing in acetone followed by ethanol using an ultrasonic bath, both the Si piece and the SiC substrate were deoxidized in dilute HF solution for 10 min, rinsed by deionized water for 1 min and then transferred onto the graphite sample holder in the crucible. The chamber was firstly vacuumed during the heating stage to outgas the crucible and then filled with pure Argon (6N, up to 900 mbar) at high temperature. Then, the crucible was heated up to the required temperature at a rate of 20 °C/min. During the heating phase, the Si piece melted on the SiC substrate forming a droplet. The experiments were kept for various times and temperatures. As mentioned above, during the liquid removing process, the liquid droplet was actually de-wetted from the SiC surface by rewetting on the removing tool. The purpose is to remove all the liquid from SiC surface as quickly as possible, without bringing any foreign phase in the sample. The ideal material for making the liquid removing tool should meet the following criteria:

 Have better wettability than SiC surface by liquid silicon at high temperature.

 High melting point and low reactivity with the silicon and carbon atoms at high temperature.

 Low solubility in the Si-C solvent to avoid introducing any impurities.

 Flexible and facile to be machined and implemented.

Initially, the bare graphite rod was used to suck the liquid silicon. It was proved not sufficient since lots of liquid was remaining on the surface. Based on the above requirements, several candidates including, molybdenum sheet, niobium sheet, tantalum sheet, tungsten sheet and zirconium sheet, porous graphite sponge, and thin graphite sheet have been tried as the sucking material (as an extension) being attached at the end of the graphite rod.

The resulting de-wetted SiC surfaces and the corresponding removing tools are showed in

++++- +++++ +---- ++++- Zirconium ++++- ----- +++++ ----- Molybdenum +++++ ++++- ++++- +++++ Niobium ++++- ++++- +++++ ++++- Tantalum +++++ +++++ +++++ +++++ Tungsten +++++ +++++ +++-- ++++-

Effect of gas injection

To limit the evaporation of liquid silicon at high temperature, all the experiments were carried out under argon pressure close to the atmospheric one (> 900 mbar). For practical reasons, the temperature was kept constant when filling the chamber with Ar, which explains the presence of a temperature plateau of a few minutes during Ar injection. During the dissolution process, these bilayer steps bunched together forming ordered step-terrace arrays with larger terraces and higher steps. This dominant feature was observed on both DIC pictures of Fig 2 .9, where the 4°off 4H-SiC Si-face surfaces were reconstructed into macrostepand-terrace structures at 1600 °C for 30 minutes. Besides that, many circular prints were observed randomly scattering on the SiC surface (Fig. 2.9a) when the argon was injected at 1100 °C during the heating stage. However, these features did not appear when the argon was injected at 1450 °C, i.e. just after the melting of silicon (Fig. For comparison, an assessment of the thickness of SiC dissolved during the equilibration between SiC and liquid Si droplet was carried out. The result is shown as Fig. 2.12. Based on the carbon solubility dataset from Scace et al [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF], the computed dissolved depth was found to be of around 143 nm at 1600 °C, within the same range as the measured height of the hillock. The calculated value was slightly smaller, but the solubility curve of carbon is still controversial to date. We used the solubility data from Scace et al. because it has been measured along the widest temperature range reported. This dataset has also been considered by some other authors as underestimating the solubility. This point will be further reviewed in chapter 6 of this thesis. Nevertheless, we think that the calculated value and the experimental values are in pretty good agreement. Therefore, it is believed that these hillocks corresponded to undissolved areas, or, in other words, areas where the liquid silicon was not in contact with SiC during the wetting, most probably because of the presence of bubbles trapped at the liquid/SiC interface. Note that this surface still might be weakly etched by the silicon vapor, which should own a quite high partial pressure inside the bubbles [START_REF] Yabuki | Development of "Sivapor etching" and "Si vapor ambient anneal" in TaC/Ta composite materials[END_REF]. A first scenario for explaining these hillocks is that they were caused by the generation of SiO(g) at the interface, due to the reaction between the Si and the native SiO2 layer on the SiC substrate [START_REF] Brewer | The Stability of SiO Solid and Gas[END_REF]:

Si(l)+SiO 2 (s)→2SiO(g) (2-1)
At room temperature, a thin oxide layer of few nanometers is usually formed on the surface of SiC wafer [START_REF] Önneby | Silicon oxycarbide formation on SiC surfaces and at the SiC/SiO2 interface[END_REF]. Even if the oxide layer was still partially present at the interface before the melting of silicon, a complete conversion into SiO(g) could not explain the amount of circular prints observed. In addition, previous works showed that in Ar of commercial purity (lower in purity than the Ar we used), there is no longer an oxide layer at the SiC surface at temperatures T>1100 °C due to active oxidation [START_REF] Jacobson | Corrosion of Silicon-Based Ceramics in Combustion Environments[END_REF]. Thus, at the Si melting point, both Si and SiC are deoxidized which means a gapless liquid/solid contact should be formed instantaneously, thus ruling out this first scenario.

A second scenario was suggested by the experimental result shown in during TSSG of SiC, the formation of voids in the grown crystal could be attributed to the entrapment of Ar bubbles at the growth front [START_REF] Mitani | Control of void formation in 4H-SiC solution growth[END_REF]. To explain this, they speculated that Ar, initially dissolved in the solvent, vaporized and formed bubbles during the crystal growth process. In our case the effect was similar but the origin of the bubbles was different. If the bubbles had come from dissolved argon, we would have observed hillocks along the whole liquid/SiC interface and yet hillocks were only found where the solid Si piece was originally located, before melting (Fig. 2.13). This suggests that the argon was already present between the two solid pieces and then entrapped at the interface upon melting. Therefore, by melting the silicon under vacuum and filling the Argon at 1450 °C, i.e. once the liquid/solid interface was formed, no more hillocks were able to be observed at the surface.

Effect of electromagnetic field

There is a dual aim for using the AC electromagnetic field (EMF) in the experiment. Firstly, it allows the inductive heating of the susceptor and subsequently the heating of sample.

Secondly, at a certain AC frequency, the EMF can be used to generate forced convective flow in the droplet. These effects are discussed in detail in this section.

When an alternative magnetic field penetrates an electric conductor, its magnitude decreases exponentially owing to the energy loss. The penetration depth (δ) defines the distance at which the magnetic field has been attenuated to 1/e of its magnitude at the conductor's surface [START_REF] Jordan | Electromagnetic Waves and Radiating Systems[END_REF]. The value of δ is expressed as:

𝛿 = = (2-2)
Where f is the frequency of the EMF (or alimenting electric current AC) (Hz), μr is the relative permeability of the material (usually =1 for conductors), μ0 is the permeability of the vacuum (= 4π ×10 -7 H•m -1 ) and σel is the electrical conductivity of the material. The electrical conductivity of tantalum and graphite as a function of temperature are given in equations (2-3) and (2-4) respectively, expressed in S•m -1 . For Ta, this was obtained from the data reported by Desai et al [START_REF] Desai | Electrical Resistivity of Selected Elements[END_REF]. For graphite, it is the optimized polynomial form, used for numerical simulation of SiC sublimation growth process [START_REF] Ariyawong | Process modeling for the growth of SiC using PVT and TSSG methods[END_REF]. The dependence of the penetration depth on temperature in graphite and Ta was calculated for different AC frequencies (Fig. 2.14). For f = 22 kHz (the frequency used in our apparatus), at 1600 °C (1873 K), the penetration depth is about 10.5 mm in graphite, which is thicker than our graphite susceptor. That means that at this AC frequency a fraction of EMF propagates through the crucible and may affect the electrically conducting sample (the droplet). In some experiments, a shield made of tantalum was considered to be introduced into the furnace to avoid the EM field having a direct effect on the drop. The numerical simulation was used again to quickly test its effect before experiments. The power volumetric loss density in the crucible, both with and without the Ta shield is evaluated in the FEM simulation and compared in Fig. 2.15. It is shown that the Ta shield dissipated a large part of the EMF through Joule losses, significantly reducing the magnetic field density applied on the droplet. Then our experiments were performed using these configurations.

Alternatively, the direct effect of the electromagnetic field on the droplet can be reduced by increasing the AC frequency which makes the penetration depth of the magnetic field smaller according to Eq. 2-2. For instance, at 280 kHz, the penetration depth in graphite was reduced to 3 mm which is thus much smaller than the thickness of the susceptor (see the red line in Fig. 2.14). Interestingly, the effect of the electromagnetic field on the droplet can be revealed through characterization of the liquid/SiC interface. To demonstrate this, the whole liquid/SiC interface topography was characterized by a mechanical profilometer, for two different experiments: in the first, the configuration is as shown in Fig. 2. 16a, while in the second the sample and graphite holder was placed in a 1 mm-thick Ta tube as shown in Fig. 2.

16b.

• In the experiment performed without Ta tube, the interface was concave, with the edge of the droplet at a higher level than the central area, and even higher than the initial SiC surface. This implies that mass transport should have occured, from the center to the edge of the droplet.

• In the experiment performed with the Ta tube, the interface was almost flat, though slightly below the initial surface level due to the SiC dissolution. This implies that mass transport was much weaker and even negligible in this case. In electrically conductive liquids subjected to monophase AC electromagnetic field, the motion of the liquid can come from the action of Lorentz forces. Their action takes place inside the liquid Si droplet near its surface and generates liquid motion along the droplet basis from its edge to the center which is compensated by the downward flow along the droplet free surface (see Fig. 2.17). The boundary setting of the droplet is also shown as an inset, indicating that the calculation was performed with the fixed shape of the droplet assuming a no-slip condition at the SiC/liquid interface and the absence of shear on the droplet free surface. Comparing the two cases in simulation results, we can see that using of the Ta shield does not change the character of the flow, i.e. the shape of the convective vortex. However, its presence makes the intensity of the fluid velocity (size of the velocity vectors) significantly weaker. Again, this has to be linked with the shielding effect of the Ta tube with respect to the electromagnetic field. Based on above results, the effect of fluid flow onto carbon distribution inside the droplet can also be calculated. For a better visualization, we used the relative supersaturation σr, defined as the relative deviation from equilibrium concentration:

𝜎 = (2-5)
Where c is the actual carbon concentration in the solvent while ceq is the equilibrium concentration which only depends on temperature. These values could be obtained by classical convection-diffusion equation and the carbon solubility curve respectively. The detailed calculations have been already described in previous work [START_REF] Lefebure | Modeling of the growth rate during top seeded solution growth of SiC using pure silicon as a solvent[END_REF]. As aforementioned, Lorentz forces acting inside the liquid Si droplet, generated convection loop moving along the droplet free surface from the center to its edges. As a consequence, a carbon transport was expected from center to the edge. However, the electromagnetic shielding strongly reduced the magnitude of the fluid velocity, thus a weaker mass transport should exist in the droplet. To demonstrate the impact of such a transport, we further display the computed relative carbon supersaturation distribution in the droplets (in Figs.

2.18a and 2.18b), and plotted its profile along the droplet radius in one image (Fig. 2.18c).

We can see that:

 Without the Ta tube, there is a significant variation of supersaturation along the droplet radius, from a negative value at the center to a positive one (σr > 0) at the very edge of the droplet. The direct consequence should be a dissolution at the center of the droplet and crystallization at the edge. This EM convective transport of carbon is believed working as the driving force for the dissolution/crystallization phenomenon observed.

 With the Ta tube, the variation of relative supersaturation along the radius was very week. Note that the value is always negative all along the radius of the liquid/SiC interface (σr < 0). In conclusion, the AC magnetic field can be used as an actuator for carbon transport and its effect was demonstrated based on a coupled approach, using experiments and numerical simulation. Without electromagnetic forces, after the initial dissolution of the SiC surface, there is almost no evolution of the interface morphology, which can be considered as being in equilibrium after a few minutes. While applying electromagnetic forces onto the liquid droplet, a stationary solute transport is generated along the interface, maintaining a constant undersaturation (dissolution) at the center of the droplet. Using the electromagnetic field thus allows switching from static to dynamic dissolution of the SiC crystal, which is ideal for studying the interface behavior, and especially the step dynamics.

Surface characterization

After the reconstruction and necessary cleaning, the SiC surfaces need to be characterized in several respects. First we will introduce the facilities used for surface morphologies characterization including differential interference contrast microscopy, scanning electron microscope, profilometer and atomic force microscope. For some methods only the general concept and purpose will be introduced while the details will be put in the appendix.

The following two sections will present the process of statistical analysis for the surface, including the step spacing, the surface roughness as well as the structural auto-correlations (fractal analysis).

Surface morphology

After rinsing and blowing, the reconstructed sample surfaces were systematically observed using an optical microscope (Axio, Zeiss) in differential interference contrast (DIC) mode.

The DIC microscope is more sensitive to height change than normal optical microscopy, which is suitable for observing the subtle step structures on the surface. The DIC image usually gives the appearance of a three-dimensional physical relief, with a contrast of strong light and dark shadows around the object when under a suitable conditions. In specific, the circular polarized light-differential interference contrast (C-DIC) method was employed in this thesis. The advantage of the C-DIC compared with the conventional DIC technique is that the DIC prism in the former can be rotated directionally in accordance with the characteristics of the object. In another word, the sample (or stage) needs not to be rotated when using the C-DIC microscopy. The mechanism of DIC as well as the specific C-DIC is further introduced in Appendix A. Moreover, the Zeiss optical microscope used in this work is also equipped with an automatic movable stage in XYZ directions, which is capable of conducting mapping on large-size samples. Nevertheless, it is not possible to get height values from the C-DIC images. For this, the global 3D topography of the reconstructed SiC/liquid interfaces were scanned with a mechanical profilometer (Talysurf 50, Taylor Hobson). The 3D profile of the surface is obtained by scanning the surface with a diamond stylus (curvature radius: 2 μm). The mechanical profiler measuring is a direct measurement technique that is independent of material properties. The force between the stylus and the surface is adjustable. It is capable of measuring the surface height in a large range from -520 μm to 520 μm with a resolution of 3.2 nm in z-axis direction. The lateral resolution is 0.5 µm along the x-axis (advancing direction) and 1 µm along the y-axis (line displacement direction). A maximum scanning area of 50 mm×100 mm can be achieved.

Another characterization technique for measuring surface height is the atomic force microscope (AFM), which is especially useful for characterizing the local areas of interest.

It is actually the most extensively used tool in this thesis to determine the specific stepand-terrace structures on the surfaces. All the AFM images were taken by using a Veeco Dimension 3100 AFM (Digital instruments) in tapping mode. The mechanism and parameters of the AFM measurement are further introduced in Appendix B.

For some samples, the scanning electron microscope (SEM) is used for morphology observation. Though being a very popular tool in material science for general morphology and element composition characterization, the SEM seems not suitable for this kind of surface. One critical reason is that small steps of several nanometres cannot be distinguished under SEM (very low contrast). Thus the surface details were mostly imaged by AFM except for the large scale areas.

Statistical analysis

For some experiments, the surface morphology and step-and-terrace structures are similar.

It is thus hard to detect potential differences with only naked eyes. To have a clearer comparison of the structures and then attempt to explore the evolution trends from them, it is necessary to extract more quantitative information from the surface morphologies.

The step spacing and roughness are two valuable parameters for investigating the surface evolution.

1) The step spacing

The steps spacing is defined as the distance between two steps, which is reversely proportional to the step density. The step spacing, together with the step density, are often used to evaluate and discuss the step bunching on the surface. Here, we not only consider the average value of step spacing, but also its detailed distribution in the whole covering range. In this thesis, the quantification of step spacing is processed in OriginPro software based on the images from C-DIC optical microscopy.

The basic data processing procedure is shown in Fig. 2) The roughness

The RMS roughness (refer to section 1.1.4) is used in this thesis to compare the roughness of SiC surfaces that own similar morphologies. The stronger surface reconstructions usually generate larger steps and thus give rise to surface roughness. In this thesis, the statistical RMS roughness values were directly computed from AFM height images (typical in 20 μm ×10 μm) being processed in Gwyddion software [START_REF] Nečas | Gwyddion: An open-source software for SPM data analysis[END_REF].

Fractal analysis

The RMS roughness value is tricky and sometimes can give misleading information. The reason is that the RMS roughness only measures the vertical properties of the surface. In another word, it assesses the average height deviation from the mean plane and does not offer any information on the lateral distribution of height. Therefore, two seemingly different surfaces might have the same RMS roughness. One example is shown in Fig.

2.20.

We might feel that surface B is a bit rougher than surface A. But the fact is that they have the same RMS values. Yet, the vertical variation of the altitude is faster for surface B than for surface A. In another respect, a "lateral roughness" is needed to reflect the correlations between different lateral positions on the surface. To put it simply, we need a parameter to describe the shape and size (width) of the bump, which is referred to as "correlation length" (usually marked as ξ in literature). This kind of investigation is also

1) Height-height correlation function (HHCF)

HHCF is a second-order surface statistical parameter that takes into account both height variation and lateral variation in distance and gives more useful information about surface in terms of roughness, grain size and fractal dimension. For quantitative analysis of SiC surfaces, one-dimensional HHCF can be evaluated based on the height profiles along the fast scanning axis in AFM measurements. Theoretically the HHCF can be obtained from the discrete AFM data using the following formulation [START_REF] Klapetek | Quantitative Data Processing in Scanning Probe Microscopy: SPM Applications for Nanometrology: Second Edition[END_REF]:

𝐻(𝑟 ) = ( ) ∑ ∑ 𝑧 , -𝑧 , (2-6)
Where zn+m and z are the height values at two points (x, y), 𝑟 = 𝑥 -𝑥 is the distance between the points, m=𝑟 /∆𝑥. Thus in one word, HHCF can be evaluated in a discrete set of values of 𝑟 separated by the sampling interval ∆x.

For a self-affine surface, the HHCF is often assumed to be Gaussian, which behaves as [START_REF] Sinha | X-ray and neutron scattering from rough surfaces[END_REF]:

𝐻(𝑟) = 2𝑤 1 -exp - (2-7)
Here, w is the interface width or commonly termed as the RMS roughness, α (0⩽α⩽1) is the roughness scaling exponent characterizing the local surface morphology, a larger α value means a smoother local surface profile. The ξ is the lateral correlation length, beyond which the surface heights are no longer correlated on average.

The representative HHCF profiles of both self-affine and mounded surface are shown in

2) Power spectral density function (PSDF)

As described above, the HHCF offers the lateral correlation length (ξ) that represents the short-range lateral features of a surface. Even though the surface heights are not significantly correlated beyond the lateral correlation length, they may also exhibit a periodic oscillation on the plateau part of HHCF profile (r ≥ ξ). In order to determine this long-range feature on the surface, the power spectral density function (PSDF), also known as the structure function, is used. The PSDF is related to a Fourier transform of the ACF, 𝐺 𝜏 , 𝜏 , which is expressed as [START_REF] Klapetek | Quantitative Data Processing in Scanning Probe Microscopy: SPM Applications for Nanometrology: Second Edition[END_REF]:

𝑊 𝐾 , 𝐾 = ∬ 𝐺 𝜏 , 𝜏 e d𝜏 d𝜏 (2-8)
Where Kx and Ky are components of the wave vector of the harmonic component of a certain spatial frequency of the boundary roughness and τ= τ x 2 +τ y 2 is the distance between the points. From the AFM measurements we usually evaluate one-dimensional PSDF, which is given by:

𝑊 (𝐾 ) = ∫ 𝑊 𝐾 , 𝐾 d𝐾 (2-9)
If the ACF follows the Gaussian function, the corresponding PSDF also behaves as Gaussian as following [START_REF] Klapetek | Atomic force microscopy analysis of morphology of the upper boundaries of GaN thin films prepared by MOCVD[END_REF]:

𝑊 (𝐾 ) = √ exp (-𝐾 𝜉 /4) (2-10)
Where σ is the RMS roughness value and ξ is the autocorrelation length, which should be the same as those obtained from the HHCF.

The presentative PSDF profiles of both self-affine and mounded surface are shown in Fig. In this thesis, one dimensional HHCF and PSDF are extracted from the AFM height pictures using Gwyddion software. The fitting of the profiles is carried out using OriginPro software. The application of these two fractal analysis methods will be further discussed for the specific experimental cases.

Conclusion

To study the equilibrium surface reconstruction of vicinal SiC surface, a sessile drop approach was proposed and designed combing experiments and numerical simulation. A high temperature induction furnace was developed, in order to apply a well-controlled environment to the SiC/liquid interface at high temperature. An isothermal environment was realized to get rid of any contribution from temperature gradients (Marangoni convection, diffusion, etc.). The liquid removing tool was carefully selected to get a clean surface. The electromagnetic field was used as an actuator for carbon transport and its effect was demonstrated by altering surface morphology and fluid dynamic in the droplet.

