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Study and optimization of gas flow and temperature distribution in a Czochralski configuration
Journal of Crystal Growth 361 (2012) 114–120
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Study and optimization of gas flow and temperature distribution in a Czochralski configuration H.S. Fang n, Z.L. Jin, X.M. Huang School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2012 Received in revised form 25 August 2012 Accepted 17 September 2012 Communicated by J.J. Derby Available online 25 September 2012
The Czochralski (Cz) method has virtually dominated the entire production of bulk single crystals with high productivity. Since the Cz-grown crystals are cylindrical, axisymmetric hot zone arrangement is required for an ideally high-quality crystal growth. However, due to three-dimensional effects the flow pattern and temperature field are inevitably non-axisymmetric. The grown crystal suffers from many defects, among which macro-cracks and micro-dislocation are mainly related to inhomogeneous temperature distribution during the growth and cooling processes. The task of the paper is to investigate gas partition and temperature distribution in a Cz configuration, and to optimize the furnace design for the reduction of the three-dimensional effects. The general design is found to be unfavorable to obtain the desired temperature conditions. Several different types of the furnace designs, modified at the top part of the side insulation, are proposed for a comparative analysis. The optimized one is chosen for further study, and the results display the excellence of the proposed design in suppression of three-dimensional effects to achieve relatively axisymmetric flow pattern and temperature distribution for the possible minimization of thermal stress related crystal defects. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Stresses A1.Fluid flows A2. Czochralski method
1. Introduction The Czochralski method is one of the most stable processes for the growth of bulk single crystals, and has virtually dominated the entire production in the microelectronic and optical industries [1,2]. The primary focus of current Cz growth is to produce massively large-size and high-quality crystals with low economic costs. Various defects, such as cracking and dislocation, related to non-uniform temperature field, significantly affect the performance of the crystals. Therefore, it is extremely important to get the detailed hot zone information in the growth configuration and to optimize it for the improvement of the crystal productivity. However, in situ measurements of the temperature within the crystal are difficult. With the development of computational technique, numerical modeling of the crystal growth process has become an ideal alternative. To improve the growth process, numerical studies of Cz crystal growth process have been conducted by many researchers. The work [1,3–10] covers many topics, such as hot zone design, defect control, modeling technique and so on, and has greatly speeded up the pace to achieve high-quality crystals in experiments and
n
Corresponding author. Tel./fax: þ 86 27 87542618. E-mail addresses: [email protected] (H.S. Fang), [email protected] (X.M. Huang). 0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.09.027
industries. The shape of the grown crystal is usually cylindrical due to the natural characteristics of the Cz technique, which is advantageous for the growth of crystals with uniform properties. An appropriate hot zone in the growth furnace is a key factor for high-quality crystal growth, and the cylindrical crystal geometry requires a correspondingly axisymmetric temperature field, which is determined by heat transport during the growth. Such an ideal hot zone could be achieved only if conduction and radiation were present with axisymmetric heating sources and thermal boundary conditions. However, in a real growth system, complex thermal boundaries drive buoyancy convection, and the gas inlet and outlet are non-axisymmetric in most furnaces, which results in the complex flow patterns. Therefore, in the regions, for example, at the top part of the furnace, where convective heat transfer cannot be neglected, temperature field is inevitable to be three-dimensional because of the convective heat transfer. The temperature of gas environment around the growing crystal, conclusively, is non-axisymmetric, which causes large thermal stress and stress-related defects in the crystal [8,11]. An example is the cracks in Yb:S-FAP laser crystals [8,12]. During the growth and cooling processes, cracks around the neck of the crystal are often observed after the crystal grows long, which is formed by the non-uniform cooling at the top part of the furnace. The above discussion shows that the gas partition, i.e., the design of gas flow and associated temperature field, is a key factor
H.S. Fang et al. / Journal of Crystal Growth 361 (2012) 114–120
affecting the crystal quality. Researchers have made many efforts to discuss the issue [13,14]. In the present paper, a coupled mathematical tool is applied to study heat transfer and gas flow during the growth stages of RF-heated Cz process. Threedimensional effects are examined on the various furnace designs with the modifications at the top part of the side insulation. Temperature distributions and gas flow patterns are presented at the different sections of the furnace, and an optimized design is proposed for a further examination to obtain a more uniform temperature distribution around the crystal. With the simple but effective design, the reduction of three-dimensional effects in the furnace can be realized to reduce crystal defects related to thermal stress, such as cracks and dislocation.
2. Mathematical models 2.1. Problem description Since developed by Jan Czochralski [2], the Cz growth method has been greatly improved in both design and technology. A typical configuration of Cz furnace is given in Fig. 1a. The system mainly includes a metal crucible, a seed rod, ZrO2-based heat insulation, and RF coil. The side insulation, usually in cylindrical shape, encloses the furnace to prevent heat loss. The schematic
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diagram is similar to the system in our previous study designed by Lawrence Livermore National Laboratory (LLNL), which is applied to grow large-size and high-quality Yb:S-FAP single crystals (see Fig. 1b) [8]. The chamber of the furnace is in inert (Nitrogen, Argon) or oxidizing atmosphere (Air) depending on the chemical components of the crystal and the crucible material. There are one gas inlet at the bottom of the furnace, two gas inlets at the top of the furnace, and one gas outlet at the central of the top furnace. It is noted here that the arrangement of gas inlet and outlet may vary for the different modified-Cz systems. The gas flowing into the furnace from the bottom plays a critical role in providing appropriate atmosphere for crystal growth, protecting the crucible from oxidation, and bringing away the evaporated component from the melt. Since the crystal growth process is extremely sensitive to the environment temperature of the crucible, the gas from the bottom inlet is strictly controlled, and the flow rate is usually low. The main purposes of the gas from the top inlets are to cool the top part of the furnace, to provide the temperature gradient for melt solidification, and to cool the growing or as-grown crystal. The gas flow rate of the top inlet is much larger than that of the bottom inlet. During the growth, thermal stress builds up inner the crystal, and dislocation generates at the growth front. As the crystal grows long, the upper part of the crystal will be in cool gas environment. The cooling of the crystal accompanies by the release of thermal stress in two ways. The slow release leads to dislocation multiplication, which generates at the growth front or exists in the seed, while the fast release results in cracking of the crystal as shown in Fig. 1b. Three-dimensional flow structure and non-axisymmetric temperature distribution severely aggravate the defects due to the non-uniform cooling. Therefore, the study and optimization of gas flow pattern and temperature distribution in a Cz configuration are extremely important for the achievement of high-quality crystal with low stress-related defects.
