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Effects of cusp-shaped magnetic field on melt convection and oxygen transport in an industrial CZ-Si crystal growth
Journal of Crystal Growth 354 (2012) 101–108
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effects of cusp-shaped magnetic field on melt convection and oxygen transport in an industrial CZ-Si crystal growth Xin Liu, Lijun Liu n, Zaoyang Li, Yuan Wang 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
abstract
Article history: Received 18 January 2012 Received in revised form 30 May 2012 Accepted 1 June 2012 Communicated by P. Rudolph Available online 12 June 2012
To clarify the effects of cusp-shaped magnetic field (CMF) on turbulent convection and oxygen transport in the melt, a set of 2D global simulations were conducted for growth of a 300 mm Czochralski-Si crystal. Turbulent melt convection was handled by a modified low-Re k–e turbulence model with additional magnetohydrodynamic (MHD) terms. Three configurations of CMF were numerically compared. Different types of melt convection and growth interface geometry emerged as CMF configuration and magnetic field strength were varied. CMF configuration was found to be crucial in reducing oxygen dissolution from the crucible wall and oxygen incorporation into the crystal. The results demonstrated that the growth interface shape and the oxygen concentration at the growth interface can be controlled effectively by applying the appropriate CMF. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Convection A1. Impurities A1. Interfaces A2. Magnetic field assisted Czochralski method
1. Introduction Currently, Czochralski (CZ) process is the dominant technique by which bulk single crystals of a wide range of electronic and photovoltaic silicon are produced. The quality, purity, and defectfree nature of crystals are the prerequisites for their technological application. Melt turbulence in industrial CZ-Si crystal growth is largely responsible for micro-defects and high concentrations of impurities in the grown crystals. Since the Si melt is electrically conductive, the application of magnetic fields to the CZ technique (MCZ) has been established as a standard method for controlling the melt convection, impurity transport and growth interface shape [1]. Magnetic fields encourage the growth of low-oxygen and dislocation-free single Si crystals by controlling the melt convection. Therefore, it is important to clarify, and later control, the action mechanism of magnetic fields on melt turbulence and impurities transport, in order to improve the crystal quality in MCZ-Si growth. For its low cost and high visualization capability, numerical simulation has become an indispensable tool in the development of efficient time- and cost-saving optimization procedures [2]. The turbulent nature of the melt flow is now a generally accepted fact of industrial CZ-Si growth. Hence, turbulence effects must be considered in model simulations if reasonable results are to be
n
Corresponding author. Tel./fax: þ 86 29 8266 3443. E-mail addresses: [email protected], [email protected] (L. Liu).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.06.004
obtained. Lipchin and Brown [3] reported that using a low-Re k–e model is preferable to the standard k–e model in CZ-Si growth modeling. Low-Re k–e turbulence models have been applied in global simulations of industrial CZ-Si growth without magnetic fields [4,5]. However, up to date global simulations including additional MHD effects have been conducted only for small-scale MCZ-Si growth [6–8]. Because of the low Reynolds number of the melt convection, turbulence has been excluded from such studies. Nevertheless, a few reports have investigated the global simulation of turbulent transport in industrial CZ-Si crystal growth under the influence of magnetic fields, which is one of the key technologies in the modern crystal growth industry. The relatively high oxygen content of CZ-Si is a certain drawback of this material for solar cell applications. The oxygen transport mechanism in CZ-Si growth has been discussed in some studies [9–12]. Accurate control of oxygen concentration in a grown crystal is of prime importance to CZ-Si crystal growth technology. Application of magnetic fields in CZ-Si crystal growth provides an effective way to control the oxygen transport in the melt, and further to control the oxygen concentration in a crystal. Since oxygen in crystals is sourced from the chemical reaction of the Si melt with the SiO2 crucible, the CMF, which affects the boundary layer flow along the crucible wall, is expected to produce low-oxygen crystals. The effects of CMF on oxygen concentration have been reported in many numerical studies [13,14]. However, such studies have tended to ignore the effects of turbulent convection on oxygen transport.
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Nomenclature ! B Magnetic flux density vector, T C Oxygen concentration, atoms/cm3 C m ,C 1 ,C 2 ,Prk ,Pre Coefficients of the k–e model CM Coefficient of the magnetic production term cp Specific heat, J/(K kg) Deff Effective diffusivity of oxygen, m2/s ! E Electric field vector, V/m fm, f1, f2, D, E Damping functions of the low-Re k–e model P,GM,GB Turbulence production terms ! J Electrical current density vector, A/m2 k Turbulent kinetic energy, m2/s2 Pr Prandtl number Sc Schmidt number Greek symbols
a
Surface tension coefficient, N/(K m)
In this work, 2D global simulations of heat transfer in an industrial CZ-Si growth were conducted. A modified low-Re k–e model with additional MHD terms was employed to handle the influence of CMF on the melt turbulence. Effects of different MCZ configurations on the melt convection and oxygen transport were analyzed. The phenomena occurring at the growth interface were also investigated by predicting the interface geometry, the oxygen concentration and the ratio of crystal growth rate and temperature gradient (Vg/Gs) [15].
2. Numerical approach and modeling The configuration and dimensions of the industrial CZ-Si crystal growth furnace are shown in Fig. 1. The CZ-Si growth system is assumed to be quasi-steady and geometrically axisymmetric. 2D global simulations are conducted assuming axial symmetry of the CMF. The global solution of heat transport in the entire system is obtained by an iterative procedure comprising a set of local iterations for all block regions, the calculation of radiative heat transfer in the furnace and a global conjugated iteration among them. The input heater power and growth interface shape are unknowns a priori and are solved during the
Thermal expansion coefficient, K 1 Turbulent dissipation rate, m2/s3 Thermal conductivity, J/(s m K) Molecular dynamic viscosity, Pa s Density, kg/m3 Electrical conductivity, S/m Electric potential, V Evaporation rate coefficient of oxygen, m/s Rotation rate, rpm
bT
e l
m r s F c O
Subscripts B C Eff M M T Ar Seg
Buoyancy Crystal side Effective value Magnetic field Melt side Turbulence Argon flow Segregation
iterative procedure. The details of the global modeling algorithm and material properties have been published in our previous research papers [16,17]. A multi-block structured grid system was used for space discretization. Finite-volume calculations were conducted for the velocity, pressure, temperature and the electromagnetic field in the melt domain. The governing equations were transformed to the general curvilinear coordination and discretized by finite volume approximations on a collocated non-orthogonal mesh. An improved pressure-based algorithm based on SIMPLEC method was adopted to couple the momentum equations and the continuity equation. The covariant velocity projections, rather than the Cartesian velocity components, were selected as the solving variables. The convective term of the governing equations were discretized with a power-law scheme. The diffusion terms in the governing equations were discretized with a central differencing scheme. The resulting non-linear algebraic equations are solved iteratively by the successive line over-relaxation (SLOR) method, with alternating direction iteration (ADI) algorithm and block correction technique to speed up the iteration.