Basically, it allowed the transition from transient surface dissolution (static) to stationary dissolution (dynamic). In addition, the temperature at which the reactor was filled with argon was found to be a critical factor in avoiding bubbles forming at the interface.

Eventually, a uniform high temperature SiC/liquid interface was successfully preserved for further ex situ observation by removing the liquid from the SiC surface at the end of the reconstruction. Standard surface characterization tools and quantitative statistical analysis were employed to analyze the reconstructed surfaces in this thesis.

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Equilibrium surface reconstruction on vicinal surface

The equilibrium surface reconstruction by using the sessile drop method is suitable for investigating the surface properties since most dynamic elements that impede the analysis Above the silicon melting point, the contacting SiC surface is subject to a continuous and rapid dissolution process. When the temperature reaches the desired set point, i.e. the end of the ramping up, a certain period is required to reach the thermodynamic equilibrium between SiC and silicon melt. To determine this required time, a series of experiments were carried out using exactly the same conditions except for the different holding times. Meanwhile, the steps meander significantly, making the steps frequently crossing with each other. However, no big difference between the DIC images can be distinguished by naked eyes. In order to rigorously study the step evolution with time, the step spacing was measured from the grey level DIC images following the processing method described in the experimental part (Section 2.3.2). The obtained results are plotted and compared in Fig. 3.1e. Again, the differences are very weak, which confirms that there is no strong evolution of step spacing with the increasing time. Nonetheless, a subtle change can be found at the beginning where the distribution of step spacing shift slightly toward the right when the time increases from 0 min to 10 min. After that, the step spacing distribution is no longer changed. In addition, the extent of step meandering seems to become slightly weaker after 10 min. Without any major changes, we can consider that a stable SiC/liquid interface with a normal-distributed step spacing has been achieved within the first 10 min, which then stops evolving with time. Another feature is that several small nanosteps of 6-8 nm in height are often observed on the terraces. After 10 min, the curved parts of the step edge have disappeared, giving rise to straight step edges. At the same time, the small nanosteps on the terrace have also disappeared. Longer holding time, e.g. 20 min, does not cause significant differences on the surface step structure. This confirms the observations from DIC microscopy that the development of the macrosteps proceeds within only 10 min. Finally, we have also checked the SiC surface reconstructed for an extreme longer time (4 hours) by AFM (see Fig. 3.

3).

The main step structure is almost the same as that after 10 min, except that the step train looks a bit straighter, i.e. less meandering, than that for shorter times. As we have mentioned, the oscillatory plateau part of the HHCF profile indicates a mound-like surface. For this kind of surface, the perfect fitting is difficult because an extra term needs to be added in the Gaussian function applied for the self-affine surface (Eq.

2-7).

In order to effectively extract information about the long-range features, the PSDF profiles were also calculated in our work. For a mounded surface, the characteristic peak position of the PSDF profile could provide the periodicity of the mounds (or grain size) on the surface. However, it was proved also quite difficult to fit the PSDF profiles in our work and thus not a clear trend was found from the fitting result. In general, the more appropriate and accurate functions are required to better fit the HHCF and PSDF profiles on our step-and-terrace like surfaces. After 10 min, the terraces become wider and dominant but the step edge are still rough.

Besides, we can also find some small steps of about 1 nm in height remaining on the terraces (Fig. 3.9a). After 20 min, smoother (sharper) step edges and the terraces with sparsely distributed small steps are observed on the surface (Fig. 3.9b). However, the same kind of small steps with similar density is also observed on the terrace after 30 min of reconstruction. From the above results, we can clearly state that the time that the C face needs to reach a stable morphology, i.e. a morphology that does not evolve significantly, is longer than the time that the Si-face needs. Generally, it would take around 20 min to get a stable surface on the C face. In summary, the general evolution trends with time of the morphology on the two polar surfaces are similar, indicating that the same mechanism might be accounted for. However, it is proved that the surface reconstruction rate is slower on the C face at least at the beginning. Combining with the much straighter steps on the C face, it seems that the steps on the C face are less likely to be influenced, suggesting a larger step stiffness. A more detailed comparison between two polar surfaces will be given in section 3.2.1.

Anyway, it seems that a holding time of 20 min is enough for both polar surfaces to reach the equilibrium. Here, the term "equilibrium" refers to the thermodynamic state when the liquid is saturated with carbon,i.e. the dissolution process and deposition process are in a balance. One can imagine that when the equilibrium is reached, no more significant evolution will be observed, whether of the step-spacing distributions or of the local step structure and configurations. Therefore, to make sure that the observed interfaces are not "under evolution", a holding time of 30 min will be systematically chosen for all the following SiC/liquid Si interface reconstruction experiments. However, it should be noticed that different kinetic pathways from non-equilibrium to equilibrium usually give rise to different surface morphologies. The different kinetic pathways can be generated for system with different external parameters or different surface structures, which will be revealed in the following sections. It is surprising to see that there is no significant differences between the morphologies as well as the distributions of step spacing. Only one difference is that the step trains become slightly straighter with increasing temperature. Such a tendency can also be observed on the C face in Fig. 3.11, whereas on this polar surface the steps trains are well arranged perpendicular to the propagating direction. This is especially pronounced at 1800 °C where the steps are very straight and perfectly parallel to each other. In addition, unlike the Si face, there exists an apparent evolution of the step spacing on the C face, in which the peak of the distribution shifts towards higher spacing when temperature rises. This phenomenon could be interpreted by the shape change of the steps: the straighter the steps are, the lower the number of crossovers between the steps, and then the larger the apparent step spacing is. The maximum reconstruction temperature used in this work is 1900 °C. The surfaces obtained at this temperature were not put together with above temperatures because the morphologies are quite peculiar. The DIC images of the surfaces are presented in Fig. 3.12. First of all, the average step spacings on both polarities are much wider than those obtained at lower temperatures, especially for the Si face. The steps are also straighter, which is consistent with the trend observed from 1600 °C to 1800 °C. One interesting feature is the presence of many small steps of variant heights on the terraces, on two polar surfaces. The height of the small steps is typically 4-5 nm (15-20 bilayers) on the Si face while it is only 0.5-0.75 nm (2-3 bilayers) on the C face. Note that these nanosteps are not necessarily moving parallel to the macrosteps; it seems that they are randomly meandering on the terrace. This phenomenon is a bit abnormal on the Si face since the small steps are not seen after 30 min at lower tempearatures. This implies that the surface reconstruction might be not completely finished even for the fast-reconstructed Si face after 30 min at 1900 °C. It is believed that a certain kind of weak driving force should be responsible for the local mass transport which is constantly proceeding at high temperatures. One proposed possibility is related to the variation of the solvent composition due to the silicon evaporation at this high temperature, which could break the equilibrium state near the interface. The small triangles formed at the step edge on the C face might support this assumption since similar features were observed when the supersaturation changed (will be talked about again in the following chapters). The reason why the nanosteps on the terrace have different step heights on two polar surfaces will be further discussed in section 3.2.1. In summary, increasing the reconstruction temperature leads to straighter step trains whatever the polarity is. Bales and Zangwill [START_REF] Bales | Morphological instability of a terrace edge during step-flow growth[END_REF] proposed that the unequal energy barrier such as the ES barrier at the step edge could cause the step meandering. According to their models, the larger ES barrier could generate the stronger meandering steps on the surface.

However, it was reported that the ES barrier has an effect on the step kinetics only in an intermediate temperature interval [START_REF] Markov | Crystal Growth for Beginners, 3rd Editio[END_REF]. For example, the ES barrier on the Cu (0001) surface only works from 120 K to 480 K, and it is practically equal to zero when the temperature is over 480 K and infinitely high when the temperature is below 120 K [START_REF] Gerlach | A novel method to determine the Ehrlich-Schwoebel barrier[END_REF].

If the ES barrier could function in our high temperature reconstruction process, the tendency of reducing step meandering, implying a decreasing ES barrier with temperature increasing. This seems to agree with the assumption that the ES barrier would finally decrease to zero over the upper temperature limitation. However, this is indeed an extreme over-simplification of the real situation. Many other effects such as step stiffness and step faceting should be considered when it comes to the complex actual step dynamic process.

Temperature ramp rate

From the above results, we could see that the macrosteps have already been formed on the surface even at the very beginning of the holding temperature. It is thus suspected that the formation of the macrosteps had started before reaching the temperature plateau, i.e.

during the temperature ramping stage. If this is true, the temperature ramp rate may have an important impact on the surface morphology and especially on the step-and-terrace structure. To address this, we carried out a series of reconstruction experiments with three different ramp rates: slow (10 °C/min), standard (20 °C/min) and fast (40 °C/min). The used temperature schemes are shown in Fig. 3.13. Note that before the gas filling, the standard ramp rate (20 °C/min) was used for all the three experiments. Although different ramp rates were applied after the gas filling, the total reconstruction time (ramping + holding) was kept identical for all the experiments. Thus the holding time was 17.5 min when using the slow ramp rate while it was 36.25 min when using the fast ramp. All the holding times were assumed as enough to reach the equilibrium according to our previous results. The morphologies of the reconstructed 4H-SiC Si face surfaces at 1700 °C using different ramp rates are shown in Fig. 3.14. Comparing the general shape of the steps when using a temperature ramp rate of 10 °C/min (Fig. 3.14a) and of 20 °C/min (Fig. 3.14b), the steps become much straighter with a faster ramping rate. However, there is no significant difference between the surfaces with the ramping rates of 20 °C/min and 40 °C/min (Figs.

b and c).

If we look into the detailed structures of the steps from AFM images, all three surfaces seem to own similar average step spacing and step height. Nevertheless, the macrosteps in Fig 3 .14d consist of two segments, which are so called "paired" macrosteps with a very narrow terrace in between. The proportion of the paired steps decreases with the increasing ramp rate and eventually only continuous macrosteps edges without any "tiny terrace" are formed such as those present in Fig. 3.14f. In addition, the slope of the macrosteps edges was also measured from the levelled up height profiles, which is slightly increasing with the increase of the ramp rate. The same experiments conducted on the C face show a stronger effect of the changing ramp rate on the surface reconstruction. As we can see from Fig. 3.15a, the morphology is quite peculiar when the surface was processed with a slower ramp rate. The overall surface is covered by high density of steps with heights of 1-2 bilayers, which appears quite similar to the surface being reconstructed at the very beginning of a given temperature (see Fig. 3.4a). In addition, these small steps are assembled (or gathered) together forming step groups of about 8 μm in width and show like ripples on the blurring surface. It is suspected that this morphology presents the early stage before the emerging of the macrosteps. One possibility is that the driving force (related to the dissolution rate) in this case is insufficient to generate macrosteps on the surface. On the contrary, using a fast ramp rate of 40 °C/min results in a very well-reconstructed surface, where both the terrace and step edge are extremely smooth without any visible small steps (Fig. 3.15f). Nevertheless, some triangle shaped structures are occasionally observed at the edge of the macrosteps. In summary, it is concluded that the SiC/Si liquid interface reconstruction is highly sensitive to the temperature ramping rate. Before reaching the holding temperature, the surface should be undergoing a dynamic-controlled reconstruction process during the temperature ramping up. During this period, the SiC/liquid interface was subject to a continuous dissolution process where the dissolution rate depends on the local undersaturation. And the latter is supposed proportional to the carbon concentration gradient close to the interface, which is determined by the ramping rate. At slow ramping rate, the dissolved carbon atoms have enough time to diffuse from the interface into the bulk solution, giving rise to a smaller carbon concentration gradient. In this way, a smaller undersaturation results at the dissolving interface. In reverse, a faster temperature increasing rate would result into a stronger undersaturation, and then turn into a faster surface reconstruction rate. The morphologies in both Fig. 3.14a and 3.15a are consistent with this hypothesis. On the other hand, this phenomenon is also more significant on the C face. It could be interpreted by its inherent slower reconstruction rate than the Si face even with the standard rate, which has been shown in section 3.1.1. As for the reason why there were no large differences between the surfaces using two faster rates, we speculated that there should exist a critical ramp rate. Over the critical rate, the surface reconstruction may be not rely on the ramp rate anymore and thus evolve in a similar route until reaching a stable state. Another reason is related to the macrostep height: in the growth from solution or melts, the rate of the step advance is inversely proportional to the step height [START_REF] Markov | Crystal Growth for Beginners, 3rd Editio[END_REF]. The receding rate of the macrosteps during solution etching should follow the same rule. Therefore once the macrosteps with certain height (a critical point) are formed, the surface structure would be difficult to evolve because it requires a large external force to move the macrosteps on the surface. The original CMP polished 4° off vicinal surface is shown in Fig. 3.16a, which appears quite smooth without any sign of steps. This is because it is very challenging to distinguish so high density of single bilayer steps on vicinal surface by using AFM whose lateral resolution is not adapted. However, the roughness (RMS) of this surface is measured at about 0.1 nm, indicating a very high quality surface [START_REF] Pan | Chemical mechanical polishing (CMP) of on-axis Si-face 6H-SiC wafer for obtaining atomically flat defect-free surface[END_REF]. Using this kind of surface, we where macroscale step bunching was observed but no graphene was found on the surface [START_REF] Bolen | Graphene formation mechanisms on 4H-SiC[END_REF]. Therefore, we think the effect of high temperature on the surface is rather limited in our work because the vacuum degree is pretty low in our furnace chamber (the ultimate vacuum is about 1×10 -3 mbar at high temperature). And another evidence is that no step bunching can be visible on our sample surface that reconstructed without silicon source. On the contrary, the surface would go through a strong reconstruction if a silicon source is added into the system. In fact the "Si-vapor etching" method has been applied to the removal of the surface damage and the post-implantation activation annealing process [START_REF] Yabuki | Development of "Sivapor etching" and "Si vapor ambient anneal" in TaC/Ta composite materials[END_REF]. In order to assess the effect of silicon vapor on the surface, the SiC surface with a silicon piece on it was annealed in a vacuum at 1400 °C for a dwell time of 5 min. Then the sample (silicon + SiC substrate) was naturally cooled down by shutting down the power.

Note that the silicon piece on top was not melted at this temperature. Fig. 3.17 shows the topographies of the two polar SiC surfaces formed by the silicon vapor etching. Obviously the Si face is already reconstructed into a hill-and-valley structure via step bunching. From the levelled up surface height profile (as inserted), the average step height is about 8 nm.

These bunched steps are well faceted since a sharp step edge is observed. On the contrary, the C face appears not being well reconstructed, which is just covered by a very high density small steps. The step heights are not uniform but they have an average value of about 1 nm, which corresponds to the unit cell height of 4H-SiC. The step height value on the C face is consistent with those reported in the literature, when the SiC surface was etched by gaseous species like H2 or HCl/H2, always yielding step bunching up to the half or one unit c-lattice parameter. However, the mechanism of further coalescence of the unit cell size steps into larger steps has not yet been well elucidated. Note these two images were taken from the area under the silicon wafer.

Another interesting phenomenon was observed during studying the SiC surface reconstructed by silicon vapor. We noticed that not only the area under the silicon wafer was reconstructed into step-and-terrace structure, but also the area out of the silicon wafer.

Besides the point beneath the Si wafer, four points outside the contact area of the Si wafer were characterized by AFM and are illustrated in 2.7 nm, 3.4 nm, 4.4 nm and 5.3 nm, respectively. This means that the surface actually becomes rougher with the increasing distance from the Si piece. By qualitatively inspecting the step spacing, we can see that the step bunching becomes stronger when moving away from the Si wafer. This trend is in fact opposite to what we intuitively expected. Indeed, the vapor pressure of silicon is expected reversely proportional to the distance from the Si piece, or from P1 to P5. In addition, P1, P2 and P3 have the similar topography but with an increasing extent of bunching. The terraces at points P4~P5 appears rough and lumpy and the step edges become more and more waving (meandering). We speculate that there is a change of the surface reconstruction mechanism between P3 and P5. A possible explanation would be a shift from a Si-rich surface reconstruction (close to the Si-piece) to a C-rich surface reconstruction (far from the Si piece). Further experiments would be necessary to verify such a hypothesis.

Effect of the intrinsic properties

From the above investigations, we already noticed that the final surface morphology not only depends on the external parameters but also strongly relies on SiC intrinsic properties.

The most significant one might be the polarity, on which very different morphologies could be obtained with the same external conditions. In this section, the influence of polarity on surface reconstruction will be specifically discussed. Then, the effect of the off-axis angle will be explored since it determines the distance between the initial steps and thus the stepstep interactions during the reconstruction process. For this, SiC substrates with three sets of off-angles, including 0.2° (on-axis), 2°, and 4°, will be adopted for conducting the SiC/Si liquid interface reconstructions. As a special case, the detailed surface morphology and structures formed on the on-axis SiC surface will be thoroughly examined.

Surface polarity

The surface polarity is one of the most important parameters that significantly influence surface morphologies. The two polar faces, having different terminal atoms and thus going through different surface relaxation process, usually give rise to different specific surface free energies. The surface free energy is about three times larger on the Si face than on C face, which has been widely adopted by researchers to interpret the distinct behaviors on two polarities in terms of minimization of the total surface free energy. However the real cases are not just related to the surface free energy, other aspects such as thermal fluctuation [START_REF] Jeong | Two-dimensional dynamical model for step bunching and pattern formation induced by surface reconstruction[END_REF] or the dynamics of the growth/etching process could also have profound influence on the final surface morphology. In this section, we summarize the relevant information from the above sections and summarize the different reconstruction features on two polar surfaces. And the possible explanation will be attempted to give at the end.

Firstly, the most obvious difference lies in the surface reconstruction rate. Despite the fact that the overall evolution trend is similar, we have clearly established that Si face is reconstructed faster than the C face. In other words, the time needed to form stable macrosteps on the Si face is shorter than the C face. This can be proven by two main observations. On the one hand, the C face treated at the 1700 °C for 0 min is nearly free of macrosteps, whereas the steps on the Si face are already strongly bunched. On the other hand, when the macrosteps and the wide terrace are formed at 10 min and even at 20 min, many small steps are still remaining on the terraces of the C face. The zoomed-in morphologies of the terraces on C face at 0 min and 10min are shown in Fig. 3.19. At 0 min, the small steps are closely packed and the step edges appear very rough. At 10 min, the large terraces between the macrosteps are covered with many small steps and narrow terraces: the lowest small terrace is set as reference height (z = 0 nm), other steps can thus be characterized by terrace height differences being a multiple of 0.25 nm. The steps with smaller terraces (right lower area) are about 2 bilayers height while those with larger terraces (left higher area) are characterized by 3 bilayers. Kimoto et al proposed that there are two kinds of terraces for 4H-SiC on which the energies cost for deposition are different and consequently two bilayer-height steps will be formed first [START_REF] Kimoto | Step bunching mechanism in chemical vapor deposition of 6H-and 4H-SiC{0001}[END_REF]. Yet, the formation of the three bilayers height steps is not common and the mechanism has not been reported. Besides the obvious difference in step spacing, other differences between two polar surfaces lie in the detailed step features including the step height and step slope. The statistical results of the step height and step slope angles, measured from the AFM images, are gathered in Fig. 3.20. The data were collected from two polar surfaces that were reconstructed with identical conditions (1700 °C for 30min). As we can see, the step height on the C face is around twice larger than that on the Si face whereas the step edge slope is smaller than the Si face. As the initial miscut direction is towards <112 0>, the terraces correspond to the {0001} basal planes whereas the macrostep edges are faceted into {112 n} planes [START_REF] Fujii | Ordering distance of surface nanofacets on vicinal 4H-SiC[END_REF]. Calculated from the interplanar angle between {0001} planes and {112 n} planes, the n value is around 30 on the Si face while around 40 on the C face, indicating that the faceting on Si face is more significant than the C face. This step slop angle is close to the nanofacets formed on the 4H-SiC vicinal surfaces either by the H2 (or HCl/H2) etching and epitaxial growth [START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF][START_REF] Tabuchi | terrace formation due to step bunching on a vicinal 4H-SiC (0001) epitaxial layer surface[END_REF][START_REF] Sawada | Magic angle and height quantization in nanofacets on SiC(0001) surfaces[END_REF]. Sawada et al proposed that surface steps are bunched to form a nanofacet with a particular angle (named as "magic facet angle") relative to the (0001) plane [START_REF] Sawada | Magic angle and height quantization in nanofacets on SiC(0001) surfaces[END_REF]. They found a magic facet angle of 15° for the (112 n) nanofacets (n=12) on the (0001) vicinal surface of 4H-SiC, which makes the total energy minimum.

In our case, the facet angle formed on the Si face is larger than this reported magic angle.

Another interesting point is that though the macrosteps on the Si seem prone to be faceted, the ultimate step heights of the macrosteps are actually lower than that on the C face.

