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Enhancement of heat transfer in Czochralski growth of silicon crystals with a chemical cooling technique
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crysgro
Enhancement of heat transfer in Czochralski growth of silicon crystals with a chemical cooling technique ⁎
Junling Ding, Lijun Liu , Wenhan Zhao Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
A R T I C L E I N F O
A BS T RAC T
Communicated by Satoshi Uda
The cost of producing single-crystalline silicon with the Czochralski method can be reduced by promoting the crystal size and/or crystal pulling rate. However, more latent heat of solidification needs to be released from the melt-crystal (m-c) interface during the crystal growth process. In this study, the C-CO2 chemical endothermic reaction is proposed as a novel and efficient cooling technique to solve this problem. Compared with the conventional gas cooling method, C-CO2 endothermic reaction method can significantly enhance the heat transfer in the crystal at the m-c interface. It was found that the heat transfer is more enhanced with a chemical reaction of smaller activation energy, and the m-c interface becomes flatter. The influence of the carbon concentration in the chemical reactive gas flow on the heat removal in the crystal at the m-c interface is also investigated. The cooling effect is significantly increased with the increase in the carbon concentration when it is small. However, when the carbon concentration in the reactive gas is high, the cooling effect just increases slightly. The research demonstrates that the proposed chemical endothermic reaction is a promising cooling technique to be applied in CZ-Si crystal growth with large size/high pulling rate.
Keywords: Heat transfer enhancement Cooling technique Czochralski method Single-crystalline silicon
1. Introduction Currently, Czochralski (CZ) growth process is the dominant technology for the production of high quality bulk single crystals for the electronic and photovoltaic applications. The urgent demand to reduce the cost of producing single-crystalline silicon puts forward a lot of challenges [1–3]. Up to now, increasing the size of single crystals and promoting the pulling rate of crystals are the two ways to reduce the crystal cost. Much works have been done on this two aspects [4–7]. However, it is worth pointing out that, with increase in the crystal size and the crystal pulling rate, huge latent heat of solidification needs to be released during the crystal growth process at the m-c interface. Therefore, it is important to enhance the heat transfer in the growing crystals. According to the cooling mechanism, the current cooling methods can be divided into two categories: physical cooling and chemical cooling. Among them, the research of physical cooling method is more mature, and the application is more extensive, including the forced air cooling and themicro channel cooling technique, etc. Different from the traditional physical cooling methods which taking away heat by changing the sensible enthalpy of the cooling medium, chemical heat sink cooling method removes the heat by chemical endothermic
⁎
reaction. Compared with the other cooling methods, the cooling capacity of the endothermic reaction increases significantly. When the reactant flows through the cooling channels, the heat can be taken away not only by radiation and convection, but also by chemical endothermic reaction. Much work has been done on the application of chemical endothermic reaction to the field of heat transfer enhancement [9–12]. For example, Cheng et al. [10] investigated the effect of using the decomposition of NH3 to improve the film cooling effectiveness of gas turbine blades. In order to meet the requirement for scramjet engine thermal protection, endothermic hydrocarbon fuel is used as an active cooling method to provide extra heat sink by thermal cracking reaction [8]. In addition, Li et al. [12] proposed the C–CO2 endothermic reaction as a chemical method to solve the instantaneous heat removal under high heat flux problem, and verified the feasibility of this technique by theoretical analysis and numerical study. Bao et al. [9] numerically studied the 3-D phenomena of fuel velocity, temperature and conversion in the cooling channels, and their effect on the utilization of fuel heat sink [9]. Overall, the chemical cooling is a very effective and promising method to remove high heat flux. In this study, we proposed to apply the chemical heat sink technique to the CZ-Si crystal growth with large crystal size/high pulling rate. It is known that a flat m-c interface or a slightly concave interface to
Corresponding author. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.jcrysgro.2016.11.036
Available online xxxx 0022-0248/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Ding, J., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.11.036
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al.
