1 Introduction

The Czochralski (CZ) growth process is the main method for single crystalline silicon preparation in microelectronic and photovoltaic industry. To meet the requirements of a high degree of integration and semiconductor devices quality, it is essential to understand the transport phenomena and the fluid flow motions in melt. Owing to the high-temperature environment and high cost, experimental investigation and in situ observation are quite difficult. Therefore, numerical analysis is considered to be a very beneficial and effective tool in analysis of CZ silicon crystal growth, especially on heat transfer and mass transport.

The crystal diameter from 150 to 400 mm requires 32-inch-quartz crucible or even more. Therefore, the crystal preparation needs a large sum of polycrystalline silicon in crucible, which means larger fluid turbulent and temperature fluctuation. The solid–liquid (SL) interface shape is a key question in large diameter crystal growth process, which can influence the crystallization rate, crystal inherent stress or even point defect behavior. Though massive efforts have been done by industry to control SL interface shape, the theory and experience from small size crystal can be adopted partially. With quartz crucible size and crystal diameter increasing, temperature distribution and melt flow indicate the character of size effect. Therefore, it is essential to investigate the effect of key processing parameters, such as crystal-crucible rotation and pulling rate, on SL interface shape.

So far, domestic and overseas scholars have done a lot of researches on melt flow and SL interface shape with different process parameters and furnace configuration. Chen et al. [1, 2] calculated oxygen concentration distribution in silicon melt for different crystal lengths with magnetic field using simulator CGSim. The hot zone structure was optimized [3–12], such as heat-shield, flow guide and sidewall insulation. The results showed that the optimized hot zone can increase the growth rate and crystal quality. The effect of gas flow on global heat transport and melt convection in CZ-Si growth was also studied [13–16], and the cause of W-shape SL interface formation was analyzed [17–19]. Wetzel et al. [20, 21] predicted the growth interface shape in industrial 300 mm CZ-Si crystal growth. Lukanin et al. [22] and Dadzis et al. [23] presented an updated version of the combined 2D/3D model of heat transfer and turbulent melt convection for industry CZ crystal growth. Then, the complex flow during CZ-Si crystal growth process was predicted [24–29]. Yi et al. studied the flow patterns within a large-scale CZ system, using 3-D time-dependent numerical model [30, 31].

The aim in this paper is to investigate the effect of pulling rate and crystal-crucible rotation rate on SL interface shape and find out a proper process route and solution. Different body lengths were considered to evaluate the variation of the optimum control parameters during CZ-Si crystal growth process.

2 Model constructions

The modeling of mass transport and heat transfer in the furnace of KAYEX-150 was analyzed by finite element method (FEM) adopting 2-D axial-symmetric model. The melt flow and heat transfer were governed by the following equations:

$$\begin{aligned} \rho_{0} \cdot (\nu \cdot {\nabla })\nu & = - \nabla p + \nabla \cdot \left( {(\mu + \mu_{\text{T}} )(\nabla \nu + \nabla^{\text{T}} \nu )} \right) \\ & \quad - \,\rho_{0} \beta_{\text{T}} (T - T_{0} )g + J \times B \\ \end{aligned}$$
(1)
$$\nabla \cdot \nu = 0$$
(2)
$$\rho_{l} c\left( {\frac{\partial T}{\partial t} + (\nu \cdot \nabla )T} \right) = \nabla \cdot ((k + k_{\text{T}} )\nabla T) + W$$
(3)
$$- k_{l} \nabla T_{l} \cdot n = - k_{s} \nabla T_{s} \cdot n - \rho_{s} \nu {}_{\text{g}}\Delta H$$
(4)
$$T_{\text{triple}} = 1685\,{\text{K}}$$
(5)

Here, \(\nu\) is velocity, \(B\) is magnetic field intensity, \(\mu\) is coefficient of the melt viscosity, \(\mu_{\text{T}}\) is coefficient of the melt viscosity under magnetic field, \(\beta_{\text{T}}\) is the coefficient of thermal expansion, \(J\) is current density, \(\rho_{0}\) is melt density, \(p\) is pressure, \(T_{0}\) is reference temperature, \(g\) is acceleration of gravity, \(k\) is thermal conductivity, \(k_{\text{T}}\) is thermal conductivity under magnetic field, \(W\) is heater power, \(\rho_{\text{l}}\) is density of melt, \(\rho_{s}\) is density of crystal, \(c\) is specific heat, \(T\) is temperature, \(T_{l}\) is melt temperature, \(T_{s}\) is crystal temperature, T triple is triple point temperature, \(\Delta H\) is crystallization enthalpy, k l is heat conductivity of melt, k s is heat conductivity of crystal, \(\nu {}_{\text{g}}\) is crystal growth rate, \(n\) is the unit outgoing normal from the melt, \(\nabla\) is differential operator.