2.2. Mathematical models
Fig. 1. (a) A schematic diagram of Cz configuration and (b) a grown crystal with cracks.
For a vertical enclosure, such as the Cz furnace enclosed by insulation, turbulent convection begins at Rah 109 [15] due to natural convection, and the transition Reynolds number for forced flow is around 2000. The Rayleigh number is calculated based on the height of the furnace chamber. The furnace enclosure of the current study has a diameter of 0.16 m and a height of 0.9 m from the crucible ring to the furnace top. The crystal size is 0.08 m in diameter. The inlet and outlet have a diameter of 0.03 m. The estimated Rah number of the growth system is around 5 109 and the Reynolds number at the outlet is about 105. Therefore, the natural convection seems to be weak turbulence, while the flow regime of forced convection is strongly turbulent. Turbulent models must be applied for the gas flow modeling. The following assumptions are made: (1) natural convection due to temperature difference in the furnace is treated by Boussinesq approximation, (2) the free surface of the melt is assumed flat, i.e., the wetting of the melt on the iridium crucible is neglected, (3) the solid/liquid (S/L) interface shape is assumed constant, i.e., the modification of the top part of the furnace only affects the cooling of the top part of the crystal. Such an assumption is reasonable when the gas flow rate of the bottom inlet keeps constant, (4) melt convection does not take into consideration, and the convective heat transfer in the melt is approximated by an enhanced thermal conductivity, and (5) during a stable Cz crystal growth process, the timeaveraged values varies by time very slowly, so the gas turbulent flow is assumed to be quasi-steady state. The governing equations
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for the Newtonian fluid and incompressible turbulence flow can be described as follows: Continuity equation: @U i ¼0 @xi
ð1Þ
Momentum equation: 0 0
Uj
@ui uj @U i 1 @p @2 U i ¼ þn g 3 bT ðTT 0 Þ @xj r @xi @xj @xj @xj
ð2Þ
Energy equation: Uj
@TU i @2 T ¼a Sr þSt @xj @xj @xj
Boussinesq hypothesis: ! @U i @U j 2 0 0 þ ui uj ¼ nt dij k 3 @xj @xi
ð3Þ
ð4Þ
The standard k e model is applied for turbulence modeling, although the suitability of the model for buoyancy flows is always in question. The turbulent kinetic energy, k, and its dissipation rate, e, are solved by the following conservation equations [15]: ! @k @ nt @k @U i @U j @U i Ui ¼ þ e ð5Þ þ nt @xi @xi sk @xi @xj @xi @xj Ui
! @e @ nt @e e @U i @U j @U i e2 ¼ nt þ c2 þc1 @xi se @xi @xi @xj @xi @xj k rk rk
ð6Þ
where U i is the time-averaged velocity, i and j denote the vector directions, r is the fluid density, p is the time-averaged pressure, n is the fluid viscosity, u0 is the fluctuation of the velocity, nt ¼ cm k2 =e is the turbulent viscosity of the fluid, and dij is the unit tensor. Sr is the radiation source term, which is calculated by the well-established view factor method and St is the turbulence source term. The constants of the turbulent model, cm ,sk ,se ,c1 , and c2 are set to 0.09, 1.0, 1.3, 1.44, and 1.92, respectively. Boundary conditions are required to solve the above governing equations. The top inlet condition is a uniform velocity distribution, and the outlet is set to be outflow where all gradients of the calculated variables are null. The bottom gas is considered as a uniform flow through the gap between the crucible and the side insulation. Other surfaces are considered as non-sliding conditions. The temperature distributions on the outer domains are achieved from two-dimensional modeling, and on the crucible wall, the heat source is given. The governing equations with their associated boundary conditions are solved using the commercial software, Fluent. The second order discretization schemes are applied to reduce numerical dissipation error. The flow at the top part of the furnace is dominated by forced convection, while it is dominated by natural convection around the crucible region. Therefore, the proportions of the viscous sub-layer and the fully turbulent layer may vary widely from cell to cell along the furnace wall. As a result, the non-equilibrium wall function is applied for the near-wall treatment of turbulence calculation instead of the standard wall function.