⎛ ∂T ⎞ ⎛ ∂T ⎞ ρc ΔHVg + ⎜λ ⎟ = ⎜λ ⎟ ⎝ ∂n ⎠m ⎝ ∂n ⎠c
(5)
where the subscript m and subscript c denote the values of crystal and melt side,ΔH is the latent heat of crystallization, Vg is the crystal pulling rate. 3. Chemical cooling model We use the C-CO2 endothermic reaction as the chemical cooling method [20,21]. The carbon and CO2 react in the form of a mixed gas. The reaction takes place inside the heat shield as shown in Fig. 1. When the reaction occurs, the temperature around the heat shield decreases, and then the heat transported from the crystal to the heat shield is enhanced by radiation and gas convection. Meanwhile, the renewable gas CO is generated. Through combustion of CO, the product-CO2 could be used again as a supplement of raw material [12]. The reaction can be approximately expressed as [21]:
Fig. 1. Configuration of an industrial-scale CZ furnace with chemical cooling technique.
C + CO2 → 2CO, Qa = 170kJ/mol
the crystal has the advantage to dislocation reduction and uniform doping [13–15]. The thermal stress near the m-c interface affects the distribution of vacancies and the formation of voids in the growing crystals [16–18]. Vanhellemont et al. [16] reported that the high level of compressive stress leads to an increase in the thermal equilibrium concentration of interstitials defects. Therefore, it is important to control the m-c interface shape and the thermal stress near the m-c interface while enhance the heat transfer in the hot zone. In this study, we proposed the application of C-CO2 endothermic reaction to the CZ-Si crystal growth. The effects of carbon types and carbon concentration in the chemical reactive gas flow on the cooling effect were investigated. We also compared the cooling effect of the proposed cooling technique with the conventional gas cooling method.
where Qa is the chemical reaction heat absorbed. The reaction rate is usually a function of the temperature. According to Arrhenius formula, reaction rate can be calculated using the following equations: Ea
k = Ae− RT
(7)
γ = k⋅c
(8)
where k is the reaction rate constant. A is the frequency factor for the reaction. Ea is the activation energy of the chemical reaction. R is the universal gas constant. T is the reaction temperature. c is the reactant concentration. γ is the reaction rate. The values of these variables were given in Ref. [21]. The heat absorbed during the endothermic reaction can be defined as:
2. Model description
Qabsorb = γ⋅Qa
2.1. Global model of heat transfer
(1)
⎞ ⎛ 2 →β (T − T ) u u ) = −∇p + ∇ ⎜ − μ∇⋅→ u ⎟ + ∇⋅(2μS ) − ρg ∇(ρ→→ 0 T ⎠ ⎝ 3
(2)
∇(ρCp → u T ) = ∇⋅(λ∇T )
(3)
(9)
we consider the chemical endothermic reaction heat through adding a source term to the energy equation. Then, the energy conservation equation for the reaction zone is rewritten as:
A schematic diagram of applying the chemical endothermic reaction in a CZ-Si crystal growth is shown in Fig. 1. The following assumptions are applied in the modeling [19]: (1) the furnace configuration is axisymmetric; (2) all radiative surfaces are diffuse-gray; (3) the growth system is quasi-steady; (4) the melt flow is incompressible; (5) natural convection satisfies the Boussinesq approximation; (6) the low Mach approximation and the ideal gas law are applied in the gas field simulation. With these assumptions, the governing equations for each zone can be written as: For the melt and argon domain:
∇⋅(ρ→ u)=0
(6)
∇(ρCp → u T ) = ∇⋅(λ∇T ) − Qabsorb
(10)
The algorithms for the global modeling of heat transfer have been described in our previous report [19]. The displacement-based thermoelastic stress model adopted to analyze the thermal stress distribution in the silicon crystal, is the same as that in our previous work [22]. All domains are fully coupled in the computation. The interface of any neighboring sub-domains satisfies the heat conservation and temperature continuity conditions. No-slip wall boundary condition is adopted in all solid surface for fluid flows. The structured/unstructured combined mesh scheme is adopted. The precisely interface tracking method is used to obtain the m-c interface shape. The temperature of the furnace wall is set to the constant ambient temperature 300 K. The argon flow rate is 33 L/min. The length and diameter of the crystal are 600 mm and 205 mm, respectively. The major thermal properties of the components in the CZ furnace are shown in Table 1 and Table 2.
