M Guillaume

Alombert-Goget Docteur-Ingénieur

Rsa Le

Rubis Sa

Invité M Thierry

I

Abstract Ti:sapphire is outstanding for producing ultra-short pulse lasers due to the presence of doping Ti 3+ ions. However, the crystal laser efficiency is decreased by the residual absorption caused by harmful Ti 4+ . The conversion mechanisms between the two Ti ions during crystal growth are not completely understood and no theoretical model is available to explain the experimental distributions of both ions in the crystals. In this thesis, a physico-chemical model is established by taking into account the diffusion and chemical reaction of point defects and the validity of the model is qualitatively checked through numerical simulation.

Ti ion concentrations in the solid phase are determined by their concentrations in the liquid phase and their segregation coefficients. FactSage software is used to thermodynamically calculate these values using a database updated with respect to experimental solubility data.

Then, the Ti ion concentrations and segregation coefficients are calculated and analyzed as function of O2 (𝑝 O 2 ). The influence of Mo and C on 𝑝 O 2 is studied and results show that graphite elements surrounding the growing crystal have much more influence on 𝑝 O 2 than the Mo crucible.

Further, a physico-chemical model is proposed to study the conversion mechanisms between Ti 3+ and Ti 4+ in the solid phase during Ti:sapphire growth. The model assumes that the conversion between Ti 3+ and Ti 4+ is controlled by the diffusion and chemical reaction of point defects, likely Al vacancies and holes. Boundary condition for the concentration of Al vacancies on the crystal surface is analyzed as function of 𝑝 O 2 and temperature. This affects the final distributions of Ti ions in grown crystal.

The proposed model is implemented in COMSOL Multiphysics software to study its relevance during Ti:sapphire growth process. Simulation of annealing experiments are performed to obtain unknown physical parameters related to the model: diffusion coefficient of Al vacancies and reaction rate constant. Then, previously calculated concentrations, segregation coefficients, boundary conditions and parameters are introduced into the numerical model.

Simulation results for a simplified growth configuration are performed. They show a qualitative agreement with the experiments. The effect of crystal growth rate, temperature gradient and 𝑝 O 2 on the Ti ion distributions is studied.

Acknowledgment

Without the kind help of my supervisor, I would not be able to achieve this thesis.

Therefore, first and foremost, I would like to give my sincere gratitude to my supervisor Prof.

Thierry Duffar. During my study road for more than 20 years, I have never met a so good supervisor! He is always kind, patient and generous to the students. The first month when I arrived in the laboratory, considering my physics background during my masters, he spent several afternoons tutoring me about the knowledge for chemistry and crystal growth. No matter how stupid questions I asked and how stupid mistakes I made, he never criticized and always explained to me again and again until I understood it. No matter when I went to his office, he always stops what he was doing and listened to me. He always corrected all the documents I wrote carefully especially for my thesis. Even if he has retired, not only did he not stop guiding me but he came to the laboratory weekly to answer my questions. I am really honored to have met him as my PhD supervisor! I thank the jury for reviewing the thesis manuscript carefully, pointing out the interesting scientific questions and sharing their rich expertise in their respective fields. Dr. Elisabeth Siebert deserves the big thanks, she offered to discuss the revision of the manuscript together after the defense to improve the quality of the whole thesis.

I would like to thank all the colleagues who helped for my research works in Grenoble especially in SIMAP laboratory and its EPM group. Abdou was very kind to share his knowledge and experiment tools with me. Jessica was generous for giving me her pure sapphire samples. Olga and Yves were very enthusiastic to help me to solve the problems about COMSOL software in the computer room. Khadija always helped me to handle with all the paper works in French and she was very kind to practice French with me. Guy was very kind to teach me how to cut Ti:sapphire crystals and always keen to help me solve all my problems in EPM. Kader, Matias, Didier, Yamina and Benjamin in EPM laboratory also helped me a lot.

Big thanks are given to Dr. Gourav Sen. Thanks to him for leaving all his crystals and samples to me and being always available to answer my questions.

Big thanks are given to associate professor Yunzhong Zhu in Sun Yat-sen University and director Jiliang Quan in the Institute of Semiconductor Materials in Guangzhou of China. I really thank them for helping us grow Ti:sapphire crystals. I thank Joëlle Calabro and Christele Poggi for teaching me how to polish sapphire samples. I thank Alexander Pisch for IV teaching me to use FactSage software. I also thank Florian Molton, Grégory Berthomé, Alexandre Crisci, Florence Robaut, Thierry Encinas, Stephane Pignard in Grenoble for helping us performing EPR, XPS, Raman, WDS, XRF measurements.

I must thank all my friends who accompanied me to pass these three years. My closest friend, Chenhao Feng in Bordeaux, we always comfort and encourage to each other since we were master students. We share my happiness and sadness with each other. We made progress together and grew together. I hope that we still have the opportunity to fight together in the future. I thank Fanshi Meng for driving me to the hospital when I was seriously sick on the ski mountain. I appreciate Dr. Xuefeng Han for sharing his rich research experience with me.

Ziyuan Dong, my other friends, Na Deng, Xinming Xing, Tao Wang (two names with the same pronunciation), Dou Quan, Yanning Hu, Bo Ren, I enjoyed the time passed with you for climbing the mountains, visiting around, having meals together. I also should thank my mathematics teacher Liang Chen, master supervisor Jean-Michel Le Floch and his mother who cared my life in France.

I thank the Chinese Scholar Council (CSC) for providing me the scholarship for pursuing the study in France.

At the end, I really appreciate my family for their endless love. Their support and encouragement is my greatest motivation to move forward.

General introduction

Ions doped crystals have attracted much attention due to their excellent properties and extensive applications. For example, Cr-doped alumina can be served as reversible thermochromic sensors, Ti-doped alumina is famous for tunable solid-state laser, V-doped Forsterite (V:Mg2SiO4) crystals can be used in quantum electronics, Yb:YAG (Yb:Y3Al5O12) crystals demonstrate significant applications in diode-pumped laser systems, LYSO:Ce, Mg single crystals have the potential for medical imaging applications, etc. In addition, ions doped crystals also have great business value. The doping ions make the crystals show a variety of attractive colors so that they are sold as beautiful jewelries, as shown in Figure 1.

Figure 1. Colored gemstones (website 1).

Above ions doped crystals are usually prepared by doping transition metal ions or rare earth ions into oxide or composite crystals. The properties of ions doped crystals are highly related to the doping active ions and host materials. However, crystals doped with transition metal ions or rare earth ions are not easy to deal with due to the problem of variable valence of dopants.

Table 1 lists some possible variable valence in ions doped crystals. The outstanding properties of ions doped crystals are always determined by a unique valence state of the dopant while the presence of other valence is detrimental. For example, this unique valence state is critically important to the laser performance of high-power solid-state lasers.

Table 1. Ions doping crystals with variable valency.

Crystals Active ion Variable valency

Ruby (Cr:Al2O3) Cr 3+ [START_REF] Kusuma | Absorption and emission properties of ruby (Cr: Al2O3) single crystal[END_REF] Cr 2+ , Cr 4+ [START_REF] Sviridov | About the possible role of Cr 2+ and Cr 4+ ions in ruby[END_REF] Ti:sapphire (Ti:Al2O3) Ti 3+ [START_REF] Moulton | Spectroscopic and laser characteristics of Ti: Al2O3[END_REF] Ti 4+ [START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF] Forsterite (V:Mg2SiO4) V 5+ [START_REF] Mackwell | High-temperature deformation of forsterite single crystals doped with vanadium[END_REF] V 3+ , V 4+ [START_REF] Veremeichik | Spectroscopy of V 4+ and V 3+ ions in a forsterite crystal[END_REF] Yb:YAG (Yb:Y3Al5O12) Yb 3+ [START_REF] Zeng | Effects of gamma-irradiation and air annealing on Yb-doped Y3Al5O12 single crystal[END_REF] Yb 2+ [START_REF] Henke | Preparation and spectroscopy of Yb 2+ -doped Y3Al5O12, YAlO3, and LiBaF3[END_REF] LYSO:Ce, Mg (Lu2(1-x)Y2xSiO3:Ce) [START_REF] Blahuta | Evidence and consequences of Ce $^{4+} $ in LYSO: Ce, Ca and LYSO: Ce, Mg single crystals for medical imaging applications[END_REF] Ce 3+ Ce 4+ Ti:sapphire is the workhorse of ultra-short pulse laser due to its ultra-intense power and excellent tunability, especially after the appearance of two Nobel Prizes in physics about the "Optical Frequency Comb" and "Chirped Pulse Amplification" techniques. Its superior performance comes from doping Ti 3+ . However, its application is limited due to the variable valence problem of Ti 3+ and Ti 4+ . The oxidized state, Ti 4+ , usually comes from the oxidation reaction during crystal growth and it cannot be totally eliminated. The mechanisms controlling the conversion between the two ions are not clear yet and no theoretical model is available for explanation. The only way to reduce harmful Ti 4+ ions is through annealing while this operation can only be conducted after the crystal growth and it is very time-consuming. Therefore, to improve the optical quality of crystal, the best choice would be to control the valence of Ti ions during crystal growth process.

RSA Le Rubis is a company growing crystals by Verneuil, EFG and Kyropoulos methods.

They also produce Ti:sapphire crystals for laser applications. Figure 2 shows a Ti:sapphire crystal grown by RSA company using Kyropoulos method. However, an opposite distributions of Ti 3+ and Ti 4+ are observed in these crystals, not only for Ti:sapphire grown by Kyropoulos method but also for crystals grown by other methods like Verneuil or Czochralski. Currently,

XI
no one is able to explain these experimental results. It could be caused by the redox reaction which is determined by the oxygen partial pressure inside the gas phase. It could result from the segregation coefficient which is related to the concentration of Ti ions in the solid and in the liquid phases at the solid-liquid interface. On the other hand, there exists a lot of point defects in the solid phase. These point defects could diffuse and react chemically with Ti ions thereby change their valence. However, no systematic study taking into account all these phenomena have been reported. Therefore, this thesis is devoted to the explanation of experimental results and understanding the conversion mechanisms between Ti 3+ and Ti 4+ ions during Ti:sapphire growth process by analyzing all above factors. Consequently, the structure of this manuscript is organized in 5 chapters:

XII

-Chapter 1 conducts the literature review for the properties and applications of sapphire and Ti:sapphire. The characteristic spectra of Ti:sapphire and their relation with crystal optical quality and the concentrations of variable valency ions are investigated. Various crystal growth techniques, solute segregation and chemistry occurring during crystal growth process are discussed.

-Chapter 2 performs thermodynamic calculations to obtain the concentrations and segregation coefficients for the ions of varying valence in Ti:sapphire. The influence of crystal growth atmosphere, crucible and furnace configuration on Ti ion concentrations and segregation coefficients of Ti ions are computed and analyzed. These calculations provide important parameters for following calculations.

-Chapter 3 discusses the point defects in pure sapphire and Ti:sapphire as well as their diffusion mechanisms and reactions. Based on this, an original physico-chemical model is proposed for understanding the conversion between Ti 3+ and Ti 4+ ions occurring during Ti:sapphire crystal growth.

-Chapter 4 deals with the numerical simulation of Ti ion behavior during annealing and crystal growth. The simulation of annealing provides necessary physical parameters for studying the physico-chemical model during crystal growth process. The influence of temperature field, crystal growth rate and oxygen partial pressure on Ti ion distributions are studied and the validity of the model is checked.

-Chapter 5 summarizes the main works of this thesis and presents perspectives for future work to further improve the results.

Chapter 1 Background

Sapphire

Alumina has eight different polymorphs: seven metastable phases (γ, δ, κ, η, ρ, χ, θ) and the thermally stable α-phase. These metastable phases can be synthesized or transformed by a variety of methods. The transformation sequence is illustrated as follows [START_REF] Wang | Phase transformation in nanometer-sized γ-alumina by mechanical milling[END_REF]:

γ -Al 2 O 3 750℃ → δ 900℃ → θ 1200℃ → α
The phase transformation can be conducted through sintering or grain growth with specific surface area changes. γ-Al2O3 has a cubic spinel structure. Evidence shows that β-Al2O3 is not an allotropic modification, but is an alkali aluminate which is assumed to be composed of 1 mole of alkali oxide to 12 moles of aluminum oxide such as Na2O.12Al2O3 and K2O.12Al2O3 [START_REF] Ridgway | The preparation and properties of So-called "Beta Alumina[END_REF]. The sapphire stable phase α is also known as corundum whose chemical formula is Al2O3. Pure sapphire is colorless and transparent. When pure sapphire is doped with other ions, it exhibits attractive colors. For example, the introduction of Fe 3+ makes sapphire becoming blue, the incorporation of Ti 3+ makes sapphire becoming pink and the existence of Ni 3+ makes sapphire showing yellow color. Therefore, mass produced colored sapphire crystals are often sold as jewelry.

Sapphire (α-Al2O3) has a trigonal structure. Its unit cell is rhombohedral but its structure is commonly represented with the multiple hexagonal cell. The most commonly used crystallographic planes of α-Al2O3, i.e. m, r, a (especially in optical fields) and c are shown in Figure 1-1 (a). In the process of epitaxial growth on sapphire one must take into account not only the lattice parameters and atomic bonds but also the electronic structure of the crystallographic planes. The r-plane has a good lattice match with silicon so that it is used in silicon-on-sapphire (SOS) technology to develop high frequency RF transceivers and switches [START_REF] Culurciello | Silicon-on-sapphire circuits and systems[END_REF]. partially covalent. For instance, Kotomin [START_REF] Kotomin | Calculations of the geometry and optical properties of F Mg centers and dimer (F 2-type) centers in corundum crystals[END_REF] showed that the ionic charges for Al and O are +2.34e and -1.56e, Nagel [START_REF] Nagel | Cluster calculation of electronic structure and quadrupole interaction in α-Al2O3[END_REF] presented their values as +2.57e and -1.88e, Batra [START_REF] Batra | Electronic structure of α-Al2O3[END_REF] showed the results of +2.7e and -1.8e. These covalent bonds probably contribute to sapphire mechanical properties while ionic bond energy contributes to the physiochemical properties of the crystal (Dobrovinskaya et al., 2009b). In this thesis, we care more about the physicochemical properties. In sapphire crystal, the cationic Al 3+ can be substituted by other impurities possibly acting as dopant, as shown in Figure 1-1 (c). A Ti 3+ ion substitutes the lattice site of Al 3+ , lying in the center of an oxygen octahedron. The central Ti 3+

is influenced by the cubic field (electrostatic field) resulting from the surrounding six O 2-and by the trigonal field caused by a trigonal distortion of the octahedron [START_REF] Byvik | Analysis of vibronic transitions in titanium doped sapphire using the temperature of the fluorescence spectra[END_REF] due to the overall crystallography. Therefore, the crystal field of Ti:sapphire can be considered as the sum of a cubic field and a trigonal field, wherein, the effect of cubic field is much larger than that of trigonal field. The properties of sapphire single crystal are listed in Thermal expansion coefficient (293-323 K) 6.6×10 -6 K -1 (‖c) ,5.0×10 -6 K -1 (⊥c)

Thermal conductivity (298 K) 32.95 W.m -1 . K -1 (‖c), 30.3 W.m -1 .K -1 (⊥c)

Resistivity (293 K) (1.2-2.9) ×10 17 Ω.m (‖c) ,5×10 16 Ω.m (⊥c) Dielectric constant (293 K) 11.53 (‖c) ,9.53 (⊥c)
Loss tangent (293 K) 8.6×10 -5 (‖c) ,3×10 -5 (⊥c)

Mohs hardness 9

Refractive index no=1.768, ne=1.760

Sapphire is a multifunctional material. Due to its prior mechanical, physical and chemical properties, it is extensively employed in optical windows, bearing and cutting tools, smartphone screen and compatible material for medical implant. In addition, it is often used as primary substrate material to produce light emitting diodes (LED) or smartphone electronic components.

It has an excellent transmission performance from ultraviolet to the infrared. All these outstanding properties of sapphire determine that it is a good host material for solid-state lasers.

Ti:sapphire

Laser efficiency and tuning range are two indexes people care most in terms of laser performance. Appropriate ions doping host crystals such as Cr 3+ , Mn 4+ , Ni 2+ , Co 2+ , V 2+ , Fe 2+ …can significantly modify crystal mechanical and optical properties, enabling them to be promising materials for solid-state lasers. For example, Tang et al. [START_REF] Tang | Growth and development of sapphire crystal for LED applications[END_REF] experimentally showed that Ti 3+ or Ti 4+ doped sapphire exhibits better fracture strength than undoped sapphire, as the defect structure resulting from the doping ions improved the sapphire fracture surface energy. Different active ions can be selected as active medium for solid-state laser according to the desired gain, energy and wavelength. Common doping ions include rare earth, transition metal or actinides. In the following, we only consider the widely used solidstate lasers based on metal transition ions. In 1982, when Moulton first time demonstrated that the optical tunability of sapphire could be extended from 660 nm to 1180 nm by using Ti 3+ as the active ion [START_REF] Moulton | Spectroscopic and laser characteristics of Ti: Al2O3[END_REF], Ti:sapphire has attracted great interest of researchers. Compared to other transition ions (see Figure 1-2), Ti 3+ possesses a large tunable bandwidth expanding from ultraviolet (UV) to infrared (IR) regions.

Figure 1-2 Laser tunable ranges of transitional metal ions in host materials. Plotted from data in [START_REF] Payne | Transition metal ion lasers other than Cr 3+[END_REF]) [START_REF] Nikolov | Cr 3+ doped NaAl (WO4)2 crystal as a new broadband laser material[END_REF] [START_REF] Boulon | Fifty years of advances in solid-state laser materials[END_REF].

Further, the peak power of Ti:sapphire laser is much higher than typical continuous-wave lasers and can produce ultra-short pulse at a level as high as petawatt (10 15 W), corresponding to tens of femtosecond regime (10 -14 s) using the mode locking technique [START_REF] Mckinnie | Ti 3+ ion concentration and Ti:sapphire laser performance[END_REF]) [START_REF] Spence | 60-fsec pulse generation from a self-mode-locked Ti: sapphire laser[END_REF]. The most known solid-state laser is, undoubtedly, Ti:sapphire due to its large tunable ability. The discovery of Nd:YAG (Y3Al5O12) laser and its extension to Nd:glass has improved the laser system from previously three energy levels (ruby) to four energy levels. This makes it easier to realize population inversion for lasing. In this table, Cr:sapphire is a three energy level system which requires higher pumping energy, while Nd:YAG, Nd:glass and Ti:sapphire belong to four energy level system, requiring lower pumping energy. Nd:YAG is more preferred in continuous and very high pulse repetition rate operation and Nd:glass is more suitable for high energy-pulsed operation such as laser fusion [START_REF] Yamanaka | Nd-doped phosphate glass laser systems for laser-fusion research[END_REF]. Peak emission cross section determines how many transitions, from the upper level to the lower level, are caused by a specific flux of photons. For low threshold and high-gain amplifier or oscillator, a large cross section value is desirable. The fluorescent lifetime, measured from the fluorescence decay, determines how much power should be applied to pump populations from lower level to higher level. Fluorescence lifetime decreases with increasing temperature because higher temperature speeds up the transition rate associated with photon induced radiation.

Ti:sapphire lasers have been extensively employed in the fields of ultra-short pulse generation [START_REF] Spence | 60-fsec pulse generation from a self-mode-locked Ti: sapphire laser[END_REF], high-power amplification [START_REF] Wall | Titanium Sapphire Lasers[END_REF], spacecraft remote sensing [START_REF] Rines | Narrowband, high-energy Ti: Al2O3 lidar transmitter for spacecraft sensing[END_REF] and bladeless eye surgery [START_REF] Soong | Femtosecond lasers in ophthalmology[END_REF]. The first petawatt scale laser operation was demonstrated in 1996 at Lawrence Livermore National Laboratory using three Ti:sapphire crystals with diameters of 85-100 mm [START_REF] Joyce | Progress in the growth of large scale Ti:sapphire crystals by the heat exchanger method (HEM) for petawatt class lasers[END_REF].

High laser pulses, 1.5 PW with a maximum output energy of 60.2 J and 10.3 PW with a maximum output energy of 339 J [START_REF] Li | 339 J highenergy Ti:sapphire chirped-pulse amplifier for 10 PW laser facility[END_REF], have been achieved from a chirped-pulse amplification Ti:sapphire laser system by Advanced Photonics Research Institute and Shanghai Institute of Optics and Fine Mechanics. At the same time, the European Union launched the Extreme Light Infrastructure project aiming to achieve a 100 PW laser facility [START_REF] Gerstner | Laser physics: extreme light[END_REF] and the French Apollo project, devoted to 10 PW laser facility [START_REF] Papadopoulos | The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics[END_REF] is under development as well. The invention of Ti:sapphire laser enabled all laser applications involved with coherent signal generation, amplification, modulation, detection and information transition to perform at frequencies million times higher, or wavelength millions times shorter, than previously. Even if the optical loss of Ti:sapphire within the lasing range limits its application, Ti:sapphire will continue to serve as the foundation of the ultrafast technology and research in the foreseeable future.

1.2.1 Characteristic optical spectra of Ti:sapphire

Absorption and emission spectra of Ti 3+

Ti:sapphire laser needs to be pumped by another laser. Typically, either argon ion laser emitting at 514 nm or frequency-doubled Nd:YAG laser emitting at 532 nm are used to realize the population inversion of doping Ti 3+ ions. Other pumping sources like flashlamps [START_REF] Lacovara | Flash-lamp-pumped Ti:Al2O3 laser using fluorescent conversion[END_REF], copper-vapor lasers [START_REF] Knowles | Efficient high-power copper-vapor-laser-pumped Ti:Al2O3 laser[END_REF], diodes, and pulsed-dye lasers are not used very often because of their low optical efficiency. the corresponding absorption and emission bands between 2 T2g and 2 Eg. The tuning range on the short-wave side is determined by the ground state absorption from 2 T2g to 2 Eg, while on the long-wave side, it is limited by the gain cross section during the emission process. The widely separated absorption and emission bands result from the strong coupling between Ti 3+ and Al2O3 lattice, which is the key point to allow ultra-short pulse generation using mode locking technique. The fluorescence lifetime of Ti 3+ is found to be 3.2 µs at room temperature and it increases to 3.8 µs at 77 K. The luminescence spectrum for Ti 3+ is measured at 730 nm after excitation at 532 nm [START_REF] Alombert-Goget | Titanium distribution profiles obtained by luminescence and LIBS measurements on Ti:Al2O3 grown by Czochralski and Kyropoulos techniques[END_REF]. [START_REF] Goossens | Ultrafast organic lasers and solid-state amplifiers[END_REF] [START_REF] Sorokin | Solid-state materials for few-cycle pulse generation and amplification[END_REF].

Apart the absorption around 500 nm, Tippins [START_REF] Tippins | Charge-transfer spectra of transition-metal ions in corundum[END_REF]) also observed another shortwavelength absorption associated with Ti 3+ in Al2O3 peaking around 180 nm. It is assigned to the charge transfer moving from an electron on one of the oxygen ions to Ti 3+ lattice site, changing Ti 3+ into Ti 2+ .

Absorption and emission spectra of Ti 4+

For laser application purpose, most crystal growth processes are conducted under reducing atmosphere so that Ti 3+ ions are expected in crystals. However, in practical crystal growth operation, it is impossible to guarantee a totally oxygen free atmosphere, leading to few amounts of Ti 4+ ions in the grown crystals.

Ti:sapphire shows a blue emission band around 420 nm after excitation with a ultraviolet light (100-400 nm). The origin of this 420 nm blue emission in titanium doped sapphire has been interpreted as due to color centers [START_REF] Summers | Thermoluminescence in single crystal alpha-Al2O3[END_REF] [START_REF] Chen | Investigation on the origin of the blue emission in titanium doped sapphire: Is F + color center the blue emission center[END_REF], or due to the charge transfer transitions associated with Ti 3+ or Ti 4+ [START_REF] Page | Role of Ti 4+ in the luminescence process of Al2O3:Si,Ti[END_REF][START_REF] Blasse | The luminescence of titanium in sapphire laser material[END_REF][START_REF] Evans | Ubiquitous blue luminescence from undoped synthetic sapphires[END_REF][START_REF] Macalik | Blue emission in Ti-sapphire laser crystals[END_REF], or due to isolated Ti 4+ ions [START_REF] Ruža | Optically detected magnetic resonance studied via the blue luminescence of Ti-doped Al2O3[END_REF]. Summers [START_REF] Summers | Thermoluminescence in single crystal alpha-Al2O3[END_REF] thought that, for emission in the range from 200 to 320 nm in Al2O3, the main emission band around 420 nm is caused by a F-center due to electron relaxation from excited state 3 P to the ground state 1 S. Based on synchrotron radiation experiments conducted in the range 100-400 nm, Chen et al. proposed that the 420 nm emission, whose corresponding excitation band lies at 244 nm, is most probably caused by a F + -type color center rather than Ti 3+ or Ti 4+ ions [START_REF] Chen | Investigation on the origin of the blue emission in titanium doped sapphire: Is F + color center the blue emission center[END_REF]. However, Reisfeld et al. suggested that the blue emission is due to a 4s → 3d transition in Ti 3+ ions [START_REF] Reisfeld | Unusual luminescence of titanium (III) in aluminumoxide[END_REF]. Page et al. pointed out that photoluminescence excitation can occur at energies lower than the band gap of Al2O3 (9.5 eV), leading to a 430 nm emission through charge transfer mechanism [START_REF] Page | Role of Ti 4+ in the luminescence process of Al2O3:Si,Ti[END_REF]. This kind et al., 1995).

of
Through the PL measurement of Ti:sapphire samples prepared under oxidation conditions,

Blasse and Verweij attributed the blue emission at 415 nm to a charge transfer transition from O 2-to Ti 4+ in [TiO6] 8-complex [START_REF] Blasse | The luminescence of titanium in sapphire laser material[END_REF]. This 415 nm emission is also observed by Evans when excited at 230 nm [START_REF] Evans | Ubiquitous blue luminescence from undoped synthetic sapphires[END_REF]. Later, the optically detected magnetic resonance experiment of Ruza et al. provided direct evidence that it is Ti 4+ that leads to the blue-green luminescence in Ti:sapphire [START_REF] Ruža | Optically detected magnetic resonance studied via the blue luminescence of Ti-doped Al2O3[END_REF]. The excitation and emission bands of Ti 4+ in titanium doped sapphire are shown in Figure 1-4 where an emission band at 420 nm is observed when pumped at 230 nm. The presence of Ti 4+ increases the optical loss of Ti:sapphire.

Optical loss

In 1986, Fahey et al. (Fahey et al., 1986a) first time demonstrated that the optical loss of Ti:sapphire is associated with a residual optical absorption in the range of 700 nm -900 nm.

The residual absorption limited its application. Through the absorption spectra measurement for several samples along crystal growth direction (parallel to the c-axis), they got the conclusion that this residual absorption intensity does not exceed 4% of the intensity of the main peak located at 490 nm. Moncorgé et al. pointed out that a residual absorption coefficient around 800 nm less than 0.03 cm -1 is extremely important for a good laser operation (Moncorgé et al., 1988). To investigate the source of this residual absorption band in Ti:sapphire, Aggarwal ratio [START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF]. Finally, their result showed that the residual absorption is largely caused by Ti 3+ -Ti 4+ pairs. This pair model is strongly supported by theoretical calculation based on the analysis of the electronic structure of Ti 3+ -Ti 4+ [START_REF] Zha | Electronic structure of the ion pair model for Ti:Al2O3[END_REF]. Recent first principles calculation shows that the residual infrared absorption is jointly dominated by the line contact Ti 3+ -Ti 3+ pair and face contact Ti 3+ -Ti 4+ pair [START_REF] Gong | Theoretical study on residual infrared absorption of Ti: sapphire laser crystals[END_REF].

Figure 1-5 Absorption (solid line) and emission (dashed line) bands for dopant ions and defects in Ti:sapphire.

Various characteristic absorption and emission bands for ions and defects in Ti:sapphire crystals are summarized in Figure 1-5. It is clear that Ti 3+ has an emission at 730 nm after being excited at 532 nm and harmful Ti 4+ has an emission at 420 nm after being excited at 230 nm.

Theses characteristic spectra not only help to identity the presence of isolated Ti ions, Ti 3+ -Ti 4+ pair and F-center but also enable to obtain the radial distribution profiles of Ti ions. This has been widely used in the characterization of Ti:sapphire samples [START_REF] Alombert-Goget | Titanium distribution profiles obtained by luminescence and LIBS measurements on Ti:Al2O3 grown by Czochralski and Kyropoulos techniques[END_REF]. Additionally, the spectra of Ti 3+ and Ti 4+ are very useful for estimating Ti ion concentrations in grown crystals.

Ti ion concentrations determination through optical spectra

As Ti doping level in Ti:sapphire is as low as a few hundred ppm in weight, currently, no analysis method is able to measure titanium ion concentrations. In reported results, titanium ion concentrations were frequently calculated using optical absorption method. Aggarwal et al.

found that the absorption coefficient of Ti:sapphire samples at 490 nm for the π polarization (E‖c) is proportional to the concentration of Ti 3+ ions [START_REF] Aggarwal | Magnetic and optical measurements on Ti:Al2O3 crystals for laser applications: Concentration and absorption cross section of Ti 3+ ions[END_REF]. It was carried out by measuring the paramagnetic moment dependence on temperature and magnetic field. This polarization experiment yields the cross-section value of Ti 3+ as 𝜎 𝑇𝑖 3+ = (9.3 ± 1.0) × 10 -20 cm 2 . Sometimes, people measure the concentration of Ti2O3 equivalent to the concentration of Ti 3+ through the absorption coefficient as:

[Ti2O3] wt.% = (0.032 ± 0.03)𝛼 𝜆 .

(1-1)

where 𝛼 𝜆 is the absorption coefficient at wavelength 𝜆. For example, the measurements in [START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF][START_REF] Strauss | Growth and characterization of Ti: Al2O3 crystals for laser applications[END_REF][START_REF] Aggarwal | Magnetic and optical measurements on Ti:Al2O3 crystals for laser applications: Concentration and absorption cross section of Ti 3+ ions[END_REF][START_REF] Sanchez | Crystal growth, spectroscopy, and laser characteristics of Ti:Al2O3[END_REF][START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF]. However, the weight percent concentration of Ti2O3 does not equal the concentration of Ti 3+ (see Annex I). According to equation (1-1), the concentration for Ti 3+ should be:

[Ti 3+ ]wt. % = 0.67 × [Ti 2 O 3 ]wt. % ≈ 0.02𝛼 𝜆 .
(1-2)

In fact, above formula was obtained on the basis of the assumption that only Ti 3+ and negligible Ti 4+ ions exist in the crystal which is never the case as will be explained in Chapter 2. In addition, the absorption coefficient in equation (1-1) was measured at λ = 490 nm. Whether it works for other absorption peaks such as 514 nm or 532 nm still needs to be checked. Especially, Aggarwal et al. [START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF] pointed out that 𝛼 532 = 𝛼 490 /1.28. DeShaze and his colleagues believed that the absorption coefficient of 1 cm -1 at 532 nm is produced by 0.07 wt.% Ti 3+ in sapphire [START_REF] Deshazer | Tunable titanium sapphire lasers[END_REF].

Another empirical formula used to compute Ti 3+ concentration [START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF][START_REF] Nizhankovskii | Optical absorption and color centers in large Ti:sapphire crystals grown by horizontally directed crystallization under reducing conditions[END_REF] is:

[Ti 3+ ] wt% = 𝛼 514
18.55 cm -1 .

(1-3) Similar to equation (1-2), this expression gives only a rough estimation, the validity of this empirical formula needs to be verified after determining the absolute concentration of Ti 3+ through other measurements.

DeShazer et al. creatively calculated Ti 3+ concentration using the Frantz-Nodvik equation for saturated green absorption at 532 nm [START_REF] Deshazer | Saturation of green absorption in titaniumdoped sapphire laser crystals[END_REF]. The concentration of Ti 3+ is related to the energy stored in the transition

𝐸 𝑠𝑡𝑜𝑟𝑒𝑑 = -𝑁 𝑇𝑖3+ ℎλ𝐿. (1-4)
where 𝑁 𝑇𝑖3+ is the concentration of Ti 3+ ions per unit volume, ℎ is Planck's constant, and L is the length of Ti:sapphire crystal. This method has a systematic error of ±20% resulting from the beam area measurement. But the cross section at 490 nm obtained by this method is 𝜎 𝑇𝑖 3+ = (6.4 ± 1.4) × 10 -20 cm 2 which is different from the result of Aggarwal et al. [START_REF] Aggarwal | Magnetic and optical measurements on Ti:Al2O3 crystals for laser applications: Concentration and absorption cross section of Ti 3+ ions[END_REF]. The two inconsistent results suggest that one of these two methods or both of them may not be so reliable.

Moskvin et al. evaluated the concentrations for both Ti 3+ and Ti 4+ by comparing their absorption coefficients at 490 nm and 220 nm with the chemical analysis data. Therefore, we suspect that they measured only the total Ti. The direct relation between ion concentrations and their corresponding absorption coefficients are given as [START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF]:

[Ti 3+ ] wt.% ≈ 0.04 ×𝛼 490 (E∥C)

[Ti 4+ ] wt.% ≈ 0.04 ×10 -4 ×𝛼 220 .

(1-5)

where 𝛼 490 and 𝛼 220 are the absorption coefficients for Ti 3+ and Ti 4+ at 490 nm and 220 nm.

But the reliability for these two relations also waits for further checking. As far as we know, the chemical analysis can only give the total concentration for a certain element but not for the absolute concentration of an ion.

In addition, Smakula's equation for Gaussian-shaped absorption bands have been reported

to give the concentration of Ti 4+ as well [START_REF] Smith | Oscillator strengths of defects in insulators-the generalization of smakula's equation[END_REF][START_REF] Nizhankovskii | Optical absorption and color centers in large Ti:sapphire crystals grown by horizontally directed crystallization under reducing conditions[END_REF][START_REF] Evans | Ubiquitous blue luminescence from undoped synthetic sapphires[END_REF]. The equation is described as below:

[Ti 4+ ] = 9 4𝜋𝑓 𝑚 𝑒 𝑐 𝑒 2 ℏ 𝑛 (𝑛 2 + 2) 2 𝛼 220 ∆ 𝐹𝑊𝐹𝑀 = 0.87 × 10 17 𝑛 𝑓(𝑛 2 + 2) 2 𝛼 220 ∆ 𝐹𝑊𝐹𝑀 (1-6)
where [Ti 4+ ] is ion concentration for Ti 4+ , f = 0.5 is the oscillator strength for the absorption at 220 nm, n is refractive index. The reliability for this theoretical result needs to be compared with the absolute concentration of Ti 4+ obtained by other methods, if any.

All above methods used to calculate the concentration of titanium ions are based on the assumption that only Ti 3+ or only Ti 4+ were present in the calibration crystals. Above expressions are either empirical formula or approximation. They are not precisely ascertained and probably varies with the crystal specimen. Therefore, all above known methods remains to be validated and a more reliable, quantitative method is required to get more precise results.

Ti:sapphire quality with respect to laser application

The so-called figure of merit (FOM) is utilized to assess the crystal laser quality. It is defined as the ratio of absorption coefficients at 532 nm and 800 nm, as shown in the expression below (Alombert-Goget et al., 2016):

FOM = 𝛼 532 𝛼 800
.

(1-7)

The two absorption coefficients at 532 nm and 800 nm are related to the concentrations of Ti 3+ and Ti 4+ . The higher the FOM value, the higher optical quality the crystal would exhibit. These two wavelengths may vary depending on the laser source setup, for instance, 𝛼 532 should be changed into 𝛼 514 when the Nd:YAG laser is replaced by an Ar-ion laser. Crystals used for lasers should have a FOM value greater than 100 (Dobrovinskaya et al., 2009a). However,

McKinnie's experiment shows that FOM values for Ti:sapphire decrease when increasing the overall titanium concentration, resulting in a stronger residual absorption in the infrared region [START_REF] Mckinnie | Ti 3+ ion concentration and Ti:sapphire laser performance[END_REF]. So, it is not easy to achieve FOM values superior to 100 for Ti:sapphire crystals containing large amounts of titanium. The residual absorption intensity, within the range 700 to 900 nm, is much lower than the main absorption of Ti 3+ (Fahey et al., 1986a).