Initially, it was believed that the step faceting is a result of the step bunching, thus the higher macrosteps (stronger step bunching) should exhibit a stronger extent of step faceting. However, the observation of higher macrosteps on the less faceted C face implies that the step bunching and step faceting are not always consistent. In addition, the two polar surfaces have shown quite different evolution scenarios, even when being reconstructed by the Si vapor before the melting of silicon. For the vapor etching, the step faceting is still stronger on the Si face while the step bunching seemed opposite to the case under liquid silicon where a stronger bunching was also seen on the Si face. Therefore, different mechanisms should be accounted for the distinct surface reconstruction features in liquid and gas environments.

To better compare the reconstruction features on two polarities, we define here two terms to describe the evolution capability of surface structures: extent of bunching (EB) and extent of faceting (EF). The term EB evaluates the ability to congregate the subordinate steps to form a larger step while EF refers to the ability to form atomically flat faceting planes (usually at the bunched step edges). From the above surface morphologies, apparently the Si face has a larger EF due to its much matured (flatter) step edge and its lower crystal plane indices. For the EB, it should be recalled that it is dependent on the reconstruction media, that is, the C face exhibits a larger EB in solution while in vapor the reverse is true. Based on our experimental results, we proposed that the EF is dominated by the thermodynamic properties of the surface (or interface) while EB is a combining result of both kinetic and energetic mechanisms.

First let us look at the step faceting, it is not difficult to understand that the EF on the Si face is always stronger than the C face from the energetic point of view. For a vicinal surface with a relatively high surface free energy (comparing to the low-index surface), the surface structures tend to be reconstructed into low-index facets to reduce the total surface free energy. For vicinal 4H-SiC surfaces, it has been reported that they were reconstructed into {0001} and {112 n} facets as a result of the minimization of the total surface free energy [START_REF] Nakagawa | Self-Ordering of Nanofacets on Vicinal SiC Surfaces[END_REF]. Our experimental results suggest that a similar mechanism could be applied in our cases. For the vicinal Si face that owns a higher surface free energy, a stronger faceting proceeds until the surface morphology has reached an equilibrium state with the environment (no matter with Si vapor or with Si solvent).

As for the step bunching, two different mechanisms are proposed to explain the steps coalescence during the etching by silicon vapor and silicon melt respectively. First let us recall that the original surfaces are composed of a train of equidistance steps, the initial step spacing of which is decided by the off-axis angle. In the presence of an ES barrier, the steps have an asymmetric step kinetic during the advancing (or receding during dissolution or sublimation) [START_REF] Schwoebel | Step Motion on Crystal Surfaces[END_REF]. This effect of ES barrier alone will lead to the equidistant distributed steps during the growing process while the local pairing of steps during the etching process. Therefore, there must exist a repulsive step-step interaction to stabilize the surface during the dissolution process. The analytical expression for the elastic potential (short-range) between steps has been given in Equa. 1-20 [START_REF] Marchenko | Elastic properties of crystal surfaces[END_REF]. The weaker EB on the C face by silicon vapor (Fig. 3.17) means a larger step repulsive interaction U(x) on the C face surface. Since the term in last bracket (a/x) for two initial polar surfaces is identical, the stiffness of steps on C face βC must be greater than that on Si face βSi. This result is consistent with a very recent study which reported that the step energy is higher on the C face [START_REF] Kawanishi | Nucleation Control of 3C-SiC Induced by the Spiral Structure of 6H-SiC[END_REF].

However, after reconstruction under Si liquid, the EB on the C face reconstructed seems to be greater than the Si face according to the higher macrosteps, which is opposite to the surface reconstructed by Si vapor. Therefore, another mechanism is needed to explain this phenomenon. First, it was found that the heights of small steps remaining on the terrace at the initial stage of the surface reconstructed at 1700 °C are in good conformity with the steps formed by the Si vapor (Fig. 3.17). In this sense, we assumed that the surface structures formed by silicon vapor would act as the new starting point for the following reconstruction under Si liquid. This assumption is reasonable since the silicon will melt and wet the surface very soon due to the high temperature increasing rate (it took less than 1 min from 1400°C to the melting point of silicon with the standard ramp rate). For this kind of starting surface, the short-range step interactions including Schwoebel pairing and elastic repulsion are not applicable anymore. Sato and Uwaha proposed that the long-range asymmetric step kinetics, which could modulate the repulsive step interaction at a long wavelength [START_REF] Sato | Morphological instability caused by asymmetry in step kinetics[END_REF]. The expression has also been given in Eq. 1-21.

In our case, the external driving force f0 is negative for the dissolution process. Thus in the case of higher undersaturation (liquid etching process), the step stiffness would be reduced significantly. As a consequence, the steps become unstable and lead to step bunching. In fact, both the Si face and C face are influenced by this asymmetric step kinetic when being dissolved by liquid silicon at high temperature. However, it is believed that the effect of asymmetric step kinetics is more significant on the C face so as to cause a stronger step bunching on this side. If this is true, the term λ--λ+ should be larger on the C face, which implies a larger ES barrier on this polar side.

When applying the asymmetric kinetic on the steps, a large undersaturation (large f0) is essential to realize a significant driving force and thus be able to sufficiently reduce the step stiffness. This effect is not considered for the Si vapor etching because the undersaturation at the interface between SiC and Si vapor might be negligible in comparison to that between SiC and Si melt. This can be verified by comparing the surface morphologies outside and inside the triple line at the droplet edge (Fig.

Off-axis angle

A full description of the surface crystalline orientation comprises two angles: one is the inplane azimuthal angle φ and the other one is the out-of-plane off-axis angle θ. For a vicinal surface, the former is usually determined by the off-axis direction, which is usually selected as <11 00> or <112 0> direction in the SiC wafer cutting process. In this chapter all the vicinal surfaces investigated were cut toward <112 0> direction.

The off-axis angle (θ) is another critical parameter for the vicinal surface. Putting the step spacing and step height L and h, respectively, we can get the step density is 1/L= (tan θ)/h.

The step-step interaction due to either an elastic interaction or a dipole-dipole interaction is generally assumed as A/(h/tan θ) 2 [START_REF] Yagi | Step bunching, step wandering and faceting: Selforganization at Si surfaces[END_REF]. Therefore, the vicinal surface with different off angles should exhibit distinct behaviors during the surface reconstruction. Here, we investigated the surface reconstruction on SiC surfaces with three different orientations, including 4° off, 2° off (both toward <112 0>) and nominal on-axis (still 0.2° off). In addition, some special structures formed on the on-axis surface of 4H-SiC and 6H-SiC will be depicted and their formation mechanism will be interpreted.

The surface morphologies of 4H-SiC Si face substrates with three different off-axis angles are shown in Fig. 3.22. From the DIC images, the step spacing is obviously much wider on the 2° off surface than on the 4° off surface, and extremely large terraces of more than 30 μm are observed on the on-axis surface. Another feature is that the steps become more and more waving when decreasing the off-angle, which results in a wide step spacing distribution as shown in Fig. 3.22d. However, the difference between the two vicinal surfaces of the 4H-SiC C face is not as significant as that of the Si face. As we can see from Fig. 3.23, the morphologies are almost the same on 4° off and 2° off vicinal surfaces, both have straight steps trains and very similar average step spacing. One small difference among them is the reduced amount of crossovers between macrosteps on the 2° off surface, which can be evidenced by the step spacing distribution histogram where the portion of smaller width (1~4 μm) decreases. In addition, the on-axis surface exhibits extremely large terrace area of more than 50 μm in width, which is even wider than that on nominal Si face. Nevertheless, this conforms to our previous conclusion that the step bunching is always stronger on the C face with the silicon liquid. However, the samples involved here only offered limited information about the impact of surface orientation since only <112 0> cut off direction was considered and actually only two points were investigated. A full-scale investigation on the dependence of crystalline orientation was carried out by using a concave-shaped surface, which will be introduced in the next chapter.

As we have already seen from DIC images, vast terraces are formed on the on-axis SiC surface. A large terrace can give rise to two-dimensional nucleation during growth if the mean free path of adatoms is shorter than the terrace width. In our case, the structure of the large terrace on the on-axis SiC surface is also quite different from that on the vicinal The formation of the triangle islands is probably due to the anisotropic growth kinetics in different orientations. The general mechanism is briefly illustrated in Fig. 3.26a: starting with a small hexagonal island which bounded by dense packed A-and B-steps, the B-step edge expends at the expense of the A-step because the advancing rate of A-step is larger than the B-step, thereby a larger triangular island is formed [START_REF] Michely | Islands, Mounds and Atoms[END_REF]. One possible origin of the different advancing rates of two steps is due to their different binding energy for adatoms. Imagining that there are two identical adatoms fluxes coming to two types of step edges, the adatoms will migrate from the edge with lower binding energy (B-step) to the one with higher binding energy (A-step). In addition, it is worth noticing that the island edges show concave shapes on the large triangular islands. This can be explained by the point effect of the diffusion: when the corners of the triangular island are formed following the above process, the point effect of diffusion leads to an increased material transport to these corners [START_REF] Michely | Islands, Mounds and Atoms[END_REF]. There exists a critical length of the triangles (Lc) as shown in Fig. 3.26b: if the edge of the triangular islands is smaller than Lc, the adatoms that migrate to the corners would be carried away by the edge diffusion; when the edge of the triangular islands exceeds Lc, the supplied adatoms at the corner due to the point effect cannot be carried away and then the growth at the corners is accelerated. Then this eventually lead to the concave shaped island edges as shown in Fig. 3.25.

Conclusion

The effect of external parameters and intrinsic properties on the reconstruction of vicinal 4H-SiC surfaces in liquid Si were carefully investigated. The equilibrium SiC/Si liquid interface was obtained fast on the Si face while relatively slower on the C face. The higher reconstruction temperature led to a straighter step edge, which was more significant on the Si face. The surface reconstruction was also strongly influenced by the temperature ramping stage as well as the silicon vapor before the melting of Si. A slower ramp rate gave rise to a weak reconstruction while the faster rate could only slightly accelerate the structure evolution. The silicon vapor was also able to change the surface step structures at lower temperature range (below the melting point of silicon) and then could influence the latter reconstruction with liquid after Si melting. The polarity was one of the most critical surface properties that generated distinct structures at the interface. The macrosteps on the Si face were faceted while those on the C face were not really faceted even if they appeared more strongly bunched. Last but not least, the crystalline orientation could also influence the surface evolution process, which will be further investigated in the next chapter.

From all the above evolution tendencies, we can obtain that: the higher surface free energy [This Page Intentionally Left Blank]

Surface reconstruction on concave-shaped surface

The surface reconstruction experiments in previous chapter were all carried out on flat vicinal surfaces, i.e. the surfaces with uniform intrinsic parameters all over the SiC surfaces.

For that way, a series of experiments are required for investigating the effect of one parameter which needs to be adjusted from experience to experience. This route seems accurate and rigorous whereas it is quite challenging to make the other experimental conditions (except the one studied) completely identical for each run especially for the high temperature experiments. Not to mention that it is usually materials-consuming and timecosting for conducting such a huge amount of experiments. In addition, it is not easy to prepare ideal smooth SiC substrates with specific crystalline orientation. Nevertheless, a very elegant way has been proposed for probing the effect of crystalline orientation, by using a concave-shaped SiC surface during H2 etching, which allowed investigating a large variety of vicinal surfaces in only one, single experiment [START_REF] Dulot | Structure and morphology of concave-shaped surfaces on 6H-SiC(0 0 0 1) after H2 etching[END_REF][140]. To our knowledge, such a kind of approach has not been applied to the SiC/liquid interface reconstruction which usually shows rather different features as compared to the vapor phase. In this chapter, the concave-shaped SiC surfaces were prepared and then reconstructed with the standard sessile drop method. The surface morphologies of two polar surfaces are shown and the dependence of crystalline orientation on the surface structures is thoroughly analyzed.

Sample preparation

In this work, the concave-shaped surfaces used for the reconstruction of the 4H-SiC/liquid silicon interface were prepared by using a ball crater grinder (Calotest CAT 2 c, Anton Paar). The instrument is designed for measuring the thickness of single or multilayer coatings, whose principle is schematically illustrated in Fig. 4.1. Generally a small crater is formed on the fixed surface after being ground with a stainless steel ball of knows geometry (R). When viewed under an optical microscope, a tapered cross-section of the film can be observed and then the thickness of the film or coating can be easily calculated.

This method can provide a precise thickness value in just one or two minutes. In our case, it was just used to create a concave pit on the single crystalline SiC surface. In our case, only one outline can thus be detected on the surface after the grinding, whose diameter is marked as r. Both the Si-and C-faces of nominal {0001} 4H-SiC wafers were used as the substrates. The actual off-axis angle of the substrates was 0.2° from the AFM checking.

The concave-shaped crater was created at the center of the substrate by using a polishing ball of 30 mm in diameter, rotating at a speed of 900 rpm. Two different diamond slurries of 0.5-1 µm and 0-0. Similarly to the procedure presented in the last chapter, a small piece of electronic-grade silicon wafer was put on the crater area at the beginning and then the SiC/Si liquid interface was formed after melting of silicon at high temperature. The sample was processed in argon gas ( ≈ 900 mbar) at 1700 °C for 30 min. At the end of the surface reconstruction, the liquid silicon was quickly removed before natural cooling down. This made the surface morphology generated at high temperature to be preserved for further ex situ characterization. The reconstructed surfaces were thoroughly characterized by DIC optical microscope and AFM. In practice, the quantitative information was mainly extracted from AFM results so as to collect a quantitative comparison between the morphologies at the area with different crystalline orientation.

Surface morphology

The surface morphologies of two polar concave-shaped surfaces after being reconstructed at 1700 °C for 30 min were scanned under the DIC microscopy and are presented in Fig. Next let us look at the effect of off angles on the surface structures along the fixed radial direction on the Si face. Here two representative azimuthal directions, <1 1 00> and <112 0>, were selected to compare the step structures. These two orientations can be distinguished according to the shape of the central part, whose six edges are perpendicular to the <11 00> directions. However, this rule is only applicable on the Si face, meaning that it is hard to judge the specific orientation on the C face. On the Si face, two distinct evolution tendencies with off angle along two directions are exhibited in Fig 4 .3.

Apparently, the steps moving along <11 00> directions are straighter and sparser. While along the <112 0> direction, i.e. in the transition areas between two <11 00> orientation areas, the steps are waving and crossing. A similar trend was found around the screw dislocation center when the 6H-SiC (0001) Si face surfaces were etched by H2 [START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF][START_REF] Wulfhekel | Regular step formation on concave-shaped surfaces on 6H-SiC[END_REF].

However, the steps revealed by the defect-selective process are usually very small (half or one unit cell height), meaning that the mechanism can be easily associated to the nature and number of dangling bonds. However, the steps formed on our surface are much larger than the unit cell, so the atomic step termination model cannot be directly applied here.

This will be further discussed in the following. both the <11 00> and <112 0> directions of the Si face, an obvious increase of the step density is observed with the increasing off angles. This tendency can be found a bit more significant along the <11 00> direction. Here the waving of steps is not clear even along the <112 0> direction because the AFM scan area is too small to show the long range step shape.

On the C face, it is difficult to distinguish the two crystalline orientations thus only one random orientation was selected. As we can see from the last column in Another interesting feature is that different surface structures appear at the central area, corresponding to the quasi-on-axis surface. As can be intuitively seen from Fig. 4.5, the on-axis area of Si face exhibits several polygons bounded by straight sides (edges) while that of the C face is quite smooth with only sparse and tiny steps propagating on it. Two hexagonal shaped hillocks (or shallow pits) are formed on the large central plane of the C face, which might be related to the screw dislocations on the surface. In general the surface structures are better developed and step edge is easier to be faceted on the Si face, which is consistent with our conclusions obtained from the standard vicinal surfaces in the previous chapter.

Discussion

The formed step-and-terrace structures are quantitatively described based on the information from the AFM results. The step density, the average step height, the macrostep angle, and the terrace width could be directly extracted from the height profiles that plotted perpendicularly to the formed macrosteps. The cross section of the macrosteps is illustrated in Fig. 4.6, which shows all the measurable step parameters. The step density is calculated as the reciprocal of the step spacing while the macrostep angle is defined as the acute angle between the macrostep edge and lower terrace. For each parameter, the values from both <11 00> direction and <112 0> direction on the Si face as well as the random direction on the C face are plotted as the function of the off angle. In addition, the parameter R, the ratio of final terrace width (L) to the original terrace width (L0), which is expressed as R = L/L0, is introduced to evaluate the extent of the step bunching with different off angle. For a vicinal surface with a uniform off angle toward a certain orientation, the L0 can be formulated as L0=1/tan θ if the height is uniformly distributed and the step height is assumed as a constant 1 (not the real case).

Then the R can be reformulated by R = L/L0 = L• tan θ. All the above quantitative results are shown together in Fig. 4.7.

First the observed features from AFM height images can be further confirmed by the step density variation trend in Fig. 4.7a. The step density is increasing with the off angle on the Si face but almost constant on the C face. On the contrary, as can be seen from Fig.

4.7c, the height of the macrosteps is seemingly independent of the off angle on the Si face. While on the C face, the step height varies following a general increasing trend with the increasing off angles. Besides, though all the terrace width continuously decreases with the increase of the off angles (Fig. 4.7b), the step edge slope (macrostep angle) is almost invariant (Fig. 4.7d). Moreover, another characteristic term, the ratio of terrace widths after and before the surface reconstruction (R), can be used to evaluate the extent of step bunching. A larger R value indicates that a stronger step bunching has occurred on the surface. Based on above evolution tendencies, generally two different scenarios of step bunching are revealed on two polar surfaces: on the Si face, the extent of bunching seems constant everywhere along a given azimuthal orientation, while on the C face, the step bunching is much stronger and is gradually enhanced with the increasing off angle. In addition, it should be noticed that on the Si face the step bunching along the <11 00> direction is always more pronounced than that along the <112 0> direction. This agrees to a certain extent with literature, in which unit cell height steps are formed along the <11 00> whereas only half unit cell high steps appear on transition areas between two <11 00> directions when the SiC surface is reconstructed by gas etching (H2 or HCl) [START_REF] Nakamura | Formation of periodic steps with a unit-cell height on 6H-SiC (0001) surface by HCl etching[END_REF][START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. As already known, the edge of the macrosteps formed at the SiC/liquid Si interface tends to be faceted into {112 m} or {11 0m} crystalline planes to minimize the total surface free energy. In our cases, the index m is always smaller on the C face due to a worse-developed macrostep edge on this polar side. This can be explained by the difference in surface free energy between both two polar faces. A surface with higher energy (Si face) will go through a stronger surface reconstruction to reduce the total surface energy. This is the reason why steeper macrostep edge (larger macrostep angle) is formed on the Si face.

For vicinal surfaces, the total surface energy can be simplified as the sum of the total terrace surface energy and the total step edge energy (any other step-step interaction is not consider), which is expressed as:

𝛾 = 𝑓 • 𝛾 + 𝑓 • 𝛾 (4-1)
Where f1 and f2 are the fractional area occupied by terrace and step edge and γ1 and γ2 are the corresponding specific free energy. The specific surface free energy of off-oriented surface parts (γ2) is always larger than the one on on-axis surfaces (γ1). It is hypothesized that whatever the polarity of the surface, the initial step configuration (step height + step edge angle) is identical. This implies that γ2 is same everywhere because it only depends on step edge angle (provided that the off-axis direction is the same). Increasing the off angle (θ) will increase the initial step density, and consequently the initial total surface energy because f2/f1 is increased. Thus a stronger faceting should occur on the surface with a higher off angle. However it seems the extent of faceting is not sensitive to the off angle on both polar surfaces as there is only weak change in the macrostep angle with off angle.

Thus we proposed that other kinetic mechanism such as elastic potential between steps (repulsive effect) should be further considered for giving a reasonable explanation.