2.1. Governing equations and turbulence model for the turbulent melt flow Applying the Boussinesq approximation for an incompressible Newtonian fluid, the Reynolds-averaged Navier–Stokes (RANS) equations of motion, energy and impurity transport for the Si melt in a crucible under the influence of a magnetic field can be written as follows: !
rU V ¼ 0, !
!
ð1Þ !
!T
!
!
r V Ur V ¼ rp þ rU½mef f ðr V þ r V Þr! g bT ðTT m Þ þ J B , ð2Þ !
rcp V UrT ¼ rUðlef f rTÞ: Fig. 1. Configuration and temperature distribution of a 300 mm CZ-Si crystal growth furnace. Isothermals are plotted at 100 K increments.
ð3Þ
The effective viscosity and effective thermal conductivity are defined as meff ¼ m þ mt, and lef f ¼ m=Pr þ mt =Prt , respectively. Here Prt is defined as a function of mt/m to correct for the thermal
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
! ! ! ! J ¼ sð E þ V B Þ,
Table 1 Physical properties of Si melt. Parameter
Value
Molecular dynamic viscosity, m Density, r Specific heat, cp Prandtl number, Pr Schmidt number, Sc Thermal expansion coefficient, bT Surface tension coefficient, a Electrical conductivity, s Melting temperature Emissivity Turbulent Prandtl number, Prt Turbulent Schmidt number, Sct
8.23 10 4 Pa s 2.53 103 kg/m3 1.04 103 J/(K kg) 0.015 10 1.36 10 4 K 1 1.5 10 4 N/(K m) 1.23 106 S/m 1685 K 0.318 f(mt/m) 0.9
convection in the boundary layer [18]: ( 2 )1 m m m Prt ¼ 0:5822 þ 0:2833 t 0:0441 t 1exp 5:165 t :
m
m
The physical properties of the Si melt are summarized in Table 1. Non-slip conditions are assumed for velocities at solid boundaries. At the melt free surface, the shear stress is balanced by the surface tension and the gas shear stress: @ut @u @T ¼ mAr Ar þ a : @t @n @n
ð5Þ
Here, n and t are the normal and tangential directions of the melt free surface, respectively. The turbulent viscosity mt is obtained using a modified low-Re k–e turbulence model [19]. The governing equations for the turbulent kinetic energy k and its dissipation rate e are: !
r V Urk ¼ rU½ðm þ mt =Prk Þrk þ ðP þGB þ GM reDÞ,
ð6Þ
!
r V Ure ¼ rU½ðm þ mt =Pre Þre þ C 1 f 1 ðP þ GB Þe=k þ GM e=kC 2 f 2 re2 =k þ E: ð7Þ Here, the turbulent eddy viscosity mt is defined as mt ¼fmCmrk2/ e. The magnetic production term GM is defined as GM ¼ sB2 kexp½C M sB2 k=ðreÞ,
ð8Þ
where B2 ¼ B2x þ B2y þ B2z and CM ¼0.025 [20]. The buoyant production term GB is defined as ! GB ¼ b g Uðmt =Prt ÞrT:
ð9Þ
The model constants ðC m ,C 1 ,C 2 ,Prk ,Pre Þ and the damping functions (fm,f1,f2,D,E) used in the turbulence model are the same as those used in Ref. [5]. Zero-value conditions are imposed for k and e at the crucible walls and the growth interface. For the melt free surface and symmetry axis, the boundary conditions are assigned as @k @e ¼ ¼ 0, @n @n
ð10Þ
where n is the direction normal to the boundaries. 2.2. Electromagnetic field and boundary conditions In order to obtain the Lorentz force exerted on the melt, the electromagnetic field was solved. This field is governed by !
rU J ¼ 0,
ð12Þ
! ð13Þ E ¼ rF, ! ! where B , E and F are the external magnetic field, the electric field and electric potential, respectively. The crucible and ambient gas are electrically insulated, while the Si crystal is an electrical conductor. Therefore, the calculation of electromagnetic field is limited to the melt and crystal domains. The boundary conditions in the presence of the magnetic field are as follows: on all of the external boundaries of the meltcrystal domain, a non-penetration condition for electrical current density is imposed, i.e. ! ! J U n ¼ 0, ð14Þ ! where n is the normal direction to the boundary surfaces. On the growth interface, ! ! ! ! ð J U n Þc ¼ ð J U n Þm : ð15Þ
m
ð4Þ
mef f
103
ð11Þ
2.3. Oxygen transport and boundary conditions The governing equation of oxygen transport in the Si melt is ! V UrC ¼ rU½Def f ðrCÞ ð16Þ where C is the oxygen concentration in the melt. Turbulent transport of impurities is incorporated into the effective diffusivity, defined as Deff ¼ m/Scþ mt/Sct. At the melt-crucible interface, an equation derived from an oxygen solubility measurement performed by Hirata and Hoshikawa [21], is applied as the boundary condition for oxygen concentration: C ¼ 3:99 1023 expð2:0 104 =TÞ ðatoms=cm3 Þ:
ð17Þ
On the melt free surface, the oxygen evaporation is incorporated based on the measurements of Togawa et al. [22]: Def f
@C ¼ cðTÞC, @n
ð18Þ
where the evaporation rate coefficient c(T) is given as
cðTÞ ¼ 5:9152 107 expð4:1559 104 =TÞ:
ð19Þ
At the growth interface, the segregation coefficient of oxygen is kseg ¼ 0.85. The boundary condition of oxygen concentration at the melt-crystal interface is given as Def f
@C m þ V g C m ð1kseg Þ ¼ 0: @n
ð20Þ
3. Results and analysis The CMF with its symmetric plane along the melt top surface and two modified CMFs (CMF-Quarter and CMF-Half) with their symmetric plane along the quarter-depth and half-depth plane in the melt were investigated, as presented in Fig. 2. The CMF distribution is described by ! ! ! ! B ¼ B0 =Rc ½x i þy j þ 2ðzH0 Þ k , ð21Þ where Rc and H0 are the radius of the crucible and the axial location of the symmetric plane of CMF, respectively. For each CMF configuration, B0 ¼0.3 T and B0 ¼ 0.4 T are assigned in global simulations. The crystal diameter is 306 mm. The diameter of the crucible inside wall is 775 mm. The melt height is 279 mm. The rotation rates of crucible and crystal are Oc ¼4 rpm and Os ¼ 15 rpm, respectively. The crystal pulling rate is 0.4 mm/
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Fig. 2. The investigated CMF configurations. (a) The CMF with its symmetric plane at the melt free surface. (b) The CMF-Quarter with its symmetric plane at the quarter-depth of the melt. (c) The CMF-Half with its symmetric plane at the halfdepth of the melt.
min. The crucible rotation rate leads to a rotational Reynolds number of Re ¼2.0 105. Turbulent transport is dominant in the melt. The argon flow rate is 5 SLPM, and the furnace pressure is 10 Torr. 3.1. CMF effects on the turbulent melt convection Fig. 3 shows the melt convection for seven cases, including the absence of magnetic field and six types of CMF. The isotherms in the melt are plotted every 2 K in the left section of these figures, while the contours of stream function are plotted every 0.01 kg/s in the right section. For the case in which CMF is absent (Fig. 3(a)), the temperature difference in the melt is about 35 K and the maximal temperature at the crucible walls is 1720 K.