Boundary conditions for the flow velocity at the solid walls were as follows: the velocity component perpendicular to the wall was set to zero, while the tangential components were equal to the solid wall velocity. The Reynolds-averaged Navier–Stokes (RANS) equations were applied to solve turbulent flow. At the outer walls of the facility, a constant temperature value was adopted as boundary conditions. The grids in the present finite element analysis were constructed as follows: auto splitting for radiative modeling uses 10 mm separations globally; crystal and melt have structured grids near the crystallization interface in 6 mm separations; the crystal and melt that are not adjacent to the interface use triangular grids also in 6 mm separations with stretching coefficient of 1.1 in Delaunay method; parts other than the crystal and melt use triangular grids in 10 mm separations with stretching coefficient of 1.25.

The drawing of Kayex-150 single crystal furnace and the temperature distribution in the growth facility obtained with the 2-D global heat computations are presented in Fig. 1. The material properties utilized in the present computations are summarized in Table 1.

Fig. 1
figure 1

Drawing of Kayex-150 single crystal furnace (left) and computational grid and temperature distributions (right) in growth system obtained with 2-D global heat computations (right)

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Table 1 Material properties used in simulation
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3 Results and discussion

The computational results in this section present the growth of 400-mm-diameter crystals with 32-inch-crucible in an industrial CZ silicon puller. The multi-block combined grid consists of about 19,200 cells and contains mismatched boundaries at SL interface. The control parameters for the crystal growth are as follows: the polycrystalline silicon charge amount is 220 kg, the crystal rotation is 8 r·min−1, the crucible rotation is 10 r·min−1, the argon flow rate is 0.002 m3·min−1, and the pressure of the furnace is 2000 Pa.

3.1 Effect of crystal size and body length on SL interface shape

During crystal growth, the melt flow in the crucible plays a crucial role in terms of crystal quality and dopant variation. Figure 2 illustrates SL interface shape at body lengths of 10, 100, 200 and 310 mm. SL interface shape switches from gull-wing shape to convex and the maximum deflection increases significantly with the crystal growth process going. Generally speaking, the flow pattern is driven by the buoyancy generated by the temperature difference within the melt, the crystal and crucible rotations, the shear stress generated by the argon flow, the capillary stress and the Marangoni stress at the melt free surface. Based on the calculations and the reported studies [13, 32], the effects of Marangoni stress, capillary stress and shear stress are ignored compared to other effects. Therefore, here it was only discussed the effects of key driving forces in the melt flow (the buoyancy, crucible-crystal rotations and crystal pull rate).

Fig. 2
figure 2

Solid–liquid interface deflection for different body lengths

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Figure 3 shows meridional cut through the melt with distribution of the stream traces for different body lengths. When the body lengths reach 10 and 100 mm, there is only one counter-clockwise vortex in the melt. It rises from the bottom along the crucible side and transports to the center at the surface and then flows to bottom. As the body length reaches 200 mm, the flow pattern in the melt starts to change. A small new vortex generates under SL interface. The two vortices have the opposite direction of movement. As the body length reaches 310 mm, the strength and size of the new vortex increase, leading to interface deflection. Overall, with the crystal growth processing, the new vortex becomes bigger and stronger. So the heat input is increased under the interface, the shape of the interface will move up.

Fig. 3
figure 3

Stream functions and streamlines for different body lengths: a 10 mm, b 100 mm, c 200 mm, and d 310 mm (Psi: pounds per square inch)

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High thermal stress (von Misses stress) in the crystal is one of the most possible reasons to generate dislocation. Figure 4 shows thermal stress distributions at the body lengths of 10, 100, 200 and 310 mm. The calculation of the thermal stress is based upon the computation data of temperature distribution. The results show that the thermal stress of the crystal is relying heavily on body length. With the crystal process going, the maximum von Misses stress rapidly increases, which is located at the SL interface and the crystal side surface. Owing to the large interface deformation, the high stress value appears at the SL interface when the crystal growth reaches 310 mm.