Figure 1-6 gives the main and residual absorption spectra of Ti:sapphire.

High power pulsed laser requires high doping concentration for active ions, but experiments [START_REF] Mckinnie | Ti 3+ ion concentration and Ti:sapphire laser performance[END_REF] reveal that excess doping level would severely degrade the FOM value. In order to get the highest FOM, the best Ti concentration is still a matter of debate as it depends on many different parameters. In addition, the optical quality of Ti:sapphire also has much to do with crystal growth method, growth atmosphere, temperature gradient, growth rate… In the next part, various crystal growth techniques will be briefly reviewed. residual absorption band [START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF].

1.3 Ti:sapphire crystal growth methods [START_REF] Zhou | Growth of large-size Ti:Al2O3 tunable laser crystal (Poster Paper)[END_REF], and micro-pulling down method (µ-PD) (Su et al., 2017) have been used to grow crystals as well, but the crystals grown by these two methods show higher dislocation density compared to the crystals grown by Kyropoulos and Czochralski methods. As this research is focused on the growth of large crystals for laser applications, we will focus on current popular bulk crystal growth methods including Verneuil, Czochralski, Kyropoulos and heat exchanger method (HEM). [START_REF] Nassau | Dr. AVL Verneuil: The man and the method[END_REF]. It allows sprayed powder to go through a hydrogen-oxygen flame and then melt and drop on the seed sticking on a crystal holder at the bottom. This crystal growth setup is shown in Figure 1-7. At present, Verneuil method is able to grow crystal as large as 4 cm in diameter and 140 cm in length. Due to the inhomogeneous temperature field in the vertical direction, crystals grown by this method contain high thermal stresses and dislocations, leading to several sub-grain boundaries. Hence, it is only suitable for producing small size, or low quality, samples. However, since this process has no crucible and has no contact with other substance, it allows to produce high purity crystals at low cost. This simple setup is often used in commercial companies to synthesize jewelry, watch bearings, scientific instruments or preparing ultra-pure raw materials for other crystal growth methods. Ti:sapphire crystals are grown from the dimensions of 1.5 mm × 20 mm [START_REF] Kurtz | Solid state reactions and defects in doped Verneuil sapphire . III , Systems -Al2O3:Fe , -Al2O3:Ti and -Al2O3 :( Fe , Ti ) defects in doped Verneuil sapphire[END_REF] and extended to 40 mm × 130 mm (Alombert-Goget et al., 2016) . It is worth mentioning that the very first Ti:sapphire lasers were grown by the Verneuil method. The features of Verneuil method are shown as below:

Advantages of Verneuil method:

1) The crystal growth process has no crucible therefore avoids the contamination from crucible;

2) Fast crystal growth rate;

3) Simple, cheap equipment and high productivity.

Disadvantages of Verneuil method:

1) Grown crystal contains many defects due to the high temperature gradient in the flame;

2) Serious loss of raw material (~30%).

3) Too small diameter for laser applications.

Czochralski method

Czochralski is one of the most important technologies used for producing single crystals from the melt. Current heating ways include inductive and resistive heating in terms of furnace structure. Inductive heating requires less energy input and is therefore used for high melting point (T > 2000 ℃) materials. Resistive heating is used commonly for semiconductor materials with the help of graphite thermal insulation elements into the furnace. Graphite elements can effectively avoid directly heating the crucible itself and then help to provide a homogeneous temperature field. It requires higher energy input. In most cases, for sapphire, Czochralski method grows crystal along the a-axis because it has a lower thermal conductivity and hence has less chance to introduce defects like cracks, residual stresses or dislocations [START_REF] Rudolph | Handbook of crystal growth: Bulk crystal growth[END_REF]. Figure 1-8 presents a Czochralski crystal growth schematic diagram. Since powder occupies a large space in crucible, a compressed block or pellet (mixing of alumina and TiO2rutile) and pure alumina crackles are more preferred for the raw material. In addition, blocks, pellets and crackles provide convenience for putting the raw material in crucible and avoid recharging further raw material which may result in fluctuating crystal growth interface. For sapphire and high melting point oxides, the crucible is typically made of refractive metal such as molybdenum, tungsten, or iridium. However, the price of iridium is much higher than for the two others. So, to reduce the cost of production, most of the companies prefer to use molybdenum crucible. To avoid oxidation damage to the crucible, the furnace is often pumped to high vacuum and flown with appropriate protective gas (generally Ar) during the whole crystal growth process. Generally speaking, flowing Ar gas increases the production cost, whatever the growth method. Czochralski growth process includes five main steps: seeding, necking, shouldering, diameter growing and tailing. The initial step of the growth is to lower down the seed and dip it into the melt. After thermal equilibrium between the seed and the melt is reached, the seed is raised and the temperature of the melt is lowered to induce the crystal growth. Before a stable growth is obtained, growth parameters, such as power, pulling rate and rotation speed should be adjusted carefully during the start stage of shouldering. In this stage, crystal diameter is increased gradually. When the growth becomes stable which is characterized by a constant diameter, the pulling rate is fixed at about several millimeters per hour depending on the temperature gradient at the solid-liquid interface. The convex interface of Czochralski sapphire growth can be flattened by lowering the radial temperature gradient or increasing the axial temperature gradient [START_REF] Perner | Liquid/solid interface profile of melt grown oxide crystals I. Czochralski growth[END_REF]. Rotating crystal at low rates (𝑤 𝑐𝑟𝑦𝑠𝑡𝑎𝑙 ≪ 5 𝑟𝑝𝑚) is also conducive to flattening the interface during Czochralski sapphire growth process. While crucible rotation results in increased interface curvature and counter rotation crucible/crystal has no flattening effect (Stelian & Duffar, 2017). Fang et al. numerically proved that the S/L interface of Czochralski grown sapphire transferred from convexity (toward the melt) to concavity when the crystal rotation rate is increased from 2.5 rpm to 18 rpm. The flat interface was found at a crystal rotation rate of 12 rpm [START_REF] Fang | To investigate interface shape and thermal stress during sapphire single crystal growth by the Cz method[END_REF]. Too high pulling rate and rotation rate could result in higher bubble density [START_REF] Li | Qualitative and quantitative bubbles defects analysis in undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF]. The pros and cons of Czochralski method are listed as below:

Advantages of Czochralski method:

1) It is possible to directly observe the crystal growth process and control growth parameters;

2) Crystal has no contact with crucible wall and is therefore less exposed to thermal stress;

3) It has a relatively fast crystal growth speed; 4) Grown crystals have high optical homogeneity.

Disadvantages of Czochralski method:

1) Grown crystals may be contaminated by crucible material;

2) The cooling of the crystal in the atmosphere upwards produce a rather high temperature gradient and consequently high dislocation density;

3) Crystal is grown from the melt and there is no shaping device to ensure a well-defined diameter. Hence diameter control is tedious [START_REF] Duffar | High-temperature conductivity and creep of polycrystalline Al2O3 doped with Fe and/or Ti[END_REF].

Kyropoulos method

Kyropoulos (Ky) growth method is designed and developed for decades. Nowadays, it is one of the important methods for producing large sapphire crystals. The main features of this method are that the crystal is grown under extremely low temperature gradient in the hot zone and the crystallization interface has a sharp convex (to the melt) shape. The crystal growth process of this method is sketched in Figure 123456789. Raw materials are melted in the crucible. Crystallization process starts when the temperature at the interface is lower than Al2O3 melting point (2050 ℃). Then crystal becomes larger due to the low temperature gradients in the radial and axial directions which are controlled by the heating power decrease. During the whole crystal growth process, a few critical operations should be addressed carefully, such as the seeding, initial expansion of shouldering, and at the end, when the last amount of melt crystalizes on the crystal and the crystal reaches the bottom of the crucible. Therefore, an automatic control system is used to prevent crystal diameter from increasing too fast. Meanwhile, the temperature should be adjusted cautiously to avoid melt super-cooling. Contrarily to the Czochralski technique, the pulling rate is very small and the growth is rather controlled by a slow cooling of the crucible and melt. Figure 2 in the "General introduction" shows a Kyropoulos crystal grown by the company RSA.

Numerical modeling suggests that the shape of the crystal-melt interface during the Kyropoulos growth of Ti-doped sapphire is significantly influenced by the internal radiative effect and forced convection. Marangoni convection further enhances the flow near the triple solid-liquid-gas point, leading to a convex (toward the melt) shaped interface with a small curvature [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF] as shown in Figure 1-9. It increases after, making the growth interface become convex-concave with a larger interface curvature. The interface curvature depends on absorption coefficient, which increases with increasing dopant concentration (Stelian et al., 2017). The curvature impacts facet formation on growth interface and subsequent defect incorporation [START_REF] Bruni | Crystal growth of sapphire for substrates for high-brightness, light emitting diodes[END_REF]. Kyropoulos furnace has a low temperature gradient, which is prone to facet formation. Therefore, a slow growth rate is required to avoid faceting and related defects. High curvature can cause large radial segregation phenomenon (section 1.4.4), impairing crystal optical quality. Large temperature gradient is prone to creating a stable growth interface [START_REF] Liu | Study on crystal-melt interface shape of sapphire crystal growth by the KY method[END_REF]. In addition, Chen et al. and Lee et al. show that a crucible with curved bottom can lower the convexity of growth interface during Kyropoulos growth of sapphire single crystal [START_REF] Chen | Effect of crucible shape on heat transport and melt-crystal interface during the Kyropoulos sapphire crystal growth[END_REF][START_REF] Lee | Effect of crucible geometry on melt convection and interface shape during Kyropoulos growth of sapphire single crystal[END_REF].

In 2009, a huge pure sapphire crystal with a diameter of 420 mm and weight of 200 kg was grown by Rubicon Technology Company in Russia [START_REF] Harris | Evolution of the sapphire industry: Rubicon Technology and Gavish[END_REF]. Since there is no or very small pulling for the crystal and the crystal has no contact with the crucible, crystals grown by Kyropoulos method have fewer thermal stresses and contamination. Therefore, Kyropoulos method not only possess the advantages of Czochralski method, but also is able to produce larger size and higher-quality sapphire crystals. Experimental results show that dislocation density in sapphire crystal grown in the c-direction by Kyropoulos method is 20 -100 times lower than that in the crystals grown by Czochralski method [START_REF] Hur | The influence of crucible and crystal rotation on the sapphire single crystal growth interface shape in a resistance heated Czochralski system[END_REF]. This method is also employed in popular sapphire doped laser crystals. In 2011, A. Nehari and his colleagues succeeded to utilize this method to grow high quality titanium doped sapphire with a diameter of 100 mm and a weight of 5 kg [START_REF] Nehari | Ti-doped sapphire (Al2O3) single crystals grown by the kyropoulos technique and optical characterizations[END_REF]. The features of Kyropoulos method are shown as follows:

Advantages of Kyropoulos method:

1) Larger crystals;

2) Crystal has no contact with crucible during crystal growth process thus reduces the stress in grown crystal;

3) Grown crystal has low dislocations compared to Czochralski method.

Disadvantages of Kyropoulos method:

1) It is easy to cause bubbles;

2) Lower growth rate compared to Cz.

Heat Exchanger Method

Aforementioned crystal growth methods, Verneuil and Czochralski, cannot meet the large size, flexibility and economy requirements for sapphire mass production market. Therefore, in 1970, a new technique -heat exchanger method (HEM), invented by Schmid and Viechnicki is initially used for the growth of large sapphire crystals based on a gradient furnace technique [START_REF] Schmid | Growth of sapphire disks from the melt by a gradient furnace technique, Army Materials and mechanics research center watertown ma[END_REF]. Later, this method was adapted for the growth of other laser crystals such as ruby [START_REF] Schmid | Large crystal sapphire optics[END_REF] and Nd:YAG [START_REF] Caslavsky | Melt growth of Nd: Y3Al5O12 (Nd: YAG) using the heat exchange method (HEM)[END_REF] crystals. The feature of the HEM method is vertical directional crystallization starting from the crucible bottom, in a very low temperature gradient. A schematic diagram of the HEM method is depicted in Figure 1-10.

Figure 1-10 Schematic diagram of Heat Exchanger Method [START_REF] Rahab | Growth by the heat exchanger method of NaBiW2O8 and Na5B2P3O13 crystals[END_REF].

The materials to be melted are contained in a molybdenum crucible, which is seated on top of a heat exchanger. A single crystal seed is centered at the bottom of the crucible in contact with the heat exchanger. The system is heated by RF-coil after evacuating the air and/or filling with desired gas. In the heating process, caution need to be taken to prevent the seed from melting because it is totally covered by a huge melt volume. The helium that flows at the crucible bottom cools down the melt and takes heat away. Then the melt starts to solidify on the seed. For a given material, temperature gradient in the liquid phase (melt) is decided by the crucible temperature, and temperature gradient in solid phase (grown crystal) is determined by the heat exchanger temperature [START_REF] Khattak | Growth of laser crystals by heat exchanger method (HEM)[END_REF]. During the whole growth process, the colder solid-liquid interface is below the hot melt, this provides a stable temperature field and weak convection. Further, both seed and crucible do not move during the growth process.

These features minimize growth striations and concentration fluctuations of the dopant. In this technique, the driving force for growth comes from the temperature gradient at the solid-liquid interface. This method can be employed for many crucible shapes and sizes such as flat shaped or conical crucibles. HEM now is a standard commercial method to produce large size sapphire up to 340 mm in diameter and 105 kg in weight, as shown in Figure 1-11. In 1987, HEM method was adapted to grow 100 mm diameter Ti:Al2O3 laser crystals [START_REF] Khattak | Growth of laser crystals by heat exchanger method (HEM)[END_REF]. In 2010, D.B.

Joyce and F. Schmid succeeded to produce a 250 mm diameter Ti:sapphire (see Figure 1 -11) and first time achieved petawatt laser operation in mode locked laser system, enabling the system to deliver pulse widths shorter than 10 fs [START_REF] Joyce | Progress in the growth of large scale Ti:sapphire crystals by the heat exchanger method (HEM) for petawatt class lasers[END_REF]. Despite HEM method possesses so many advantages, it is not always selected by crystal growers because of its very low growth rate and because the Mo crucible would be destroyed after each run. The features of HEM method are listed as follows:

Advantages of HEM method:

1) No need to move crystal, crucible and heating zone therefore reduces vibrations and convection;

2) Grown crystal possesses less cracks and dislocations due to low thermal stress.

Disadvantages of HEM method:

1) Slow crystal growth speed;

2) The crucible should be peeled of the crystal and then is destroyed.

Due to the availability of crystal growth equipment and crystal growth skills of operators, all

the samples discussed in this thesis are grown either by Kyropoulos or Czochralski techniques.

Solute segregation during crystal growth

Ti:sapphire laser efficiency is related to the concentration ratio of Ti 3+ and Ti 4+ . The higher the ratio, the better the optical properties would be. Ti ion concentrations in the solid phase depend on the concentration in the liquid phase and their segregation coefficients during crystal growth.

Equilibrium segregation coefficient

As crystal solidification from the melt is a slow process (typical growth rates are about 1-10 µm.s -1 ), it is assumed that there is an equilibrium between the solid and the liquid at crystal growth interface. For a raw material intentionally doped with impurities, large ionic radii differences between host material and dopants can cause a difference in concentration on both sides of the interface. The equilibrium segregation coefficient 𝑘 0 is defined as the ratio of solute concentrations in solid (𝐶 𝑠 * ) and liquid (𝐶 𝐿 * ) at solid-liquid interface as shown in equation (1-8).

𝑘 0 = 𝐶 𝑠 * 𝐶 𝐿 * . (1-8)
When solidus and liquidus are straight lines originating from the pure solvent (see Figure 1-12), 𝑘 0 is equilibrium segregation coefficient. Depending on the solute character, 𝑘 0 can be classified into three cases: 𝑘 0 > 1, 𝑘 0 = 1 and 𝑘 0 < 1. Here, 𝑘 0 = 1 results in a uniform distribution of solute in grown crystal. A segregation coefficient with 𝑘 0 > 1 is called positive segregation and with 𝑘 0 < 1 it is called negative segregation. For most of the single crystals and dopants, 𝑘 0 < 1. In the following discussing, 𝑘 0 is assumed to be less than unity.

Consider a crystal solidification process along horizontal direction and crystal length is L.

Under the so-called "equilibrium solidification", it is believed that there is infinitely fast diffusion in the solid phase and infinitely fast diffusion in the liquid phase. Assume the initial concentration of the solute in the liquid is 𝐶 0 . Figure 1-13 (a) shows the solute segregation at the beginning of solidification. At beginning, the solute concentration in the solid is 𝑘 0 𝐶 0 which is smaller than the initial homogeneous liquid concentration 𝐶 0 . When solidification proceeds, at temperature T* (Figure 1-13 (b)), the diffusions in solid and liquid phases are complete therefore the solute concentrations in the solid (𝐶 𝑠 ) and in liquid (𝐶 𝐿 ) are uniform, i.e. 𝐶 𝑠 = 𝐶 𝑠 * and 𝐶 𝐿 = 𝐶 𝐿 * . Under any given temperature, the sum of solute in the solid and in the liquid always equals the initial solute content:

𝐶 𝑠 𝑓 𝑠 + 𝐶 𝐿 𝑓 𝐿 = 𝐶 0 , (1-9)
where 𝑓 𝑠 and 𝑓 𝐿 are solid and liquid fractions, and 𝑓 𝑠 + 𝑓 𝐿 = 1. It needs to pay attention that the material is homogeneous only before and after the solidification but not during the solidification process.

Longitudinal segregation

In practice, equilibrium solidification does not occur in crystal growth from the melt because the diffusion in the solid is very slow: generally, this diffusion is neglected.

When only diffusion exists in the melt, the solute starts to solidify with the concentration of 𝑘 0 𝐶 0 , which is smaller than the initial melt concentration 𝐶 0 . For species with negative segregation coefficient (𝑘 0 <1), the solute is rejected back from the solid, resulting in higher solute concentration in the melt. Therefore, solute concentration in the solid increases to 𝐶 0 , as shown in Figure 1-14. Simultaneously, liquid concentration becomes from 𝐶 0 /𝑘 0 to 𝐶 0 . Since both solute concentration in the solid at solid-liquid interface and solute concentration in bulk melt are 𝐶 0 , solute effective segregation coefficient in this case is When strong convection exists in the melt, the solute in the liquid stays homogeneous all along the growth. The solidification at the beginning and at temperature T* are shown in Figure 1-15. The solute in the solid formed at the beginning stage is 𝑘 0 𝐶 0 . With the advancing of crystallizing interface, solute in liquid becomes richer and correspondingly the solid formed at later stage contains higher solute concentration. Since there is no diffusion in solid, then the solute in the solid at solid-liquid interface does not equal to the solute in the solid formed in early stage, i.e. 𝐶 𝑠 ≠ 𝐶 𝑠 * . So finally, the solute distribution in the solid is progressively increased along the crystal growth direction. In uniform liquid phase, the relation 𝐶 𝐿 * = 𝐶 𝐿 is still valid.

𝑘 𝑒𝑓𝑓 = 𝐶 𝑠 * 𝐶 𝐿 = 1,
According to the solute conservation, the solute increased in the solid should equal to the solute lost in the liquid, i.e.

(1 -𝑓 𝑠 )𝑑𝐶 𝐿 = (𝐶 𝐿 -𝐶 𝑠 * )𝑑𝑓 𝑠 .

(1-10)

Performing integration for above equation and combining with the boundary condition

𝐶 𝑠 * = 𝑘 0 𝐶 0 | 𝑓 𝑠 =0 , (1-11)
the solute in the solid at solid-liquid interface can be written as:

𝐶 𝑠 * = 𝑘 0 𝐶 0 (1 -𝑓 𝑠 ) (𝑘 0 -1) . (1-12)
Or rewrite in terms of liquid concentration and liquid fraction: 1) .

𝐶 𝐿 = 𝐶 0 𝑓 𝐿 (𝑘 0 -
(1-13)

Equations (1-12) and(1-13) are termed as Scheil-Gulliver equation or nonequilibrium lever law.

It needs to be stressed that this equation is only valid when below assumptions are satisfied:

(1) No (negligible) diffusion in solid phase but complete homogenization in liquid phase (uniform solute in liquid);

(2) There is an equilibrium at the solid-liquid solidification interface;

(3) Solidus and liquidus lines are straight in the phase diagram;

(4) Segregation coefficient 𝑘 0 is constant. [START_REF] Flemings | Solidification processing[END_REF].

The Scheil-Gulliver equation is often used to predict the solute distribution along longitudinal direction in crystal grown from the melt. Especially for crystals grown by Cz and

Ky methods because the rotation and high temperature at the bottom of the liquid lead to strong convection thereby homogeneous solute distribution in the liquid.

Boundary layer and effective segregation coefficient

When the convective mixing in the liquid is low and the growth rate is higher than the solute diffusion rate in the liquid phase, the solute rejected from the solid does not have enough time to be homogenized in the liquid. This is the most important practical case in crystal growth from the melt. Then a large amount of solute, rejected by the solid, will enrich in front of the solid-liquid interface, forming a boundary layer in the liquid. Denote the diffusion boundary layer thickness in the melt in front of the growth interface as δ (see Figure 1-16). For small growth rate, δ mainly depends on crystal rotation rate 𝜔 (Burton et al., 1953 a):

𝛿 = 1.6𝐷 𝐿 1/3  𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 1/6 𝜔 -1/2 . (1-14)
where 𝜈 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 is kinematic viscosity of the melt and 𝐷 𝐿 is the solute diffusion coefficient in the liquid. Here, 𝐷 𝐿 is assumed to be independent of concentration in the dilute solution.

During crystal growth, natural or forced convection may exist in the liquid phase. This convection tends to level the solute concentration in the liquid. A common hypothesis is that the solute inside the boundary layer in the liquid is only transported by diffusion while outside of this region, the solute in the liquid is transported by convection [START_REF] Flemings | Solidification processing[END_REF]. The stronger the convection, the more homogeneous the solute would be. After a while, a dynamic equilibrium in the boundary layer is reached: when the interface is advancing, the solute content entering the diffusion boundary layer equals to the solute exiting from it.

Figure 1-16 Solute segregation in solidification with diffusion and weak convection in the liquid (adapted from [START_REF] Flemings | Solidification processing[END_REF]).

Note that the solute concentration within this boundary layer is higher than the concentration in the melt. This pushes the solute diffusion into the melt. In this case, the liquid phase is not uniform, so the Scheil-Gulliver model is not suitable for describing the solute distribution during solidification any more. An effective segregation coefficient, 𝑘 𝑒𝑓𝑓 , is defined as the ratio of solute concentration in the solid at solid-liquid interface and the solute concentration in the bulk liquid, as following

𝑘 𝑒𝑓𝑓 = 𝐶 𝑠 * 𝐶 𝐿 . (1-15)
Solute conservation in the system gives a differential equation similar to (1-10) which gives, after integration: 1) .

𝐶 𝑠 * = 𝑘 𝑒𝑓𝑓 𝐶 0 (1 -𝑓 𝑠 ) (𝑘 𝑒𝑓𝑓 -
(1-16)

𝐶 𝐿 = 𝐶 0 𝑓 𝐿 𝑘 𝑒𝑓𝑓 -1 . (1-17)
The effective segregation coefficient can be written as the function of equilibrium segregation coefficient through the well-known BPS equation (Burton et al., 1953a;Burton et al., 1953b)

𝑘 𝑒𝑓𝑓 = 𝑘 0 𝑘 0 +(1-𝑘 0 )𝑒 -(𝑉 𝑔𝑟𝑜𝑤𝑡ℎ 𝛿/𝐷 𝐿 ) , (1-18)
Equation (1-18) suggests that convection can affect the solute distribution via affecting the boundary layer thickness. There exist two extreme cases for equation (1-18):

(1) When 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ → 0 or 𝛿 → 0, 𝑘 𝑒𝑓𝑓 → 𝑘 0 . In this situation, crystal solidification rate is close to a quasi-static state process or there is a strong convection in the liquid phase which suggests that the liquid is "completely mixed".

(2) When 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ → ∞ or 𝛿 → ∞, 𝑘 𝑒𝑓𝑓 → 1. In this case, the mass transport in liquid is only diffusion controlled. The convection in the liquid is much smaller than the growth rate and the boundary layer is a purely diffusive one. The solute approaches to a uniform distribution in the grown crystal, equal to C0. We can consider the melt has "No mixing" due to the purely diffusive, slow movement of the solute.

However, in practical crystal growth process, when convection is not very strong, the liquid is "partially mixed" and the effective segregation coefficient occurs between these two extreme cases, i.e. 𝑘 0 < 𝑘 𝑒𝑓𝑓 < 1. Theoretical solute distribution profiles in solid for above three cases are plotted in Figure 1-17.

Figure 1-17 Solute axial distribution profile in crystal with "No mixing", "Partial mixing" and "Complete mixing" in melt [START_REF] Stefanescu | Science and engineering of casting solidification[END_REF].

Radial segregation

Effect of interface curvature

Solute radial homogeneity is of significant importance for crystals because most of the crystals are manufactured as wafers which are cut perpendicular to the crystal growth direction.

Different factors can influence the radial segregation during crystal growth, for instance, curved growth interface, convection, variations in boundary layer thickness, facets and magnetic fields [START_REF] Carlberg | A review of radial segregation in crystal growth during microgravity[END_REF][START_REF] Garandet | Convection related radial segregation in an idealized horizontal Bridgman configuration; the quasi diffusive regime limit[END_REF][START_REF] Coriell | Lateral solute segregation during undirectional solidification of a binary alloy with a curved solid-Liquid interface[END_REF]. Many papers have studied the solid-liquid interface shape during the early stage of crystallization process [START_REF] Lee | Effect of crucible geometry on melt convection and interface shape during Kyropoulos growth of sapphire single crystal[END_REF][START_REF] Fang | Study of melt convection and interface shape during sapphire crystal growth by Czochralski method[END_REF][START_REF] Chen | Effect of crucible shape on heat transport and melt-crystal interface during the Kyropoulos sapphire crystal growth[END_REF][START_REF] Liu | Study on crystal-melt interface shape of sapphire crystal growth by the KY method[END_REF][START_REF] Wu | Phase transformation of TiO2 precipitates in sapphire (α-Al2O3) induced by the loss of coherency[END_REF] or the interface effect on titanium distribution (Stelian et al., 2017) through numerical modeling and experiment. In most of crystal growth processes, radial segregation is mainly determined by growth interface curvature and convection. The thermal conductivity difference between the solid and the liquid causes curved growth interface [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF].

Radial segregation

The horizontal solidification model presented in section 1.4.2 cannot be used to explain the radial segregation. Therefore, we take Figure 1-18, the Kyropoulos crystal growth process as an example to illustrate the details. The solid-liquid interface A (dashed line) is solidified first and interface B (solid line) is solidified after. Assume any point on solid-liquid interface A has a solute concentration of 𝐶 𝑠𝐴 * and a solidified fraction 𝑓 𝐴 . Assume 𝐶 𝑠𝐵 * and 𝑓 𝐵 are the solute concentration and solidified fraction at interface B. Then for a wafer cut transversely, the maximum radial segregation would be the solute concentration difference between the edge and the center (two blue dots in Figure 1-18). According to the "partial mixing" equation (1-16),

𝐶 𝑠𝐴 * = 𝑘 𝑒𝑓𝑓 𝐶 0 (1 -𝑓 𝐴 ) 𝑘 𝑒𝑓𝑓 -1 𝐶 𝑠𝐵 * = 𝑘 𝑒𝑓𝑓 𝐶 0 (1 -𝑓 𝐵 ) 𝑘 𝑒𝑓𝑓 -1 . (1-19)
Therefore, the radial segregation is [START_REF] Olson | Handbook of crystal growth[END_REF])

∆𝐶(𝑓) = 𝐶 𝑠𝐵 * -𝐶 𝑠𝐴 * = 𝑘 𝑒𝑓𝑓 𝐶 0 (𝑘 𝑒𝑓𝑓 -1)(1 -𝑓 𝐴 ) 𝑘 𝑒𝑓𝑓 -2 ∆𝑓.
(1-20)

where ∆𝑓 = 𝑓 𝐵 -𝑓 𝐴 . Denote total crystal length as 𝐿 and interface deflection between two interfaces as ∆𝑍. Z is directional to the crystallization direction. Then ∆𝑓 can be written as the function of interface deflection as:

∆𝑓 = ∆𝑍 𝐿 .
(1-21)

In the low curvature range, radial segregation is linearly increasing with the solidified length. Coriell et al. analytically relate radial segregation with interface curvature [START_REF] Coriell | Lateral solute segregation during undirectional solidification of a binary alloy with a curved solid-Liquid interface[END_REF] in the diffusive case

∆𝐶 = 𝑉 𝐷 𝐿 (𝑘 𝑒𝑓𝑓 -1)𝐶 𝐿 [𝑤(𝑥) -𝑤(0)]. (1-22)
where 𝑤(𝑥) denotes the shape of solid-liquid interface by

𝑤(𝑥) = 𝐿 2 24𝑅 - 𝑥 2 2𝑅
.

(1-23)

where 𝑅 is the radius of interface curvature. This relation is calculated based on the assumption that there is no convection (full diffusion) in the liquid. In this case, even if the interface is planar, radial segregation still exist. With the moving of the interface, radial segregation will be controlled by the diffusion boundary layer thickness [START_REF] Olson | Handbook of crystal growth[END_REF]. For doped crystals such as Ti:sapphire, it has a higher absorption coefficient which is contributing to effective thermal conductivity. Therefore, Ti:sapphire exhibits larger interface curvature compared with pure sapphire during the growth process [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF].

Figure 1-18 Radial segregation at solid-liquid interface.

Effect of melt convection

In addition to heat transfer interface curvature, natural convection driven by buoyancy, forced convection driven by crystal rotation and Marangoni convection driven by surface tension contribute to the interface shape. The modeling result of C. Stelian et al. for Ti:sapphire grown by Kyropoulos method proved that the Marangoni flow can strengthen the buoyancy flow along crystal axis, resulting in larger interface curvature [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF], i.e. larger radial segregation. The intensity of Marangoni convection is determined by the temperature gradient at free melt interface (Stelian et al., 2017).

Radial segregation is found to be sensitive to growth conditions such as temperature gradient, growth velocity and so on [START_REF] Chang | Radial segregation induced by natural convection and melt/solid interface shape in vertical Bridgman growth[END_REF]. Therefore, for the sake of obtaining minimum radial segregation, crystal pullers should find appropriate growth parameters and keep the growth interface as flat as possible. Through numerical simulation, Lan and his coworkers found that rotating the system about the growth axis may be a good way to control axial and radial segregations for crystal growth with a concave interface [START_REF] Lan | Three-dimensional analysis of heat flow, segregation, and interface shape of gradient-freeze crystal growth in a centrifuge[END_REF].

Growth striations

Growth striations, as can be seen by naked eyes in Figure 1-19, are considered as an inherent problem in crystal growth, especially in doped crystals. The profile of Ti element is consistent with the solid-liquid interface as shown in Figure 1-9. These striations are induced by growth interface oscillatory movement due to convection or growth rate variations [START_REF] Scheel | Historical aspects of crystal growth technology[END_REF][START_REF] Yeckel | Effect of accelerated crucible rotation on melt composition in highpressure vertical Bridgman growth of cadmium zinc telluride[END_REF]. Kim et al. systematically studied the influence of thermal behavior of the melt on the interface morphology and microscopic growth rate of growing crystal [START_REF] Kim | Crystal growth from the melt under destabilizing thermal gradients[END_REF]. The convection period results in growth interface exhibiting particular regularity [START_REF] Zhu | In situ visualization of the quasi-periodic crystal growth interface fluctuation by growth interface electromotive force spectrum in a Czochralski system[END_REF]. The subtle changes of growth interface are hardly to be measured by normal sensors, but it is feasible through the in-situ growth interface electromotive force spectrum. It not only allows to reveal the interface fluctuation but also enables to reveal striations formation mechanism and perform real-time feedback control for the interface. Zhu et al. quantitatively relate the pattern of electromotive force to the distribution of striations in Czochralski crystal growth [START_REF] Zhu | In-situ detection of growth striations by crystallization electromotive force measurement during Czochralski crystal growth[END_REF]. Another type of striations, perpendicular to the solid-liquid interface, are due to interface destabilization. The criterion for stable growth interface is given as [START_REF] Tiller | The redistribution of solute atoms during the solidification of metals[END_REF]:

∆𝑇 𝐿 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ > -𝑚 𝐿 𝐶 𝐿 (1-𝑘 𝑒𝑓𝑓 ) 𝑘 𝑒𝑓𝑓 𝐷 𝐿 . (1-24)
where ∆𝑇 𝐿 is the temperature gradient in the melt at the interface and 𝑚 𝐿 is the slope of liquidus line in the phase diagram. From this criterion, one can get the conclusion that stable growth interface can be maintained by either increasing the temperature gradient or decreasing the growth velocity. Sharp and his colleagues found that titanium concentration determines when the melt destabilization would begin during the crystallization run [START_REF] Sharp | Instability in the growth of Ti: Al2O3 single-crystal fibres[END_REF]. It was shown that once titanium present in the melt exceed 5.8 wt% the melt would be susceptible to a wavy disturbance in the melt free surface and the temperature fluctuation. However, as can be seen in equation (1-24), this critical value is likely to change for other growth rate or temperature gradient. To obtain high quality crystals, considerable efforts have been made to minimize the striations. General approach includes growing crystal under microgravity condition (space experiment), using accelerated crystal/crucible rotation technique [START_REF] Camp | Resistivity striations in germanium crystals[END_REF][START_REF] Choi | Suppression of temperature oscillation in Czochralski convection by superimposing rotating flows[END_REF][START_REF] Scheel | Theoretical and technological solutions of the striation problem[END_REF], or applying external magnetic field in the metallic melts [START_REF] Choe | Growth striations and impurity concentrations in HMCZ silicon crystals[END_REF].

Ti segregation coefficient in sapphire

The segregation coefficient for most of the ions are smaller than one, therefore, the dopants are usually rejected back to the melt during the solidification of impurity doped crystal grown from the melt. The major challenge for growing Ti:sapphire is the ionic radii difference between the host ion Al 3+ (0.53 Å in tetrahedral site and 0.675 Å in octahedral site) and the active ion Ti 3+ (0.81 Å in octahedral site) [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF]. This can result in severe longitudinal segregation phenomenon, impairing the optical properties of the crystal. The effective segregation coefficient of Ti has been discussed in different literature. [START_REF] Kokta | Growth of oxide laser crystals[END_REF] In this table, Ti segregation coefficient spans in the range 0.05 ~ 0.23. The variation may come from different crystal growth conditions. Therein, 𝑘 𝑇𝑖 𝑒𝑓𝑓 ≈ 0.2 always appears in Ti:sapphire crystals grown by HEM, Ky, Cz and µ-PD techniques [START_REF] Ning | Growth and characterization of large-scale Ti:sapphire crystal using heat exchange method for ultra-fast ultra-high-power lasers[END_REF][START_REF] Nehari | Ti-doped sapphire (Al2O3) single crystals grown by the kyropoulos technique and optical characterizations[END_REF][START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF][START_REF] Nehari | Chemical segregation of titanium in sapphire single crystals grown by micro-pulling-down technique: Analytical model and experiments[END_REF]. At the same time, the dependence of Ti concentration in solid on solidified fraction in all these papers are fitted using partial mixing equation (1-16), so that a 𝑘 𝑇𝑖 𝑒𝑓𝑓 coefficient is obtained. In addition, effective 𝑘 𝑇𝑖 𝑒𝑓𝑓 = 0.2 is frequently employed in Ti:sapphire crystal growth simulation process. Hu et al. and Nehari et al. found that their experimental Ti segregation profiles match well with the theoretical profile calculated by taking 𝑘 𝑇𝑖 𝑒𝑓𝑓 = 0.2 into account. Above analysis indicates that 0.2 is a reliable value for the Ti effective segregation coefficient in sapphire and this value does not seem to be very sensitive to the particular crystal growth method.