Along the c-axis of 4H-SiC, the hexagonal SiC bilayers are arranged in two groups of double bilayers. Within each group, the stacking of two bilayers has the same orientation, but from one group to the other, the lattice is rotated by 60°. As a consequence, the step edges on the vicinal 4H-SiC change their nature from one group to the other. For the steps arranged perpendicular to the <11 00> direction (Fig. 4.8a), the character of every two steps alternate between SN and SD step edges, i.e. the bonding at the step edge itself shows either one (SN) or two (SD) dangling bonds [START_REF] Pechman | Interactions of Br with Si(111)-7×7: Chemisorption, step retreat, and terrace etching[END_REF][START_REF] Ramachandran | Preparation of atomically flat surfaces on silicon carbide using hydrogen etching[END_REF]. The alternative frequency is associated with the off angle along one in-plane (azimuthal) orientation. It is assumed that the dissolving process proceeds following the "reverse" step flow mechanism where the steps are receding rather than advancing. Then the step receding rate is presumed depending on the bonding condition of the step edge, i.e. the number of the dangling bonds at two types of steps. More dangling bonds at the edge signifies that the outmost atoms are easier to be detached and ejected to the lower terrace or solvent. Therefore the SD steps should be receding faster than the SN steps, i.e. the latter is caught up by the former, and eventually the bunched steps result. However, when the direction deviates from <11 00> direction (Fig. 4.8b), the total dangling bonds number at the step edge varies with the changing azimuthal angle. The step edge can be regarded as the mixture of the SN and SD steps, and the equal number of the SN and SD steps (50% SN+50% SD) is fulfilled for all the step edges when reaching the <112 0> direction [START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. In this sense, all the initial steps would recede at a homogeneous rate and thus no step bunching should be found. However, this is a very ideal case. Tabuchi et al proposed that the orthogonal misorientation could also give rise to different dangling bond densities between two adjacent steps, which lead to a difference in step advancing velocity and then result in the pairing of half unit-cell height steps [START_REF] Tabuchi | terrace formation due to step bunching on a vicinal 4H-SiC (0001) epitaxial layer surface[END_REF].

By analogy with the kink growth and step growth proposed by Kimoto et al [START_REF] Kimoto | Nucleation and step motion in chemical vapor deposition of SiC on 6H-SiC{0001} faces[END_REF], we proposed that there might be two modes during the surface dissolving process, i.e. "kink dissolution" and "step dissolution". The dissolution follows the "step dissolution" mode along the <11 00> direction while follows the "kink dissolution" mode along the <112 0> direction. The different dissolving rates along two directions result in a six-fold symmetric shape on the surface. Another feature of the steps in <112 0> direction is that they are much more meandering than that in the <11 00> direction. It was reported that the "Kink Ehrlich-Schwoebel Effect", which results from the different adatoms incorporation rate when they arrive at a kink site from different direction, can also lead to step meandering [START_REF] Pierre-Louis | Edge Diffusion during Growth: The Kink Ehrlich-Schwoebel Effect and Resulting Instabilities[END_REF]. In this sense, the higher kink densities at the step edge along the <112 0> direction might enhance this effect and thus cause a stronger meandering in this direction. Another possible reason is that the better faceted macrosteps along the <11 00> direction are stiffer so as their shape is not prone to be changed during the movement [START_REF] Akutsu | Profile of a Faceted Macrostep Caused by Anomalous Surface Tension[END_REF]. Nevertheless, the above explanation on the effect of azimuthal angles is only applicable on the Si face where six-fold symmetry is found on the surface. As for the C face, the steps evolution seems not sensitive to the azimuthal angles, implying that the bonding condition is not a critical factor for the step bunching on the C face. This might be validated by the special feature of bunched macrosteps on the C face. As we can see from Fig. 4.9, the macrosteps formed on the C face are actually not very well developed (faceted) because amounts of nanosteps of 1-2 bilayer in height are observed on the top part of the macrosteps. This indicates that the dangling bonds model is not applicable for the C face, otherwise the nanosteps will not exist along all the direction after the step bunching.

However, due to limited experiments on the concave-shaped surface, the specific mechanism of the step bunching on the C face is still not clear. The off-axis angle is around 2° at this area. The top part of this macrostep is covered by high density small steps of 1~2 bilayer in height.

Conclusion

The concave-shaped SiC surfaces have been applied to the study of the effect of crystalline orientation on surface reconstruction in liquid silicon. The reconstructed Si face revealed a six-fold symmetric structure which was not observed on the C face. The detailed characterizations demonstrated that step bunching became gradually weaker from the <11 00> direction to <112 0> direction on the Si face. A step termination model was used to explain the change of the step density and shapes along the azimuthal direction. For any fixed azimuthal direction, the extent of step bunching was enhanced on C face while almost invariant on the Si face when increasing the off angles. The extent of step faceting was more significant on the Si face. The special structure of macrosteps formed on C face indicated a specific bunching feature on this polarity. More experiments are needed to clearly understand the impact of crystalline orientations on the surface reconstruction.

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Surface reconstruction out of equilibrium

From near-equilibrium surface reconstruction works, we have grasped the general evolution features of the SiC/ liquid silicon interface under various conditions. Based on those results, some information about surface properties and effects of external parameters have been obtained. However, for the solution growth process, the crystal/liquid interface undergoes conditions which slightly or sometimes significantly deviate from the equilibrium. To get a better picture of the step dynamics in relation to the bulk SiC growth process, it is of great importance to compare the surface under growth conditions with the results of the previous chapters. In the first part of this chapter, the growth is carried out directly on the surfaces that were reconstructed under the equilibrium condition. The growth is achieved by controllably cooling down the droplet on top of the SiC substrate, where the as-reconstructed SiC surface could be regarded as the "initial surface" for later growth. In the second part, a specific experiment called "sandwich" reconstruction will be explored. For two surfaces with small difference in properties, the high temperature surface reconstructions sometimes result into similar surface morphologies which make the analysis difficult or even impossible. Similarly, reproducibility issue of high temperature experiments could lead to wrong conclusions. In order to address this problem and realize a direct comparison of the two surfaces, the sandwich-type surface reconstructions were employed to ensure an identical environment for two surfaces. The detailed experimental process and the primary results will be presented in the second part. This approach is also categorized as an out-of-equilibrium process because the liquid is not able to be removed during the operations.

Growth on as-reconstructed surface

For the experiments carried out in the previous chapters, the silicon melt was removed at the end of the reconstruction process in order to prevent crystallization during the cooling down period. This is suitable for the ex situ study of the high temperature equilibrium crystal/liquid interface. Tuning the interface morphology has been used to improve the quality of the bulk crystals. For example, Murayama et al proposed a two-step SiC solution growth method for dislocation reduction, consisting of the sequential growth on first a Si face, and then on a C face [START_REF] Murayama | Two-step SiC solution growth for dislocation reduction[END_REF]. The growth on the Si face was firstly carried out to reduce the threading dislocations by the movement of macrosteps outside the crystal. Secondly, the growth on the C face was conducted on the seed prepared in the first step, in order to reduce the density of basal plane dislocations and keep a smooth growth surface. In our work, we carried out the growth on the previously reconstructed SiC surface by directly cooling down the liquid at a controlled rate. Both the vicinal 4H-SiC {0001} Si face and C face were considered and the different growth scenarios are revealed as follows. At last, the models are given to elucidate the difference growth behaviors on two polarities.

Growth on as-reconstructed Si face

As already discussed, the vicinal Si face is prone to be reconstructed than the C face and the macrostep edges are well faceted after the surface reconstruction. Here we presumed that adding a cooling down period will promote the further movement of the macrosteps but in an opposite direction. The growth is carried out directly on the as-reconstructed surface by exerting a controlled cooling down on the SiC surfaces with the droplet on top.

The used temperature regimes are plotted in Fig. 5.1. During the cooling from 1700 °C to 1500 °C, two different cooling rates of 4 °C/min and 10 °C/min were applied, and then the liquid droplet was removed instantly when the temperature reached to 1500 °C.

According to the carbon solubility and the sample configuration, the growth rates during the cooling down are calculated as 0.55 μm/h and 1.36 μm/h, respectively. Two different cooling down rates, 4 °C/min and 10 °C/min were applied.

The obtained surface morphologies and step structures on the 4° off Si face are demonstrated in Fig. 5.2. As can be obviously seen, larger terraces are formed on the surface when the cooling down period is added, indicating that there exists a further step movement during cooling down. The average terrace width is about twice as that on the surface without the cooling down. No big difference in terrace width can be found on two surfaces with different cooling rates. However, it should be noticed from the AFM images that when using a smaller cooling rate, the formed macrosteps are usually discontinuous, i.e. they are often composed of two or three segments (Fig. 5.2 f). The detailed step structures on the surfaces with an additional cooling down period are shown in Fig. 5.3. In both cases, the macrosteps are higher and they own larger step slope angles (30~32°) than the macrosteps on standard surface without cooling down. This implies that further step movements must have occurred after the equilibrium reconstruction process (dissolution). With a slower cooling rate, the macrosteps edge is usually composed of two parts. In other words, there exists a narrow terrace which separates the whole macrostep, which can be clearly seen from the height profile (Fig.

5.3c).

The discontinuous parts are remarked with the red arrows in the AFM amplitude image and on the corresponding profile as well. When the faster cooling rate of 10 °C/min was applied, the narrow terrace in the middle of the macrosteps is disappeared (Fig. 5.3 f). This means the "singular" macrosteps with smooth and continuous edges are obtained when using a faster cooling rate. This might be due to the larger supersaturation generated in the liquid during a faster cooling down, which facilitates the further evolution of the macrostep edge.

Growth on as-reconstructed C face

The macrosteps formed on the C face by equilibrium reconstruction were found to be less faceted, though in average they are higher than those on the Si face. Due to the weak interaction between the smaller unit steps in the macrosteps, it is expected that the macrosteps on the C face can be disassembled backward if the steps propagate in the opposite direction. The same cooling down scheme was applied on the 4° off 4H-SiC C face, the obtained surface morphologies and step structures are presented in Fig. 5.4.

Comparing with the surface morphology without cooling down period, the general step shape does not vary too much when an extra cooling down period was added. However, the average step height is measured at more than 500 nm on the C face with the additional cooling down, which is a bit larger than the surface without cooling down period. Moreover, it should be noticed that the macrosteps edges have been obviously expanded after cooling down. This is more significant when the smaller cooling rate is used, giving rise to terraces even narrower than the step edge extent (Figs. 5.4 c and f). Correspondingly, the slope angles of the macrosteps are about 10~11° and 6~8° for the surface cooled with the rates of 10 °C/min and 4 °C/min, respectively. The detailed structures of the macrosteps edges on the C face after cooling down are shown in Fig. 5.5. The macrostep area is obviously composed of large amounts of small steps. As shown in the enlarged image (Fig. 5.6), the small steps are characterized as having a height of 8~15 nm. These small steps at the macrosteps edges are a bit denser when the cooling rate increases to 10 °C/min. For this case, the small steps are too dense to be distinguished but some bigger protrusions can be observed at the macrostep edge (Figs.

d and e).

Comparing to the macrosteps formed with a slower cooling rate, it is however hard to determine which one corresponds to the early stage of the growth or which one has grown further. On one hand, the larger cooling rate exerts a larger supersaturation at the SiC/ liquid interface, which might promote the growth process. On the other hand, the growth might be insufficient as a result of the shorter growth time due to the faster cooling rate (especially for the C face). Here we speculated that the structures are more developed in Figs. 5.5 a and b based on the further expanded macrostep area. The underlying hypothesis is that the growth time, rather than the supersaturation, is more important for the surface reconstruction on the C face. On the Si face, the macrosteps formed at the end of equilibrium reconstruction have been reconstructed into certain crystalline planes though with high miller indices. During the cooling down period, the growth of the surfaces will follow the Wulff construction theory.

The proposed process is schematically illustrated in Fig. 5.7. The crystal planes at the macrostep edge propagate faster because they own a higher surface free energy. According to the Wulff 's principle, the final crystal is prone to be covered by low energy planes which advance with a slower moving speed. As a result, the larger terrace area is formed after the additional cooling down period. When the cooling down is faster, the higher supersaturation should enhance the amounts the adatoms supplying to the surface and also accelerate the growth of the slower planes. This might be the reason why the very narrow terrace disappeared when a higher cooling down rate was adopted. A quite distinct growth behavior has been revealed on the C face, which can be briefly illustrated in Fig. 5.8. As we can see, just after the equilibrium reconstruction, the macrosteps are not well faceted (only bundled) and show as rough step edges (Fig. 5.8a).

The rough macrosteps edges have numbers of kink positions where the adatoms are prone to be attached to them. Therefore when the supersaturation is applied to the steps, the small steps (with a height of 8-15 nm) can separate from the bundled steps and advance independently following the step flow growth mechanism. As a consequence, the macrostep is gradually expanding and simultaneously its edge is less sharp than the beginning. It is reasonable to say that the growth on the C face costs less energy than the Si face because the moving unit is much smaller. In addition, we assumed that the step advancing rate on the C face should be independent of the supersaturation because the heights of the small steps appear similar on two surfaces with different cooling rates (see

Figs. 5.5 b and e).

The density of the smaller steps at the macrostep edge is higher when using a large cooling rate probably just because the growth time is relatively shorter.

Sandwich-type surface reconstruction

In this section, another configuration called "sandwich" is proposed to perform surface reconstruction for two combined surfaces in a shared liquid environment. In this way, the critical surface properties that control the structure evolution can be unraveled since all the external parameters are identical in one and the same experimental run. As a preliminary exploration in this thesis, two polarities of SiC, i.e. the Si face and the C face, were used to build sandwich-type samples with different combinations. The step-and-terrace structures on the reconstructed surfaces were carefully characterized from several aspects after the sample reconstruction. The mass transport phenomenon among two involved surfaces was also expected to occur in the sandwich-type sample, which will be carefully examinated by comprehensively comparing the morphologies in different combinations.

Experimental process

The sample preparation process of the sandwich-type samples is schematically illustrated in Fig. 5.9a. The studied two SiC surfaces were both contacted with the thin electronic grade Si wafer. The assembled samples were put on a graphite holder, which were surrounded by a 1 mm thick tantalum tube for shielding the electromagnetic field. The surface reconstruction was performed in the aforementioned induction furnace at a given temperature holding for a certain time. The temperature was manually increased to 1100 °C beyond which the temperature was automatically controlled with a ramping up rate of 20 °C/min. Like before, the furnace chamber was vacuumed before 1450 °C and then was filled with highly pure Ar gas to a pressure at around 900 mbar. The samples were naturally cooled down with a rate of about 40 °C/min, by simply switching the power off. One point to note is that the liquid silicon between two SiC substrates was not removed before the cooling down.

After surface reconstruction, the sample was cut into two half parts perpendicular to the macrostep edges (see Fig.

Results and Discussion

The sandwich-type samples were cut and the cross sections were specially prepared to observe the interfaces between SiC substrates and the silicon layer. One of the typical cross sectional stacks is demonstrated in Fig. 5.11. The thickness of the SiC substrates is about 350 µm and that of the inner molten silicon is about 120 µm. Thus the total sample thickness is about 820 µm, which is very thin compared to the overall crucible configuration. According to our simulation in chapter 2, the sandwich-type samples are also believed being in an isothermal environment. The morphologies of the cross section of the reconstructed sandwich-type samples are shown in Fig. 5.12. For all the Si face surfaces (in Figs. It has been displayed in the last section that the C face tends to be de-bunched due to the growth during the cooling down period. In that case, the macrostep edges expand and the surface is gradually covered by smaller steps. We think the similar step behaviors happen on the C face in the Si-C combination. Besides, the extent of growth in the Si-C sandwich-type structure is generally stronger even though the growth time is very short due to the fast natural cooling rate. At the same time, we have noticed that the macrosteps edges of the C face in the Si-C combination is quite different from that in the C-C combination.

Hence, we believe that there should exist a mass transport between two polar surfaces when being reconstructed in the shared liquid silicon. It is proposed that the surface reconstruction in sandwich-type structures generally comprises two stages, which is schematically illustrated in Fig. 5.15. At the initial stage (after silicon melting), both the vicinal Si face and C face are quickly dissolved into the silicon melt and the surface steps are bunched into macrosteps. The macrostep edge on Si face is prone to be faceted while the C face is more like a shallow hill-and-valley. After a certain time, the equilibrium state should be obtained when the liquid silicon is saturated with carbon atoms. This first stage gives a sharp macrosteps profile on the Si face whereas a less undulant step-and-terrace structure on the C face (Fig. 5.15a). Next, the mass transport is expected to be occurring between two polar surfaces due to their different properties. From the energetic point of view, the Si face has a higher surface free energy so as to undergo a stronger reconstruction than the C face in order to reduce the total surface free energy. In this sense, the macrosteps could probably continue to be rearranged though the dissolution process has reached equilibrium. Along with this process, more carbon atoms are ejected into the liquid and give rise to the supersaturation of carbon in the liquid. On the other side, there are numerous kinks at the macrosteps edges on the C face, especially for the vicinal surface off towards <112 0> direction [START_REF] Kimoto | Nucleation and step motion in chemical vapor deposition of SiC on 6H-SiC{0001} faces[END_REF]. A small supersaturation could lead to the kink growth at the macrosteps edges on the C face. The growth proceeds by firstly forming protrusions which can be then developed into subordinate steps at the macrosteps edge. Nevertheless, the nucleation of the protrusions seems not uniform on the surface and thus growth morphology at different stages can be observed on the surface. As an extreme case, the macrosteps has been completely replaced by the new steps and terraces as shown in Fig. The process described above is only considered occurring before the cooling down of liquid silicon. In the last section of this chapter, we have proved the occurrence of growth during the cooling down period. However, it is presumed that the growth during cooling down is limited and negligible in this case as the natural cooling rate is pretty fast (about 40 °C/min). Hence we infer that the growth occurred at the macrostep edges is majorly resulting from the mass transport from the Si face to the C face. This could be supported by the fact that no obvious growth feature was observed on the C-face in the C-C combination processed in the same experimental run.

5.14d.

Conclusion

In this chapter, the surface reconstructions out of equilibrium were investigated. The cooling down experiments manifested two different kinds of macrosteps had formed on two polarities during the equilibrium reconstruction. The macrostep formed on the Si face was faceted and stable, which cannot be decomposed during the growth (the evolution is irreversible). However, macrosteps on the C face was de-bunched first and propagated by moving the smaller steps, implying reversibility of bunching-de-bunching for the macrosteps on the C face.

The sandwich-type reconstruction experiments were implemented to compare the relative activity of two involved surfaces. Two polar surfaces of 4H-SiC were compared with this configuration. The growth on the C face was confirmed by peculiar structure feature, which resulted from the mass transport generated between the Si face and the C face. In addition, this method is versatile, which would be powerful in distinguishing or comparing two surfaces that are only slightly different from each other

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Carbon solubility determination

The solubility of an element in the liquid phase is an important parameter for both the metallurgical process and solution growth process. Carbon is one of the most common impurities in silicon ingots and electronic-grade silicon wafers. For instance, it can come from the graphite components that are used as the crucible and insulation in the furnace.

The dissolved carbon would precipitate as carbides during cooling down, which is detrimental to the solar cell operation and electronic devices properties [START_REF] Kolbesen | Carbon in silicon: Properties and impact on devices[END_REF]. The thermodynamic of carbon in liquid silicon is important for understanding and preventing the formation of silicon carbides. On the contrary, a larger carbon solubility in liquid is desired for the solution growth process of SiC, in order to reach fast growth regimes. For example, the Si-Cr and Si-Fe solvents have been used for the solution growth of SiC due to their much higher C solubility than pure silicon [START_REF] Kawanishi | Availability of Cr-rich Cr-Si solvent for rapid solution growth of 4H-SiC[END_REF][START_REF] Kawanishi | Measurement and Thermodynamics of Carbon Solubilities in Molten Si-Fe, Si-Ni, and Si-Cr-Fe Alloys at 2073 K[END_REF].

However, it is usually challenging to measure the low solubility of light elements like C, O, and S in high temperature molten metals and the results always suffer from controversies when different methods and procedures were applied. For instance, the carbon solubility in liquid silicon equilibrated with SiC varies by three orders of magnitude while comparing the determinations from Scace et al [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF] and Dolloff [START_REF] Dolloff | Research study to determine the phase equilibrium relations of selected metal carbides at high temperatures[END_REF]. Based on the work of previous chapters, we proposed a novel methodology for the determination of the solubility curve based on the sessile drop approach. In the following, we will first review the previous works in terms of the used method and procedures, then our specific methodology will be introduced. Last but not least, we will discuss the advantage and disadvantages of our method through comparing the results obtained by conventional methods. The probable source of errors will be thoroughly analyzed and the possible development will also be given.

Introduction of previous solubility studies

The solubility of carbon has been extensively studied in both solid silicon [START_REF] Kleykamp | The Constitution of the Silicon-Carbon System[END_REF][148] and liquid silicon [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF][START_REF] Dolloff | Research study to determine the phase equilibrium relations of selected metal carbides at high temperatures[END_REF][START_REF] Hall | Electrical contacts to silicon carbide[END_REF][START_REF] Oden | Phase equilibria in the Al-Si-C system[END_REF][START_REF] Yanaba | Solubility of carbon in liquid silicon equilibrated with silicon carbide[END_REF][START_REF] Dalaker | Time and Temperature Dependence of the Solubility of Carbon in Liquid Silicon Equilibrated with Silicon Carbide and Its Dependence on Boron Levels[END_REF]. In this work, we only address the measurement of the solubility in liquid silicon. Generally there are two kinds of methodologies for the determination of carbon solubility:

1) Combustion coupled with IR absorption spectroscopy. This technique measures the carbon content in the quenched samples. For this, a very fast quenching is required to prevent the carbide precipitation during solidification or particles generated from the eutectic step because they will not be considered during the later analysis.

