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공학박사 학위논문

Development of numerical models

for Czochralski sapphire single

crystal growth system

쵸크랄스키 사파이어 단결정

성장 시스템에 대한 수치 모델 개발

2015년 2월

서울대학교 대학원

재 료 공 학 부

허 민 재

Development of numerical models

for Czochralski sapphire single

crystal growth system

쵸크랄스키 사파이어 단결정 성장 시스템에 대한 수치 모델 개발

지도교수 이 경 우

이 논문을 공학박사 학위논문으로 제출함

2015년 2월

서울대학교 대학원

재 료 공 학 부

허 민 재

허민재의 공학박사 학위논문을 인준함

2015년 2월

i

Abstract

Currently sapphire single crystal mass production for LED substrate

mostly takes place through the Kyropoulos method. Other mass production

methods include HEM and VHGF methods. However, all of these methods are

limited by their low yield, as they involve a-axis crystal growth.

However, the Czochralski (CZ) method involves crystal growth in the

c-axis direction, so that there is less loss due to coring. This means that the yield

is higher than other growth methods.

In spite of sapphire single crystals can be obtained with good

throughput through the CZ method with growth in the c-axis direction, the CZ

method is avoided in mass production because of issues with crystal quality.

The quality of sapphire single crystals used as substrates for LED

production are largely influenced by two defects: dislocation density and

bubbles trapped in the crystal. High dislocation densities can lead to substrate

fracture during GaN deposition and diminished LED efficiency. And the

presence of bubble defects affect the optical performance and mechanical

properties of the crystals, thus limiting their utilization in the components.

Therefore in the present study, we developed numerical models for

Czochralski (CZ) sapphire single crystal growth system to investigate

improving growth conditions to enable higher-quality crystal growth.

We calculate decreased convexity and thermal gradient at the crystal

front (CF) through the use of an additional heater in an induction heated CZ

system. Changes in the CF shape with the use of an additional heater were found

ii

through changes in the melt flow direction and hot-zone temperature

distribution, and in comparison with previous crystal growth methods, this was

found to result in lower absolute values for the thermal gradient at the CF as

well as smaller deviations according to location. Moreover, using additional

heater, power consumption deceased.

In addition, we develop a solute concentration model by which the

location of bubble formation in CZ growth is calculated, and the results are

compared with experimental results. The model was used to predict that under

growth conditions involving an additional heater, bubbles would be trapped at

the crystal peripheral edges. This is expected to be of great value in improving

crystal quality.

We calculated the influence of both crystal and crucible rotation to

reduce dislocation density in a resistance heated CZ system. Compared to a

configuration with no crystal or crucible rotation, rotating the crystal and

crucible in the same direction results in a lower variation of the thermal gradient

depending on radial location, but this is accompanied by undesirable convexity.

In contrast, rotating the crystal and crucible in opposite directions results in

both a lower thermal gradient variation with radial location, and improved

convexity.

Keywords : sapphire, single crystal growth, numerical analysis,

Czochralski method, global model.

Student number : 2010-20644

iii

List of Figures

Fig. 1-1. Finished sapphire substrate market forecast.

Fig. 1-2. Diameter trends in sapphire substrates for LEDs.

Fig. 1-3. Schematic diagram of Verneuil method.

Fig. 1-4. Schematic diagram of zone melting method.

Fig. 1-5. Schematic diagram of Kyropoulos method.

Fig. 1-6. Schematic diagram of HEM furnace.

Fig. 1-7. Schematic diagram of VHGF method.

Fig. 1-8. Schematic diagram of EFG method.

Fig. 1-9. The steps of CZ method.

Fig. 1-10. Schematic diagram of induction heated CZ method.

Fig. 1-11. Schematic diagram of resistance heated CZ method.

Fig. 1-12. Comparison of the CZ method and Kyropoulos method.

Fig. 2-1. Physical phenomena in induction heated CZ system.

Fig. 2-2. Physical phenomena in resistance heated CZ system.

Fig. 2-3. Computational boundary cell.

Fig. 3-1. Configurations of the induction heated CZ system and meshes.

iv

Fig. 3-2. Temperature and flow results of global model.

Fig. 3-3. Induction heating result of global model.

Fig. 3-4. Geometries and meshes of the solute concentration model. (a)

Curved CF model and (b) flat CF model.

Fig. 3-5. Comparison of the CF shape for calculated and experimentally

measured results. (a) Shouldering stage and (b) body growth stage.

Fig. 3-6. Measurement positions for thermal gradient.

Fig. 3-7. Thermal gradient variation with radial position at CF.

Fig. 3-8. (a) Hatched line indicates a heating region by induction heating;

dotted line indicates additional heater position. (b) Melt flow pattern

and CF shape in the presence of induction heating only. (c) Melt flow

pattern and CF shape in the presence of induction heating and

additional heating.

Fig. 3-9. Slip systems in sapphire.

Fig. 3-10. Thermal gradient that normal to growth direction at CF.

Fig. 3-11. Thermal gradient that parallel to growth direction at CF.

Fig. 3-12. Etch pit dislocation density (EPD) for vertical and horizontal

section of crystal grown by induction heating only.

Fig. 3-13. Temperature distribution according to growth direction in crystal.

Fig. 3-14. Melt flow and oxygen concentration distribution in solute

v

concentration model: (a) Curved CF model and (b) flat CF model.

Fig. 3-15. Oxygen concentration distribution at CF.

Fig. 3-16. A crystal grown in conditions identical to curved CF model and

bubbles therein.

Fig. 3-17. Cellular structure of sapphire.

Fig. 3-18. A ratio of heterogeneous nucleation energy to homogeneous

nucleation energy depend on cosθ.

Fig. 3-19. Forces acting on bubble.

Fig. 3-20. FT change according to melt flow velocity and bubble radius.

Fig. 3-21. Magnified view of the top region in Fig. 9 illustrating the impact of

stronger drag force.

Fig. 3-22. Concentration increase rate at solute pile-up region according to

concentration at solute pile-up region.

Fig. 4-1. Configurations of the resistance heated CZ system and meshes.

Fig. 4-2. Temperature and flow in resistance heated CZ system.

Fig. 4-3. Cooling water flow rate and outlet water temperature.

Fig. 4-4. Change in the crystal growth interface shape with crucible and

crystal rotation in the same direction.

Fig. 4-5. Change in the crystal growth interface shape with crucible and

crystal rotation in opposite directions.

vi

Fig. 4-6. Convexity changes according to crystal rotation and crucible

rotation.

Fig. 4-7. Comparison of y axis direction velocity for 0x0rpm and n15x0rpm.

Fig. 4-8. Comparison of y axis direction velocity for n15x0rpm and

n15x15rpm.

Fig. 4-9. Convexity changes according to crystal rotation and crucible

rotation.

Fig. 4-10. Comparison of z axis direction velocity for n15x15rpm and counter

rotation c15x15rpm.

Fig. 4-11. Comparison of y axis direction velocity for n15x15rpm and counter

rotation c15x15rpm.

Fig. 4-12. Thermal gradient variation with radial location for (a) n0 (b) n10

(c) c10.

Fig. 4-13. Thermal gradient comparison of no rotation condition with lowest

thermal gradient variation condition.

Fig. 4-14. Thermal gradient that normal to growth direction at CF.

Fig. 4-15. Thermal gradient that parallel to growth direction at CF.

Fig. 5-1. Thermal gradient variations for induction heated CZ, resistance

heated CZ and Kyropoulos system.

vii

List of Tables

Table 2-1. Physical properties used in numerical model.

Table 3-1. Oxygen flux at CF segment in curved CF model.

Table 3-2. Convexities according to coil position.

viii

Contents

Abstract ………………………………………………………………. i

List of Figures ………………………………………………………. iii

List of Tables ……………………………………………………….. vii

Chapter 1. Introduction ……………………………………………... 1

1. 1 Single crystal sapphire for LED substrate ………………………….. 1

1. 2 Growth methods ………..................................................................... 3

1. 2. 1 Verneuil Method …………………………………………...... 3

1. 2. 2 Zone melting method ………………………………………... 4

1. 2. 3 Kyropoulos method ………………………………………..... 5

1. 2. 4 Heat-Exchange Method (HEM) …………………………….. 6

1. 2. 5 Vertical/Horizontal Gradient Freezing (VHGF method) ……. 8

1. 2. 6 Edge-defined Film fed Growth (EFG method) ……………... 9

1. 2. 7 Czochralski method ………………………………………... 10

1. 3 Problems in CZ method ………………………………………….... 15

1. 3. 1 Dislocations in crystal ……………………………………… 15

1. 3. 2 Bubbles in crystal …………………………………………... 15

1. 4 Previous studies for the sapphire single crystal growth …………… 16

1. 4. 1 Dislocations in sapphire single crystal ……………………... 16

ix

1. 4. 2 Bubbles in sapphire single crystal ………………………….. 17

1. 5 Goals of the research ………………………………………………. 19

Chapter 2. Numerical modeling …………………………………… 22

2. 1 Considered physical phenomena in CZ system …………………… 22

2. 2 Numerical calculation ……………………………………………... 24

2. 2. 1 Flow ………………………………………………………... 24

2. 2. 2 Heat transfer ………………………………………………... 27

2. 2. 3 Radiation …………………………………………………… 28

2. 2. 4 Electromagnetic field ………………………………………. 30

2. 2. 5 Turbulence Module ………………………………………… 33

2. 2. 6 Solidification ………………………………………………. 37

2. 2. 7 Boundary conditions ……………………………………….. 38

2. 3 Physical properties used in numerical model ……………………… 41

Chapter 3. Induction heated CZ system …………………………... 42

3. 1 Numerical modeling ………………………………………………. 42

3. 1. 1 Global modeling …………………………………………… 42

3. 1. 2 Solute concentration modeling …………………………….. 46

3. 2 Global model results and analysis …………………………………. 52

3. 2. 1 Model verification through comparison with experimental

results ……………………………………………………... 52

3. 2. 2 Decreased temperature gradient at CF by use of additional

x

heater ……………………………………………………… 55

3. 2. 3 Decreased CF convexity by use of additional heater ………. 66

3. 3 Solute concentration model results and analysis …………………... 68

3. 3. 1 Verification of solute concentration model by comparison with

experimental results ………………………………………. 68

3. 3. 2 Calculation of likelihood of bubble movement in CZ growth

…………………………………………………………….. 76

3. 3. 3 Calculation of bubble entrapment location in crystal growth

using additional heater ……………………………………. 82

3. 4 Summary ………………………………………………………….. 85

Chapter 4. Resistance heated CZ system ………………………….. 87

4. 1 Numerical modeling ………………………………………………. 87

4. 2 Results and analysis ……………………………………………….. 91

4. 2. 1 Change in convexity with crystal/crucible rotation ………... 91

4. 2. 2 Variation in thermal gradient at the CF with crystal and crucible

rotation conditions ……………………………………….. 103

4. 3 Summary ………………………………………………………… 110

Chapter 5. Conclusions …………………………………………… 111

References …………………………………………………………. 114

Korean abstract …………………………………………………… 124

Acknowledgement ………………………………………………… XX

1

Chapter 1. Introduction

1. 1 Single crystal sapphire for LED substrate

Single crystal sapphire substrate is good for use in Light Emitting

Diode (LED) due to high temperature resistance, high strength, high chemical

stability, good electrical insulation, low dielectric loss and low price. Sapphire

substrate in LED lighting helps prevent stray currents caused by radiation from

spreading to nearby circuit elements. Its crystal structure also allows LED lights

to have a wider beam angle. [1]

