Adjustment of oxygen transport phenomena for Czochralski silicon crystal growth

Abstract

During silicon crystal growth, oxygen, a well-known major impurity, affects the final silicon wafer's mechanical and electrical properties. This study focused on regulation of discharge of different concentrations of oxygen from the quartz crucible into the silicon melt while considering the crucible angular speed and the friction at the melt–crucible interface. The three-dimensional transient governing equations for heat transfer, fluid flow, and impurity transportation in the Czochralski (CZ) puller were solved numerically. The oxygen solvation equation representing the crucible to silicon melt was modified to evaluate the accuracy of oxygen concentration calculations during the CZ process. Experimental measurements using the Fourier-transform infrared (FTIR) technique were used to confirm the simulation results. The results demonstrate that the crucible angular speed affects the oxygen concentration near the crucible wall and therefore in the silicon ingot. The proposed modifications for evaluating oxygen concentration offer a more comprehensive understanding of the oxygen dynamics during the CZ crystal growth.

Highlights

• •Oxygen distribution is influenced by crucible speed, and existing equations may not accurately predict oxygen concentration changes.
• •A modified equation considering crucible-melt interactions and crucible quality yielded more accurate results, aligning with experimental data.
• •The ratio between the crucible-melt contact surface (Acru) and silicon melt free surface (Asurf) decreased as the crystal length increased, leading to lower oxygen concentration.

1. Introduction

The Czochralski (CZ) method is used to produce the monocrystalline or single crystal silicon wafers for photovoltaic or semiconductor applications [[1], [2], [3], [4], [5]]. Presently, this method is the most widely used technique for crystal growth worldwide [6]. Oxygen is one of the most important impurities during silicon crystal growth. Oxygen concentration affects the silicon melt's thermal properties, heat transfer characteristics, and crystal growth rates and therefore plays an important role in controlling the stability and rate of crystal growth during the CZ process [[7], [8], [9], [10]]. Additionally, in producing high-quality silicon wafers, it is necessary to minimize oxygen impurities because oxygen impurities can cause light-induced degradation and negatively affect solar cell performance [11,12]. Haunschild et al. [13] investigated the impact of oxygen concentration on silicon solar cells and accurately forecasted oxygen-induced losses of over 4% from PL images of multicrystalline silicon wafers. These losses manifest as ring-like structures with reduced intensity. By comparing with QSSPC-based measurements, PL-based quality control enables early identification of critical wafers during production, thereby boosting yield and efficiency in industrial solar cell process. Basnet et al. [14] investigated silicon heterojunction solar cells on high-quality wafers. Their study confirmed that pre-fabrication treatments notably enhance bulk quality and reduce defects in upgraded metallurgical-grade silicon, resulting in increased cell efficiencies of up to 21.2% for UMG-Cz and 22.7% for electronic-grade cells. Li et al. [15] investigated the impact of oxygen-related thermal donors (TDs) on silicon heterojunction (SHJ) solar cell performance. Their study revealed that even a moderate concentration of TDs, around 10¹⁵ cm⁻³, significantly reduces SHJ solar cell efficiency by approximately 1% absolute. They found that TDs notably diminish the carrier lifetime of n-type silicon substrate without affecting carrier mobility. Deep level transient spectroscopy (DLTS) identified a specific energy level associated with TDs, contributing to reduced carrier lifetime and solar cell efficiency. Basnet et al. [16] examined electronic quality in Ga-doped silicon wafers from three RCz-Si ingots. As-grown, iron concentrations ranged from 3 × 10⁹ cm⁻³–2 × 10¹⁰ cm⁻³, increasing from seed to tail due to metal accumulation. Iron–gallium pairs weren't dominant recombination centers, suggesting oxygen precipitates or other transition metals play a role. Tabula rasa treatment increased iron concentration, possibly due to dissolved iron-decorated oxygen precipitates. Phosphorus diffusion gettering reduced iron's negative impact. However, subsequent ingots showed decreasing effective lifetimes, suggesting limitations from other recombination centers, potentially related to oxygen precipitates and transition metals.