According to equation (1-16), effective segregation coefficient values of Ti like 0.1 or between 0.1 and 0.2 could be caused by different crystal growth conditions such as a faster growth rate or rotation rate that could decrease boundary layer thickness . H.M. Dess [START_REF] Dess | Melt growth of refractory oxide single crystals[END_REF] pointed out that slow growth rate and rotation rate tend to increase the segregation of dopants, while fast growth rate and rotation rate tend to decrease dopant segregation. Su et al.

( Su et al., 2017) also proved that high pulling rate can prevent the back diffusion of Ti into the melt in their simulation results. In addition, the simulation model of Nehari et al. [START_REF] Nehari | Chemical segregation of titanium in sapphire single crystals grown by micro-pulling-down technique: Analytical model and experiments[END_REF] clearly suggest that fast growth rate results in a stationary concentration, however, slow growth rate gives rise to Scheil-Gulliver like segregation. The values lie on the last three lines of the table are reported as well. But there is no more information about crystal growth details that can be dug out from the published literature.

At the very beginning of solidification, solid fraction 𝑓 𝑠 = 0 and titanium concentration in the liquid equals to its initial concentration 𝐶 0 . In this case, the effective segregation coefficient equals to the equilibrium segregation coefficient. So, the equilibrium segregation coefficient can be evaluated from the ratio of Ti concentration in the solid at the beginning of solidification to its initial concentration. It follows from the expression in equation (1-16) that

𝑘 0 = 𝐶 𝑠 * 𝐶 0 .
(1-25)

Strauss et al. [START_REF] Strauss | Growth and characterization of Ti: Al2O3 crystals for laser applications[END_REF] estimated Ti equilibrium segregation coefficient to be 0.079 and 0.087 in sapphire doped with 0.5 wt.% Ti2O3 and 0.3 wt.% Ti2O3 grown by gradient freeze technique. In practical, it is difficult to find the exact position where the melt starts to solidify.

Before solidifying, the bottom of the seed is melted a bit which dilutes the Ti concentration in the melt. So, Ti incorporated into the crystal at the very beginning is lower than its initial concentration.

Currently, all the measured values are for effective segregation coefficients of total Ti, no studies concerning Ti 3+ and Ti 4+ ions segregation phenomenon have been considered. If the segregation coefficients for both of these two ions could be obtained, it would enable us to compute the concentration of Ti ions incorporated into the solid phase.

1.5 Distributions of Ti 3+ and Ti 4+ in sapphire crystals

Numerous large size Ti:sapphire single crystals have been grown by Czochralski [START_REF] Fukuda | Growth of bubble-free Ti-doped Al2O3 single crystal by the czochralski method[END_REF], Kyropoulos [START_REF] Nehari | Ti-doped sapphire (Al2O3) single crystals grown by the kyropoulos technique and optical characterizations[END_REF] and HEM [START_REF] Joyce | Progress in the growth of large scale Ti:sapphire crystals by the heat exchanger method (HEM) for petawatt class lasers[END_REF] techniques, but there are obstacles for producing high quality crystals. Due to the influence of ion segregation, growth interface curvature, convection, and growth velocity as discussed in section 1.4, it is impossible to get homogeneous Ti distribution along longitudinal and radial directions.

The redox reactions governed by the growth atmosphere makes it not easy to control the valence of titanium (Ti 3+ and Ti 4+ ) during the growth [START_REF] Alombert-Goget | Scattering defect in large diameter titanium-doped sapphire crystals grown by the Kyropoulos technique[END_REF].

Table 1-5 Ion radii of Al 3+ , Ti 3+ and Ti 4+ in octahedral sites [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF].

Ions Al 3+ Ti 3+ Ti 4+
Radius (nm) 0.068 0.081 0.056

Even if the radius of Ti 4+ is smaller than Ti 3+ (see Table 12345), the solubility of Ti 3+ in sapphire is higher than Ti 4+ [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF][START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF][START_REF] Roy | Solubilities of magnesia, titania, and magnesium titanate in aluminum oxide[END_REF].

This is certainly linked to the fact that the substitution of Ti 4+ for Al 3+ requires the formation of aluminum vacancy or oxygen interstitial for charge compensation while the substitution between Ti 3+ and Al 3+ does not need any extra charge compensation (refer to Chapter 3). So, for a same amount of Ti 3+ and Ti 4+ in the liquid, more Ti 3+ is expected in grown crystals than In addition, it was found that titanium ion concentration profiles may vary along the crystal length. Figure 1-22 gives the radial luminescence profiles of Ti 3+ cut from different positions along a Kyropoulos Ti:sapphire crystal. Both top and middle samples show higher Ti 3+ concentration at the edge than at the center due to the curvature of growth interface. The reason causing the distribution profile in the bottom sample is not clear. It may be associated with crystal growth condition changes when the S/L interface is close to the crucible bottom but more evidence needs to be provided.

Also, there are other concentration distribution profiles for Ti 3+ and Ti 4+ reported by Alombert-Goget and his colleagues in Ti:sapphire single crystals grown by Verneuil and Czochralski methods. They are presented in Figure 1-23. By comparing, it is found that the inverse distribution for Ti 3+ and Ti 4+ is only observed in crystal grown by Kyropoulos method.

In Verneuil or Czochralski crystals, Ti 3+ and Ti 4+ have the same distribution profile. Currently, the mechanisms causing the opposite distribution profiles for Ti 3+ and Ti 4+ and the reasons why the profile varies with sample position are not clearly understood. The inhomogeneous distribution of Ti 3+ ions, the existence of undesired Ti 4+ ions and the induced defects resulting from heterogeneous ion doping make it difficult to produce high quality Ti:sapphire crystals for laser operation.

Crystal optical quality dependence on titanium ion concentrations

Depending on crystal growth conditions, such as growth atmosphere, crucible material or aluminum oxide dissociation at elevated temperature, desirable Ti 3+ may be oxidized to Ti 4+ in grown crystals. As discussed in section 1.2.3, Ti 3+ and Ti 4+ concentrations can affect the FOM value by influencing the absorption coefficients at 532 nm and 800 nm. Therefore, FOM value represents the indirect measurement of ion concentration ratio of Ti 3+ and Ti 4+ .

In 1980, Moskvin and his colleagues carefully analyzed the concentration distribution profiles for Ti 3+ , Ti 4+ and total titanium in a series of Ti:sapphire samples doped with varying titanium concentration [START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF]. Through optical absorption, luminescence and chemical measurements of samples annealed under oxygen and vacuum conditions, they obtained the conclusion that the intensity of the residual absorption and the luminescence at 410 nm region is determined not only by the concentration of Ti 4+ and cationic vacancies, but also by the concentration ratio of [Ti 4+ ]/[Ti 3+ ]. Numerous Ti:sapphire crystals grown under different atmosphere have shown concentration ratio of [Ti 4+ ]/[Ti 3+ ] ranging from 0.01 to 0.25 [START_REF] Sidelnikova | Charge state of the activator in Ti:sapphire crystals grown by HDC method[END_REF]Fahey et al., 1986b;[START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF][START_REF] Nizhankovskii | Optical absorption and color centers in large Ti:sapphire crystals grown by horizontally directed crystallization under reducing conditions[END_REF][START_REF] Wong | Charge-exchange processes in titanium-doped sapphire crystals. I. Charge-exchange energies and titanium-bound excitons[END_REF]Matsunaga et al., 2003;[START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF][START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF]. These results are calculated based on the empirical formulas given in section 1.2.2, so they did not show a consistency. For example, Wong et to Ti 3+ starts from the edge of the crystal. However, this observation is in contradiction with the result of Moskvin et al. , who surprisingly found the same drop luminescence profile for Ti 4+ at 410 nm at the edge of sample annealed in oxygen atmosphere compared to the sample annealed in vacuum [START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF]. The intrinsic mechanisms for this luminescence drop at the edge is not completely understood yet. Therefore, investigating the conversion mechanisms between Ti 3+ and Ti 4+ is another task deserving to work on.

1.7 Chemistry of Ti:sapphire under different phases 1.7.1 Gas phase Ti:sapphire crystal has been grown under various atmospheres such as Ar, He [START_REF] Strauss | Growth and characterization of Ti: Al2O3 crystals for laser applications[END_REF], N2 [START_REF] Lacovara | Growth, spectroscopy, and lasing of titanium-doped sapphire[END_REF], H2 [START_REF] Deng | Investigation on improvement of laser quality of tunable Al2O3:Ti 3+ crystals[END_REF], N2+H2 [START_REF] Kokta | Effects of growth conditions and post-growth thermal treatment on the quality of titanium-doped sapphire[END_REF], CO + H2 and Ar + CO + H2 [START_REF] Sidelnikova | Charge state of the activator in Ti:sapphire crystals grown by HDC method[END_REF]. Growth atmosphere directly contributes to the oxygen partial pressure (𝑝 O 2 ) level in the gas phase. Additionally, to the oxygen originating from filled gas, dissociative evaporation can release O2 to the gas phase as well. The dissociative evaporation often occurs before Al2O3 solidification point is reached.

The thermal dissociation reactions of Al2O3 in the melt are concluded as follows [START_REF] Bagdasarov | The problems of high-temperature crystallization[END_REF]:

Al 2 O 3 ↔ AlO + AlO 2 Al 2 O 3 ↔ Al 2 O + 2O Al 2 O 3 ↔ Al 2 O 2 + 𝑂 AlO ↔ Al + O AlO 2 ↔ AlO + O AlO 2 ↔ Al + 2O Al 2 O ↔ AlO + Al Al 2 O ↔ 2Al + O Al 2 O 2 ↔ Al 2 O + O Al 2 O 2 ↔ 2Al + 2O Al 2 O 2 ↔ 2AlO 2O ↔ O 2 (1-26)
When the crystal is grown inside a furnace equipped with graphite heating elements, below reactions can also occur:

C + 1 2 O 2 ↔ CO C + O 2 ↔ CO 2 1 2 C + 1 2 CO 2 ↔ CO CO + 1 2 O 2 ↔ CO 2 (1-27)
These reactions can seriously decrease the 𝑝 O 2 inside the furnace. Under high temperature and strong reducing atmosphere, the main component of the gas will be CO. Then CO has a large possibility to react with Al2O3 melt and its dissociation products, as shown in equation

(1-28) (Sidelnikova et al., 2015a). These reactions will facilitate the evaporation of Al2O3 and hence release more oxygen into the gas phase.

Al 2 O 3 + CO ↔ 2AlO + CO 2 1 2 Al 2 O 3 + CO ↔ 1 2 Al 2 O + CO 2 1 3 Al 2 O 3 + CO ↔ 2 3 Al + CO 2 AlO + CO ↔ Al + CO 2 Al 2 O + CO ↔ 2Al + CO 2 1 2 O 2 + CO ↔ CO 2 (1-28)
Since CO is a very strong oxygen getter, it can decrease the 𝑝 O 2 and may create more oxygen vacancies at the same time in the crystal.

In case of large melt surface, titanium can also evaporate from the melt, leading to titanium loss. The titanium evaporation rate depends on 𝑝 O 2 in the system, the composition of growth atmosphere and melt superheating [START_REF] Nizhankovskiy | Optical homogeneity of Ti:sapphire crystals grown by horizontal directional solidification[END_REF]. It was found that melt evaporation rate could be decreased by increasing the inert gas pressure (Ar) (Sidelnikova et al., 2015a). Therefore, it is important to control 𝑝 O 2 during crystal growth process.

However, even under vacuum condition, the oxygen partial pressure is still too high to completely suppress the residual absorption at 800 nm (section 1.2.1.3 Optical loss). Through thermodynamic equilibrium calculations, Uecker et al. proposed that 2 × 10 -8 bar < 𝑝 O < 2.3 × 10 -7 bar must be fulfilled at 2050 ℃ to maintain the +3 state for titanium [START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF]. This requirement can be met if the crystal is grown under Ar/CO mixture atmosphere.

Liquid phase

As explained in section 1.4, the main factors determining titanium ion distributions in solid phase are the concentration of ions in the melt and the effective distribution coefficient of each species. Since all titanium ions entering the grown crystal come from the melt, it is necessary to consider the physico-chemical reactions in the melt. These reactions even govern crystallization kinetics and crystal structure. Unfortunately, the knowledge for these physico-chemical reactions is so far poorly developed. In melt, various complex situations can occur.

The dissociation reactions of Al2O3 in section 1.7.1 can affect the melt interaction with the crucible wall, causing crucible destruction and melt pollution. In the case of Ti:sapphire grown in a molybdenum crucible, for example, the reaction below can occur [START_REF] Bagdasarov | The problems of high-temperature crystallization[END_REF]:

Al 2 O + MoO 2 ↔ Al 2 O 3 + Mo (1-29)
When the 𝑝 O 2 is increased in the melt or in the gas phase, molybdenum inclusions may form [START_REF] Vyshnevskiy | Foreign phase inclusions in Ti-sapphire grown in a carbon-containing medium[END_REF]. These inclusions will degrade crystal quality. In the melt, TiO2 has the following dissociation reaction:

TiO 2 ↔ Ti 4+ + 2O 2- (1-30)
Then the chemical equilibrium between Ti 3+ and Ti 4+ can occur [START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF]:

2Ti 3+ + 1 2 O 2 ↔ 2Ti 4+ + O 2- (1-31) 1.7.3 Solid phase
Many theoretical and experimental studies have been reported for the solid phase. The case for solid-state is much more complicated because it involves crystal growth, growth atmosphere, species diffusion and chemical reaction effects on a number of point defects. For consistency, the detailed discussion for solid phase chemistry is presented in Chapter 3.2.

1.8 Thermal treatment

Carbothermal reduction

To further improve crystal quality, either raw materials or more often grown crystals are subjected to a thermal treatment. Following contents briefly introduce these two cases.

The ionic defects are usually noted by Kröger-Vink notation [START_REF] Kröger | Relations between the concentrations of imperfections in crystalline solids[END_REF]. For Ti:sapphire, the reduction degree of the conversion from Ti 4+ to Ti 3+ is determined by the equilibrium constant of the following reaction (see Chapter 3 for detailed discussion):

3Ti Al × + 3 4 O 2 + Al Al × ↔ 3Ti Al • + V Al ′′′ + 1 2 Al 2 O 3 . (1-32)
Therefore, Ti 4+ (≡ Ti Al • ) can be reduced either through decreasing oxygen partial pressure or through increasing the content of aluminum vacancy (V Al ′′′ ). Aluminum vacancy concentrations can be improved by means of carbothermal reduction of the raw material, namely, carbon is introduced in the form of fine-dispersed powder of graphite or aluminum carbide (Al4C3) [START_REF] Nizhankovskiy | Influence of crystal growth conditions and carbothermal treatment on activator charge state in Ti: sapphire[END_REF]. It was found that the introduction of carbon into the raw material can decrease the content of Ti 4+ in grown crystal by 3 times. It gives a similar result as preliminarily heating the raw material (Al2O3+TiO2) in Ar+CO+H2 atmosphere for 20 hours at a temperature of 2000 K, leading to reduce the relative content of Ti 4+ ions in grown crystal approximately by 1.5 times. Thus, this carbothermal reduction method allows to decrease the residual absorption in the region of 800 nm.

Post growth annealing

Grown crystals inevitably contain thermal stresses and inclusions that impair crystal properties. The post-thermal annealing treatment has been applied after various crystal have been grown to reduce the dislocation density, to decrease inclusions, to improve the optical homogeneity and to optimize the residual absorption [START_REF] Kokta | Effects of growth conditions and post-growth thermal treatment on the quality of titanium-doped sapphire[END_REF]Alombert-Goget et al., 2016;[START_REF] Guiping | Dislocation lines in doped-Ti sapphire crystal and the effect of annealing[END_REF]Danileiko et al., 1985). For Ti:sapphire, annealing is the most widespread method for reducing Ti 4+ ions present in as-grown crystals.

Since Ti 3+ is sensitive to oxygen partial pressure, annealing atmosphere plays an important role in determining annealed crystal optical quality. Previously studies of Evans and Kokta revealed that annealing in vacuum or reducing atmosphere leads to converting the Ti 4+ present in as-grown crystals to Ti 3+ [START_REF] Evans | Optical properties of the F + center in crystalline Al2O3[END_REF][START_REF] Kokta | Effects of growth conditions and post-growth thermal treatment on the quality of titanium-doped sapphire[END_REF]. For example, Evans showed that there is about 13% Ti 4+ present in as-grown crystal while only 2% in H2 annealed crystal. This can be observed by the enhanced absorption band of Ti 3+ at 490 nm. It is assumed that the total titanium in crystal does not change after annealing treatment. This annealing process helps to decrease the residual absorption without significantly changing the main absorption. Therefore, annealing treatment can improve the FOM value of Ti:sapphire (Alombert-Goget et al., 2016). Figure 1-24 shows the FOM values of Ti:sapphire samples before and after annealing treatment. However, annealing in oxidizing atmosphere would convert Ti 3+ into undesired Ti 4+ and therefore increases residual absorption and simultaneously decreases main absorption [START_REF] Aggarwal | Residual infrared absorption in as-grown and annealed crystals of Ti:Al2O3[END_REF]. During annealing process, the temperature needs to be taken into account as TiO2 precipitates might form in the range 1200-1350 ℃ (Xiao et al., 1997). By annealing vertical Bridgman Ti 3+ :sapphire crystal under air for 24 hours, Tang et al.

found that the pink color of crystal became lighter and nearly transparent (see Figure 1-25). It indicates that Ti 4+ ions result from the oxidation of Ti 3+ . Ti 4+ is assigned as a defect causing impurity therefore may increase crystal optical loss if titanium concentration is increased [START_REF] Kokta | Effects of growth conditions and post-growth thermal treatment on the quality of titanium-doped sapphire[END_REF]. Kokta studied how the residual absorption loss at 830 nm varies with titanium concentration for as-grown and annealed crystals. The experiment result presented in Figure 1-26 shows that annealing can effectively decrease Ti 4+ concentration in as-grown crystals. The reversible annealing process showing that Ti ions do not significantly move during the successive treatments [START_REF] Keig | Influence of the valence state of added impurity ions on the observed color in doped aluminum oxide single crystals[END_REF][START_REF] Cox | Chemical reactions and mass transport processes in donor and acceptor doped Al2O3 crystals[END_REF][START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF]. This has been experimentally

proved by the concentration variation of Ti 3+ in crystals reduced from oxidized state and oxidized from reduced state [START_REF] Kurtz | Solid state reactions and defects in doped Verneuil sapphire . III , Systems -Al2O3:Fe , -Al2O3:Ti and -Al2O3 :( Fe , Ti ) defects in doped Verneuil sapphire[END_REF]. Therefore, post annealing under reducing atmosphere is frequently used to decrease the Ti 4+ ions in Ti:sapphire single crystals. The V Al ′′′ (Optical) concentration is obtained from the charge compensation relation: during annealing at 1600 ℃. The unit of 𝑝 O 2 is atm [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF].

[Ti Al • ] = 3[V Al ′′′ ]. The V Al ′′′ (
Even if annealing can further convert Ti 4+ to Ti 3+ hence improve crystal optical quality, the annealing time is proportional to the square of the sample thickness [START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF].

Therefore, due to the low diffusion coefficients of ions, electrons, defects… in sapphire, annealing treatment is time-consuming, especially for large samples. It is impractical for the mass production of samples with large diameters. Consequently, it is of great significance to decrease the initial Ti 4+ content during the crystal growth step to obtain high quality Ti:sapphire crystals.

Conclusion

Through the investigation about literature devoted to Ti:sapphire, it is concluded that the optical quality of the crystal is determined by the ratio of Ti 3+ /Ti 4+ and homogeneity of Ti 3+ .

However, no theory is available to understand the opposite distributions of Ti 3+ and Ti 4+ observed in some experiments. By reviewing many details such as crystal growth methods, longitudinal and radial segregations during crystal growth, Ti ion distributions in grown crystals, the chemistry occurring in the gas, liquid and solid phases (reported to Chapter 3 for convenience), we concluded that in order to know the Ti ion concentrations in the solid phase and understand the conversion mechanisms between two Ti ions, we need to:

• Investigate the factors affecting concentration distribution of Ti ions during Ti:sapphire growth process: in the gas phase, liquid phase, then at the interface and finally in the solid;

• Study the influence of oxygen partial pressure on titanium ion concentrations and their segregation coefficients;

• Propose a model to explain the charge conversion mechanism between Ti 3+ and Ti 4+ that takes into account crystal growth, crystal growth atmosphere and temperature distribution;

• Study proposed model and its consequences, probably through numerical simulation;

• Grow crystals, measure Ti ion distributions in grown crystals and compare with simulation results to validate the model. All these will be studied one after another in the following chapters.

Chapter 2 Thermodynamics of Ti:sapphire solidification

Introduction

Obtaining the concentrations of Ti 3+ and Ti 4+ ions in sapphire after growth requires to trace back to the solidification process, especially the situation in the liquid, then at the solidliquid interface. Titanium ions are incorporated into the crystal depending on two key indexes: solute solubility and segregation coefficient. The segregation phenomenon involves both solid and liquid phases and solubility involves phases such as Ti2O3, TiO2, TiO that may precipitate in the crystal. Thus, the phase diagrams of Al2O3-Ti2O3-O2 and Al2O3-TiO2-O2 under reducing and oxidizing atmospheres must be taken into account. In this chapter, the experimental precipitation limits of TiO2 and Ti2O3 are collected and used to optimize the phase diagrams calculated by FactSage TM software. The influence of oxygen partial pressure on the concentrations of Ti 3+ and Ti 4+ in the solid and liquid phases, as well as the segregation coefficients of Ti 3+ , Ti 4+ and total Ti are studied. The appropriate oxygen partial pressure range for obtaining Ti 3+ is assessed on the basis of equilibrium thermodynamic calculations. The reactions (1-26) to (1-31) in the gas and liquid phases all occur simultaneously. They should be taken into account all together. This can be done only by a software such as FactSage. The factors influencing 𝑝 O 2 in crystal growth atmosphere are analyzed including filling gas and furnace materials.

Gibbs energy and thermodynamic properties

Gibbs free energy is a state function of a system related to entropy and enthalpy. For a pure element or a pure compound such as Al2O3, Gibbs energy at temperature T is described by

𝐺 T 0 = 𝐻 T 0 -𝑇𝑆 𝑇 0 , (2-1)
where 𝐻 T 0 and 𝑆 𝑇 0 are the enthalpy and entropy of the pure substance at temperature T. Without intermediate phase change before temperature T, with respect to 298 K, they are defined as:

𝐻 T 0 = ∆𝐻 298 K 0 + ∫ 𝐶 𝑝 (𝑇)𝑑𝑇 𝑇 298 𝐾 , 𝑆 T 0 = 𝑆 298 K 0 + ∫ 𝐶 𝑝 𝑇 𝑑𝑇 𝑇 298 𝐾 , (2-2)
where 𝐶 𝑝 is the heat capacity. 𝐺 298 K 0 is considered as the standard Gibbs energy of a pure substance at 298 K and 1 atm. In case of intermediate phase change, the latent heat should be added. For a chemical reaction between pure compounds, the Gibbs energy change can be expressed as ∆𝐺 = ∆𝐻 -𝑇∆𝑆.

(2-3)

The sign of each term impacts the state of the reaction as listed in Table 2-1. A chemical reaction always proceeds for the aims of lowering the total Gibbs energy of the system. At the beginning, when reaction takes place, the Gibbs energy becomes more negative until a minimum energy is reached. The reaction proceeds spontaneously when ∆𝐺 < 0 and vice versa. ∆𝑆 denotes the degree of disorder of the system which is introduced by the energy exchange (∆𝐻). Crystal growth from the melt is an example because the atoms change from disorder state in the liquid phase to the ordered state in the solid phase. For a chemical reaction, the Gibbs energy change can be written as

∆𝐺 = ∆𝐺 °+ 𝑅𝑇𝑙𝑛𝐾 (2-4)
where K is the reaction quotient. When the reaction is at equilibrium, ∆𝐺 = 0 so that

𝐾 𝑒𝑞 = 𝑒 -∆𝐺 °𝑅𝑇 (2-5)
where 𝐾 𝑒𝑞 is the equilibrium constant of the reaction. For an elementary reaction of the form

𝑎𝐴 + 𝑏𝐵 ↔ 𝑐𝐶 + 𝑑𝐷 (2-6)
The equilibrium constant can be written as

𝐾 𝑒𝑞 = [𝐶] 𝑐 • [𝐷] 𝑑 [𝐴] 𝑎 • [𝐵] 𝑏 (2-7)
Optionally, the concentration of the substance can be replaced with the partial pressure, mole numbers, mole fraction, etc.

For solutions, the thermodynamic activities ought to be used in the expression of 𝐾 𝑒𝑞 . The activity is defined as

𝑎 𝑖 = 𝛾 𝑖 • [𝑖] (2-8)
where 𝛾 𝑖 is the dimensionless activity coefficient of a component 𝑖 and [𝑖] is the concentration of component 𝑖. For ideal solution, 𝛾 𝑖 = 1. The activity coefficient describes how away a solution deviates from the ideal behavior. It is related to temperature and in most cases can be described by the Henrian relation

log 10 𝛾 𝑖 = 𝐴 𝑇 + 𝐵 (2-9)
where A and B are constants, 𝑇 is temperature with the unit of Kelvin. The activity of a gas equals the partial pressure with respect to the total pressure and is sometimes referred as fugacity. The activity of a pure solid or a pure liquid equal to unity. Further, the activity is related to the chemical potential, 𝜇 𝑖 , of a substance as

𝜇 𝑖 = 𝜇 𝑖 0 + 𝑅𝑇𝑙𝑛𝑎 𝑖 , (2-10)
where 𝜇 𝑖 0 is the molar chemical potential of species 𝑖 under standard conditions. Chemical potential is useful for predicting the tendency of a substance to change under various conditions (temperature, pressure, phase, concentration…).

The equilibrium constant is also important on the kinetic point of view. The kinetics of reaction (2-6) is described by the reaction rates as follows:

𝑅 𝑓 = 𝑘 𝑓 [𝐴] 𝑎 • [𝐵] 𝑏 𝑅 𝑟 = 𝑘 𝑟 [𝐶] 𝑐 • [𝐷] 𝑑 (2-11)
where 𝑅 𝑓 and 𝑅 𝑟 are the forward and reverse reaction rate, and 𝑘 𝑓 and 𝑘 𝑟 are the forward and reverse reaction rate constants. At equilibrium, the forward reaction rate equals the reverse reaction rate and they link chemical kinetics with chemical equilibrium through Through above discussed 𝛾 𝑖 , [i], 𝐶 𝑝 (𝑇), ∆𝐻 etc, it is possible to compute ∆𝐺 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 and 𝐾 𝑒𝑞 for a system at given temperature and pressure. However, as ∆𝐺 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 for all the chemical reactions (1-26) to (1-28) need to be solved together, it is necessary to use a dedicated software with a thermodynamic database taking into account all reactions, to calculate the minimum Gibbs energy change ∆𝐺 𝑚𝑖𝑛 .

FactSage thermodynamic software

FactSage TM is a well-known commercial software devoted to thermochemical calculations such as multiple phase diagrams, substance compositions, complex phase

𝐾 𝑒𝑞 = 𝑘 𝑓 𝑘 𝑟 . (2-12)
equilibria, dynamics and processes in a reactor. The databases in FactSage TM are developed based on built-in mathematical models for the Gibbs energy of each phase and is optimized using available thermodynamic and phase equilibrium data from literature (temperatures, pressures, heat capacity, solubilities, entropies, enthalpies, chemical potential, activities, etc.).

Then the final models and databases are used to estimate the properties of multicomponent systems. The applications of FactSage TM span gases, oxides, organic compounds, metallic alloys, solutions, plasmas and aqueous systems… There exists 4928 compounds and 7046 phases in the database used by FactSage 8.0 software. In addition, it permits to create and save a private database according to the requirements of the users. The detailed introduction about its database can be found in [START_REF] Bale | FactSage thermochemical software and databases-recent developments[END_REF][START_REF] Bale | FactSage thermochemical software and databases[END_REF][START_REF] Bale | Recent developments in Factsage thermochemical software and databases[END_REF] and on the interface of the software. modules are the hearts of FactSage software. "Equilib" module can be used to calculate the equilibrium conditions (temperature, activity, composition of the component) of a multiphase or a multicomponent system. As long as the conditions of a system are given, necessary data are automatically extracted from the database to compute the equilibrium by minimizing the total Gibbs energy. For instance, the remaining chemical composition of components can be computed as the function of temperature, pressure and the number of moles of each element.

The "Phase Diagram" module is used to calculate the binary, ternary and multicomponent phase diagram sections where the axes can be various combinations of temperature (T or 1/T), pressure (P, logP or RTln(P)), total volume, chemical potential, composition, etc. "Equilib" and "Phase Diagram" modules have access to both compound and solution databases, therefore, it provides the possibility to study the solidification process in casting or crystal growth. Modules like "Reaction", "Predom" and "EpH" are also useful for mass action law, compound phase diagram and pH calculations but they exceed the scope of this thesis. Other thermochemistry software like Thermo-Calc or Pandat also perform similar calculations but they are preferred for calculations in alloy industry field while FactSage TM is also good at thermodynamic equilibrium calculations for oxides at high temperatures. In this thesis, oxides (FToxid) and pure substances (FactPS) database are selected to perform the thermodynamic calculations using FactSage TM software.

2.2 Ti solid solubility and phase diagrams 2.2.1 Solubility of Ti 3+ and Ti 4+ in sapphire Ti:sapphire is usually grown by doping TiO2 into alumina raw material. Depending on the crystal growth atmosphere, Ti 3+ and/or Ti 4+ can be formed. However, this process could be accompanied by the formation of some precipitates when the solid solubility limits of Ti 3+ and Ti 4+ are reached. Therefore, the solubility of Ti 3+ or Ti 4+ are critically important for computing their concentrations in sapphire.

The solution of Ti 4+ occurs by substituting the host ion Al 3+ which is accompanied by the formation of aluminum vacancies [START_REF] Petot-Ervas | Transport properties of titanium-doped αalumina: experimental results[END_REF]. The details are discussed in Chapter 3. The authors found that this solution mechanism is much stronger if 𝑛 × Ti 4+ -V Al ′′′ clusters are taken into account. Winkler et al. measured the solid solubility of Ti 4+ to be 0.25 mole% -0.3 mole% for samples fired in air at 1300 ℃ using diffuse reflectance spectra [START_REF] Winkler | Solid solution of titanium dioxide in aluminum oxide[END_REF]. When the oxygen partial pressure is too high, precipitates such as rutile (TiO2) will form inside the crystal as shown in the Ellingham phase diagram of Al2O3-TiO2 system (see Figure 23456789). Therefore, K. Hamano et al. inferred the Ti 4+ solubility by detecting the amount of Al2TiO5 or rutile and then comparing with the amount of initial TiO2 [START_REF] Hamano | Effects of TiO2 on sintering of alumina ceramics[END_REF]. Since Ti 4+ lacks sufficient solubility in α-Al2O3, it has no influence on the lattice parameter of α-Al2O3 [START_REF] Mckee | Aluminum oxide-titanium oxide solid solution[END_REF][START_REF] Roy | Solubilities of magnesia, titania, and magnesium titanate in aluminum oxide[END_REF]), up to date, no measurable Ti 4+ is observed in Ti:sapphire single crystal with X-Ray diffraction technique. But C.-S. Hwang et al. reported Ti 4+ solid solution to be 0.04 wt% -1.5 wt% at 1300 -1700 ℃ by comparing the intensity ratio of X-ray diffraction peaks of rutile and Al2TiO5 with that in sintered samples [START_REF] Hwang | Microstructure and mechanical strength of TiO2doped Al2O3 ceramics fired in vacuum atmosphere[END_REF]. The samples, containing 0 -4 wt% TiO2 are sintered at 1500-1700 ℃ in vacuum first and then annealed at 1300 ℃ in air. In addition, the sintering rate of α-Al2O3 is found to increase with TiO2 concentration. The concentration which produces the maximum sintering rate is considered as the Ti 4+ solubility limit. From which, R.D. Bagley et al. indicate that the Ti 4+ solubility at 1405, 1463 and 1520 ℃ are 0.39, 0.59 and 0.78 mole% [START_REF] Bagley | Effect of TiO2 on initial sintering of Al2O3[END_REF].

Table 2-2 Lattice parameters of pure and Ti:sapphire crystals at 295 K (Phillips et al., 1980;[START_REF] Heuer | Oxygen and aluminum diffusion in α-Al2O3: How much do we really understand[END_REF]. The Ti concentration is 1 × 10 -3 but the unit is not specified. There may exist some uncertainties for part of the solubility data. For example, under reducing atmosphere, Ti 3+ should be expected while Roy and Coble ascribed their obtained solubility data to Ti 4+ [START_REF] Roy | Solubilities of magnesia, titania, and magnesium titanate in aluminum oxide[END_REF]). The solubility of Ti 4+ given by Winkler's group is based on a polycrystalline Al2O3 doped with TiO2 but not a single crystal [START_REF] Winkler | Solid solution of titanium dioxide in aluminum oxide[END_REF].

Even though, their results are still considered as a reference by most researchers. Ohta and Morita got the solubility of Ti 3+ in Al2O3 at 1600 ℃ by measuring the linear relation between the lattice spacing and the solid solubility [START_REF] Ohta | Thermodynamics of the Al2O3-SiO2-TiOX oxide system at 1873 K[END_REF]. However, their results are obtained in the ternary Al2O3-SiO2-TiOx system but not in the pure Al2O3-Ti2O3 system. To make the thermodynamic calculation results more reliable, we assume that Ti 3+ solid solubility is obtained under reducing atmosphere (low 𝑝 O 2 ) and Ti 4+ solid solubility is obtained under oxidizing atmosphere (high 𝑝 O 2 ). These experimental solubility data will be further discussed in the phase diagrams computed by FactSage software. are excluded because the 𝑝 O 2 for crystal growth is too high to obtain metal phase. TiO only appears in the system when 𝑝 O 2 < 10 -30 bar (see Figure 23456789). The solidus and solvus lines of corundum show a good agreement with the Ti 3+ solid solubility data except the results of Belon et al [START_REF] Belon | Study of the system Al2O3-Ti2O3 (Formation of eutectic mixture of titanium oxide-aluminum oxide system with a particular solubility limit)[END_REF]. They mixed Al2O3 with reduced Ti2O3 under vacuum and reported the precipitation limit of Ti2O3 in Al2O3 to be 12.5 mole% at 1700 ℃. All other reported precipitation limit of Ti2O3 in Al2O3 are no more than 4 mole%, as shown in the enlarged Figure 2-4. When Ti 3+ exceed its solubility limit coarse Ti2O3 precipitates with an average size of about 500 nm are formed inside the grains [START_REF] Yasuda | Nanoprecipitation in Al2O3-3 mol% Ti2O3 due to Oxidation[END_REF]. Additionally to the phase called "ilmenite" (i.e. Ti2O3, the real mineral ilmenite being FeTiO3), several other titanium oxides, Magnéli phases (TinO2n-1, 4≤n≤20), are also present in binary Al2O3-Ti2O3 system. These Magnéli phases are considered as stoichiometric compounds in the temperature range 1200 -1300 ℃. The crystal structures of the phases with 4≤n≤10 can be deduced from the rutile structure by crystallographic shear and the structures for phases with n >10 can be determined from other shear operations [START_REF] Eriksson | Critical evaluation and optimization of the thermodynamic properties and phase diagrams of the MnO-TiO2, MgO-TiO2, FeO-TiO2, Ti2O3-TiO2, Na2O-TiO2, and K2O-TiO2 systems[END_REF]. The binary phase diagram of Al2O3-TiO2 system in air has been reported by Bunting, Lang, Goldberg, Mills and Lejus (Bunting, 1933;[START_REF] Lang | The system beryllia-alumina-titania: phase relations and general physical properties of three-component porcelains[END_REF][START_REF] Goldberg | Contribution to study of systems formed by alumina and some oxides of trivalent and tetravalent metals especially titanium oxide[END_REF][START_REF] Mills | Slag atlas. VDEh 2nd Edition[END_REF][START_REF] Lejus | Sur quelques composes nouveaux formes par le rutile TiO2 avec des oxydes de metaux trivalents et tetravalents[END_REF]. Meanwhile, the Al-O-Ti system has also been studied by Seifer, Das, and Ilatovskaia [START_REF] Seifert | Phase equilibria and diffusion paths in the Ti-Al-O-N system[END_REF][START_REF] Das | The Al-O-Ti (Aluminum-oxygen-titanium) system[END_REF][START_REF] Ilatovskaia | Thermodynamic description of the Ti-Al-O system based on experimental data[END_REF]. These studies indicate that Al2TiO5 intermediate compound appears in Al2O3-TiO2 phase diagram in the temperature range of 1200 -1860 ℃ when the solubility limit is reached. This Al2TiO5 can further decompose into other compounds when exceeding a certain temperature range but the composition of these compounds and decomposition temperature are disputed. Wartenburg and Bunting reported a eutectic temperature of 1850 ℃ with 28 wt% TiO2 (33 mole%) and another eutectic temperature at 1715 ℃ with 80 wt% TiO2 (84 mole%). The study of Lang et al. thought there exists two forms of Al2TiO5. The low-temperature form β-Al2TiO5 is stable from room temperature to 750 ℃ and from about 1300 ℃ to 1820 ℃. Additionally, it is found that precipitated rutile (if formed) could convert to β-Al2TiO5 when the temperature is higher than 1450 ℃:

α -Al 2 O 3 + TiO 2 (rutile) → β -Al 2 TiO 5 .
Al2TiO5 exists as a second phase, suppressing the grain growth of alumina [START_REF] Yokoi | Preparation of coated alumina powders and their microstructure development during heating[END_REF][START_REF] Hwang | Microstructure and mechanical strength of TiO2doped Al2O3 ceramics fired in vacuum atmosphere[END_REF]. The high-temperature form α-Al2TiO5 is stable from 1820 ℃ to its melting to 1841 ℃. In our study, Al2O3 solid solubility in TiO2 [START_REF] Slepetys | Solid solution of aluminum oxide in rutile titanium dioxide[END_REF] is not considered in this study.