The c-plane (0001) of sapphire has a large mismatch with GaN

compared with other substrate materials (SiC, GaN, ZnO). However, sapphire

has properties which enable it to endure the high temperatures and high

reactivity of Metal-Organic Chemical Vapor Deposition (MOCVD). It is also

available for mass production and relatively inexpensive. As a result,

approximately 90% of LEDs currently being produced make use of a sapphire

substrate. [2]

With the continued growth of the LED market, sapphire production

output is also increasing. According to a Yole Dévelopment market forecast,

sapphire production is rising steadily each year, and its growth is projected to

continue in the future. [3] In addition, efforts to increase substrate sizes are

ongoing in order to decrease production costs, and it appears that 6 inch

substrate will be the norm by 2016. [4]

2

Fig. 1-1. Finished sapphire substrate market forecast. [3]

Fig. 1-2. Diameter trends in sapphire substrates for LEDs. [4]

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

% o

f to

tal

surfa

ce p

rocess

ed

Year

8inch 6inch 4inch 3inch 2inch

3

1. 2 Growth methods

1. 2. 1 Verneuil Method

This method is the first usable sapphire growth technology [5] and this

method is still in use. [2] The raw materials are added to the top chamber of the

furnace. Oxygen and hydrogen are blown into the cabin for combustion, where

a high temperature is achieved. Liquid droplets of materials form single crystal

at the tip. This method could provide a high growing speed. However the

quality of the crystal produced is limited by the irregular temperature

distribution and cooling velocity. [6]

Fig. 1-3. Schematic diagram of Verneuil method. [7]

4

1. 2. 2 Zone melting method

The zone melting method is widely used for material purification and

we can control the concentration of impurities through this method. However,

zone melting also turned out to be applicable as a crystal growth method, since

in the process of purification single crystals often are formed. [8]

A sample to be refined is placed in a trough along which a heating ring

can be passed in a slow and regular manner. If it is required to grow a single

crystal, a seed is placed at the left end of the boat. The heating ring is activated

at one end of the trough and enough heat applied to melt the disc of the material

immediately within the ring. The ring is then moved slowly towards the other

end of the trough. As it moves, sufficient heat is applied in the ring to melt the

new material moving under the ring, while the previously-molten material left

behind cools and solidifies. The ring is gradually moved to the other end of the

trough and the heat turned off, as one cycle of refining is finished. [8, 9]

Fig. 1-4. Schematic diagram of zone melting method. [9]

5

1. 2. 3 Kyropoulos method

The Kyropoulos process was developed by Spyro Kyropoulos to grow

single crystals of alkaline halides. To initiate growth a cooled seed or shaft is

dipped into the melt and the furnace temperature slowly cooled to encourage

the downward growth of the crystal into the melt. Submerged growth like this

allows the crystal to growth in a shallower thermal gradient than the

Czochralski process. The shallower thermal gradient is important for thermally

sensitive scintillator and semiconductor crystals that are prone to fracture. [10]

The Kyropoulos method is used to grow sapphire crystals with a

diameter exceeding 350mm and a weight larger than 80kg. The ratio of the

diameter to the height may change within the interval of 3:1 to 1:3. [11]

Fig. 1-5. Schematic diagram of Kyropoulos method. [10]

6

1. 2. 4 Heat-Exchange Method (HEM)

Heat-Exchange Method (HEM) is actively used for production of

sapphire crystals of a large scale. This method is characterized by a high level

of process automatization. In these conditions a well developed hot-zone design

and process recipe are becoming key factors to grow large weight and high

quality sapphire boules. [12]

In HEM, a sapphire seed crystal is placed at the bottom of a

molybdenum crucible which is then loaded with pure alumina crackle, a

byproduct of the Verneuil process. The furnace is evacuated and resistively

heated to melt the crackle while keeping the seed just below its melting point

by passing helium gas through the heat exchanger beneath the center of the

crucible. Heat and vacuum help purify the alumina by vaporizing some

impurities. After partial melting of the seed, helium flow is increased to cool

the seed and initiate crystallization of alumina onto the seed. The furnace is

held at constant temperature during growth of the crystal, which proceeds out

from the seed in 3 dimensions.

After crystallization is complete, the furnace temperature and the

helium flow are decreased and the boule is slowly annealing in situ. The long

slow cool down produces sapphire of high crystal quality. [13]

7

Fig. 1-6. Schematic diagram of HEM furnace. [14]

8

1. 2. 5 Vertical/Horizontal Gradient Freezing (VHGF method)

The VHGF method is developed by Sapphire Technology Company

(STC). This method can grow single crystal of the high quality of which

dislocation density is low. Without any limitation in size (diameter and length)

of single crystal or no limitation in shape, this method can produce sapphire

single crystal of the high quality. [15]

Fig. 1-7. Schematic diagram of VHGF method. [16]

9

1. 2. 6 Edge-defined Film fed Growth (EFG method)

The EFG method is a single-crystal growing technique for

synthesizing pure, single crystal materials. It is a technique whereby a die for

growing the crystal is placed on the surface or a matrix is dipped into the melt.

special shapes include sapphire ribbons, sapphire single crystals with

predefined diameter and sapphire tubes. [17]

Fig. 1-8. Schematic diagram of EFG method. [17]

10

1. 2. 7 Czochralski method

In the Czochralski (CZ) method, a single crystal is pulled from the

melt. The steps of CZ method are illustrated in Fig. 1-9. The first step is to pull

a single crystal with the same crystallographic orientation of a small single

crystalline seed crystal out of melt. The melt temperature is kept constant

roughly above the melting point. A single crystalline seed crystal with the

desired crystal orientation is immersed into the melt and acts as a starting point

for the crystal formation supported by the heat transfer from the melt to the

already grown crystal. The seed crystal is slowly pulled out of the melt, where

the pull speed determines the crystal diameter. During crystal growth, the

crystal rotates in order to improve the homogeneity of the crystal and its dopant

concentration. [18, 19]

This method has had nearly one hundred years’ history, whereas

currently is still the most widely used method to fabricate single crystal

materials, especially large semiconductor and metallic materials. It can produce

very high quality crystals. [19]

The CZ method can be categorized depending on whether it is heated

by induction heating (Fig. 1-10.) or resistance heating (Fig. 1-11.). Each

method has its pros and cons. Induction heating requires less energy input and

is therefore easier to heat to high temperatures, but it does not allow for crucible

rotation. In contrast, resistance heating involves high energy input, but it allows

for crucible rotation and favorable for large size crystal growth. [20]

The primary advantage of the CZ method in sapphire single crystal

11

growth for LED substrates is the high yield. Currently sapphire single crystal

mass production mostly takes place through the Kyropoulos method. Other

mass production methods include HEM and VHGF methods. However, all of

these methods are limited by their low yield, as they involve a-axis crystal

growth. Use as an LED substrate requires a coring process, and this results in a

large unused amount for crystals grown in the a-axis, leading to a low yield.

Moreover, increased substrate sizes for lower production costs will drop yields

even further.

However, the CZ method involves crystal growth in the c-axis

direction, so that there is less loss due to coring. This means that the yield is

higher and there is no loss of yield even for larger substrates. (Fig. 1-12.)

Because of these advantages, the CZ method will become more desirable as a

mass production method in the sapphire single crystal growth market.

12

Fig. 1-9. The steps of CZ method. [21]

13

Fig. 1-10. Schematic diagram of induction heated CZ method. [22]

Fig. 1-11. Schematic diagram of resistance heated CZ method. [21]

14

Fig. 1-12. Comparison of the CZ method and Kyropoulos method. [2]

15

1. 3 Problems in CZ method

In spite of sapphire single crystals can be obtained with good

throughput through the CZ method with growth in the c-axis direction, the CZ

method is avoided in mass production because of issues with crystal quality.

Issues of crystal quality can be largely categorized into high dislocation density

and bubbles in the crystal.

1. 3. 1 Dislocations in crystal

The first problem is high dislocation density. Sapphire c-axis growth

using the CZ method leads to high dislocation density within the crystal which

is 20-100 times higher than for crystals grown by the Kyropolous method. High

dislocation densities can lead to substrate fracture during GaN deposition and

diminished LED efficiency. [23-29]

1. 3. 2 Bubbles in crystal

The second problem is bubbles in crystal. Bubbles in sapphire single

crystals are, along with dislocations, an important defect. The presence of

macroscopic and microscopic bubble defects affect the optical performance and

mechanical properties of the crystals, thus limiting their utilization in the

components. Their presence decreases the geometrical yield which describes

the maximum theoretical quantity of usable core that can be extracted from a

given boule, reduces the transparency and induces surface defects during

16

substrate polishing. They can also be initiation sites for cracks propagation. [30,

31]

1. 4 Previous studies for the sapphire single crystal growth

Sapphire single crystal growth occurs at a high temperature, above

2300K. This makes it difficult to directly observe the growth process or

measure interior physical phenomena. Moreover, single crystal growth is a slow

process, so that experimentation requires significant time and cost. As a result,

various studies have been performed using numerical analysis, which provides

important information on physical phenomena related to the growth process

while overcoming limitations of time and cost.

1. 4. 1 Dislocations in sapphire single crystal

Many previous studies have been presented on improving growth

conditions to suppress dislocation formation in the crystal. Variables dealt with

in these studies include hot-zone design, crucible shape, coil position, and

crystal rotation rate.