The heat transfer equation for the heater is written as :
∇⋅(λ∇T ) + Q = 0
4. Results and discussions
(4)
u is the velocity, ρ is the density, μ is the dynamic viscosity, T is where → the temperature, T0 is the reference temperature, S is the strain rate g is the gravity tensor, βT is the thermal expansion coefficient, → acceleration vector, λ is the thermal conductivity, Cp is the heat capacity, Q is the heat generation rate per unit volume. The heat flux at the crystal-melt interface satisfies the following equation:
It is known that the activation energy of the C-CO2 reaction has a great effect on the reaction rate [20]. Meanwhile, the activation energy varies widely depending on the type of carbon: 150 ± 8 kJ/mol for petro coke, 171 ± 6 kJ/mol for coke breeze and 209 ± 16 kJ/mol for graphite. In addition, according to the Eq. (8), the reaction rate may be influenced by the concentration of reactants. Therefore, in the next sections, we focus our attention on the effects of the type and 2
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al.
Table 1 Thermal properties of components in the CZ furnace. Components
Density
Heat capacity
Thermal conductivity
Emissivity factor
(kg/m3)
(J/(kg K))
(W/(m K))
–
Insulation
2101
−317 +4.03T−2.4×10−4T2
0.45
Graphite Furnace wall Quartz crucible Silicon crystal Argon gas
2300 7900 2650 2330
+5.0×10−7T3 2019 477 1232 1059 520.64
Silicon melt
2530
1000
0.334 −1.83×10−4T +2.15×10−7T2 −1.89×10−11T3 90 15 4 22 0.01 +2.5×10−5T 64
0.8 0.6 0.85 0.3
0.3
concentration of carbon on the cooling effect of crystal. Fig. 2. Temperature gradient distributions in the crystal at the m-c interface with the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
5. Effect of the activation energy We first carried out a series of simulations to investigate the effects of the activation energy of different types of carbon on the enhancement of heat transfer in the hot zone of the furnace. In this series of simulations, the concentration of carbon in the chemical reactive gas flow is kept the same as 300 g/m3. Fig. 2 shows the axial and radial temperature gradient distribution in the silicon crystal at the m-c interface. It indicates that the chemical endothermic reaction has a great more influence on the axial temperature gradient than the radial temperature gradient. The C-CO2 endothermic reaction mainly enhance the axial heat transfer, while slightly weaken the heat transfer in the radial direction. We can also find that when the activation energy is 143 kJ/mol, the increase in the axial temperature gradient is more significant and it is increased by 37% compared to the gas cooling method. For the cases of activation energy equals to 143 kJ/mol and 247 kJ/mol, the heat flux removal in the crystal at the m-c interface are 3114 J and 3857 J, respectively, while it is 3000 J for the gas cooling case. It indicates that the heat transfer in the crystal at the m-c interface are enhanced by 3.8% and 28.7% compared to the conventional gas cooling technique. From Fig. 3, it is found that the m-c interface becomes flatter by using the C-CO2 endothermic reaction compared to the gas cooling method. With the decrease in the activation energy, the interface becomes flatter. Fig. 4 shows the thermal stresses in the growing crystals at the m-c interface. When the activation energy is 247 kJ/mol, the thermal stress near the m-c interface is always smaller than that of the gas cooling method. However, for the case in which the activation energy is 143 kJ/mol, the thermal stress near the interface is greater than that of the gas cooling method in the central area, while it is smaller in the peripheral regions of the crystal.