The solid solubility of Ti 4+ is so small that it is not observable in Figure 2 In the commercial FactSage software and database, the solubility data of Ti 4+ in the solid phase is lacking. This means that it cannot be used to calculate the concentrations of Ti 3+ and Ti 4+ in the solid and liquid phases and corresponding segregation coefficients. Therefore, the phase diagrams (i.e. the solid database) must be optimized to reproduce the experimental solubility data for Ti 3+ and Ti 4+ ions.

Optimized Al2O3-TiO2 system

To improve the database for thermodynamic calculations, Ti into the solid phase and 𝑝 total = 1 bar.

Al2O3-TiO2 Ellingham phase diagram

Titanium valence in the crystal is function of temperature and oxygen partial pressure.

There exist large temperature gradients in crystal growth setup. In Kyropoulos crystal growth setup, temperature on the crucible wall is higher than that in the center of the melt. precipitates as rutile at temperatures below 1350 ℃ [START_REF] Hamano | Effects of TiO2 on sintering of alumina ceramics[END_REF]. The presence of other phases may change the local chemical equilibrium and therefore the concentrations of Ti 3+ , Ti 4+

and corresponding defects that determine the transport properties. When temperature is lower than 600 ℃, substitutional Ti 3+ cannot be obtained through adjusting 𝑝 O 2 .

𝑝 O 2 during Ti:sapphire growth

When the oxygen partial pressure in the growth atmosphere increases or decreases, the reaction (1-32)

3Ti Al × + 3 4 O 2 ↔ 3Ti Al • + V Al ′′′ + 3 2 O O × .
will proceed forward or reverse. This oxidizes the Ti 3+ to Ti 4+ or converts the oxidized Ti 4+ (≡ Ti Al • ) back to Ti 3+ (≡ Ti Al × ). Therefore, 𝑝 O 2 is a critical parameter in determining titanium valence and concentration in grown crystals.

Filling gas

The oxygen partial pressure inside the furnace at any time is determined by crystal growth conditions (temperature, atmosphere, furnace structure and materials). Figure 2-10 shows a sketch of Ky method growing Ti:sapphire. The crystal is grown under flowing Ar gas.

But even though for the purest commercial Ar gas, it always contains small amount of O2 and water vapor. According to the purity certificate, pure gas should contain less than 5 ppm O2 and 5 ppm H2O (g). Apart from this, the furnace cannot be perfectly sealed. Tiny O2 and water vapor in the air still have a chance to enter in the furnace. Thus, it is assumed that 10 ppm O2 and 10 ppm H2O (g) impurities are contained inside the incoming gas. Raw material, composed of pure alumina crackles and Ti doped Al2O3 powder, is melted in a Mo crucible and the crystal is grown along radial and downward directions. Graphite elements are used as casing, susceptor and insulation for uniformly heating the crucible. The gas (Ar + 10 ppm O2 + 10 ppm H2O (g))

exists everywhere inside the furnace. Except for specific study purpose, Ti:sapphire is grown under pure inert gas or reducing gas to obtain sufficient Ti 3+ ions for lasing purpose. Reported Ti:sapphire crystals have been grown under pure Ar, N2, and H2 atmospheres [START_REF] Lacovara | Growth, spectroscopy, and lasing of titanium-doped sapphire[END_REF][START_REF] Deng | Investigation on improvement of laser quality of tunable Al2O3:Ti 3+ crystals[END_REF]. In contrast to pure inert gases, mixtures of inert gas with small amount of reducing gas in the growth or annealing processes contributes to further improve crystal quality. The oxygen partial pressure under different gas compositions is plotted in Figure 2-11.

By taking into account the existence of trace impurities in the system (10 ppm O2 and 10 ppm H2O (g)), pure Ar and N2 atmospheres give rise to 𝑝 O 2 as high as 10 -5 and 10 -7 bar. The mixtures of 99.34 % Ar + 0.6% CO + 0.06 % H2, 95% Ar + 5% CO and 95% Ar + 5% H2 (Alombert-Goget et al., 2016) produce a wider oxygen partial pressure range, i.e. 10 -55 ≤ 𝑝 O 2 < 10 -8 bar.

S. Ganschow and co-workers experimentally showed that Ti:sapphire grown by Czochralski technique under N2 atmosphere exhibits a broad and intense infrared absorption compared to crystal grown under the mixture of 95% Ar + 5% CO [START_REF] Ganschow | Application of predominance diagrams in melt growth of oxides[END_REF]. They found that under the optimized atmosphere, the absorption coefficient of Ti 3+ is increased by a factor of 2 and yielded a FOM > 100. Pure H2 atmosphere exhibits the lowest oxygen partial pressure with the range of 10 -57 ≤ 𝑝 O 2 < 10 -13 bar. However, due to safety reason, pure H2 gas is rarely used and it is frequently mixed (a few percent) with inert gases.

Low oxygen partial pressure not only has the advantage of obtaining desired Ti 3+ ions but is also useful for suppressing the bubbles resulting from the thermal dissociation of molten alumina [START_REF] Li | Bubbles propagation in undoped and Titanium ( Ti 3+ ) -doped sapphire crystals grown by Czochralski ( Cz ) technique[END_REF][START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF]. It is helpful in decreasing the precipitates in Ti:sapphire as well [START_REF] Clarke | Effect of oxygen partial pressure on precipitation in titanium-doped aluminum oxide[END_REF].

Crucible materials

Mo, Ir and W are three typical crucible materials used in Ti:sapphire crystal growth. The trace impurities O2 and H2O in the growth atmosphere can oxidize the crucible material and even heavily destroy the crucible. Mo crucible is easy to be oxidized with the formation of MoO(g) and MoO3(g). The reactions between molybdenum and oxygen are given as follows [START_REF] Borodin | Production of sapphire tubes for high-pressure sodium lamps using the stepanov method at high rates of growth[END_REF]:

Mo(s) + O(g) → MoO(g) 2Mo(s) + 3O 2 (g) → 2MoO 3 (g) (2-13)
The Mo crucible reaction with water vapor occur through:

Mo + 2H 2 O ↔ MoO 2 + 2H 2 MoO 2 + H 2 O ↔ MoO 3 + H 2 (2-14)
Tiny amount of Mo may dissolve in molten alumina [START_REF] Labelle | Apparatus for growing crystalline bodies from the melt[END_REF][START_REF] Jindra | Multiple growth of profiled sapphire crystals[END_REF], but until now, no molybdenum or molybdenum oxide has been observed inside reported Ti:sapphire crystals. Therefore, the contact reaction between Mo crucible and alumina melt is negligible.

As strong oxygen getter materials, W and Ir can, for example, give the following reactions [START_REF] Wendel | Thermodynamics and kinetics of tungsten oxidation and tungsten oxide sublimation in the temperature interval 200-1100 C[END_REF]): gives a 𝑝 O 2 of 8.87 × 10 -8 bar, while, without Mo crucible (only Al2O3 + gas), FactSage gives a 𝑝 O 2 of 6.22 × 10 -6 bar. This means that even with a casing made of oxides, the Mo crucible still controls the 𝑝 O 2 in the liquid. However, most of the time, graphite casing is used.

W(s) + O 2 (g) ↔ WO 2 (s) 2 3 W(s) + O 2 (g) ↔ 2 3 WO 3 (s) (

Graphite elements

The graphite elements inside the furnace also can react with O2(g) and H2O(g). The reactions between C and O2 have been discussed in (1-27) in Chapter 1 and the reactions between C and H2O are:

C + H 2 O ↔ CO + H 2 C + 2H 2 O ↔ CO 2 + 2H 2 C + 2H 2 ↔ CH 4 (2-17)
The produced reducing gases like CO, H2 and CH4 contributing to decreasing 𝑝 O 2 . The loss of C may affect the homogeneous heating of crucible. Therefore, to prevent the oxidation of graphite, a few amounts of CO or CO2 are often added into the inlet gas. Figure 2-13 shows the 𝑝 O 2 when furnace gas reacts with 95% Ar + 5% CO, 95% Ar + 5%

CO2 and 95% Ar + 2.5% CO + 2.5% CO2. Among them, the mixture of 95% Ar + 5% CO presented the lowest oxygen partial pressure from 10 -45 to 10 -9.46 bar. The mixture of 95% Ar + 5% CO2 did not change the 𝑝 O 2 so much. The equimolar mix of CO and CO2 possesses the largest range of 𝑝 O 2 , i.e. 10 -45 < 𝑝 O 2 < 10 -3.13 bar which is large enough to tune the 𝑝 O 2 for 10 -20 < 𝑝 O 2 < 1 bar during Ti:sapphire growth process. In the calculation, the trace impurities of 10 ppm O2 and 10 ppm H2O (g) in the system are considered.

Except the oxygen reacting with crucible and graphite elements, other oxygen sources like the evaporation of components from the decomposition of melt (section 1.7), the chemical reactions between decomposed components and gas, reactions between melt and crucible can also affect the 𝑝 O 2 inside the furnace. For the aim of simplifying thermodynamic calculations and the numerical modeling in Chapter 4, we only consider the oxygen source coming from filled gas and the oxygen consumed by graphite elements and Mo. It may be possible that oxygen in the melt could diffuse through the Mo crucible. This effect is also not considered in present study.

The influence of C and Mo on 𝑝 O 2

In practical crystal growth run, the gas inside furnace does not react with Mo or graphite separately but react with both simultaneously or successively. Therefore, in order to determine the influence of Mo and graphite on 𝑝 O 2 , the 𝑝 O 2 inside furnace is computed by taking into account the reactions between Mo and O2 and C and O2 as well as the trace impurities 10 ppm O2 and 10 ppm H2O. The result is shown in Figure 2-14. In a practical crystal growth setup, the crucible is located at the center of the furnace and is surrounded by the graphite elements (see

Ti ion concentrations calculated by FactSage

The 𝑝 O 2 in the gas phase can affect titanium valence during the crystal growth thereby the concentrations of Ti 3+ and Ti 4+ ions present in the crystals. Considering Ti:sapphire growth process, the equilibrium thermodynamic calculation results from FactSage should contain the co-existence of solid and liquid phases and do not contain other phases.

The 𝑝 O 2 at the melting point of sapphire at 2053 ℃ is critical for crystal solidification. For a sapphire crystal doped with 0.1 wt% (0.1 mole% or 37.5 mol.m -3 ) TiO2 growing under Ar atmosphere, the equilibrium concentrations of Ti 3+ and Ti 4+ in the solid and liquid phases are computed based on the previously optimized database (section 2.2.3) with FactSage software.

For 𝑝 O 2 = 1.1×10 -20 bar where only liquid phase exists, the ratio of [Ti 4+ ]Liquid /[Ti 3+ ]Liquid is calculated to be 4.76×10 -4 . Therefore, the initial concentration of Ti 3+ and Ti 4+ in the liquid phase are 37.482 mol.m -3 and 0.018 mol.m -3 , respectively. To study how titanium ion concentrations are influenced during solidifying process, 𝑝 O 2 is varied and the results are plotted in Figure 2-16. The trace impurities (10 ppm O2 and 10 ppm H2O) in the system are taken into account in the calculation. In liquid phase, when 𝑝 O 2 is smaller than 10 -7 bar, Ti 3+ concentration is larger than the concentration of Ti 4+ while beyond this pressure, Ti 4+ concentration becomes the larger one. In the solid phase, the concentration of Ti 4+ becomes larger than Ti 3+ when 𝑝 O 2 ≥1×10 -4 bar. Mohapatra and Kröger also observed this reverse concentration phenomenon (see Figure 1-27) by annealing Ti:sapphire crystal at 1600 ℃ under oxygen, air, nitrogen and mixtures of CO and CO2 [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF] atmospheres. Therefore, when crystal is grown or annealed under a given temperature, desired titanium ion concentrations can be adjusted through 𝑝 O 2 . In addition, when 𝑝 O 2 is low (𝑝 O 2 < 10 -20 bar) in Figure 2-16, there still can observe the presence of [Ti 4+ ]S rather than absolutely zero. When 𝑝 O 2 is high, say, 1 bar, the concentration of [Ti 3+ ]S is only one-order of magnitude lower than [Ti 4+ ]S so that its concentration cannot be neglected. Therefore, no matter 𝑝 O 2 is very high or very low, Ti 3+ and Ti 4+ always co-exist in grown crystals. Consequently, reported empirical formula in section 1.2.2 used to estimate the concentrations of Ti 3+ and Ti 4+ through absorption coefficient are not reliable because all these formulas are obtained on the basis of the assumption that "only Ti 3+ or only Ti 4+ ion exist in the calibration crystal".

In the liquid phase, below reaction can occur:

TiO 1.5 + 1 4 O 2 ↔ TiO 2 .
(2-18)

Since ∆𝐺 = ∆𝐺 0 + 𝑅𝑇𝑙𝑛𝐾 𝑒𝑞 and ∆𝐺 = 0 at equilibrium, the equilibrium constant for this reaction is

𝐾 𝑒𝑞 = [TiO 2 ] [TiO 1.5 ]•𝑝 O 2 1 4 = exp (- ∆𝐺 °𝑅𝑇 ). (2-19)
where ∆𝐺 0 is the standard molar Gibbs energy given by FactSage database:

∆𝐺 °= -202.262 + 0.055 • 𝑇 (kJ/mol).

(2-20)

At t=2053 ℃, the Gibbs energy of reaction (2-18) is calculated to be -74.86 kJ. mol -1 using the "Reaction" module in FactSage. Defining the ratio of [Ti 4+ ] and [Ti 3+ ] in the liquid phase as Knowing the Ti ion concentrations in the solid and liquid phases, their segregation coefficients can be calculated. Titanium has been reported with a small segregation coefficient lower than unity (see Table 1-

[Ti 4+ ] L [Ti 3+ ] L = 𝛼. ( 2
𝑘 𝑇𝑖3+ = [Ti 3+ ] S [Ti 3+ ] L , (2-24) 𝑘 𝑇𝑖4+ = [Ti 4+ ] S [Ti 4+ ] L , (2-25)
It needs to be noted that the total Ti segregation coefficient does not equal the sum of the segregation coefficients of Ti 3+ and Ti 4+ but is described as

𝑘 𝑇𝑖 = [Ti 3+ ] S +[Ti 4+ ] S [Ti 3+ ] L +[Ti 4+ ] L
.

(2-26)

According to the titanium concentrations obtained in section 2.4, the segregation coefficients of Ti 3+ , Ti 4+ and total Ti at melting point are computed and shown in Figure 2-18.

It is found that 𝑘 𝑇𝑖3+ > 𝑘 𝑇𝑖 > 𝑘 𝑇𝑖4+ . When 𝑝 O 2 is very small (reducing atmosphere), most titanium is present as Ti 3+ . It follows from equation (2-26) that

𝑘 𝑇𝑖 = [Ti 3+ ] s + [Ti 4+ ] s [Ti 3+ ] L + [Ti 4+ ] L ≈ [Ti 3+ ] s [Ti 3+ ] L = 𝑘 𝑇𝑖3+ .
(2-27)

Similarly, when 𝑝 O 2 is very high (oxidizing atmosphere), most titanium is present as Ti 4+ . Then 𝑘 𝑇𝑖 can be approximated as

𝑘 𝑇𝑖 = [Ti 3+ ] s + [Ti 4+ ] s [Ti 3+ ] L + [Ti 4+ ] L ≈ [Ti 4+ ] s [Ti 4+ ] L = 𝑘 𝑇𝑖4+ .
(2-28)

Therefore, under given temperature, the increase of 𝑝 O 2 leads to 𝑘 𝑇𝑖 changing gradually from the maximum value 𝑘 𝑇𝑖3+ to the minimum value 𝑘 𝑇𝑖4+ . This can be seen on Figure 2 -30) The relation between 𝑘 𝑇𝑖_𝑡ℎ𝑒𝑜𝑟𝑦 and 𝑝 O 2 is also plotted in Figure 2-18. It is observed that the FactSage results are very close to the theoretical deduction. This suggests that the concentrations calculated by FactSage are reliable.

𝑘 𝑇𝑖_𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑘 𝑇𝑖3+ +48•𝑘 𝑇𝑖4+ 𝑝 O 2 1/4 1+48•𝑝 O 2 1/4 . ( 2
In addition, it is noted that the segregation coefficient of Ti 3+ calculated by FactSage is much smaller than reported experimental values in the range of 0.05 ~ 0.23 (Table 1234). This is because the reported experimental values are the effective segregation coefficient for total Ti while what we calculated are thermodynamic equilibrium segregation coefficients of Ti 3+ . In experiments, these values of effective segregation coefficient for total Ti are sometimes referred as Ti 3+ segregation coefficient, based on the assumption that all the Ti is present as Ti 3+ and nearly no Ti 4+ is present in the crystal. But in practical crystal growth process, there are always Ti 4+ ions present in the grown crystals, as shown by the results on Figure 2-16. In addition, the reported values vary upon crystal growth techniques and conditions, depending on rotation and pulling rates. So, the exact thermodynamic segregation coefficients of Ti 3+ , Ti 4+ and Ti, could not be determined precisely from experiments. By the way, the segregation literature shows that the effective segregation coefficient is necessarily larger than the thermodynamic one (𝑘 0 < 𝑘 𝑒𝑓𝑓 < 1).

Equilibrium segregation coefficient of Ti

The effective segregation coefficient of Ti has been reported by many investigators (see Table 1-4) but there are very few papers discussing about the equilibrium segregation coefficient of Ti. According to known conditions, the equilibrium segregation coefficient of Ti can be deduced from the effective segregation coefficient. Re-write the boundary layer thickness and BPS equations presented in Chapter 1 for Cz or Ky crystal growth process:

𝛿 = 1.6𝐷 𝐿 1/3  𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 1/6 𝜔 -1/2 , 𝑘 𝑒𝑓𝑓 = 𝑘 0 𝑘 0 +(1-𝑘 0 )𝑒 -(𝑉 𝑔𝑟𝑜𝑤𝑡ℎ 𝛿/𝐷 𝐿 ) ,
it is obtained that

𝑘 𝑒𝑓𝑓 = 𝑎 𝑎+(1-𝑎)𝑒 -(𝑏×𝑉 𝑔𝑟𝑜𝑤𝑡ℎ ) .
(2-31)

where 𝑎 = 𝑘 0 and 𝑏 = 1.6𝐷 𝐿 -2

3 𝜈 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 [START_REF] Strauss | Growth and characterization of Ti: Al2O3 crystals for laser applications[END_REF]. Therefore, it is concluded that the equilibrium segregation coefficient of Ti in sapphire is around 0.08. This value is also close to the thermodynamically calculated value 𝑘 0 = 0.045 for 𝑝 O 2 =10 -20 bar at 2053 ℃. An unoptimized liquidus line and some inconsistent rotation rates can be possible reasons for the deviation. Strictly speaking, the calculated 𝑘 0 is not completely the equilibrium segregation coefficient but the BPS equation extrapolated to 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ = 0 [START_REF] Müller | Convection and inhomogeneities in crystal growth from the melt[END_REF]. Even though, it still can be considered as a reference for the equilibrium segregation coefficient of Ti in sapphire.

It is possible that if the mismatch between Ti 4+ solid solubility and the Al2O3-TiO2 phase diagram could be improved, the calculated 𝑘 0 value may be closer to the reported value. With the help of technical support from FactSage (Dr. Tatjana Jantzen), we tried to adjust the solidus and liquidus lines in the phase diagram based on the private database from Dr. Jantzen and the experimental data in Belon's paper. However, we got 𝑘 0 = 0.3 which is heavily away from the reported value 0.087. It is even larger than the reported effective segregation coefficient of Ti in sapphire. This is unreasonable but it should be noted that Belon's experimental data are quite different from others (see Figure 2.4). Therefore, we concluded that the results calculated by using the database in FactSage 8.0 and the introduced Herian activity coefficient for TiO2 is the most reliable one for us. Additionally, the obtained Ti 3+ , Ti 4+ and the ratio for [Ti 4+ ] L /[Ti 3+ ] L can be used for the numerical modeling of the concentration distributions of Ti 3+ and Ti 4+ in grown Ti:sapphire single crystal. 2-3) and the red line is the fitting of the data.

Conclusion

This chapter mainly focuses on the thermodynamic calculations for Ti ion concentrations in the solid and liquid phases and their thermodynamic equilibrium segregation coefficients. The works achieved in this chapter are summarized as below:

1.

Chapter 3 Physico-chemical model for Ti valence state conversion

Introduction

Ti ion distributions and concentrations in the grown crystal mainly depend on the solidliquid interface shape and the segregation coefficients of the solutes. Ti ion concentrations in the solid phase can be calculated as function of solidified fraction (crystal length) by the Scheil-Gulliver equation (1-12) using the effective segregation coefficients of Ti 3+ and Ti 4+ and their initial concentrations in the liquid phase, as obtained in Chapter 2. The small segregation coefficients make Ti ions to present inhomogeneous distribution in grown crystal. Figure 3-2 (a) shows the expected distribution profiles of Ti 3+ and Ti 4+ along axial direction with the proceeding of crystal growth. From crystal top to the bottom, Ti ion concentrations increase with crystal length and their profiles follow the profile of Scheil-Gulliver law. For the same crystal length, the concentration of Ti 4+ should be lower than the concentration of Ti 3+ if 𝑝 O 2 < 10 -4 bar (see Figure 2345678910111213141516). In the radial direction, the distribution profiles of Ti 3+ and Ti 4+ are expected to follow the profile of solid-liquid interface as shown in This suggests that Ti ion concentrations in the solid phase not only come from the incorporation during crystal growth, they are also affected by the diffusion of Ti ions as well as point defect reactions in the solid phase. In addition, the concentration of point defects is highly related to the oxygen partial pressure in growth atmosphere. Meanwhile, the oxygen partial pressure impacts the charge conversion between Ti 3+ and Ti 4+ , leading to the variation of their concentrations and distributions. Therefore, to explain the opposite distribution profiles for Ti 3+

and Ti 4+ and investigate the conversion mechanisms between two Ti ions, a theoretical physicochemical model needs to be established.

Consquently, this chapter proposes a physico-chemical model taking into account the effect of diffusion and chemical reaction of point defects for the conversion mechanisms between Ti 3+ and Ti 4+ ions during Ti:sapphire growth process. Some physical parameters related to the proposed model are analyzed. A method for determining these parameters is suggested. The relation between the boundary concentration of point defects and oxygen partial pressure is also given.

Point defects

Point defects in crystals are responsible for optical absorption, mechanical strength, ionic conductivities and mass transport of substances. Point defects in ionic crystals comprise ionic defects (vacancies, interstitials and substitutionals) and electronic defects (electrons e -, and holes h • ).

Kröger-Vink notation [START_REF] Kröger | Relations between the concentrations of imperfections in crystalline solids[END_REF]) is used to describe the effective charges and lattice positions of point defect species in crystals. It is particularly useful for describing the defect reactions occurring in ionic crystals. The standard notation 𝐴 𝐵 𝑌 denotes that species A replaces the position of host lattice B and produces Y effective charges. A can be vacancy; B can be interstitials. Positive effective charge is represented by point, negative effective charge is represented by coma, and cross means null charge, as shown in reactions in section 3.2.1. For example, Ti Al • means that Ti species is placed in an aluminum site and the effective charge is +1 because Ti 4+ replaces Al 3+ .

Point defects in pure sapphire

The intrinsic defects in pure sapphire contain Schottky and Frenkel defects which are produced through the following reactions [START_REF] Dienes | Shell-model calculation of some point-defect properties in α-Al2O3[END_REF][START_REF] Moon | Defect clustering in H, Ti: -Al2O3[END_REF]:

Schottky reaction: 0 (null) Al 2 O 3 → 2V Al ′′′ +3V O •• (3-1)
Here, the 0 (null) designates the perfect crystal.

Cation Frenkel reaction:

Al Al × Al 2 O 3 → Al i ••• + V Al ′′′ (3-2)
Anion Frenkel reaction:

O O × Al 2 O 3 → O i ′′ + V O •• (3-3)
Interstitial reaction:

2Al Al × + 3O o × ↔ 2Al i ••• + 3O i ′′ (3-4)
These reactions produce so-called "thermodynamic defects" because they occur spontaneously at a given temperature due to the thermodynamic lattice vibrations. It is noticed that Schottky reaction produces cation and anion vacancies from stoichiometric Al2O3 while Frenkel reaction simultaneously produce interstitials and vacancies for cations or anions. In general, Schottky defects are mainly formed on crystal surface while Frenkel defects are formed within the whole crystal volume [START_REF] Kao | Dielectric phenomena in solids[END_REF]. Under thermodynamic equilibrium, the concentration of Schottky defects can be calculated by the equilibrium constant:

𝐾 𝑆𝑐ℎ𝑜𝑡𝑡𝑘𝑦 = [V Al ′′′ ] 2 [V O •• ] 3 . (3-5) Note that [V Al ′′′ ] and [V O •• ] denote the concentrations of V Al ′′′ and V O •• .
Assume the Gibbs free energy to form one mole of "Al2O3" vacancies is ∆𝐺 𝑓 , it is obtained that:

[V Al ′′′ ] 2 [V O •• ] 3 = 𝑒 - ∆𝐺 𝑓 𝑘 𝐵 𝑇 . (3-6)
It follows from Schottky reaction and charge equilibrium that 3

[V Al ′′′ ]= 2[V O •• ].
The thermal equilibrium concentration of intrinsic defects is obtained as

[V Al ′′′ ] = 2/3[V O •• ] = 2 3 × ( 9 4
) [START_REF] Cox | Chemical reactions and mass transport processes in donor and acceptor doped Al2O3 crystals[END_REF].

The incorporation of Ti 4+ involves the formation of extra defects to keep the charge balance. In both cases, titanium substitutes aluminum lattice site forming Ti Al × (= Ti 3+ ) and Ti Al • (= Ti 4+ ).

The compensation mechanism for Ti 4+ in Al2O3 is highly related to the defect energy which will be discussed in more details in the following.

Defect energies

Shell-model, first principle and other methods have been used to calculate the defect energies in pure or Ti-doped sapphire [START_REF] Dienes | Shell-model calculation of some point-defect properties in α-Al2O3[END_REF]Matsunaga et al., 2003a;Matsunaga et al., 2003b;[START_REF] Jacobs | Defect energies for pure corundum and for corundum doped with transition metal ions[END_REF]. Reported formation and migration energies of various defects in Al2O3 are shown in Table 3-1. It is noted that the formation of vacancies requires lower energy than interstitials thereby vacancies are more likely to be formed.

Table 3-1 Theoretical defect formation and migration energies per defect in sapphire single crystal [START_REF] Dienes | Shell-model calculation of some point-defect properties in α-Al2O3[END_REF]. [START_REF] Hollenberg | Effect of oxygen partial pressure on the creep of polycrystalline Al2O3 doped with Cr, Fe, or Ti[END_REF] in favor of Frenkel disorder is inconvincible because their work was conducted in polycrystalline Al2O3 instead of a single crystal. In addition, it is noted that the defect energies in Ti:sapphire calculated by Mohapatra et al. are lower than those in pure sapphire (all other values). This is consistent with the results of Matsunaga et al. who studied the defect structures in pure and titanium-doped sapphire [START_REF] Matsunaga | Theoretical study of defect structures in pure and titanium-doped alumina[END_REF]. From defect formation energies, it is predicted that divalent ions are charge compensated by oxygen vacancies and tetravalent ions are compensated by cation vacancies [START_REF] Atkinson | Accommodation of impurities in α-Al2O3, α-Cr2O3 and α-Fe2O3[END_REF][START_REF] Harding | Experiment and theory of diffusion in alumina[END_REF]. But the compensation mechanism for Ti Al • (≡Ti 4+ ) in sapphire is strongly disputed. The Ti Al • has been reported to be accompanied either by O i ′′ (Phillips et al., 1980;[START_REF] Kröger | Defect related properties of doped alumina[END_REF][START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF] or [START_REF] Roy | Solubilities of magnesia, titania, and magnesium titanate in aluminum oxide[END_REF] or V Al ′′′ , as shown below [START_REF] Powers | Titanium effects on microstructure development in alumina[END_REF][START_REF] Lagerlof | Lattice diffusion kinetics in undoped and impurity-doped sapphire (α-Al2O3): A dislocation loop annealing study[END_REF][START_REF] Reddy | Oxygen diffusion in sapphire[END_REF][START_REF] Rasmussen | Effect of dopants on the defect structure of single-crystal aluminum oxide[END_REF]:

Defect

V O •• (
Ti 2 O 3 Al 2 O 3 → 2Ti Al × + 3O O × (3-8) TiO 2 Al 2 O 3 → Ti Al • + 2O O × + V Al ′′′ + V O •• (3-9) 2TiO 2 Al 2 O 3 → 2Ti Al • + 3O O × + O i ′′ (3-10) 3TiO 2 Al 2 O 3 → 3Ti Al • + 6O O × + V Al ′′′ (3-11)
S.K. [START_REF] Mohapatra | The dominant type of atomic disorder in α-Al2O3[END_REF] found that their experimental values of formation energy per defect are consistent with the theoretical values computed by G.J. Dienes et al [START_REF] Dienes | Shell-model calculation of some point-defect properties in α-Al2O3[END_REF]. This enabled them to get the conclusion that Schottky disorder is energetically more favorable in α-Al2O3. Therefore, there is a large possibility that Ti 4+ might be compensated by V Al ′′′ or V O •• . Further, the electron paramagnetic resonance (EPR) measurements of γ-irradiated Ti doped sapphire samples has confirmed that Ti Al • is compensated by V Al ′′′ [START_REF] Cox | Chemical reactions and mass transport processes in donor and acceptor doped Al2O3 crystals[END_REF]. This means that when 3Ti Al • is incorporated, one V Al ′′′ is created at the same time. At present, most researchers prefer aluminum vacancies compensation mechanism, which is therefore considered in this thesis. [START_REF] Manning | Diffusion kinetics for atoms in crystals[END_REF].

Ion and point defect diffusion

to move from one interstitial site to neighboring interstitial site. Therefore, it requires lower energy and possesses a higher diffusion rate. The formation and migration energies of

V Al ′′′ ,V O •• , Al i ••• and O i ′′ are shown in Table 3-1.
The interstitialcy diffusion occurs for atoms, in a normal lattice site, being pushed into an interstitial site and make the initial interstitial site to occupy a normal lattice site. This diffusion is possible for an atom with large radius. In sapphire, vacancy diffusion and interstitial diffusion, as shown in Figure 3-3, are most frequently studied. The presence of crystal imperfection such as dislocations and grain boundaries can also affect the diffusion process. The diffusion along dislocations and grain boundaries are called pipe diffusion and grain boundary diffusion, respectively. However, we are dealing with single crystals (no grain boundary) with dislocation densities lower than 4 × 10 4 cm -2 [START_REF] Sen | Kyropoulos growth and characterisation of titanium doped sapphire[END_REF].

Therefore, no pipe diffusion and grain boundary diffusion are expected in Ky Ti:sapphire samples. where the direction of vector 𝑱 describes the rate of transfer species per unit volume and has the unit of mol. m -1 . s -1 , 𝐷 is diffusion coefficient of the species with the unit of m 2 . s -1 and [𝑖]

Diffusion equation

is the concentration of diffusing species 𝑖 per unit volume with the unit of mol.m -3 . The diffusion coefficient 𝐷 is independent of time and species concentration and is just related to temperature. The negative sign suggests that species diffusion occurs in the direction opposite to the concentration gradient increase. When species concentration [i](x,y,z) does not change with time, the diffusion process is called steady state diffusion.

If the species concentration changes with time, matter conservation requires that the concentration change rate over time must be equal to the negative divergence of the flux, i.e.

∂[i] ∂t = 𝐷∇ 2 [𝑖]. (3-14)
where ∇ 2 is known as Laplace operator. Equation (3-14) is called Fick's second law [START_REF] Pelleg | Diffusion in ceramics[END_REF] or the diffusion equation. If there exist motion with respect to the reference frame and chemical reaction, convection and reaction will also contribute to the flux of species. Then a convection-diffusion equation needs to be considered. For a dilute species, it will be:

∂[i] ∂t = 𝐷∇ 2 [𝑖] -𝑽∇[𝑖] + 𝑅 𝑖 . (3-15)
where 𝑽 is the velocity field and 𝑅 𝑖 is the reaction rate of the species.

Diffusion coefficients

The diffusion coefficients of ions and point defects play an important role in sapphire properties such as sintering, creep and oxide film growth. Their values can be measured through tracer diffusion and annealing experiments. The self-diffusion of Al and O in sapphire have been studied for six decades.

The diffusion coefficient of oxygen, Do, is often measured by gas exchange technique using 18 O as the tracer [START_REF] Cawley | Oxygen tracer diffusion in single-crystal alumina[END_REF][START_REF] Prot | Self-diffusion in α-Al2O3 II. Oxygen diffusion in 'undoped' single crystals[END_REF], [START_REF] Nakagawa | Oxygen pipe diffusion in sapphire basal dislocation[END_REF]. Y.

Oishi and W.D. Kingery reported that Do in the high temperature range 1650 < T < 1780 ℃ in pure sapphire is [START_REF] Oishi | Self-diffusion of oxygen in single crystal and polycrystalline aluminum oxide[END_REF]:

𝐷𝑜 = 0.19𝑒𝑥𝑝 (- 635.968 kJ/mol 𝑅𝑇 ) m 2 . s -1 . (3-16)
Depending on samples type, the measured Do show inconsistency among each other. Do in polycrystalline alumina is reported to be two orders of magnitudes larger than that in sapphire single crystal. In addition, the sample preparation process is also important. For instance, samples prepared by crushing introduced higher dislocation density thereby exhibit higher oxygen diffusivity than those in samples prepared by slicing and polishing [START_REF] Oishi | Self-diffusion of oxygen in single crystal alumina[END_REF].

The discrepancy may also result from the temperature measurement.

Only few papers reported Al diffusion because there is only one radioactive isotope, 26 Al, whose long half-time (7.2×10 5 years) results in a very low activity [START_REF] Gall M Le | Self-diffusion in α-Al2O3 I. Aluminium diffusion in single crystals[END_REF].