Lu et al. calculated melt temperature change and power consumption

depending on coil location and growth stage in the induction heated CZ method.

[32] Lee et al. employed numerical analysis to calculate crystal front (CF) shape,

melt flow and flow velocity variation depending on crucible shape in the

17

Kyropoulos method, and analyzed thermal gradient variation depending on

location. [33] Tavakoli et al. performed various studies on induction heating,

calculating variations that arise from consideration of eddy currents that occur

at the coil for the Induction heated CZ method in numerical models [34] and

reporting on the influence of the driving current frequency on induction heating.

[35] Studies also investigated changes in induction heating depending on

crucible shape and position [36, 37] as well as changes depending on coil

geometry. [38] Studies reported on the crystal growth process include

investigations of the influence coil position has on melt and gas flow [39] as

well as heat transport and fluid flow changes depending on growth stage. [40]

Fang et al. set the melt’s marangoni flow and surface tension as variables,

investigating the effect of changes in those values on the CF shape. [41]

Moreover, the thermal stress in the crystal was calculated depending on changes

in the coil position, growth stage and crystal radius. [42] Demina et al. used

CGSim software to perform studies on the induction heated CZ method and the

Kyropoulos method. They tackled the problem of the appearance of a remelting

zone in actual experiments through thermal optimization of the Hot-zone, and

presented numerical analysis results on their findings. [43] Later studies

employed 3D unsteady numerical modeling to investigate melt instability and

thermal patterns. [44] Chen et al. used the finite-element method to study CF

shape variation as well as flow and temperature distributions in the melt

depending on the growth stage for the Kyropoulos method. [45]

18

1. 4. 2 Bubbles in sapphire single crystal

Research on bubble formation in the crystal has primarily been

centered on the EFG method. This is because the equipment is relatively small

and growth is easy, making lab-scale experimentation possible.

Nicoara et al. used STHAMAS 3D software for 3D numerical

modeling of the EFG process and investigated the influence of melt flow on

bubble formation. [46] They later presented experimental results on bubble

formation in sapphire crystal using central capillary channel (CCC) shaper and

annular capillary channel (ACC) shaper. STHAMAS 3D and FIDAP software

were used to calculate impurity concentration in the melt. Comparison of the

calculated results with experimentally obtained results led to the conclusion that

bubble formation takes place at locations where impurity concentration is high.

[47]

Bunoiu et al. also studied bubble formation phenomena in sapphire

single crystal growth through the EFG method. [48-51] The 2010 report ‘Gas

bubbles in shaped sapphire’ presented a solute segregation theory which can

accurately explain experimental bubble formation results in the EFG method,

along with both experimental and calculated results. The solute segregation

theory proposes that differences in oxygen (or carbon monoxide) solubility

between the melt and crystal leads to segregation at the CF, and with continued

occurrence of this phenomenon along with melt flow, solute pile-up arises at

specific locations. At locations where the pile-up creates an oversaturation of

solute, bubble nuclei are formed, which, if grown past a certain size, lead to a

19

radiation cooling effect so that the bubble is trapped within the crystal. [51]

Studies have also been reported on bubbles entrapped in sapphire

single crystal grown by the CZ method. Li et al. set crystal growth direction,

pulling rate and crystal rotation rate in the CZ method as variables and

performed experiments for 15cases, finding that a higher pulling rate results in

more bubbles being trapped in the crystal. [30] Recent studies have found

through sapphire and Ti-doped sapphire growth by the CZ method that in

addition to pulling rate, crystal rotation rate increase also has an effect on

bubble entrapment in the crystal. [31] However, these studies are limited to

experimentally obtained trends, with no theoretical approach regarding bubble

formation.

1. 5 Goals of the research

To summarize previous studies, efforts to lower the dislocation density

in crystals grown by the induction heated CZ method have involved research to

make the CF shape flat or to lower the thermal gradient at the CF. While most

studies set factors such as coil position, crucible shape and crystal rotation rate

as variables and thereby presented methods for improvement, major

enhancements have thus far been limited.

Research on bubbles has also been largely limited to the EFG method,

and studies regarding the CZ method, which has different growth conditions,

20

have consisted entirely of experimental observations on general trends.

There has been nearly no reported research on the resistance heated

CZ method, which will be highly competitive when sapphire substrate

enlargement leads to substrates larger than 8inches dominating the market in

the future.

Therefore in the present study, we employ an additional heater for the

induction heated CZ method to enable higher-quality crystal growth. The use

of an additional heater alters the melt flow pattern, and thereby enables control

of CF shape as well as the thermal gradient at the CF. In addition, we develop

a solute concentration model by which the location of bubble formation in CZ

growth is calculated, and the results are compared with experimental results.

We further calculate the location of bubble entrapment corresponding with the

use of an additional heater.

We also investigate conditions for growth of higher-quality sapphire

single crystals by the resistance heated CZ method. The resistance heated CZ

method, unlike the induction heated CZ method, enables crucible rotation. We

analyze through numerical simulation the effect of simultaneous rotation of the

crucible and crystal on the CF shape and thermal gradient at the CF. [20]

Many studies have been performed to explore the influence of crystal

and crucible rotation in a resistance heated CZ system on CF shape as well as

flow and temperature patterns within the melt. [52-63] Hintz et al. investigated

oscillation in the melt resulting from crystal rotation with a fluid with Pr =7

[55], and Noghabi et al. studied the influence of crucible and crystal rotation on

21

crystal growth interface geometry for the case of silicon single crystal growth.

[61] The effect of forced convection caused by crucible and crystal rotation

varies depending on the melt properties, as does the CF shape, which is why

the present study is concerned with the variables above for the sapphire single

crystal growth process. [20]

22

Chapter 2. Numerical modeling

The investigation of temperature and flow field using experimental

measurements in CZ system is quite difficult because of the high temperature

of the melt and limitation according to their nature. So, the numerical

simulation has become quite popular since it could obtain more detailed data.

At the same time, the development of computers and algorithms has been able

to perform huge amount of the calculation and complicated problems in a

sufficiently short time.

In this study, numerical simulation was performed to investigate the

temperature distribution and flow field in a CZ system. We use CFD-ACE+ for

numerical analysis. [64]

2. 1 Considered physical phenomena in CZ system

We take into account physical phenomena such as flow, heat transfer,

radiation, electric current, electromagnetic field, turbulence and solidification

in global model. [20] In solute concentration model, we take into account flow,

heat transfer, radiation, solute transfer and diffusion of solute in melt.

23

Fig. 2-1. Physical phenomena in induction heated CZ system.

Fig. 2-2. Physical phenomena in resistance heated CZ system.

24

2. 2 Numerical calculation

2. 2. 1 Flow

Mass conservation

Conservation of mass requires that the time rate of change of mass in

a control volume be balanced by the net mass flow into the same control volume

(outflow - inflow). This can be expressed as:

0)(

Vt

(2-1)

The first term on the left hand side is the time rate of change of the density

(mass per unit volume). The second term describes the net mass flow across the

control volume’s boundaries and is called the convective term. [65]

Momentum conservation

Newton’s second law states that the time rate of change of the

momentum of a fluid element is equal to the sum of the forces on the element.

The x-component of the momentum equation is found by setting the

rate of change of x-momentum of the fluid particle equal to the total force in

the x-direction on the element due to surface stresses plus the rate of increase

of x-momentum due to sources:

Mxzxyxxx Szyx

puV

t

u

)()(

)( (2-2)

25

Similar equations can be written for the y- and z-components of the momentum

equation:

My

zyyyxyS

zy

p

xvV

t

v

)()(

)( (2-3)

Mzzzyzxz S

z

p

yxwV

t

w

)()(

)(

(2-4)

In these equations, p is the static pressure and tij is the viscous stress tensor.

[65]

Navier-Stokes equations

The momentum equations, given above, contain as unknowns the

viscous stress components tij, therefore a model must be provided to define the

viscous stresses.

In Newtonian flows, the viscous stresses are proportional to the

deformation rates of the fluid element. The nine viscous stress components (of

which six are independent for isotropic fluids) can be related to velocity

gradients to produce the following shear stress terms:

)(3

22

Vu

x

uxx (2-5)

)(3

22

Vu

y

vyy (2-6)

)(3

22

Vu

z

wzz (2-7)

26

x

v

y

uyxxy (2-8)

x

w

z

uzxxz (2-9)

y

w

z

vzyyz (2-10)

Substitution of the above shear stress terms into the momentum equations

yields the Navier-Stokes equations:

MxSx

w

z

u

zx

v

y

u

y

Vx

u

xx

puV

t

u

)(3

22)(

)(

(2-11)

MySy

w

z

v

zV

x

v

y

x

v

y

u

xy

pvV

t

v

)(3

22

)()(

(2-12)

MzSVy

w

zx

w

z

v

y

x

v

y

u

xz

pwV

t

v

)(3

22

)()(

(2-13)

By rearranging these equations and moving the smaller contributions of the

viscous stress terms to the momentum source term, we can rewrite the Navier-

Stokes equations in a more useful form:

27

MxSux

puV

t

u

)()()(

(2-14)

MySvy

pvV

t

v

)()()(

(2-15)

MzSwz

pwV

t

w

)()()(

(2-16)

[65]

2. 2. 2 Heat Transfer

Heat transfer processes are computed by solving the equation for the

conservation of energy. This equation can take several forms and CFD-ACE+

numerically solves the energy equation in the form known as the total enthalpy

equation. This form is fully conservative and is given in equation (2-17).

hzxyxxz

zyyyxy

zxyxxx

eff

Sz

u

y

u

x

u

z

u

y

u

x

u

z

u

y

u

x

u

t

pTkhV

t

h

)()()(

)()()(

)()()(

)()()(

00

(2-17)

Where h0 is the total enthalpy and is defined as

28

222

02

1wvu

pih

(2-18)

Where i is the internal energy and is a function of the state variables r and T, P

is the static pressure and tii is the viscous stress tensor. keff is the effective thermal

conductivity of the material. In laminar flow, this will be the thermal

conductivity of the fluid, k. In turbulent flows:

t

pt

eff

CKk

(2-19)

Where σt is the turbulent Prandtl number.

Sh contains terms for additional sources due to reactions, radiation,

spray, body forces, etc. [65]

2. 2. 3 Radiation

In this study, we use a Sn Discrete Ordinate Method (SnDOM) and

gray model for radiation calculation.

