- Email: [email protected]
Influence of gas flow on thermal field and stress during growth of sapphire single crystal using Kyropoulos method
RARE METALS Vol. 2 5 , Spec. Issue, Dec 2006, p . 260
Muence of gas flow on thermal field and stress during growth of sapphire single crystal using Kyropoulos method U Jinquan” , SV Xiaoping” , NA M y h ’ ) , YANG Hail),U Jianmin” , W Ymqi2’, and MI Jimjm2’
.
1 ) Reijing Guojing Infrared Optical Technology Co. ,Ltd , General Research Institute for Non-ferrous Metals, Beijing 1 0 8 8 , China 2) The 205 Research Institute of China Weapon Industry, Xi‘an 710065, China (Received u)06-08-18)
A M : The professional modeling software package CrysVUn was employed to study the process of a large sapphire single crystal p w t h using K p p u l o s method. The influence of gas pressure on thermal field, solid-liquid interface shape, gas velocity field and von Mises stress were studied for the first time. It is found that the root of the seed melt when gas pressure equals to one atmosphere or more than one atmosphere, especially during the seeding period, this result is consistent with the experimental observation, and this paper presents three ways to solve this problem. The temperature gradient and stress decreases significantly es the gas pressure increases. The convexity of the solid-liquid interface slightly increases when the gas pressure increases. Numerical analysis was used to optimize the hot zone design.
Key words:
1.
gas convection; thermal field; von Mises stress; sapphire single crystal; numerical simulation
Intruduction
Sapphire single crystals are widely used in a variety of modem high-tech applications, from commercial and military optical systems to high power laser components because of the favorable combination of excellent optical, mechanical, thermal and chemical characteristics { 1-21. The high crystal perfection, low reactivity, and appropriate unit cell size make sapphire an excellent substrate in the semiconductor industry for blue light-emitting diodes and diode lasers [ 3] . Large columnar sapphire crystals can be grown by various techniques. A technique commonly used for growing sapphire single crystals is the K p p o u l o s method that provides the growth of high quality crystals of large size up to @ 300 mm. The growth occurs at very high temperatures (the melting temperature is 2043 “c ) [41. This practically eliminates a possibility of accurate experimental investigations inside the hot area[ 51. So, the numerical simulation Correspoading aathor: LI Jinquan
is an effective tool for analyzing processes that take place in the reactor. In particular, the simulation allows evaluating such important parameters as temperature and heat flux distributions in growing crystal and in other parts of the setup. Lukanina et al. [ 4 ] had suggested a 3D numerical model which had been developed on the basis of the CGSim package for the simulation of heat transfer in the furnace for growing large sapphire crystals by horizontal directional crystallization method (HDC ) . The model had been used to improve the presented growth setup, to reduce the energy losses, to control temperature gradients in the crystal, and to design a new HDC growth setup for the p w t h of corundum single crystals of larger size. The thermal fields at various positions of the crystal container and the effect of various setup units and their design on the temperature distribution were analyzed. Lu and Chen[ 6-81 used FIDAP to simulate the HEM crystal growth process in a cylindrical crucible, both the energy input and
E-mail : sapphimubetrate@ yahoo. corn. cn
Li J . Q . et ul . , Influence of gas flow on sapphire single crystal using K y r o p ~ ~method l~s the energy output were modeled under convection boundary conditions. They found that the contact angle was obtuse before the solid-melt interface touched the sidewall of the crucible, and that hot spots always appeared. The effects of various thermal fields on the shape of interface and the increasing rate of the crystal size were investigated. Influence of the crucible geometry on the shape of the melt-crystal interface was also studied. Brandon S .et al . [ 91 had examined the growth of cylindrical sapphire single crystals numerically, using a GSM system. They considered quasi-steady-state solutions for the melt-crystal interface. The sensitivity of the thermal fields to the parameter values was markedly dependent on the diameter of the crucible. They found that the effect of the furnace gradients on the interface in an intermediate size crucible was larger than h e interface in crucibles with different diameters. Floricica Barvinschi et al. [ 101 employed the finite element software FIDAP to simulate the Verneuil process for the sapphire crystal growth. A steady-state numerical model of the Verneuil growth process had been built and gave results in qualitative agreement with some experimental observations. Borodin A . V. et al. [ 11 ] had simulated the pressure distribution caused by a hydrodynamic stream of the melt in meniscus and capillary regions to define and analyze the forces that a crystal experiences during the Stepanov (EFG) growth. The velocity fields and the corresponding pressure distribution in the capillary and in the meniscus were described by Stokes equations and the appropriate class of functions. Generally, the furnace was charged with argon as a protective gas during sapphire single crystal growth by Kyropoulos method. And gas pressure was an important parameter during Kyropoulos-sapphire single crystal growth. Until now, there are few researchers who have simulated the Kyropoulos process. The effect of the gas pressure and flow on the Kyropoulos crystal growth has not been considered. Hence, in the present study, we employ software CrysVUn [ 12-15] ( a finite-volume code developed at the Fraunhofer Institute in Erlangen ) to study the
261
correlation between the gas pressure and the thermal field, solid-liquid interface shape, gas velocity field as well as vonMisesStress during Kyropoulos sapphire crystal growth.