Understanding and regulating the oxygen transport process during crystal growth is essential for optimizing the crystal quality and yield. Liu et al. [8] found that oxygen impurity comes from the dissolution of the quartz crucible. During CZ growth, part of dissolved oxygen segregates into growing crystals, and the remaining oxygen evaporates on the free surface (the melt-argon gas interface) [17]. Various researchers have proposed different equations to determine the dissolved oxygen concentration based on the crucible temperature [7,8,11,[18], [19], [20]]. However, these equations do not incorporate the effects of crucible angular speed, crucible-melt friction, and/or the quality of the crucible coating. It is crucial to consider these factors as they influence the release of oxygen from the crucible and affect the oxygen distribution within the silicon melt and final silicon wafers. In this study, we attempted to regulate the oxygen concentration discharged from the quartz crucible into the melt. The equations for calculating dissolved oxygen concentrations were modified by addition of a new term that included crucible angular speed and crucible quality. The simulation results were then compared with experimental measurements based on the Fourier-transform infrared (FTIR) technique to validate the proposed modifications and assess their effectiveness in predicting oxygen dynamics during crystal growth. By obtaining a better understanding of the oxygen transport process, we can improve the control over crystal quality and optimize the CZ crystal growth process for producing high-quality monocrystalline silicon for semiconductor applications.

2. Method and modelling

2.1. Governing equations

The schematic depiction in Fig. 1 shows the potential oxygen transitions in the CZ processes, emphasizing the significant interactions between the species in the melt, gas, and solid surfaces.

Fig. 1.

Oxygen reaction during Czochralski (CZ) silicon growth. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

The walls of the quartz crucible serve as the main sources of oxygen during the CZ process. During the process, the oxygen that has been partially dissolved may then be transported into silicon ingots or evaporate as argon gas from the melt free surface.

The governing energy equation and naiver stocks equations applied to the CZ puller are [[21], [22], [23], [24]]:

$$\frac{\partial T}{\partial t}+v_r\frac{\partial T}{\partial r}+v_z\frac{\partial T}{\partial z}+\frac{v_{\phi }}{r}\frac{\partial T}{\partial \phi }=\alpha \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}T}{{\partial \phi }^2}+\frac{{\partial }^{2}T}{{\partial z}^2}\right)+\frac{\dot{q}}{\rho c_p}+q$$ (1)

$$\mathrm{\rho }\left(\frac{\partial u_r}{\partial t}+u_r\frac{\partial u_r}{\partial r}+\frac{u_{\phi }}{r}\frac{\partial u_r}{\partial \phi }+u_z\frac{\partial u_z}{\partial z}-\frac{{u_{\phi }}^2}{r}\right)=\mu \left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u_r}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{u}_r}{\partial {\phi }^2}+\frac{{\partial }^2{u}_r}{\partial z^2}-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_{\phi }}{\partial \phi }\right]$$ (2)

$$\left(\frac{\partial u_{\phi }}{\partial t}+u_r\frac{\partial u_{\phi }}{\partial r}+\frac{u_{\phi }}{r}\frac{\partial u_r}{\partial \phi }+u_z\frac{\partial u_{\phi }}{\partial z}+\frac{u_r{u}_{\phi }}{r}\right)=\mu \left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u_{\phi }}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{u}_{\phi }}{\partial {\phi }^2}+\frac{{\partial }^2{u}_{\phi }}{\partial z^2}-\frac{u_{\phi }}{r^2}+\frac{2}{r^2}\frac{\partial u_{\phi }}{\partial \phi }\right]$$ (3)

$$\left(\frac{\partial u_z}{\partial t}+u_r\frac{\partial u_z}{\partial r}+\frac{u_z}{r}\frac{\partial u_z}{\partial \phi }+u_z\frac{\partial u_z}{\partial z}\right)=-\frac{\partial P}{\partial z}+\mu \left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u_z}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2{u}_z}{\partial {\phi }^2}+\frac{{\partial }^2{u}_z}{\partial z^2}\right]+\rho {\beta \left(T-T_m\right)g}_z$$ (4)

T represents temperature, $\alpha$ is the thermal diffusivity, $\rho$ is the density, $c_p$ is the specific heat, and $u_r,u_z\text{,}$ and $u_{\phi }$ are the melt or argon gas velocity components in the r, z, and $\phi$ directions, respectively. Also, $\beta$ is thermal expansion coefficient, $\mu$ is silicon viscosity, $T_m$ is silicon melting temperature, and $\dot{q}$ is the heat generation, which is just applied to top and side heaters.