Fig. 3. The melt-crystal interface shapes for the cases with the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
observed that the heat transfer is more enhanced with the increase in the carbon concentration when it is small. However, the cooling effect just increases lightly with the increase in the carbon concentration when it is large. Fig. 6 shows the axial and radial temperature gradient distribution in the growing silicon crystal at the m-c interface for different carbon concentrations in the chemical reactive gas flow. It is found that the carbon concentration mainly influences the axial temperature gradient in the crystal at the m-c interface, while the radial temperature gradient almost keeps unchanged. Fig. 7 shows that the m-c interface shape changes with the carbon concentration. As shown in the figure, the m-c interface shape becomes
6. Effect of the reactant concentration The effects of carbon concentration in the chemical reactive gas flow in the cooling technique on the heat transfer in the growing crystal were investigated. In the simulations, the activation energy is 143 kJ/ mol. The heat transfer in the growing crystal at the m-c interface is changed with the concentration of carbon as shown in Fig. 5. It can be Table 2 Thermal properties of fluids in the system. Fluids
Melting heat (J/kg)
Thermal expansion coefficient (K−1)
Dynamic viscosity (kg/(m s))
Melting temperature (K)
Silicon Melt Argon gas
1,411,000
0.00014
0.0007 8.466×10−6+5.365×10−8T-8.68×10−12T2
1685
3
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al.
Fig. 6. Temperature gradient distributions in the crystal at the m-c interface for cases with different carbon concentrations in the chemical reactive gas flow. Fig. 4. Thermal stress distributions in the crystal at the m-c interface for the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/ mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
Fig. 7. The m-c interface shapes for cases with different carbon concentrations.
Fig. 5. The heat removal from the m-c interface versus the carbon concentration in the chemical reactive gas flow.
flatter with the increase in the carbon concentration. Fig. 8 shows that the thermal stress in the growing crystals at the m-c interface increases with the increase in the carbon concentration. The thermal stress has the same changing trend as the heat removal from the m-c interface.
7. Conclusions The C-CO2 endothermic reaction was proposed as a chemical cooling method to enhance the heat transfer in the CZ-Si crystal growth process. The effects of the activation energy of the chemical reaction and the concentration of reactant on the enhancement of heat transfer in the crystal at the m-c interface is investigated in detail. The results show that the C-CO2 endothermic reaction can significantly enhance the heat transfer in the growing crystal at the m-c interface and makes the interface flatter. With the decrease in the activation energy, the heat transfer is more enhanced and the m-c interface shape becomes flatter. The axial temperature gradient in the silicon crystal at the m-c interface increase with the increase in the carbon concentration in the chemical reactive gas flow. The cooling effect is significantly
Fig. 8. Thermal stress distributions in the crystal at the m-c interface for cases with different carbon concentrations.
4
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al. [5] [6] [7] [8] [9] [10]
increased with the increase in the carbon concentration when it is small. However, when the carbon concentration is high, the cooling effect just increases lightly. It demonstrates that the proposed chemical endothermic reaction is a promising cooling technique to be applied in CZ-Si crystal growth with large size/high pulling rate.
[11]
Acknowledgments
[12] [13] [14] [15]
This work was supported by the National Natural Science Foundation of China (No. 51676154) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JZ014).
[16] [17] [18]
References
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[1] G. Bye, B. Ceccaroli, Sol. Energy Mater. Sol. Cells 130 (2014) 634–646. [2] G. Coletti, I. Gordon, M.C. Schubert, W. Warta, et al., Sol. Energy Mater. Sol. Cells 130 (2014) 629–633. [3] G. Müller, J. Friedrich, J. Cryst. Growth 266 (2004) 1–19. [4] Z. Lu, S. Kimbel, J. Cryst. Growth 318 (2011) 193–195.