A considerable contribution of convective heat exchange is observed under the growth interface and close to the crucible side wall. An obvious vortex is formed under the crystal growth interface, which is known as the Ekman layer due to the counter rotation of crystal and crucible. This vortex results in a highly concave growth interface to the crystal side, which is unfavorable to crystal quality. The counter-clockwise vortex in the central region of the melt, induced by centrifugal forces competing with the radial pressure gradient, is referred to as the Taylor– Proudman cell. A buoyancy-driven flow cell is observed near the side wall of the crucible. These flow structures resemble those given by Raufeisen et al. [23], who conducted direct numerical simulation (DNS) for melt turbulence in a cylindrical crucible of 340 mm in diameter. Our large eddy simulation (LES) for melt convection in the large diameter crucible of 300 mm CZ-Si crystal growth also revealed analogous flow structures [24,25]. The convection is intense in the melt domain due to the complex interaction between the thermal buoyancy, the surface tension and the crucible/crystal rotations. The stream function ranges from 0.33 kg/s to 0.03 kg/s. For a CMF of 0.3 T, the melt convection evolves into different structures, as shown in Fig. 3(b). Because of the suppression effect of CMF, the range of stream function decreases notably, indicating that the melt convection becomes weak. Since the magnetic field intensity is very small near the growth interface, the Ekman layer is retained though it becomes much thinner. Accordingly, the deformation of the growth interface is much reduced, relative to the case of no magnetic field. The shear flow in this layer leads to a W-shaped growth interface geometry, which is known as a source of dislocation generation [26]. The CMF suppresses the buoyancy convection near the crucible wall, a region of high magnetic field intensity. The Taylor–Proudman cell extends from the central part to the outer region of the melt. The applied CMF reverses the flow direction at the melt free surface. The temperature difference in the melt decreases to 24 K. This is due to the increase in the effective dynamic viscosity of the silicon melt under the influence of magnetic field, which is verified in experiments [27], since the effective thermal conductivity is proportional to the effective dynamic viscosity. The decrease of the maximal temperature is favorable to reduce the dissolution of oxygen from the crucible wall. Fig. 3(c) shows the effect of increasing the CMF strength to 0.4 T. The flow pattern is similar to that of the former case (CMF ¼0.3 T), but the strength of flow vortex is enhanced. Because of the increase in the magnetic strength near the crucible, the buoyancy cell becomes weaker and the Taylor–Proudman cell strengthens. The temperature difference in the melt decreases with the increase of the magnetic strength. To achieve a more optimal growth condition, the CMF was modified by descending the symmetric plane of the CMF to the quarter-depth and the half-depth of the melt. The results for the CMF-Quarter cases of 0.3 T and 0.4 T are presented in Fig. 3(d) and (e), respectively. For the CMF-Quarter cases, the axial magnetic field near the growth interface suppresses the Ekman layer convection in the crystallization zone. This suppression effect leads to a less concave growth interface. Meanwhile, the horizontal magnetic field near the crucible side wall damps the buoyancy convection. The flow intensity and the temperature difference in the melt decreases with the increase of the magnetic strength. Especially for the CMF-Quarter of 0.4 T, the melt convection in the meridian plane is very weak. The results for the CMF-Half cases of 0.3 T are presented in Fig. 3(f). Similar to the corresponding CMF-Quarter case, the melt convection in the crystallization zone is weak due to the local damping effect of the axial magnetic field near the growth interface. Because the intensity of the magnetic field near the
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
105
Fig. 3. Comparison between the melt convection fields. In the left section, isothermals are plotted at 2 K temperature increments. In the right section, stream function (c) contours are plotted at 0.01 kg/s increments: (a) without CMF, (b) with CMF 0.3 T, (c) with CMF 0.4 T, (d) with CMF-Quarter 0.3 T, (e) with CMF-Quarter 0.4 T, (f) with CMFHalf 0.3 T, and (g) with CMF-Half 0.4 T.
crucible wall decreases when the CMF is relocated, the buoyancy cell is stronger than that of the corresponding CMF case and CMFQuarter case. The vortex near the melt free interface is dominated by the shear stress induced by the argon gas flow. The growth interface becomes almost flat in contrast to the CMF case, CMFQuarter case and the case of no magnetic field. The temperature gradient at the growth interface reduces. When the CMF-Half is increased to 0.4 T (Fig. 3(g)), the range of the stream function is smaller than that for CMF-Half 0.3 T and the corresponding CMF case with 0.4 T. 3.2. CMF effects on the oxygen transport Turbulent transport of oxygen impurity in the melt was investigated in different CMF configurations. The oxygen concentration distributions for the seven cases are displayed in Fig. 4. In the absence of CMF (Fig. 4(a)), maximal oxygen dissolution occurs at the location of the maximal temperature at the crucible wall. Most of the oxygen is evaporated from the melt free surface. Only a small amount is transported to the crystallization zone. The oxygen concentration at the growth interface is highly
homogeneous in the radial direction due to the Ekman layer convection close to the growth interface. A small gradient of oxygen concentration is found in the core region of the melt due to the Taylor–Proudman flow. For the CMF of 0.3 T, the maximal temperature on the crucible wall decreases, resulting in a decrease in oxygen dissolution on the crucible wall. Due to the suppression of the melt convection close to the crucible wall, less oxygen can be transported from the region near the crucible wall to the core region of the melt, as shown in Fig. 4(b). The oxygen concentration gradient in the core region increases owing to different flow structure relative to the case without CMF. The oxygen concentration at the growth interface decreases from 7 1017 atoms/cm3 with CMF absent to 6 1017 atoms/cm3. If we continue to increase the strength of the CMF to 0.4 T, the oxygen dissolution on the crucible wall decreases constantly, as shown in Fig. 4(c). The oxygen concentration at the growth interface is smaller than that in the 0.3 T CMF case. In the CMF-Quarter cases, the maximal temperature at the crucible wall is higher than that in the corresponding CMF cases, resulting in the increase of oxygen dissolution on the crucible
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Fig. 4. Comparison between the oxygen distributions in the melt. Contours are plotted at 0.5 1017 atoms/cm3 increment of oxygen concentration: (a) without CMF, (b) with CMF 0.3 T, (c) with CMF 0.4 T, (d) with CMF-Quarter 0.3 T, (e) with CMF-Quarter 0.4 T, (f) with CMF-Half 0.3 T, and (g) with CMF-Half 0.4 T.