Therefore, it is difficult to measure the diffusivity of aluminum, 𝐷 𝐴𝑙 , in sapphire compared to the measurement of Do. P. Fielitz et al. measured 𝐷 𝐴𝑙 in pure sapphire single crystal for 1280 < T < 1500 ℃ to be [START_REF] Fielitz | 26Al tracer diffusion in titanium doped single crystalline α-Al2O3[END_REF]:

𝐷 𝐴𝑙 = 2.1 × 10 -6 𝑒𝑥𝑝 (- (362±10 ) kJ/mol 𝑅𝑇 ) m 2 . s -1 . (3-17)
Other reported values are either measured in polycrystalline aluminum oxide or in impure or polluted sapphire [START_REF] Paladino | Aluminum ion diffusion in aluminum oxide[END_REF], [START_REF] Fielitz | Self-diffusion in high-purity α-Al2O3: Comparison of Ti-doped, Mg-doped and undoped single crystals[END_REF].

In sapphire, it is believed that oxygen diffuses through oxygen vacancies or oxygen interstitials and aluminum diffuses through aluminum vacancies or aluminum interstitials [START_REF] Heuer | Oxygen self-diffusion in corundum (alpha-Al2O3): a conundrum[END_REF]. Doping with aliovalent impurities could significantly change the concentration of point defects thereby the diffusion coefficient of the atoms. It has been reported

that Mg-doping increases Do for 100 times and Ti-doping decreases Do for a factor of 5 or 50 compared to undoped sapphire [START_REF] Lagerlof | Lattice diffusion kinetics in undoped and impurity-doped sapphire (α-Al2O3): A dislocation loop annealing study[END_REF][START_REF] Haneda | Oxygen self-diffusion in magnesium-or titanium-doped alumina single crystals[END_REF][START_REF] Reddy | Oxygen diffusion in sapphire[END_REF]. By comparing the diffusivity of 26 Al tracer in undoped sapphire and Ti:sapphire , P.

Fielitz et al. believed that Ti-doping has a small influence on 𝐷 𝐴𝑙 [START_REF] Fielitz | 26Al tracer diffusion in nominally undoped single crystalline α-Al2O3[END_REF]. By codoping of divalent and tetravalent ions (MgO and TiO2) with various contents, it was experimentally shown that oxygen diffusion occurs via oxygen vacancy when dopants are mainly divalent and oxygen diffuses via oxygen interstitials when dopants are mainly tetravalent.

It is widely accepted that 𝐷 𝐴𝑙 ≫ 𝐷 𝑂 [START_REF] Paladino | Aluminum ion diffusion in aluminum oxide[END_REF]. But this is in conflict with previous results of M.L. Gall et al. who thought that the diffusion coefficients of aluminum and oxygen are of similar magnitude (𝐷 𝐴𝑙 ≈ 5𝐷 𝑂 ) [START_REF] Gall M Le | Self-diffusion in α-Al2O3 I. Aluminium diffusion in single crystals[END_REF]. Directly comparing the reported values for 𝐷 𝑂 and 𝐷 𝐴𝑙 has no sense because nearly all the values are measured in different samples and under different conditions. This has been clearly demonstrated in the systematic study of oxygen and aluminum diffusion in α-Al2O3 by A.H. Heuer [START_REF] Heuer | Oxygen and aluminum diffusion in α-Al2O3: How much do we really understand[END_REF].

They listed most of reported values for 𝐷 𝑂 and 𝐷 𝐴𝑙 and discussed the contradiction among the values. Hopefully, P. Fielitz et al. simultaneously measured the tracer diffusions of 18 O and 26 Al in the same Ti:sapphire sample. They demonstrated that aluminum diffusion is orders of magnitude higher than oxygen diffusion in Ti:sapphire single crystals [START_REF] Fielitz | 26Al tracer diffusion in titanium doped single crystalline α-Al2O3[END_REF].

Especially, they pointed out that the diffusion coefficient of aluminum does not equal the diffusion coefficient of aluminum vacancies. They proposed the relation:

𝐷( 26 Al) ≅ 𝑓 • 𝐷 V Al ′′′ • [V Al ′′′ ], (3-18)
where 𝑓 is a correlation factor often in the order of 1, and

[V Al ′′′ ] is the concentration of V Al ′′′ .
Figure 3-4 summarizes the reported diffusion coefficients of Al and V Al ′′′ in undoped sapphire and Ti-doped sapphire. The results of Fielitz et al. in 2020 show that 𝐷 𝐴𝑙 in Ti:sapphire is higher than that in undoped sapphire and this value increases with Ti doping concentration until 1000 ppmw. Particularly, the plot clearly shows that the diffusion coefficient of V Al ′′′ is higher than the self-diffusion coefficient of Al. The measured 𝐷 𝐴𝑙 value of Shi et al. [START_REF] Shi | Modelling the elements reaction-diffusion behavior on interface of Ti/ Al2O3 composite prepared by hot pressing sintering[END_REF] is high because their samples are prepared by hot pressing sintering from Ti/Al2O3 composite rather than by cutting a single crystal, therefore, the presence of grain boundaries may lead to measured results larger than exact values. Ti 3+ in sapphire makes the crystal exhibiting pink color. This pink color becomes colorless after the crystal is annealed under oxidizing atmosphere due to the oxidation from Ti 3+ to Ti 4+ .

The color boundary between pink and colorless is sometimes called color front. During the oxidizing annealing process, the color front moves from sample edge to the center with annealing time. Taking advantage of this feature, some authors measured the diffusion coefficients of Ti 3+ and Ti 4+ by annealing Ti:sapphire samples in air or oxygen atmosphere at high temperatures and observing the movement of the color front. Other methods like heat sintering or put the pure sapphire into chemical reagent are also reported [START_REF] Ruzeng | Sapphire diffusion treatment and the behaviour of iron and titanium[END_REF][START_REF] Shi | Modelling the elements reaction-diffusion behavior on interface of Ti/ Al2O3 composite prepared by hot pressing sintering[END_REF]. But the obtained values are less reliable compared to the values measured in Ti:sapphire crystals. Some literature measured the Ti diffusion coefficient without mentioning the experimental atmosphere so that it is difficult to judge if the measured diffusion coefficient is for Ti 3+ or Ti 4+ ions. However, all above reported diffusion coefficient values for Ti 3+ , Ti 4+ and aluminum vacancies in Ti:sapphire are measured without taking into account the point defect reactions.

So, these values cannot be directly used to study the concentrations and distributions of Ti 3+

and Ti 4+ where both diffusion and point defect reactions occur. In this case, two situations can happen. When the reaction rate constant is very small, the concentration variation caused by chemical reaction can be neglected. Then the Ti ion concentrations variation is mainly determined by diffusion thereby the diffusion coefficients for Ti 3+ , Ti 4+ and point defects are very close to reported values. But when the point defect reaction is faster than species diffusion, i.e., the reaction rate constant is very large, then the reported diffusion coefficient values may be not reliable anymore and further investigations are needed. The detailed studies for these two situations and the determination for diffusion coefficient and reaction rate constant will be deeply discussed in Chapter 4.

Ti valence state conversion model during crystal growth

Introduction

Ti ions and point defects not only diffuse but also participate in solid phase chemical reactions. These point defects mostly result from Ti entering the Al2O3 matrix. From the defect energy comparison in section 3.2.3, it is favored that Ti 4+ is compensated by V Al ′′′ . Therefore, the conversion between Ti 3+ and Ti 4+ may be related to V Al ′′′ . Since the concentration of intrinsic V Al ′′′ is very small [START_REF] Lagerlöf | The defect chemistry of sapphire (α-Al2O3)[END_REF], all V Al ′′′ are coming from the compensation for Ti 4+ .

Some papers [START_REF] Akul'onok | Diffusion of point defects participating in solid-phase chemical reactions (trap diffusion): Demonstration for the Ti 3+ → Ti 4+ transition in corundum[END_REF][START_REF] Akulenok | Diffusion of intrinsic defects in dielectric and semiconductor crystals with impurities[END_REF][START_REF] Maier | Mass transport in the presence of internal defect reactions-Concept of conservative ensembles: III, Trapping effect of dopants on chemical diffusion[END_REF] [START_REF] Akul'onok | Diffusion of point defects participating in solid-phase chemical reactions (trap diffusion): Demonstration for the Ti 3+ → Ti 4+ transition in corundum[END_REF][START_REF] Akulenok | Diffusion of intrinsic defects in dielectric and semiconductor crystals with impurities[END_REF][START_REF] Maier | Mass transport in the presence of internal defect reactions-Concept of conservative ensembles: III, Trapping effect of dopants on chemical diffusion[END_REF]. They were interested in solid phase chemical reactions during annealing Ti:sapphire samples with different doping levels. This provides reference for establishing a theoretical model involving diffusion and chemical reaction of point defects.

Proposed physico-chemical model

The pioneering works suggest that the [Ti 3+ ]/[Ti 4+ ] ratio in crystals is highly related to the surrounding atmosphere [START_REF] Nizhankovskii | Optical absorption and color centers in large Ti:sapphire crystals grown by horizontally directed crystallization under reducing conditions[END_REF], [START_REF] Moskvin | Color centers and luminescence in corundum crystals containing titanium[END_REF][START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF][START_REF] Eigenmann | Valence states, redox reactions and biparticle formation of Fe and Ti doped sapphire[END_REF]. Additionally, by annealing samples at 1600 ℃ in various atmospheres, it was found that the concentrations of Ti 3+ and Ti 4+ are related to the oxygen partial pressure [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF]. However, our thermodynamic calculation results in Chapter 2 show that the segregation coefficients of Ti 3+ and Ti 4+ have no significant change with 𝑝 O 2 (section 2.5.1). Therefore, according to the Scheil-Gulliver theory, the same radial distribution profiles for Ti 3+ and Ti 4+ should be expected in the same crystal, as shown in Figure 3-2(b).

However, this is not consistent with the Ti ion distributions observed in grown crystals.

Considering the decreased concentration profile of Ti 4+ at the periphery of the Ky-grown crystal in Figure 1-20 -Figure 1-22, and the low 𝑝 O 2 crystal growth atmosphere caused by the Mo crucible and the massive use of graphite elements in the furnace (Chapter 2), it is assumed that the decrease of Ti 4+ concentration near the crystal periphery is caused by the reducing reaction of Ti 4+ into Ti 3+ . Numerous articles reported that the global chemical reaction

Ti Al • + 1 3 V Al ′′′ + 1 6 Al 2 O 3 ↔ Ti Al  + 1 3 Al Al × + 1 4 O 2 (3-19)
between Ti 4+ and Ti 3+ occurs in bulk crystal [START_REF] Hollenberg | Effect of oxygen partial pressure on the creep of polycrystalline Al2O3 doped with Cr, Fe, or Ti[END_REF][START_REF] Cox | Chemical reactions and mass transport processes in donor and acceptor doped Al2O3 crystals[END_REF][START_REF] Savrun | The effect of titanium doping on the high-temperature rhombohedral twinning of sapphire[END_REF]. Some annealing experiments have shown that this reaction is reversible [START_REF] Keig | Influence of the valence state of added impurity ions on the observed color in doped aluminum oxide single crystals[END_REF][START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF]. Based on this global reaction, reported experimental results and trap diffusion model, we propose that the global reaction consists in two local reactions: reaction (3-20) [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF] V Al ′′′ . In the general case, as species have different diffusion coefficients, an electric field produced between the fast moving and slow moving carriers. This affects the diffusion process in such a way that fast carriers are slow down and slow carriers are accelerated. It is therefore assumed that these aluminum vacancies are transported with associated electronic holes to maintain local charge neutrality all along the diffusion path [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF]

1 3 V Al ′′′ + 1 6 Al 2 O 3 ↔ 1 3 Al Al × + 1 4 O 2 + e -.
(3-22)

Ti Al

• + e -↔ Ti Al  .

(3-23)

Or the global reaction (3-24) in which Ti Al • is compensated by O i ′′ [START_REF] Kröger | Defect related properties of doped alumina[END_REF]: This equation can also be considered as the combination of two local reactions which occur on the crystal surface and in the crystal bulk, as shown in (3-25) and (3-26):

2Ti Al • + O i ′′ ↔ 2Ti Al  + 1 2 O 2 + V i × .
O i ′′ ↔ 1 2 O 2 + V i × + 2e -. (3-25) 2Ti Al • + 2e -↔ 2Ti Al  . (3-26)
Anyhow, the real compensating point defect for Ti Al • would not change the basic mechanism for the valence state conversion between the two Ti ions. Only parameters such as diffusion coefficients of point defects or reaction rates for local reactions may change. For the sake of simplicity, in the following text, it will always be made reference to V Al ′′′ and h • , while we should keep in mind that the real diffusion defects could be others.

In addition, the point defects inside the crystal could form clusters (or defect complexes or defect associations). Two most likely neutral clusters have been reported to be (2 Ti Al [START_REF] Pletka | Solid solution hardening of sapphire (α-Al2O3)[END_REF][START_REF] Lagerlof | Lattice diffusion kinetics in undoped and impurity-doped sapphire (α-Al2O3): A dislocation loop annealing study[END_REF][START_REF] Moon | Defect clustering in H, Ti: -Al2O3[END_REF]. The formation of the clusters is determined by the binding energies. According to the reported binding energies of various clusters for incorporating TiO2 into sapphire [START_REF] Lagerlöf | The defect chemistry of sapphire (α-Al2O3)[END_REF], clusters are only easily formed at low temperature. The concentrations of point defects are rather low, so the probability for them to combine together to form clusters is very low.

• O i ′′ ) × and (3 Ti Al • V Al ′′′ ) × (
When the temperature is too high, formed clusters may break up and decompose into isolated point defects. The extent to which point defects will form clusters or clusters will break into isolated point defects is highly related to the binding energy. It is possible that the isolated point defects V Al ′′′ and 3h • in the model combines into a cluster but no paper is referring to that.

Atkinson et al. pointed that clustering does not change the compensation mechanism for the substitution of dopants [START_REF] Atkinson | Accommodation of impurities in α-Al2O3, α-Cr2O3 and α-Fe2O3[END_REF]. In fact, no matter it is cluster or isolated point defect, only the values of diffusion coefficients will be affected. In addition, the concentration of isolated point defect V Al ′′′ on crystal surface is helpful to study the influence of temperature field and 𝑝 O 2 on the Ti ion distributions, as described in section 3.6. Therefore, the joint diffusion of V Al ′′′ and 3h • in our model is not considered to be a cluster in this thesis.

Mathematical model

It can be seen from equation (3-19) that the concentrations of Ti 3+ and Ti 4+ are highly related to the concentration of V Al ′′′ . Crystal growth, species diffusion and point defect reactions lead to the concentrations of Ti 3+ and Ti 4+ in the crystal varying with time. Therefore, three transient diffusion equations for Ti 3+ , Ti 4+ and V Al ′′′ must be taken into account. The electronic holes produced by equation (3-21) immediately diffuse away with the diffusion of V Al ′′′ so that their concentration need not to be explicitly calculated. Because the reference axis is attached to the seed, the velocity V is zero. According to diffusion-reaction equation (3-15), the governing equations for Ti 3+ , Ti 4+ and V Al ′′′ are:

∂ [Ti Al  ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙  (𝑇)∇ 2 [Ti Al  ] + 𝑅 𝑇𝑖 𝐴𝑙  , (3-27) ∂ [Ti Al • ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙 • (𝑇)∇ 2 [Ti Al • ] + 𝑅 𝑇𝑖 𝐴𝑙 • , (3-28) ∂ [V 𝐴𝑙 ′′′ ] ∂t = 𝐷 𝑉 𝐴𝑙 ′′′ (𝑇)∇ 2 [V Al ′′′ ] + 𝑅 𝑉 𝐴𝑙 ′′′ .
(3-29)

where 𝐷 𝑇𝑖 𝐴𝑙  , 𝐷 𝑇𝑖 𝐴𝑙 • , 𝐷 𝑉 𝐴𝑙 ′′′ , 𝑅 𝑇𝑖 𝐴𝑙  , 𝑅 𝑇𝑖 𝐴𝑙 • and 𝑅 𝑉 𝐴𝑙 ′′′ are the diffusion coefficients and reaction rates of Ti 3+ , Ti 4+ and V Al ′′′ in sapphire. The square brackets denote the concentration of species.

The concentration of intrinsic V Al ′′′ in sapphire is very small compared to that introduced by Ti doping [START_REF] Lagerlöf | The defect chemistry of sapphire (α-Al2O3)[END_REF]. Consequently, it is assumed that the concentration

relation [Ti Al • ] ≈ 3[V Al ′′′ ]
is valid in the crystal, due to the compensation mechanism. We believe that this compensation mechanism is only limited by the reaction rate in equation (3-21).

It is assumed that this concentration relation for Ti Al • and V Al ′′′ is valid everywhere in the crystal because crystal growth is a very slow process. When

([Ti Al • ] -3[V Al ′′′ ] 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 ) which is also the number of involved h • . Herein, the [V Al ′′′ ] 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 = [V Al ′′′ ] 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 -[V Al ′′′ ] 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 .
Assuming that the two local reactions occurring on crystal surface and bulk crystal are elementary reactions, the reaction rates for Ti Al  , Ti Al • and V Al ′′′ can be written as:

𝑅 𝑇𝑖 𝐴𝑙 • = -𝑘 𝑐𝑔 • [Ti Al • ] • ( [Ti Al • ] -3[V Al ′′′ ] 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 ), (3-30)
𝑅 𝑇𝑖 𝐴𝑙  = -𝑅 𝑇𝑖 𝐴𝑙 • , (3-31)
𝑅 𝑉 𝐴𝑙 ′′′ = 0.

(3-32)

where 𝑘 𝑐𝑔 is the forward reaction rate constant of reaction (3-21), with 𝑘 𝑐𝑔 > 0.

The [Ti Al • ]
term in equation (3-30) denotes the probability of Ti Al • to appear in reaction (3-21). Since V Al ′′′ inside the crystal only diffuses but does not participate to the chemical reaction, its reaction rate is zero. The diffusion coefficients and reaction rate constant are function of temperature.

Dimensionless equations

The trap diffusion reported by Akul'onok et al. [START_REF] Akul'onok | Diffusion of point defects participating in solid-phase chemical reactions (trap diffusion): Demonstration for the Ti 3+ → Ti 4+ transition in corundum[END_REF][START_REF] Akulenok | Diffusion of intrinsic defects in dielectric and semiconductor crystals with impurities[END_REF] is the only case where point defect diffusion and chemical reactions are both considered. They were focusing on the annealing of Ti-doped sapphire. It was assumed that the point defects react with motionless impurities, i.e. the diffusion coefficient of Ti ions are considered to be neglected. A diffusion process along x-direction involving a second-order chemical reaction is described by:

𝜕[V Al ′′′ ] 𝜕𝑡 = 𝐷 V Al ′′′ 𝜕 2 [V Al ′′′ ] 𝜕𝑥 2 -𝑘 𝑎𝑛𝑛 [V Al ′′′ ] [Ti Al  ]. (3-33) 𝛽 ∂𝜑 ∂𝜏 = ∂ 2 𝜑 ∂𝑦 2 + 𝑒 -𝜑 .
(3-45)

with the initial and boundary conditions as:

𝜑(𝜏 = 0, y > 0) = 0, (3-46) ∂𝜑 ∂𝜏 (𝜏 = 0, y > 0) = 0, (3-47)
𝜑(𝜏 > 0, y = 0) = 𝜏.

(3-48)

It needs to be pointed out that the second term 𝑒 -𝜑 in equation (3-45) is written as (𝑒 -𝜑 -1)

by mistake in Akul'onok's paper. From equations (3-38) -(3-44), it is seen that the solution for 𝜑 (or for 𝑢 and 𝜈 ) is determined only by three independent parameters

( 𝑘 𝑓 [Ti Al  ] 𝑡0 𝐷 V Al ′′′ ) 1/2 , 𝑘 𝑎𝑛𝑛 [V Al ′′′ ] 𝑡=0 and [V Al ′′′ ] 𝑡=0 [Ti Al  ] 𝑡=0
. Among them, the unknown parameters 𝑘 𝑎𝑛𝑛 , [V Al ′′′ ] 𝑡=0 and 𝐷 V Al ′′′ characterize the trap diffusion of point defects and cannot be independently measured from experiments. The only value that can be directly measured from experiment is impurity

concentration [Ti Al  ] t>0,x>0
. It is very difficult to get an analytical solution for equation (3-45)

and determine the three parameters for 𝑘 𝑎𝑛𝑛 , [V Al ′′′ ] 𝑡=0 and 𝐷 V Al ′′′ through known conditions.

However, this can be solved by the characteristics of trap diffusion without solving equation (3-45). The color front (see section 3.3.3) displacement, 𝑋 𝐵 , resulting from point defect diffusion and reaction can be demonstrated as:

𝑋 𝐵 = √ 𝐷 𝑓 𝑡. (3-49)
with the reaction front displacement coefficient to be:

𝐷 𝑓 = 2 [V Al ′′′ ] 𝑡=0 [Ti Al  ] 𝑡=0 𝐷 V Al ′′′ .
(3-50)

The reaction front displacement can be measured by observing the color boundary motion in annealed Ti:sapphire samples. Generally, authors [START_REF] Fielitz | 26Al tracer diffusion in titanium doped single crystalline α-Al2O3[END_REF][START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF] measured 𝐷 V Al ′′′ by this way, but this analyze shows that in reality they measured 𝐷 𝑓 . The

coefficient 𝐷 𝑓 is different from the diffusion coefficient of aluminum vacancies 𝐷 V Al ′′′ found in literature because 𝐷 𝑓 is determined by [V Al ′′′ ] 𝑡=0 and [Ti Al  ] 𝑡=0 while 𝐷 V Al ′′′ is independent of them.
In fact, the measured diffusion coefficient concerns the point defect which diffuses,

whatever it is. It could be V Al ′′′ , h • , or V Al ′′′ -3h • ....
Hopefully, two annealing experiments are detailed in the literature. Through the numerical simulation of annealing at different temperatures, with respect to the annealing experimental data, it is therefore possible to determine the Arrhenius equations for 𝐷 V Al ′′′ and 𝑘 𝑎𝑛𝑛 . Details about the annealing simulation are discussed in Chapter 4. The third unknown parameter [V Al ′′′ ] 𝑏 can be computed through the equilibrium constants of other point defect reactions, which will be explained in the following part. Then, the obtained three parameters can be applied to our proposed model to perform numerical simulations for crystal growth.

Point defect concentration on the crystal surface

The boundary concentration for aluminum vacancies on the crystal surface is related to the 𝑝 O 2 as shown in the local reaction (3-20). The equilibrium constant for surface reaction (3-20) is written as:

𝐾 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 𝑝 O 2 1 4 [V Al ′′′ ] 𝑏 1/3 •[O O × ] 1/2 •[h • ]
.

(3-51) [START_REF] Libowitz | Defect equilibria in solids[END_REF]. The intrinsic excitation reaction occurs inside sapphire:

As O is an intrinsic constituent, [O O × ] ≈ 1 (
0 ↔ e -+ h • (3-52)
with the equilibrium constant as [START_REF] Mohapatra | The dominant type of atomic disorder in α-Al2O3[END_REF]:

𝐾 𝑖 = [e -] • [h • ]. (3-53)
The point defect reaction involving electrons and V Al ′′′ has been reported as [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF]:

3 4 O 2 + 3e -↔ 3 2 O O × + V Al ′′′ (3-54)
with the equilibrium constant as:

𝐾 e = [V Al ′′′ ] 𝑏 [e -] 3 •𝑝 O 2 3/4 .
(3-55)

From the equilibrium constant expressions of (3-51), (3-53) and (3-55), it is obtained that

𝐾 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 𝐾 𝑒 -1/3 𝐾 𝑖 . (3-56)
𝐾 𝑖 can be estimated from the optical band gap at 300 K [START_REF] Mohapatra | The dominant type of atomic disorder in α-Al2O3[END_REF]:

𝐾 𝑖 = 2.28 × 10 31 𝑇 3 exp ( 𝛽 𝑖 𝑘 𝐵 ) • exp [- (9.9 + 298𝛽 𝑖 )eV 𝑘 𝐵 𝑇 ] cm -6
= 2.28 × 10 31 𝑇 3 exp (17.39 -10.35eV 𝑘 𝐵 𝑇 ) cm -6 (3-57)

where 𝛽 𝑖 = 1.5 × 10 -3 eV/K, 𝛽 𝑇𝑖 = 5 × 10 -4 eV/K . 𝛽 𝑖 and 𝛽 𝑇𝑖 are constants describing respectively the temperature dependence of the band gap of Al2O3 and the temperature dependence of the separation between the Ti level and the conduction band. In addition, the equilibrium constant 𝐾 e is reported to be [START_REF] Mohapatra | Defect structure of α-Al2O3 doped with titanium[END_REF]: those resulting from doping. However, there is neither precipitation nor plastic deformation observed in our Ti:sapphire crystals. The ambipolar diffusion means the ion and electronic defects diffuse in the opposite direction. But in our proposed model, we believe that the ion and electronic defects diffuse together along the same direction because in this case the charge neutrality condition can be satisfied during the whole diffusion and point defect reaction process.

So, the reason causing the discrepancy between the 0.095 and 0.018 power laws is still not clear.

According to FactSage support team, Dr. Tatjana Jantzen, charged point defects are not considered in FactSage software. Therefore, the V Al ′′′ concentration relation with 𝑝 O 2 which could be calculated by FactSage (through 1/3 [Ti 4+ ]S) is not reliable and cannot be used for the numerical simulation.

In Therefore, theoretical calculated aluminum vacancies concentration (3-60) will be used for crystal growth process simulation as presented in Chapter 4. Clearly, further studies, theories and experiments, are needed to give better boundary condition relations.

Conclusion

This chapter proposes a theoretical model to explain the observed opposite distributions of Ti 3+ and Ti 4+ and understand the conversion mechanisms between Ti 3+ and Ti 4+ ions in the solid phase of Ti:sapphire during its growth. The main results are summarized as follows:

1. Considering that Ti doping could give rise to point defects, it is suspected that the Ti ions conversion may be related to point defects. Therefore, four types of point defects mechanisms, Schottky, Frenkel (cation and anion) and Interstitial are presented. As the formation of point defects is determined by their formation energies, they are discussed.

By comparing the defect formation energy, it is found that, in sapphire, aluminum vacancies are more likely to be formed to compensate Ti 4+ ions which sets the fundamentals for the following proposed physico-chemical model.

2. Diffusion of point defects in crystal can affect Ti ion concentrations. Therefore, vacancy, interstitial and interstitialcy diffusion mechanisms in ionic crystals are discussed.

Among them, vacancy and interstial diffusions most frequently occur in sapphire. Then, reported diffusion coefficients for Ti ions and aluminum vacancies are shown and compared. The results show that Ti-doping decreases Do for a factor of 5 or 50 compared to those in undoped sapphire. In addition, it is found that 𝐷 𝐴𝑙 > 𝐷 𝑂 and 𝐷 𝐴𝑙 in Ti:sapphire is higher than that in undoped sapphire and this value increases with Ti doping concentration. Especially important is that the diffusion coefficient of V Al ′′′ is higher than the self-diffusion coefficient of Al, i.e. 𝐷 V Al ′′′ > 𝐷 𝐴𝑙 . However, there exists conflicts for diffusion coefficient values on literature and they did not take into account simultaneous reaction rate. Therefore, further investigations are needed and they will be studied in Chapter 4.

Based on the investigations about point defect diffusion and chemical reaction, we

propose a physico-chemical model, that the conversion between Ti 3+ and Ti 4+ could be controlled by jointly diffusing V Al ′′′ -h • from crystal bulk towards crystal surface and the chemical reaction of h • inside the crystal. Consequently, some mathematical equations are defined to describe quantitatively the diffusion and chemical reaction of point defects, especially the reaction rate for each species is carefully analyzed. process. However, currently, no direct evidence can be shown to prove the reliability for these different relations under high or low 𝑝 O 2 . Further studies, theories and experiments, are needed. In the next step, the temperature function of 𝐷 V Al ′′′ and 𝑘 𝑐𝑔 will be determined through numerical simulation of annealing experiments. These parameters will further be implemented into numerical modeling of crystal growth to study the validity of the proposed physico-chemical model.

Chapter 4 Numerical simulation of Ti ion behavior during annealing and crystal growth

Introduction

In the previous chapter, a physico-chemical model has been proposed to explain the conversion mechanism between Ti 3+ and Ti 4+ during Ti:sapphire crystal growth. Next step consists in checking the validity of the model and compare with experimental results, if any.

The spatial distributions for Ti ions and point defects in the solid phase during Ti:sapphire growth process can be obtained by solving the diffusion-reaction equations of the proposed model. This requires to know all the physical parameters related to point defect diffusion and chemical reactions.

Therefore, annealing experiments are analyzed with the help of numerical simulation to determine the physical parameters of diffusion coefficient of aluminum vacancies and reaction rate constant. Following this, the determined parameters are applied in the physico-chemical model to study the distributions of Ti ions. In addition, the influences of temperature field, crystal growth rate and oxygen partial pressure are also studied. All calculations in this chapter are performed with COMSOL Multiphysics software.

As explained in section 3.6, there are several boundary condition equations for aluminum vacancy concentration that are not consistent but will be used in this chapter. Therefore, we are not expecting reliable results, we rather describe a methodology for tackling such phenomena.

4.2 Finite element method and COMSOL software 4.2.1 Finite element method (FEM) Partial differential equations (PDEs) allow to describe the change of a system over space and time. Nearly all the laws of physics can be described by PDEs. In practical engineering and research, many physical or chemical problems involving multiphysics fields cannot be carried out due to the restricted time, cost, and feasibility in experiments. Therefore, these problems can only be studied through mathematical methods. This requires to translate the real-world physical model into a well-posed mathematical model which consists of one or several PDEs together with some initial and boundary conditions. The most common PDEs are conservation equations of energy, momentum and mass. These PDEs usually cannot be solved with analytical methods but they can be approximated by discretization methods using numerical model equations, which can be solved using numerical tools. The discretization of mathematical model give rise to a numerical model. The solutions of numerical model equations are therefore an approximation of the real solution to the mathematical model. The finite element method (FEM) is used to compute such approximations.

FEM is a computational method that subdivides the studied object into many small but finite-size elements. The physics of one element is approximately described by a finite number of degrees of freedom (DOFs). Each element is accompanied with a set of characteristic equations for describing the physical properties and boundary conditions. Then all the equations which describe all the elements are solved simultaneously to study the physical process and phenomenon or predict the behavior of the object.

The FEM introduces test functions which are defined through a computational mesh. For each mesh element, a number of local test functions are defined. These test functions are loworder polynomials. The discretization of system induces looking for an approximate solution which is described by the linear combination of a group of test functions. Consequently, these combined test functions can be written in the form of system matrix. Then, the solution of the PDEs can be approximated by solving the algebraic equations with matrix analysis. The denser the mesh, the closer the approximated solution approaches to the actual solution.

Newton method is usually used to search for potential solutions, which is only reliable when the approximated solution is close enough to the true numerical solution. The difference between the approximated solution and true numerical solution is called error. When an estimated error tolerance is reached, convergence occurs. In this case, the approximated solution can be considered as the solution of PDEs.

Brief introduction to COMSOL Multiphysics software

The discretized equations must be solved and they are too heavy to be solved manually.

Therefore, many commercial FEM softwares have been developped as tools to solve these equations, for example, Hyperworks, ABAQUS, ANSYS and the COMSOL Multiphysics software used in this thesis. Each software has its own characteristics. For example, Hyperworkds and its main product -Hypermesh is famous for pretreatment; ABAQUS is wellknown for its powerful convergence for nonlinear models; ANSYS has a better convergence for fluid dynamic calculations than COMSOL Multiphysics; But, COMSOL has the unique multiphysics coupling and analyzing capabilities such as electromagnetic, structural mechanics, acoustics, fluid flow, heat transfer and chemical engineering. It enables to simulate design, preparation, and processes in fields such as engineering, manufacturing and scientific research.

Especially, it is the only software that puts all the governing equations (PDEs), initial and boundary conditions on the operational interface, which could help users to better relate the mathematical model with the physical model they want to study. It has been extensively used in the simulation for crystal growth [Kakimoto et al., 2015]. Anyhow, COMSOL is just a "tool" to solve the mathematical equations of models, the most important thing is to understand the model itself and the principles hiden on the back of the model.

In our model, we would like to study the species behaviour during crystal growth, the effect of chemical reaction and the influence of temperature field. Therefore, COMSOL software is used in this thesis to study these problems involving multiphysics fields. The details for the settings and operations in COMSOL will be explained later in conjunction with our physico-chemical model.

Physical parameters determination

The diffusion coefficient of V Al ′′′ and the reaction rate constant for point defect reaction Sample thickness is L [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF].

The works of T.P. Jones et al. [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF] is the only one presenting annealing experimental data. Cubic samples are used in the annealing experiment. However, all measured XB are small compared to L, thereby a simplified 2D geometry (Figure 4-3) is built for the study. The governing equations are the same as proposed model, i.e.

∂ [Ti Al

 ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙  (𝑇)∇ 2 [Ti Al  ] + 𝑅 𝑇𝑖 𝐴𝑙  , ∂ [Ti Al • ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙 • (𝑇)∇ 2 [Ti Al • ] + 𝑅 𝑇𝑖 𝐴𝑙 • , ∂ [V 𝐴𝑙 ′′′ ] ∂t = 𝐷 𝑉 𝐴𝑙 ′′′ (𝑇)∇ 2 [V Al ′′′ ] + 𝑅 𝑉 𝐴𝑙 ′′′ .
The reaction rates in annealing (oxidizing reaction) are different from crystal growth (reducing reaction) process. In annealing process, the reactions occurring on crystal surface and inside bulk crystal proceed along the reverse direction of equations (3-20) and (3-21), i.e.

1 4 O 2 ↔ 1 3 V Al ′′′ + 1 2 O O × + h • . (4-1) Ti Al  + h • ↔ Ti Al • . (4-2)
Therefore, the reaction rates in reaction (4-2) are:

𝑅 𝑇𝑖 𝐴𝑙  = -𝑘 𝑎𝑛𝑛 • [Ti Al  ] • (3[V Al ′′′ ] -[Ti Al • ]), (4-3) 𝑅 𝑇𝑖 𝐴𝑙 • = -𝑅 𝑇𝑖 𝐴𝑙  , ( 4

Grids

COMSOL solves the governing equations with finite element method which means that the geometric domain is divided into many small discrete cells to perform the computation and then approximate the solutions over the larger domain. The classification of grids consists in structured grids like quadrilateral and unstructured grids like triangles. The structured grids produce less cells therefore shorter computation time. A mapped quadrilateral mesh is particularly effective for 2D simulations with regular shapes. Therefore, it is used in the annealing simulation for the rectangle domain. Mesh sequence order also has a large influence to the final mesh result. Because the after-mesh element sizes are built based on the foregoing mesh operations. Therefore, the area which requires the highest resolution should be meshed with very fine grids.

Mesh size is critically important for the accuracy of simulation results and is highly related to the physical model. Too large mesh size may lead to inaccurate results or convergence problem while too small size leads to very long computation time. In diffusion-reaction model, the mesh size should be smaller than the diffusive length:

𝐿 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 = 2√𝐷 • ∆𝑡. (4-10)
where 𝐷 is the diffusion coefficient of species and ∆𝑡 is time step of each calculation. A compromise mesh size should be selected by taking into account both the precision of calculation results and the computation time. A mapped mesh with the maximum mesh size of 3 × 10 -3 cm and the minimum mesh size of 3 × 10 -6 cm is used in annealing simulation. The number of elements is 70474.

Numerical procedures

Transient solver is selected to solve the time-dependent diffusion-reaction equations with a time step of 1000 s. The simulated time is the same as the annealing experiment. The relative tolerance is set to 10 -4 and maximum number of iterations are set to be 100 so that the results can converge properly during the computation. Increasing the iteration number and decreasing the relative tolerance works well for some of the convergence problems. Refining the mesh is another effective method for improving the convergence.