2.
Modeling
Over two decades of development, modeling is now an essential research field in crystal growth. Its impact is not only on the better understanding of fundamental phenomena, but also on the much faster process improvement and innovation of crystal growth technologies [ 161 . Two mathematical procedures were implemented in the software system CrysVUN for global thermal simulation of crystal growth processes. The forward simulation can calculate the temperature distribution as long as the heater powers are given, like in experiments. The inverse simulation was able to calculate the powers of an arbitrary number of heaters in a crystal growth configuration in order to obtain a prescribed temperature distribution in a growing crystal[ 171. In the present study, we employed the standard K-epsilon gas turbulent model to simulate the gas convection using the inverse simulation in the numerical modeling, and set the temperature of the controlled point (0, 0.54) equal to 2316 K , i .e . , the solid-liquid interface should pass through this point. To simplify the problem, melt convection was not taken into account. Because the growth velocity was very small, generally from 0. 5-5 mm h - ' , the pseudo-stationary assumption was used. Table 1 shows the thermo-physical of sapphire properties taken into account and some different non-dimensional numbers of the problem. Argon physical data, including thermal diffusivity, specific heat and viscosity used in the simulation of gas convection are list in Table 2. In addition to the simulations of thermal field, gas flow field, the stress field was also calculated in the present study. The analysis of thermo-elastic stress in the crystal was quite important for predictions on the quality of sapphire single crystal . The stress - strain relation for a
262
RARE METALS, Vo1.25, Spec. Issue, Dee2006
Table 1. Sapphire tbermo-physical data used Value
Description
Thermal conductivity of crystal/(W-K-lvn-')
References
- 27.4+ 590.57 * exp(- TI%?.51) + 24.49 * mp(-T/6696.83) +
19.92A~exp(-T/6441.34,)
Thermal conductivity of melt/(W*K-'*rn-l) 3.5 Melt point/K 2316
0.9 1.1x106
Emissivity
Latent heat/(J.kg-')
0.1-6 pn for crystal; 1-5 flm for melt
Transparency -2.
Argunpbysicpldetallsed Description
T % e ddiffusivity/(rn' * s I ) Specific heat/( J * (kg. K )
References
Value
)
Vkwity/( kg * m - I * s - ) or ( N * s) e r n -
1.553406e-018* T * T * T * T-1.523237e-014 * T* T* T+ 1.23372le-010* T * T + 5.101993e-008 * T-7.m469e-006 5.m3e+m -1.0075&+018*T * T * T * T+8.881353e-015*T * T * T-3. 008134e-O11*T * T+7.512290e-008*T+2.823345e-006
thermo-elastic anisotropic solid body was taken as described by Lambropoulos[ 23-24] .
(1) Here a was the thermal expansion coefficient, ?'& the reference temperature for the relaxed body and E,,,,,., the strains. c,, were the elastic material constants in the Voigt notation. A very important scalar for the discussion of stress in solid bodies, especially for crystal growth was the von Mises stress, which was computed from the distinct stress components. In cylindrical coordinates, it was given by
(2) As for sapphire single crystal, the stress coefficients[ 251 C,,= 496 x 109 Pa, C12= 164 x 109 Pa, C,,= 115 x 109 Pa, C33 = 498 x 109 Pa, C, = 148 x 109 Pa.
3.
[all [221
[221
Results and discussion
Because gas pressure has a direct impact on temperature field and gas flow field, its a very important parameter during Kyropoulos processes for sapphire single crystal growth. To investigate the effects of gas pressure, it had been changed from 0 to 106 Pa. Fig. 1 illustrates the temperature distribution and gas vector fields during the initial Kyropoulos growth for different gas pressure. There was no gas convection in vacuum systems. For the lower gas pressure, the gas velocity was much slower and the gas flow was less turbulent than the higher pressure, and its effects on thermal fields were not obvious. The result shows that the temperature fields change significantly when the pressure equals to or higher than 16 Pa (about one atmosphere). All the isothermals were parabola before the pressure equal to lo5 Pa, it was also found that the isothermals above the melt was close to the seed, the gas vector fields show that the gas flow was moving upw a d , and the flow was much more turbulent
Li J . Q . et al . , Influence of gas flow on sapphire single crystal using Kyropodos method
around the comer. With the increase of gas pressure, the isothermals which were higher than the melt point of sapphire were appearing in the regions above the seed. When the gas pressure equaled to lo6Pa ( Fig. 1 ( f ) ) , the seed and the isothermals of melt point were nearly touch with each other. It had a direct relationship with gas vector field, because tie temperature of argon gas around the crucible was very high, and the gas would take heat flux with high temperature upward. Fig. 2 showed that the seed did not melt during the initial Kyropoulos growth, as the pressure equaled to 16 Pa. But the melt-solid interface would move upward during the initial seeding process, which meant that the isother-
263
mals which were higher than the melt point would be much closer to the seed, and because of many actual factors the gas velocity field should be unstable and fluctuant in a relatively large range during the crystal growth experiments. This property of the gas turbulent would greatly increase the possibility of seed melting. It was noticeable that the corner between the water-cooling rod and the seed was orthogonal (Fig. 2( b) ) . We could find from the gas vector fields in Fig. 2. That the gas convection strengthened around the comer, the heat exchange between the gas and the seed quickened at the same time, and the gas transferred the heat of high energy to the seed, which made the root of the seed easily melt during the initial
Fig. 1. Isotherm& and gas flow vector fields for various gas pressure (Temperature fields in the systems are Td=2316K, A T = l K ) : (a)vacuum; ( b ) p = l x l @ P a ; ( c ) p = l x l @ P a ; ( d ) p = l x l d P a ; ( e > p = 3 x 1 d Pa; (f) p = 1 x 106 Pa.