system, no crucible rotation is considered during our simulation. Crystal rotation rate is low (5 rpm) for many growth system, which has less effect on the gas flow around the crystal. The determination of the S/L interface is done by a home-built code [4,10,16] in a twodimensional modeling. The region as marked in Fig. 1a is chosen for the three-dimensional calculation. Two-dimensional geometry and interface shape are directly revolved to form the three-dimensional simulation zone. Temperature values on the boundaries and heat sources on the crucible walls are imported from the twodimensional result, which means axisymmetric thermal conditions are imposed on the three-dimensional configuration. With such setups, three-dimensional effects of the furnace will be weakened compared to a real growth system. However, the qualitative information can be still effective for a real furnace analysis. Fig. 2 illustrates five different furnace designs for a comparative analysis. Design I is the current furnace configuration, which is shown in Fig. 1 as a two-dimensional view. Design II has inclined side insulation with the convergent space from the middle to the top of the furnace. Design IV is upside down of design II. Design III is also has a vertical enclosure, but the diameter of the furnace space is smaller than that of design I. Design V has a particularly serrated shape of the insulation wall. The growth stages can be characterized by the length of the crystal and by the height of the melt. The predicted temperature distributions of the furnace with the different crystal lengths are presented in Fig. 3. The Cz growth process includes seeding, necking/shouldering, stable growth, and tailing stages. The short crystal in Fig. 3a represents the starting of the stable growth stage. One can see that temperature in the furnace is completely non-axisymmetric. However, the environmental temperature of the short crystal is uniform, since it is located in the crucible, where the effect of top gas flow is insignificantly. Therefore, in the following study the interface shape could be approximately assumed as constant, as an assumption mentioned above, when the different designs are adopted. In other words, the change of the design at the top part of the furnace is assumed not to significantly affect the temperature condition in the crucible. Fig. 3a also indicates that when the crystal is short, its rotation effect could be neglected, which tells that the crystal rotation will not change the results of the qualitative analysis in the following discussion. In other words, the conclusions will be the same for the cases with a strong or weak crystal rotation. As the crystal grows longer (see Fig. 3b), the gas temperature around the crystal, especially around the seed and shoulder regions, becomes highly irregular. The cracking at the upper part of the crystal as shown in Fig. 1b is caused by the complex gas cooling condition. During growth, a short crystal is rarely failed by cracking, while a long
3. Results and discussion The gas is nitrogen at ordinary pressure, and the flow rates of the top and bottom of the furnace are about 100 l/min and 10 l/min, respectively. Thermal conductivity of the crystal is considered by the Rosseland model, an effective value of 30 W/m-K. Other properties of the material are given in the Ref. [8]. Since crucible is static in many growth systems, including the real Yb:S-FAP Cz growth
Fig. 2. Five different furnace designs used in the simulation.
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Fig. 3. Three-dimensional simulation results of temperature field and velocity magnitude distribution of design I: (a) with a short crystal and (b) with a long crystal.
Fig. 4. (a) Streamlines and (b) temperature field of the different designs at (x, z) plane.
one is often with problems of seed broken and critical cracks. The experimental observations are coincident with the analysis of the simulation results. It is further noted that the isothermal lines in the melt are asymmetric due to the effect of asymmetric gas flow around the crucible. The magnitude of the velocity is also presented in Fig. 3. One can see that the maximum velocity magnitude region is caused by the forced convection, and is around the inlet and outlet regions, that is, at the top part of the furnace. The buoyancy convection around the crucible region is much weaker than the forced convection. In the following figures, the velocity field will only show the qualitative information of the flow structure, and the quantitative information can be obtained from Fig. 3. To examine the effects of the different designs, comparative analysis of the vertical sections is presented in Figs. 4 and 5. It is noted that the S/L interface shape keeps the same for each design due to the assumption that the modification of the furnace enclosure only affects the cooling of the top part of the crystal. The streamlines in (x, z) and (y, z) planes are constructed by the corresponding components of the velocity vector, for example, x-velocity and z-velocity are applied on the (x, z) plane to draw the streamlines. From Fig. 4a, one can see that the patterns of the vortices of the different designs have common characteristics. The vortices inner the crucible caused by natural convection and those at the top of the furnace resulting from the forced convection of the gas from the top inlets are similar in all designs. The figures further tell that there exists recirculation flow around the right
shoulder of the crystal. It is noted that the words of ‘‘left’’ and ‘‘right’’ here are adopted for the convenient analysis of the twodimensional sections. Design I has the minimal vortex, while design II has the largest one. Others have similar vortex structure. It is concluded that the gradually convergent furnace enclosure design (design II) strengthens the convection around the crystal. The arrows of the streamlines indicate that the gas flows up at the (x, z) section. Temperature distributions in Fig. 4b reveal that in design I the cold gas can deeply penetrate into the middle part of the furnace, and significantly reduce the environment temperature of the crystal. Designs II, III, and V have similar cold temperature regions around the crystal, i.e., the left cold region is like an arm stretching from the crucible ring to the seed along the crystal surface, and the right cold region is similar to the left one but much weaker. Both design I and design IV exhibit a long cold zone on the left surface and a short one on the right surface. The diverse gas patterns and temperature distributions in Fig. 4 depict that the furnace space design is efficient and important to organize the gas partition in the Cz growth system. The streamlines in Fig. 5a show that compared to other designs, design II has a uniquely large vortex around the left shoulder, and a particularly strong one in the upper right of the furnace space. One can find from the direction of the streamlines that gas flowing from the top inlet mixes with that flowing up from the right part of the furnace. The mixing of the two gas flows leads to the particular flow patterns of design II. Other designs, however, have similar vortex distributions, i.e., one vortex around
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Fig. 7. (a) Gas flow and (b) temperature field of the different designs at z ¼0.35 m.
Fig. 8. (a) Gas flow and (b) temperature field of the different designs at z ¼0.45 m.
Fig. 5. (a) Streamlines and (b) temperature field of the different designs at (y, z) plane.
Fig. 6. (a) Velocity vector and (b) temperature field of the different designs at z ¼0.25 m.