Radiative transfer equation

The integro-differential radiative heat transfer equation for an

emitting-absorbing and scattering gray medium can be written as:

4''',

4

,,

drI

rkIrIkrI b

(2-20)

29

Where Ω is the direction of propagation of the radiation beam, I is the radiation

intensity which is a function of both position (r) and direction (Ω), κ and σ are

the absorption and scattering coefficients respectively, Ib is the intensity of

black body radiation at the temperature of the medium and Φ is the phase

function of the energy transfer from the incoming Ω' direction to the outgoing

direction Ω. The term on the left-hand side represents the gradient of the

intensity in the specified direction Ω. The three terms on the right-hand side

represent the changes in intensity due to absorption and out-scattering, emission

and in-scattering, respectively.

The boundary condition for solving the above equation (2-20) may be

written as:

0'',',

nb drInrIrI

(2-21)

Where I is the intensity of radiant energy leaving a surface at a boundary

location, ε is the surface emissivity, ρ is the surface reflectivity, and n is the unit

normal vector at the boundary location. [65]

Discrete ordinate method

In the discrete ordinate method, equation (2-20) and equation (2-21)

are replaced by a discrete set of equations for a finite number of ordinate

directions. The integral terms on the right hand side of equation (2-20) is

approximated by a summation over each ordinate. The discrete-ordinate

equations may then be written as:

30

Mm

IW

FIk

Ikz

I

y

I

x

I

mmmmm

b

mm

mm

mm

m

,1

4''''

(2-22)

In the previous equations, m and m' denote the outgoing and incoming

directions, respectively. For a direction represents the associated weight while

α, β, and γ represent the direction cosines corresponding to the x, y and z

coordinates respectively. Equation (2-22) represents M coupled partial

differential equations for M intensities, Im.

For the gray model, the subscript should be dropped from the above

equation and becomes unity. [65]

2. 2. 4 Electromagnetic field

We solve Maxwell’s equations using the magnetic vector potential A.

The derivation begins with Faraday’s Law and Ampère’s Law (with

displacement current) respectively in differential form and assuming mks units:

t

BE

(2-23)

Jt

DH

(2-24)

Using the definition of the magnetic vector potential, B = ∇xA and the

constitutive relation, B = µH, equation (2-23) and equation (2-24) become:

31

0

t

AE (2-25)

Jt

DA

1 (2-26)

Using equation (2-25) and the vector identity ∇ x ∇ ϕ = 0 (where ϕ is any scalar)

the electric field is defined as:

t (2-27)

Where ϕ is the electric potential and -∇ ϕ is the electrostatic field. Using the

vector identity ∇ x ∇x V = ∇( ∇•V) - ∇2V (where V is any vector), assuming

∇(1/µ) = 0, and assuming the current density J includes conductive (σE),

convective (σuxB), and other (Js) components, equation (2-26) becomes:

SJuxtD 21 (2-28)

Assuming the Coulomb gauge (∇•A = 0), writing the permeability as µ = µrµo,

and using the constitutive relation D = εE, equation (2-28) becomes:

S

ro

JBuEt

EA

21 (2-29)

Finally, substituting the expression for the electric field in equation (2-27) into

equation (2-29) and writing the permittivity as ε = εrεo, yields:

Soror

ro

JBut

A

tt

AA

2

221

(2-30)

32

In the frequency domain equation (2-30) becomes:

Soror

ro

JBuAjjAA~~~~~~~1 22

(2-31)

~~2 oror jA : Displacement current

Aj~

: Eddy current

~

: Electric conduction

Bu~

: Convective current

SJ~

: Specified current

[65]

AC single frequency

We choose the AC single frequency option. The AC single frequency

option solves the complete form of the ac sinusoidal steady state form of

equation (2-31).

The possible source currents include specified currents, displacement

currents, currents calculated from the solution of the electric conduction

problem or eddy currents. [65]

Induction heating (coupling with heat transfer)

A time varying magnetic field generates a time varying electric field

33

which generates eddy current in conductive materials (iridium crucible). The

flow of current through the conductor generates heat and is called inductive

heating. The inductive heating can be thought of as joule heating (J·E) with the

conductive currents (J = σE) generated by the time varying field . So assuming

the electric field is due only to the time varying magnetic field and the current

is only a conduction current the time averaged expression for joule heating

becomes

2

2 ~

2

1*

~~Re

2

1AEJ r (2-32)

Where σr is the real component of conductivity and ω is the radian frequency

(2πf). [65]

2. 2. 5 Turbulence Module

Theory introduction

For more than a century the preferred approach in the treatment of

turbulent flows is to predict macroscopic statistics using the RANS formalism.

Introduced by Reynolds in 1895, it involves a simple decomposition of the

instantaneous fields in mean values and fluctuations via an averaging operation.

The issue of turbulence modeling arises from the need to represent turbulent or

Reynolds stresses, which are additional unknowns introduced by averaging the

Navier-Stokes equations. The proportionality parameter is called the turbulent

or eddy viscosity, and is expressed phenomenologically or obtained from

transport equations. Unlike its laminar counter-part, the turbulent viscosity is

34

not a property of the fluid but rather a characteristic of the flow.

Within the framework of RANS modeling, various models differ in

the way the turbulent viscosity is calculated. These models are typically

categorized by the number of additional transport equations to be solved.

Almost all the models in this simulation involve solutions of two extra transport

equations. One is for the turbulent kinetic energy, k, and the other are for the

rate of dissipation, e, or the specific rate of dissipation, w.

Based upon the way the near-wall viscous sublayer is handled, these

models are further classified into high-Reynolds-number and low-Reynolds-

number models. Here the qualifier Reynolds-number refers to the local

turbulent Reynolds number:

2

Rek

t (2-33)

It will be shown that Ret is proportional to the ratio of the eddy viscosity to

molecular viscosity, v. High Reynolds models are designed for regions where

the eddy viscosity is much larger than the molecular viscosity and, therefore,

cannot be extended into the near-wall sublayers where viscous effects are

dominating. The standard wall-function model is used to bridge the gap

between the high-Reynolds-number regions and the walls or to connect

conditions at some distance from the wall with those at the wall. Low-

Reynolds-number models are designed to be used in the turbulent core regions

and the near-wall viscous sublayers. [65]

35

Standard k-e model

Sapphire melt has a low Reynolds number (100~200) in CZ growth

condition. Therefore we can assume that sapphire melt flow is laminar flow.

But gas in the CZ system has a high Reynolds number. So we use the standard

k-e model for turbulence calculation. In the model, the turbulent viscosity is

expressed as:

2kCvt (2-34)

The transport equations for k and e are,

jk

i

j

j

j x

k

xPku

xk

t

)()( (2-35)

j

i

j

j

j

xx

k

PC

k

PCu

xt

)()(2

21

(2-36)

With the production term P defined as:

m

m

j

iij

m

m

i

j

j

it

x

uk

x

u

x

u

x

u

x

uP

3

2

3

2 (2-37)

The five constants used in this model are:

36

The standard k-e model is a high Reynolds model and is not intended

to be used in the near-wall regions where viscous effects dominate the effects

of turbulence. Instead, wall functions are used in cells adjacent to walls.

Adjacent to a wall the non-dimensional wall parallel velocity is obtained from

vyuyu (2-38)

vyyEyu ln1

(2-39)

Where:

v

uyy

u

uu

2141

kCu

4.0k 0.9E for smooth walls

Here yv+ is the viscous sublayer thickness obtained from the

intersection of equation (2-38) and equation (2-39). The production and

dissipation terms appearing in the turbulent kinetic energy transport equation

are computed for near wall cells using:

y

u

u

uu

y

uu

y

uP w

2 (2-40)

1.31.01.921.440.09 1.31.01.921.440.09

C 1C2C k

37

y

kCuuu

y

uu23

(2-41)

Similarly for heat transfer if we define a non-dimensional temperature,

w

pw

q

ucTTT

(2-42)

Then the profiles of temperature near a wall are expressed as :

TyyuT (2-43)

Tt yyuT P (2-44)

Where P+ is a function of the laminar and turbulent Prandtl numbers (σ and σt)

given by Launder and Spaulding as:

4121

14sin

4

t

tk

AP (2-45)

Here yT+ is the thermal sublayer thickness obtained from the intersection of

equation (2-43) and equation (2-44). Once T+ has been obtained, its value can

be used to compute the wall heat flux if the wall temperature is known, or to

compute the wall temperature if the wall heat flux is known.

2. 2. 6 Solidification

The present numerical model reflects the solidification of liquid and

the melting of solid by varying the viscosity based on a melting temperature of

Tm=2327K. Each cell is assigned the viscosity, thermal conductivity, heat

38

capacity, and radiation properties of the melt for a temperature higher than Tm,

and a viscosity of 20 (kg/m∙sec) and the properties of sapphire crystal at

temperatures below Tm. This method does not take into account latent heat, but

as the model calculates an equilibrium state between the melt and the crystal

for given conditions, the latent heat does not enter into the equations, allowing

for more efficient calculation of the equilibrium state. Moreover, sapphire

single crystal growth rate is very low. The ratio of (latent heat)/(conduction heat

released through the crystal) is 0.05 when the growth rate is 1mm/h. This result

shows that the latent heat is negligible, even disregarding radiation heat release

through the crystal.

Moreover, the model described above allows for stable calculations as

no adjustment of the grid is required to reflect changes in the crystal growth

interface. [20]

2. 2. 7 Boundary conditions

The general treatment of boundary conditions in the finite-volume

equations is discussed in this section. A control cell adjacent to the west

boundary of the calculation domain is shown in Fig. 2-3.

39

Fig. 2-3. Computational boundary cell.