wall, as shown in Fig. 4(d) for 0.3 T and Fig. 4(e) for 0.4 T. However, for the CMF-Quarter of 0.3 T, the oxygen concentration at the growth interface is smaller than that in the corresponding CMF case. This indicates that the flow structures in the CMFQuarter cases are in favor of the oxygen evaporation from the melt free surface. If we continue to increase the strength to 0.4 T (Fig. 4(e)), the oxygen concentration at the growth interface is higher than that in the corresponding CMF case due to the weakened melt convection. In the CMF-Half cases, the oxygen dissolution on the crucible wall decrease in contrast to the corresponding CMF cases and CMF-Quarter cases, as shown in Fig. 4(d) for 0.3 T and Fig. 4(e) for 0.4 T. Since the convection under the growth interface is very weak, the radial oxygen concentration distribution becomes less homogeneous. These results indicate that, for a given magnetic field strength, the CMF-Half reduces the oxygen dissolution from crucible to the melt and the oxygen incorporation into the crystal. However, the increase of the axial magnetic strength in the crystallization zone leads to a less homogeneous distribution of oxygen in the radial direction at the growth interface. Incorporation of oxygen into crystals can be controlled in three ways: dissolution from the crucible, convection in the melt and evaporation at the melt free surface. The dissolution and amount of incorporated oxygen in the three cases are compared in Table 2. Only about 1–2% of the dissolved oxygen is incorporated into the grown crystal, which is consistent with the measurements [9,11]. With CMF absent, the percentage of oxygen incorporated into the
Table 2 Oxygen dissolution from crucible and incorporation into crystal. Cases
Dissolution (1017 atoms/cm)
Incorporation (1017 atoms/cm)
Percentage (%)
CMF¼ 0.0 T CMF¼ 0.3 T CMF¼ 0.4 T CMF Quarter ¼0.3 T CMF Quarter ¼0.4 T CMF Half¼ 0.3 T CMF Half¼ 0.4 T
288973 276012 268721 283413 274767 269808 265508
5755 4254 3273 3975 3835 3922 3450
1.99 1.54 1.22 1.40 1.39 1.45 1.30
grown crystal is maximized. As CMF strength is increased, the oxygen dissolution from the crucible decreases, as does the oxygen incorporation into the crystal. This tendency is attributable to the suppression effects of CMF on the melt convection. With the same strength of 0.3 T, the CMF-Quarter evaporates the most percentage of oxygen from the melt free interface. The results also show that CMF-Half is more effective at reducing oxygen dissolution than CMF and CMF-Quarter. In Fig. 5 we compare the oxygen concentration at the growth interface between the seven cases. The cases with 0.3 T and 0.4 T are presented in the left section and right section, respectively. The oxygen concentrations at the growth interfaces are in the range 3.5–8.5 1017 atoms/cm3. This range is consistent with that measured in real CZ-Si grown crystals [1,2]. The absence of CMF yields the highest concentration and the most radial
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
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Ekman layer convection. In the CMF-Half case, the oxygen concentration close to the crystal rim is decreased, and an inhomogeneous distribution in the radial direction emerges. The lowest oxygen concentration at the interface is attained for the CMF case of 0.4 T. 3.3. CMF effects on growth interface dynamics
Fig. 5. Oxygen distribution at the growth interface.
In Fig. 6(a) we compare the growth interface shape between the seven cases. The most concave growth interface shape is obtained in the absence of CMF. The W-shaped growth interfaces shape is obtained in the CMF cases. These three interface shapes are all undesirable for CZ-Si growth, since a non-flat interface can lead to stresses, cracks and core phenomena, etc. Less concave growth interface shape is obtained in the CMF-Quarter cases and the CMF-Half cases. An almost flat shape of the growth interfaces is obtained in the CMF-Half case with 0.3 T. The concentration of point defects in a growing crystal can be expressed as a function of Vg/Gs at the growth interface, where Vg is the crystal growth rate and Gs is the temperature gradient in the crystal at the growth interface. The value of Vg/Gs correlates with the growth interface shape. Too large or too small Vg/Gs values result in micro-defects in the Si crystal. For the reference Vg/Gs of about 1.34 10 3 cm2/min K, the so-called ‘‘perfect silicon’’ is attainable [28]. In Fig. 6(b) we compare the Vg/Gs values at the growth interface for the seven cases investigated. The smallest Vg/Gs is obtained in the absence of CMF, and the radial inhomogeneity is obvious. When CMF is applied, larger Vg/Gs results and more radial inhomogeneity emerges than for the other cases. When CMF-Quarter is applied, the Vg/Gs value becomes closer to the reference value. Application of CMF-Half renders the Vg/Gs distribution more homogeneous in the radial direction. In the CMF-Half case of 0.4 T, Vg/Gs closely approaches to the reference line. Therefore, to obtain homogeneous distribution of Vg/Gs at the melt–crystal interface, the melt convection should be controlled in the vicinity of the growth interface.
4. Conclusions Comparison investigations were carried out to study the effects of CMF on the melt convection and turbulent transport of oxygen in the growth of a 300 mm CZ-Si crystal. MHD modifications were incorporated to model the effects of CMF on the melt turbulence. It was found that the melt convection evolves into different structures under different CMF configurations. Furthermore, application of CMF can decrease oxygen concentration at the crystal growth interface. A flatter shape of the growth interface can be obtained in an industrial CZ-Si crystal growth by applying a CMF with its symmetric plane under the melt free surface. This indicates that the flatness of the growth interface shape can be controlled by adjusting the symmetry plane of CMF.