264
RARE M E T U , Vol. 2 5 , Spec. Issue, Dee 2006
Fig.2. The root ofthe seed melted during initial seeding period at p = l e P a ( a ) , the binding structure used at present(b) , improved binding strueture(c), and the seed was all right with the improved binding structure (d).
seeding period. This was consistent with the experimental observation ( Fig. 2 ( a ) ) . It is very disadvantageous for the whole crystal growtb process that the root of the seed melted during seeding period. The experiment.. may be abortive due to awful condition of the seeds. The melting of the seed root had direct relationship with the gas turbulent around the orthogonal corner. We should decrease the gas convection at the corner between the rod and the seed to avoid this problem. Hence, this paper brings forward three methods in the present study. 1) Increasing the space between the CNcihle, melt and the upper heat shield will lower the heat exchange between the gas and the seed. But the space should not be too large, or the function of the upper heat shield would be weakened. 2) Reducing the gas pressure is also an effective way to solve this problem. 3) Improced the binding structure ( Fig. 2 ( c ) ) , just changing the orthogonal angle to an obtuse angle. Experiments show that the problem was well solved (Fig. 2( d ) ) . Fig. 3 plotted the axial temperature distribution in crystal and melt versus different gas
pressure. All the curves intersected at the same point, which was the controlled point ( 0, 0.54) in the inverse simulation. The temperature distribution of the crystal was below this point, and the parts above the point were the melt temperature distribution. The slopes of these curves were regarded as the axial temperature gradients. Its obvious that the temperature gradients decrease significantly as the gas pressure increases.
z s o + . . . , . . . . . . a a ou a a 0% 0s) a a
. . , . I
OM
ow
om
2 axis in sapphire (melt & crystal)
Fig.3. Calculated temperature distribution in sap pbire melt &crystal of axis.
Li J .Q .et uJ . , Intluence of gas flow on sapphire single crystal Using Kyropoulos method
The vonMisesStress and thermal fields of the sapphire single crystals are present in Fig.4. It can be clearly seen that all the meltsolid interface are convex. The convexity was defined by D =max ZIR (3) max Z was the height of the interface in the z direction and R was the radius of the crystal. When the convexity equals to 1, it means that the shape of the melt-solid interface is hemispherical. The convexity of the interface is an important growth parameter that affects the quality of the crystal, The formation of crystal
265
facets can be prevented by lower convexity at the interface, except for a certain orientation of the crystal seed [ 261. Higher quality crystals may be grown when there was less convexity at the melt-solid interface. Fig. 4 shows that the shape of melt-solid interface is a parabola when gas pressure equals to lo6 Pa (Fig. 4 ( f ) ) , but the shape is nearly a triangle in vacuum systems. Hence, although the maximum values of the convexity does not change significantly with the variation of gas pressure, the average convexity of the whole crystal slightly increases as the pressure increases.
Fig.4. Influence of gas pressure on kmpemtme field and thermal stress u,(nmximum of von Mises stress factor): (a) vacuum; (b) p = 1 x d Pa; (c) p = 1 x la' Pa; (d) p = 1 x 10s Pa; (e) p = 3 x l@Pa, (f) p = 1 x 106 Pa.
big stress may crack, e . g., sapphire crystal easily cracked using Vemeuil method. The maximum stress mainly distributes in the region connecting with the seed and the edge of the crystal during the seeding period (Fig. 4) . Fig. 5 illustrates that the von Mises stress decreases significantly when the gas pressure increases, but after the gas pressure equates to 3 x 16 Pa, the von Mises stress decreases slightly.
4. Pressure/io.'Pa Fig.5.
Effects of gas pressure on von Mises stress.
The analysis of thermoelastic stress in the crystal was quite important for predictions on the quality of the grown material. Sapphire crystal of
Conclusions
The effect of gas pressure on the thermal field, solid-liquid interface shape, gas velocity field and von Mises stress during the application of the Kyropoulos-sapphire single crystal growth were investigated numerically using the CrysVUn software. The results show that the root of
RARE METALS , Vol. 2 5 , Spec. Issue , Dee 2006
266 the Beed melt when gas pressure equals to one
atmosphere or more than one atmosphere, especially during the seeding period, this result is consistent with the experimental observation. The melting of the seed root has direct relationship with the gas turbulent around the orthogonal comer, and this paper presents three ways to decrease the gas convection at the comer between the rod and the seed. The experiment had validated that the improved binding structure could resolve it well, but the other two methods have not yet been confirmed by comparing with experiments. It is found that the temperature gradient and stress decrease significantly as the gas pressure increases. The convexity of the solid-liquid interface slightly increases when the gas pressure increases. The further work will be related to optimizing the thermal insulation system, in particularly, the geometry of the upper heat shield systems.
