Numerical Investigation of the Influence of Material Property of a Crucible on Interface Shape in a Unidirectional Solidification Process Hiroaki Miyazawa,† Lijun Liu,‡,§ and Koichi Kakimoto*,‡
CRYSTAL GROWTH & DESIGN 2009 VOL. 9, NO. 1 267–272
Graduate School of Engineering, Kyushu UniVersity, 6-1, Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan, RIAM, Kyushu UniVersity, 6-1, Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan, and T School of Energy and Power Engineering, Xi’an Jiaotong UniVersity, Xi’an, Shaanxi 710049, China ReceiVed April 28, 2008; ReVised Manuscript ReceiVed September 18, 2008
ABSTRACT: We carried out calculations to investigate the influence of thermal conductivity of the wall of a crucible on melt-crystal interface shape using three-dimensional global analyses. It was found that thermal conductivity of the wall of a crucible has significant influence on the melt-crystal interface shape due to modification of the amount of outgoing heat flux through the wall of a crucible. The results indicate that we should control not only heater power, growth velocity, and melt flow but also thermophysical properties of the wall of a crucible in order to reduce deformation of the melt-crystal interface.
1. Introduction One of the most important defects of degradation of minority carrier lifetime and open-circuit voltage in silicon crystals for solar cells is dislocations generated during crystal growth by the unidirectional solidification method.1 Generation of dislocations mainly depends on thermal stress distribution in a growing crystal.2-4 Therefore, reduction of thermal stress in the crystal is a critical issue to improve the quality of silicon ingots. It is important to optimize the melt-crystal (m-c) interface shape in order to reduce the thermal stress in the solidification process since the m-c interface shape has a great effect on the magnitude of thermal stress in the crystals.2,5 Formation of a flat m-c interface reduces such thermal stress. To realize such an m-c flat interface, heat flux through the interface should flow perpendicular to the interface, i.e., in the growth direction. Due to recent developments in computer technology and computation techniques, numerical simulation has become a powerful tool for investigation of an m-c interface shape in a solidification process.3,6 Since conduction, convection in the melt and radiative heat transfer in a growth furnace have effects on m-c interface shape,7 global modeling that takes into account radiative, conductive and convective heat transfer in a furnace is a key technique for investigation and optimization of the m-c interface shape. There have been several works on twodimensional (2D) global analyses of a unidirectional solidification process with cylindrical crucibles;8,9 however, a square crucible is used to obtain a square-shaped crystal, which is commonly used for solar cells. When square crucibles are used, the configuration of the furnace becomes asymmetric, and heat transfer in the furnace consequently becomes three-dimensional. Three-dimensional (3D) global modeling is therefore necessary for investigation of an m-c interface shape with square crucibles. In this study, we carried out calculations to investigate m-c interface shapes with two different materials for the wall of the crucible different thermal conductivities in order to determine * Corresponding author. Telephone: +81-92-583-7741. Fax: +81- 92-5837743. E-mail:
[email protected]. † Graduate School of Engineering, Kyushu University. ‡ RIAM, Kyushu University. § T School of Energy and Power Engineering, Xi’an Jiaotong University.
Figure 1. Configuration and computation grid of a unidirectional solidification furnace with cylindrical crucibles.
the influence of material property of the wall of a crucible on m-c interface shape by controlling heat flux flow near the m-c interface.
2. Model Description and Computation Method Figure 1 shows the configuration and dimensions of the unidirectional solidification furnace used for 2D global analysis with cylindrical crucibles. The domains of all components in the furnace are subdivided into a number of block regions shown in the left part of Figure 1. The melt, a crystal, crucibles, pedestals, thermal shields, and three heaters are labeled as 1, 2, 3-4, 5-6, 7-10, and 11-13, respectively. Each block is then discretized by structured grids as shown in the right part of Figure 1. The following assumptions are made for the 2D global
10.1021/cg800435d CCC: $40.75 2009 American Chemical Society Published on Web 11/19/2008
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Figure 2. 3D computation grid used for the 3D global model.