2) Gravimetry method. It measures either the "weight loss" of carbon source after dissolution [START_REF] Hall | Electrical contacts to silicon carbide[END_REF], or the "weight gain" if all the dissolved SiC precipitates during cooling down [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF]. Actually this strategy will also be applied in our new approach but in a different format.

The experimental works on determining the carbon solubility in liquid silicon are compiled in Table 6.1. There are lots of discrepancies among the solubility values in previous investigations due to the different methods and (or) specific procedures used. The error can come from the material of the container, the partial pressure of oxygen in the atmosphere, the surface removal process and so on. One special case is the paper of Doloff [START_REF] Dolloff | Research study to determine the phase equilibrium relations of selected metal carbides at high temperatures[END_REF], where he gave much higher carbon solubility than others. In his work he also measured the eutectic temperature (Liquid = SiC(s) + Si(s)), about 10 °C below the pure Si melting point, which was obviously unreasonable. Considering there was no experimental details in his work, it should be thus regarded as just an outlier or more likely being in error. Another special case is the paper from Durand and Duby, where they critically reviewed the published data without bringing new measurements [START_REF] Durand | Carbon solubility in solid and liquid silicon -A review with reference to eutectic equilibrium[END_REF]. They disregarded all the experimental works using combustion IR absorption method and even the results of Scace just because their samples were sandblasted. They derived the temperature dependent carbon solubility function based on only the limited results points of Dash and Hall [START_REF] Hall | Electrical contacts to silicon carbide[END_REF] and Kleykamp et al [START_REF] Kleykamp | The Constitution of the Silicon-Carbon System[END_REF]. Nevertheless, their discussion on the position of the eutectic point (temperature and composition) was highly relevant, using a data optimization approach. The obtained solubility values were close to the Si-C diagram published by Gröbner et al [START_REF] Gröbner | Thermodynamic calculation of the ternary system Al-Si-C[END_REF] at high temperature, while exist a significant deviation at low temperature where the data of Scace and Slack were adopted. But both Gröbner and Durand's papers are assessments based on already published datasets. In addition, the carbon solubility value at the melting point (1414 °C) from Durand's law was higher than any other works (except Doloff 's). So in summary it is hard to accept their data as the real carbon solubility.

Next we will generally review again the previous experimental works on carbon solubility.

The purpose is not in order to underestimate the previous results, but to better assess and choose the most "relevant" data. On the other hand, this can help us scrutinize our own determination approach and help ameliorate our solubility results.

Firstly let us look at the works based on gravimetry method, which is also the methodology used in our measurement. In Hall's work, the small SiC crystals were floating on the silicon melt for a given time. Then the SiC crystal was removed from the melt at the end of the heating and cleaned from silicon residues [START_REF] Hall | Electrical contacts to silicon carbide[END_REF]. The biggest issue of their works was from the crucibles which were both oxide materials (SiO2 and Al2O3). The oxygen present in the crucibles could diffuse into the melt at high temperature and then constantly react with the dissolved carbon to form volatile CO or CO2. This effect, pointed out by some authors afterwards, gave rise to an overestimation of the carbon solubility. It was also time dependent because the dissolving of carbon could proceed as long as there was oxygen in the system. The next year (1958), Scace and Slack measured the carbon solubility over a wide temperature range 1560 to 2900°C using a different kind of gravimetry method [START_REF] Scace | Solubility of carbon in silicon and germanium[END_REF].

The dense graphite containers were used to load silicon powder, which also worked as the carbon source for the liquid solution. The experiments were operated in a high pressure atmosphere of 35 atm, which was intended to prevent the evaporation of the silicon. The keeping time at the desired temperature was set as 1.5 to 2 min. It was assumed that the dissolved SiC at high temperature had completely crystallized as SiC upon cooling and was enclosed in the frozen silicon slug. Then the graphite container was burned away at 1100°C

in an oxygen atmosphere. The silicon slug was slightly sandblasted (a layer less than 25 μ) to remove the SiC formed in the wall of graphite container and adhered to the slug surface. The Si ingot was etched by HF/HNO3 (1/3) acidic solution. The SiC residues were finally dried and weighed. The authors has recognized that the major source of error came from the sandblasting process, where the carbon segregation close to the surface was lost. Though this could give rise to a lower carbon solubility, they thought the error was not significant because the layer was thin. They estimated that 90% of SiC was still retained in the slug. This point is highly disputable now. We also speculate that another possibility of having lower solubility is related to the holding time, which is a bit too short to reach to equilibrium from our experience.

Concerning the combustion analysis method, which is widely adopted in the literature, it should not be completely discounted like Durand et al suggested. Actually we also took a page from this method because the liquid removal process has a similar function with the quenching. As we have mentioned, the quenching should be carefully performed to avoid any crystallization from liquid. Otherwise the SiC layer close to the crucible surface would act as a growth substrate for the SiC crystallization during the cooling down. Therefore the carbon solubility might be underestimated because the outside layers were usually removed before the combustion analysis [START_REF] Oden | Phase equilibria in the Al-Si-C system[END_REF]. In the recent work of Dalaker, the silicon melt was extracted by a quartz tube at different temperatures during one experimental run [START_REF] Dalaker | Time and Temperature Dependence of the Solubility of Carbon in Liquid Silicon Equilibrated with Silicon Carbide and Its Dependence on Boron Levels[END_REF]. The quartz tube with the sample inside was etched by HF and the samples were then analyzed by using the conventional combustion method. They claimed that there was no SiC precipitated on the quartz tube with a rapid quenching. And they studied the evolution of carbon solubility at the given temperature with time from 3 min to 158 min, showing no dependence of carbon concentration on the holding time. This indicates the equilibrium state has been established very fast (in just a few minutes), which conforms to our previous experiments. In order to get a statistically relevant solubility value, they measured a very large number of samples, in a very small temperature range. However, their results suffered from a huge dispersion.

In conclusion, our novel method can be regarded as a hybridization of the two conventional methods, in which the "weight loss" and "sample quenching" were combined.

The specific procedure will be thoroughly described in the following section.

Methodology 6.2.1. Sample elaboration

A modified sessile droplet method was implemented to determine the carbon solubility in pure liquid silicon in a wide temperature range of 1500 °C~2200 °C. The experimental apparatus and the procedure are almost the same with the sessile drop method described in chapter 2. The only different point is that the high purity argon gas (6N) was filled up to 900 mbar at 1100°C rather than at 1450 °C. The purpose of this modification is to introduce the inert gas at the interface between the SiC and Si wafer, which can impede the contact between the SiC surface and the Si liquid. The specific purpose of this will be further explained in the following sections. Note that the bubbles only existed at the periphery of the contacting area between solid Si and SiC (see Fig. 6.1), implying that they cannot influence the overall dissolving process.

In all the experiments for carbon solubility determination, the 4° off 4H-SiC (0001) Si face substrates were used as the carbon source because the Si face/liquid has been proved to be faster reaching the equilibrium during dissolution. Identical electronic-grade silicon cubes with sides of 2 mm were used as the source of silicon melts in each experiment. The deviation of the silicon cube weight is about 2% due to the width of the cutting saw.

According to our previous studies, the keeping time was set as 30 min for low temperatures and 10 min for high temperatures over 2000 °C. Since the tantalum sheet at the end of the removing tool reacted dramatically with liquid silicon at high temperatures, it was replaced by the graphite sheet when the temperature was over 1900 °C. It is worth mentioning again that all the samples were all surrounded by a 1 mm thick tantalum tube to shield the electromagnetic field, which has the same function as discussed in the previous chapters.

Sample characterization and parameters measurement

The fundamental step of this approach, as essentially similar to the weight loss method, is to correctly calculate the amount of SiC dissolved into the liquid. In our process, instead of removing the floating SiC like the work of Hall, we remove the liquid droplet from the SiC substrate at the end of the dissolution process. As the surface is "frozen" to room temperature, the loss of the SiC substrate could be obtained by characterizing the dissolved parts of the substrate surface. Two values are required to calculate the volume loss: the contact area (Sc ) and the dissolution depth (∆H). The former can be easily measured from the optical microscope, while for the latter two different routes are considered.

The first way to get the dissolution depth is by calculating the average height along the whole contact area. First the 3D topography images of the dissolved surface are measured by a mechanical profilometer, as shown in Fig. 6.2a. The integral area of the height profile with respect to the initial surface level is extracted from the height profile passing along the droplet diameter. Then the average dissolution depth (ΔHavg) is expressed as the integral area (Sc) divided by the droplet diameter (D): ΔHavg = Sc /D. Noticing that the edge of the droplet (around the triple line) is even higher than the non-contacting surface.

And this is usually only seen in one side of the droplet, which was believed resulting from a combination of the step bunching and ridge effect [START_REF] Kaplan | A review of wetting versus adsorption, complexions, and related phenomena: the rosetta stone of wetting[END_REF]. The edge effect will be discussed at the end of this chapter. Another method to assess the dissolution depth is from the intentionally introduced bubbles at the interface. The unwetted area under the bubbles is characterized by the presence of hillocks at the dissolved surface. The hillocks were analyzed by AFM and a typical picture is shown in Fig. 6.3a. The top of the hillock is covered by small steps while the surroundings are already significantly reconstructed into macrosteps and wide terraces.

As explained in section 2.2.2, it is suspected that the top of the hillock is almost not dissolved (or just weakly etched by the gases in the bubble). In this sense, the dissolution extent can be reflected by the height difference between the top and surrounding of the hillocks. In practice, two height profiles, crossing and not crossing the hillocks, are extracted with the same starting and ending positions (same x-axis values). Then the difference of the mean heights of the mesa and that of the surrounding area are approximately regarded as the dissolution depth ∆Hafm of the overall surface. For each sample surface, at least three hillocks from different areas are counted to reduce the random error. duration is also applicable for the slightly lower temperature, i.e. for 1500 °C. Considering that the reaction rate is normally exponentially proportional to the temperature, a holding time of 30 min should be long enough for investigating the carbon solubility over the lower temperature range (1500 °C ~1900 °C). For the higher temperature range (2000~2200 °C), the holding time was shortened (10 min) in order to limit the potential effect of Si evaporation. It is worth noting that we could not detect any weight loss of the Si droplet after holding the sample 10 min at 2200 °C. But this does not guarantee that there is no evaporation as the sample is so small that the weight loss could not be detected below the limit of the analytical balance (resolution of 0.1 mg). The SiC surfaces dissolved by liquid silicon from 1500 °C to 2200 °C are shown in Fig. 6.5. The silicon solvent has been removed by tantalum or graphite sheets at the end of the dissolution, meaning that the dissolved surface was like being "frozen" from high temperature to room temperature. The effect of the cooling down of the liquid was thus completely excluded. Though a shorter keeping time was applied at the ultra-high temperatures, the shrinking of the triple lines due to silicon evaporation is still visible at 2100 °C and 2200 °C. It is quite difficult to define the real contacting area in Fig. 6.5h

where multiple triple lines were overlapped. For this situation, only the final imprint of triple lines (smallest one) was utilized for the measurement of the contact area. Generally, the contact areas apparently expand with the temperature rising. The liquid/SiC interface area was plotted as a function of temperature as shown in Fig. 6.6. It seems that the contact area increasing rate is faster when the temperature is higher than 1900 °C. Since the same amount of silicon source is used, a larger contact area implies a larger work of adhesion as well as a smaller contact angle. This trend is consistent with the result in the literature [START_REF] Nikolopoulos | Wettability and interfacial energies in SiC-liquid metal systems[END_REF]. The amount of Si used is kept the same for each experiment.

In order to get the quantitative information about the dissolution depth, the dissolved surfaces were characterized by the profilometer, the topographies of which are present in Fig. 6.7. As a general rule, the dissolution process is enhanced with the temperature increase, which can be confirmed from the increasing contrast between the contact area and non-contact area. To better compare the extent of dissolution, the height profiles along the lines crossing the center of the prints were extracted and are compared in Fig 6 .8. We can see that the interfaces evolve slowly at low temperatures but become very significant after 2000 °C. Subsequently, the average dissolution depth at each temperature was calculated from the height profiles using the method described in Fig. 6.2. The results are listed in As aforementioned, we took benefit from the presence of bubbles trapped at the SiC/liquid interface, which cause many hillocks on the SiC surface. The hillocks were first observed under a high magnification DIC optical microscope; their morphologies are shown in Fig. 6.9. Note that the steps propagate (recede) in the direction from bottom to top in all the photos. Apparently, the hillocks act as pinning points of the surface steps, making them gradually accumulating at the bottom of the hillocks. Since no incoming steps arrive further on this side, a relatively larger terrace is generated at one side of the hillock.

For the same reason, a large triangle area is formed in front of the hillock, however it disappears (is covered by macrosteps) when the macrosteps from two sides meet with each other (Fig. 6.9d). Apparently, the pinning effect becomes more significant with the temperature rising because of the enhanced dissolution as well as the stronger step movement. Another feature is that the top surfaces of the hillocks appear very smooth only at low temperatures, while they become rough, covered with small steps at higher temperatures. This indicates that there exists also a gas/solid surface reconstruction in the bubble, especially at high temperatures. Note that when samples were processed at even higher temperature, i.e. from 1900°C to higher temperatures, no more hillock was found on the surfaces. One possibility is that the bubbles were expelled from the silicon droplet at high temperatures, another possibility is the hillocks were etched away by the gas species in the bubbles at high temperatures due to the stronger reaction between Si-rich gases and SiC substrate. The hillocks were further characterized by AFM to measure their height; the results are presented in Fig. 6.10. The AFM images show a very smooth top surface giving a RMS roughness of 2 nm (from a 2 μm×2 μm area at 1500 °C), proving again that the top of the hillock is only weakly reconstructed. The vicinity of the hillocks has been reconstructed into the typical structure composed of macrosteps and wide terraces. It is estimated that the effect of the silicon vapor is negligible at the lower temperature range, we can assume that the top of the hillock is only slightly dissolved and can be regarded as having the same height as the initial surface. Therefore, the dissolution depth, marked as ΔHafm, can be defined as the height difference between the top of the hillocks and its surrounding. The specific method has been described in Section 6.2.2. The calculation results ΔHafm together with the average height ΔHavg from Profilometer are listed in Table 6-2. As we

𝑛 = ∆ (6-1)
Where ρ is the density of the SiC material (ρ = 3.21 g•cm -1 ); Sc is the contact area between SiC and the liquid Si droplet; ∆𝐻 is the dissolution depth and M is the molar mass of SiC (M = 40.096 g•mol -1 ).

Assuming that all the lost SiC is completely dissolved into liquid silicon and that no precipitation happens during the operation, the carbon content in the liquid, i.e. the carbon solubility at a given temperature is equal to:

𝑋 = 𝑋 = (6-2)
𝑛 can be calculated from the weight of the Si piece assumed that there is no loss during the dissolution. The carbon solubility is calculated by using both the average dissolution depths ΔHavg from profilometer and local dissolving depth ΔHafm from AFM; the obtained solubility results are listed in Table 6-3. As we can see, the values obtained using local height are always higher than those using average height. This might be related to the inhomogeneous shape of the contacting interface. Due to the concave shape of the dissolution crater, the centre area where the Si cube was placed (and also where the bubbles were located) is generally lower than other contacting areas. Therefore, the AFM local measurement tends to overestimate the dissolution depth, as compared to that from the integral value. Another point we should notice is that the results from AFM are closer and closer to the values from Profilometer, indicating the interface became more homogenous with the temperature increasing. This could be verified from Figs. 6.5 and 6.7, in which the imprints of the silicon squares are less distinct at the higher temperatures.

In order to compare with previous works in the literature, our results are plotted together with some of the previous carbon solubility data in Fig. 6.11. All the previous data are fitted with the Arrhenius function and are extrapolated into the temperature range from 1414 °C to 2200 °C. Generally speaking, our solubility results are close to the data of Hall and Yanaba at low temperature range. At high temperature range, it is found they are in good accordance with the result from Durand. This is unexpected and quite surprising because there was no experimental work in Durand's paper and the data points they used for analysis were very limited. On the other hand, our results are much larger than the data of Scace and Slack, the latter has been widely adopted in the Si-C binary phase diagram to determine the liquidus due to its wide temperature range [START_REF] Camassel | SiC materials: a semiconductor family for the next century[END_REF][START_REF] Hall | Electrical contacts to silicon carbide[END_REF]. The fitting line is shown in Fig. 6.11 as the green line. By extrapolating to the melting point of silicon we can obtain the carbon solubility at T=Tm=1683K is equal to 0.0102 mol %, which is in a reasonable range [START_REF] Kleykamp | The Constitution of the Silicon-Carbon System[END_REF][START_REF] Suherlan | Temperature determination of the Si-SiC eutectic fixed point using thermocouples[END_REF]. Next the dissolution energy of carbon is calculated by using our results of carbon solubility.

In the silicon solvent, the equilibrium reaction between liquid silicon and SiC can be expressed as:

𝑆𝑖 ( ) + 𝐶 ( ) = 𝑆𝑖𝐶 ( ) (6-4)
The standard change of the Gibbs energy of chemical reactions is

∆𝐺 = -𝑅𝑇𝑙𝑛𝐾 (6-5)
Where R is the gas constant, K1 is the equilibrium constant of the reaction. Here: The reaction to form SiC from Si melt and solid carbon is expressed as:

𝐾 = = = (
𝑆𝑖 ( ) + 𝐶 ( ) = 𝑆𝑖𝐶 ( ) (6-8)
We have known the standard Gibbs free energy change of this reaction [START_REF] Kubaschewski | Materials Thermochemistry[END_REF]: The dissolution energy at the melting point calculated from above last expression is 80.9 KJ/mol, which is however larger than 20.4 KJ/mol and 23.2 KJ/mol obtained in the works of Yanaba et al [START_REF] Yanaba | Solubility of carbon in liquid silicon equilibrated with silicon carbide[END_REF] and Dalaker et al [START_REF] Dalaker | Time and Temperature Dependence of the Solubility of Carbon in Liquid Silicon Equilibrated with Silicon Carbide and Its Dependence on Boron Levels[END_REF] respectively.

∆𝐺 = -113000 -11.4𝑇𝑙𝑜𝑔𝑇 + 75.7𝑇 (

Evaluation of our approach

In general, a good approach for determining the solubility by weight loss method should meet the following criteria: g) Precise and accurate measurement method.

As for our method, most of them are satisfied: our experiments were carried in a wellcontrolled furnace; the temperature difference is less than 0.1 °C along the solid/liquid interface, as shown in chapter 2. The selected holding times in our work were adapted as the surface morphologies on Si face did not evolve anymore after 10 min (chapter 3).

Dalaker et al. also reported that no dependence of the carbon levels on the settlement time (ranging from 3 to 158 min) could be detected [START_REF] Dalaker | Time and Temperature Dependence of the Solubility of Carbon in Liquid Silicon Equilibrated with Silicon Carbide and Its Dependence on Boron Levels[END_REF]. Therefore the equilibrium should be established rapidly (in a range of a few minutes). In the most common case, the dissolution rate should increase with increasing temperature. Then, at higher temperature, the required holding time should be shorter for reaching the equilibrium state. That is why the samples were held only 10 min for the high temperature range.

One of the advantages of our method is that no cooling down period is involved because the liquid silicon was removed from the surface at the end of the dissolution. Thus theoretically there should not be any precipitation which occurs during cooling down.

However, some irregular structures were observed on the SiC surfaces dissolved at temperatures over 2000 °C. As we can see from the DIC images in Fig 6 .12, some protrusions were formed at the edges of the macrosteps, on top of which sometimes there existed small tiny particles. Note the macrosteps receded in direction from top to bottom while it seems the protrusions expanded in an exactly opposite direction. When the temperature was increased to 2200 °C, the protrusion was not seen any more but the steps edges became much wider (Fig. 6.12c). To exclude the possibility that the structures were from impurities, the surface was further characterized by SEM and the element composition of the protrusions and the other areas were analyzed by EDS. As we can tell from the results in Fig. 6.12e, both areas only comprise silicon and carbon elements. Even if EDS is only semi-quantitative for light elements, their ratios are close to that of SiC (1:1).

Therefore, it is considered that this protrusions were nucleated from the liquid before the removal. A possible interpretation for this phenomenon is due to the silicon evaporation in the high temperature range. The evaporation of silicon broke the equilibrium state and gave rise to the supersaturation in the liquid. As a result, the SiC particles were deposited on the SiC/liquid interface. It should be emphasized here that these protrusions have the different formation mechanism as the aforementioned hillocks (in Fig. 6.9) though they look similar. The major difference can be found at the front (end) of the protrusions (hillocks): there is a significant accumulation of steps at the end of the hillocks while it is not the case for the protrusions here. The SiC precipitation stage associated to the silicon evaporation slightly reduces the measured dissolution thickness, and then give the underestimated carbon solubility values.