Fig. 4
figure 4

Von Mises stress distribution for different body lengths: a 10 mm, b 100 mm, c 200 mm, and d 310 mm (Svm: stress of von mises)

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3.2 Effect of crystal-crucible rotation rates on SL interface shape

The region of maximum SL interface deformation will become the stress concentration point. It means that the intensity of the thermo-elastic stress concentration is in proportion to the degree of interface deflection. Therefore, it is essential to investigate how the crystal and crucible rotation rates affect maximum interface deflection. Figure 5 illustrates the effect of crystal size on SL interface shape. The SL interface deflection increases from 7.4 to 51.3 mm with an increase in crystal size from 150 to 400 mm. As the crystal size gets larger, the maximum SL interface shape deformation rapidly increases.

Fig. 5
figure 5

Solid–liquid interface deflection with different crystal diameters

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Figure 6 illustrates the combined effects of crystal and crucible rotation rates on the maximum interface deflection. It can be seen that the SL interface shape deflection is smaller for higher crucible rotation rates. The minimum point of interface deflection is closely related to the combinations of crystal and crucible rotation rates. It means that improving the quality in terms of the behavior of the melt flow or the shape of the SL interface can be achieved through the selected combinations of crystal and crucible rotation rates. For current furnace geometry and control parameters, the minimum interface deflection appears when the crystal rotation rates are between 4 and 8 r·min−1 and crucible rotation rate is 15 r·min−1 for 100 mm body length. The maximum deformation mostly increases with body length increasing. However, the minimum value still exists, as shown in Fig. 6. The curve for crucible rotation rate of 5 r·min−1 shows parabolic trend with a distinct minimum point. The curves for crucible rotation rates of 10 and 15 r·min−1 are nearly flat. After reaching the minimum point, the effect of crystal rotation on the interface is less significant at higher crucible rotation rates. This is more distinct for 310 mm body length, as shown in Fig. 6 (and also for 400 mm body length, although not shown here).

Fig. 6
figure 6

Maximum solid–liquid interface deflection for lengths of a 100 mm and b 310 mm with various combinations of crystal and crucible rotation rates

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3.3 Effect of pulling rate on SL interface shape

In a CZ furnace, the crystallization takes place at the SL interface. This interface is also called the crystallization front. The SL interface shape directly influences the crystalline perfection and the impurity distribution. The convex shape (viewed from the crystal side) is prone to generate dislocations. As the pulling rate increases, the SL interface gradually changes to the concave shape, as shown in Fig. 7.

Fig. 7
figure 7

Solid–liquid interface deflection for lengths of a 100 mm and b 310 mm with various pulling rates

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The growth rate of crystal, everyplace at the interface, is equal to the axial pulling rate at body growth. During the process of crystallization, the latent heat will release at the crystallization front. The faster the crystal we pull from melt, the more the latent heat will release. The heat power reduces by 4.2 kW with an increase in the pull rate from 0.35 to 1.00 mm·min−1. The change in the heater power leads to the temperature field fluctuation in CZ furnace.

The temperature in the furnace decreases because of the reducing heater power. With an increase in pulling rate, more latent heat is released at the crystallization front and must be conducted away into crystal. This is accomplished partly through decreasing heat power and partly by increasing interface deflection. If the heat input is increased, the shape of the SL interface will move up. If the heat input is decreased, the SL interface will move down closer to the melt level. The interface height increases by 8 mm with an increase in the pull rate from 0.35 to 1.00 mm·min−1.

4 Conclusion

In this study, a set of numerical simulations were carried out to investigate the change rules of melt flow and SL interface geometry for different control parameters. With crystal diameter increasing, the SL interface deflection increases accordingly. The SL interface deformation and the maximum thermal stress continue to increase with the crystallization going. The stress concentration mainly locates at the crystallization front and the crystal side surface. The SL interface shape is sensitive to both crucible and crystal rotation rates. The optimal combination of crystal and crucible rotations slightly changes during the crystal growth process. The maximum SL interface shape deflection increases with the crystal rotation rate accelerating and decreases with the crucible rotation rate speeding. The lowest SL interface deflection can be realized through optimizing the combination of crucible and crystal rotation. In addition, reducing pulling rate can flat SL interface shape and subsequently add energy-consuming.