Acknowledgments Fig. 6. Comparison between growth interfaces evolving under different CMFs: (a) growth interface shape and (b) Vg/Gs distribution at the growth interface.
homogeneity of oxygen concentration at the growth interface due to the intensive convection under the growth interface. The oxygen concentrations for the 0.3 T cases are higher than that for the 0.4 T cases. The CMF cases and the CMF-Quarter case of 0.3 T exhibit more homogeneous oxygen distribution at the growth interface than other cases due to the existence of the
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effects of cusp-shaped magnetic field on melt convection and oxygen transport in an industrial CZ-Si crystal growth Xin Liu, Lijun Liu n, Zaoyang Li, Yuan Wang 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
abstract
Article history: Received 18 January 2012 Received in revised form 30 May 2012 Accepted 1 June 2012 Communicated by P. Rudolph Available online 12 June 2012
To clarify the effects of cusp-shaped magnetic field (CMF) on turbulent convection and oxygen transport in the melt, a set of 2D global simulations were conducted for growth of a 300 mm Czochralski-Si crystal. Turbulent melt convection was handled by a modified low-Re k–e turbulence model with additional magnetohydrodynamic (MHD) terms. Three configurations of CMF were numerically compared. Different types of melt convection and growth interface geometry emerged as CMF configuration and magnetic field strength were varied. CMF configuration was found to be crucial in reducing oxygen dissolution from the crucible wall and oxygen incorporation into the crystal. The results demonstrated that the growth interface shape and the oxygen concentration at the growth interface can be controlled effectively by applying the appropriate CMF. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Convection A1. Impurities A1. Interfaces A2. Magnetic field assisted Czochralski method
1. Introduction Currently, Czochralski (CZ) process is the dominant technique by which bulk single crystals of a wide range of electronic and photovoltaic silicon are produced. The quality, purity, and defectfree nature of crystals are the prerequisites for their technological application. Melt turbulence in industrial CZ-Si crystal growth is largely responsible for micro-defects and high concentrations of impurities in the grown crystals. Since the Si melt is electrically conductive, the application of magnetic fields to the CZ technique (MCZ) has been established as a standard method for controlling the melt convection, impurity transport and growth interface shape [1]. Magnetic fields encourage the growth of low-oxygen and dislocation-free single Si crystals by controlling the melt convection. Therefore, it is important to clarify, and later control, the action mechanism of magnetic fields on melt turbulence and impurities transport, in order to improve the crystal quality in MCZ-Si growth. For its low cost and high visualization capability, numerical simulation has become an indispensable tool in the development of efficient time- and cost-saving optimization procedures [2]. The turbulent nature of the melt flow is now a generally accepted fact of industrial CZ-Si growth. Hence, turbulence effects must be considered in model simulations if reasonable results are to be
n
Corresponding author. Tel./fax: þ 86 29 8266 3443. E-mail addresses: [email protected], [email protected] (L. Liu).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.06.004
obtained. Lipchin and Brown [3] reported that using a low-Re k–e model is preferable to the standard k–e model in CZ-Si growth modeling. Low-Re k–e turbulence models have been applied in global simulations of industrial CZ-Si growth without magnetic fields [4,5]. However, up to date global simulations including additional MHD effects have been conducted only for small-scale MCZ-Si growth [6–8]. Because of the low Reynolds number of the melt convection, turbulence has been excluded from such studies. Nevertheless, a few reports have investigated the global simulation of turbulent transport in industrial CZ-Si crystal growth under the influence of magnetic fields, which is one of the key technologies in the modern crystal growth industry. The relatively high oxygen content of CZ-Si is a certain drawback of this material for solar cell applications. The oxygen transport mechanism in CZ-Si growth has been discussed in some studies [9–12]. Accurate control of oxygen concentration in a grown crystal is of prime importance to CZ-Si crystal growth technology. Application of magnetic fields in CZ-Si crystal growth provides an effective way to control the oxygen transport in the melt, and further to control the oxygen concentration in a crystal. Since oxygen in crystals is sourced from the chemical reaction of the Si melt with the SiO2 crucible, the CMF, which affects the boundary layer flow along the crucible wall, is expected to produce low-oxygen crystals. The effects of CMF on oxygen concentration have been reported in many numerical studies [13,14]. However, such studies have tended to ignore the effects of turbulent convection on oxygen transport.
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Nomenclature ! B Magnetic flux density vector, T C Oxygen concentration, atoms/cm3 C m ,C 1 ,C 2 ,Prk ,Pre Coefficients of the k–e model CM Coefficient of the magnetic production term cp Specific heat, J/(K kg) Deff Effective diffusivity of oxygen, m2/s ! E Electric field vector, V/m fm, f1, f2, D, E Damping functions of the low-Re k–e model P,GM,GB Turbulence production terms ! J Electrical current density vector, A/m2 k Turbulent kinetic energy, m2/s2 Pr Prandtl number Sc Schmidt number Greek symbols
a
Surface tension coefficient, N/(K m)
In this work, 2D global simulations of heat transfer in an industrial CZ-Si growth were conducted. A modified low-Re k–e model with additional MHD terms was employed to handle the influence of CMF on the melt turbulence. Effects of different MCZ configurations on the melt convection and oxygen transport were analyzed. The phenomena occurring at the growth interface were also investigated by predicting the interface geometry, the oxygen concentration and the ratio of crystal growth rate and temperature gradient (Vg/Gs) [15].
2. Numerical approach and modeling The configuration and dimensions of the industrial CZ-Si crystal growth furnace are shown in Fig. 1. The CZ-Si growth system is assumed to be quasi-steady and geometrically axisymmetric. 2D global simulations are conducted assuming axial symmetry of the CMF. The global solution of heat transport in the entire system is obtained by an iterative procedure comprising a set of local iterations for all block regions, the calculation of radiative heat transfer in the furnace and a global conjugated iteration among them. The input heater power and growth interface shape are unknowns a priori and are solved during the
Thermal expansion coefficient, K 1 Turbulent dissipation rate, m2/s3 Thermal conductivity, J/(s m K) Molecular dynamic viscosity, Pa s Density, kg/m3 Electrical conductivity, S/m Electric potential, V Evaporation rate coefficient of oxygen, m/s Rotation rate, rpm
bT
e l
m r s F c O
Subscripts B C Eff M M T Ar Seg
Buoyancy Crystal side Effective value Magnetic field Melt side Turbulence Argon flow Segregation
iterative procedure. The details of the global modeling algorithm and material properties have been published in our previous research papers [16,17]. A multi-block structured grid system was used for space discretization. Finite-volume calculations were conducted for the velocity, pressure, temperature and the electromagnetic field in the melt domain. The governing equations were transformed to the general curvilinear coordination and discretized by finite volume approximations on a collocated non-orthogonal mesh. An improved pressure-based algorithm based on SIMPLEC method was adopted to couple the momentum equations and the continuity equation. The covariant velocity projections, rather than the Cartesian velocity components, were selected as the solving variables. The convective term of the governing equations were discretized with a power-law scheme. The diffusion terms in the governing equations were discretized with a central differencing scheme. The resulting non-linear algebraic equations are solved iteratively by the successive line over-relaxation (SLOR) method, with alternating direction iteration (ADI) algorithm and block correction technique to speed up the iteration.