Inside the furnace, the radiation heat transfer is also applied on all surfaces. During transient simulation, the crystal front shape is determined using the moving mesh approach. In this method, the movement of each mesh is estimated at each time step using the movement due to temperature gradient difference and growth rate as shown below [21,25]:

$$\Delta X=\left({v_{Pull+}v}_{growth}\right)\Delta t\text{,}$$ (5-a)

$v_{growth}$ is the growth velocity due to temperature gradient difference in crystal and melt, which can be determined by the equation shown below [25]:

$$v_{growth}=\frac{k_s\nabla T_s-k_l\nabla T_l}{{\rho }_s\Delta H}\text{,}$$ (5-b)

in which $\Delta H$ denotes the solidification latent heat, and s and l denote solid and liquid phases, respectively, on the silicon.

The following mass transportation equation can be applied to calculate the oxygen concentration in silicon ingot, silicon melt. and argon flow [26]:

$$\frac{\partial C}{\partial t}+v_r\frac{\partial C}{\partial r}+v_z\frac{\partial C}{\partial z}+\frac{v_{\phi }}{r}\frac{\partial C}{\partial \phi }=D\left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}C}{{\partial \phi }^2}+\frac{{\partial }^{2}C}{{\partial z}^2}\right)$$ (6)

in which C is the oxygen concentration, and D is the diffusion coefficient of oxygen in silicon melt, silicon ingot, or argon gas.

2.2. Oxygen incorporation into silicon melt

Different equations have been proposed by various researchers to determine the oxygen concentration that dissolves from the crucible wall into the melt. All researchers believe that oxygen will dissolve in the molten silicon based on the reaction shown below:

$${{SiO}_2}_{solid}\iff {Si}_{solid}+2O_{liquid}\text{,}$$ (7)

According to Matsuo et al. [20], the boundary conditions for the oxygen concentration on a crucible's wall is as follows:

$$C_{\mathrm{o}}=0.5\times {10}^{23}\frac{\mathit{exp}\left(-3200/T-8.19\right)}{1-\mathit{exp}\left(-3200/T-8.19\right)}\left[\frac{atom}{cm3}\right]$$ (8)

in which T is the temperature of crucible wall. Huang et al. [18] defined the following relationship between oxygen content and quarts crucible wall temperature:

$$C_O=1.32\times {10}^{19}\mathit{exp}\left(\frac{-3.20x{10}^3}{T}\right)\left[\frac{atom}{cm3}\right]\text{,}$$ (9)

Helming et al. [27] also suggest the following relationship between oxygen content and crucible wall temperature:

$$C_O=3.99\times {10}^{23}\mathit{exp}\left(-\frac{2.0x{10}^4}{T}\right)\left[\frac{atom}{cm3}\right]\text{,}$$ (10)

The oxygen concentration dissolves into the silicon melt in all of the aforementioned relationships simply because of the temperature of the crucible wall without consideration of the silicon melt–crucible interaction or the crucible angular speed. Additionally, the coating and quality of the crucible have no impact on the amount of dissolved oxygen. Previous research has shown that the dissolution of oxygen is not solely dependent on temperature. Sabatino et al. [28] conducted experiments using two different crucibles sourced from different vendors. They discovered that the quality of the crucible significantly influences the dissolution of oxygen into the silicon melt. Additionally, they observed that the top region of the crucible, which experiences lower temperatures compared to the middle and bottom areas near the heater, exhibited the highest formation of bubbles, indicating a higher rate of oxygen dissolution. This phenomenon occurs because the silicon melt speed is highest at this point due to its larger radius. Wang et al. [29] discovered that interactions between silicon (Si) and quartz crucible surface cause surface irregularities. These irregularities fill grooves during solidification, leading to adhesion at the interface between silicon and silicon dioxide. Sangiorgi and Muolo [30] documented a notable adhesion strength between silicon (Si) and silicon dioxide (SiO2) attributed to the involvement of chemical bonding, signifying a reactive wetting regime. Additionally, the erosion groove on the crucible is detected by Richer et al. [31] and it was reported that this erosion is caused by erosion through the silicon melt. For this situation, we modified the equations shown above with new non-dimensional term FR as shown below:

$$FR=\frac{f{R\left(z\right)}^2\omega }{\alpha }$$ (11)

in which f is the empirical coefficient depends on crucible coating and quality, R is the crucible radius at any contact point with silicon melt, $\omega$ is the angular crucible speed, and $\alpha$ is the thermal diffusivity of silicon melt next to crucible wall. The empirical coefficient 'f' can be determined by comparing a set of experimental data with simulations using different values of 'f'. It should be noted that 'f' depends on the quality and age of the crucible, as well as any potential coatings on the quartz. Additionally, the density of bubbles in the outer layer of the crucible can play a significant role in the final release of oxygen between the quartz and the melt. The FR term is added into equation (8) as shown below:

$$C_O=\text{FR}\times 0.5\times {10}^{23}\frac{\mathit{exp}\left(-3200/T-8.19\right)}{1-\mathit{exp}\left(-3200/T-8.19\right)}\left[\frac{atom}{cm3}\right]\text{,}$$ (12)

Equation (8) was chosen because it is favored by more researchers over other equations. This new proposed equation not only provides greater flexibility by including the constant f to account for the coating and quality of the crucible, but it also considers the friction between the melt and the crucible, which is dependent on crucible angular speed.

2.3. SiO-evaporation from silicon melt

For the transport of oxygen impurities from silicon melt to crystal, the segregation effect is considered using the equation shown below for the silicon melt–crystal interface [32]:

$$k_0{C}_{liquid}=C_{solid}\text{,}$$ (13-a)

$${\left({kD}_{liquid}\frac{\partial C_{liquid}}{\partial n}\right)}_{\text{melt}}={\left(D_{solid}\frac{\partial C_{solid}}{\partial n}\right)}_{\text{crystal}}\text{,}$$ (13-b)

in which $C_{liquid}$ and $C_{solid}$ are the oxygen concentrations in the melt and crystal, D is the oxygen diffusivity is silicon crystal and melt and $k_0$ is the segregation coefficient.

2.4. Oxygen evaporation from the silicon melt to argon gas

The boundary condition for the oxygen evaporation at molten silicon free surface can be determined according to Smirnov and Kalaev [33]:

$${\mathrm{C}}_{\mathrm{O}}={\mathrm{C}}_{\text{Sio}}\frac{m_O}{m_{\mathit{Si}}}\frac{m_{\mathit{Ar}}}{m_{\mathit{SiO}}}2.4\times {10}^{‐9}{\mathrm{P}}_{\text{Ar}}\exp \left(\frac{34470}{\mathrm{T}}‐17.62\right)\text{,}$$ (14)

in which $m_{\mathrm{O}},m_{\text{Si}},m_{\text{Ar}}$, and $m_{\text{Sio}}$ are the molar masses of oxygen, silicon, argon, and silicon oxide (SiO), respectively, and $P_{\text{Ar}}$ is the CZ puller's argon pressure. The concentration flow of SiO in the argon and oxygen in the melt is conserved at the free surface based on the equation shown below:

$$\frac{{\left(\rho D\right)}_0}{m_O}\frac{\partial C_O}{\partial n}=\frac{{\left(\rho D\right)}_{SiO}}{m_{SiO}}\frac{\partial C_{SiO}}{\partial n}\text{,}$$ (15)

3. Results

A CZ process was simulated and experimentally performed for a silicon ingot with a diameter of 200 mm diameter and length of 700 mm. The walls of the CZ furnace were considered to be at room temperature. The argon gas pressure and volume were 15 Torr and 50 L/s, and the crystal growth rate was set at 1 mm/min. The initial melt silicon was about 250 kg and located inside a SQC series 20 inch crucible. For this crucible, the friction coefficient, f, was considered to be 0.001. The crystal rotated with a constant speed of 5 rpm in different directions. The crucible rotation speed, $\omega \text{,}$ was varied to investigate the effects of different oxygen concentration on the ingot. A mesh sensitivity analysis was performed, and the maximum mesh size was fixed at 0.1 mm with a mesh growth rate of 0.9 and a curvature factor of 0.3. To maintain the courant number below 0.5, the maximum time step was restricted.