A fictitious boundary node labeled B is shown. The finite-volume equation for

node P will be constructed as:

Saaaa SSNNEEpp (2-46)

Coefficient wa is set to zero after the links to the boundary node are

incorporated into the source term S in its linearized form,

ppu SSS (2-47)

All boundary conditions using this numerical simulation are implemented in

this way. [66]

Fixed value boundary condition

If the boundary value is fixed as ϕB, the source term is modified as:

40

BWUU aSS (2-48)

WPP aSS (2-49)

Zero-flux boundary conditions

At zero-flux boundaries, such as adiabatic walls for heat and

symmetric boundaries for any scalar variables, the boundary link coefficients

are simply set to zero without modifying source terms. [66]

41

2. 3 Physical properties used in numerical model

description value(unit)

Melt viscosity 0.057 (𝑃𝑎 ∙ 𝑠) [67]

Melt density 3030(𝐾𝑔 𝑚3)⁄ (at 2327K) [67]

Melt thermal conductivity 2.02(𝑊 𝑚 ∙ 𝐾)⁄ (at 2027K) [67]

Melt heat capacity 1260(𝐽 𝐾𝑔 ∙ 𝐾)⁄ [67]

Melt emissivity 0.33 [33]

Crystal density 3970(𝐾𝑔 𝑚3)⁄ (at 2327K) [68]

Crystal thermal conductivity 5.6(𝑊 𝑚 ∙ 𝐾)⁄ (at 1700K) [68]

Crystal heat capacity 1313(𝐽 𝐾𝑔 ∙ 𝐾)⁄ (at 1700K) [68]

Crystal emissivity 0.869 [33]

Solubility of melt 7.8 × 10−10(𝑘𝑚𝑜𝑙 𝑚3)⁄

(estimated)

Solubility of crystal 7.8 × 10−11(𝑘𝑚𝑜𝑙 𝑚3)⁄

(estimated)

Diffusivity of oxygen in melt 1.5 × 10−9(𝑚2 𝑠⁄ ) [69]

Table 2-1. Physical properties used in numerical model.

42

Chapter 3. Induction heated CZ system

3. 1 Numerical modeling

3. 1. 1 Global modeling

In the present study, we developed a 2D axis-symmetric global

numerical model of a 4 inch boule. Calculations were performed using CFD-

ACE+ software. As the sapphire single crystal growth process is a slow one,

calculations were performed assuming steady-state conditions. [20, 70]

Argon and nitrogen mixed gas keep flowing in chamber during overall

growth process. We set inlet and outlet pressure to 1200 Pa. Temperature of

inlet gas is 298K and inlet gas velocity is 0.42 cm/s. Chamber wall boundary

condition is set to isothermal condition (298K).

Fig. 3-2. and 3-3. shows temperature, flow and induction heating

result. There is a dramatic difference in temperature inside and outside the hot-

zone, creating a strong argon gas flow outside the hot-zone. Through induction

heating, iridium crucible side is mainly heated.

43

Fig. 3-1. Configurations of the induction heated CZ system and meshes.

44

Fig. 3-2. Temperature and flow results of global model.

45

Fig. 3-3. Induction heating result of global model.

46

3. 1. 2 Solute concentration modeling

Geometry and boundary conditions

Based on the solute segregation theory, in the present study we created

a solute concentration model which predicts bubble location and is applicable

at the CF with the CZ method.

Fig. 3-4. shows the structure of the solute concentration model. Since

bubble formation is a phenomenon which occurs within the melt, the model

geometry was created based on a CF shape obtained through calculated and

experimentally measures results, and the bottom, side wall, and free surface are

given temperature conditions taken from the global model. The solute

segregation phenomenon caused by solubility differences between the crystal

and melt during the crystal growth process is simulated in the model by

applying oxygen flux at the CF. (as the equipment for experimentation and

modeling in the present study does not contain materials with carbon content,

calculations are performed for oxygen bubbles, rather than carbon monoxide

bubbles.)

47

Fig. 3-4. Geometries and meshes of the solute concentration model. (a) Curved

CF model and (b) flat CF model.

48

Calculation of oxygen concentration in melt and oxygen flux at CF

In order to calculate oxygen flux at the CF, it is necessary to determine

the melt and crystal oxygen solubility ( LC , SC ) and segregation coefficient

( LS CCk ). However, as there are no known values for LC or SC , the

values are calculated using measured results of bubble amounts trapped in

crystals grown by the EFG method. Regardless of growth conditions, the

amount of bubbles trapped in crystals grown by the EFG method is fixed at

7.0∙10-10kmol/m3. [51] When melt supersaturated with oxygen moves to the free

surface, it forms an equilibrium with the atmosphere and has a value of LC .

However, the free surface is very narrow and the melt flow is weak for the EFG

method, so that oxygen out-diffusion through the free surface is negligible.

Therefore, it can be assumed that all bubbles trapped in the crystal are due to

the difference between LC and SC . This is shown in the equation below:

LS

SL

CkC

CC

10107

A value of 0.1 is taken from a separate study for the segregation coefficient. [48]

49

311

31010

/10778.7

/10778.79.0

107

mkmolC

mkmolC

S

L

Calculating oxygen flux at the CF from the result above gives

smkmol

A

VkC

vAVrA

L 216

2

/1095.11

CFat flux oxygen

,

A = cross-section area of crystal, V = volume of growing crystal per second

(m3/s), v =growth rate (1mm/h)

In most cases in CZ, unlike in EFG, the CF shape is not flat, and

instead demonstrates a curved shape which extends into the melt. (Fig. 3-4. a.)

As the crystal growth rate over unit area differs depending on the CF slope, the

CF must be divided into multiple sectors depending on the slope and designated

an appropriate corresponding oxygen flux. Differences in oxygen flux for each

section are shown in Table 3-1. As shown in Fig. 3-4. b. , a flat CF due to the

use of additional heater negates the flux value differences for each section.

50

Initial conditions for the solute concentration model

The initial conditions for the solute concentration model are taken

from steady state results of calculations taking into account only flow and heat,

without oxygen concentration. The initial oxygen concentration of the melt is

set as the saturated concentration ( LC ). [51] Oxygen flux at the CF

continuously and gradually raises the melt oxygen concentration, and as

supersaturated melt reaches equilibrium at the free surface with the saturated

concentration, the boundary condition for the free surface is set as the LC . In

the present study we performed both steady state and transient calculations for

curved and flat CF geometries, with transient calculations performed for a

period of 24 hours following the initial conditions.

51

CF_# Flux (10-16

kmol/m2s)

CF_1 0.612

CF_2 0.769

CF_3 0.952

CF_4 1.190

CF_5 1.291

CF_6 1.393

CF_7 1.517

CF_8 1.633

CF_9 1.735

CF_10 1.835

CF_11 1.911

CF_12 1.935

CF_13 1.943

Table 3-1. Oxygen flux at CF segment in curved CF model.

52

3. 2 Global model results and analysis

3. 2. 1 Model verification through comparison with experimental

results

In the present study we choose to compare the CF shape for calculated

and experimentally measured results in order to evaluate the model's

effectiveness. This method is relatively costly, but it is judged to be the most

appropriate in evaluating the model, as it provides direct confirmation of

whether the CF shape, which is of primary importance in this study, is properly

reflected in the model. To verify the accuracy of the model, we performed

comparisons for two stages, the shouldering stage and the body growth stage.

Calculated results for both the shouldering stage and body growth stage are in

very good agreement with experimental results. (Fig. 3-5.) This demonstrates

that the model used in the present study accurately reflects the actual system.

53

Fig. 3-5. Comparison of the CF shape for calculated and experimentally

measured results. (a) Shouldering stage and (b) body growth stage.

54

Fig. 3-6. Measurement positions for thermal gradient.

55

3. 2. 2 Decreased temperature gradient at CF by use of additional

heater

Sapphire growth in the c-axis direction results in a high dislocation

density within the crystal, which is a leading cause of diminished LED

efficiency. Therefore, the production of higher quality LEDs requires achieving

lower dislocation density in the sapphire crystal. There are many causes of

dislocation formation during sapphire single crystal growth, but one of the most

important is dislocation formation through thermal stress at the CF. Sapphire

has a very narrow plastic zone, so that apart from dislocations originating in the

seed, the ones formed during crystal growth are mostly caused by thermal stress

in a narrow range near the CF directly after crystallization. [71] Therefore, in

order to decrease dislocation density within the crystal, it is important to

decrease thermal stress at the CF. The present study explores a way to decrease

thermal stress at the CF by employing an additional heater.

In most cases, sapphire growth by the CZ method involves a

maintained form of the CF extending below the melt as in Fig. 3-5. b.

Calculating the temperature gradient depending on location at the CF results in

values of 10~32K/cm. (Fig. 3-7.) Compared to Kyropoulos method sapphire

growth, which results in a CF temperature gradient of 0~5K/cm [43] , these are

very high values. Additionally, existing growth conditions using the CZ method

results in high values not only for the absolute values for the temperature

gradient, but also for the deviation of temperature gradient value according to

radial direction location at the CF. In short, CZ method growth conditions, in

56

comparison with Kyropoulos method, results in higher thermal gradient

absolute values and deviation according to location, so that a greater thermal

stress is unavoidable. The greater temperature gradient for the CZ method can

be attributed to the larger hot-zone size and a structure which results in a greater

temperature variation in the vertical direction. Therefore, the production of

higher quality crystals will require an improved control of hot-zone temperature

distribution to decrease thermal gradient values at the CF as well as minimize

deviation according to location.

57

Fig. 3-7. Thermal gradient variation with radial position at CF.

58

The most important factor to influence temperature gradients at the CF

during sapphire growth is the melt flow. Because sapphire has a relatively high

prandtl number, convection has a major impact on the temperature distribution

of melt. The prandtl number can be defined as follows:

k

Cv P

ratediffusion thermal

ratediffusion viscousPr

A higher prandtl number indicates greater heat transfer by convection

than by conduction within a fluid. Sapphire melt has a prandtl number of 35,

much higher than values of 7 for water, or 0.009 for silicon melt. [20, 72]

Therefore, in order to change the temperature gradient at the CF, the melt flow

must first be changed.

Generation of heat during induction heating mostly takes place at the

crucible side, as shown by hatched line in Fig. 3-8. a. In the presence of

induction heating only, heat is released from the side to create natural

convection as shown in Fig. 3-8. b. With melt convection formed as in Fig. 3-

8. b, melt heated to a high temperature flows from the crystal periphery to the

center, and is slowly cooled, so that a relatively higher temperature gradient is

found at the crystal periphery. Fig. 3-7. graphically shows differences in the

normal temperature gradient depending on location at the CF for the cases of

induction heating only and the application of an additional heater. By altering

the hot-zone temperature distribution and melt convection direction by use of

59

the additional heater, the temperature gradient absolute value is lowered and the

deviation depending on location is also greatly reduced.