analysis: (1) the geometry of the furnace configuration is axisymmetric, (2) radiative heat transfer is modeled as diffusegray surface radiation, (3) the melt flow in the crucible is laminar and incompressible, and (4) the effect of gas flow in the furnace is negligible. Conductive heat transfer in all solid components, radiative heat exchange between all diffusive surfaces in the furnace and the Navier-Stokes equations for the melt flow in the crucible are coupled and solved iteratively by a finite volume method in a steady condition. The governing equations for melt convection in a crucible are as follows:
∇·b V)0
(1) bT
b· ∇b b + ∇ V )] - Fg FV V ) - ∇ p + ∇ · [µ(∇V bβT(T - Tm) (2) b · ∇ T ) ∇ · (k ∇ T) FcV
(3)
where b V, F, p, µ, b g, βT, c and k are melt velocity, melt density, melt pressure, melt viscosity, gravitational acceleration, thermal expansion coefficient, heat capacity and thermal conductivity, respectively. The melt flow was solved by a finite volume method. The shape of an m-c interface is obtained by using a dynamic interface tracking method.10 3D global analysis with square crucibles requires large computational resources because of the huge number of 3D structured grids. To overcome this difficulty, a 2D/3D mixed
Figure 4. The m-c interface shape and temperature distribution of the melt and crystal obtained by 3D global analysis with two different thermal conductivities of the wall of a crucible: (a) 53 [W/mK] and (b) 0.267 [W/mK].
discretization scheme is employed to reduce the requirement of computational resources.10,11 The domains in the central area of the furnace, in which the configuration and heat transfer are
Figure 3. Schematics of melt, crystal, and a crucible, the crucible made of carbon (a) and carbon felt (b).
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nonaxisymmetric, are discretized in a 3D way. A local 3D computational grid in the domains in the central area of the furnace is established as shown in Figure 2. The other block regions that are away from the central area of the furnace, in which the configuration and heat transfer are axisymmetric, are discretized in a 2D way. The 2D computational grid in other block regions of the furnace is the same as that used in the 2D global model as shown in Figure 1. The assumptions used in the 3D global analysis are (2), (3), and (4) stated above. The computation method used for 3D global analysis is the same as that used for 2D global analysis. The calculation of view factors in the radiation modeling is a key issue of the model with such a space discretization scheme. Let ∂V 2 and ∂V 3 denote the radiative surfaces that fall in the 2D domain and 3D domain of a furnace, respectively. For any two radiative surface elements X and Xi, the view factor Ke(X,Xi) between them is deduced and calculated as follows: When Xi ∈ ∂V 2,
Ke(X, Xi) ) KC(x bc, b x ic ) when Xi ∈ ∂V 3 and X ∈ ∂V 2,
(4)
Ke(X, Xi) )
(θi2 - θi1) bc, b x ic ) KC(x 2π
(5)
and when Xi ∈ ∂V 3 and X ∈ ∂V 3,
Ke(X, Xi) )
∫θθ
i2
i1
K(x bc, b x i∗) dθ∗
(6)
where b xc′ and b xic′ are the respective circumferential projections in a meridional plane of the geometrical centers of X and Xi. The azimuthal angle range (θi1,θi2) is covered by the radiative surface element Xi ∈ ∂V 3 and b xi* ) (ric′ cos θ*, ric′ sin θ*, zic′). The axisymmetric view factor KC(x bc′, b xic′) is defined by the integral
KC(x bc, b x ic ) )
∫02π K(xbc, bx i∗) dθ∗
(7)
It is calculated by taking into account all objects in view and all hidden objects in the furnace with a view and hidden algorithm.12 We obtained temperature and flow distributions in the furnace in 2D configuration and then we used the distributions as an initial configuration for the 3D calculation.
Figure 5. Temperature (a), velocity (b) and particle paths (c) in the case with large thermal conductivity (53 [W/mK]) of the wall in planes of D-o-Z (l) and X-o-Z (s).
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Figure 6. Temperature (a), velocity (b), and particle paths (c) in the case with small thermal conductivity (0.267 [W/mK]) of the wall in planes of D-o-Z (l) and X-o-Z (s).