Parameter estimations

Since the diffusion coefficients of ions in the solid phase are very small (about three orders of magnitude lower than that in the liquid phase) and their values are smaller than the diffusion coefficient of point defect, in the preliminary study it is assumed that the diffusion coefficients for Ti 3+ and Ti 4+ are zero. Their values will be considered later. No value for the diffusion coefficient of V Al ′′′ or the reaction rate constant is reported as a reference. According to the annealing experiment of [START_REF] Akul'onok | Diffusion of point defects participating in solid-phase chemical reactions (trap diffusion): Demonstration for the Ti 3+ → Ti 4+ transition in corundum[END_REF] as shown in Figure 4-4, it was deduced that the square of color boundary movement is proportional to the initial Ti concentration ([Ti] initial , known parameter), the V Al ′′′ concentration on crystal surface and the diffusion coefficient of V Al ′′′ :

𝐷 𝑓 = 2 [V Al ′′′ ] 𝑏 [Ti] initial 𝐷 V Al ′′′ .
which has been shown in equation . Therefore, linearly fitting the 𝐷 𝑓 as function of annealing time 𝑡 enables to estimate the possible order of magnitude of 𝐷 𝑉 𝐴𝑙 ′′′ at T=1500 ℃ through the slope of fitted line. On the other hand, through dimensionless equations discussed in section 3.5, the reaction rate constant can be estimated through the product of characteristic annealing time and concentration of aluminum vacancies on the boundary, i.e. 𝑡 ≈ 𝑘 𝑎𝑛𝑛 • [V Al ′′′ ] 𝑏 . Then start the simulation for annealing experiment with 𝐷 𝑉 𝐴𝑙 ′′′ = 9 × 10 -12 m 2 . s -1 , 𝑘 𝑎𝑛𝑛 = 5 × 10 -7 m 3 . mol -1 . s -1 and adjust their values by observing the positions of the color boundaries at different times until the simulated color boundary positions are close to the experimental positions. Finally, the 𝐷 V Al ′′′ and 𝑘 𝑎𝑛𝑛 at T=1500 ℃ can be determined. According to the rule of thumb, the value of 𝐷 V Al ′′′ increased about one-order of magnitude when the temperature is increased for about 100 ℃. We assume that this rule works for 𝑘 𝑎𝑛𝑛 as well.

Then estimated values of 𝐷 V Al ′′′ and 𝑘 𝑎𝑛𝑛 at T=1600 ℃ can be obtained and be applied to start the simulation. The same adjustment for these values is performed as before until the appropriate values are found. These operations are repeated for higher temperatures until all the required values under different temperatures are obtained. taken from the annealing experiment of Akul'onok et al [START_REF] Akul'onok | Diffusion of point defects participating in solid-phase chemical reactions (trap diffusion): Demonstration for the Ti 3+ → Ti 4+ transition in corundum[END_REF].

Results for annealing simulation

Figure 4-5 shows the color boundary positions for Ti:sapphire sample annealed for 280000 s, 440000 s, and 630000 s at T=1400 ℃. It can be observed that simulated color boundary positions agree well with the experimental positions (three thin lines inside the sample). In addition, it is found that the position of the color boundary is mainly affected by 𝐷 𝑉 𝐴𝑙 ′′′ and that the value of reaction rate constant 𝑘 𝑎𝑛𝑛 has a rather low influence on it. When 𝐷 𝑉 𝐴𝑙 ′′′ and kann vary for a small range, the simulated results have no significant change. So, the obtained values have an acceptable error range. For example, at T=1400 ℃, it is obtained that 𝐷 𝑉 𝐴𝑙 ′′′ = (9.5 ± 0.5) × 10 -13 m 2 . s -1 and 𝑘 𝑎𝑛𝑛 = (2.5 ± 0.5) × 10 -6 m 3 . mol -1 s -1 . Both 𝐷 𝑉 𝐴𝑙 ′′′ and 𝑘 𝑎𝑛𝑛 increase with temperature. It is assumed that the reaction rate for Ti 4+ in annealing is the same as that in crystal growth process. It follows from the reaction rate expression in annealing and physico-chemical model that:

𝑘 𝑎𝑛𝑛 • [Ti Al  ] • (3[V Al ′′′ ] -[Ti Al • ]) = -𝑘 𝑐𝑔 • [Ti Al • ] • ( [Ti Al • ] -3[V Al ′′′ ]), (4-13)
Therefore,

𝑘 𝑐𝑔 = 𝑘 𝑎𝑛𝑛 • [Ti Al  ]/ [Ti Al • ]. (4-14)
The Arrhenius relation for 𝐷 𝑉 𝐴𝑙 ′′′ and 𝑘 𝑐𝑔 will be applied to the simulation of crystal growth for further studies.

Crystal growth model basic assumptions

In practical crystal growth process, it is difficult to grow a crystal with a fixed radius.

The incorporation of Ti ions, the diffusion and point defect reaction processes can be affected by many factors. For example, the diffusion coefficient of point defect and reaction rate constant change with temperature, the solid-liquid interface shape can affect the radial distributions of incorporated Ti ions, the liquid phase might not be completely mixed, elements might evaporate from the solid and liquid phase, the 𝑝 O 2 of surrounding atmosphere might be different depending on the position inside the furnace, etc. However, not all of these are important to our proposed model. As explained in Chapter 3, the key parameters related to the model are diffusion coefficient of point defect, the reaction rate constant, which are function of temperature, and the concentration of point defect on crystal surface which is function of temperature and 𝑝 O 2 . Therefore, except temperature and 𝑝 O 2 , all other factors can be considered later. To simplify the modelling and computing processes to study proposed physico-chemical model during crystal growth, a few hypotheses are made: 1) Ti:sapphire crystal is grown with a constant growth rate of 3 mm. h -1 ;

2) The crystal is an axisymmetric cylinder which has a flat top surface;

3) The solid-liquid interface is assumed to be curved in order to introduce radial segregation of Ti and its shape does not change during the whole crystal growth process;

4) The initial concentration of species in the solid (seed) is homogeneous;

5) The liquid phase is homogeneous and its concentration changes with time following the Scheil-Gulliver law. This is possible when strong convection is present in the melt; 6) No Ti evaporates from the solid or liquid surfaces;

7) The evaporation of alumina from solid and liquid phase is not considered; 8) 𝑝 O 2 is the same everywhere around the crystal and liquid;

9) The reaction occurring on crystal surface is very fast [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF]

∂ [Ti Al  ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙  (𝑇)∇ 2 [Ti Al  ] + 𝑅 𝑇𝑖 𝐴𝑙  , ∂ [Ti Al • ] ∂t = 𝐷 𝑇𝑖 𝐴𝑙 • (𝑇)∇ 2 [Ti Al • ] + 𝑅 𝑇𝑖 𝐴𝑙 • , ∂ [V 𝐴𝑙 ′′′ ] ∂t = 𝐷 𝑉 𝐴𝑙 ′′′ (𝑇)∇ 2 [V Al ′′′ ] + 𝑅 𝑉 𝐴𝑙 ′′′ .
The final goal of the simulation is to obtain the spatial distributions of Ti 3+ , Ti 4+ and V Al ′′′ with time during Ti:sapphire crystal growth.

Initial conditions

In Ky crystal growth process, the liquid phase can be considered to be homogeneous when there is a strong convection in the melt. This has been proved by the numerical simulation results of Stelian et al. (Stelian et al., 2017). The solute concentrations in the liquid phase vary with time following the Scheil-Gulliver equation. According to the segregation theory (section 1.4.2 in Chapter 1), the solute concentrations in the solid phase at the solid-liquid interface at any time can be calculated from the concentrations in the liquid phase by: (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) where 𝑘 𝑇𝑖 3+ and 𝑘 𝑇𝑖 4+ are the segregation coefficients of Ti 3+ and Ti 4+ . The subscripts "S" and "L" represent species concentrations in the solid or liquid phases, which are determined by 𝑝 O 2 (section 2.4). The segregation coefficients of Ti 3+ and Ti 4+ have been computed to be 0.046 and 0.011 through the thermodynamic calculations in Chapter 2. Taking into account the influence of Mo crucible, graphite elements and trace impurities (10 ppm O2 + 10 ppm H2O) inside the furnace, the 𝑝 O 2 is calculated to be 1.1 × 10 -20 bar at T=2054 ℃. According to the total Ti concentration of 37.5 mol/m 3 and

[Ti Al  ] S = [Ti 3+ ] 𝑆 = 𝑘 𝑇𝑖 3+ [Ti 3+ ] L, (4-15) [Ti Al • ] S = [Ti 4+ ] 𝑆 = 𝑘 𝑇𝑖 4+ [Ti 4+ ] L , (4-16) [V Al ′′′ ] S = 1 3 [Ti Al • ] S .
[Ti 4+ ] L [Ti 3+ ] L
= 4.76 × 10 -4 , the initial concentrations of Ti 3+ and Ti 4+ in the liquid phase are calculated to be 37.482 mol.m -3 and 0.018 mol.m -3 . Assuming that the seed is homogeneous and in equilibrium with the initial liquid. The initial concentrations of Ti 3+ and Ti 4+ in the solid phase (seed) are:

[Ti 3+ ] S_initial = 𝑘 𝑇𝑖 3+ [Ti 3+ ] L = 1.724 mol. m -3 , (4-18)

[Ti 4+ ] S_initial = 𝑘 𝑇𝑖 4+ [Ti 4+ ] L = 1.98 × 10 -4 mol. m -3 , (4-19)
According to the charge compensation relation, the initial concentration of V Al ′′′ in the solid phase is

[V Al ′′′ ] S_initial = 1 3 [Ti Al
• ] S = 6.6 × 10 -5 mol. m -3 . (4-20)

Boundary conditions

Since no significant Ti evaporation was observed at crystal surface [START_REF] Sharp | Instability in the growth of Ti: Al2O3 single-crystal fibres[END_REF][START_REF] Nizhankovskiy | Optical homogeneity of Ti:sapphire crystals grown by horizontal directional solidification[END_REF], the boundary conditions for Ti 3+ and Ti 4+ at the top and lateral surfaces are considered to be zero flux:

𝜕 [Ti Al × ] 𝜕𝑛 = 0, (4-21) 𝜕 [Ti Al • ] 𝜕𝑛 = 0, (4-22)
where n is the normal direction to the surface. At solid-liquid interface, [Ti 3+ ] Liquid and [Ti 4+ ] Liquid vary with crystal growth time 𝑡, following the Scheil-Guilliver equation so that

[Ti Al  ] S at t = 𝑘 𝑇𝑖 3+ [Ti 3+ ] L at t=0 • (1 + 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ • 𝑡 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ • 𝑡 𝑡𝑜𝑡𝑎𝑙 ) 𝑘 𝑇𝑖 3+ -1 , (4-23) [Ti Al • ] S at t = 𝑘 𝑇𝑖 4+ [Ti 4+ ] L at t=0 • (1 + 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ • 𝑡 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ • 𝑡 𝑡𝑜𝑡𝑎𝑙 ) 𝑘 𝑇𝑖 4+-1 , (4-24) [V Al ′′′ ] S at t = 1 3 [Ti Al • ] S at t . (4-25)
where 𝑡 𝑡𝑜𝑡𝑎𝑙 is the total crystal growth time, 𝑉 𝑔𝑟𝑜𝑤𝑡ℎ = -𝑉 𝑧 is crystal growth rate and the negative sign means that crystal grows along minus z direction. Considering that the computational geometry varies with time due to crystal growth, the "Deformed Geometry" mesh is used in the simulation. To better track the crystal growth process and observe the concentration change at solid-liquid interface, the geometry is divided into two parts. The crystal on the top of liquid phase is filled with mapped mesh while the crystal submerged in the melt is meshed with boundary layer to better observe the segregation phenomenon occurring at solid-liquid interface. Figure 4-9 shows the grids for crystal growth at time 50000 s. With the proceeding of crystal growth, a larger area is filled by mapped mesh.

The lateral boundary and the area close to the submerged part are filled will a denser grid. Different mesh sizes are tried and compared. Figure 4-10 presents the radial concentration distributions of Ti 3+ along z=-50 mm calculated by "finer", "fine" and "coarse" mesh sizes. No significant difference (< 0.01%) is observed among the simulation results for three types of mesh sizes. It is the same for different types of time step 10 s, 100 s and 1000 s. Therefore, based on the mesh resolution, calculation time and precision of the results, "Fine" mesh size and 100 s time step are selected to perform the calculation. The grid has a maximum mesh size of 0.795 mm and a minimum mesh size of 0.0045 mm. The total element numbers are 15635. However, with COMSOL software it may happens that during iterations the concentration of Ti 4+ becomes negative. The reaction term continues to consume the species, finally leading to negative concentration. This could be avoided by manually applying a constraint on reaction rate as "Ri*max(0, cTi4)" instead of "Ri" in COMSOL. when a variable temperature field is considered. The temperature field distribution in the crystal growth process has been well studied [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF][START_REF] Qi | Effect of internal radiation on heat transfer during Ti:sapphire crystal growth process by heat exchanger method[END_REF] and the focal point of our simulation is to check the validity of proposed model. Therefore, instead of simulating the temperature field in the whole crystal growth set up, we assume that the temperature linearly changes from crystal solid-liquid interface to the crystal top.

Variable temperature field is added into the model using "Heat Transfer in Solid" interface and it is coupled into "Transport of Diluted Species" physical interface to observe its influence on species diffusion and chemical reaction. The governing equation for heat transfer is:

𝜌𝐶 𝑝 𝜕𝑇 𝜕𝑡 = 𝑘∇ 2 𝑇 (4-26)
where 𝜌, 𝐶 𝑝 and 𝑘 are the density, heat capacity and the thermal conductivity of sapphire. For simplicity, they are supposed as constants.

The boundary conditions for heat transfer are shown in Figure 4-12. It has been reported

in the literature that the temperature on crystal solid-liquid interface is about 200 K higher than the temperature on crystal top [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Ti-doped sapphire crystals[END_REF]. Therefore, the temperature gradient κ is estimated to be 2244 K.m -1 . It is assumed that the lateral crystal surface is insulated and the temperature on the solid-liquid interface equals sapphire melting point.

Figure 4-12 Boundary conditions for heat transfer.

The boundary condition of V Al ′′′ , the diffusion coefficient 𝐷 𝑉 𝐴𝑙 ′′′ and the reaction rate constant 𝑘 𝑐𝑔 are function of temperature. Therefore, their expressions in Table 4-2 with constant temperature T0 have to be replaced with the variable T. The mesh size and time step are the same as before.

The concentration distributions of Ti 3+ , Ti 4+ and V Al ′′′ in grown crystals for isothermal (T=2273.15 K) and varying temperatures are shown in Figure 4-13. Compared to isothermal case, the varying temperature field with the temperature gradient of 2244 K.m -1 has no significant influence on the concentration of Ti 3+ . The varying temperature field decreases a bit the concentrations of Ti 4+ and V Al ′′′ along radial direction. This is because the temperature gradient gives a higher temperature to the solid-liquid interface thereby speeds up the chemical reaction occurring inside the crystal, leading to more Ti 4+ is converted into Ti 3+ . The growth rate has no observable influence on the concentration of Ti 3+ , so only the results of Ti 4+ are presented and V Al ′′′ shows similar profiles as Ti 4+ . In axial direction Figure 4-17 (a)), for the same z-coordinate, crystal grown by the rate of Vz= 1 mm.h -1 contributes to reducing the content of Ti 4+ . This is because a slow growth rate keeps the solid at high temperature and Ti 4+ has enough time to be converted to Ti 3+ under reducing crystal growth atmosphere. When the growth rate is increased to 10 mm.h -1 , the profile of Scheil-Gulliver is changed a bit at the beginning stage of crystal growth. More harmful Ti 4+ is incorporated into the crystal because it does not have enough time to be reduced to Ti 3+ . It is the same for radial direction (Figure 4-17(b)). Slow growth rate helps to get less Ti 4+ . In addition, a more homogeneous distribution is observed for Ti 4+ with the growth rate of 1 mm.h -1 . Therefore, a slow growth rate should be used during Ti:sapphire crystal growth to reduce the residual absorption caused by Ti 4+ . The simulation results for different 𝑝 O 2 are shown in Figure 4-18. When 𝑝 O 2 increases from 1.1 × 10 -20 atm to 10 -15 atm, the concentrations of Ti 3+ and Ti 4+ have no significant change. But when 𝑝 O 2 increases to medium (10 -5 atm) and air (0.21 atm) atmospheres, the concentration of Ti 3+ decreases heavily and the concentration of Ti 4+ increases a lot simultaneously. Especially in the Figure 4-18(d), the concentration of Ti 4+ is even higher than the concentration of Ti 3+ from r=0 to r=12 mm. It could be caused by the high concentration ratio of [Ti 4+ ]L/[Ti 3+ ]L in the liquid phase as shown in Table 4-3. Therefore, it is concluded that 𝑝 O 2 has more influence through the liquid phase than through the solid phase. To determine the main factor, more thermodynamic calculations and experiments are needed, for example, characterizing the Ti ion concentrations in grown Ti:sapphire samples and then deduce the ratio of ions in the liquid phase and compare with the thermodynamic calculation results.

Conclusion

Chapter 5 Conclusions and perspectives

Conclusions

The optical quality of Ti:sapphire is highly related to the valence and homogeneity of doping ions. Therefore, in order to understand the conversion mechanisms between Ti 3+ and Ti 4+ during Ti:sapphire crystal growth process, and to explain the observed experimental phenomenon, below works are achieved:

1. Literature review showed that the optical quality of Ti:sapphire is determined by the ratio of Ti 3+ /Ti 4+ in grown crystals. Ti ion concentrations in grown crystals are related to the initial concentrations and segregation coefficients of Ti ions in the liquid phase, and the possible point defect reactions in the solid phase. Therefore, all these processes are studied step by step and are analyzed in the points below.

2. The goal of studying liquid phase is to obtain Ti ion initial concentrations, segregation coefficients and their relations with 𝑝 O 2 to make preparation for calculating their concentrations in the solid phase. Therefore, thermodynamic calculations are performed to obtain these values using FactSage software. But the lack of solubility data of Ti 4+ in the database of the software imposed to optimize the database according to reported precipitation limit of TiO2 and Ti2O3. Since Ti ion valences and concentrations depend on . This means that the total segregation coefficient includes the contributions of both 𝑘 𝑇𝑖3+ and 𝑘 𝑇𝑖4+ . Consequently, the common optical absorption methods used to determine Ti ion concentrations may not be so reliable because they are based on the assumption that only Ti 3+ or Ti 4+ existed in the calibration crystal.

3. The diffusion and possible point defect reactions in the solid phase are investigated.

Analyzing the formation and migration energy of point defects, allowed to conclude that V Al ′′′ is the most favorable compensator for Ti 4+ in Ti:sapphire. As Ti:sapphire is usually grown under very low 𝑝 O 2 , this leads to O2 production at the crystal surface with decrease of V Al ′′′ . Then V Al ′′′ diffuses from the bulk, where Ti 4+ disappear and produce Ti 3+ and h • .

Therefore, the conversion between Ti 3+ and Ti 4+ during Ti:sapphire growth is controlled by joint diffusion of V Al ′′′ -h

Perspectives for future work

Ti:sapphire is a well-studied laser crystal and it is also a practically stoichiometric crystal, which has simplified the problem (intrinsic V Al ′′′ are negligible). These allowed us to find a lot of information about Ti:sapphire crystals, especially studies about the diffusion and chemical reaction of point defects. We have been lucky to get reference data from reported annealing experiments to calculate the diffusion coefficient of point defect and the reaction rate constant through numerical simulation. We obtained the boundary conditions for point defect through reported equilibrium constants of various point defect reactions occurring in Ti:sapphire. The three parameters are the key for studying the proposed physico-chemical model in this thesis.

Currently, the model has been successfully built and have been qualitatively verified but due to the restrictions of time and availability of resources, more studies still remain to be performed and some results remain to be further checked and improved:

1. The reference data we used to obtain the diffusion coefficient of point defect, chemical reaction constant and boundary concentration on crystal surface have been published more than 4 decades ago. Due to the limit of knowledge, devices and precision of the measurement at that time, the reference data may not be so reliable. So, to check the validity of these three parameters, new annealing experiments inspired from published paper [START_REF] Jones | Defect diffusion in single crystal aluminum oxide[END_REF] should be performed with advanced device and measured with higher precision. Especially, it is important to extend the measurement range for temperature and 𝑝 O 2 to improve the reliability of the experimental data.

2. About our proposed model, the simulation results should be compared with experiments.

All previous experimental data are only qualitative so that we cannot quantitatively compare the results with our model. Therefore, a few crystals should be grown under the same condition as numerical simulation using advanced technologies. Then Ti ion concentrations and distributions should be carefully analyzed and compared with simulation results. However, there is no characterization technique presently available for absolute measurement of the concentrations of Ti 3+ and Ti 4+ (see Annex III).

Actually, in the frame of this thesis, some crystals have been grown by Czochralski method under various 𝑝 O 2 as shown in Annex II and a few characterization methods for Ti ions have been tried as shown in Annex III. Unfortunately, an accident (samples were lost by the delivery company) prevented the analysis of these crystals.

3. Numerical simulation results presented in this thesis are obtained on the basis of many assumptions to simplify the model and calculations. A more complex model close to real crystal growth process needs to be considered, taking into account irregular crystal geometry, solid-liquid phase transition, varying solid-liquid interface shape, the evaporation of alumina from the solid and liquid phases, the heat transfer from convection and radiation, the gas convection effect on its local composition, etc. Therein, the gas convection is the most important one. Anyhow, because all of this is now routinely simulated, often with dedicated commercial software, there should be no difficulty to implement our model to more complex crystal growth simulations.

4. Our model could be applied to study dopant ions with variable valence in other crystals.

To realize this, it is required to perform a literature review about the reported point defects and compensation mechanisms inside this crystal. In addition, the three key parameters: diffusion coefficient, reaction rate constant and the boundary concentration of point defects should be known. We have described from which kind of experiments those parameters could be obtained and how. It should be mentioned that there is no stoichiometric issues for Ti:sapphire while there may be stoichiometric problems for other crystals, which might complicate the studies.

1 ppmw Ti=0.0829 mol.m -3 .

(3) The concentration of Ti2O3 with respect to total atoms in atomic percent is during crystal growth so that all the melted material is lost. The growth under reducing atmosphere has been tried for three times but each time the crucible cracked due to inhomogeneous temperature field. Finally, due to the cost and time limit, we have had to stop to grow the crystal under Ar + 2% Vol H2 atmosphere.

We received the two crystals and sent them to CristalInnov company for cutting and polishing. Unfortunately, the samples have been lost when they send us back the package.

Therefore, we could not compare our simulation results with experimental results.

Annex III Measurements for determining Ti ion concentrations

It was planned to analyze the crystals grown in Annex II and check the validity of the optical absorption method discussed in Chapter 1. While the crystals were grown in China, in parallel, we tried various techniques to search for a reliable measurement which could give us the concentrations of both Ti 3+ and Ti 4+ at ppm level. Some samples grown by RSA company using Ky method with Ti concentration around a few hundred ppmw are taken for these measurements.

Electron spin resonance (EPR), X-ray photoelectron spectroscopy (XPS), Raman spectroscopy, wavelength-dispersive X-ray spectroscopy (WDS), inductive coupled plasma mass spectrometry (ICPMS), X-ray fluorescence (XRF) and microwave measurements are tried to measure the quantitative concentrations of Ti 3+ , Ti 4+ and total Ti in Ti:sapphire samples.

Among them, the first two techniques can detect the concentration of ions and the remaining four can detect the concentration of total element.

EPR measurement

EPR is a non-destructive device which can measure ions with unpaired electron on outer orbit using the paramagnetic property. Ti 3+ has an extra electron on its 3d orbit, then, in principle, this technique can be used to measure Ti 3+ concentration in our samples. The samples are hold inside a quartz tube and put inside a furnace to heat for 20 minutes at 2000 ℃. Then the samples are taken out from the furnace and throw into cold water quickly so that they can be thermally quenched into small sands. Two pieces of pure sapphire blocks are used to grain the Ti:sapphire sands into powder for EPR measurement. The measurement is conducted by engineer Florian Molton in Chemistry building in Grenoble. Same quantity of sample powder is scaled (132.4 mg) and put in a tube with the diameter of 3 mm and height of 30 mm. This tube is inserted inside EPR device which has been cooled down to 100 K using liquid nitrogen. The magnetic field is swept from 3000-4000 Gauss with a step of 0.5 Gauss. for EPR signal can give the area of the signal which is proportional to ion concentration. After two times integrating within 3412-3604 Gauss (dark area), the peak intensities for disc 2 and disc 3 are 4.022 × 10 8 and 4.013 × 10 8 , respectively. There is only 0.2% difference between these two samples.

However, the signal is too small to get precise result. To get a more reliable result, a series of samples with different concentrations need to be measured and at least one sample with known Ti 3+ concentration to calibrate the spectrum. This is difficult to be realized at ppm level concentration. It is noticed that one sharp peak appears around 3100 Gauss. This is probably caused by crystal structure because it shifts when the sample tube is moved.

XPS measurement

XPS measurement is able to determine ion's concentration through the characteristic peak of element on each orbit according to their binding energy. Sample A with the highest Ti concentration around 400 ppmw is measured. This measurement is conducted by Grégory Berthomé in SIMAP-research building. The sample is cleaned with Acetone and Ethanol in ultrasound bath for 20 minutes and the surface of the sample is carefully cleaned with plasma.

The X-ray incidents to the surface of the sample have a detect depth of 10 nm. elements are also detected inside the sample. The theoretical binding energy of titanium should be found around 464 or 458 eV (within the blue square area). However, no characteristic peak for titanium is observed. This may be because the detect limit for XPS is 1 at% and the total titanium concentration in our sample is only around 400 ppmw (0.08 at%) which is much lower than the detection limit of XPS. Thus, the titanium ion concentration in our sample is too low to be detected through XPS measurement.

Raman measurement

Raman spectrum method is tried to measure the total Ti concentration. Three Ky samples with total titanium concentrations of about 2 ppmw, 8 ppmw and 400 ppmw are measured, wherein, 2 ppmw sample is pure sapphire without Ti doping. This measurement is conducted by Alexandre Crisci in SIMAP laboratory.

WDS measurement

WDS measurement is applied to measure the total Ti concentration. Since Al2O3 is not a conductive material, all the samples are coated with carbon layer on the surface (the thickness is about 15 nm) to conduct away the electrons injected from WDS device. Because the electrons remain on the surface will repulse the new injected electrons and hence affect the measurement result. The electron beams injected by WDS device has the area of 1×1 µm and these beams go through the depth of 1 µm. Therefore, each detect point is like a cube with the volume of 1 µm 3 . Three samples from crystal K-06-05-15 and five samples from crystal K07-04-13 are measured. All the samples have a width of 10 mm and a thickness of 1 mm. Each sample is measured for 10 times to get the mean value. The measurement results for two crystals are shown in The two times measurement results show that WDS measurement is reproducible. The total Ti concentration in crystal K-06-05-15 is around 200 ppmw and total Ti concentration in crystal K07-04-13 is around 300 -500 ppmw. However, the measurement results are still not precise enough because the intensity peak is very close to the background limit and the precision for WDS device is only 0.02 wt% (200 ppmw). Therefore, more calibration experiments like ICPMS or XPS still need to be conducted to compare the result.

ICPMS

ICPMS gives total Ti concentration in weight percent. The measurement results for two crystals are shown in Table III We assume that Ti concentration for sample 3C is close to the value of sample 3A, i.e. 110 ppmw.

Table III-4 Ti concentration measured by ICPMS in disc 1 for crystal K-07-04-13

Ti concentration in disc 2 in crystal k-07-04-13 is not measured. But from the sample position in the bulk crystal, it can be obtained that Ti concentration is disc 2 is higher than that in disc 1. Therefore, it is approximated that Ti concentration in sample A in disc 2 for crystal K-07-04-13 is higher than 400 ppmw.

XRF measurement

Total Ti concentration measured by XRF-PANALYTICAL device is shown in Table II-5.

This technic is a surface measurement. Samples with the highest concentration were selected. The Ti concentrations measured by XRF is higher than the values measured by WDS but the precision and reproducibility of XRF measurement still needs to be checked.

Microwave measurement

During my master, through the cooperation between Center for Gravitational Experiment The measured concentration is 3 times higher than the results measured by ICPMS and WDS except the sample 2. And the estimated total Ti concentration is even higher than the doping concentration. So, this method might not be so reliable and still needs to be improved.

Optical absorption measurement

This method has been discussed in the section 1.2.2 in the thesis. Ti ion concentrations can be determined through the relation:

[Ti 3+ ] wt.% ≈ 0.04 × α490 (E∥C)

[Ti 4+ ] wt.% ≈ 0.04 ×10 -4 × α220.

While this is only an empirical formula. Since our samples are lost, we cannot perform experiment to check its validity. But from the analysis of the segregation coefficient of total Ti (equation (2-32)), it is concluded that this optical absorption method may not so reliable.

Conclusions

From the measurements of EPR and XPS it is concluded that Ti 3+ or Ti 4+ ion concentrations in Ky crystals with the doping level of a few hundred ppm in weight cannot be obtained using these two techniques. We only can measure the total Ti concentration with techniques such as WDS, ICPMS or XRF. Due to the lattice vibration of Al2O3 lattice, Raman technique is not able to measure total Ti concentration for a few hundred ppmw level. Below

Tabel III-6 and Table III The values measured by these techniques did not show consistency. However, it is difficult to judge which technique is more reliable. We failed to measure the concentration of Ti 3+ and Ti 4+ by microwave, EPR and XPS techniques. To get more precise conclusion, a crystal with higher Ti concentration should be grown for calibration and the same sample should be analyzed by these techniques and the re-of the techniques should be verified.

For crystals grown in China, it was planned to use XRF and WDS to measure the longitudinal and radial distributions of Ti, respectively. And then ICPMS will be used to check the validity of these measurements. In addition, optical absorption method will be used to measure the longitudinal and radial profiles for Ti 3+ and Ti 4+ ions. However, due to the loss of the samples, these measurements have not been possible.

Résumé

Le saphir dopé Ti présente des propriétés exceptionnelles pour la production de lasers à

Figure 2 .

 2 Figure 2. 26 kg Ti:sapphire crystal grown by Kyropoulos technique by RSA company (Alombert-Goget et al., 2016).

  Exchanger Method .................................................................................. 1.4 Solute segregation during crystal growth ................................................................... 1.4.1 Equilibrium segregation coefficient ................................................................ 1.4.2 Longitudinal segregation ................................................................................. 1.4.3 Boundary layer and effective segregation coefficient ..................................... 1.4.4 Radial segregation ........................................................................................... 1.4.5 Growth striations ............................................................................................. 1.4.6 Ti segregation coefficient in sapphire ............................................................. 1.5 Distributions of Ti 3+ and Ti 4+ in sapphire crystals ...................................................... 1.6 Crystal optical quality dependence on titanium ion concentrations ........................... 1.7 Chemistry of Ti:sapphire under different phases ....................................................... 1.7.1 Gas phase ......................................................................................................... 1.7.2 Liquid phase .................................................................................................... 1.7.3 Solid phase ...................................................................................................... 1.8 Thermal treatment ...................................................................................................... 1.8.1 Carbothermal reduction ................................................................................... 1.8.2 Post growth annealing ..................................................................................... 1.9 Conclusion .................................................................................................................. Chapter 2 Thermodynamics of Ti:sapphire solidification .................................................. 2.1 Introduction ...........................................................................................................

2. 5 . 1

 51 Segregation coefficients of Ti 3+ and Ti 4+ ......................................................... 2.5.2 Equilibrium segregation coefficient of Ti ....................................................... 2.6 Conclusion ..................................................................................................................Chapter 3 Physico-chemical model for Ti valence state conversion .................................. 3.1 Introduction ................................................................................................................ 3.2 Point defects ............................................................................................................... 3.2.1 Point defects in pure sapphire .......................................................................... 3.2.2 Point defects in Ti doped sapphire .................................................................. 3.2.3 Defect energies ................................................................................................ 3.3 Ion and point defect diffusion ..................................................................................... 3.3.1 Diffusion mechanisms ..................................................................................... 3.3.2 Diffusion equation ........................................................................................... 3.3.3 Diffusion coefficients ...................................................................................... 3.4 Ti valence state conversion model during crystal growth .......................................... 3.4.1 Introduction ..................................................................................................... 3.4.2 Proposed physico-chemical model .................................................................. 3.4.3 Mathematical model ...................................................................................... 3.5 Dimensionless equations .......................................................................................... 3.6 Point defect concentration on the crystal surface ..................................................... 3.7 Conclusion ................................................................................................................ XV Chapter 4 Numerical simulation of Ti ion behavior during annealing and crystal growth ....

  Figure 1-1 (b) shows the atom arrangement inside an α-Al2O3 hexagonal multiple cell. Oxygen ions nearly form a hexagonal close-packed structure and aluminum ions are located in two-thirds of the octahedral interstices. When sapphire is fully ionic crystal, the ionic charges for Al and O should be +3 and -2. There are some papers or books pointing out by computing the effective charges of Al and O that the chemical bound between Al-O is

Figure 1

 1 Figure 1-1 (a) Crystallographic planes of sapphire: m plane is {11 ̅ 00}, r plane is {11 ̅ 02}, a plane is {112 ̅ 0}, c plane is {0001}; (b) Al2O3 atomic arrangement; (c) Coordination environment of Ti:sapphire (website 2, website 3).

  Figure 1-3 (a) shows the transition levels in Ti 3+ . Because of the anisotropic properties of Ti:sapphire, light polarized in different directions could result in different absorption coefficients. The absorption and emission bands of Ti 3+ are measured with respect to the c-axis of the crystal, namely, with the maximum electric field polarized parallel to the c-axis (π polarization).Figure 1-3 (b) presents

Figure 1 -

 1 Figure 1-3 Ti 3+ lasing mechanism and corresponding absorption and fluorescence spectra: (a) Ti 3+ lasing mechanism; (b) π polarized absorption and fluorescence spectra for Ti:sapphire at room temperature[START_REF] Goossens | Ultrafast organic lasers and solid-state amplifiers[END_REF][START_REF] Sorokin | Solid-state materials for few-cycle pulse generation and amplification[END_REF].

  transition does not result in the formation of stable species, but transient species like Ti 3+ -O - where the d-electron of Ti 3+ and the hole of O -are strongly coupled and form a triplet. However, Allombert-Goget et al. thought the emission of Ti 4+ at 430 nm occurs after the energy transfer from F + -center (Alombert-Goget et al., 2016).

Figure 1 - 4

 14 Figure 1-4 Emission and excitation bands of Ti 4+ in titanium doped sapphire at 300 K (Wong

  et al. studied the dependence of residual absorption coefficient on the main absorption coefficient for partially oxidized Ti:sapphire samples with constant total Ti concentration [Ti] but different [Ti 4+ ]/[Ti 3+ ] ratio and for as-grown samples with varied [Ti] but similar [Ti 4+ ]/[Ti 3+ ]

  Figure 1-6 Room temperature absorption spectra for as-grown sample of Ti:Al2O3 measured with light parallel (π) or perpendicular (σ) to the crystal c-axis: (a) main absorption band; (b)

Figure 1 - 7

 17 Figure 1-7 Schematic diagram of Verneuil crystal growth method.

Figure 1 - 8

 18 Figure 1-8 Schematic diagram for Czochralski crystal growth.

Figure 1 - 9

 19 Figure 1-9 Schematic diagram of Kyropoulos method. The crystal in the middle is the experimental picture of a crystal quickly removed from the melt in order to show the interface curvature.

Figure 1 -

 1 Figure 1-11 Pure sapphire (left) and Ti:sapphire (right) crystals grown by the heat exchanger method (Joyce & Schmid, 2010).

Figure 1 -

 1 Figure 1-12 Equilibrium solidification phase diagram at solid-liquid interface.

Figure 1 -

 1 Figure 1-13 Solute segregation in equilibrium solidification: (a) start stage of solidification; (b) solidification at temperature T*.

Figure 1 -

 1 Figure 1-14 Solute segregation in solidification with diffusion and no convection.

Figure 1 -

 1 Figure 1-15 Solute segregation in solidification with diffusion and strong convection in the liquid: (a) start stage of solidification; (b) solidification at temperature T*[START_REF] Flemings | Solidification processing[END_REF].