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L.J. Liu, X. Liu, Y. Wang, Int. J. Heat Mass Transf. 55 (2012) 53–60. V. Kalaev, A. Sattler, L. Kadinski, J. Cryst. Growth 413 (2015) 12–16. R. Assaker, N. VandenBogaert, F. Dupret, J. Cryst. Growth 180 (1997) 450–460. W. Ning, P. Yu, Z. Jin, J. Aerospace Eng. 227 (2012) 1780–1794. W. Bao, S.L. Zhang, J. Qin, W.X. Zhou, K.L. Xie, Energy 67 (2014) 149–161. K.Y. Cheng, W. Chen, S.Q. Liang, Y.X. Guo, X.L. Huai, X.H. Gui, Chinese Sci. Bull. 57 (2012) 3652–3659. Y. Feng, J. Qin, S. Zhang, W. Bao, Y. Cao, H.Y. Huang, Int. J. Heat Mass Transf. 91 (2015) 520–531. J. Li, G.M. He, C. Zeng, Y.M. Liu, Heat Mass Transfer 49 (2013) 1299–1305. L.J. Liu, K. Kakimoto, J. Cryst. Growth 310 (2008) 306–312. L.J. Liu, T. Kitashima, K. Kakimoto, J. Cryst. Growth 275 (2005) E2135–E2139. K. Nakamura, S. Maeda, S. Togawa, T. Saishoji, J. Tomioka, Electrochem. Soc. Proc. 4218 (2000) 31–43. J. Vanhellemont, Physica B: Condensed Matter 407 (2012) 3009–3012. K. Nakamura, R. Suewaka, B. Ko, ECS Solid State Letters (3) (2014) N5–N7. K. Tanahashi, M. Kikuchi, T. Higashino, N. Inoue, Y. Mizokawa, J. Cryst. Growth 210 (2000) 45–48. L.J. Liu, K. Kakimoto, Int. J. Heat Mass Transf. 48 (2005) 4481–4491. M. Zamalloa, D. MA, T.A. Utigard, ISIJ Int. 35 (1995) 458–463. E. Mon, N.R. Amundson, Ind. Eng. Chem. Fundam. 17 (1978) 313–321. W.H. Zhao, L.J. Liu, L. Sun, A.N. Geng, J. Cryst. Growth 401 (2014) 296–301.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crysgro
Enhancement of heat transfer in Czochralski growth of silicon crystals with a chemical cooling technique ⁎
Junling Ding, Lijun Liu , Wenhan Zhao Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
A R T I C L E I N F O
A BS T RAC T
Communicated by Satoshi Uda
The cost of producing single-crystalline silicon with the Czochralski method can be reduced by promoting the crystal size and/or crystal pulling rate. However, more latent heat of solidification needs to be released from the melt-crystal (m-c) interface during the crystal growth process. In this study, the C-CO2 chemical endothermic reaction is proposed as a novel and efficient cooling technique to solve this problem. Compared with the conventional gas cooling method, C-CO2 endothermic reaction method can significantly enhance the heat transfer in the crystal at the m-c interface. It was found that the heat transfer is more enhanced with a chemical reaction of smaller activation energy, and the m-c interface becomes flatter. The influence of the carbon concentration in the chemical reactive gas flow on the heat removal in the crystal at the m-c interface is also investigated. The cooling effect is significantly increased with the increase in the carbon concentration when it is small. However, when the carbon concentration in the reactive gas is high, the cooling effect just increases slightly. The research demonstrates that the proposed chemical endothermic reaction is a promising cooling technique to be applied in CZ-Si crystal growth with large size/high pulling rate.
Keywords: Heat transfer enhancement Cooling technique Czochralski method Single-crystalline silicon
1. Introduction Currently, Czochralski (CZ) growth process is the dominant technology for the production of high quality bulk single crystals for the electronic and photovoltaic applications. The urgent demand to reduce the cost of producing single-crystalline silicon puts forward a lot of challenges [1–3]. Up to now, increasing the size of single crystals and promoting the pulling rate of crystals are the two ways to reduce the crystal cost. Much works have been done on this two aspects [4–7]. However, it is worth pointing out that, with increase in the crystal size and the crystal pulling rate, huge latent heat of solidification needs to be released during the crystal growth process at the m-c interface. Therefore, it is important to enhance the heat transfer in the growing crystals. According to the cooling mechanism, the current cooling methods can be divided into two categories: physical cooling and chemical cooling. Among them, the research of physical cooling method is more mature, and the application is more extensive, including the forced air cooling and themicro channel cooling technique, etc. Different from the traditional physical cooling methods which taking away heat by changing the sensible enthalpy of the cooling medium, chemical heat sink cooling method removes the heat by chemical endothermic
⁎
reaction. Compared with the other cooling methods, the cooling capacity of the endothermic reaction increases significantly. When the reactant flows through the cooling channels, the heat can be taken away not only by radiation and convection, but also by chemical endothermic reaction. Much work has been done on the application of chemical endothermic reaction to the field of heat transfer enhancement [9–12]. For example, Cheng et al. [10] investigated the effect of using the decomposition of NH3 to improve the film cooling effectiveness of gas turbine blades. In order to meet the requirement for scramjet engine thermal protection, endothermic hydrocarbon fuel is used as an active cooling method to provide extra heat sink by thermal cracking reaction [8]. In addition, Li et al. [12] proposed the C–CO2 endothermic reaction as a chemical method to solve the instantaneous heat removal under high heat flux problem, and verified the feasibility of this technique by theoretical analysis and numerical study. Bao et al. [9] numerically studied the 3-D phenomena of fuel velocity, temperature and conversion in the cooling channels, and their effect on the utilization of fuel heat sink [9]. Overall, the chemical cooling is a very effective and promising method to remove high heat flux. In this study, we proposed to apply the chemical heat sink technique to the CZ-Si crystal growth with large crystal size/high pulling rate. It is known that a flat m-c interface or a slightly concave interface to
Corresponding author. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.jcrysgro.2016.11.036
Available online xxxx 0022-0248/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Ding, J., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.11.036
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al.