the right shoulder of the crystal, and one around the upper left of the furnace space, which are caused by the mixing of the downwards gas and upwards gas from the left part of the furnace. Temperature distribution on the vertical section (y, z) is given in Fig. 5b. In all designs except design II, the cold gas from the top inlets flows down to the crucible ring region on the right, and the right part of the crystal is critically cooled down. Design I is found to have the coldest temperature region around the crystal surface. Temperature distribution of design II is relatively uniform due to the factor that the strong vortex formed at the middle part of the furnace hinders the cold gas flowing downwards. As discussed
above, the vertical section (y, z) has very different gas flow pattern and temperature distribution from those of the section (x, z), which displays the non-axisymmetric effects of the furnace. Temperature distributions and velocity vectors of the cross sections in the furnace are presented in Figs. 6–8. Since the vertical sections and cross sections both can show the magnitude information of the variables, the legends in Figs. 3–5 can help the reader to know the magnitudes of the velocity. If the same scale of the velocity is used in Figs. 6–8, the vectors become out of order due to the large difference of the velocity magnitude at the different sections of the various designs. Therefore, the vectors in the following figures have different scale, and only present a qualitative analysis. The temperature distributions have the same legend as that in Fig. 3. The positions of the three chosen sections for analysis are marked in the leftmost picture of Fig. 4b by arrows with the z-axis coordinates of 0.25 m (Fig. 6), 0.35 m (Fig. 7), and 0.45 m (Fig. 8), respectively. From Fig. 6, one can see that the crystal temperature of design I is much lower than that of others due to the cooling of the cold environment. Design III also leads to a low-temperature environment around the crystal. Therefore, it can be concluded that the vertical enclosure of the Cz growth furnace is unfavorable for the achievement of axisymmetric hot zone. Designs IV and V have similar temperature distributions at the cross section with obvious cold regions around the crystal. Design II with the furnace enclosure of circular cone can provide relatively uniform temperature field around the crystal. The velocity vectors given in Fig. 6b indicate that design II has a distinguishing flow field from others, which has also been predicted in Fig. 5a. Crystal growth requires appropriate temperature gradients at the solidification front, which determines the growth rate. The temperature distribution at the cross section, z ¼0.35 m, plays a critical role of obtaining such temperature gradients, since most of the heat is lost through the section to the cold top of the furnace. From Fig. 7a, one can know that design I has the lowest
H.S. Fang et al. / Journal of Crystal Growth 361 (2012) 114–120
temperature at the section, which means that the temperature gradients at the growth front are the largest in all designs. Designs III and V have high temperature distributions at the section, and the growth rate will be slow. Especially, for crystals requiring large temperature gradients they may not work well. Design II has a moderate averaged temperature at the section, and can achieve a moderate growth rate consequently. The velocity field is obviously three-dimensional from the view of the vectors in Fig. 7b. Fig. 8 illustrates temperature distributions and velocity vectors at the cross section with the z-axis coordinate of 0.45 m. Due to the effect of top gas inlet and outlet, the gas flow pattern and temperature distribution are extremely non-uniform. However, the temperature field at the section has fewer effects on the crystal growth process than others do, since it has a long distance from the growing crystal. The top region of the furnace is affected directly by the top gas inlet and outlet, and all examined designs have the same conditions. As a result, temperature distributions and velocity vectors of all designs are very close at the section. From the above analysis, we can conclude that design II is able to provide a more uniform temperature distribution around the crystal and to obtain a moderate growth rate. Thus, the design is more promising to achieve crystals with less thermal stress induced dislocation. Then, numerical study is further performed to optimize the furnace structure based on it. The top gas inlet and outlet are significant for the gas partition in the furnace, and the designs discussed above have two symmetric gas inlets and one gas outlet. To further improve the hot zone, more inclined side insulation and more uniform gas inlets are investigated, and the results of temperature distributions are illustrated in Fig. 9. With more inclined side insulation (see Fig. 9a), the temperature around the crystal gets much more uniform than that of the previous design. Four gas inlets are also adopted with the more inclined enclosure as shown in Fig. 9b, which leads to the most
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relatively axisymmetric temperature field in the entire furnace. With the optimized design, it is obvious that thermal stress related defects due to the three-dimensional effects of the furnace geometry could be well minimized.
4. Conclusion During the Czochralski crystal growth, thermal stress related defects are dislocation and cracks, which are mainly caused by the non-uniform temperature field during the growth and cooling processes. In the paper, numerical simulation is conducted to study heat transfer and gas flow during a RF-heated Czochralski crystal growth system. Analysis of the three-dimensional effects in the furnace indicates that the general design is hard to obtain axisymmetric temperature field, and thus is unfavorable for the uniform cooling of the crystal. The top cold gas flows deeply down the crucible region, which accounts for the cold environment of the growing crystal, and leads to the cracks around the seed and shoulder of the crystal. Therefore, several furnace designs, with the modification of the furnace enclosure, are proposed for a comparative analysis. It is found that the design with circular cone enclosure (Design II) is the most promising for the achievement of the relatively axisymmetric temperature field with a moderate temperature gradient. More uniform inlets and more inclined side insulation enclosure can further improve the performance of the optimized design for the suppression of the three-dimensional effects. It should be noted that seal and evacuation of the furnace space might become more challenging due to the increase of the gas inlets. Conclusively, with the proposed simple but an effective design of the furnace enclosure, it is convenient to achieve more axisymmetric flow pattern and temperature distribution, and may be helpful to minimize thermal stress related defects during the Cz crystal growth.
Acknowledgment The work is supported in part by the National Natural Science Foundation of China (NSFC51106059), and in part by the Research Fund of the Doctoral Program of Higher Education of China for young scholars (20110142120065). References
Fig. 9. Temperature distributions of further improved furnaces based on design I: (a) with more inclined enclosure and (b) with more inclined enclosure and four gas inlets.
[1] C.D. Brandle, Czochralski growth of oxides, Journal of Crystal Growth 264 (2004) 593–604. [2] P.E. Tomaszewski, Jan Czochralski—father of the Czochralski method, Journal of Crystal Growth 236 (2002) 1–4. [3] J.J. Derby, R.A. Brown, On the dynamics of Czochralski crystal growth, Journal of Crystal Growth 83 (1987) 137–151. [4] H. Zhang, V. Prasad, D.F. Bliss, Modeling of high pressure, liquid-encapsulated Czochralski growth of InP crystals, Journal of Crystal Growth 169 (1996) 250–260. [5] R. Assaker, N. van den Bogaert, F. Dupret, Time-dependent simulation of the growth of large silicon crystals by the Czochralski technique using a turbulent model for melt convection, Journal of Crystal Growth 180 (1997) 450–460. ¨ ¨ [6] D. Vizman, O. Grabner, G. Muller, Three-dimensional numerical simulation of thermal convection in an industrial Czochralski melt: comparison to experimental results, Journal of Crystal Growth 233 (2001) 687–698. [7] C. Qisheng, D. Guyu, A. Ebadian, V. Prasad, Numerical Study on Flow Field and Temperature Distribution in Growth Process of 200 mm Czochralski Silicon Crystals, Journal of Rare Earths 25 (Supplement 2) (2007) 345–348. [8] H.S. Fang, S.R. Qiu, L.L. Zheng, K.I. Schaffers, J.B. Tassano, J.A. Caird, H. Zhang, Optimization of the cooling profile to achieve crack-free Yb: S-FAP crystals, Journal of Crystal Growth 310 (2008) 3825–3832. [9] M.S. Kulkarni, Defect dynamics in the presence of nitrogen in growing Czochralski silicon crystals, Journal of Crystal Growth 310 (2008) 324–335. [10] H. Fang, L. Zheng, H. Zhang, Y. Hong, Q. Deng, Reducing melt inclusion by submerged heater or baffle for optical crystal growth, Crystal Growth & Design 8 (2008) 1840–1848.