References : [ l ] Schmid F., Kkattak C.P., and Felt M . D . , Roducing large sapphire for optical applications. Am. Ceramic Soc. Bull., 1994. "2): 39. i23 Savage J . A, Reparation and properties of hard crystelline materials for optical applications: a review. J.Crystal G w h , 1991, lW(33): 698. [3: Miyazawa S., Optical crystals sunived in information technolo~y system. O p E k t r o n . Rev., 2003, l l ( 2 ) : 77. [4] Lukanina M . A . , Hodosevitch K . V . , KJ&V V . V ., et al , 3D numerical simulation of heat transfer during horizontal direct cqaallimtion of conindun s i d e crystals. J o d of Crystal Growth, 2006, 330. [5] Dobmvinskaya E., Iitvinov L.L., and Pishik V, Encyciopedie of Sapphire (in Russian), institute of
.
m:
monocrystals, M O V , m. [6] Lu C.W., and Chen J.J., Numerical computation of sapphire crystal p w t h using heat exchanger method. J.Crystal Gzowth. 2001,225: 274. [7; L u C . W . , C h e n J . J . , a n d H u L . J . , Anumerical investigation of the the& distribution effects in a heat-exchanger-methcd crystal growth system. Modelh+qSbnul. Moter. Sci. E n g . , aoo2, 10: 147. [8: 1 x C . W . . andChenJ.J., Influenceofthecrucible geometry on the shape of the melt-crystal interface during grow& of sapphire crystal using a heat exchanger method. J . Crystal 6!rowth, 2004, 266:
239. i91 Brandon S., and &it
S., Interface s h a p and
thermal fields during the pdient solidification
method p w t h of sapphire single crystals. Journol
of Crystal Growth , 1996,167: 190. [ 101 Floricica Baninschi, Jean-buis %taller, Thieny Duffsr, et a l . , Modelling of Vemeuil process for the sapphire crystal growth. Journal of Crystal Growth, 1999, Wl99:239. [ l l ] BorodinA.V.,BorodinV.A.,andZhdanovA.V., Simulation of the pressure distribution in the melt for sapphire ribbon growth by the Stepanov (EFG) technique. Journal of Crystal Growth, 1999, 198:
220. [12] K u n M . , PusztaiA.,andMullerG.,Development of a new powerful computer code CrysWN + + especidy designed for fast simulation of bulk crystal p w t h processes. Proceedings of SPIE, 2003,
5078. [I31 K u n M., and MuK ller G . , Control of thermal conditions during crystal growth by inverse modeling. J o d of Crystal W h , 2000, ##I: 341. [ 141 Muller G . , and Birkmann B . , Optimization of VGFgrowth of GaAs crystals by the aid of numerical modelling. J o d of Crystal Growth, 2W2, Z37239: 1745. 151 Backofen R., Kun M . , and MuK Uer G . , Process modeling of the industrial VGF p w t h process using the software package CrysVUN + + . Journal of Cryslal &moth, 2OO0, 211: 202. 161 Jan C . W . , Imaishi N . , and Kakimoto K., &face. Journal of Crystal Growth, 2004,266: 1. [17] manual of CrysWn [ 181 Iingart Yu.K., Radiation-Conductive Heat W e r in Optical Crystals, Crystal Growth Fipipment: Internal Report, 1992. [I91 Weast R . C . , and Astle M.J., Eds., CRC Handbook of Chemistry and Physics, 59th ed. Boca Raton, n, CRC Ress, 1978-1979. 201 Kestin J . , Knierim K . , Mason E.A., et ol . , Equilibrium and transport properties of the noble gases and their mixtures at low density. J . Phys . C h . Ref.Data, 1984, W(1): 229. 211 Ho C . Y . , Powell R . W . , and Lley P.E., Thermal conductivity of the elements. J . Phys . Chem . Ref. Dara, 1972, 1: 279. [22] McBride B.J., Gordon S. , and Reno M.A., 'Ihermodynamic Data for F i i Reference Elements NASA Technical Paper. Jan., 1 M : 3287. [231 Marc Hans Emst Hainke, Computation of Convection and AUoy Solidification with the Software Package CrysWn, PhD thesis, University ErlangenNumberg, 2004. [24] Lambropoulos J .C . , The isotropic assumption during the Czochralski growth of single semiconductors crystals. J . Ciyst. Growth, 1987, 84: 349. [U]http://mvw. mt-berlin. com/frames-cryst/descriptions/sapphire.htm [26] Viechnicki D . , and Schmid F . , Crystal p w t b using the heat exchanger method (HEM). Journal of Crystal Gowth, 1974, 26: 162.