⎛ ∂T ⎞ ⎛ ∂T ⎞ ρc ΔHVg + ⎜λ ⎟ = ⎜λ ⎟ ⎝ ∂n ⎠m ⎝ ∂n ⎠c
(5)
where the subscript m and subscript c denote the values of crystal and melt side,ΔH is the latent heat of crystallization, Vg is the crystal pulling rate. 3. Chemical cooling model We use the C-CO2 endothermic reaction as the chemical cooling method [20,21]. The carbon and CO2 react in the form of a mixed gas. The reaction takes place inside the heat shield as shown in Fig. 1. When the reaction occurs, the temperature around the heat shield decreases, and then the heat transported from the crystal to the heat shield is enhanced by radiation and gas convection. Meanwhile, the renewable gas CO is generated. Through combustion of CO, the product-CO2 could be used again as a supplement of raw material [12]. The reaction can be approximately expressed as [21]:
Fig. 1. Configuration of an industrial-scale CZ furnace with chemical cooling technique.
C + CO2 → 2CO, Qa = 170kJ/mol
the crystal has the advantage to dislocation reduction and uniform doping [13–15]. The thermal stress near the m-c interface affects the distribution of vacancies and the formation of voids in the growing crystals [16–18]. Vanhellemont et al. [16] reported that the high level of compressive stress leads to an increase in the thermal equilibrium concentration of interstitials defects. Therefore, it is important to control the m-c interface shape and the thermal stress near the m-c interface while enhance the heat transfer in the hot zone. In this study, we proposed the application of C-CO2 endothermic reaction to the CZ-Si crystal growth. The effects of carbon types and carbon concentration in the chemical reactive gas flow on the cooling effect were investigated. We also compared the cooling effect of the proposed cooling technique with the conventional gas cooling method.