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[11] H.S. Fang, S. Wang, L. Zhou, N.G. Zhou, M.H. Lin, Influence of furnace design on the thermal stress during directional solidification of multicrystalline silicon, Journal of Crystal Growth 346 (2012) 5–11. [12] K.I. Schaffers, J.B. Tassano, A.B. Bayramian, R.C. Morris, Growth of Yb: S-FAP [Yb3þ :Sr5(PO4)3 F] crystals for the Mercury laser, Journal of Crystal Growth 253 (2003) 297–306. [13] V.V. Kalaev, I.Y. Evstratov, Y.N. Makarov, Gas flow effect on global heat transport and melt convection in Czochralski silicon growth, Journal of Crystal Growth 249 (2003) 87–99.
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Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Study and optimization of gas flow and temperature distribution in a Czochralski configuration H.S. Fang n, Z.L. Jin, X.M. Huang School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2012 Received in revised form 25 August 2012 Accepted 17 September 2012 Communicated by J.J. Derby Available online 25 September 2012
The Czochralski (Cz) method has virtually dominated the entire production of bulk single crystals with high productivity. Since the Cz-grown crystals are cylindrical, axisymmetric hot zone arrangement is required for an ideally high-quality crystal growth. However, due to three-dimensional effects the flow pattern and temperature field are inevitably non-axisymmetric. The grown crystal suffers from many defects, among which macro-cracks and micro-dislocation are mainly related to inhomogeneous temperature distribution during the growth and cooling processes. The task of the paper is to investigate gas partition and temperature distribution in a Cz configuration, and to optimize the furnace design for the reduction of the three-dimensional effects. The general design is found to be unfavorable to obtain the desired temperature conditions. Several different types of the furnace designs, modified at the top part of the side insulation, are proposed for a comparative analysis. The optimized one is chosen for further study, and the results display the excellence of the proposed design in suppression of three-dimensional effects to achieve relatively axisymmetric flow pattern and temperature distribution for the possible minimization of thermal stress related crystal defects. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Stresses A1.Fluid flows A2. Czochralski method
1. Introduction The Czochralski method is one of the most stable processes for the growth of bulk single crystals, and has virtually dominated the entire production in the microelectronic and optical industries [1,2]. The primary focus of current Cz growth is to produce massively large-size and high-quality crystals with low economic costs. Various defects, such as cracking and dislocation, related to non-uniform temperature field, significantly affect the performance of the crystals. Therefore, it is extremely important to get the detailed hot zone information in the growth configuration and to optimize it for the improvement of the crystal productivity. However, in situ measurements of the temperature within the crystal are difficult. With the development of computational technique, numerical modeling of the crystal growth process has become an ideal alternative. To improve the growth process, numerical studies of Cz crystal growth process have been conducted by many researchers. The work [1,3–10] covers many topics, such as hot zone design, defect control, modeling technique and so on, and has greatly speeded up the pace to achieve high-quality crystals in experiments and
n
Corresponding author. Tel./fax: þ 86 27 87542618. E-mail addresses: [email protected] (H.S. Fang), [email protected] (X.M. Huang). 0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.09.027
industries. The shape of the grown crystal is usually cylindrical due to the natural characteristics of the Cz technique, which is advantageous for the growth of crystals with uniform properties. An appropriate hot zone in the growth furnace is a key factor for high-quality crystal growth, and the cylindrical crystal geometry requires a correspondingly axisymmetric temperature field, which is determined by heat transport during the growth. Such an ideal hot zone could be achieved only if conduction and radiation were present with axisymmetric heating sources and thermal boundary conditions. However, in a real growth system, complex thermal boundaries drive buoyancy convection, and the gas inlet and outlet are non-axisymmetric in most furnaces, which results in the complex flow patterns. Therefore, in the regions, for example, at the top part of the furnace, where convective heat transfer cannot be neglected, temperature field is inevitable to be three-dimensional because of the convective heat transfer. The temperature of gas environment around the growing crystal, conclusively, is non-axisymmetric, which causes large thermal stress and stress-related defects in the crystal [8,11]. An example is the cracks in Yb:S-FAP laser crystals [8,12]. During the growth and cooling processes, cracks around the neck of the crystal are often observed after the crystal grows long, which is formed by the non-uniform cooling at the top part of the furnace. The above discussion shows that the gas partition, i.e., the design of gas flow and associated temperature field, is a key factor
H.S. Fang et al. / Journal of Crystal Growth 361 (2012) 114–120
affecting the crystal quality. Researchers have made many efforts to discuss the issue [13,14]. In the present paper, a coupled mathematical tool is applied to study heat transfer and gas flow during the growth stages of RF-heated Cz process. Threedimensional effects are examined on the various furnace designs with the modifications at the top part of the side insulation. Temperature distributions and gas flow patterns are presented at the different sections of the furnace, and an optimized design is proposed for a further examination to obtain a more uniform temperature distribution around the crystal. With the simple but effective design, the reduction of three-dimensional effects in the furnace can be realized to reduce crystal defects related to thermal stress, such as cracks and dislocation.