Muence of gas flow on thermal field and stress during growth of sapphire single crystal using Kyropoulos method U Jinquan” , SV Xiaoping” , NA M y h ’ ) , YANG Hail),U Jianmin” , W Ymqi2’, and MI Jimjm2’
.
1 ) Reijing Guojing Infrared Optical Technology Co. ,Ltd , General Research Institute for Non-ferrous Metals, Beijing 1 0 8 8 , China 2) The 205 Research Institute of China Weapon Industry, Xi‘an 710065, China (Received u)06-08-18)
A M : The professional modeling software package CrysVUn was employed to study the process of a large sapphire single crystal p w t h using K p p u l o s method. The influence of gas pressure on thermal field, solid-liquid interface shape, gas velocity field and von Mises stress were studied for the first time. It is found that the root of the seed melt when gas pressure equals to one atmosphere or more than one atmosphere, especially during the seeding period, this result is consistent with the experimental observation, and this paper presents three ways to solve this problem. The temperature gradient and stress decreases significantly es the gas pressure increases. The convexity of the solid-liquid interface slightly increases when the gas pressure increases. Numerical analysis was used to optimize the hot zone design.
Key words:
1.
gas convection; thermal field; von Mises stress; sapphire single crystal; numerical simulation
Intruduction
Sapphire single crystals are widely used in a variety of modem high-tech applications, from commercial and military optical systems to high power laser components because of the favorable combination of excellent optical, mechanical, thermal and chemical characteristics { 1-21. The high crystal perfection, low reactivity, and appropriate unit cell size make sapphire an excellent substrate in the semiconductor industry for blue light-emitting diodes and diode lasers [ 3] . Large columnar sapphire crystals can be grown by various techniques. A technique commonly used for growing sapphire single crystals is the K p p o u l o s method that provides the growth of high quality crystals of large size up to @ 300 mm. The growth occurs at very high temperatures (the melting temperature is 2043 “c ) [41. This practically eliminates a possibility of accurate experimental investigations inside the hot area[ 51. So, the numerical simulation Correspoading aathor: LI Jinquan
is an effective tool for analyzing processes that take place in the reactor. In particular, the simulation allows evaluating such important parameters as temperature and heat flux distributions in growing crystal and in other parts of the setup. Lukanina et al. [ 4 ] had suggested a 3D numerical model which had been developed on the basis of the CGSim package for the simulation of heat transfer in the furnace for growing large sapphire crystals by horizontal directional crystallization method (HDC ) . The model had been used to improve the presented growth setup, to reduce the energy losses, to control temperature gradients in the crystal, and to design a new HDC growth setup for the p w t h of corundum single crystals of larger size. The thermal fields at various positions of the crystal container and the effect of various setup units and their design on the temperature distribution were analyzed. Lu and Chen[ 6-81 used FIDAP to simulate the HEM crystal growth process in a cylindrical crucible, both the energy input and
E-mail : sapphimubetrate@ yahoo. corn. cn
Li J . Q . et ul . , Influence of gas flow on sapphire single crystal using K y r o p ~ ~method l~s the energy output were modeled under convection boundary conditions. They found that the contact angle was obtuse before the solid-melt interface touched the sidewall of the crucible, and that hot spots always appeared. The effects of various thermal fields on the shape of interface and the increasing rate of the crystal size were investigated. Influence of the crucible geometry on the shape of the melt-crystal interface was also studied. Brandon S .et al . [ 91 had examined the growth of cylindrical sapphire single crystals numerically, using a GSM system. They considered quasi-steady-state solutions for the melt-crystal interface. The sensitivity of the thermal fields to the parameter values was markedly dependent on the diameter of the crucible. They found that the effect of the furnace gradients on the interface in an intermediate size crucible was larger than h e interface in crucibles with different diameters. Floricica Barvinschi et al. [ 101 employed the finite element software FIDAP to simulate the Verneuil process for the sapphire crystal growth. A steady-state numerical model of the Verneuil growth process had been built and gave results in qualitative agreement with some experimental observations. Borodin A . V. et al. [ 11 ] had simulated the pressure distribution caused by a hydrodynamic stream of the melt in meniscus and capillary regions to define and analyze the forces that a crystal experiences during the Stepanov (EFG) growth. The velocity fields and the corresponding pressure distribution in the capillary and in the meniscus were described by Stokes equations and the appropriate class of functions. Generally, the furnace was charged with argon as a protective gas during sapphire single crystal growth by Kyropoulos method. And gas pressure was an important parameter during Kyropoulos-sapphire single crystal growth. Until now, there are few researchers who have simulated the Kyropoulos process. The effect of the gas pressure and flow on the Kyropoulos crystal growth has not been considered. Hence, in the present study, we employ software CrysVUn [ 12-15] ( a finite-volume code developed at the Fraunhofer Institute in Erlangen ) to study the
261
correlation between the gas pressure and the thermal field, solid-liquid interface shape, gas velocity field as well as vonMisesStress during Kyropoulos sapphire crystal growth.