where Qa is the chemical reaction heat absorbed. The reaction rate is usually a function of the temperature. According to Arrhenius formula, reaction rate can be calculated using the following equations: Ea
k = Ae− RT
(7)
γ = k⋅c
(8)
where k is the reaction rate constant. A is the frequency factor for the reaction. Ea is the activation energy of the chemical reaction. R is the universal gas constant. T is the reaction temperature. c is the reactant concentration. γ is the reaction rate. The values of these variables were given in Ref. [21]. The heat absorbed during the endothermic reaction can be defined as:
2. Model description
Qabsorb = γ⋅Qa
2.1. Global model of heat transfer
(1)
⎞ ⎛ 2 →β (T − T ) u u ) = −∇p + ∇ ⎜ − μ∇⋅→ u ⎟ + ∇⋅(2μS ) − ρg ∇(ρ→→ 0 T ⎠ ⎝ 3
(2)
∇(ρCp → u T ) = ∇⋅(λ∇T )
(3)
(9)
we consider the chemical endothermic reaction heat through adding a source term to the energy equation. Then, the energy conservation equation for the reaction zone is rewritten as:
A schematic diagram of applying the chemical endothermic reaction in a CZ-Si crystal growth is shown in Fig. 1. The following assumptions are applied in the modeling [19]: (1) the furnace configuration is axisymmetric; (2) all radiative surfaces are diffuse-gray; (3) the growth system is quasi-steady; (4) the melt flow is incompressible; (5) natural convection satisfies the Boussinesq approximation; (6) the low Mach approximation and the ideal gas law are applied in the gas field simulation. With these assumptions, the governing equations for each zone can be written as: For the melt and argon domain:
∇⋅(ρ→ u)=0
(6)
∇(ρCp → u T ) = ∇⋅(λ∇T ) − Qabsorb
(10)
The algorithms for the global modeling of heat transfer have been described in our previous report [19]. The displacement-based thermoelastic stress model adopted to analyze the thermal stress distribution in the silicon crystal, is the same as that in our previous work [22]. All domains are fully coupled in the computation. The interface of any neighboring sub-domains satisfies the heat conservation and temperature continuity conditions. No-slip wall boundary condition is adopted in all solid surface for fluid flows. The structured/unstructured combined mesh scheme is adopted. The precisely interface tracking method is used to obtain the m-c interface shape. The temperature of the furnace wall is set to the constant ambient temperature 300 K. The argon flow rate is 33 L/min. The length and diameter of the crystal are 600 mm and 205 mm, respectively. The major thermal properties of the components in the CZ furnace are shown in Table 1 and Table 2.
The heat transfer equation for the heater is written as :
∇⋅(λ∇T ) + Q = 0
4. Results and discussions
(4)
u is the velocity, ρ is the density, μ is the dynamic viscosity, T is where → the temperature, T0 is the reference temperature, S is the strain rate g is the gravity tensor, βT is the thermal expansion coefficient, → acceleration vector, λ is the thermal conductivity, Cp is the heat capacity, Q is the heat generation rate per unit volume. The heat flux at the crystal-melt interface satisfies the following equation:
It is known that the activation energy of the C-CO2 reaction has a great effect on the reaction rate [20]. Meanwhile, the activation energy varies widely depending on the type of carbon: 150 ± 8 kJ/mol for petro coke, 171 ± 6 kJ/mol for coke breeze and 209 ± 16 kJ/mol for graphite. In addition, according to the Eq. (8), the reaction rate may be influenced by the concentration of reactants. Therefore, in the next sections, we focus our attention on the effects of the type and 2
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
J. Ding et al.
Table 1 Thermal properties of components in the CZ furnace. Components
Density
Heat capacity
Thermal conductivity
Emissivity factor
(kg/m3)
(J/(kg K))
(W/(m K))
–
Insulation
2101
−317 +4.03T−2.4×10−4T2
0.45
Graphite Furnace wall Quartz crucible Silicon crystal Argon gas
2300 7900 2650 2330
+5.0×10−7T3 2019 477 1232 1059 520.64
Silicon melt
2530
1000
0.334 −1.83×10−4T +2.15×10−7T2 −1.89×10−11T3 90 15 4 22 0.01 +2.5×10−5T 64
0.8 0.6 0.85 0.3
0.3
concentration of carbon on the cooling effect of crystal. Fig. 2. Temperature gradient distributions in the crystal at the m-c interface with the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
5. Effect of the activation energy We first carried out a series of simulations to investigate the effects of the activation energy of different types of carbon on the enhancement of heat transfer in the hot zone of the furnace. In this series of simulations, the concentration of carbon in the chemical reactive gas flow is kept the same as 300 g/m3. Fig. 2 shows the axial and radial temperature gradient distribution in the silicon crystal at the m-c interface. It indicates that the chemical endothermic reaction has a great more influence on the axial temperature gradient than the radial temperature gradient. The C-CO2 endothermic reaction mainly enhance the axial heat transfer, while slightly weaken the heat transfer in the radial direction. We can also find that when the activation energy is 143 kJ/mol, the increase in the axial temperature gradient is more significant and it is increased by 37% compared to the gas cooling method. For the cases of activation energy equals to 143 kJ/mol and 247 kJ/mol, the heat flux removal in the crystal at the m-c interface are 3114 J and 3857 J, respectively, while it is 3000 J for the gas cooling case. It indicates that the heat transfer in the crystal at the m-c interface are enhanced by 3.8% and 28.7% compared to the conventional gas cooling technique. From Fig. 3, it is found that the m-c interface becomes flatter by using the C-CO2 endothermic reaction compared to the gas cooling method. With the decrease in the activation energy, the interface becomes flatter. Fig. 4 shows the thermal stresses in the growing crystals at the m-c interface. When the activation energy is 247 kJ/mol, the thermal stress near the m-c interface is always smaller than that of the gas cooling method. However, for the case in which the activation energy is 143 kJ/mol, the thermal stress near the interface is greater than that of the gas cooling method in the central area, while it is smaller in the peripheral regions of the crystal.