However, we think that this impact is negligibly small at low temperatures and still weak at high temperatures. Only when the protrusions are significantly developed and cover the whole surface can its influence be detected. That is the possible reason why the deviation of the data from the fitting line can only be seen at 2200 °C. However, another possible reason for the deviation is related to the shrinking contacting area, which in fact also results from the Si evaporation.

Another issue in our approach is the inhomogeneous SiC/liquid interface. The variation of dissolution depth from center to the edge of surface crater made the data extracted from the local hillocks unreliable. Several factors can contribute to this phenomenon. First the temperature contribution can be excluded because the variation of temperature is less than 0.1 °C (at 1600 °C) between the center and the periphery. Besides, even the temperature is slightly higher at the edge, meaning a stronger dissolution should occur. This is contrary to our observation where the central part was higher (dissolved more) than the periphery. The mass transport driven by Marangoni convection and electromagnetic convection can also be excluded because of the small temperature gradient and the surrounding magnetic shield. Last, one might consider the influence of the ridge effect around the triple line where the interfacial reconstruction can be enhanced [START_REF] Chatain | Spreading of metallic drops[END_REF]. The configuration of ridges could be altered by the interface diffusion depending on whether it is receding or advancing liquid fronts. The receding liquid will leave a concave-shaped ridge while the advancing drop can result in a sharp peak [START_REF] Saiz | Ridging effects on wetting and spreading of liquids on solids[END_REF]. This is consistent with our results where two different shapes formed at two sides of each surface height profile.

However, we think the ridge effect comes from the local force balance, thus its effect should only influence limited space. The edge of the contacting area and its width are shown in Fig. 6.13. The morphology inside the rings at the periphery is of great difference with the inner part of the droplet. In the temperature range 1700~2200 °C, edge rings have an average width of < 500 μm and seems independent of temperature. We speculated the ridge effect might not just function on the periphery surface directly, it could probably cause (or change) the vortex flow in the droplet and then contribute to the mass transport in the central part. In conclusion, the exact mechanism for the inhomogeneous interface still need to be studied.

Conclusion

In this chapter, a new experimental approach based on the sessile drop method was adopted to measure the carbon solubility in liquid silicon over a large temperature range from 1500 °C to 2200 °C. Two dissolution depth determination methods were used to calculate the weight loss of SiC substrate. In general, our results were in a reasonable range of the reported data, which help further clarify the carbon solubility discrepancy in the literature.

The experimental and analytical conditions could be further improved to reduce the errors.

For example, a more homogenous interface or direct volume integral from 3D height maps could provide more reliable data. Increasing the argon pressure could reduce the evaporation of Si at high temperatures. Anyway, much more accurate data could be obtained in an optimized procedure. As no crucible is needed, this approach would be applicable to solubility determination in various other chemical systems.

General conclusion and perspectives

Solution growth is a promising method for growing ultimate quality SiC crystals necessary for the development of reliable high power electronic devices. One critical issue for solution growth is the morphological instabilities at the growth front, where complex phenomena such as step bunching, micro-faceting and solvent trapping frequently occur during the growth process. To further develop the SiC solution growth, a comprehensive understanding and perfect control of the high temperature interface between the SiC crystal and the liquid phase is critical. Yet, their mechanisms are still unclear because it is difficult to investigate the interface in high temperature conditions where multiple physicalchemical processes are involved and strongly coupled. This thesis work aims to provide more information on the morphological evolution at the SiC/liquid interface by conducting SiC surface reconstructions under various conditions at and out of thermodynamic equilibrium.

As the core method in this thesis, a sessile drop approach was firstly designed combing experiments and numerical simulation, with the main target of ensuring a well-controlled environment to the SiC/liquid interface at high temperature, i.e. in the 1400-2200 °C temperature range. An isothermal environment was realized to get rid of any contribution from temperature gradients (Marangoni convection, diffusion, etc.). A tool was carefully designed to remove the liquid at high temperature, in order to "freeze" the surface at the end of the interfacial reconstruction. The electromagnetic field, used for induction heating, was then explored as an actuator for the carbon transport, allowing to shift from transient surface dissolution (reaching equilibrium) to stationary dissolution (keeping the deviation from equilibrium) of the SiC crystal. In addition, the temperature at which the reactor was filled with argon was found to be a critical factor in avoiding bubbles forming at the interface. Eventually, a uniform high temperature SiC/liquid interface was successfully preserved for further ex situ observation by removing the liquid from the SiC surface at the end of the reconstruction.

The equilibrium reconstructions of vicinal SiC {0001} surfaces in liquid silicon were conducted under various conditions. The effects of external parameters (time, temperature, temperature ramp rate) and of the intrinsic properties (polarity and crystalline orientation) were systematically investigated. Generally, the equilibrium SiC/liquid silicon interface was reached very fast on the Si face while relatively slowly on the C face. Under identical conditions, the macrosteps formed on the Si face appeared more meandering than those on the C face. Higher reconstruction temperature led to straighter macrostep edges, which was more significantly found on the Si face than on the C face. The surface reconstruction was also strongly influenced by the temperature ramp rate as well as the presence of silicon vapor before the melting of silicon. A sluggish temperature ramp rate gave rise to a much weaker reconstruction while a fast rate could only slightly accelerate the structure evolution.

The silicon vapor was also able to change the surface step structures in lower temperature range (< Tm) and then influenced the later reconstruction with liquid after Si melting.

The influence of crystalline orientations on the SiC surface reconstruction was comprehensively explored by utilizing concave-shaped SiC {0001} surfaces. The reconstructed surfaces revealed a six-fold symmetric structure on the Si face while only a concentric and isotropic structure was observed on the C face. The extent of step bunching was more pronounced on the C face and enhanced with the increasing off angle while it was almost invariant on the Si face. On the Si face, the step bunching became gradually weaker from the <1 1 00> direction to <11 2 0> direction, which could be explained by the variation of the kink density along the azimuthal direction. The special edge of macrostep formed on the C face indicated a specific step bunching feature on this polar surface. In addition, the faceting was independent of the off angle on both polarities of SiC {0001} surfaces.

From the surface reconstruction in equilibrium, the polarity seemed to be the most critical parameter that generated distinct structures at the SiC/liquid interface. The slower surface reconstruction process and straighter macrosteps on the C face indicated a larger step stiffness. A higher surface free energy gave rise to more pronounced faceting on the Si face, while a higher ES barrier led to stronger step bunching on the C face. The increasing straightness of the steps with the increasing temperature implied that the ES barrier on the SiC surface might be reversely proportional to the temperature.

Out of equilibrium reconstruction experiments were carried out to further investigate the macrostep movements during the SiC growth process. The growth was performed directly after the initial equilibrium reconstruction. We could evidence that the macrosteps formed on the Si face were stable and did not decompose while those formed on the C face were unstable and prone to de-bunch. Differently said, total reversibility of bunching-de-bunching of the macrosteps occurred on the C face while the transition to a faceted surface made the morphological evolution of the Si face irreversible.

Sandwich-type reconstruction experiments were also explored in this thesis in order to compare the relative activity of two different surfaces. When two different polarities were compared with this approach, the observed morphologies suggested that growth occurred on the C face. This implied a mass transport from the Si face to the C face, which was driven by the difference in surface energetics.

Finally, to clarify the wide discrepancy in the literature, the carbon solubility in liquid silicon over a wide temperature range (1500 °C ~2200 °C) was re-evaluated by using the modified sessile drop method. The results were compared with previous data in literature and all the possible sources of errors were analyzed.

As research work is never done, we want to draw out a few perspectives, as some questions are still waiting to be addressed.

1) Further characterization of the step structures

The bunched macrosteps on the two polarities always showed different features. The AFM images revealed that the macrosteps edges on the Si face were faceted into high index crystal planes while those on the C face were not faceted. Due to the limited lateral resolution of the AFM, the detailed structures of macrosteps edges could not be distinguished. Therefore, further characterization of macrosteps by using high resolution transmission electron microscopy (HRTEM) is required to better compare the step edge structures at the atomic level.

2) Real-time observation of the surface reconstruction

The surface reconstruction at high temperature is quite fast (almost finish in a few minutes) especially for the Si face. Therefore, it is challenging to study the step evolution with time by using conventional routes. One possible way to study this is using real-time microscopy.

Our collaborators (IIS, The University of Tokyo) have developed a unique optical high temperature microscope, which has realized the observation of the high temperature SiC/liquid interface [START_REF] Kawanishi | Real-time observation of the interface between SiC and a liquid alloy and its application to the dissolution behavior of SiC at 1573 K[END_REF]. Besides simple optical microscopy observations, the system is also fitted with a laser interferometer so as to measure the height profile of the interface.

The use of such a system could help provide both qualitative and quantitative information on interface kinetics.

Figure 1 .

 1 Figure 1.1 (a) the tetrahedral structure of Si and C atoms in SiC and (b) the occupation sites (A, B, and C) in the hexagonal close-packed system.

Figure 1 . 2

 12 Figure 1.2 The stacking sequences of three main polytypes: 3C, 4H and 6H. The Si atoms and C atoms are in blue and brown colors, respectively. Note that the structures are projected along <110> direction (for 3C-SiC) or along <112 0> directions (for 4H-and 6H-SiC)

Figure 1 . 3

 13 Figure 1.3 Two kind of step terminations for steps perpendicular to [11 00] direction. SN indicates steps with one dangling bond and SD with two dangling bonds. Plan viewed from [0001] direction.

Figure 1 . 4

 14 Figure 1.4 The structure of 4H-SiC crystal projected in the [112 0] direction. Two polar faces: Si face (top side) and C face (bottom side) are shown with different terminating atoms.

Figure 1 . 6

 16 Figure 1.6 Schematic description of the sublimation growth process. This sublimation reactor concept is the current "state-of-the-art" industrial growth technique for SiC bulk single crystals;

Figure 1 . 8

 18 Figure 1.8 Schematic drawing of contact angles and interfacial energies. a) good wetting when θ < 90° and b) bad wetting θ > 90°.

Figure 1 . 9

 19 Figure 1.9 Schematic illustrations of a) step configurations on a 6H-SiC (0001) face and b) the step motion during H2 etching on surfaces tilted toward <11 00> and <112 0> [69,70].

Nakajima

  et al. summarized the anisotropic step bunching on vicinal 6H-SiC(0001) surface induced by H2 etching[START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF], as shown in Fig 1.10a. There are three types of step bunching which can appear on the surface: (1) straight six-bilayer steps, (2) straight six-bilayer steps and nanofacets, and (3) zigzag three-bilayer steps[START_REF] Nakajima | Step control of vicinal 6H-SiC(0001) surface by H2 etching[END_REF]. They proposed that the crystal orientation dependence of the step height comes from the different etching rates of two kinds of 3-bilayer steps (ACB steps and ABC steps). When the miscut direction is <11 00>, there exist two kinds of step edges (denoted as α and β step edges), which have different dangling bonds and thus lead to different etching rates. The faster 3-bilayer steps among them could catch the slower ones, forming higher steps of six bilayers. While on the vicinal surface towards <112 0>, the two steps of three bilayers have the similar number of dangling bonds (50%α+50%β), thus resulting in same etching rate. This explains why three bilayer steps are retained on the surface off towards <112 0>.

Figure 1 .

 1 Figure 1.10 (a) Diagram of step structure of H2 etched 6H-SiC (0001) surface. (b) The transformation of the step edge configuration on three-bilayer steps [69].

Figure 1 .

 1 Figure 1.11 Surface free energy curve as a function of vicinal angle. The dashed line indicates the surface energy will increase monotonically with the vicinal angle if there is no step bunching [67].

Figure 1 .

 1 Figure 1.12 Dissolution morphologies of on-axis SiC {0001} faces with pure Si melt. The first row shows 4H-SiC surfaces and the second raw 6H-SiC. For each raw: (a)(d) Si face 1540 °C; (b)(e) Si face 1840 °C and (c)(f) C face 1740 °C [11].

  directly through the SiC substrate, thanks to the wide bandgap of SiC and almost total reflection of liquid alloys. The observed interface morphologies between 4H-SiC (0001) Si face and Fe-Si alloy are shown in Fig.1.13. At the initial stage of dissolution, lots of hexagonal pits are observed, which originate from the threading screw dislocations (TSDs) or threading edge dislocations (TEDs) existing in the SiC substrate[START_REF] Kawanishi | Real-time observation of the interface between SiC and a liquid alloy and its application to the dissolution behavior of SiC at 1573 K[END_REF]. After holding some time, the dissolution continues in the lateral direction and the interface becomes smoother. The interface also becomes smoother with increasing temperature, as a result of further preferential dissolution in the lateral direction. At the same time, the dissolution thickness of SiC can be measured from interference patterns (Fig.1.13e). As we can see, the dissolution in Fe-Si alloy proceeds very quickly (almost finished after only 20 s), regardless of the initial SiC contents in the solvent[START_REF] Kawanishi | Real-time observation of the interface between SiC and a liquid alloy and its application to the dissolution behavior of SiC at 1573 K[END_REF].

Figure 1 .

 1 Figure 1.13 Morphology of the interface between 4H-SiC (0001) Si face and Fe-Si alloy: (a) just after melting at 1573 K (b) after 15min at 1573 K, (c)(d) initial stage at 1673 K and 1773 K and (e) dissolution thickness of SiC with time from interference patterns [74].

Figure 1 .

 1 Figure 1.14 Nomarski reflection optical micrographs of the 6H-SiC growth facet. (a) (b) Si face growth facet. Smooth growth facet and concentric circular giant steps; (c) C face growth facet. A clear hexagonal symmetry around the c-axis with ridges in the six equivalent <112 0> directions was detected [79].

Figure 1 .

 1 Figure 1.15 DIC images of surface morphology taken from the grown crystal using (a) Si solvent on the Si face, (b) Si solvent on the C face, (c) Si-Al 5 mol% solvent on the Si face, and (d) Si-Al

Figure 1 .

 1 Figure 1.16 AFM images of the terrace area between the macrosteps on crystals grown with (a) Si0.5Cr0.5 and (b) Si0.46Cr0.5Al0.04 solvents [60].

Figure 1 .

 1 Figure 1.17 Cross-sectional optical micrograph of LPE layers on off-axis 4H-SiC substrate: (a) irregular layer grown under large temperature gradient and (b) homogeneous layer grown under small temperature gradient. The top surface morphology under a large temperature gradient is shown on the right [82].

Figure 1 .

 1 Figure 1.18 The cross section of a monatomic step in a surface and the hypothetical potential associated with the diffusion of an atom over such a surface step.

Figure 1 .

 1 Figure 1.19 The terrace-step-kink (TSK) model of a thin film surface and illustration of all kind of Ehrlich-Schwoebel (ES) barriers [92]. Atoms travelling over steps must overcome the 2D ES barrier. Atoms moving along a step must overcome the 1D ES barrier, across facets the 3D ES barrier.

1 )

 1 Asymmetric step kinetics Based on the ES barrier, Schwoebel et al. proposed a model to study the step motion during growth. They assumed that the capture probability of absorbed atoms around a step is directionally dependent [95]. Due to the different potential energies, atoms adsorbed in the regions to the left (a) or right (b) of the step will have different probabilities of moving into position c at the step (see Fig. 1.18). The probability of an adsorbed atom at position a moving into the step is called γ-and the probability for an atom adsorbed at position b moving into the step is designated γ+. According to their model, when γ+>γ-, initial steps with arbitrary distribution tend toward a sequence of uniformly spaced steps.

Figure 1 .

 1 Figure 1.20 Two kinds of possible terraces on 4H-SiC {0001} surfaces. Open and closed circles represent Si atoms and C atoms, respectively [9].

Figure 1 .

 1 Figure 1.21 Side view of the SN and SD steps and two different routes of adatoms diffusing over the steps.

  of the SD steps. The fast growth or etching of SD leads to the development of triangular protrusions, the edges of which are aligned along the <112 0> direction and correspond to energetically stable SN steps (right column of Fig. 1. 20). Then a preferential etching occurs at the apex of these triangles, eventually leading to a train of full unit-cell-height steps.

Figure 1 .

 1 Figure 1.22 Illustration of evolution process for steps perpendicular to the [112 0] direction (left) and [11 00] direction (right) [105].

Fig. 1 . 23 ,

 123 where two different cases were considered. The original step is in form of the small amplitude sinusoid. The isoconcentration lines are shown on the terrace, the number on them indicating the increasing adatom concentration when moving away from the step edge (Fig.1.23a). As we can notice, on the lower terrace the isoconcentation lines are packed and closed to the step at the convex parts, while they are sparse and separated away at the concave parts of the step. Therefore, compared to the straight step, the contribution of atoms from the lower terrace to the convex part (A) is enhanced whereas the contribution to the concave part (B) is reduced. The atom contribution from the upper terrace are exactly opposite, which is enhanced at the concave part (B) and reduced at the concave part (A). Thus overall there is no change of the total adatom supplement and thus the moving rate is equal everywhere, meaning no change of the shape of the step during the propagation. However, in most cases, the flux of the adatoms from the upper terrace is weaker than the lower terrace (Fig.1. 23b). One of the reason is due to the existence of the energy barrier at the edge (such as ES barrier). Now the instability from the lower terrace cannot be compensated by the upper terrace. As a consequence, the amplitude of the step will increase gradually and lead to an extreme instable morphology.

Figure 1 .

 1 Figure 1.23 Top view of vicinal surfaces in the vicinity of a perturbed step edge. The dashed lines are isoconcentration lines around the step (1) for the case of equal adatoms flux toward the step from both bounding terraces and (2) for the case when the supply of atoms from the upper terrace is greatly reduced. Note that the rate of advance of different parts of the step is the same in the equal flux case but differs in the unequal flux case [111].

  surface physical-chemistry properties can be revealed from the systematically and strictly controlled surface reconstruction. On the other hand, this work may shed some light on the origin of interface instability and the possible strategies of steering the step movement during the SiC solution growth. This thesis spreads out as follows: first of all, development and optimization of the standard sessile drop method are carried out based on experiments together with numerical simulations, the aim is to create an equilibrium surface reconstruction environment without external disturbances. Using the optimized procedure, a series of reconstruction experiments are performed on different vicinal SiC surfaces under various conditions. Besides the standard commercial vicinal surfaces, specially designed surfaces such as concave-shaped surfaces are also investigated. The careful characterizations of surface morphologies and step structures are carried out ensuring comprehensive descriptions of the surface structures evolution during surface reconstruction. In addition, out of equilibrium SiC/liquid interface reconstructions are also studied, which provide new perspectives to understand the step movement and mass transport during the solution growth process. Last but not least, the carbon solubility in pure Si solvent is revisited based on our sessile drop method where two kinds of information are used for the calculations. The advantages and disadvantages of our measurement approach are discussed by comparing with the previous works in literature.

Figure 2 . 1

 21 Figure 2.1 Photo of the set-up used for surface reconstruction experiments. It is mainly composed of three parts: (1) the heating unit, (2) the reactor, and (3) the controller.

Figure 2 . 2

 22 Figure 2.2 The configuration of the top flange of the setup.

Figure 2 . 3

 23 Figure 2.3 Schematic drawing of the experimental apparatus used for the interface reconstruction experiments. The heating zone (external surfaces of the graphite insulation) is 300 mm high with a diameter of 130 mm. Red arrows indicate the locations of the temperature measurement.

Figure 2 . 4

 24 Figure 2.4 Two-dimensional axisymmetric model of the set-up in COMSOL Multyphysics software.

Figure 2 . 5

 25 Figure 2.5 The computed temperature distribution. a) in the whole crucible; b) focus on the temperature distribution in the droplet. Temperature isothermal line step: 0.01 °C.

Figure 2 . 6

 26 Figure 2.6 The schematic illustration of the sessile drop method used for surface reconstruction investigation.

Figure 2 . 7

 27 Figure 2.7 The typical temperature and pressure profiles of the sessile drop method used for SiC surface reconstruction. Here the surface is reconstructed at 1600 °C for 30 min.