2.
Modeling
Over two decades of development, modeling is now an essential research field in crystal growth. Its impact is not only on the better understanding of fundamental phenomena, but also on the much faster process improvement and innovation of crystal growth technologies [ 161 . Two mathematical procedures were implemented in the software system CrysVUN for global thermal simulation of crystal growth processes. The forward simulation can calculate the temperature distribution as long as the heater powers are given, like in experiments. The inverse simulation was able to calculate the powers of an arbitrary number of heaters in a crystal growth configuration in order to obtain a prescribed temperature distribution in a growing crystal[ 171. In the present study, we employed the standard K-epsilon gas turbulent model to simulate the gas convection using the inverse simulation in the numerical modeling, and set the temperature of the controlled point (0, 0.54) equal to 2316 K , i .e . , the solid-liquid interface should pass through this point. To simplify the problem, melt convection was not taken into account. Because the growth velocity was very small, generally from 0. 5-5 mm h - ' , the pseudo-stationary assumption was used. Table 1 shows the thermo-physical of sapphire properties taken into account and some different non-dimensional numbers of the problem. Argon physical data, including thermal diffusivity, specific heat and viscosity used in the simulation of gas convection are list in Table 2. In addition to the simulations of thermal field, gas flow field, the stress field was also calculated in the present study. The analysis of thermo-elastic stress in the crystal was quite important for predictions on the quality of sapphire single crystal . The stress - strain relation for a
262
RARE METALS, Vo1.25, Spec. Issue, Dee2006
Table 1. Sapphire tbermo-physical data used Value
Description
Thermal conductivity of crystal/(W-K-lvn-')
References
- 27.4+ 590.57 * exp(- TI%?.51) + 24.49 * mp(-T/6696.83) +
19.92A~exp(-T/6441.34,)
Thermal conductivity of melt/(W*K-'*rn-l) 3.5 Melt point/K 2316
0.9 1.1x106
Emissivity
Latent heat/(J.kg-')
0.1-6 pn for crystal; 1-5 flm for melt
Transparency -2.
Argunpbysicpldetallsed Description
T % e ddiffusivity/(rn' * s I ) Specific heat/( J * (kg. K )
References
Value
)
Vkwity/( kg * m - I * s - ) or ( N * s) e r n -
1.553406e-018* T * T * T * T-1.523237e-014 * T* T* T+ 1.23372le-010* T * T + 5.101993e-008 * T-7.m469e-006 5.m3e+m -1.0075&+018*T * T * T * T+8.881353e-015*T * T * T-3. 008134e-O11*T * T+7.512290e-008*T+2.823345e-006
thermo-elastic anisotropic solid body was taken as described by Lambropoulos[ 23-24] .
(1) Here a was the thermal expansion coefficient, ?'& the reference temperature for the relaxed body and E,,,,,., the strains. c,, were the elastic material constants in the Voigt notation. A very important scalar for the discussion of stress in solid bodies, especially for crystal growth was the von Mises stress, which was computed from the distinct stress components. In cylindrical coordinates, it was given by
(2) As for sapphire single crystal, the stress coefficients[ 251 C,,= 496 x 109 Pa, C12= 164 x 109 Pa, C,,= 115 x 109 Pa, C33 = 498 x 109 Pa, C, = 148 x 109 Pa.
3.
[all [221
[221
Results and discussion
Because gas pressure has a direct impact on temperature field and gas flow field, its a very important parameter during Kyropoulos processes for sapphire single crystal growth. To investigate the effects of gas pressure, it had been changed from 0 to 106 Pa. Fig. 1 illustrates the temperature distribution and gas vector fields during the initial Kyropoulos growth for different gas pressure. There was no gas convection in vacuum systems. For the lower gas pressure, the gas velocity was much slower and the gas flow was less turbulent than the higher pressure, and its effects on thermal fields were not obvious. The result shows that the temperature fields change significantly when the pressure equals to or higher than 16 Pa (about one atmosphere). All the isothermals were parabola before the pressure equal to lo5 Pa, it was also found that the isothermals above the melt was close to the seed, the gas vector fields show that the gas flow was moving upw a d , and the flow was much more turbulent
Li J . Q . et al . , Influence of gas flow on sapphire single crystal using Kyropodos method
around the comer. With the increase of gas pressure, the isothermals which were higher than the melt point of sapphire were appearing in the regions above the seed. When the gas pressure equaled to lo6Pa ( Fig. 1 ( f ) ) , the seed and the isothermals of melt point were nearly touch with each other. It had a direct relationship with gas vector field, because tie temperature of argon gas around the crucible was very high, and the gas would take heat flux with high temperature upward. Fig. 2 showed that the seed did not melt during the initial Kyropoulos growth, as the pressure equaled to 16 Pa. But the melt-solid interface would move upward during the initial seeding process, which meant that the isother-
263
mals which were higher than the melt point would be much closer to the seed, and because of many actual factors the gas velocity field should be unstable and fluctuant in a relatively large range during the crystal growth experiments. This property of the gas turbulent would greatly increase the possibility of seed melting. It was noticeable that the corner between the water-cooling rod and the seed was orthogonal (Fig. 2( b) ) . We could find from the gas vector fields in Fig. 2. That the gas convection strengthened around the comer, the heat exchange between the gas and the seed quickened at the same time, and the gas transferred the heat of high energy to the seed, which made the root of the seed easily melt during the initial
Fig. 1. Isotherm& and gas flow vector fields for various gas pressure (Temperature fields in the systems are Td=2316K, A T = l K ) : (a)vacuum; ( b ) p = l x l @ P a ; ( c ) p = l x l @ P a ; ( d ) p = l x l d P a ; ( e > p = 3 x 1 d Pa; (f) p = 1 x 106 Pa.