Fig. 3. The melt-crystal interface shapes for the cases with the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
observed that the heat transfer is more enhanced with the increase in the carbon concentration when it is small. However, the cooling effect just increases lightly with the increase in the carbon concentration when it is large. Fig. 6 shows the axial and radial temperature gradient distribution in the growing silicon crystal at the m-c interface for different carbon concentrations in the chemical reactive gas flow. It is found that the carbon concentration mainly influences the axial temperature gradient in the crystal at the m-c interface, while the radial temperature gradient almost keeps unchanged. Fig. 7 shows that the m-c interface shape changes with the carbon concentration. As shown in the figure, the m-c interface shape becomes
6. Effect of the reactant concentration The effects of carbon concentration in the chemical reactive gas flow in the cooling technique on the heat transfer in the growing crystal were investigated. In the simulations, the activation energy is 143 kJ/ mol. The heat transfer in the growing crystal at the m-c interface is changed with the concentration of carbon as shown in Fig. 5. It can be Table 2 Thermal properties of fluids in the system. Fluids
Melting heat (J/kg)
Thermal expansion coefficient (K−1)
Dynamic viscosity (kg/(m s))
Melting temperature (K)
Silicon Melt Argon gas
1,411,000
0.00014
0.0007 8.466×10−6+5.365×10−8T-8.68×10−12T2
1685
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Fig. 6. Temperature gradient distributions in the crystal at the m-c interface for cases with different carbon concentrations in the chemical reactive gas flow. Fig. 4. Thermal stress distributions in the crystal at the m-c interface for the chemical cooling method (including two types of carbon. case 1, the activation energy is 143 kJ/ mol; case 2, the activation energy is 247 kJ/mol) and the gas cooling technique.
Fig. 7. The m-c interface shapes for cases with different carbon concentrations.
Fig. 5. The heat removal from the m-c interface versus the carbon concentration in the chemical reactive gas flow.
flatter with the increase in the carbon concentration. Fig. 8 shows that the thermal stress in the growing crystals at the m-c interface increases with the increase in the carbon concentration. The thermal stress has the same changing trend as the heat removal from the m-c interface.
7. Conclusions The C-CO2 endothermic reaction was proposed as a chemical cooling method to enhance the heat transfer in the CZ-Si crystal growth process. The effects of the activation energy of the chemical reaction and the concentration of reactant on the enhancement of heat transfer in the crystal at the m-c interface is investigated in detail. The results show that the C-CO2 endothermic reaction can significantly enhance the heat transfer in the growing crystal at the m-c interface and makes the interface flatter. With the decrease in the activation energy, the heat transfer is more enhanced and the m-c interface shape becomes flatter. The axial temperature gradient in the silicon crystal at the m-c interface increase with the increase in the carbon concentration in the chemical reactive gas flow. The cooling effect is significantly
Fig. 8. Thermal stress distributions in the crystal at the m-c interface for cases with different carbon concentrations.
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increased with the increase in the carbon concentration when it is small. However, when the carbon concentration is high, the cooling effect just increases lightly. It demonstrates that the proposed chemical endothermic reaction is a promising cooling technique to be applied in CZ-Si crystal growth with large size/high pulling rate.
[11]
Acknowledgments
[12] [13] [14] [15]
This work was supported by the National Natural Science Foundation of China (No. 51676154) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JZ014).
[16] [17] [18]
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