2.1. Governing equations and turbulence model for the turbulent melt flow Applying the Boussinesq approximation for an incompressible Newtonian fluid, the Reynolds-averaged Navier–Stokes (RANS) equations of motion, energy and impurity transport for the Si melt in a crucible under the influence of a magnetic field can be written as follows: !
rU V ¼ 0, !
!
ð1Þ !
!T
!
!
r V Ur V ¼ rp þ rU½mef f ðr V þ r V Þr! g bT ðTT m Þ þ J B , ð2Þ !
rcp V UrT ¼ rUðlef f rTÞ: Fig. 1. Configuration and temperature distribution of a 300 mm CZ-Si crystal growth furnace. Isothermals are plotted at 100 K increments.
ð3Þ
The effective viscosity and effective thermal conductivity are defined as meff ¼ m þ mt, and lef f ¼ m=Pr þ mt =Prt , respectively. Here Prt is defined as a function of mt/m to correct for the thermal
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
! ! ! ! J ¼ sð E þ V B Þ,
Table 1 Physical properties of Si melt. Parameter
Value
Molecular dynamic viscosity, m Density, r Specific heat, cp Prandtl number, Pr Schmidt number, Sc Thermal expansion coefficient, bT Surface tension coefficient, a Electrical conductivity, s Melting temperature Emissivity Turbulent Prandtl number, Prt Turbulent Schmidt number, Sct
8.23 10 4 Pa s 2.53 103 kg/m3 1.04 103 J/(K kg) 0.015 10 1.36 10 4 K 1 1.5 10 4 N/(K m) 1.23 106 S/m 1685 K 0.318 f(mt/m) 0.9
convection in the boundary layer [18]: ( 2 )1 m m m Prt ¼ 0:5822 þ 0:2833 t 0:0441 t 1exp 5:165 t :
m
m
The physical properties of the Si melt are summarized in Table 1. Non-slip conditions are assumed for velocities at solid boundaries. At the melt free surface, the shear stress is balanced by the surface tension and the gas shear stress: @ut @u @T ¼ mAr Ar þ a : @t @n @n
ð5Þ
Here, n and t are the normal and tangential directions of the melt free surface, respectively. The turbulent viscosity mt is obtained using a modified low-Re k–e turbulence model [19]. The governing equations for the turbulent kinetic energy k and its dissipation rate e are: !
r V Urk ¼ rU½ðm þ mt =Prk Þrk þ ðP þGB þ GM reDÞ,
ð6Þ
!
r V Ure ¼ rU½ðm þ mt =Pre Þre þ C 1 f 1 ðP þ GB Þe=k þ GM e=kC 2 f 2 re2 =k þ E: ð7Þ Here, the turbulent eddy viscosity mt is defined as mt ¼fmCmrk2/ e. The magnetic production term GM is defined as GM ¼ sB2 kexp½C M sB2 k=ðreÞ,
ð8Þ
where B2 ¼ B2x þ B2y þ B2z and CM ¼0.025 [20]. The buoyant production term GB is defined as ! GB ¼ b g Uðmt =Prt ÞrT:
ð9Þ
The model constants ðC m ,C 1 ,C 2 ,Prk ,Pre Þ and the damping functions (fm,f1,f2,D,E) used in the turbulence model are the same as those used in Ref. [5]. Zero-value conditions are imposed for k and e at the crucible walls and the growth interface. For the melt free surface and symmetry axis, the boundary conditions are assigned as @k @e ¼ ¼ 0, @n @n
ð10Þ
where n is the direction normal to the boundaries. 2.2. Electromagnetic field and boundary conditions In order to obtain the Lorentz force exerted on the melt, the electromagnetic field was solved. This field is governed by !
rU J ¼ 0,
ð12Þ
! ð13Þ E ¼ rF, ! ! where B , E and F are the external magnetic field, the electric field and electric potential, respectively. The crucible and ambient gas are electrically insulated, while the Si crystal is an electrical conductor. Therefore, the calculation of electromagnetic field is limited to the melt and crystal domains. The boundary conditions in the presence of the magnetic field are as follows: on all of the external boundaries of the meltcrystal domain, a non-penetration condition for electrical current density is imposed, i.e. ! ! J U n ¼ 0, ð14Þ ! where n is the normal direction to the boundary surfaces. On the growth interface, ! ! ! ! ð J U n Þc ¼ ð J U n Þm : ð15Þ
m
ð4Þ
mef f
103
ð11Þ
2.3. Oxygen transport and boundary conditions The governing equation of oxygen transport in the Si melt is ! V UrC ¼ rU½Def f ðrCÞ ð16Þ where C is the oxygen concentration in the melt. Turbulent transport of impurities is incorporated into the effective diffusivity, defined as Deff ¼ m/Scþ mt/Sct. At the melt-crucible interface, an equation derived from an oxygen solubility measurement performed by Hirata and Hoshikawa [21], is applied as the boundary condition for oxygen concentration: C ¼ 3:99 1023 expð2:0 104 =TÞ ðatoms=cm3 Þ:
ð17Þ
On the melt free surface, the oxygen evaporation is incorporated based on the measurements of Togawa et al. [22]: Def f
@C ¼ cðTÞC, @n
ð18Þ
where the evaporation rate coefficient c(T) is given as
cðTÞ ¼ 5:9152 107 expð4:1559 104 =TÞ:
ð19Þ
At the growth interface, the segregation coefficient of oxygen is kseg ¼ 0.85. The boundary condition of oxygen concentration at the melt-crystal interface is given as Def f
@C m þ V g C m ð1kseg Þ ¼ 0: @n
ð20Þ
3. Results and analysis The CMF with its symmetric plane along the melt top surface and two modified CMFs (CMF-Quarter and CMF-Half) with their symmetric plane along the quarter-depth and half-depth plane in the melt were investigated, as presented in Fig. 2. The CMF distribution is described by ! ! ! ! B ¼ B0 =Rc ½x i þy j þ 2ðzH0 Þ k , ð21Þ where Rc and H0 are the radius of the crucible and the axial location of the symmetric plane of CMF, respectively. For each CMF configuration, B0 ¼0.3 T and B0 ¼ 0.4 T are assigned in global simulations. The crystal diameter is 306 mm. The diameter of the crucible inside wall is 775 mm. The melt height is 279 mm. The rotation rates of crucible and crystal are Oc ¼4 rpm and Os ¼ 15 rpm, respectively. The crystal pulling rate is 0.4 mm/
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Fig. 2. The investigated CMF configurations. (a) The CMF with its symmetric plane at the melt free surface. (b) The CMF-Quarter with its symmetric plane at the quarter-depth of the melt. (c) The CMF-Half with its symmetric plane at the halfdepth of the melt.
min. The crucible rotation rate leads to a rotational Reynolds number of Re ¼2.0 105. Turbulent transport is dominant in the melt. The argon flow rate is 5 SLPM, and the furnace pressure is 10 Torr. 3.1. CMF effects on the turbulent melt convection Fig. 3 shows the melt convection for seven cases, including the absence of magnetic field and six types of CMF. The isotherms in the melt are plotted every 2 K in the left section of these figures, while the contours of stream function are plotted every 0.01 kg/s in the right section. For the case in which CMF is absent (Fig. 3(a)), the temperature difference in the melt is about 35 K and the maximal temperature at the crucible walls is 1720 K.