264
RARE M E T U , Vol. 2 5 , Spec. Issue, Dee 2006
Fig.2. The root ofthe seed melted during initial seeding period at p = l e P a ( a ) , the binding structure used at present(b) , improved binding strueture(c), and the seed was all right with the improved binding structure (d).
seeding period. This was consistent with the experimental observation ( Fig. 2 ( a ) ) . It is very disadvantageous for the whole crystal growtb process that the root of the seed melted during seeding period. The experiment.. may be abortive due to awful condition of the seeds. The melting of the seed root had direct relationship with the gas turbulent around the orthogonal corner. We should decrease the gas convection at the corner between the rod and the seed to avoid this problem. Hence, this paper brings forward three methods in the present study. 1) Increasing the space between the CNcihle, melt and the upper heat shield will lower the heat exchange between the gas and the seed. But the space should not be too large, or the function of the upper heat shield would be weakened. 2) Reducing the gas pressure is also an effective way to solve this problem. 3) Improced the binding structure ( Fig. 2 ( c ) ) , just changing the orthogonal angle to an obtuse angle. Experiments show that the problem was well solved (Fig. 2( d ) ) . Fig. 3 plotted the axial temperature distribution in crystal and melt versus different gas
pressure. All the curves intersected at the same point, which was the controlled point ( 0, 0.54) in the inverse simulation. The temperature distribution of the crystal was below this point, and the parts above the point were the melt temperature distribution. The slopes of these curves were regarded as the axial temperature gradients. Its obvious that the temperature gradients decrease significantly as the gas pressure increases.
z s o + . . . , . . . . . . a a ou a a 0% 0s) a a
. . , . I
OM
ow
om
2 axis in sapphire (melt & crystal)
Fig.3. Calculated temperature distribution in sap pbire melt &crystal of axis.
Li J .Q .et uJ . , Intluence of gas flow on sapphire single crystal Using Kyropoulos method
The vonMisesStress and thermal fields of the sapphire single crystals are present in Fig.4. It can be clearly seen that all the meltsolid interface are convex. The convexity was defined by D =max ZIR (3) max Z was the height of the interface in the z direction and R was the radius of the crystal. When the convexity equals to 1, it means that the shape of the melt-solid interface is hemispherical. The convexity of the interface is an important growth parameter that affects the quality of the crystal, The formation of crystal
265
facets can be prevented by lower convexity at the interface, except for a certain orientation of the crystal seed [ 261. Higher quality crystals may be grown when there was less convexity at the melt-solid interface. Fig. 4 shows that the shape of melt-solid interface is a parabola when gas pressure equals to lo6 Pa (Fig. 4 ( f ) ) , but the shape is nearly a triangle in vacuum systems. Hence, although the maximum values of the convexity does not change significantly with the variation of gas pressure, the average convexity of the whole crystal slightly increases as the pressure increases.
Fig.4. Influence of gas pressure on kmpemtme field and thermal stress u,(nmximum of von Mises stress factor): (a) vacuum; (b) p = 1 x d Pa; (c) p = 1 x la' Pa; (d) p = 1 x 10s Pa; (e) p = 3 x l@Pa, (f) p = 1 x 106 Pa.
big stress may crack, e . g., sapphire crystal easily cracked using Vemeuil method. The maximum stress mainly distributes in the region connecting with the seed and the edge of the crystal during the seeding period (Fig. 4) . Fig. 5 illustrates that the von Mises stress decreases significantly when the gas pressure increases, but after the gas pressure equates to 3 x 16 Pa, the von Mises stress decreases slightly.
4. Pressure/io.'Pa Fig.5.
Effects of gas pressure on von Mises stress.
The analysis of thermoelastic stress in the crystal was quite important for predictions on the quality of the grown material. Sapphire crystal of
Conclusions
The effect of gas pressure on the thermal field, solid-liquid interface shape, gas velocity field and von Mises stress during the application of the Kyropoulos-sapphire single crystal growth were investigated numerically using the CrysVUn software. The results show that the root of
RARE METALS , Vol. 2 5 , Spec. Issue , Dee 2006
266 the Beed melt when gas pressure equals to one
atmosphere or more than one atmosphere, especially during the seeding period, this result is consistent with the experimental observation. The melting of the seed root has direct relationship with the gas turbulent around the orthogonal comer, and this paper presents three ways to decrease the gas convection at the comer between the rod and the seed. The experiment had validated that the improved binding structure could resolve it well, but the other two methods have not yet been confirmed by comparing with experiments. It is found that the temperature gradient and stress decrease significantly as the gas pressure increases. The convexity of the solid-liquid interface slightly increases when the gas pressure increases. The further work will be related to optimizing the thermal insulation system, in particularly, the geometry of the upper heat shield systems.