2. Mathematical models 2.1. Problem description Since developed by Jan Czochralski [2], the Cz growth method has been greatly improved in both design and technology. A typical configuration of Cz furnace is given in Fig. 1a. The system mainly includes a metal crucible, a seed rod, ZrO2-based heat insulation, and RF coil. The side insulation, usually in cylindrical shape, encloses the furnace to prevent heat loss. The schematic
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diagram is similar to the system in our previous study designed by Lawrence Livermore National Laboratory (LLNL), which is applied to grow large-size and high-quality Yb:S-FAP single crystals (see Fig. 1b) [8]. The chamber of the furnace is in inert (Nitrogen, Argon) or oxidizing atmosphere (Air) depending on the chemical components of the crystal and the crucible material. There are one gas inlet at the bottom of the furnace, two gas inlets at the top of the furnace, and one gas outlet at the central of the top furnace. It is noted here that the arrangement of gas inlet and outlet may vary for the different modified-Cz systems. The gas flowing into the furnace from the bottom plays a critical role in providing appropriate atmosphere for crystal growth, protecting the crucible from oxidation, and bringing away the evaporated component from the melt. Since the crystal growth process is extremely sensitive to the environment temperature of the crucible, the gas from the bottom inlet is strictly controlled, and the flow rate is usually low. The main purposes of the gas from the top inlets are to cool the top part of the furnace, to provide the temperature gradient for melt solidification, and to cool the growing or as-grown crystal. The gas flow rate of the top inlet is much larger than that of the bottom inlet. During the growth, thermal stress builds up inner the crystal, and dislocation generates at the growth front. As the crystal grows long, the upper part of the crystal will be in cool gas environment. The cooling of the crystal accompanies by the release of thermal stress in two ways. The slow release leads to dislocation multiplication, which generates at the growth front or exists in the seed, while the fast release results in cracking of the crystal as shown in Fig. 1b. Three-dimensional flow structure and non-axisymmetric temperature distribution severely aggravate the defects due to the non-uniform cooling. Therefore, the study and optimization of gas flow pattern and temperature distribution in a Cz configuration are extremely important for the achievement of high-quality crystal with low stress-related defects.
2.2. Mathematical models
Fig. 1. (a) A schematic diagram of Cz configuration and (b) a grown crystal with cracks.
For a vertical enclosure, such as the Cz furnace enclosed by insulation, turbulent convection begins at Rah 109 [15] due to natural convection, and the transition Reynolds number for forced flow is around 2000. The Rayleigh number is calculated based on the height of the furnace chamber. The furnace enclosure of the current study has a diameter of 0.16 m and a height of 0.9 m from the crucible ring to the furnace top. The crystal size is 0.08 m in diameter. The inlet and outlet have a diameter of 0.03 m. The estimated Rah number of the growth system is around 5 109 and the Reynolds number at the outlet is about 105. Therefore, the natural convection seems to be weak turbulence, while the flow regime of forced convection is strongly turbulent. Turbulent models must be applied for the gas flow modeling. The following assumptions are made: (1) natural convection due to temperature difference in the furnace is treated by Boussinesq approximation, (2) the free surface of the melt is assumed flat, i.e., the wetting of the melt on the iridium crucible is neglected, (3) the solid/liquid (S/L) interface shape is assumed constant, i.e., the modification of the top part of the furnace only affects the cooling of the top part of the crystal. Such an assumption is reasonable when the gas flow rate of the bottom inlet keeps constant, (4) melt convection does not take into consideration, and the convective heat transfer in the melt is approximated by an enhanced thermal conductivity, and (5) during a stable Cz crystal growth process, the timeaveraged values varies by time very slowly, so the gas turbulent flow is assumed to be quasi-steady state. The governing equations
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for the Newtonian fluid and incompressible turbulence flow can be described as follows: Continuity equation: @U i ¼0 @xi
ð1Þ
Momentum equation: 0 0
Uj
@ui uj @U i 1 @p @2 U i ¼ þn g 3 bT ðTT 0 Þ @xj r @xi @xj @xj @xj
ð2Þ
Energy equation: Uj
@TU i @2 T ¼a Sr þSt @xj @xj @xj
Boussinesq hypothesis: ! @U i @U j 2 0 0 þ ui uj ¼ nt dij k 3 @xj @xi
ð3Þ
ð4Þ
The standard k e model is applied for turbulence modeling, although the suitability of the model for buoyancy flows is always in question. The turbulent kinetic energy, k, and its dissipation rate, e, are solved by the following conservation equations [15]: ! @k @ nt @k @U i @U j @U i Ui ¼ þ e ð5Þ þ nt @xi @xi sk @xi @xj @xi @xj Ui
! @e @ nt @e e @U i @U j @U i e2 ¼ nt þ c2 þc1 @xi se @xi @xi @xj @xi @xj k rk rk
ð6Þ
where U i is the time-averaged velocity, i and j denote the vector directions, r is the fluid density, p is the time-averaged pressure, n is the fluid viscosity, u0 is the fluctuation of the velocity, nt ¼ cm k2 =e is the turbulent viscosity of the fluid, and dij is the unit tensor. Sr is the radiation source term, which is calculated by the well-established view factor method and St is the turbulence source term. The constants of the turbulent model, cm ,sk ,se ,c1 , and c2 are set to 0.09, 1.0, 1.3, 1.44, and 1.92, respectively. Boundary conditions are required to solve the above governing equations. The top inlet condition is a uniform velocity distribution, and the outlet is set to be outflow where all gradients of the calculated variables are null. The bottom gas is considered as a uniform flow through the gap between the crucible and the side insulation. Other surfaces are considered as non-sliding conditions. The temperature distributions on the outer domains are achieved from two-dimensional modeling, and on the crucible wall, the heat source is given. The governing equations with their associated boundary conditions are solved using the commercial software, Fluent. The second order discretization schemes are applied to reduce numerical dissipation error. The flow at the top part of the furnace is dominated by forced convection, while it is dominated by natural convection around the crucible region. Therefore, the proportions of the viscous sub-layer and the fully turbulent layer may vary widely from cell to cell along the furnace wall. As a result, the non-equilibrium wall function is applied for the near-wall treatment of turbulence calculation instead of the standard wall function.