Figure 1 -

 1 Figure 1-19 Growth striations of Ti element mapped by LIBS measurements on a 3 cm ×1 cm Ti:sapphire sample. The resolution of this map is 50 µm (the measurement was conducted by Guillaume Alombert-Goget).

  and coworkers observed titanium segregation along axial and radial directions in Ti:sapphire single crystals grown by Kyropoulos technique at RSA company in Grenoble. The attractive sanguineous color results from the contribution of incorporated Ti 3+ ions. Crystal color becomes darker from top to bottom (see Figure2in General introduction) reflecting the axial segregation of titanium due to Scheil-Gulliver law. The radial segregations for Ti 3+ and Ti 4+ are estimated by the luminescence measurement of a disc cut transversely from the crystal. The results are shown in Figure1-20. The measurement interval on crystal diameter is 1 mm. The luminescence of Ti 3+ is measured at 730 nm after excitation at 532 nm and the luminescence of Ti 4+ is measured at 420 nm after excitation at 256 nm. Ti 3+ exhibits similar distribution profile as crystal growth interface as shown in Figure1-9. Therefore, Ti 3+ ions present higher concentration at the edge but lower concentration at the center. Out of expected, Ti 4+ ions did not follow the profile of growth interface but show a totally opposite distribution profile compared with Ti 3+ . The similar opposite distribution profile for Ti 4+ is also observed in another Kyropoulos Ti:sapphire crystal as presented in Figure1-21. The luminescence intensity of Ti 4+ increases first from the center of the crystal and suddenly decreases at the edge of the crystal. It needs to stress that this luminescence intensity measurement cannot give absolute concentration for Ti 3+ and Ti 4+ ions, but just the relative distribution profiles for each ion.

Figure 1 -

 1 Figure 1-20 Luminescence intensity profiles for Ti 3+ and Ti 4+ along diameter (Alombert-Goget et al., 2016).

Figure 1 -

 1 Figure 1-21 Luminescence intensity profiles for Ti 3+ (a) and Ti 4+ (b) along radius in a Kyropoulos Ti:sapphire sample (Alombert-Goget et al., 2017).

Figure 1 -

 1 Figure 1-22 Radial luminescence profiles of Ti 3+ in samples cut from the top (a), middle (b)and bottom (c) of Ti:sapphire crystal grown by Kyropoulos technique(Stelian et al., 2017).

Figure 1 -

 1 Figure 1-23 Ti 3+ (a, c) and Ti 4+ (b, d) luminescence profiles in Ti:sapphire grown by Czochralski ((a) and (b)), and Verneuil methods ((c) and (d))(Alombert-Goget et al., 2016).

Figure 1 -

 1 Figure 1-24 FOM values of 1 cm 3 Kyropoulos Ti:sapphire samples before and after annealing (95% Ar + 5% H2) at 1300 ℃ for 73 h (Alombert-Goget et al., 2016).

Figure 1 -

 1 Figure 1-25 (a) Ti 3+ :sapphire single crystal; (b) Ti 4+ :sapphire (annealing at 1600 ℃ in the air for 24 h from Ti 3+ :sapphire) (Tang et al., 2013).

Figure 1 -

 1 Figure 1-26 Absorption loss at 830 nm in as-grown and annealed Ti:sapphire crystals as a function of titanium concentration[START_REF] Kokta | Effects of growth conditions and post-growth thermal treatment on quality of titanium doped sapphire[END_REF].

Figure 1 -

 1 Figure 1-27 The concentrations of Ti 3+ ≡ Ti Al × , Ti 4+ ≡ Ti Al • , and V Al ′′′ dependence on 𝑝 O 2

Figure 2 - 1

 21 Figure 2-1 Main interface of FactSage 8.0 software.

  𝑝 O 2 is low in the atmosphere, most Ti 4+ ions are reduced to Ti 3+ and Ti2O3 precipitates could form when 𝑝 O 2 is extremely small. Coarse precipitates of Ti2O3 are found to be formed in Al2O3 matrix after annealing at 1100 ℃ in the reduced atmosphere of Ar-5%H2.The presence of disolved Ti 3+ can change the lattice constant or unit cell volume of Al2O3 structure. Table2-2 shows the lattice parameters of pure sapphire and Ti:sapphire crystals after annealing at 1600 ℃ in air for 24 hours. The lattice parameters increase with elevating temperature. These allow W.D. McKee et al. to report that the solid solubility of Ti 3+ in Al2O3 are 1.0, 1.8, and 2.5 mole% at 1400, 1600 and 1700 ℃ by heating alumina and titanium dioxide in H2 atmosphere. These values are obtained by measuring the change in unit-cell dimensions of Al2O3 as function of Ti 3+ content through X-Ray diffraction. Using the same technique, A.J.Strauss et al. measured the Ti 3+ solubility to be 0.44 mole% at 1300 ℃ and 2.9 mole% at 1750 ℃ by annealing known mixtures of Al2O3 and Ti2O3 powders in an Ar or Ar-H2 atmosphere.Reported solid solubility of Ti 3+ and Ti 4+ in alumina are summarized in Figure2-2. It is noted that the solubility of Ti2O3 reported by Belon et al. is much higher than the values reported by others. Both solubilities of Ti 3+ and Ti 4+ increase with elevating temperature. It is observed that Ti 3+ is more soluble in α-Al2O3 than Ti 4+ . This is reasonable because no extra charge compensator is required for the incorporation of Ti 3+ while one aluminum vacancies is created to compensate the incorporation of every three Ti 4+ . As these solubility data have a messy distribution, it is impossible to obtain a functional expression between these experimental solubility data and temperature through fitting.

Figure 2 - 2

 22 Figure 2-2 Solid precipitation limit of Ti2O3 and TiO2 in α-Al2O3. Black symbols represent the precipitation of Ti2O3 and red symbols represent the precipitation of TiO2.

  2.2.2 Al2O3-Ti2O3 and Al2O3-TiO2 systems Under reducing atmosphere, most titanium in the system is present as Ti 3+ . The phase diagram of Al2O3-Ti2O3 system at 𝑝 O 2 = 10 -15 bar accompanied with the experimental solubility data of Ti 3+ from the literature is shown in Figure2-3. It can be seen that the eutectic temperature is 1642 ℃. This eutectic temperature increases with decreasing oxygen partial pressure. As we consider Ti:sapphire growth process, all the metal phases(Al, Ti, TiAl,TiAl3)

Figure 2 - 3

 23 Figure 2-3 Calculated phase diagram of Al2O3-Ti2O3 system at 𝑝 O 2 = 10 -15 bar and 𝑝 total = 1 bar.

Figure 2 - 4

 24 Figure 2-4 The enlarged details of Figure 2-3 for Ti2O3/(Al2O3+ Ti2O3) = 0 -0.2.

  point 1860 ℃. Panda and Jung demonstrated the optimized melting and decomposition temperatures of Al2TiO5 are 1860 ℃ and 1279 ℃[START_REF] Panda | Coupled experimental study and thermodynamic modeling of the Al2O3-Ti2O3-TiO2 System[END_REF].Kato et al. determined the decomposition temperature of β-Al2TiO5 to be 1280 ℃[START_REF] Kato | Decomposition temperature of β-Al2TiO5[END_REF]. In present study, we denote both forms of α-Al2TiO5 and β-Al2TiO5 as Al2TiO5. The calculated phase diagram of Al2O3-TiO2 system in air is shown in Figure2-5. The melting point and decomposition temperature of Al2TiO5 in the phase diagram are calculated to be 1846 and 1264 ℃, the two eutectic temperatures locate at 1701 and 1835 ℃, as annotated in Figure 2-5. However, Panda and Jung believed that Al6Ti2O13 is the stable compound at high temperature between 1797 and 1857 ℃. In the commercial FactSage 8.0 software, Al6Ti2O13 lacks in the solid phase database. So, only Al4TiO8 compound appears in the phase diagram at the temperature range from 1803

  Figure 2-5 is further zoomed in Figure 2-6 and the solubility data of Ti 4+ from the literature are plotted. Only in the small area on the left, titanium ions are incorporated into Ti:sapphire without the existence of other phases. However, this does not fit with the experimental results.

  4+ is introduced into the solid phase by adding TiO2 into the solid solution through the Henrian relation given by expression (2-9). The solidus and solvus lines in Al2O3-TiO2 change with A, B parameters and with the oxygen partial pressure. TiO2 is selected as an ideal solution in the solid product first and then merged into the solid solution. By trying and error, it is found that A= 4360 and B=0gives a good agreement with the experimental solubility data. The optimized phase diagram of Al2O3-TiO2 is presented in Figure2-7 along with the experimental solubility data of Ti 4+ ions.Compared to Figure2-5, the solvus line is observable now and the solubility data of Ti 4+ distributed well on the solvus line (the result is zoomed in Figure2-8). The phase diagram of Al2O3-Ti2O3 system has nearly no change compared to before. Now the database in FactSage software is complete and it can be used for thermodynamic calculations.

Figure 2 - 5

 25 Figure 2-5 Calculated phase diagram of the Al2O3-TiO2 system in air and 𝑝 total = 1 bar.

Figure 2 -

 2 Figure 2-6 Ti 4+ solubility in enlarged Al2O3-TiO2 system in air and 𝑝 total = 1 bar.

Figure 2 - 7

 27 Figure 2-7 Phase diagram of Al2O3-TiO2 system in air after imported Ti 4+ (Ideal-1) into the solid phase and 𝑝 total = 1 bar.

Figure 2 - 8

 28 Figure 2-8 Enlarged phase diagram of Al2O3-TiO2 system in air after imported Ti 4+ (Ideal-1)

  For aCzochralski crystal growth process, the temperature on the earlier grown part (top of the crystal) is lower than the temperature of the bottom part which is close to the solid-liquid interface. The temperature varies drastically during crystal cooling process. Therefore, for the aim of getting desired valence for Ti 3+ , oxygen partial pressure must be carefully controlled to fit the phase diagram at any temperature. At high temperature, it is possible that O2 gas is decomposed into O atoms, but on the point view of experiments, 𝑝 O 2 has much more physical meaning than 𝑝 O and it is easier to compare the results with experiments. Therefore, in the thermodynamic calculations, 𝑝 O 2 is used to describe the oxygen partial pressure.As introduced in section 2.1.2, the axes of the phase diagram can also be plotted with oxygen partial pressure and temperature, which is the well-known Ellingham phase diagram.Based on the optimized Al2O3-TiO2 phase diagram, its corresponding Ellingham phase diagram is plotted in Figure2-9. Red dash line denotes the melting point of sapphire at 2053 ℃. For the solidification to occur, 𝑝 O 2 should be larger than 10 -20 bar. "M2O3+Ideal-1 + Slag-liq" is the coexistence of solid and liquid phases. The solid phase containing Ti 3+ and Ti 4+ ions, without the formation of other phase, can only be found within the shaded area (Corundum +Ideal-1).At high temperature (T>1000 ℃), for 𝑝 O 2 lower than 10 -20 bar, other phase like TiO may appear.At low temperature, either rutile (TiO2) or ilmenite (Ti2O3) precipitates are contained in the crystal. This has been observed in the sintering experiment ofHamano et al., that excess TiO2

Figure 2 - 9

 29 Figure 2-9 Ellingham type predominance diagram of Al2O3-TiO2 system and 𝑝 total = 1 bar. M2O3 (Corundum) is Al2O3 + Ti 3+ , Ideal-1 is Ti 4+ solute, Rutile and Ilmenite are TiO2 and Ti2O3 precipitates.

Figure 2 -

 2 Figure 2-10 Kyropoulos crystal growth sketch for Ti:sapphire. The crystal picture has been obtained by suddenly extracting a crystal from the melt (Stelian et al., 2017).

Figure 2 -

 2 Figure 2-11 𝑝 O 2 varies with temperature under different filling gases. The results are calculated by taking into account trace impurities of 10 ppm O2 and 10 ppm H2O in the system.

  influence of crucible materials, the corresponding oxygen partial pressure above different crucibles (still with 10 ppm traces O2 and 10 ppm H2O) is computed and plotted in Figure 2-12. It is observed that Ir crucible resulted in the highest 𝑝 O 2 inside the furnace. Mo and W exhibited the same 𝑝 O 2 and the 𝑝 O 2 of Mo is slightly lower when T > 2000 ℃. Considering the recycling rate, cost and Ti:sapphire growth conditions (reducing atmosphere), Mo crucible is often selected by crystal growers than the other two. It has been used for a long time in the RSA Company in Grenoble to grow Ti:sapphire crystals by Kyropoulos or EFG techniques.

Figure 2 -

 2 Figure 2-12 𝑝 O 2 inside furnace for different crucible materials.

Figure 2 -

 2 Figure 2-13 𝑝 O 2 for Ar mixed with pure CO, pure CO2 and the equimolar mixture of both.

Figure 2 -

 2 Figure 2-10). So, the filling gas should contact first with outer graphite elements and then contact with the crucible. The joint action of C + Mo decreased 𝑝 O 2 for 13 orders of magnitude compared to Mo action alone. It is 2 orders of magnitude higher than the case where furnace gases react only with C. By calculating, it is found that reaction with Mo first then with C gives the same 𝑝 O 2 as C alone. These suggest that graphite elements may have much more influence on 𝑝 O 2 than Mo crucible.

Figure 2 -

 2 Figure 2-14 C and Mo influence on 𝑝 O 2 .

Figure 2 -

 2 Figure 2-15 The influence of C and Mo on titanium valence state.

Figure 2 -

 2 Figure 2-16 Titanium ion concentration dependence on 𝑝 𝑂 2 at 2053 ℃. The data points are calculated from 1mole Al2O3 + 10 -3 mole TiO2 + 1 mole Ar + 10 -5 mole O2 +10 -5 mole H2O.

  Figure 2-17 The relation between [Ti 4+ ]L/[Ti 3+ ]L and oxygen partial pressure at 2053 ℃.

  4) in sapphire resulting in more titanium remaining in the melt than what is incorporated into the crystal. This is proofed by the calculation results already shown in Figure 2-16, i.e. [Ti 3+ ]L > [Ti 3+ ]S and [Ti 4+ ]L > [Ti 4+ ]S. The segregation coefficients of titanium ions are defined as the ratio of Ti concentration in the solid phase to the concentration in the liquid phase:

Figure 2 -

 2 Figure 2-18 Titanium segregation coefficients dependence on 𝑝 𝑂 2 at 2053 ℃. The 𝑘 𝑇𝑖_𝐹𝑎𝑐𝑡𝑆𝑎𝑔𝑒 is calculated from FactSage by equation (2-26) and 𝑘 𝑇𝑖_𝑡ℎ𝑒𝑜𝑟𝑦 is calculated by equation (2-30).

Figure 2 -

 2 Figure 2-19 Ti effective segregation coefficient dependence on crystal growth rate. The scatter points are the experimental data (Table2-3) and the red line is the fitting of the data.

  Figure 3-1 shows the solidliquid interface shape (dashed blue line) of Ti:sapphire grown using Ky technique by the company RSA. The crystal has been quickly pulled out of the melt after shoulder growth in order to measure the interface shape. During crystal growth process with strong mixing, Ti ions are incorporated into the crystal following the U-shape profile of solid-liquid interface. The numerical simulations and experimental results of C. Stelian et al. have already proved that the U-shape radial profile of Ti in Ky crystal is caused by the high curvature of the growth interface (Stelian et al., 2017).

Figure 3 - 1

 31 Figure 3-1 Ti:sapphire solid-liquid interface submerged in the melt.

Figure 3

 3 Figure 3-2 (b) due to the segregation of Ti ions. The curves in Figure 3-2 (a) and (b) do notrepresent the absolute concentrations but only the profiles because the calculated segregation coefficient of Ti 4+ is much lower than Ti 3+ . Put it simply, the curve of Ti 4+ located at the top of the crystal only means that it should has the same U-shape profile as Ti 3+ but does not mean that its concentration is high. However, as can been seen from Figure1-20, the experimentally observed Ti ion profiles exhibit totally opposite distributions along radial direction which are not consistent with the expected results (see Figure3-2(b)). Currently, no theoretical model is available to explain this phenomenon.

Figure 3 - 2

 32 Figure 3-2 Expected distribution profiles for Ti 3+ and Ti 4+ along axial (a) and radial (b) directions.

Formation

  the average formation energy for Schottky, Frenkel and interstitial reactions in equations (3-1)-(3-4) can be computed by the sum of defect energies divided by the defect number.

  Figure 3-3 Vacancy and interstitial diffusion mechanisms[START_REF] Manning | Diffusion kinetics for atoms in crystals[END_REF]).

  Doping ions and point defects inside inhomogeneous solid will diffuse and their concentration would become homogeneous when the diffusion time is long enough. The flux for diffusing species passing through a unit volume along different directions of a 3D volume during the unit time is proportional to the concentration gradient, ∇[𝑖], as described by Fick'

  equation (3-12) into equation (3-13) gives

Figure 3 - 4

 34 Figure 3-4 Diffusion coefficients of V Al ′′′ and Al in undoped sapphire and Ti:sapphire. The red thick line represents the analytic expression of D(T) while other thin lines are just used to connect the data points.

  Figure 3-5 shows the measured diffusion coefficients of Ti 3+ , Ti 4+ and Ti (unknown atmosphere) in literature. Similar to the case of 𝐷 𝐴𝑙 and 𝐷 𝑂 , the difference in sample types, measurement conditions and techniques leads to disagreement among reported values.

Figure 3 - 5

 35 Figure 3-5 Diffusion coefficients of Ti 3+ and Ti 4+ in Ti:sapphire. Black, red and pink colors denote respectively the diffusion coefficients of Ti 3+ , Ti 4+ and Ti (measured under unknown atmosphere).

  Figure 3-6 Schematic diagram of the proposed physico-chemical model.

  could participate to reaction (3-21). Consequently, the reaction rate of reaction (3-21) is proportional to

  log 10 [V Al ′′′ ] 𝑏 = log 10 𝐴 1 + 𝑚 • log 10 𝑝 O 2 , (3-63) Parameters 𝐴 1 and 𝑚 can be determined through the linear fitting with equation (3-63). The relation between log 10 [V Al ′′′ ] 𝑏 and log 10 𝑝 O 2 is fitted in Figure 3-7 by referring to reported experimental data.

Figure 3

 3 Figure 3-7 V Al ′′′ concentration relation with 𝑝 O 2 . The data points are taken from Figure 1-27.

4.

  Some physical parameters need to be determined before solving the mathematical equations for proposed model. By analogy with the "trap diffusion" model, converting the defined mathematical equations into dimensionless equation, showed that only three parameters: reaction rate constant 𝑘 𝑐𝑔 , thermodynamic equilibrium concentration of aluminum vacancies on crystal surface [V Al ′′′ ] 𝑏 and diffusion coefficient 𝐷 V Al ′′′ are important for the solution results. Through literature investigation, it is found that 𝐷 V Al ′′′ and 𝑘 𝑐𝑔 can be determined by numerical simulation of annealing experiments. 5. The thermodynamic concentration of V Al ′′′ on crystal surface is shown to be function of 𝑝 O 2 and temperature. Considering that annealing experiments occur under high 𝑝 O 2 while crystal growth occurs under low 𝑝 O 2 , we consider aluminum vacancies concentration on crystal surface to be [V Al ′′′ ] 𝑏 = 14.46 • 𝑝 O

( 3 -

 3 21) are unknown. Therefore, these two physical parameters must be determined first before solving the governing equations for proposed model. It has been pointed out in the section 3.5 of Chapter 3 that the Arrhenius equations for these two parameters can be determined through the numerical simulation of annealing experiments. The main process for solving the governing equations for proposed physico-chemical model is shown in the chart Figure 4-1. In the figure, kann(T) and kcg(T) represent the reaction rate constant of point defect reaction during annealing and crystal growth (in proposed model) which are function of temperature.

Figure 4 - 1

 41 Figure 4-1 Chart for solving the governing equations of physico-chemical model.

Figure 4 - 2

 42 Figure 4-2 Relative concentration of Ti 3+ in Ti:sapphire sample after annealing under air.

  Figure4-2 shows the relative concentration of Ti 3+ in their Ti:sapphire sample after annealing under air. Through annealing experiment, they determined the Arrhenius equation of aluminum vacancies diffusion. Their diffusion coefficients are obtained without taking into account the point defect reaction but only diffusion. Therefore, their measured diffusion coefficient for aluminum vacancies cannot be directly applied to the proposed physico-chemical model. But it is possible to simulate the annealing process by considering both aluminum vacancy diffusion and point defect reaction with respect to their experimental data. After this, the kcg can be deduced from kann using the relation of reaction rate for the same species in the redox reaction (3-21).4.4.2 Mathematical modelFrom the annealing experiment, it is known that all samples have the height of 10 mm are annealed under air ( 𝑝 O 2 = 0.21 atm). Samples with different thicknesses, temperatures, annealing times and color boundary positions are given in the reported paper. The total Ti concentration is 0.05 wt% (=500 ppmw = 41.46 mol.m -3 ).

  Figure 4-3 Simplified 2D geometry for Ti:sapphire sample in annealing simulation.

Figure 4 - 4

 44 Figure 4-4 Dependence of the square of the color boundary on crystal annealing time under 𝑝 𝑂 2 =1 atm for Ti:sapphire samples at T=1500 ℃ with different concentrations. The data are

Figure 4 - 5

 45 Figure 4-5 Color boundaries of Ti:sapphire after annealing for 280000 s, 440000 s and 630000 s at T=1400 ℃. The thin lines around 0.1 cm represent the experimental XB, for the three different times. Sample thickness is 0.63 cm.

Figure 4 - 6

 46 Figure 4-6 Diffusion coefficient of 𝑉 𝐴𝑙 ′′′ variation with temperature.

Figure 4 - 7

 47 Figure 4-7 Reaction rate constant variation with temperature for reaction

  so that the 𝑝 O 2 released by surface reaction is in equilibrium with surrounding atmosphere; 10) Boundary condition can be represented by the equation (3-60), likely to be valid for low 𝑝 O 2 (~10 -20 atm) and which takes into account temperature effect; 11) Enthalpy of point defect reactions is negligible. 4.6 Crystal growth model implementation in COMSOL 4.6.1 Simulation domain Considering the complexity of crystal growth (temperature field, growth rate, phase transition, mass transportation and chemical reactions…), a local model which only focuses on the solid phase during Ti:sapphire growth will be considered. Figure 4-8 shows a geometrically simplified domain for the simulation. The seed is shown in the geometry and it is assumed that it has the diameter of crystal. During crystal growth, crystal length increases along minus z direction with the speed of 3 mm. h -1 and the diameter of 15 mm is constant. These values are taken by referring to Ti:sapphire crystal growth experiments described in Annex II. Only the top part of the crystal (green area) is exposed to the surrounding atmosphere where point defect concentration on crystal surface will be affected by the 𝑝 O 2 . The governing equations in the simplified geometry for Ti 3+ , Ti 4+ and V Al ′′′ have been discussed in equations (3-27)-(3-29) in Chapter 3 as:

Figure 4 - 8

 48 Figure 4-8 Computational domain at the beginning of the simulation.

Figure 4 -

 4 Figure 4-10 Ti 3+ (a) and Ti 4+ (b) radial concentration distributions (z = -50 mm) comparison for fine, finer and coarse mesh size.

Figure 4 -

 4 Figure 4-11 shows the 2D plots and radial distributions of Ti 3+ , Ti 4+ , and V Al ′′′ in Ti:sapphire grown crystal. The axial concentration distributions of species can be observed in figures (a), (c) and (e). In axial direction, species concentrations increase from crystal top to the bottom. This is consistent with Scheil-Gulliver behavior. The radial concentration distributions of species are shown in figures (b), (d) and (f).Figure 4-11 (b) shows that the concentration of

   shows that the concentration of Ti 3+ increases radially from the crystal center to the periphery. This profile is consistent with the curved solid-liquid interface as shown in Figure1-23. The concentration of Ti 4+ increases first like Ti 3+ but the chemical reaction starting from the crystal surface leads to Ti 4+ and V Al ′′′ to be consumed. Therefore, the concentrations of Ti 4+ and V Al ′′′ both decrease at the periphery, as shown in Figure 4-11 (d) and (f)). These distribution profiles are consistent with the expected effect of the chemical reaction. Further, the radial concentration distribution profiles of Ti 3+ and Ti 4+ (Figure 4-11 (b) and (d)) are in qualitative agreement with the experimental results observed in the Ky Ti:sapphire crystal (Figure 1-20).This suggests that the proposed model has the potential to explain the experimental results and understand the conversion mechanism between Ti 3+ and Ti 4+ ions during Ti:sapphire growth process. The point defect reaction, leading to Ti 4+ being continuously converted to Ti 3+ , is realized in COMSOL through the reaction rates appearing in diffusion-reaction equations (3-27)-(3-29).

Figure 4 -

 4 Figure 4-11 2D plot and radial distribution of Ti 3+ (a, b), Ti 4+ (c, d) and 𝑉 𝐴𝑙 ′′′ (e, f) concentrations (mol.m -3 ) at T=2000 ℃ along z=-80 mm at t=106900 s.

Figure 4 -

 4 Figure 4-13 Concentration distributions of Ti 3+ (a), Ti 4+ (b) and 𝑉 𝐴𝑙 ′′′ (c) at z=-80 mm under varying (κ=2244 K.m -1 ) temperature field and isothermal temperature T0=2273.15 K.

Figure 4 -

 4 Figure 4-14 The color map shows the temperature field (left) and concentration distribution of 𝑉 𝐴𝑙 ′′′ for κ=224.4 K.m -1 , 2244 K.m -1 and κ=22440 K.m -1 at t=106900 s.

Figure 4 -

 4 Figure 4-15 Radial concentration distributions of Ti 3+ (a) and Ti 4+ (b) at t=106900 s for κ=224.4 K.m -1 , 2244 K.m -1 , and 22440 K.m -1 along z=-80 mm.

Figure 4 -

 4 Figure 4-16 The influence of 𝐷 𝑉 𝐴𝑙 ′′′ (𝑇), 𝑘 𝑐𝑔 (𝑇) and [𝑉 𝐴𝑙 ′′′ ] 𝑏 (𝑇) (denoted as cVac_b(T) in COMSOL software) on the concentration distributions of Ti 3+ (a) and Ti 4+ (b) along z=-80 mm at t=106900 s.

Figure 4 -

 4 Figure 4-17 The axial (a) and radial (b, at z = -80 mm) distributions of Ti 4+ for crystal growth rate at 1 mm.h -1 , 3 mm.h -1 and 10 mm.h -1 . The temperature gradient is 2244 K.m -1 .

Figure 4 -

 4 Figure 4-18 Radial concentration distributions of Ti 3+ and Ti 4+ along z=-80 mm under 𝑝 O 2 =1.1×10 -20 atm (a), 10 -15 atm (b) and 10 -5 atm (c) and 0.21 atm (d).

  71 × [Ti 2 O 3 ]wt. % (4) The concentration of Ti 3+ with respect to Al2O3 lattice represents the proportion of Al 3+ is substituted by Ti 3+ . Then the concentration of Ti 3+ with respect to Al2O3 lattice in concentration of Ti2O3 respect to total atoms in mole percent is[Ti 2 O 3 ]mole% = [Ti 2 O 3 ]at. % ≈ 0.71 × [Ti 2 O 3 ]wt. % (6)The concentration of Ti 3+ respect to total atoms in mole percent is[Ti 3+ ]mole% = 2 × [Ti 2 O 3 ]mole% = 1.42 × [Ti 2 O 3 ]wt. %Similar calculation can be carried out for TiO2 doped sapphire. The conversion relationships among different units are summarized in TableI-2.

Figure II- 2

 2 Figure II-2 Insight of Cz crystal growth furnace.

Figure

  Figure II-3 Ti:sapphire crystals grown under different atmospheres.

Figure III- 1

 1 Figure III-1 Sample positions located in crystal k06-05-15.

Figure III- 2

 2 Figure III-2 Sample positions located in crystal k07-04-13.

Figure

  Figure III-1 and Figure III-2 show Ky crystals grown by RSA company along C-axis and Aaxis together with sample positions located in the crystals. Samples are taken from these two crystals for measurements. The initial Ti concentration in crystal k06-05-15 is 0.2 wt% and in crystal k07-04-13 is 0.15%.

Figure

  Figure III-3 EPR signal of disc 2 (at the bottom of sample E) and disc 3 (near the periphery) powder in crystal K07-04-13

Figure

  Figure III-4. XPS identified elements for sample A in crystal k07-04-13

Figure

  Figure III-5 Raman signals for pure sapphire (2 ppmw) and samples with titanium concentrations of 8 ppmw and 400 ppmw.

  in Huazhong University of Science and Technology in China and Grenoble-INP, UGA. Ti ion concentrations in some samples are also measured with Microwave technique at 4 K. This microwave technique could determine Ti ions through the ESR frequency and g-factor using the whispering gallery mode. The Ti ion concentrations are shown in Table III-6.

  impulsions ultra-courtes en raison de la présence d'ions dopants Ti3+ . Cependant, l'efficacité du laser est diminuée par l'absorption résiduelle causée par les ions Ti 4+ . Les mécanismes de conversion entre les deux valences du Ti pendant la croissance du cristal ne sont pas complètement compris et aucun modèle théorique n'est disponible pour expliquer les distributions expérimentales des deux ions dans les cristaux. Dans cette thèse, un modèle physico-chimique est établi en prenant en compte la diffusion et la réaction chimique des défauts ponctuels et la validité du modèle est vérifiée par simulation numérique.Les concentrations en ions Ti dans la phase solide sont déterminées par leurs concentrations dans la phase liquide et leurs coefficients de ségrégation. Tout d'abord, le logiciel FactSage est utilisé pour calculer thermodynamiquement ces valeurs en utilisant une base de données actualisée par rapport aux données expérimentales de solubilité. Ensuite, les concentrations en ions Ti (Fig.1) et les coefficients de ségrégation (Fig.2) sont calculés et analysés en fonction de la pression partielle d'O2 ( 𝑝 O 2 ). On constate que l'augmentation de 𝑝 O 2 entraîne un changement progressif du coefficient de ségrégation du Ti, 𝑘 𝑇𝑖 , de 𝑘 𝑇𝑖3+ vers 𝑘 𝑇𝑖4+ . L'influence, sur la pression partielle d'oxygène dans le four, 𝑝 O 2 , du creuset en Mo et des éléments du four en graphite est étudiée. Les résultats montrent que les éléments en graphite entourant le cristal lors de la croissance ont beaucoup plus d'influence sur 𝑝 O 2 que le creuset en Mo (Fig.3).

Fig. 1

 1 Fig. 1 Dépendance des concentrations d'ions Ti par rapport à 𝑝 O 2 à 2053 ℃.

Fig. 2

 2 Fig. 2 Dépendance des coefficients de ségrégation par rapport à 𝑝 O 2 à 2053 ℃.

Fig. 3

 3 Fig. 3 Diagramme de prédominance de type Ellingham du système Al2O3-TiO2. "O2 (Mo)" montre le 𝑝 O 2 ; "O2 (C)" montre le 𝑝 O 2 dans le cas d'une atmosphère en équilibre avec du graphite; "O2 (C+Mo)" (triangles gris) montre le 𝑝 O 2 pour le gaz de remplissage réagissant avec le C et le gaz résultant réagissant ensuite avec le Mo.

Fig. 4

 4 Fig. 4 Diagramme schématique du modèle physico-chimique proposé.

Fig. 5

 5 Fig. 5 Variation du coefficient de diffusion de V Al ′′′ avec la température.

Fig. 6

 6 Fig. 6 Variation de la constante de vitesse de réaction en fonction de la température pour la reaction : Ti 3+ + h • ↔ Ti 4+ .

  Abstract ....................................................................................................................................... I Acknowledgment ....................................................................................................................... III Nomenclature ............................................................................................................................. V General introduction .................................................................................................................. IX Chapter 1 Background ............................................................................................................. 1.1 Sapphire ........................................................................................................................ 1.2 Ti:sapphire .................................................................................................................... 1.2.1 Characteristic optical spectra of Ti:sapphire ..................................................... 1.2.2 Ti ion concentrations determination through optical spectra .......................... 1.2.3 Ti:sapphire quality with respect to laser application ....................................... 1.3 Ti:sapphire crystal growth methods ...........................................................................

XIII Contents 1.3.1 Verneuil method .............................................................................................. 1.3.2 Czochralski method ......................................................................................... 1.3.3 Kyropoulos method .........................................................................................

  Gibbs energy and thermodynamic properties .................................................. 2.1.2 FactSage thermodynamic software .................................................................. 2.2 Ti solid solubility and phase diagrams ....................................................................... 2.2.1 Solubility of Ti 3+ and Ti 4+ in sapphire ............................................................. 2.2.2 Al 2 O 3 -Ti 2 O 3 and Al 2 O 3 -TiO 2 systems.............................................................. 2.2.3 Optimized Al 2 O 3 -TiO 2 system ......................................................................... 2.2.4 Al 2 O 3 -TiO 2 Ellingham phase diagram ............................................................. 2.3 𝑃 O 2 during Ti:sapphire growth ................................................................................... 2.3.1 Filling gas ........................................................................................................ 2.3.2 Crucible materials ............................................................................................ 2.3.3 Graphite elements ............................................................................................

..... XIV 2.1.1 2.3.4 The influence of C and Mo on 𝑝 O 2 .................................................................. 2.4 Ti ion concentrations calculated by FactSage ............................................................ 2.5 Ti ion segregation coefficients ...................................................................................

  ............................................................................................................................................ 4.1 Introduction .............................................................................................................. 4.2 Finite element method and COMSOL software ....................................................... 4.2.1 Finite element method (FEM) ....................................................................... 4.2.2 Brief introduction to COMSOL Multiphysics software ................................ 4.3 Physical parameters determination ........................................................................... 4.4 Numerical simulation of annealing .......................................................................... 4.4.1 Experiment statement .................................................................................... 4.4.2 Mathematical model ...................................................................................... 4.4.3 Initial and boundary conditions ..................................................................... 4.4.4 Grids ..............................................................................................................

4.4.5 Numerical procedures .................................................................................... 4.4.6 Parameter estimations .................................................................................... 4.4.7 Results for annealing simulation ................................................................... 4.5 Crystal growth model basic assumptions ................................................................. 4.6 Crystal growth model implementation in COMSOL ............................................... 4.6.1 Simulation domain ......................................................................................... 4.6.2 Initial conditions ............................................................................................ 4.6.3 Boundary conditions ...................................................................................... 4.6.4 Grid and solver settings ................................................................................. 4.7 Preliminary results for isothermal configuration ...................................................... 4.7.1 Physical parameters ....................................................................................... 4.7.2 Concentration distributions of Ti ions ........................................................... 4.8 The influence of temperature .................................................................................... 4.8.1 Varying temperature field .............................................................................. 4.8.2 The effect of temperature gradient ................................................................ 4.8.3 The dominant factor among three temperature related physical parameters . 4.9 The influence of crystal growth rate ......................................................................... 4.10 The influence of 𝑝 O 2 ............................................................................................... 4.11 Conclusion .............................................................................................................. Chapter 5 Conclusions and perspectives ............................................................................ 5.1 Conclusions .............................................................................................................. 5.2 Perspectives for future work ..................................................................................... References .............................................................................................................................

Table 1

 1

-1. Sapphire has a good thermal conductivity, high hardness and is insoluble to any acid or alkali except for hydrofluoric acid (HF), phosphoric acid (H3PO4) or potassium hydroxide (KOH) heated up to 300 ℃. Its wear, shocking and vibration resistance further extend its applications. Sapphire is an anisotropic material. When temperature changes, temperature related parameters such as lattice parameters, thermal expansion coefficient, thermal conductivity, dielectric constant, loss tangent…would change as well.

Table 1 -

 1 1 Sapphire single crystal properties(Dobrovinskaya et al., 2009b)(Sarukura et al.,

	2015).

Table 1

 1

	-2

Table

1

-2 Spectroscopy properties of Cr:sapphire, Nd:YAG, Nd:glass and Ti:sapphire

[START_REF] Moulton | Spectroscopic and laser characteristics of Ti: Al2O3[END_REF]

[START_REF] Ghatak | Lasers fundamentals and applications-second Edi[END_REF]

[START_REF] Birnbaum | Laser emission cross section of Nd:YAG at 1064 nm[END_REF]

[START_REF] Svelto | Principles of lasers[END_REF][START_REF] Jelı Ńek | Waveguiding pulsed laser deposited Ti:sapphire layers on quartz[END_REF]

.