Figure 3 shows schematics of the melt, crystal, crucible and pedestal used for 2D and 3D global analyses. In this calculation, we use two kinds of material for the wall of a carbon crucible with two different thermal conductivities labeled 4 in Figures 1 and 2 to modify direction of heat flux near the m-c interface. Thermal conductivities of the walls of carbon crucibles made of carbon and felt, shown in parts a and b of Figure 3, were set to 53 and 0.267 W/mK, respectively. The heater power distributions of heater 1 and heater 2 were set to 60% and 40%, respectively.
3. Results and Discussion Parts a and b of Figure 4 show the m-c interface shape and temperature distribution of the melt and crystal obtained by 3D global analysis with the two different thermal conductivities of the wall of a crucible. The m-c interface shape with high thermal conductivity shown in Figure 4a becomes more convex
to the melt than that with low thermal conductivity shown in Figure 4b because the direction of heat flux at the m-c interface with high thermal conductivity tilts toward the outside through the vertical wall of the crucible. Figures 5 and 6 show distributions of temperature, velocity and particle paths in the cases with two different thermal conductivities of the wall in planes of D-o-Z and X-o-Z shown in Figure 4, respectively. In the case with large thermal conductivity shown in Figures 5 and 6 (l-a) to (l-c), flow velocity near the m-c interface and deformation of the m-c interface are larger than those with low thermal conductivity shown in Figures 5 and 6 (s-a) to (s-c). The flow patterns in the two cases are also different. Due to the difference in thermal conductivities of the wall of a crucible in the two cases, heat flux through the wall of a crucible becomes larger in the case with a larger thermal conductivity than that with a small thermal conductivity as
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Figure 7. Outgoing heat flux through the crystal and melt with larger thermal conductivity (a) and small thermal conductivity (b) [W/m2].
conductivity shown in Figure 7a, the heat flux at the m-c interface tilts toward the outside. The bottom of the melt in the crucible is cooled and then the crystal grows from the bottom of the crucible. Therefore, most of the heat transfer is from the top of the melt to the bottom of the crystal through the m-c interface. On the other hand, there is also outgoing heat flux through the vertical wall, the magnitude of which is larger with large thermal conductivity of the wall of a crucible than with small thermal conductivity. The tilt of heat flux causes deformation of the m-c interface because heat flux flows almost perpendicular to the m-c interface, which can be assumed to be an isothermal surface of melting temperature. Therefore, it was found that the outgoing heat flux through the vertical wall of a crucible has significant influence on deformation of the m-c interface. Flow velocity of the melt near the edge of the m-c interface in the case with large thermal conductivity of the wall of a crucible is larger than that in the case with small thermal conductivity. This is because that the temperature gradient in the radial direction near the edge of the m-c interface becomes larger in the case with large thermal conductivity of the wall of a crucible due to large deformation of the m-c interface. Therefore, reduction of flow velocity of the melt near the edge of the m-c interface is required to reduce deformation of the m-c interface as shown in Figures 5 and 6. In order to reduce the magnitude of deformation of the m-c interface, the outgoing heat flux through the vertical wall of the crucible must be reduced. The gradient of temperature near a crucible wall at the m-c interface becomes large in the case with large thermal conductivity of the wall of a crucible, and the m-c interface in the case with large thermal conductivity therefore becomes more convex to the melt than that in the case with small thermal conductivity as shown in Figure 8.
4. Conclusion
Figure 8. Deflection of the m-c interface with large thermal conductivity (a) and small thermal conductivity (b).
shown in Figure 7. A positive value of heat flux shown in Figure 7 is defined as outgoing heat flux through the vertical wall of the melt and crystal. Since large outgoing heat flux is obtained around the m-c interface in the case with a large thermal
The results of 3D global analysis with square crucibles having large thermal conductivity showed that deformation of the m-c interface near the corner of a crucible becomes larger than that in the case with small thermal conductivity of a crucible. It was found that thermal conductivity of a crucible has significant influence on the amount of outgoing heat flux through the wall of the crucible. Therefore, we should control not only heater power, growth velocity and melt flow but also thermo-physical properties of the wall of a crucible in order to reduce deformation of the m-c interface.
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Acknowledgment. This work was supported by a NEDO project, a Grant-in-Aid for Science Research (B) 19360012 and a Grant-in-Aid for the creation of innovation through businessacademy-public sector from the Japanese Ministry of Education, Science, Sports and Culture.
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