References : [ l ] Schmid F., Kkattak C.P., and Felt M . D . , Roducing large sapphire for optical applications. Am. Ceramic Soc. Bull., 1994. "2): 39. i23 Savage J . A, Reparation and properties of hard crystelline materials for optical applications: a review. J.Crystal G w h , 1991, lW(33): 698. [3: Miyazawa S., Optical crystals sunived in information technolo~y system. O p E k t r o n . Rev., 2003, l l ( 2 ) : 77. [4] Lukanina M . A . , Hodosevitch K . V . , KJ&V V . V ., et al , 3D numerical simulation of heat transfer during horizontal direct cqaallimtion of conindun s i d e crystals. J o d of Crystal Growth, 2006, 330. [5] Dobmvinskaya E., Iitvinov L.L., and Pishik V, Encyciopedie of Sapphire (in Russian), institute of
.
m:
monocrystals, M O V , m. [6] Lu C.W., and Chen J.J., Numerical computation of sapphire crystal p w t h using heat exchanger method. J.Crystal Gzowth. 2001,225: 274. [7; L u C . W . , C h e n J . J . , a n d H u L . J . , Anumerical investigation of the the& distribution effects in a heat-exchanger-methcd crystal growth system. Modelh+qSbnul. Moter. Sci. E n g . , aoo2, 10: 147. [8: 1 x C . W . . andChenJ.J., Influenceofthecrucible geometry on the shape of the melt-crystal interface during grow& of sapphire crystal using a heat exchanger method. J . Crystal 6!rowth, 2004, 266:
239. i91 Brandon S., and &it
S., Interface s h a p and
thermal fields during the pdient solidification
method p w t h of sapphire single crystals. Journol
of Crystal Growth , 1996,167: 190. [ 101 Floricica Baninschi, Jean-buis %taller, Thieny Duffsr, et a l . , Modelling of Vemeuil process for the sapphire crystal growth. Journal of Crystal Growth, 1999, Wl99:239. [ l l ] BorodinA.V.,BorodinV.A.,andZhdanovA.V., Simulation of the pressure distribution in the melt for sapphire ribbon growth by the Stepanov (EFG) technique. Journal of Crystal Growth, 1999, 198:
220. [12] K u n M . , PusztaiA.,andMullerG.,Development of a new powerful computer code CrysWN + + especidy designed for fast simulation of bulk crystal p w t h processes. Proceedings of SPIE, 2003,
5078. [I31 K u n M., and MuK ller G . , Control of thermal conditions during crystal growth by inverse modeling. J o d of Crystal W h , 2000, ##I: 341. [ 141 Muller G . , and Birkmann B . , Optimization of VGFgrowth of GaAs crystals by the aid of numerical modelling. J o d of Crystal Growth, 2W2, Z37239: 1745. 151 Backofen R., Kun M . , and MuK Uer G . , Process modeling of the industrial VGF p w t h process using the software package CrysVUN + + . Journal of Cryslal &moth, 2OO0, 211: 202. 161 Jan C . W . , Imaishi N . , and Kakimoto K., &face. Journal of Crystal Growth, 2004,266: 1. [17] manual of CrysWn [ 181 Iingart Yu.K., Radiation-Conductive Heat W e r in Optical Crystals, Crystal Growth Fipipment: Internal Report, 1992. [I91 Weast R . C . , and Astle M.J., Eds., CRC Handbook of Chemistry and Physics, 59th ed. Boca Raton, n, CRC Ress, 1978-1979. 201 Kestin J . , Knierim K . , Mason E.A., et ol . , Equilibrium and transport properties of the noble gases and their mixtures at low density. J . Phys . C h . Ref.Data, 1984, W(1): 229. 211 Ho C . Y . , Powell R . W . , and Lley P.E., Thermal conductivity of the elements. J . Phys . Chem . Ref. Dara, 1972, 1: 279. [22] McBride B.J., Gordon S. , and Reno M.A., 'Ihermodynamic Data for F i i Reference Elements NASA Technical Paper. Jan., 1 M : 3287. [231 Marc Hans Emst Hainke, Computation of Convection and AUoy Solidification with the Software Package CrysWn, PhD thesis, University ErlangenNumberg, 2004. [24] Lambropoulos J .C . , The isotropic assumption during the Czochralski growth of single semiconductors crystals. J . Ciyst. Growth, 1987, 84: 349. [U]http://mvw. mt-berlin. com/frames-cryst/descriptions/sapphire.htm [26] Viechnicki D . , and Schmid F . , Crystal p w t b using the heat exchanger method (HEM). Journal of Crystal Gowth, 1974, 26: 162.