system, no crucible rotation is considered during our simulation. Crystal rotation rate is low (5 rpm) for many growth system, which has less effect on the gas flow around the crystal. The determination of the S/L interface is done by a home-built code [4,10,16] in a twodimensional modeling. The region as marked in Fig. 1a is chosen for the three-dimensional calculation. Two-dimensional geometry and interface shape are directly revolved to form the three-dimensional simulation zone. Temperature values on the boundaries and heat sources on the crucible walls are imported from the twodimensional result, which means axisymmetric thermal conditions are imposed on the three-dimensional configuration. With such setups, three-dimensional effects of the furnace will be weakened compared to a real growth system. However, the qualitative information can be still effective for a real furnace analysis. Fig. 2 illustrates five different furnace designs for a comparative analysis. Design I is the current furnace configuration, which is shown in Fig. 1 as a two-dimensional view. Design II has inclined side insulation with the convergent space from the middle to the top of the furnace. Design IV is upside down of design II. Design III is also has a vertical enclosure, but the diameter of the furnace space is smaller than that of design I. Design V has a particularly serrated shape of the insulation wall. The growth stages can be characterized by the length of the crystal and by the height of the melt. The predicted temperature distributions of the furnace with the different crystal lengths are presented in Fig. 3. The Cz growth process includes seeding, necking/shouldering, stable growth, and tailing stages. The short crystal in Fig. 3a represents the starting of the stable growth stage. One can see that temperature in the furnace is completely non-axisymmetric. However, the environmental temperature of the short crystal is uniform, since it is located in the crucible, where the effect of top gas flow is insignificantly. Therefore, in the following study the interface shape could be approximately assumed as constant, as an assumption mentioned above, when the different designs are adopted. In other words, the change of the design at the top part of the furnace is assumed not to significantly affect the temperature condition in the crucible. Fig. 3a also indicates that when the crystal is short, its rotation effect could be neglected, which tells that the crystal rotation will not change the results of the qualitative analysis in the following discussion. In other words, the conclusions will be the same for the cases with a strong or weak crystal rotation. As the crystal grows longer (see Fig. 3b), the gas temperature around the crystal, especially around the seed and shoulder regions, becomes highly irregular. The cracking at the upper part of the crystal as shown in Fig. 1b is caused by the complex gas cooling condition. During growth, a short crystal is rarely failed by cracking, while a long
3. Results and discussion The gas is nitrogen at ordinary pressure, and the flow rates of the top and bottom of the furnace are about 100 l/min and 10 l/min, respectively. Thermal conductivity of the crystal is considered by the Rosseland model, an effective value of 30 W/m-K. Other properties of the material are given in the Ref. [8]. Since crucible is static in many growth systems, including the real Yb:S-FAP Cz growth
Fig. 2. Five different furnace designs used in the simulation.
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Fig. 3. Three-dimensional simulation results of temperature field and velocity magnitude distribution of design I: (a) with a short crystal and (b) with a long crystal.
Fig. 4. (a) Streamlines and (b) temperature field of the different designs at (x, z) plane.
one is often with problems of seed broken and critical cracks. The experimental observations are coincident with the analysis of the simulation results. It is further noted that the isothermal lines in the melt are asymmetric due to the effect of asymmetric gas flow around the crucible. The magnitude of the velocity is also presented in Fig. 3. One can see that the maximum velocity magnitude region is caused by the forced convection, and is around the inlet and outlet regions, that is, at the top part of the furnace. The buoyancy convection around the crucible region is much weaker than the forced convection. In the following figures, the velocity field will only show the qualitative information of the flow structure, and the quantitative information can be obtained from Fig. 3. To examine the effects of the different designs, comparative analysis of the vertical sections is presented in Figs. 4 and 5. It is noted that the S/L interface shape keeps the same for each design due to the assumption that the modification of the furnace enclosure only affects the cooling of the top part of the crystal. The streamlines in (x, z) and (y, z) planes are constructed by the corresponding components of the velocity vector, for example, x-velocity and z-velocity are applied on the (x, z) plane to draw the streamlines. From Fig. 4a, one can see that the patterns of the vortices of the different designs have common characteristics. The vortices inner the crucible caused by natural convection and those at the top of the furnace resulting from the forced convection of the gas from the top inlets are similar in all designs. The figures further tell that there exists recirculation flow around the right
shoulder of the crystal. It is noted that the words of ‘‘left’’ and ‘‘right’’ here are adopted for the convenient analysis of the twodimensional sections. Design I has the minimal vortex, while design II has the largest one. Others have similar vortex structure. It is concluded that the gradually convergent furnace enclosure design (design II) strengthens the convection around the crystal. The arrows of the streamlines indicate that the gas flows up at the (x, z) section. Temperature distributions in Fig. 4b reveal that in design I the cold gas can deeply penetrate into the middle part of the furnace, and significantly reduce the environment temperature of the crystal. Designs II, III, and V have similar cold temperature regions around the crystal, i.e., the left cold region is like an arm stretching from the crucible ring to the seed along the crystal surface, and the right cold region is similar to the left one but much weaker. Both design I and design IV exhibit a long cold zone on the left surface and a short one on the right surface. The diverse gas patterns and temperature distributions in Fig. 4 depict that the furnace space design is efficient and important to organize the gas partition in the Cz growth system. The streamlines in Fig. 5a show that compared to other designs, design II has a uniquely large vortex around the left shoulder, and a particularly strong one in the upper right of the furnace space. One can find from the direction of the streamlines that gas flowing from the top inlet mixes with that flowing up from the right part of the furnace. The mixing of the two gas flows leads to the particular flow patterns of design II. Other designs, however, have similar vortex distributions, i.e., one vortex around
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Fig. 7. (a) Gas flow and (b) temperature field of the different designs at z ¼0.35 m.
Fig. 8. (a) Gas flow and (b) temperature field of the different designs at z ¼0.45 m.