Analyzing changes in the thermal gradient, keeping in mind the crystal

structure and slip system of sapphire, reveals a closer look into the effects of

the additional heater. Sapphire has a hexagonal crystal structure, with a basal

slip system in the c-plane and prismatic slip system in the a, m-plane. (Fig. 3-

9.) CZ systems, which grow the crystal in the C-axis, involves thermal stress

(dT/dx) in the direction of growth acting on the prismatic slip system, and

thermal stress (dT/dy) perpendicular to the growth direction acting on the basal

slip system. Using an additional heater makes the CF flat and the value for

dT/dy approaches 0, but employing only induction heating results in a larger

value for all regions excluding the crystal center. (Fig. 3-10.) dT/dx for the case

of induction heating is lower at the crystal center compared to when the

additional heater is used, but the value drastically increases at the outer parts of

the crystal. (Fig. 3-11.) Larger values of dT/dx and dT/dy can lead to greater

formation of dislocations, the effects of which combine to form the overall

dislocation density in the crystal. For the case of induction heating only, values

of dT/dx, and dT/dy at the crystal center are low, but the dT/dy value increases

at the location 0.015, and the dT/dx value increases at the locations 0.03, 0.045.

In short, it can be assumed that high dislocation densities would be formed at

any region apart from the crystal center (0.0).

Fig. 3-12. shows the dislocation density of a crystal grown using only

induction heating. The brighter regions represent higher dislocation density,

and the figure illustrates that the dislocation density is higher at the crystal’s

60

outer regions. This dislocation density distribution demonstrates similar

tendencies to results from our calculations for induction-only heating.

Employing an additional heater brings the dT/dy value close to 0, and the dT/dx

value within a range of 5~13 (K/cm), which on average is lower than for the

case of induction heating only.

Fig. 3-13. shows the temperature distribution inside the crystal

depending on crystal growth direction. This distribution is similar to the thermal

history which the crystal undergoes during growth. For the case of induction

heating, the crystal center remains in relatively high temperature for an

extended period. Sapphire undergoes annealing at temperatures above 2000K

in order to lower the dislocation density and residual stress. Longer annealing

times increase the effect. [73] Based on this, we can predict that for the case of

induction heating only, the annealing effect will be greatest at the crystal center.

However, temperature distribution in the crystal during growth can change

depending on the hot-zone design, and the period during which the crystal

remains at over 2000K depends on the pulling rate. Moreover, the thermal

gradient value at the CF is higher in the crystal outer region when only

induction heating is used, compared to when an additional heater is employed,

and as the thermal history of the crystal during growth is similar to the

additional heater, a relatively high dislocation density is developed.

Therefore, the use of additional heater lowers thermal stress in the

plastic zone, and crystal growth under such conditions can be predicted to result

in lower dislocation density than before.

61

Fig. 3-8. (a) Hatched line indicates a heating region by induction heating; dotted

line indicates additional heater position. (b) Melt flow pattern and CF shape in

the presence of induction heating only. (c) Melt flow pattern and CF shape in

the presence of induction heating and additional heating.

62

Fig. 3-9. Slip systems in sapphire. [74, 75]

63

Fig. 3-10. Thermal gradient that normal to growth direction at CF.

Fig. 3-11. Thermal gradient that parallel to growth direction at CF.

64

Fig. 3-12. Etch pit dislocation density (EPD) for vertical and horizontal section

of crystal grown by induction heating only.

65

Fig. 3-13. Temperature distribution according to growth direction in crystal.

66

3. 2. 3 Decreased CF convexity by use of additional heater

Much like the thermal gradient at the CF, the flatness of the CF

geometry during crystal growth is also of great importance in determining the

crystal quality. In short, a flatter CF shape results in improved crystal quality.

[76, 77] To evaluate the flatness, we implement the concept of convexity as

outlined by Chen et al. [76] The convexity refers to the difference in height

between the highest and lowest points of the crystal which is growing within

the melt.

)min()max( hh ZZD

Several studies in the past have mentioned raising the crystal rotation

rate in order to alter the melt flow direction or the CF shape.[56, 61, 78, 79]

Generally, an increase crystal rotation rate results in a stronger forced

convection. The centrifugal force of forced convection forms flow from the

crystal center to the outer regions, which results in decreased convexity.

In the present study, we employ an additional heater instead of forced

convection to create a reverse direction natural convection and thereby alter the

CF shape and temperature distribution. The advantage of this method is that

convexity can be decreased even at lower crystal rotation rates so that more

stable growth is made possible, which is predicted to be of even greater

advantage in the future when the diameter of the grown crystal is increased.

67

The additional heater was simulated in the numerical model by applying heat

flux at the bottom of crucible, and the results of the simulation are shown in

Fig. 3-8. As mentioned earlier, sapphire melt demonstrates strong heat transfer

by convection, making a change in the convection imperative to alter the CF

shape. The heat provided by the additional heater creates a force which lifts the

melt from the bottom of crucible to the crystal, resulting in a natural convection

which opposite that which was observed before, and thereby creating a flat CF

shape.

To compare against the effects of the additional heater, we calculated

the convexity change depending on coil position. Lowering the coil below the

standard position (0mm) by 40mm results in a decreased convexity by

approximately 8mm. In contrast, the use of an additional heater decreases

convexity by approximately 85mm, so the effect of altering the coil position

can be said to relatively small.

Coil position -40mm +40mm 0mm 0mm + additional heater

Convexity 81mm 118mm 89mm 4mm

Table 3-2. Convexities according to coil position.

68

3. 3 Solute concentration model results and analysis

3. 3. 1 Verification of solute concentration model by comparison with

experimental results

Fig. 3-14. shows the flow results of the solute concentration model as

well as the overall oxygen concentration distribution in the melt. Comparing

Fig. 3-14. a. and Fig. 3-14. b. , the solute pile-up location is governed by the

melt convection direction, and melt which has a decreased oxygen

concentration from passing the free surface is found to regain a higher oxygen

concentration near the CF.

69

Fig. 3-14. Melt flow and oxygen concentration distribution in solute

concentration model: (a) Curved CF model and (b) flat CF model.

70

Of the total concentration distribution, the concentration of the CF part is shown

graphically in Fig. 3-15. The solid line shows results for the curved CF and the

dashed line corresponds with results for the flat CF. Steady state results are

shown in red and blue, respectively, and transient results are given in 1 hour

increments from hour 1 to hour 24. The oxygen concentration for the melt is

given an initial condition value of LC and as calculation progresses, oxygen

flux at the CF continuously raises the oxygen concentration. For the curved CF

model, the melt flow is directed from the free surface to the crystal so that melt

with a concentration lowered to LC at the free surface passes by the CF. As a

result, the oxygen concentration is lower near the free surface but rises with

greater proximity to the crystal center due to pile-up phenomena, and the region

of highest oxygen concentration is formed near the crystal center. This trend is

also found in transient results.

Interpreting curved CF result based on the solute segregation theory,

bubbles are formed at the crystal center region and trapped in the crystal. Fig.

3-16. shows photographs of a crystal grown in conditions identical to curved

CF model as well bubbles within the crystal. Experimental results also reveal

bubbles having formed and been trapped near the crystal center. We can

therefore conclude that the numerical model based on the solute segregation

theory can explain experimental results for the CZ method as well.

In curved CF steady results, the degree of oxygen supersaturation at

the crystal center region is 3.7%. Generally, this is a low level for homogeneous

nucleation to occur. However, earlier research has shown that bubbles form by

71

heterogeneous nucleation, in which case the nucleation site’s shape factor

determines the nucleation free energy. For the case of a curved CF shape, it is

likely that a cellular structure will form in the crystal, as in Fig. 3-17. The

formation of a cellular structure results in a relatively high nucleation site shape

factor value, which decreases the bubble’s nucleation free energy. Hence,

bubble nucleation can occur regardless of a low degree of supersaturation.

72

Fig. 3-15. Oxygen concentration distribution at CF.

73

Fig. 3-16. A crystal grown in conditions identical to curved CF model and

bubbles therein.

74

Fig. 3-17. Cellular structure of sapphire. [51]

75

Fig. 3-18. A ratio of heterogeneous nucleation energy to homogeneous

nucleation energy depend on cosθ.

76

3. 3. 2 Calculation of likelihood of bubble movement in CZ growth

While the solute concentration model results are in good agreement

with experimental findings, it is necessary to consider the possibility that the

bubbles trapped near the center of the crystal were actually formed elsewhere

and transported to their final location by the melt flow. Previous solute

segregation models have focused on bubble formation in the EFG method.

Unlike the CZ method, the EFG method does not involve crystal or

crucible rotation, and the primary causes for the flow are capillarity and the

Marangoni effect at the melt-crystal-gas triple point, and as a result the flow is

relatively much weaker. Therefore, it is very unlikely in the EFG method that

the melt flow might move the bubbles.

However in the CZ method, the natural convection is stronger and the

crystal and/or crucible are rotated, so the melt flow velocities are much higher

than for the EFG method, and could therefore move the bubbles to a point at

which they may be entrapped in the crystal center region.

Bubble nucleation within the melt mostly likely occurs

heterogeneously, as this involves a relatively lower activation energy. Therefore,

bubble may nucleate at the CF or crucible surface and undergo growth.

Bubbles in the melt are subjected to three primary forces: Buoyancy

force ( BF ), drag force ( DF ) caused by the melt, and in the case of crucible

and/or crystal rotation, centrifugal force ( CF ). These are defined by the

following equations:

77

22

2

21

2

1

)(

VrmrF

ACvF

gVF

C

DD

B

Where V is the bubble volume, ,1 is the melt density, 2 is the gas

density, g is gravity, v is the velocity of the bubble relative to the melt, DC

is the drag coefficient(0.47), A is the bubble cross section area, r is the

distance from the rotational axis, and is the melt angular velocity.

If a bubble is formed at a location not near the crystal center, it is

subjected to forces as shown in Fig. 3-19. For the bubble to be transported to

the crystal center region, the sum of the forces must be in a direction similar to

the melt flow. Since the bubble volume and melt flow velocity are important

variables in calculating the sum of the three forces, the sum of FB and FD is

calculated using these two as variables. (Fig. 3-20. , 3-21.) FC is not included

in the calculation because under the given experimental condition of crystal

rotation at 0.2rpm, it is negligibly small at 10-4~10-5 times the value of the other

two forces.

Calculations were performed for the variables in ranges of bubble

radius 0~800μm and melt velocity 0~0.08m/s. The vertical axis of Fig. 3-20.

marked FT is defined as FD-FB and it is shown that a larger bubble radius makes

FB more dominant. What this indicates is that the drag force exerted by the melt

78

cannot move the bubble to the crystal center region.