Table 1

 1

		-3 Crystal growth methods classification
		Nutrient Phase	Crystal growth methods
		Gas	Sublimation, Chemical vapor deposition
		Solution	Hydrothermal (Wang et al., 2009)
	Liquid	Small melt amounts Verneuil, Stepanov, Zone melting methods
		Large melt amounts Kyropoulos, Czochralski, HEM, Bridgman,
			EFG, Vertical gradient freeze
		Solid	Horizontal directed crystallization
	1.3.1 Verneuil method
	Verneuil (or flame fusion) method is the first commercial method invented by French
	chemist Verneuil for growing gemstones

  Table 1-4 summarizes the segregation coefficient of Ti in Ti:sapphire single crystals grown by different techniques.

	0.21	Gradient freeze	2-4 mm/h	W	Strauss et al. 1987 (Strauss
		technique			et al., 1987)
	0.15	Horizontal	0.5-2 mm/h	Mo	Nizhankovskiy et al. 2010
		directional			(Nizhankovskiy et al.,
		solidification			2010)
	0.12 -	Horizontal	1-1.5 mm/h	Mo	Nizhankovskiy et al. 2012
	Table 1-4 Summary of Ti effective segregation coefficient in Ti:sapphire crystal grow by 0.15 directional (Nizhankovskiy et al.,
		solidification	different techniques.	2012)
	𝑘 𝑇𝑖 𝑒𝑓𝑓 < 0.2	Crystal growth Unknown	Growth rate -	Crucible -	Reference Dong et al. 2004 (Dong &
		method			Deng, 2004)
	0.23 0.14	HEM Unknown	--	Mo -	Ning et al. 2015 (Ning et Kokta 1985b (Kokta,
					al., 2015) 1985b)
	< 1 0.1	HEM Unknown (Cz ?) 0.25 mm/h -	Mo Ir	Schmid and Khattak 1985 Kokta 2007
					(Schmid & Khattak, 1985)
	0.2	Ky	-	Mo	Nehari et al. 2011 (Nehari
					et al., 2011)
	0.21	Ky	-	Mo	Hu et al. 2012 (Hu et al.,
					2012)
	0.2	Ky	-	Mo	Boulon et al. 2012 (Boulon,
					2012)
	0.17 -	Cz	1 mm/h	Ir	Uecker et al. 2005 (Uecker
	0.2				et al., 2005)
	0.12	Cz	0.38 mm/h	Ir	Kokta 1986 (Kokta, 1986)
	0.11	Cz	0.30-2.54 mm/h -	Kokta 1985a (Kokta,
					1985a)
	0.05	Cz	0.25-0.05 mm/h Ir	Fielitz et al. 2008 (Fielitz et
					al., 2008)
	0.10	Cz	1 mm/h	Ir	Kimura and Kitamura 1993
					(Kimura & Kitamura, 1993)
	0.2	µ-PD	0.72 mm/h	Ir	Nehari et al. 2014 (Nehari
					et al., 2014)
	0.1	Floating zone	1-3 mm/h	-	Kimura et al. 1987 (Kimura
					et al., 1987)

  al. believed that [Ti 4+ ]/[Ti 3+ ] < 0.5 while Nizhankovskii et al. thought that [Ti 4+ ]/[Ti 3+ ] < 0.1. Mohapatra et al. obtained that [Ti 4+ ]/[Ti 3+ ] equals to 0.25. Most of the researchers believe that there are more Ti 3+ in Ti:sapphire than Ti 4+ .

However, the recent study of Moulton's group gives a different result. They found that [Ti 4+ ]/[Ti 3+ ] ≈ 0.066 in Ti2O3 doped sapphire while [Ti 4+ ]/[Ti 3+ ] ≈ 2.1 in TiO2 doped sapphire

[START_REF] Moulton | Characterization of absorption bands in Ti:Sapphire crystals[END_REF]

. Both Ti2O3 and TiO2 doped crystals are grown under Ar atmosphere in an Ir crucible. The only difference is that TiO2 doped crystal experienced an annealing treatment under an unknown atmosphere. Currently, no precise value is given for the concentration ratio of Ti 4+ and Ti 3+ in Ti:sapphire single crystals. In fact, no matter the raw material is Ti2O3 or TiO2, both Ti 3+ and Ti 4+ ions will exist in grown crystals. As shown in Chapter 2, the ratio of [Ti 4+ ]/[Ti 3+ ] in grown crystal depends on the 𝑝 O 2 in growth atmosphere but not on the raw material.

Annealing is a time-consuming process and is impractical for large size crystals (see section 1.8.2 "post growth annealing"). Growing crystals directly under reducing atmosphere is the best choice. Alombert-Goget et al. suggested that Kyropoulos technique has an advantage to reduce the proportion of Ti 4+ in Ti:sapphire during its growing process compared to Czochralski technique

[START_REF] Alombert-Goget | Titanium distribution profiles obtained by luminescence and LIBS measurements on Ti:Al2O3 grown by Czochralski and Kyropoulos techniques[END_REF]

. They thought the conversion from Ti 4+

Table 2 -

 2 1 Physical meaning of ∆H, ∆S and ∆G.

	Sign	∆𝐻 (J. mol -1 )	∆𝑆 (J. mol -1 . K -1 )	∆𝐺 (J. mol -1 )
	Positive	Endothermic	Entropy increases	Nonspontaneous
			(disorder)	(reverse) reaction
	Negative	Exothermic	Entropy decreases	Spontaneous reaction

  For instance,Sidelnikova et al. reported that the Ti:sapphire grown under the mixtures of 99.34 % Ar + 0.6% CO + 0.06 % H2 can lower the [Ti 4+ ]/[Ti 3+ ] ratio[START_REF] Sidelnikova | Charge state of the activator in Ti:sapphire crystals grown by HDC method[END_REF]. The results of Uecker et al. suggest that the mixtures of 95% Ar + 5% CO is efficient in maintaining titanium at +3 state[START_REF] Uecker | Czochralski growth of Ti:sapphire laser crystals. Optically Based Biological and Chemical Sensing, and Optically Based Materials for Defence[END_REF]. Alombert-Goget et al. showed that the FOM value of

	Ti:sapphire crystal is increased after annealing the crystal under 95% Ar + 5% H2 atmosphere
	at 1300 ℃ for 73 hours (Alombert-Gogetet al., 2016).

  -18. The variation of 𝑝 O 2 has no significant effect on 𝑘 𝑇𝑖3+ and 𝑘 𝑇𝑖4+ . Their value variations are no

	more than 0.4% from 𝑝 O 2 =10 -20 to 10 -1 bar. Further, from equations (2-22) and (2-26) it is
	obtained that
	𝑘 𝑇𝑖 =	𝑘 𝑇𝑖3+ +𝛼•𝑘 𝑇𝑖4+ 1+𝛼	.	(2-29)
	At 2053 ℃, it follows from equations (2-19), (2-20) and (2-22) that the theoretical value for
	𝑘 𝑇𝑖 is:

Table 2

 2

	-3 Ti effective segregation coefficient measured in Ti:sapphire grown under different
			growth and rotation rates.
	Crystal growth	Effective	Growth rate	Rotation rate	Reference
	method	k_Ti	(m.s -1 )	(rpm)
	Cz	0.20	2.78 × 10 -7	10-15 *	(Uecker et al., 2005)
	Cz	0.12	1.06 × 10 -7	5	(Kokta, 1986)
	Cz	0.11	8.47 × 10 -8	5	(Kokta, 1985a)
			-
			7.06 × 10 -7

* value obtained from private communication.

  Through calculating the reaction of C and Mo with furnace gases, it is found that graphite elements may have much more influence on 𝑝 O 2 than Mo crucible. By plotting the results in Al2O3-TiO2 Ellingham phase diagram it is concluded that 𝑝 O 2 needs to be controlled during the crystal growth and cooling process. Especially, at sapphire melting temperature Tm = 2053 ℃, 𝑝 O 2 should not be lower than 10 -20 bar. 3. The influence of 𝑝 O 2 on Ti 3+ and Ti 4+ , concentrations in the solid and liquid phases are calculated. At sapphire melting temperature 2053 ℃, by adjusting 𝑝 O 2 , it is found that [Ti 4+ ]L > [Ti 3+ ]L for 𝑝 O 2 >10 -7 bar and [Ti 4+ ]S > [Ti 3+ ]S for 𝑝 O 2 >10 -4 bar. By comparing Ti ion concentrations at very high and very low 𝑝 O 2 , it is concluded that reported empirical formulas used to compute Ti ion concentration through absorption coefficient are not reliable. In addition, the relation between [Ti 4+ ]L/[ Ti 3+ ]L and 𝑝 O 2 is

	The experimental precipitation limit of TiO2 and Ti2O3 from published literature are collected to optimize the phase diagram of Al2O3-TiO2 and to reproduce the experimental data. This ensures the reliability of the database for thermodynamic calculations. The plotted Ellingham phase diagram for Al2O3-TiO2 system shows that 𝑝 O 2 plays an important role in determining Ti valence state. 2. The 𝑝 O 2 comes from different filling gases are calculated and compared. The results showed that pure H2 exhibits the lowest oxygen partial pressure. In addition, the mixture of Ar gas with reducing gases such as Ar + 5% H2 or Ar + 5% CO or 99.34 Ar + 0.6% CO + 0.06% H2 shows lower 𝑝 O obtained to be log 10 [Ti 4+ ] L [Ti 3+ ] L = 0.25 log 10 𝑝 O 2 + 1.68.

2 than that in Ar. By analyzing the structure of crystal growth setup it is found that Mo crucible and graphite heating elements both can affect the 𝑝 O 2 inside the furnace, therefore titanium valence state and concentration.

4. The thermodynamic equilibrium segregation coefficients of Ti 3+ and Ti 4+ are computed to be 𝑘 𝑇𝑖3+ = 0.046 and 𝑘 𝑇𝑖4+ = 0.010. Additionally, it is found that the increase of 𝑝 O 2 leads to 𝑘 𝑇𝑖 changing gradually from 𝑘 𝑇𝑖3+ to 𝑘 𝑇𝑖4+ .The calculation results show that 𝑘 𝑇𝑖3+ > 𝑘 𝑇𝑖 > 𝑘 𝑇𝑖4+ . Through the relation between effective segregation coefficient and equilibrium segregation coefficient and reported experimental data, the equilibrium segregation coefficient of total Ti in sapphire is fitted to be 𝑘 0 = 0.085. It has a reasonable agreement with the thermodynamically calculated equilibrium segregation coefficient, with 𝑘 0 = 0.046, at low 𝑝 O 2 .

  For isovalent doping, for example, when Ti 3+ is incorporated into Al2O3, Ti 3+ will replace previous Al 3+ lattice site. Since Ti 3+ and Al 3+ have the same valence state, no extra defects are required for charge compensation. Heterovalent doping of sapphire leads to the formation of extrinsic defects to maintain the charge neutrality. Therefore, there are certain relations between the dopant concentration and the compensating defect concentration. Donor-dominated crystals are compensated by aluminum vacancies V Al ′′′ or oxygen interstitials O i ′′ and acceptor-dominated crystals are compensated by aluminum interstitials Al i ••• or oxygen vacancies V O ••

	1/5 𝑒	-5𝑘 𝐵 𝑇 . ∆𝐺 𝑓	(3-7)
	Therefore, the energy to form aluminum or oxygen vacancies,	∆𝐺 𝑓 5	, equals the average energy
	per defect.
	3.2.2 Point defects in Ti doped sapphire

  Table 3-2 compiles the average defect energies in literature.

	Except the results of Lagerlöf et al.(Lagerlöf & Grimes, 1998), all other computed results show

that Schottky defects possess the lowest energy. Therefore, it is widely accepted that Schottky disorder is the main disorder mechanism in sapphire, i.e. V Al ′′′ and V O •• are the dominant point defects in sapphire. Dienes's values are accepted by most researchers. Hollenber and Gordon's opinion

Table 3 -

 3 2 Theoretical defect formation energies for various defect reactions in sapphire.

		Dienes	Catlow (Catlow et al.,	Lagerlöf	Mohapat	Katsuyuki
	Defects	and Moon (Dienes et al., 1975;	1982) Empirical potential empirical Non-potential	(Lagerlöf & Grimes, 1998)	ra (Mohapa tra & Kröger, 1978)	(Matsunag a, Tanaka, et al., 2003)
		Moon &
		Phillips,
		1991)
	Schottky	5.74	4.18	5.14	5.17	3.83	4.01
	(eV)
	Oxygen	7.0	3.79	8.27	4.87	-	6.52
	Frenkel
	(eV)
	Aluminum	9.95	5.22	7.09	6.59	4.45	4.95
	Frenkel
	(eV)
	Interstitial	10.62	4.54	10.44	-	-	-
	(eV)

  already reported the variable valence state problem of Ti in sapphire by studying the diffusion and chemical reactions of Ti ions and point defects in annealing experiments which conditions are different from the practical crystal growth process. None of their theories can be used to explain the observed experimental results. In addition, no one have considered both the effect of point defects diffusion and chemical reactions and applied them in the crystal growth process. Luckily,

E.M. Akulono'k et al. and J. Maier proposed chemical models for understanding what is called the "trap diffusion" of point defects (

  , i.e. one V Al ′′′ and three h • diffuse jointly. Therefore, the diffusion of V Al ′′′ should be considered even if it did not appear in reaction (3-21) because it is the slowest species limiting the diffusion. In our model, we believe that the conversion between

	Ti Al • and Ti Al × is essentially determined by joint diffusion and chemical reaction of point defects,
	namely, V Al ′′′ and h

• . It needs to be pointed out that the proposed model can be applied for other compensation mechanism or other similar global reactions that can be split in 2 local equations. For instance, the global reaction (3-19) can also be split in a local reaction (3-22) occurring on the crystal surface and a local reaction (3-23) occurring inside the crystal:

  conclusion, we only have the experimental power law relation [V Al ′′′ ] 𝑏 = 14.46 • which are not consistent to each other's. The fitting results of "conductivity" is obtained from annealing experimental data, therefore, relation (3-64) will be considered for the numerical simulation of annealing which is under a high 𝑝 O 2 . However, this relation is obtained at 1600 ℃ and does not take into account temperature effect. It is not known whether this relation is applicable to the crystal growth process which occurs under a low 𝑝 O 2 (10 -20 bar).

	𝑝 O 2 0.095 mol. m -3 and the theoretical relation [V Al ′′′ ] 𝑏 = 3.17 × 10	-23 4 • 𝑇	9 8 • exp (-2.45	eV 𝑘 𝐵 𝑇	) •
	3
	𝑝 O 2 16 mol. m -3

Table 4 -

 4 1 Comparison of color boundary in annealing simulation and experiment.

	Temperature	Time	XB simulation	XB experiment
	(℃)	(s)	(cm)		(cm) (Jones et al.,
					1969)
		2.8 × 10 5	0.079		0.08
		4.4 × 10 5	0.098		0.10
		6.3 × 10 5	0.118		0.12
		1.79 × 10 5	0.119		0.12
		2.42 × 10 5	0.141		0.14
		1.60 × 10 4	0.081		0.08
		2.44 × 10 4	0.117		0.120
		8 × 10 3	0.077		0.08
		1.2 × 10 4	0.119		0.12
		1.6 × 10 4	0.119		0.16
		4.6 × 10 3	0.120		0.12
		7.2 × 10 3	0.145		0.15
	The aluminum vacancy diffusion coefficient and reaction rate constant values under different
	temperatures are presented in Figure 4-6 and Figure 4-7. By fitting the simulated data, the
	Arrhenius equations for 𝐷 𝑉 𝐴𝑙 ′′′ and 𝑘 𝑎𝑛𝑛 are obtained as:
		𝐷 𝑉 𝐴𝑙 ′′′ = 0.09 × 𝑒 -(351±16) kJ/mol 𝑅𝑇	m 2 . s -1 .	(4-11)
	𝑘 𝑎𝑛𝑛 = 3.4 × 10 5 × 𝑒 -(362±50) kJ/mol 𝑅𝑇	m 3 . mol -1 s -1 .	(4-12)

  Boundary conditions for V Al ′′′ on the top and lateral surfaces of crystal are function of 𝑝 O 2 . As no experimental data under low 𝑝 O 2 is available, theoretical value taken from equation (3-60) is selected for simulation: It needs to be addressed that this boundary condition is different from the one used in annealing

	1
	part with [V Al ′′′ ] 𝑏 = 14.46 • 𝑝 O 2 10 mol. m -3 in equation (4-9). It is chosen because it takes into
					3
	account temperature effect and is supposed to be valid at low 𝑝 O 2 . This 𝑝 O 2 16 power law gives
	very low concentration of V Al ′′′ on crystal surface (see Table 4-3), it allows to obviously observe
	the reduction phenomenon of Ti 4+ thereby making easier understanding the model results.
	Besides, this expression would correspond to the simulation of crystal growth configuration
	where liquid phase is in equilibrium with a given 𝑝 O 2 while the solid lateral surface is in
	equilibrium with a much lower 𝑝 O 2 thereby a much lower concentration for V Al ′′′ . By applying
	all above initial and boundary conditions in COMSOL, the spatial concentration distributions
	of Ti 3+ , Ti 4+ , and V Al ′′′ at any time and any position during Ti:sapphire growth can be obtained
	by solving the three governing equations.
	4.6.4 Grid and solver settings
	[V Al ′′′ ] 𝑏 = 3.17 × 10	-23 4 • 𝑇	9 8 • exp (-2.45	eV 𝑘 𝐵 𝑇	) • 𝑝 O 2

  The implicit time step solver BDF is selected because it has extra robustness and is more stable for transport applications. The relative tolerance is set as 1×10 -4 .Physical parameters used in the simulation are presented in Table4-2. To simplify the preliminary calculations and result analysis, the temperature field is considered as isothermal at 2000 ℃ and the oxygen partial pressure (1.1 × 10 -20 bar) is obtained from the joint effect of Mo crucible and graphite elements at T=2054 ℃ as calculated in section 2.3.4. The concentration ratio [Ti Al• ] 𝐿 / [Ti Al  ] 𝐿 in the liquid phase and their segregation coefficients are obtained from the thermodynamic calculations in Chapter 2. This ratio depends only on 𝑝 O 2 and temperature. The influence of variable temperature field and oxygen partial pressure will be considered in the following calculations. The diffusion coefficients of Ti 3+ and Ti 4+ are taken from reported values at high temperature. The expressions of 𝐷 𝑉 𝐴𝑙 ′′′ and 𝑘 𝑎𝑛𝑛 are obtained from

	Table 4-2 Parameters and variables used in the simulation. 4.7.2 Concentration distributions of Ti ions
	Description	Value (Unit)
	Isothermal temperature, T0	2273.15 K
	Sapphire melting point,Tm	2326.15 K
	4.7 Preliminary results for isothermal configuration Oxygen partial pressure, 𝑝 O 2 1.1 × 10 -20 bar
	4.7.1 Physical parameters Concentration ratio of [Ti Al • ] L / [Ti Al  ] L 4.76 × 10 -4
	Segregation coefficient of Ti 3+ in	0.046
	sapphire, 𝑘 𝑇𝑖 3+
	Segregation coefficient of Ti 4+ in	0.011
	sapphire, 𝑘 𝑇𝑖 4+
	Ti 3+ diffusion coefficient in sapphire,	1 × 10 -14 (m 2 .s -1 ) (Hickey et al., 1996)
	𝐷 𝑇𝑖 𝐴𝑙 
	Ti 4+ diffusion coefficient in sapphire,	6.57 × 10 -13 (m 2 .s -1 ) (Ruzeng et al., 2005)
	previous annealing simulation results (section 4.4.7). As explained in section 4.6.3, the 𝐷 𝑇𝑖 𝐴𝑙 •
	theoretical value taking from (3-60) is selected for the boundary condition in the crystal growth simulation. V Al ′′′ diffusion coefficient in sapphire, 0.09 × 𝑒 (351±16) kJ/mol -𝑅𝑇 0 (m 2 .s -1 )
	𝐷 𝑉 𝐴𝑙 ′′′
	Reaction rate constant in annealing,	3.4 × 10 5 × 𝑒	-(362±50) kJ/mol 𝑅𝑇 0	m 3 . mol -1 s -1
	𝑘 𝑎𝑛𝑛
	Reaction rate constant in crystal	𝑘 𝑎𝑛𝑛 • [Ti Al  ]/ [Ti Al • ]
	growth, 𝑘 𝑐𝑔
	V Al ′′′ concentration on crystal surface,	3.17 × 10	-23 4 • 𝑇 0	9 8
	[V Al ′′′ ]b			× 𝑒	-2.45eV 𝑘 𝐵 𝑇 0 • 𝑝 O 2 3 16 mol. m -3
	Total crystal growth time	118800 s

Table 4

 4 S_initial whatever the value of 𝑝 O 2 . This means that the crystal growth always occurs under reducing conditions. But this strongly depends on which expression is used for [V Al ′′′ ]b. As explained in section 3.6, there exist many possible expressions for [V Al ′′′ ] b so that more experiments are needed to get a reliable one.

	-3. It is found that [V Al ′′′ ]b, as computed by equation (3-60) is always smaller than
	[V Al ′′′ ]

Table 4 -

 4 3 The concentrations of Ti ion and boundary concentration of 𝑉 𝐴𝑙 ′′′ under each 𝑝 𝑂 2 at

	2054 ℃.

  Pointed out the procedure to use annealing simulation to obtain the diffusion coefficient of V Al ′′′ and the reaction rate constant to prepare the study of crystal growth in COMSOL. . A few assumptions are made to simplify the calculations for the crystal growth. Physical parameters obtained from annealing simulation are applied into the physico-chemical model. Then, the details about the computational domain, initial and boundary conditions, grids and solver settings in COMSOL are presented. Preliminary concentration distributions of Ti 3+ , Ti 4+ and V Al ′′′ are studied under isothermal field and found to be qualitatively consistent with experimental results. This means that the proposed model has the potential to explain the experimental results and understand the conversion mechanism between Ti 3+ and Ti 4+ ions during Ti:sapphire growth process. The influence of 𝑝 O 2 =1.1 × 10 -20 , 10 -15 , 10 -5 and 0.21 atm in determining Ti ion concentrations are studied. Generally, with the increase of 𝑝 O 2 , Ti 3+ decreased and Ti 4+ increased due to the oxidation from Ti 3+ to Ti 4+ . Therefore, low 𝑝 O 2 should be maintained during Ti:sapphire crystal growth. In addition, the initial concentration of aluminum vacancy in the solid is found to be always larger than that on crystal surface which means that the crystal growth always occurs under reducing conditions. This is consistent with our model. For 𝑝 O 2 =0.21 atm, Ti 4+ exhibited higher concentration than Ti 3+ which shows that the 𝑝 O 2 has more influence through Ti ion concentrations in the liquid phase than through the solid phase. But this strongly depends on which expression of aluminum vacancies is chosen for the solid boundary condition. More experiments are needed to get a better knowledge on this point. It should be remembered that these results are obtained with various aluminum vacancies boundary conditions. There is no ascertained reaction taking simultaneously into account large temperature and 𝑝 O 2 ranges. Clearly, very different results could be obtained with different boundary condition. This clearly shows that further experimental and theoretical works is necessary to validate a correct boundary equation.

	5.
	Annealing simulation is performed at different temperatures to determine these two
	parameters. The initial and boundary conditions for annealing simulation are carefully
	analyzed. By referring to the experimental color boundary positions and annealing time,
	it is obtained that the Arrhenius expressions of the diffusion coefficient of V Al ′′′ and the
	reaction rate constant in annealing are:
	𝐷 𝑉 𝐴𝑙 ′′′ = 0.09 × 𝑒 -(351±16) kJ/mol 𝑅𝑇	m 2 . s -1 .
	𝑘 𝑎𝑛𝑛 = 3.4 × 10 5 × 𝑒 -(362±50) kJ/mol 𝑅𝑇	m 3 . mol -1 s -1 .
	3. Linear temperature field is introduced into the model and coupled with previous species
	diffusion and chemical reaction process. It is found that Ti 3+ , Ti 4+ and V Al ′′′ show higher
	concentration under linear temperature field compared to that under isothermal field. In
	addition, by varying the temperature gradient to be 224.4 K.m -1 , 2244 K.m -1 and 22440
	K.m -1 , it is found that low temperature gradient helps to improve the conversion
	efficiency from Ti 4+ to Ti 3+ , getting more Ti 3+ and less Ti 4+ along radial direction. By
	fixing two of the three parameters of 𝐷 V Al ′′′ (𝑇), kcg(T) and [V Al ′′′ ]b to isothermal field and
	another one to varying temperature field, it is concluded that the temperature field has
	more influence through 𝐷 V Al ′′′ (𝑇) and has very small influence through kcg(T) and [V Al ′′′ ]b .
	4. Different crystal growth rates are simulated to study its effect on Ti ion concentrations
	in grown crystal. It is found that slow growth rate contributes to reducing the content of
	Ti 4+ in both axial and radial directions. Therefore, low crystal growth rate should be
	used during Ti:sapphire crystal growth.

This chapter implemented the proposed physico-chemical model in COMSOL software to study its validity. All the initial and boundary conditions in COMSOL are given and some preliminary results are presented. The influence of temperature field, crystal growth rate and 𝑝 O 2 on final results are also analyzed. The main works achieved in this chapter include:

1.

2

  𝑝 O 2 in surrounding atmosphere, all the factors which can affect 𝑝 O 2 during crystal growth are analyzed including the filling gas, Mo crucible and graphite elements inside the furnace.It is found that mixing Ar gas with reducing gas like H2 or CO contributes to obtaining lower 𝑝 O 2 . By calculating the reaction of Mo crucible and graphite elements with furnace gas and combine with Al2O3-TiO2 Ellingham phase diagram, it is showed that graphite elements may have much more influence on 𝑝 O 2 than Mo crucible. At sapphire meltingpoint, it is found that [Ti 4+ ]L > [Ti 3+ ]L for 𝑝 O 2 >10 -7 bar and [Ti 4+ ]S > [Ti 3+ ]S for 𝑝 O 2 >10 -4bar. Consequently, the thermodynamic equilibrium segregation coefficients of Ti 3+ and Ti 4+ are calculated to be 𝑘 𝑇𝑖3+ = 0.046 and 𝑘 𝑇𝑖4+ = 0.010. Their relation is found to be 𝑘 𝑇𝑖3+ > 𝑘 𝑇𝑖 > 𝑘 𝑇𝑖4+ . The theoretical segregation coefficient of Ti is obtained to be the function of 𝑝 O 2 as: 𝑘 𝑇𝑖_𝑡ℎ𝑒𝑜𝑟𝑦 =

	𝑘 𝑇𝑖3+ +48•𝑘 𝑇𝑖4+ •𝑝 O 2 1/4
	1+48•𝑝 O 2 1/4

  • from crystal bulk towards crystal surface and the chemical reaction of h • inside the crystal. So, the diffusion coefficients of ions and point defects from literature are collected to make preparation for studying their movement inside the crystal. The results show that 𝐷 V Al ′′′ > 𝐷 𝐴𝑙 . The reported values did not consider point defect reactions so that they remain to be checked. Consequently, the mathematical equations, chemical reaction rate, initial and boundary conditions for proposed model are carefully analyzed and presented to study the model. Especially, the boundary concentration for V Al ′′′ is calculated to be function of temperature and 𝑝 O 2 , i.e. These relations are then applied in the model implemented in COMSOL for crystal growth process together with calculated initial and boundary conditions. Under isothermal field, the simulation results show that the concentration distributions of Ti 3+ and Ti 4+ are qualitatively consistent with experimental profiles. It means that the proposed model has the potential to explain the experimental results and understand the conversion mechanism between Ti 3+ and Ti 4+ ions during Ti:sapphire growth process. After introducing a linear temperature field and varying the temperature gradient, it is obtained that low temperature gradient helps to improve the conversion efficiency from Ti 4+ to Ti 3+ , getting more Ti 3+ and less Ti 4+ along radial direction. It is shown that the varying temperature field has more influence through 𝐷 V Al ′′′ (𝑇) and has very small influence through kcg(T) and [V Al ′′′ ] 𝑏 . In addition, low growth rate is also found helping to reduce the content of Ti 4+ in both axial and radial directions. The study for 𝑝 O 2 =1.1×10 -20 , 10 -15 , 10 -5 and 0.21 atm indicates that, as expected, low 𝑝 O 2 contributes to get more Ti 3+ and less Ti 4+ therefore should be used during Ti:sapphire crystal growth.

	[V Al ′′′ ] 𝑏 = 3.17 × 10	-23 4 • 𝑇	9 8 • exp (-2.45	eV 𝑘 𝐵 𝑇	3 ) • 𝑝 O 2 16 mol. m -3 .
	However, this relation is deduced from literature within certain range of temperature and
	𝑝 O 2 , so it should be verified by further experiments on the whole range of temperature and
	𝑝 O 2 .
	4. The proposed model is implemented in COMSOL software to study the Ti ion behavior
	during crystal growth process through numerical simulation. However, the lacking
	parameters for diffusion coefficient of aluminum vacancies and reaction rate constant of
	point defect reaction in the model have to be determined first. Therefore, numerical
	simulation of annealing is conducted to determine the lacking parameters by taking into
	account both point defect diffusion and chemical reaction by referring to reported
	experimental data. It is obtained that:
	𝐷 𝑉 𝐴𝑙 ′′′ = 0.09 × 𝑒 -(351±16) kJ/mol 𝑅𝑇	m 2 . s -1 .
	𝑘 𝑎𝑛𝑛 = 3.4 × 10 5 × 𝑒 -(362±50) kJ/mol 𝑅𝑇	m 3 . mol -1 . s -1 .

Table I -

 I 2 Dopants and concentration conversion among different units Ti 3+ ]wt. % = 0.67 × [Ti 2 O 3 ]wt. % [Ti 4+ ]wt. % = 0.6 × [TiO 2 ]wt. % [Ti 2 O 3 ]at. % ≈ 0.71 × [Ti 2 O 3 ]wt. % [TiO 2 ]at. % ≈ 1.28 × [TiO 2 ]wt. % [Ti 3+ ]at. % = 2 × [Ti 2 O 3 ]at. % [Ti 4+ ]at. % = [TiO 2 ]at. % [Ti 2 O 3 ]mole% ≈ 0.71 × [Ti 2 O 3 ]wt. % [TiO 2 ]mole% ≈ 1.28 × [TiO 2 ]wt. % [Ti 3+ ]mole% = 1.42 × [Ti 2 O 3 ]wt. % [Ti 4+ ]mole% = 1.28 × [TiO 2 ]wt. % Above conversion among different units suggests that different unit gives rise to different concentration result. Therefore, great care should be taken during ion concentration calculation.

		Ti 2 O 3			TiO2
	[Ti 2 O 3 ]wt. % =	𝑚 Ti2O3 𝑚 Al2O3 + 𝑚 Ti2O3	× 100	[TiO 2 ]wt. % =	𝑚 TiO2 𝑚 Al2O3 + 𝑚 TiO2	× 100
	[

  Table III-1 and Table III-2.Table III-1 WDS measurement for Ti concentration in crystal K-06-05-15

	Samples		Ti ppm(wt)-first time		Ti ppm(wt)-second time
	1C		213.7		242
	2C		213.1		242
	3C		266.5		274
	Table III-2 WDS measurement for Ti concentration in disc 2 of crystal K07-04-13
	Samples	A	B	C	D	E
	Average Ti	485	364	425	359	446
	concentration
	(ppmw)

186

  -3 and Table III-4. This measurement results are obtained from the work of Dr. Gourav Sen. Table III-3 Ti concentration measured by ICPMS for crystal K-06-05-15

	Sample	ICP-MS (ISA-Lyon)
	level		Ti ppm(wt)
			Average
	1	1A	71	57
		1B	43
	2	2A	130	80.5
		2B	31
	3	3A	110	112
		3B	114

Table III -

 III 5 Ti concentration measured by XRF for crystals K-06-05-15 and K-07-04-13

	Samples	XRF (ppmw)
	K06-05-15-3C	315.3
	K07-04-13-A	546.7
	Samples	Ti
	1	398
	2	290
	3	315.5
	4	419
	5	307

Table III -

 III 6 Ti concentration measured by Microwave technique for crystal K-06-05-15

	Sample	Microwave (Wuhan)
	level		Ti ppm (wt)
		Ti 3+	Ti 4+	Ti
	1	832	319	1151
	2	138	159	297
	3	1336	562	1898

  -7 compares the total Ti concentration measured by different techniques.Table III-6 Ti ions and total Ti concentration measured by different techniques

	Table III-7 Comparison of total Ti concentration measured by different techniques
		Samples	WDS (ppmw)	ICPMS (ppmw)		XRF (ppmw)
	K06-05-15-3C	270.3		110		315.3
	K07-04-13-A		485		> 400		546.7
	Sample	ICP-MS (ISA-Lyon)	WDS (CMTC)	Microwave (Wuhan)
	level	Ti ppm(wt)		Ti ppm(wt)		ppm(wt)
			Average 1 st time	2 nd time	Ti 3+	Ti 4+	Ti
	1	1A	71	57	213.7		832	319	1151
		1B	43		(1C)	242
	2	2A	130	80.5	213.1		138	159	297
		2B	31		(2C)	242
	3	3A	110	112	266.5		1336	562	1898
		3B	114		(3C)	274

mol. m -3 .

𝜕 [Ti Al

 -34)where [V Al ′′′ ] and [Ti Al  ] are the concentrations of vacancy and impurity at time t, 𝐷 V Al ′′′ is the diffusion coefficient of aluminum vacancies and 𝑘 𝑎𝑛𝑛 is the reaction rate constant in annealing.

It needs to be noted that the model of Akul'onok et al. was studied for annealing process which means that no crystal growth (Ti ion incorporation) process is considered. In addition, the reaction rate for [V Al ′′′ ] is taking into account in Akul'onok's model while it is considered to be zero in our model (equation (3-32)). However, their methodology for solving the problem can still be extended to study our model. The only difference is that the

and (3-34) represent the total concentration of aluminum vacancies while it should be equal to

According to the modeling of Akul'onok et al, the homogeneous initial impurity concentration is [Ti Al  ] 𝑡=0 and the equilibrium concentration of aluminum vacancies on crystal surface is [V Al ′′′ ] 𝑏 . The initial conditions are:

The boundary condition is:

(3-37)

Therefore, it is obtained that

The relation between aluminum vacancy concentration on crystal surface and oxygen pressure exponent 3/16 has been reported by J.

Performing logarithmic calculation, it is obtained that would be involved in reaction (4-2). As before, V Al ′′′ only diffuses but does not participate to the point defect reaction so that its reaction rate is zero. COMSOL Multiphysics software is used as a tool to solve the diffusion-reaction equations.

Initial and boundary conditions

It is assumed that Ti 4+ is negligible inside the Ti:sapphire samples before annealing.

Therefore, the initial concentrations for Ti 3+ , Ti 4+ and V Al ′′′ in the geometry domain are:

Annex I Ion concentration unit conversion

In most of the impurity doped laser crystals, different units for dopant or incorporated ion concentrations, like wt.%, at.%, ppmw, etc. are used in the literature, depending on the favor of the authors. This makes readers feel confused when comparing the results from different papers.

This part lists all the necessary formulas useful for conversion from one concentration unit to another one.

Ti:sapphire single crystal is usually grown by mixing Ti2O3 or TiO2 into aluminum oxide powder. The molar mass for Al2O3, Ti2O3 and TiO2 are listed in Table I-1. The atomic molar masses for Al, Ti, O atoms are 27 g.mol -1 , 48 g.mol -1 and 16 g.mol -1 .

Different concentration results are calculated as follows.

(1) The concentration of Ti2O3 with respect to total weight in weight percent is calculated as

It should be noted that the Ti2O3 and TiO2 in this part do not represent the precipitates. People used the concentration of [Ti 2 O 3 ] to be equivalent to the concentration of [Ti 3+ ].

(2) The concentration of Ti 3+ with respect to total weight in weight percent is

In addition, 1 wt.%=10000 ppmw (part per million in weight).

Annex II Ti:sapphire growth under different atmospheres using

Cz method

To further study how 𝑝 O 2 influence the concentrations of Ti 3+ and Ti