Fig. 2 . 8

 28 Fig.2.[START_REF] Pechman | Interactions of Br with Si(111)-7×7: Chemisorption, step retreat, and terrace etching[END_REF] As we can see, almost all the materials could "suck" the liquid Si droplet except the Zr sheet (not shown) because it reacted dramatically with the liquid. The graphite sponge and the graphite sheet seem to be very attractive candidates since they do not bring any foreign element into Si-C solvent. However, they were not practical because both of them are very fragile, and were proved can be easily broken during the operation. The molybdenum sheet and tungsten sheet were also partly dissolved maybe because the sheets used were too thin. As for niobium and tantalum sheets, both of them have no obvious loss but the surface dewetted by the niobium sheet is not as specular as that by the tantalum sheet. In order to have a clearer comparison of the possible tool materials, their applying effects of the tested materials were assessed from four criteria, which are summarized in the Table 2-1. Through an overall consideration, we eventually selected tantalum (Ta) sheet as the removing tool which offered the best compromise in terms of liquid removal rate and integrity of the SiC surface after the sucking process.

Figure 2 . 8

 28 Figure 2.8 The enlarged end of the liquid removing tool and the SiC surfaces after being dewetted with different sucking materials.

Fig 2 . 9

 29 shows two different temperature/argon injection profiles, the first (a) with Ar filling at 1100 °C and the second one (b) with Ar filling at 1450 °C, i.e. respectively before and after the melting of Si. The DIC images of the substrate surface obtained in these experiments are shown as the insets to the graphs. Apparently the temperature at which Ar was injected in the reactor had a very strong effect on the interface morphology, as will be discussed hereafter.

Figure 2 . 9

 29 Figure 2.9 Two different temperature profiles for 4° off 4H-SiC Si-face surfaces reconstructed at 1600 °C for 30 minutes. The main difference is the temperature at which the reactor is filled with Ar: a) at 1100 °C, b) at 1450 °C (just after Si melting). The insets are the corresponding surface morphologies observed using DIC optical microscopy.

Figure 2 .

 2 Figure 2.10 AFM image of the 0.2° off 4H-SiC surface after CMP and the height profile along the line 1. The average step height is about 0.25 nm.

  2.9b). An AFM picture of one of the circular prints is shown in Fig.2.11. The height profile along the white line is plotted and placed below the image.The circular hillock is characterized by a flat top surface, with an appearance similar to the pristine surface. All around, the surface has evolved significantly, generating a train of steps with width in the micrometer range. The hillock thus appears as a discontinuity of the interface morphological evolution. In addition, the height difference between the top of the hillock and the average altitude of the surrounding was measured at about 220 nm.

Figure 2 .

 2 Figure 2.11 (a) AFM picture showing the hillock formed by a bubble trapped at the liquid/solid interface upon melting. (b) Height profile along the hillock. This sample corresponds to a vicinal 4H-SiC Si-face treated at 1600 °C for 30 minutes.

Figure 2 .

 2 Figure 2.12 Calculated thickness of SiC dissolved as a function of temperature, for a droplet diameter of 4.0 mm at equilibrium.

Fig 2 . 9

 29 and involved the entrapment of argon at the interface during melting. Mitani et al. demonstrated that

Figure 2 .

 2 Figure 2.13 Schematic illustration showing the position of hillocks formed at the interface. The hillocks can only be found where the solid silicon piece was originally located before melting.

σ

  Ta (T)=2.3923•10 7 •e T 116 +7.1739•10 6 •e T 338 +2.6234•10 6 •e T 1572 +4.9657•10 5 (2-3) σ C (T)=-6.519•10 -9 •T 4 +6.091•10 5 •T 3 -1.915•10 -1 •T 2 +2.242•10 2 •T+3.247•10 4 (2-4)

Figure 2 .

 2 Figure 2.14 Penetration depth as a function of temperature in graphite and tantalum calculated from equation (2-2). For graphite, two different frequencies (22 kHz and 280 kHz) were considered for the calculation.

Figure 2 .

 2 Figure 2.15 computed volumetric loss density into and around the sample (a) without and (b) with a Ta tube (shield). Representations are given for half of the inner part of the crucible as it is considered as axisymmetric.

Figure 2 .

 2 Figure 2.16 3D topography of the whole liquid/solid interaction area, mapped with a mechanical profilometer. These two samples were identical and were treated using the same conditions (1600 °C and 30 min) except (a) without Ta shield and (b) with a Ta shield. (c) The plotting of the measured surface heights along the white lines in two cases.

Figure 2 .

 2 Figure 2.17 Mass transport as computed in the droplet. The white arrows correspond to magnitude of the fluid velocity. Left side illustrates the result without EM shielding and right side with an EM shielding (Ta tube); Note here that only a half droplet is presented. The setting boundary condition of the droplet is shown as an inset.

Figure 2 .

 2 Figure 2.18 Mass transport and the related distribution of supersaturation, as computed in the droplet. The color scale corresponds to the relative carbon supersaturation (color) and the black arrows to magnitude of the fluid velocity. (a) Without EM shielding and (b) with an EM shielding (Tantalum tube); Note here that only a half droplet is presented. (c) Comparison of the profiles of the relative supersaturation along the radius of the droplet, near the SiC/liquid interface.

  photo is transformed into a grey level image with the value range of 0-255. Then the grey level profiles along the lines (approximately perpendicular to the step edges) are extracted.In this example, the step edges are brighter under C-DIC and thus generate the peaks in the profile. All the peaks are detected and spatially indexed. Then the step spacing can be simply calculated by differencing the two adjacent peak centers. Last the distribution of step spacing is obtained using a typical bin size of 0.5 μm. To get a statistically relevant data sample, about 500 step spacing values from several lines in different representative areas were extracted and analyzed for each sample.

Figure 2 .

 2 Figure 2.19 The basic procedure of analysing the step spacing in OriginPro software

  called fractal analysis in mathematics. The most commonly used functions to define the correlation length are autocorrelation function (ACF), height-height correlation function (HHCF) and power spectral density function (PSDF). The former two functions, ACF and HHCF, are similar and the only difference is that HHCF use the power of the distance between points instead of just the distance used in ACF. Actually the shape of the HHCF is simply an inversion and vertical translation of the autocorrelation function. The PSDF is the Fourier transform of the ACF, which is more useful for the mounded surface. Both the HHCF and PSDF are adopted in this thesis, their features and applications will be introduced separately in the following.

Figure 2 .

 2 Figure 2.20 Two surfaces with the same RMS roughness value but give different feelings about surface roughness. The surface B is pressed from surface A in lateral direction so they have the same amount of deviation from the mean plane (yellow dashed line).

Fig. 2 . 21 .

 221 Fig.2.21. As we can see, initially the H(r) increases rapidly with increasing r following a power law dependence. It then converts gradually into a plateau feature. The starting point of the plateau represents the correlation length (ξ), meaning two points that have lateral distance more than ξ are no longer correlated or badly correlated. When the r is small the profile is almost linear, the slope of this part gives roughness exponent representing the high frequency roughness or local morphology of structures. Lastly, the saturated plateau part could give us the RMS roughness value. For some surfaces, there exists an oscillatory in the plateau part (Fig.2.21b), this feature usually indicates a mound-like morphology.

Figure 2 .

 2 Figure 2.21 Representative height-height correlation function obtained for (a) a self-affine surface and (b) a mounded surface. The plots are on a log-log scale, which give the height-height correlation function a linear behavior for small r with slope 2α [127].

2. 22 .

 22 They show two distinct parts in the profiles: (i) a high-frequency regime with a frequency-dependent decaying branch and (ii) a low spatial frequency regime. For the mounded surface, there is a characteristic frequency peak in the low frequency part. The roughness scaling exponent α can be obtained by fitting the high-frequency part with the power law function. The peak that emerged on the PSDF of mounded surface can be used to measure the periodicity of the mounds (λ)𝜆 = 2𝜋𝑘(2-11)In addition, the FWHM of the characteristic frequency peak can provide the lateral correlation length ξ (FWHM≈ξ -1 ).

Figure 2 .

 2 Figure 2.22 Representative power spectral density function (PSD) for (a) a self-affine surface and (b) a mounded surface. They exhibit no characteristic peak in the profile of self-affine surface; for mounded surface, the peak is located at κ = κm, and the full width at half maximum (FWHM) is inversely proportional to the lateral correlation length ξ [127].

3 . 1 .

 31 have been excluded. The vicinal surface provides an initial homogeneous surface step structure allowing the accurate characterization and comparison for the reconstructed morphologies. In this chapter, the results of surface reconstruction by using the standard sessile drop method are demonstrated, in which the equilibrium between vicinal SiC surface and pure liquid silicon were established at high temperatures. The first section presents the influence of the external parameters including temperature, holding time, temperature ramping rate and silicon vapor as well. The second section shows the effects of the intrinsic properties of the SiC material including surface polarity and crystalline orientation (only off angle is referred to in this chapter). In each part, the specific phenomena are thoroughly discussed and reasonable interpretations are proposed. In particular, the possible relationships between the influencing factors and the experimental results are partially established in this chapter. Effect of external parameters 3.1.1. Time

Fig. 3 . 1

 31 Fig. 3.1 shows the 4° off 4H-SiC Si face substrates reconstructed under liquid silicon at 1700 °C for different times with an interval of 10 min. The step-and-terrace morphology is present in all the pictures, where the macrosteps have typical step spacing of 1-2 μm.

Figure 3 . 1

 31 Figure 3.1 DIC images of the reconstructed 4° off 4H-SiC Si face at 1700 °C for various holding times: (a) 0 min; (b) 10 min; (c) 20 min and (d) 30 min. The distribution histograms of the step spacing in each image are shown in (e).

Figure 3 . 2

 32 Figure 3.2 AFM images of the reconstructed 4° off 4H-SiC Si-face at 1700 °C held for different times. (a) 0 min (b) 10 min and (c) 20 min. The height profiles at the right hand side correspond to the white lines in the AFM images. Two white arrows in (a) indicate two small steps of 8 nm on the terrace area, which are not seen in (b) and (c).

Figure 3 . 3

 33 Figure 3.3 AFM images of the reconstructed 4° off 4H-SiC Si-face at 1700 °C held for 4 hours.

Figure 3 . 4

 34 Figure 3.4 DIC images of the reconstructed 4° off 4H-SiC C face at 1700 °C for various holding times: (a) 0 min; (b) 10 min; (c) 20 min and (d) 30min. The distributions of the step spacing in each image are shown as histograms in (e). Note that the x-axis ranges are 6 µm for the Si face and 20 µm for the C face.

Figure 3 . 5

 35 Figure 3.5 AFM amplitude images of the 4° off 4H-SiC C face reconstructed at 1700 °C for different holding times: (a) 0 min, (b) 10 min, (c) 20 min, and (d) 30 min.

Figure 3 . 6

 36 Figure 3.6 The calculated HHCF profiles (dotted lines) and their fittings (solid lines) of the reconstructed 4° off 4H-SiC C face at 1700 °C for different holding times.

Figure 3 . 7

 37 Figure 3.7 The time dependence of surface roughness (ω) and the correlation length (ξ) obtained from the fitting of the HHCF profiles.

Figure 3 . 8

 38 Figure 3.8 AFM images of the reconstructed 4° off 4H-SiC C face at 1700 °C held for different times: (a) 0 min (b) 10 min and (c) 20 min. The height profiles at the right correspond to the white lines in the left images.

Figure 3 . 9

 39 Figure 3.9 AFM amplitude images that show the small steps remaining on the terrace of C face reconstructed at 1700 °C for: (a) 10 min and (b) 20 min.

3. 1 . 1 .

 11 TemperatureIt is expected that a stronger surface reconstruction should occur when the temperature is higher because the carbon solubility in silicon is exponentially proportional to the temperature. Vicinal SiC surfaces were reconstructed under Si droplets at different temperatures from 1600 °C to 1900 °C for the defined holding time of 30 min. Again, both Si-and C-faces were considered. The surface morphologies and the corresponding step spacing distributions of the reconstructed 4H-SiC Si face are presented in Fig.3.10.

Figure 3 .

 3 Figure 3.10 DIC images of the 4° off 4H SiC (0001) Si face surfaces reconstructed 30 min at (a) 1600 °C, (b) 1700 °C and (c) 1800 °C. The distributions of the step spacing in each image are shown as histograms in (d).

Figure 3 .

 3 Figure 3.11 DIC images of the 4° off 4H SiC (0001 ) C face surfaces reconstructed 30 min at (a) 1600 °C, (b) 1700 °C and (c) 1800 °C. The distributions of the step spacing in each image are shown as histograms in (d).

Figure 3 .

 3 Figure 3.12 DIC and AFM images in amplitude mode of the 4° off 4H-SiC surfaces reconstructed at 1900 °C for 30 min. (a) Si face and (b) C face. The local steps details are exhibited at the bottom where the nanosteps of 4-5 nm and 0.5-0.75 nm are observed on the terraces of Si face and C face respectively.

Figure 3 .

 3 Figure 3.13 Three temperature regimes with different ramp rates. Here the total time (ramping stage + holding stage) after filling the gas is kept identical for each set of experiment.

Figure 3 .

 3 Figure 3.14 The surface morphology of the 4° off 4H-SiC Si face surface reconstructed at 1700 °C using different temperature ramping rates: (a) 10 °C/min, (b) 20 °C/min and (c) 40 °C/min. The AFM amplitude images below show the corresponding step-and-terrace structure details on the reconstructed surfaces.

Figure 3 .

 3 Figure 3.15 The surface morphology of the 4° off 4H-SiC C face reconstructed using different temperature ramping rates: (a) 10 °C/min, (b) 20 °C/min and (c) 40 °C/min. The AFM amplitude images (d) (e) (f) show the corresponding local step-and-terrace details of above surfaces.

3. 1 . 3 .

 13 Silicon vapor etchingNow we have established that the SiC/liquid interface is continuously evolving during the temperature ramping up period as well as the early holding stage after reaching the targeted temperature. It is now interesting to check what is the surface morphology just before being contacted with liquid Si, whether or not the high temperature and (or) silicon vapor can change the SiC surface morphology. As mentioned in the last chapter, the initial vicinal SiC surface was processed by CMP, and consists of single bilayer (0.25 nm) steps. Thus the theoretical step spacing on the 4° off vicinal surface is about 0.25 nm/tan4°=3.57 nm.

  performed short-time (5 min) annealing experiments in vacuum without adding silicon source for both the Si face and the C face at 1400 °C. The obtained Si face and C face were then characterized and are presented in Figs. 3.16 b and c, respectively. The voids formed on the surface are usually attributed to the preferential sublimation of silicon in vacuum at high temperature. The evolution of the surface is more significant on the Si face than on the C face, with the RMS roughness values of 610 pm and 285 pm respectively. Bolen et al processed the 4H-SiC surface at 1400 °C for 10 min in high-vacuum (1.3×10 -6 mbar),

Figure 3 .

 3 Figure 3.16 The initial SiC surface just after CMP and acid etching (a) and the SiC surfaces annealed at 1400 °C for 5 min in a vacuum without silicon pieces on top: (b) the Si face and (c) the C face.

Figure 3 .

 3 Figure 3.17 AFM images of the vicinal 4H-SiC surfaces reconstructed by Si vapor at 1400 °C for 5 min (a) the Si face (b) the C face. Insets are the height profile of AA' and BB' lines in the images.

Fig 3 . 18 .

 318 The interval between the Si wafer edge and the points P2~P4 is about 0.5 mm. The point P5 is located at the edge of the SiC wafer, which is about 8 mm far from the Si wafer. The AFM height images at each location are shown around. The RMS roughness values of P1-P5 are measured at 1.1 nm,

Figure 3 .

 3 Figure 3.18 The AFM height images of different areas on vicinal 4H-SiC Si face surface reconstructed with the silicon wafer at 1400 °C for 5 min. As remarked in the illustration diagram, point 1 (P1) is beneath the Si wafer while other points P2 , P3 and P4 is about 0.5 mm, 1 mm and 1.5 mm far from the silicon wafer, respectively; P5 is near the edge of the SiC wafer.

Figure 3 .

 3 Figure 3.19 AFM height images of the terrace area on the C face reconstructed at 1700 °C for (a) 0 min and (b) 10 min.

Figure 3 .

 3 Figure 3.20 Statistical results of the macrostep height and macrostep slope on the two polar surfaces. The values were collected from the reconstructed surfaces at 1700 °C for 30min.

  3.21). It could be clearly seen that the extent of step bunching is stronger inside the droplet which is related to the different undersaturation in two situations. If we zoom into the triple line in this picture, we could observe that groups of 5-8 step trains on the left are further bunched to the larger steps on the right, which confirms that the reconstruction under liquid is indeed based on the reconstructed surface by Si vapor before the melting of silicon. In addition, we have observed different surface morphologies by heating the sample with different temperature ramp rates, which have already proved that undersaturation indeed plays an important role in surface reconstruction in liquid.

Figure 3 .

 3 Figure 3.21 SEM images of the surface morphology close to the triple line (dash line). This was captured from the 4° off 4H-SiC Si face surface reconstructed at 1700 °C for 30 min.

Figure 3 .

 3 Figure 3.22 DIC images of the 4H-SiC Si face with different off angles towards <112 0> direction reconstructed at 1700 °C for 30 min. (a) 4° (b) 2° and (c) 0.2° (nominal on-axis), (d) shows the distributions of step spacing of the 4° and 2° off surfaces. The step spacing on the on-axis surface is too large to be compared with the vicinal surfaces. Note that some irregular islands or shallow pits are observed on the wide terrace of the on-axis surface.

Figure 3 .

 3 Figure 3.23 DIC image of the 4H-SiC C face with different off angles towards <112 0> direction reconstructed at 1700 °C for 30 min. (a) 4° (b) 2° and (c) 0.2° (nominal on-axis), (d) shows the distributions of step spacing of the 4° and 2° off surfaces. The step spacing on the on-axis surface is too large to be compared with the vicinal surfaces.

  surfaces. The AFM images of the terrace area on the on-axis 4H-SiC Si and C faces are shown in Fig.3.24. On the Si face, many small irregular islands of about one bilayer in height are randomly distributed on the terrace. However, a different morphology is observed on the on-axis C face. The super large terraces are divided into many narrower terraces with an equidistance width of about 2 μm. And there are no scattered islands on the terrace. The steps are also characterized by a constant height, which is equal to the unit cell height of 4H-SiC.

Figure 3 .

 3 Figure 3.24 AFM height images of the terrace area on the on-axis 4H-SiC surfaces reconstructed at 1700 °C for 30 min. (a) on the Si face and (b) on the C face.

Figure 3 .

 3 Figure 3.25 The triangular-shaped islands formed on the terrace of the on-axis 6H-SiC Si face reconstructed at 1700 °C for 30 min. (a) on the terrace between spiral steps; (b) on a large terrace.

Figure 3 .

 3 Figure 3.26 (a) Illustration of triangular shaped island growing from a small hexagonal adatom island when the step advancement speed for the A-step VA is significantly larger than B-Step VB,(b) schematic sketch of island shape evolution. The islands develop from an irregular or hexagonal (stage-1) shape via a well-defined triangular (stage-2) shape to a concave (stage-3) shape[START_REF] Michely | Islands, Mounds and Atoms[END_REF]

  on the Si face leads to stronger faceting. The higher ES barrier on the C face gives rise to stronger step bunching. The step stiffness on the C face is larger than that on the Si face providing they have identical configurations. Different external forces exerted on surface steps (by liquid or vapor phase) can cause different extents of surface reconstruction on vicinal SiC surface.

  2 µm, were used sequentially for obtaining a relatively smooth surface of the craters (RMS roughness was measured at < 5 nm for a central area of 20 μm by 20 μm). The final diameter of the crater was about 2.3 mm, as exampled in Fig.4.1. The maximum off-angle relative to the initial surface (nominal surface) can be calculated from the geometry, which is equal to 4.3°. Considering the original off-angle (0.2°), the obtained concave-shaped surface owned the in-plane azimuthal angle (φ) varying from 0° to 360° and the out-of-plane polar angle (θ) from -4.5° to 4.1°. The prepared concave-shaped substrates were then washed in acetone and deionized water for 10 min separately, by using an ultrasonic cleaner.

Figure 4 . 1

 41 Figure 4.1 Schematic illustration of the surface crater preparing process by using the calotest machine. The off angle can be calculated from the relationship sin (θ) = r/R where r is the diameter of the crater viewed from top and R is the diameter of the grinding ball (R=30 mm).

4. 2 .

 2 For both cases, the concaved surfaces have been reconstructed into clear step-andterrace structures and the step spacing and shape are changing with both the radial and azimuthal directions. It is worth noting that on the Si face, the morphology and size distribution of steps reflect the six-fold symmetry of the 4H-SiC crystalline structure, through the development of faceting. However, the surface structure is quite homogenous on the C face, exhibiting a perfectly concentric morphology. In addition, the central parts of the crater have a hexagonal shape and a circular shape on the Si face and C face, respectively.

Figure 4 . 2

 42 Figure 4.2 DIC pictures of two concave-shaped 4H-SiC surfaces reconstructed at 1700°C for 30 min. (a) the Si face and (b) the C face. A six-fold distribution can be clearly seen on the Si face.