Fig. 3-21. is a magnified view of the region in Fig. 3-20. for smaller

bubble radii, and it helps illustrate the impact of stronger drag force. To better

observe the sections for which FT has a positive value, the sections in which the

value is negative are instead all designated a value of 0. Results show that the

bubble radius must be below 50μm and the melt velocity higher than 0.03m/s

in order for the drag force to exceed the buoyancy force. However, considering

that flow velocity near the CF is below 0.005m/s, as well as the bubble radius,

the flow velocity near a bubble growing at the CF is expected to be much lower.

Moreover, since the FD direction changes depending on the CF slope,

it becomes even less likely that the sum of the forces will point toward the

crystal center region. As a result, these two figures show that even in the

presence of crystal rotation in CZ growth, (and even if the rotation rate is high

at above 5rpm) it is unlikely that a bubble would be transported by the melt

flow.

79

Fig. 3-19. Forces acting on bubble.

80

Fig. 3-20. FT change according to melt flow velocity and bubble radius.

81

Fig. 3-21. Magnified view of the top region in Fig. 9 illustrating the impact of

stronger drag force.

82

3. 3. 3 Calculation of bubble entrapment location in crystal growth

using additional heater

As the solute concentration model (curved CF model) can explain

experimental results, we applied it to calculate bubble formation locations in

crystal growth using an additional heater (flat CF model). Crystal growth with

an additional heater involves a melt convection direction opposite that of

previous CZ growth, and its CF shape is also flat. As such, a corresponding

model is shown in Fig. 3-4. b., and the boundary conditions are taken from

temperature results obtained from the global model.

The use of additional heater results in melt convection directed from

the crystal center region to the free surface, so that solute pile-up occurs at the

crystal periphery. Bubble nucleation occurs when the oxygen concentration

exceeds a certain value, and the bubbles are trapped in the crystal after growth.

Bubble growth process was not calculated in the present study, but it was

possible to predict the degree of bubble entrapment by analyzing the rate at

which oxygen concentration increases in the region of solute pile-up.

Fig. 3-22. shows the rate of oxygen concentration increase over time

at the region of solute pile up (curved CF : at radius 0.5mm, flat CF : 48.51mm)

from the transient results in Fig. 3-15. For example, when the maximum

oxygen concentration at the CF is 8.038∙10-10(kmol/m3), the maximum oxygen

concentration increases at a rate of 7.2∙10-13(kmol/m3hr) for the curved CF,

and 4.1∙10-13(kmol/m3hr) for the flat CF. A trend is observed in which a higher

oxygen concentration brings out-diffusion at the free surface and flux at the CF

83

closer to equilibrium, so that the pile-up rate is decreased. Also, for a given

oxygen concentration at a pile-up location, the concentration increase rate is

lower for the flat CF than for the curved CF. The pile-up region for the curved

CF is at the crystal center and therefore its area is very small, whereas the pile-

up region for the flat CF is at the crystal periphery so that its area is much wider.

Consequently, as the oxygen concentration must increase in a much wider

region for the flat CF, the rate of concentration increase is naturally lower, given

a certain oxygen concentration. A lower rate of concentration increase indicates

a smaller bubble size or a lower frequency of its entrapment. Hence, if

additional heaters are used to change the melt convection direction and flatten

the CF shape, the intervals between bubbles are lengthened or the bubble size

is decreased compared to a curved CF. In addition, the location of bubble is

moved from the crystal center to the crystal periphery.

Bubbles trapped at the outer edges of the crystal can be eliminated

through polishing or coring and therefore this is expected to be of great

importance in improving crystal quality.

84

Fig. 3-22. Concentration increase rate at solute pile-up region according to

concentration at solute pile-up region.

85

3. 4 Summary

In this chapter, we developed a numerical model of an induction-

heated CZ method sapphire growth system, and verified the accuracy of the

model through comparison with experimentally obtained results. Moreover, we

calculated changes in the CF shape corresponding with the use of an additional

heater. Changes in the CF shape with the use of an additional heater were found

through changes in the melt flow direction and hot-zone temperature

distribution, and in comparison with previous crystal growth methods, this was

found to result in lower absolute values for the thermal gradient at the CF as

well as smaller deviations according to location.

Regarding the formation of bubbles, a major defect in sapphire growth,

we developed a solute concentration model through the solute segregation

theory mostly researched in relation to the EFG method. The model is found to

be applicable also to the CZ method despite very different CF shapes and melt

velocities. The numerical model results were found to be in good agreement

with experimental results in terms of the locations of bubble entrapment. The

model was used to predict that under growth conditions involving an additional

heater, bubbles would be trapped at the crystal peripheral edges. This is

expected to be of great value in improving crystal quality. In conclusion, the

advantages of an additional heater in crystal growth can be summarized as

follows:

- Decreased thermal gradient (for both absolute value and deviations between

different locations)

86

- Decreased convexity (flat CF shape)

- Decreased energy needed for crystal growth (Numerically, approximately

10%)

- Bubble formation location moved to outer edges of crystal

87

Chapter 4. Resistance heated CZ system

4. 1 Numerical modeling

We developed a 2D axis-symmetric global numerical model of a 4

inch boule. Calculations were performed using CFD-ACE+ software. As the

sapphire single crystal growth process is a slow one, calculations were

performed assuming steady-state conditions. [20, 70]

Argon gas keep flowing in chamber during overall growth process.

We set inlet and outlet pressure to 1200 Pa. Temperature of inlet gas is 298K

and inlet gas velocity is 0.1 m/s. Chamber wall boundary condition is set to

external heat release condition. Ambient temperature is 298K and external heat

transfer coefficient is 8.36 W/m2K. In this model, we calculate cooling water

together to check safety of equipment.

88

Fig. 4-1. Configurations of the resistance heated CZ system and meshes.

89

Fig. 4-2. Temperature and flow in resistance heated CZ system.

90

Fig. 4-3. Cooling water flow rate and outlet water temperature.

91

4. 2 Results and analysis

4. 2. 1 Change in convexity with crystal/crucible rotation

In the present study, we show rotation in the same direction for the

crystal and crucible as 'n crucible rotation rate x crystal rotation rate,' and

rotation in opposite directions for the crystal and crucible as 'c crucible rotation

rate x crystal rotation rate.'

Fig. 4-4. and 4-5. show the variation in crystal growth interface shape

for crystal and crucible rotation in identical and opposite directions,

respectively. As seen in Fig. 4-4, the effect of crystal rotation appears

differently depending on the presence of crucible rotation when the crucible

and crystal are rotated in the same direction. Fig. 4-6. graphically illustrates the

convexity based on the results shown in Fig. 4-4.

In the case of n0, for which there is no crucible rotation, higher

rotation rates for the crystal do not result in a major variation of the convexity.

However, in the presence of crucible rotation as in n5, n10, and n15, a higher

crystal rotation generally results in a greater convexity.

92

Fig. 4-4. Change in the crystal growth interface shape with crucible and crystal

rotation in the same direction.

93

Fig. 4-5. Change in the crystal growth interface shape with crucible and crystal

rotation in opposite directions.

94

Fig. 4-6. Convexity changes according to crystal rotation and crucible rotation.

95

These trends arise because the forced convection caused by the

rotation of the crucible and crystal have a large influence on the CF shape.

Because sapphire has a relatively high prandtl number, convection in the melt

has a major impact on the CF shape.

Therefore, heat transfer within sapphire melt is dominated by

convection, and the CF shape is strongly influenced by the melt convection.

Hence, the CF shape in a sapphire melt growth system may be explained in

terms of the melt convection.

Convexity changes :

higher crucible rotation → lower convexity (no crystal rotation)

When there is no crystal rotation, a higher crucible rotation results in

a lower convexity value. This phenomena can be explained in terms of the melt

flow as shown in Fig. 4-7. To emphasize the influence of crucible rotation, we

compare the cases of crucible rotation at 0rpm and 15rpm.

The color scale in Fig. 4-7. represents the y direction velocity within

the melt. This particular parameter is shown because the main heating region is

at the crucible side wall, and when only the crucible is rotated, the y direction

flow of high temperature heated melt from the crucible wall towards the crystal

has a major impact on the CF shape.

Only natural convection occurs for the case of 0x0rpm, whereas

rotational flow in the melt caused by crucible rotation becomes distinctly

96

apparent for the case of n15x0rpm.

Here it is worth noting the y direction velocity at the free surface

region and below the CF. Flow velocities in these regions are much higher for

the case of n15x0rpm than for 0x0rpm, revealing a decreased convexity as hot

melt from the crucible side wall flows rapidly towards the crystal. Such flow

patterns are observed to arise from crucible rotation because the melt near the

crucible floor takes on rotation momentum and a centrifugal force, but there is

no corresponding rotation momentum acting on the free surface region, so that

natural convection and convection by centrifugal force are combined, leading

to a high y direction flow velocity at the free surface and below the CF.

97

Fig. 4-7. Comparison of y axis direction velocity for 0x0rpm and n15x0rpm.

98

Convexity changes :

higher crystal rotation → higher convexity (with n crucible rotation)

The increased convexity which arises when crystal rotation is added

to crucible rotation can also be analyzed in terms of the melt flow. The color

scale in Fig. 4-8. indicates the y direction velocity for the cases of n15x0rpm

and n15x15rpm.

When the crystal is rotated, rotation momentum is transferred to the

melt below the CF, leading to a centrifugal force. This force acts in the +y

direction, opposite the –y direction flow formed at the free surface due to the

crucible rotation. Therefore, flow towards the crystal caused by the crucible

rotation and flow towards the crucible side wall caused by the crystal rotation

meet below the crystal and the y direction flow in this region becomes close to

0.

As a result, the CF shape becomes elongated downward, leading to a

higher convexity value. This phenomenon becomes increasingly strong as the

crystal rotation rate is increased as 5, 10, and 15rpm, leading to greater

convexity values with higher crystal rotation rates.

99

Fig. 4-8. Comparison of y axis direction velocity for n15x0rpm and n15x15rpm.

100

The convexity values for various conditions when the crucible and

crystal are rotated in opposite directions are shown in Fig. 4-9. Unlike the case

of corotation, there is little variation in the convexity with crystal rotation rate.

Nonetheless, the trend of lower convexity values with increased crucible

rotation rates is observed regardless of the crystal rotation conditions, the

explanation for which has been given above.