Fig. 5. (a) Streamlines and (b) temperature field of the different designs at (y, z) plane.
Fig. 6. (a) Velocity vector and (b) temperature field of the different designs at z ¼0.25 m.
the right shoulder of the crystal, and one around the upper left of the furnace space, which are caused by the mixing of the downwards gas and upwards gas from the left part of the furnace. Temperature distribution on the vertical section (y, z) is given in Fig. 5b. In all designs except design II, the cold gas from the top inlets flows down to the crucible ring region on the right, and the right part of the crystal is critically cooled down. Design I is found to have the coldest temperature region around the crystal surface. Temperature distribution of design II is relatively uniform due to the factor that the strong vortex formed at the middle part of the furnace hinders the cold gas flowing downwards. As discussed
above, the vertical section (y, z) has very different gas flow pattern and temperature distribution from those of the section (x, z), which displays the non-axisymmetric effects of the furnace. Temperature distributions and velocity vectors of the cross sections in the furnace are presented in Figs. 6–8. Since the vertical sections and cross sections both can show the magnitude information of the variables, the legends in Figs. 3–5 can help the reader to know the magnitudes of the velocity. If the same scale of the velocity is used in Figs. 6–8, the vectors become out of order due to the large difference of the velocity magnitude at the different sections of the various designs. Therefore, the vectors in the following figures have different scale, and only present a qualitative analysis. The temperature distributions have the same legend as that in Fig. 3. The positions of the three chosen sections for analysis are marked in the leftmost picture of Fig. 4b by arrows with the z-axis coordinates of 0.25 m (Fig. 6), 0.35 m (Fig. 7), and 0.45 m (Fig. 8), respectively. From Fig. 6, one can see that the crystal temperature of design I is much lower than that of others due to the cooling of the cold environment. Design III also leads to a low-temperature environment around the crystal. Therefore, it can be concluded that the vertical enclosure of the Cz growth furnace is unfavorable for the achievement of axisymmetric hot zone. Designs IV and V have similar temperature distributions at the cross section with obvious cold regions around the crystal. Design II with the furnace enclosure of circular cone can provide relatively uniform temperature field around the crystal. The velocity vectors given in Fig. 6b indicate that design II has a distinguishing flow field from others, which has also been predicted in Fig. 5a. Crystal growth requires appropriate temperature gradients at the solidification front, which determines the growth rate. The temperature distribution at the cross section, z ¼0.35 m, plays a critical role of obtaining such temperature gradients, since most of the heat is lost through the section to the cold top of the furnace. From Fig. 7a, one can know that design I has the lowest
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temperature at the section, which means that the temperature gradients at the growth front are the largest in all designs. Designs III and V have high temperature distributions at the section, and the growth rate will be slow. Especially, for crystals requiring large temperature gradients they may not work well. Design II has a moderate averaged temperature at the section, and can achieve a moderate growth rate consequently. The velocity field is obviously three-dimensional from the view of the vectors in Fig. 7b. Fig. 8 illustrates temperature distributions and velocity vectors at the cross section with the z-axis coordinate of 0.45 m. Due to the effect of top gas inlet and outlet, the gas flow pattern and temperature distribution are extremely non-uniform. However, the temperature field at the section has fewer effects on the crystal growth process than others do, since it has a long distance from the growing crystal. The top region of the furnace is affected directly by the top gas inlet and outlet, and all examined designs have the same conditions. As a result, temperature distributions and velocity vectors of all designs are very close at the section. From the above analysis, we can conclude that design II is able to provide a more uniform temperature distribution around the crystal and to obtain a moderate growth rate. Thus, the design is more promising to achieve crystals with less thermal stress induced dislocation. Then, numerical study is further performed to optimize the furnace structure based on it. The top gas inlet and outlet are significant for the gas partition in the furnace, and the designs discussed above have two symmetric gas inlets and one gas outlet. To further improve the hot zone, more inclined side insulation and more uniform gas inlets are investigated, and the results of temperature distributions are illustrated in Fig. 9. With more inclined side insulation (see Fig. 9a), the temperature around the crystal gets much more uniform than that of the previous design. Four gas inlets are also adopted with the more inclined enclosure as shown in Fig. 9b, which leads to the most
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relatively axisymmetric temperature field in the entire furnace. With the optimized design, it is obvious that thermal stress related defects due to the three-dimensional effects of the furnace geometry could be well minimized.
4. Conclusion During the Czochralski crystal growth, thermal stress related defects are dislocation and cracks, which are mainly caused by the non-uniform temperature field during the growth and cooling processes. In the paper, numerical simulation is conducted to study heat transfer and gas flow during a RF-heated Czochralski crystal growth system. Analysis of the three-dimensional effects in the furnace indicates that the general design is hard to obtain axisymmetric temperature field, and thus is unfavorable for the uniform cooling of the crystal. The top cold gas flows deeply down the crucible region, which accounts for the cold environment of the growing crystal, and leads to the cracks around the seed and shoulder of the crystal. Therefore, several furnace designs, with the modification of the furnace enclosure, are proposed for a comparative analysis. It is found that the design with circular cone enclosure (Design II) is the most promising for the achievement of the relatively axisymmetric temperature field with a moderate temperature gradient. More uniform inlets and more inclined side insulation enclosure can further improve the performance of the optimized design for the suppression of the three-dimensional effects. It should be noted that seal and evacuation of the furnace space might become more challenging due to the increase of the gas inlets. Conclusively, with the proposed simple but an effective design of the furnace enclosure, it is convenient to achieve more axisymmetric flow pattern and temperature distribution, and may be helpful to minimize thermal stress related defects during the Cz crystal growth.
Acknowledgment The work is supported in part by the National Natural Science Foundation of China (NSFC51106059), and in part by the Research Fund of the Doctoral Program of Higher Education of China for young scholars (20110142120065). References
Fig. 9. Temperature distributions of further improved furnaces based on design I: (a) with more inclined enclosure and (b) with more inclined enclosure and four gas inlets.
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