A considerable contribution of convective heat exchange is observed under the growth interface and close to the crucible side wall. An obvious vortex is formed under the crystal growth interface, which is known as the Ekman layer due to the counter rotation of crystal and crucible. This vortex results in a highly concave growth interface to the crystal side, which is unfavorable to crystal quality. The counter-clockwise vortex in the central region of the melt, induced by centrifugal forces competing with the radial pressure gradient, is referred to as the Taylor– Proudman cell. A buoyancy-driven flow cell is observed near the side wall of the crucible. These flow structures resemble those given by Raufeisen et al. [23], who conducted direct numerical simulation (DNS) for melt turbulence in a cylindrical crucible of 340 mm in diameter. Our large eddy simulation (LES) for melt convection in the large diameter crucible of 300 mm CZ-Si crystal growth also revealed analogous flow structures [24,25]. The convection is intense in the melt domain due to the complex interaction between the thermal buoyancy, the surface tension and the crucible/crystal rotations. The stream function ranges from 0.33 kg/s to 0.03 kg/s. For a CMF of 0.3 T, the melt convection evolves into different structures, as shown in Fig. 3(b). Because of the suppression effect of CMF, the range of stream function decreases notably, indicating that the melt convection becomes weak. Since the magnetic field intensity is very small near the growth interface, the Ekman layer is retained though it becomes much thinner. Accordingly, the deformation of the growth interface is much reduced, relative to the case of no magnetic field. The shear flow in this layer leads to a W-shaped growth interface geometry, which is known as a source of dislocation generation [26]. The CMF suppresses the buoyancy convection near the crucible wall, a region of high magnetic field intensity. The Taylor–Proudman cell extends from the central part to the outer region of the melt. The applied CMF reverses the flow direction at the melt free surface. The temperature difference in the melt decreases to 24 K. This is due to the increase in the effective dynamic viscosity of the silicon melt under the influence of magnetic field, which is verified in experiments [27], since the effective thermal conductivity is proportional to the effective dynamic viscosity. The decrease of the maximal temperature is favorable to reduce the dissolution of oxygen from the crucible wall. Fig. 3(c) shows the effect of increasing the CMF strength to 0.4 T. The flow pattern is similar to that of the former case (CMF ¼0.3 T), but the strength of flow vortex is enhanced. Because of the increase in the magnetic strength near the crucible, the buoyancy cell becomes weaker and the Taylor–Proudman cell strengthens. The temperature difference in the melt decreases with the increase of the magnetic strength. To achieve a more optimal growth condition, the CMF was modified by descending the symmetric plane of the CMF to the quarter-depth and the half-depth of the melt. The results for the CMF-Quarter cases of 0.3 T and 0.4 T are presented in Fig. 3(d) and (e), respectively. For the CMF-Quarter cases, the axial magnetic field near the growth interface suppresses the Ekman layer convection in the crystallization zone. This suppression effect leads to a less concave growth interface. Meanwhile, the horizontal magnetic field near the crucible side wall damps the buoyancy convection. The flow intensity and the temperature difference in the melt decreases with the increase of the magnetic strength. Especially for the CMF-Quarter of 0.4 T, the melt convection in the meridian plane is very weak. The results for the CMF-Half cases of 0.3 T are presented in Fig. 3(f). Similar to the corresponding CMF-Quarter case, the melt convection in the crystallization zone is weak due to the local damping effect of the axial magnetic field near the growth interface. Because the intensity of the magnetic field near the
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
105
Fig. 3. Comparison between the melt convection fields. In the left section, isothermals are plotted at 2 K temperature increments. In the right section, stream function (c) contours are plotted at 0.01 kg/s increments: (a) without CMF, (b) with CMF 0.3 T, (c) with CMF 0.4 T, (d) with CMF-Quarter 0.3 T, (e) with CMF-Quarter 0.4 T, (f) with CMFHalf 0.3 T, and (g) with CMF-Half 0.4 T.
crucible wall decreases when the CMF is relocated, the buoyancy cell is stronger than that of the corresponding CMF case and CMFQuarter case. The vortex near the melt free interface is dominated by the shear stress induced by the argon gas flow. The growth interface becomes almost flat in contrast to the CMF case, CMFQuarter case and the case of no magnetic field. The temperature gradient at the growth interface reduces. When the CMF-Half is increased to 0.4 T (Fig. 3(g)), the range of the stream function is smaller than that for CMF-Half 0.3 T and the corresponding CMF case with 0.4 T. 3.2. CMF effects on the oxygen transport Turbulent transport of oxygen impurity in the melt was investigated in different CMF configurations. The oxygen concentration distributions for the seven cases are displayed in Fig. 4. In the absence of CMF (Fig. 4(a)), maximal oxygen dissolution occurs at the location of the maximal temperature at the crucible wall. Most of the oxygen is evaporated from the melt free surface. Only a small amount is transported to the crystallization zone. The oxygen concentration at the growth interface is highly
homogeneous in the radial direction due to the Ekman layer convection close to the growth interface. A small gradient of oxygen concentration is found in the core region of the melt due to the Taylor–Proudman flow. For the CMF of 0.3 T, the maximal temperature on the crucible wall decreases, resulting in a decrease in oxygen dissolution on the crucible wall. Due to the suppression of the melt convection close to the crucible wall, less oxygen can be transported from the region near the crucible wall to the core region of the melt, as shown in Fig. 4(b). The oxygen concentration gradient in the core region increases owing to different flow structure relative to the case without CMF. The oxygen concentration at the growth interface decreases from 7 1017 atoms/cm3 with CMF absent to 6 1017 atoms/cm3. If we continue to increase the strength of the CMF to 0.4 T, the oxygen dissolution on the crucible wall decreases constantly, as shown in Fig. 4(c). The oxygen concentration at the growth interface is smaller than that in the 0.3 T CMF case. In the CMF-Quarter cases, the maximal temperature at the crucible wall is higher than that in the corresponding CMF cases, resulting in the increase of oxygen dissolution on the crucible
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Fig. 4. Comparison between the oxygen distributions in the melt. Contours are plotted at 0.5 1017 atoms/cm3 increment of oxygen concentration: (a) without CMF, (b) with CMF 0.3 T, (c) with CMF 0.4 T, (d) with CMF-Quarter 0.3 T, (e) with CMF-Quarter 0.4 T, (f) with CMF-Half 0.3 T, and (g) with CMF-Half 0.4 T.