Convexity changes :

higher crystal rotation → higher convexity (with c crucible rotation)

That there is a slight change in convexity despite greater crystal

rotation rates when the crucible and crystal are rotated in opposite directions

can be explained through Fig. 4-10. and 4-11. Fig. 4-10. shows the z-direction

velocity which arises from crucible and crystal rotation.

For n15x15rpm, the region below the crystal also takes on rotational

momentum, as the crucible and crystal are moving in the same direction.

Therefore, the y direction flow decreases below the crystal growth interface as

outlined earlier, and the convexity increases.

In contrast, for the case of c15x15rpm, the rotational momentum by

crystal and crucible rotation cancel each other out in the melt beneath the CF,

since the crystal and crucible are rotating in opposite directions. Therefore, the

melt does not take on a centrifugal force, and the –y direction flow created near

the free surface by the crucible rotation is maintained below the CF, and the

101

convexity is decreased.

Fig. 4-9. Convexity changes according to crystal rotation and crucible rotation.

102

Fig. 4-10. Comparison of z axis direction velocity for n15x15rpm and counter

rotation c15x15rpm.

Fig. 4-11. Comparison of y axis direction velocity for n15x15rpm and counter

rotation c15x15rpm.

103

4. 2. 2 Variation in thermal gradient at the CF with crystal and

crucible rotation conditions

In addition to the convexity, yet another factor which plays a major

role in determining the crystal quality is the thermal gradient at the CF. Sapphire

has a narrow plasticity zone, so that dislocations and residual stresses are

formed in the region of the CF. As this is governed by the thermal gradient

perpendicular to the CF, in this study we calculate the thermal gradient for a

thickness of 1mm perpendicular the CF, at the crystal center (0mm) and at radial

locations 15mm, 30mm, and 45mm from the center, for each of the rotational

conditions. While it has been reported that the plasticity zone for sapphire in

melt growth is 0.5mm [80] , in the present study we measure the thermal

gradient for a thickness of 1 mm because of the cell size.

A number of prevailing tendencies arise in common for the thermal

gradients in all of the 25 cases calculated in the present study. Some of the

results are shown in Fig. 4-12.

104

105

Fig. 4-12. Thermal gradient variation with radial location for (a) n0 (b) n10 (c)

c10.

106

Firstly, for crystal rotation rates up to 10rpm, the largest thermal

gradient arises at 0.015m. However, this tendency disappears for a crystal

rotation rate of 15rpm. For crystal rotation at 15rpm, the thermal gradient at

0.015m decreases to values similar to those for 0.0m and 0.03m, and lesser

variations of the thermal gradient value with radial location are observed.

Secondly, when the crystal is rotated at 15rpm, the variation between

values of thermal gradient for each of the 4 radial locations decreases compared

to other conditions, and the absolute value itself also decreases. The absolute

value of the thermal gradient has a direct impact on stress imposed on the

crystal, but variation in the value with radial location can also have a significant

impact on the formation of dislocations. This is because variations in the

thermal gradient with radial location results in varying thermal expansion

between radial locations, leading to shear stress within the crystal.

Finally, higher rotation rates for the crucible result in higher absolute

values for the thermal gradient and greater variations from location to location.

As shown in Fig. 4-7, rapid movement of high temperature melt heated at the

crucible wall towards the crystal causes a larger thermal gradient at the crystal

growth interface.

There is not yet a definite maximum value of the thermal gradient at

the CF known to exist for sapphire melt growth. However, it is possible to

predict that smaller local variations in the absolute value of the thermal gradient

will result in a lower frequency of dislocation formation. Hence, Fig. 4-13.

shows that thermal gradient comparison of no rotation condition (n0x0) with

107

lowest thermal gradient condition (c10x15). In comparison with no rotation

condition, c10x15 condition was found to result in lower absolute values for

the thermal gradient at the CF as well as smaller deviations according to

location. (Fig. 4-13.) Analysis results of thermal gradient along the slip system

direction also reveals similar tendencies as dT/dn. (Fig. 4-14. , Fig. 4-15.)

108

Fig. 4-13. Thermal gradient comparison of no rotation condition with lowest

thermal gradient variation condition.

109

Fig. 4-14. Thermal gradient that normal to growth direction at CF.

Fig. 4-15. Thermal gradient that parallel to growth direction at CF.

110

4. 3 Summary

The CF shape has a major influence on the quality of the final product

in sapphire melt growth. As the CF shape is governed by flow within the melt,

the crucible and crystal rotation which is dealt with in the present study may

help to determine the final crystal quality. In the past research on CZ crystal

growth had focused on induction heated systems, which are limited by an

inability to rotate the crucible. Through the model presented in this study, we

were able to successfully simulate and analyze the influence of both crystal and

crucible rotation on sapphire crystal growth by the CZ method.

Compared to a configuration with no crystal or crucible rotation,

rotating the crystal and crucible in the same direction results in a lower variation

of the thermal gradient depending on radial location, but this is accompanied

by undesirable convexity. In contrast, rotating the crystal and crucible in

opposite directions results in both a lower thermal gradient variation with radial

location, and improved convexity.

111

Chapter 5. Conclusions

In this study, we developed numerical models for an induction heated

CZ and resistance heated CZ system.

We used an additional heater to reduce dislocation density and bubble

defect in an induction heated CZ system. We calculated changes in the CF shape

corresponding with the use of an additional heater. Changes in the CF shape

with the use of an additional heater were found through changes in the melt

flow direction and hot-zone temperature distribution, and in comparison with

previous crystal growth methods, this was found to result in lower absolute

values for the thermal gradient at the CF as well as smaller deviations according

to location. Moreover, using additional heater, power consumption deceased

approximately 10%.

Regarding the formation of bubbles, a major defect in sapphire growth,

we developed a solute concentration model through the solute segregation

theory mostly researched in relation to the EFG method. The model is found to

be applicable also to the CZ method despite very different CF shapes and melt

velocities. The numerical model results can explain experimental results in

terms of the locations of bubble entrapment. The model was used to predict that

under growth conditions involving an additional heater, bubbles would be

trapped at the crystal peripheral edges. This is expected to be of great value in

improving crystal quality.

We calculated the influence of both crystal and crucible rotation to

reduce dislocation density in a resistance heated CZ system. Compared to a

112

configuration with no crystal or crucible rotation, rotating the crystal and

crucible in the same direction results in a lower variation of the thermal gradient

depending on radial location, but this is accompanied by undesirable convexity.

In contrast, rotating the crystal and crucible in opposite directions results in

both a lower thermal gradient variation with radial location, and improved

convexity. In conclusion, when the crucible rotate 10rpm and crystal counter-

rotate 15rpm, thermal gradient and convexity are decreased.

As shown in Fig. 5-1, thermal gradient and convexity are both higher

in the resistance heated CZ system. However, taking into account increased

wafer sizes in the future, the resistance heated CZ system is worthy of further

continued research. We believe that conditions for the growth of higher-quality

crystal growth may be found through thermal optimization of the hot-zone.

113

Fig. 5-1. Thermal gradient variations for induction heated CZ, resistance heated

CZ and Kyropoulos system.

114

References

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2013,December 5, Available from:

http://www.ledinside.com/outlook/2013/12/sapphire_substrate_to_do

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초 록

현재 LED기판용 사파이어 단결정은 대부분 Kyropoulos법

으로 양산이 이루어 지고 있으며 일부는 HEM법과 VHGF법을 통해

생산된다. 하지만 이 방법들은 결정을 a축으로 성장시키기 때문에

수율이 낮다는 한계점이 있다.

하지만 Czochralski (CZ) 법으로 사파이어를 c축 성장시키

면 코어링 과정에서 손실되는 부분이 적기 때문에 다른 성장 법 보

다 높은 수율을 얻을 수 있다.

CZ법으로 c축 성장시킨 사파이어 잉곳은 높은 수율을 가짐

에도 불구하고 결정의 품질이 떨어진다는 문제점 때문에 현재까지

양산 법으로 사용하지 못하고 있다.

LED기판용 사파이어 단결정의 품질은 크게 두 가지 요인에

영향을 받는다. 첫 번째는 전위 밀도이고 두 번째는 결정 내 기공이

다. 결정이 높은 전위 밀도를 가지면 GaN증착 과정에서 기판이 깨

지거나 LED의 광효율이 감소하는 문제가 있다. 결정 내 기공은 결

정의 광학적 특성과 기계적 물성을 저하시키는 원인이 되고 제품으

로서 가치를 떨어뜨린다.

따라서 본 연구에서는 사파이어 단결정 성장용 CZ 시스템

을 모사할 수 있는 수치모델을 개발하여 결정의 품질을 올릴 수 있

는 성장 방법을 연구했다.

125

유도가열 CZ 시스템에서 추가히터를 넣었을 때 결정 성장

계면에서의 온도기울기와 convexity의 감소를 계산했다. 결정 성장

계면의 변화는 추가히터를 통한 hot-zone의 온도분포 변화와 멜트

의 대류방향 변화에 기인한다. 추가히터는 결정 성장 계면에서의 온

도기울기 절댓값 감소뿐 아니라 위치에 따른 온도기울기 값의 편차

도 감소시킨다. 또한 에너지 소비도 줄일 수 있다.

또한 CZ 시스템에서 사파이어 성장 시 결정 내 기공 생성

위치를 계산하기 위해 용질 농도 계산 모델을 개발했으며 계산결과

와 실험결과를 비교했다. 용질 농도 계산 모델을 이용해 추가히터가

있는 성장 조건에서 기공이 생성될 위치를 계산했으며 계산 결과

기공이 결정의 외곽에 잡힐 것으로 예측됐다. 이는 결정의 품질 향

상에 긍정적인 영향을 미칠 것으로 본다.

저항가열 CZ 시스템에서 결정과 도가니의 동시회전이 결정

성장에 미치는 영향을 계산했다. 회전이 없는 경우와 비교해 결정과

도가니가 같은 방향으로 회전할 경우 결정 성장 계면에서 온도기울

기는 감소하지만 convexity가 증가하는 경향을 보였다. 반면 결정

과 도가니를 반대로 회전시킨 경우 결정 성장 계면에서의 온도기울

기와 convexity가 동시에 감소하는 효과가 있음을 확인했다.

주요 단어 : 사파이어, 단결정 성장, 수치해석, Czochralski법, 글로벌

126

모델

학 번 : 2010-20644