Figure 4 . 3

 43 Figure 4.3 DIC photos of the concave-shaped 4H-SiC Si face along: (a) the <11 00> direction and (b) the <112 0> direction.

Fig. 4 . 4 ,

 44 the step density on the C face only slightly increases with the increasing off angles. Comparing with any direction on the Si face, the step density on C face is obviously much lower. Combined with the DIC microscopy photos, we primarily conclude that the surface structures on the C face are only weakly dependent on the surface orientation including azimuthal angle and off angle.

Figure 4 . 4

 44 Figure 4.4 AFM images of the step-and-terrace structures on the concave shaped surface. (a)~(h): 4H-SiC Si face, and (i)~(l): 4H-SiC C face. The height scales are 300 nm and 500 nm for the Si face and C face, respectively. Be aware of the different scanning lengths on the Si face and C face, which are 20 μm and 40 μm, respectively.

Figure 4 . 5

 45 Figure 4.5 DIC images of the central area formed on the concave shaped surface after the reconstruction. (a) the Si face and (b) the C face.

Figure 4 . 6

 46 Figure 4.6 Scheme of a vicinal surface with parameter definition: H: macrostep height, L: terrace width, θ: off-axis angle, and β: macrostep angle.

Figure 4 . 7

 47 Figure 4.7 The effect of off-axis angle on the: (a) step density, (b) terrace width (c) step height, (d) macrostep angle and (e) the extent of the step bunching (R). R is defined as the ratio of final terrace width to original terrace width.

Figure 4 . 8

 48 Figure 4.8 The different step configurations along two azimuthal directions. (a) Steps on surface off toward [11 00] direction (b) steps on surface off toward [112 0] direction. Note that for surface off towards [11 00] direction, two kinds of step edge are formed with different numbers of dangling bond. Only one kind of step edge is formed when cutting off towards [112 0] direction.

Figure 4 . 9

 49 Figure 4.9 AFM amplitude image of the macrostep formed on the concave-shaped 4H-SiC C face.

Figure 5 . 1

 51 Figure 5.1 The temperature regimes for the experiments with the additional cooling down period.

Figure 5 . 2

 52 Figure 5.2 DIC and AFM amplitude images of 4° off 4H-SiC Si face with an additional cooling down period from 1700 °C to 1500 °C: (b) (e) at a cooling rate of 10 °C/min, (c) (f) at a cooling rate of 4 °C/min. Note that (a) and (d) show the sample where the liquid was removed before the cooling down as a comparison.

Figure 5 . 3

 53 Figure 5.3 AFM height and amplitude images showing the step structure on 4° off 4H-SiC Si face with different cooling rates. (a) (b) 4 °C/min and (d) (e) 10 °C/min. The height profiles along the white lines crossing the steps are plotted in (c) and (f). Note the steps are composed of two parts as shown with red arrows in (b) and (c) when using a smaller cooling rate.

Figure 5 . 4

 54 Figure 5.4 The DIC and AFM images of 4° off 4H-SiC C face surfaces with an additional cooling down period from 1700 °C to 1500 °C: (b) (e) at a cooling rate of 10 °C/min, (c) (f) at a cooling rate of 4 °C/min (a) (d) show the sample where the liquid was removed before cooling down as a comparison.

Figure 5 . 5

 55 Figure 5.5 AFM height and amplitude images showing the step structure on 4H-SiC C face with different cooling rates. (a) (b) 4 °C/min and (d) (e) 10 °C/min. The height profiles along the white line crossing the macrosteps are plotted in (c) and (f). Note the macrosteps are composed of smaller steps of 8~15 nm in height.

Figure 5 . 6

 56 Figure 5.6 The small steps at the edge of the macrosteps on the C face with the cooling rate of 4 °C/min. The small steps are measured at a height of 8~15 nm.

Figure 5 . 7

 57 Figure 5.7 Schematic illustration of the movement of macrostep edge on the Si face during growth. (a) The reconstructed surface without cooling down, (b) after adding a cooling down period. The large grey arrows remark the steps advancing direction, whose lengths are proportional to the advancing rates. The red arrows mark the atom flux from liquid to the step edge.

Figure 5 . 8

 58 Figure 5.8 Schematic illustration of the movement of macrosteps on the C face during growth. (a) The reconstructed surface without cooling down, (b) after adding a cooling down period. The macrosteps with lots of kinks are de-bunched with an exerted supersaturation. The white arrows remark the steps advancing direction while the red arrows indicate the atom flux from liquid.

  5.9b). The direction of the macrostep edges was determined under DIC optical microscopy in advance as the 4H-SiC wafer is semi-transparent for white light. For half of the sandwich-type samples, the cross sections were prepared by standard metallographic sample processing, and then was observed by using DIC microscope and SEM. Another half was etched by HF/HNO3 (1:1) acid solution to remove the silicon layer between them. Then the surface morphology and step structures on the exposed surfaces were characterized by the AFM.In the previous chapters, it has been proved that the polarity of SiC plays an important role in surface reconstruction. Therefore in this section, two polar surfaces, i.e. the Si face and C face of 4H-SiC substrates were primarily studied in the sandwich structure. Both surfaces were commercial {0001} vicinal surfaces with a 4° off angle towards <112 0> direction. Three possible combinations by two polar surfaces were prepared and processed in one run of high temperature reconstruction experiment. The configuration of three sets of samples and the holder is presented in Fig.5.10. The samples were treated at 1700 °C for 1h and a series of ex situ characterizations was carried out after cooling to room temperature.

Figure 5 . 9

 59 Figure 5.9 Schematic drawing of (a) the preparation process and (b) the characterization process of sandwich-type samples.

Figure 5 .

 5 Figure 5.10 Photo of one experimental run for three combinations between two polar surfaces of 4H-SiC. ① Si-Si combination; ② Si-C combination; ③ C-C combination.

  5.12 a and b), the clear zig-zag boundaries are present at the SiC (Si face)/silicon interfaces, indicating the well-developed (faceted) step-and-terrace structures are formed at the interface. However, for the C face (in Figs. 5.12 b and c), almost flat boundaries, which exhibit only weak and blurring oscillation, are generated at the SiC (C face)/silicon interfaces. These can also be reflected by the macrostep angles, which are in the ranges of 31-33° and 8-10° on the Si face and the C face, respectively. On the other hand, there seem no obvious differences either between two Si face surfaces in the Si-Si combination or between the Si face in Si-Si combination and the one in Si-C combination based merely on the cross sectional images. Similar scenarios can be revealed on the C face when comparing the two groups where the C face is referred to.

Figure 5 .

 5 Figure 5.11 DIC photo of a typical cross section of the sandwich-type samples. The total thickness of the sample is less than 1 μm.

Figure 5 .

 5 Figure 5.12 DIC photos of the cross section of the three sandwich-type samples: (a) Si-Si combination (b) Si-C combination, and (c) C-C combination. At the bottom are the SEM photos (SE detector) showing steps on (d) Si face/silicon interface and (e) C face/silicon interface.

Figure 5 .

 5 Figure 5.13 AFM amplitude images of the reconstructed SiC surface in different sandwich-type structures: (a) Si face of the Si-C combination, (b) C face of the Si-C combination, (c) one Si face of the Si-Si combination, and (d) one C face of the C-C combination.

Figure 5 .

 5 Figure 5.14 AFM amplitude images of the reconstructed SiC C face surfaces. (a) C face in the C-C combination; (b) (c) (d) different configurations formed on the C face in the Si-C combination. It is assumed they are at different stages of the growth process.

Figure 5 .

 5 Figure 5.15 Schematic illustration of the surface evolution in the Si-C combination sandwich-type surface reconstruction. The red arrows indicate the step moving direction while the white arrows show the flux of carbon atoms in the liquids.

Figure 6 . 1

 61 Figure 6.1 DIC images of the SiC/liquid interface kept for 30 min at 1500 °C. Note the circular hillocks can only be found at the area where the silicon piece were initially located. The remarks S/S indicates the initial contact between solid SiC and solid silicon, the L/S means having contact between liquid silicon and solid SiC. The edge of solid silicon piece is marked with dash lines.

Figure 6 . 2

 62 Figure 6.2 Illustration of the process for getting the average dissolution depth. (a) the 3D topography of the SiC surface being dissolved at 1900 °C for 30 min. (b) the height profile of the red line crossing the center of the erosion crater. The average dissolution depth is defined as the integral area divided by the droplet diameter: ΔHavg = S/D.

Figure 6 . 3 6 . 3 .

 6363 Figure 6.3 Illustration of the process for getting dissolution depth from hillocks. (a) AFM height image of the hillocks formed on the SiC surface dissolved by silicon at 1700 °C for 30 min. (b)The corresponding height profiles of the lines 1&2 in (a), the relative height ∆H is the difference of the average height between the top and the surrounding of the hillock.

Figure 6 . 4

 64 Figure 6.4 Height profile of the SiC surface after dissolution by the liquid silicon at 1600 °C for 10 min, 30 min and 120 min.

Figure 6 . 5

 65 Figure 6.5 DIC pictures of the SiC surfaces after being dissolved by liquid Si: (a)~(e) from 1500 °C to 1900 °C for 30 min; (f)~(h) from 2000 °C to 2200 °C for 10 min.

Figure 6 . 6

 66 Figure 6.6 Size of the contact area of the Si droplet on a SiC surface as a function of temperature.

Figure 6 . 7

 67 Figure 6.7 3D topography images of the SiC surface after being dissolved by liquid Si: (a)~(e) from 1500 °C to 1900 °C for 30 min; (f)~(h) from 2000 °C to 2200 °C for 10 min.

Figure 6 . 8

 68 Figure 6.8 Height profile of the SiC surface after dissolution by the liquid Si at 1500 °C~2200 °C. All the profiles were extracted from the lines crossing the center of the droplet along the slow scanning directions.

Figure 6 . 9

 69 Figure 6.9 DIC images of the hillocks formed on the SiC surface dissolved at different temperatures: (a) 1500 °C, (b) 1600 °C, (c) 1700 °C and (d) 1800 °C.

Figure 6 .

 6 Figure 6.11 Carbon solubility (as mole fraction) as a function of temperature in literature and in our work. The dash lines indicate extrapolations of the experimental measurement range (solid lines). The green triangles presents the results using the average depth while the red dots are obtained using the height of the hillocks. The fitting of the results using the average depth is shown as the green line.

6 - 6 )

 66 Thus the change of the Gibbs energy of reaction (6-4) can be obtained:

6 - 9 )

 69 By subtracting reaction (6-4) from reaction (6-8), we can get the standard Gibbs energy change of the carbon dissolution in silicon

a)

  An isothermal condition and no temperature fluctuation at solid/liquid interface. b) Be able to attain the equilibrium state in a limited time. c) No precipitation during the whole dissolving process. d) Low-level partial pressure of oxygen. e) No mass transport at the boundaries (such as free surface and solid-liquid-gas triple lines) except solid/liquid interface. f) Stable and constant wetting area (≈constant silicon amount).

Figure 6 .

 6 Figure 6.12 DIC images showing the surface morphology of the dissolved SiC surface at (a) 2000 °C (b) 2100 °C and (c) 2200 °C. (d) SEM picture of the dissolved surface at 2000 °C. The EDS results of point 1 and area 2 are shown together in (e), for both cases the element ratio Si : C is similar and close to 1:1.

Figure 6 .

 6 Figure 6.13 DIC images showing the edges of the droplet prints on SiC substrates dissolved by silicon melt at (a) 1700 °C, (b) 1800 °C, (c) 1900 °C, and (d) 2000 °C.

Figure A. 1 F- 1 F 2 Figure F. 1 F- 3 Figure F. 2

 112132 Figure A.1 Optical diagram of differential interference contrast (DIC) according to Nomarski [1]: 1, light source; 2, collector; IE, input intensity; 3, polarizer; 4, flat glass; 5, DIC prism (modified Wollaston prism1); 6, objective exit pupil; 7, objective; ''blue'' and ''green'', image and twin image of the reflected wave front and the wave front deformed by the object; 8, specimen; 9, analyzer; IA, output intensity; 10, tube lens; 11, intermediate image plane[162].

Figure F. 3

 3 Figure F.3 Images DIC de la face C du 4H-SiC 4° off, reconstruite à 1700 °C pour différents temps de maintien : (a) 0 min ; (b) 10 min ; (c) 20 min et (d) 30 min. Les histogrammes de distribution de l'espacement des marches dans chaque image sont présentés dans (e).

Figure F. 4

 4 Figure F.4 Images DIC de deux surfaces concaves de 4H-SiC reconstruites à 1700°C pendant 30 min. (a) la face Si et (b) la face C. La symétrie d'ordre 6 est clairement visible sur la face Si.

Figure F. 5

 5 Figure F.5 Différentes configurations de bords de marches le long de deux directions azimutales. (a) Marches sur la surface en direction de [11 00]. (b) Marches sur la surface en direction de [112 0].

Figure F. 6 6 Figure F. 7

 667 Figure F.6 Résultats statistiques de la hauteur et de l'angle du macrostep sur les deux surfaces polaires. Les valeurs ont été recueillies sur les surfaces reconstruites à 1700 °C pendant 30 min.

Figure F. 8

 8 Figure F.8 Images d'amplitude AFM des surfaces de la face C du 4H-SiC reconstruites. (a) la face C dans la combinaison C-C ; (b) (c) (d) différentes configurations formées sur la face C dans la combinaison Si-C. Il est supposé qu'elles sont à différents stades du processus de croissance.

Table 1 -

 1 1 Surface free energies of the two polar faces for 3C-, 4H-, and 6H-SiC crystal. Note that the unit of the values is erg•cm -2 .

Table 1 -

 1 2 Major physical properties of common semiconductor materials[START_REF] Camassel | SiC materials: a semiconductor family for the next century[END_REF].

	Properties/materials	Si	GaN	3C-SiC 4H-SiC 6H-SiC
	Bandgap at 300 k [eV]	1.1	3.4	2.3	3.2	3.0
	Breakdown field [MV•cm -1 ]	0.3	3.3	1.4	2.8	3.0
	Electron Mobility at 300 K[cm 2 •V -1 •s -1 ]	1100	900	750	800	370
	Saturated drift electron velocity
		1	2.7	2.5	2	2
	[×10 7 cm•s -1 ]
	Thermal conductivity [W•cm -1 •K -1 ]	1.5	1.3	5	4.9	4.9
	Maximum operating temperature [K]	600	1930	1200	1720	1580
	1.1.3. SiC crystal growth

Table 2 -

 2 [START_REF] Kimoto | Fundamentals of Silicon Carbide Technology[END_REF] The performance of different removing tools (the "+" donates good and "-"donates poor).

	Materials	Wetting	Stability	Flexibility	Cleanliness
	Graphite
	(sponge+sheet)

Table 6 -

 6 1 Experiments on solubility of carbon in silicon liquid.

	Year	Researcher	container	Temperature	Atmosphere	Solubility determination	Ref.
	1958	Hall&Dash	Quartz or Alundum	1414, 1520, 1600, and 1725 °C	Argon	Weight loss of silicon carbide	[149]
					Argon
	1959	Scace&Slack	Graphite	1560~2900 °C		Weight of SiC residue after etching the Si	[43]
					(35 atmos)
	1960	Dolloff	Graphite	1500~2600 °C	unknown	Wet chemical analysis (Details unknown)	[146]
	1987	Oden&McCune	Graphite	1700~2150 °C	Argon (1 atmos)	Combustion IR. Diamond grinding of the outer surface of the ingot	[150]
	1993	Kleykamp & Schumacher	ZrO2 or ThO2	1420, 1600, 1800 and 2830 °C	unknown	X-ray spectrometer	[147]
	1997	Yanaba et al	SiC	1450~1600 °C	Ar-CO (19:1) 1 atm	Combustion IR. Quenched melt with SiC crucible	[151]
	2009 Dalaker&Tangstad	Graphite	1414~1559 °C	Argon	Combustion IR. Extracted liquid every 3 min in one run by quartz tube	[152]

Table 6 -

 6 2.

Table 6 -

 6 3 Carbon solubility calculated by using average dissolution depth from profilometer (XC-1) and using local dissolving depth from atomic force microscopy (XC-2).

	Temperature [°C]	1500	1600	1700	1800	1900	2000	2100	2200
	XC-1
	[×10 -5 mol fraction]	24.69	47.96	127	449	615	1025	1798	2168
	XC-2
	[×10 -5 mol fraction]	47.61	62.71	227	473	/	/	/	/

Acknowledgement

in AFM measurements. Thanks to Francine Roussel and Joelle Masson . In addition, Dr. Jean-Marc Dedulle and Dr. Yves Du Terrail Couvat are greatly acknowledged for their help on COMSOL simulations. This work was supported by a French-Japanese partnership (CNRS-JSPS), thus I should present my gratitude to our collaborators at the University of Tokyo. Besides the numerous discussion and help from Yoshikawa sensei and Dr. Sakiko Kawanishi (now at Tohoku University

Using the above dissolution depth data, the amount of lost SiC (𝑛 , mol) can be calculated easily from:

3) Surface reconstruction with different solvent This work only tackled the surface reconstruction with pure silicon largely because of the simplicity of the Si-SiC system. However, most solvents used for solution growth are Sibased alloys instead of pure Si, which aims to obtain higher carbon solubility and then fast growth rates. Therefore, studying the SiC surface reconstruction in other Si-based liquid alloys would be more useful for the future development of solution growth approach. It could especially help go deeper into understanding the interaction between the solvent and the SiC surface at high temperature.

4) Numerical simulation of step movement.

The initial stage of the surface reconstruction is very intriguing because it has a profound impact on the following surface reconstruction. Unfortunately, due to the high inertness of our device, done on purpose, it is impossible to get the relevant information in our experimental approach due to the practical reasons (necessary time for injecting gas and removing liquid). As a complementary approach, the kinetic Monte Carlo (KMC) method can be used to explore the kinetic behavior of sequences of arbitrary fundamental transitions, and it has been successfully used for the study of the step-flow growth [START_REF] Battaile | The Kinetic Monte Carlo method: Foundation, implementation, and application[END_REF].

Basically, the KMC model needs to consider all kinds of the transport of species, such as diffusion, reaction and desorption of adatoms on the surface, and the attachment or detachment of adatoms to or from step edges [START_REF] Stout | Modeling surface kinetics and morphology during 3C, 2H, 4H, and 6H-SiC (111) step-flow growth[END_REF]. For example, KMC would be an interesting tool to study the effect of the ES barrier on the step bunching process. or phase difference), the two paths will be recombined into the same linearly polarized light as it is before the light splitting. In this case the analyser will not detect any contrast.

Appendix

However if the specimen generates some path difference, the two images with a small lateral offset will interfere with one another. Thus the impression of an image in relief will be created, and the maximum contrast can only be obtained when the phase boundaries lie perpendicular to the direction of the split. As literally, the tip contacts the sample surface during the scanning when the contact mode is employed. The deflection between the cantilever and the sample is maintained constant ("set point") by the feedback loop. According to Hooke's law, the force between the tip and sample need also to be constant. When the surface height changes, the scanner will move vertically to maintain a constant "setpoint" of deflection, the distance of this moving will be stored in the computer to form a topographic image of the sample surface.

For the tapping mode, the tip slightly "taps" on the sample surface during the scanning, oscillating with a frequency at or near the resonance frequency of the cantilever. The tip is only in intermittent contact with the surface, only at the bottom point of the oscillation.

The oscillation amplitude is maintained constant by the feedback loop. In order to keep the "setpoint" of the amplitude, the vertical position of the scanner will change according to the surface height. The tip-sample is in a repulsive interaction is constant in tapping mode.

Comparing these two modes, the scan speed of the contact mode is slightly higher than the tapping mode. And usually the contact mode has a higher resolution than tapping mode [START_REF] Eaton | Atomic Force Microscopy[END_REF]. Nevertheless, actually both modes are capable of getting atomic resolution images under certain conditions such as ultra-high vacuum [START_REF] Giessibl | Advances in atomic force microscopy[END_REF]. However, the tapping mode has higher lateral resolution on most surfaces. The tapping mode can also minimize the frictional forces that are present in the contact mode, thus cause less damage to the soft samples.

In this thesis, the tapping mode was employed to obtain the topography of the reconstructed SiC surfaces. All the measurements were performed on a Veeco Dimension Appendix 162 3100 AFM (Digital Instruments) using a commercial etched single crystal silicon probe in the ambient environment. For each manipulation, the frequency of oscillation which produces the largest response (resonance frequency) was determined by using the automatic cantilever tuning. The height and amplitude data were collected simultaneously, the latter is the derivative of the height and can provide a clearer edge of structures. All the captured data were further processed and analysed using the Gwyddion software.

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