wall, as shown in Fig. 4(d) for 0.3 T and Fig. 4(e) for 0.4 T. However, for the CMF-Quarter of 0.3 T, the oxygen concentration at the growth interface is smaller than that in the corresponding CMF case. This indicates that the flow structures in the CMFQuarter cases are in favor of the oxygen evaporation from the melt free surface. If we continue to increase the strength to 0.4 T (Fig. 4(e)), the oxygen concentration at the growth interface is higher than that in the corresponding CMF case due to the weakened melt convection. In the CMF-Half cases, the oxygen dissolution on the crucible wall decrease in contrast to the corresponding CMF cases and CMF-Quarter cases, as shown in Fig. 4(d) for 0.3 T and Fig. 4(e) for 0.4 T. Since the convection under the growth interface is very weak, the radial oxygen concentration distribution becomes less homogeneous. These results indicate that, for a given magnetic field strength, the CMF-Half reduces the oxygen dissolution from crucible to the melt and the oxygen incorporation into the crystal. However, the increase of the axial magnetic strength in the crystallization zone leads to a less homogeneous distribution of oxygen in the radial direction at the growth interface. Incorporation of oxygen into crystals can be controlled in three ways: dissolution from the crucible, convection in the melt and evaporation at the melt free surface. The dissolution and amount of incorporated oxygen in the three cases are compared in Table 2. Only about 1–2% of the dissolved oxygen is incorporated into the grown crystal, which is consistent with the measurements [9,11]. With CMF absent, the percentage of oxygen incorporated into the
Table 2 Oxygen dissolution from crucible and incorporation into crystal. Cases
Dissolution (1017 atoms/cm)
Incorporation (1017 atoms/cm)
Percentage (%)
CMF¼ 0.0 T CMF¼ 0.3 T CMF¼ 0.4 T CMF Quarter ¼0.3 T CMF Quarter ¼0.4 T CMF Half¼ 0.3 T CMF Half¼ 0.4 T
288973 276012 268721 283413 274767 269808 265508
5755 4254 3273 3975 3835 3922 3450
1.99 1.54 1.22 1.40 1.39 1.45 1.30
grown crystal is maximized. As CMF strength is increased, the oxygen dissolution from the crucible decreases, as does the oxygen incorporation into the crystal. This tendency is attributable to the suppression effects of CMF on the melt convection. With the same strength of 0.3 T, the CMF-Quarter evaporates the most percentage of oxygen from the melt free interface. The results also show that CMF-Half is more effective at reducing oxygen dissolution than CMF and CMF-Quarter. In Fig. 5 we compare the oxygen concentration at the growth interface between the seven cases. The cases with 0.3 T and 0.4 T are presented in the left section and right section, respectively. The oxygen concentrations at the growth interfaces are in the range 3.5–8.5 1017 atoms/cm3. This range is consistent with that measured in real CZ-Si grown crystals [1,2]. The absence of CMF yields the highest concentration and the most radial
X. Liu et al. / Journal of Crystal Growth 354 (2012) 101–108
107
Ekman layer convection. In the CMF-Half case, the oxygen concentration close to the crystal rim is decreased, and an inhomogeneous distribution in the radial direction emerges. The lowest oxygen concentration at the interface is attained for the CMF case of 0.4 T. 3.3. CMF effects on growth interface dynamics
Fig. 5. Oxygen distribution at the growth interface.
In Fig. 6(a) we compare the growth interface shape between the seven cases. The most concave growth interface shape is obtained in the absence of CMF. The W-shaped growth interfaces shape is obtained in the CMF cases. These three interface shapes are all undesirable for CZ-Si growth, since a non-flat interface can lead to stresses, cracks and core phenomena, etc. Less concave growth interface shape is obtained in the CMF-Quarter cases and the CMF-Half cases. An almost flat shape of the growth interfaces is obtained in the CMF-Half case with 0.3 T. The concentration of point defects in a growing crystal can be expressed as a function of Vg/Gs at the growth interface, where Vg is the crystal growth rate and Gs is the temperature gradient in the crystal at the growth interface. The value of Vg/Gs correlates with the growth interface shape. Too large or too small Vg/Gs values result in micro-defects in the Si crystal. For the reference Vg/Gs of about 1.34 10 3 cm2/min K, the so-called ‘‘perfect silicon’’ is attainable [28]. In Fig. 6(b) we compare the Vg/Gs values at the growth interface for the seven cases investigated. The smallest Vg/Gs is obtained in the absence of CMF, and the radial inhomogeneity is obvious. When CMF is applied, larger Vg/Gs results and more radial inhomogeneity emerges than for the other cases. When CMF-Quarter is applied, the Vg/Gs value becomes closer to the reference value. Application of CMF-Half renders the Vg/Gs distribution more homogeneous in the radial direction. In the CMF-Half case of 0.4 T, Vg/Gs closely approaches to the reference line. Therefore, to obtain homogeneous distribution of Vg/Gs at the melt–crystal interface, the melt convection should be controlled in the vicinity of the growth interface.
4. Conclusions Comparison investigations were carried out to study the effects of CMF on the melt convection and turbulent transport of oxygen in the growth of a 300 mm CZ-Si crystal. MHD modifications were incorporated to model the effects of CMF on the melt turbulence. It was found that the melt convection evolves into different structures under different CMF configurations. Furthermore, application of CMF can decrease oxygen concentration at the crystal growth interface. A flatter shape of the growth interface can be obtained in an industrial CZ-Si crystal growth by applying a CMF with its symmetric plane under the melt free surface. This indicates that the flatness of the growth interface shape can be controlled by adjusting the symmetry plane of CMF.
Acknowledgments Fig. 6. Comparison between growth interfaces evolving under different CMFs: (a) growth interface shape and (b) Vg/Gs distribution at the growth interface.
homogeneity of oxygen concentration at the growth interface due to the intensive convection under the growth interface. The oxygen concentrations for the 0.3 T cases are higher than that for the 0.4 T cases. The CMF cases and the CMF-Quarter case of 0.3 T exhibit more homogeneous oxygen distribution at the growth interface than other cases due to the existence of the
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