Fig. 2 shows the temperature distribution during the crystal growth process. According to the simulation, the maximum CZ puller temperature roughly reached 1525 °C. The thermal shield, which is also called the heat shield, created a hot region in its lower area and a cooling region at the upside region. These two hot and cold areas provided free energy for crystal growth. Because the main focus of this study was oxygen concentration, additional details about temperature distribution can be find in our previous study [21].

Fig. 2.

Temperature distribution during the CZ silicon crystal growth for crucible speed 10 rpm and crystal lengths (a) 175 mm, (b) 350 mm, (c) 525 mm, and 700 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

The oxygen concentration during the process is demonstrated in Fig. 3 for crystal lengths 175 mm, 350 mm, 525 mm, and 700 mm. As a general trend, the oxygen concentration close to the crucible wall is very high. As the melt surface was approached, lower oxygen concentrations were detected because oxygen evaporated from the free surface. Due to high value of the partition coefficient, $k_0=0.9$, on the crystal's front side, most of the oxygen atoms that reached the melt–solid interface were segregated into the silicon crystal. A concentration homogeneity was found at the melt–crystal interface.

Fig. 3.

Oxygen concertation in the melt and silicon ingot at a crucible speed of 10 rpm and crystal lengths of (a) 175 mm, (b) 350 mm, (c) 525 mm, and (d) 700 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

The oxygen concentration at the crystal centreline is depicted in Fig. 4. As can be observed, the oxygen concentration decreased from 21.4 to 16.3 Ppma as a function of crystal length. Since the melt inside the crucible declined as the crystal grew, is less surface contact between the melt and the crucible was present, which results in less oxygen being dissolved into melt. On the other hands, the silicon melt free surface did not change. Under these conditions, the ratio between the crucible–melt contact surface, Acru and silicon melt free surface, Asurf, decreased, which resulted in less dissolved oxygen and higher oxygen evaporation into argon gas. Therefore, as shown in Fig. 4, when the crystal length increased, the oxygen concentration decreased. The Acru/Asurf ratio as a function of crystal length is shown in Fig. 5. As can be observed, initially the Acru/Asurf ratio was about 3.43. Under these conditions, the melt height inside the crucible was about 270 mm. Due to the large melt–crucible interface, the oxygen concentration was high inside the melt. During this time, the average oxygen concertation in the silicon melt was determined to be about 44 Ppma. During the crystal's growth, the melt transferred into the crystal while less melt remained inside the crucible. For example, at a crystal length of 700 mm, the melt height inside the crucible was 150 mm and the Acru/Asurf declined to 2.36. Under these conditions, the average oxygen concentration in the melt was about 34 Ppma.

Fig. 4.

Oxygen concentration at the crystal's centerline at a crucible speed of 10 rpm.

Fig. 5.

The melt–crucible interface and melt free surface ratio during crystal growth.

Although the main source of oxygen was the crucible wall, the oxygen concentration and amount of oxygen transported cannot explain the amount of oxygen released from the crucible wall. The studies showed that melt flow also plays important role in the final oxygen distribution as the silicon melt is the oxygen transporter from the crucible wall to the crystal front or melt-free surface. Fig. 6 shows the melt flow streams at different crystal lengths; the color of the melt streams indicates the melt speed. As the results show, when the silicon melt amount was large in the crucible, the Taylor–Proudman vortex formed in the bottom and center of the crucible. At the same time, on the upper melt region and next to crucible wall, buoyancy forces caused the formation of counter rotating vortex, a phenomenon known as buoyancy–thermocapillary [32]. Under these conditions, the oxygen released from crucible wall underwent transfer to the melt-free surface directly via the buoyancy vortex. On the crucible bottom but the released oxygen has high chance to mix with silicon melt and transfer to crystal front. During the crystal growth and decrease in the melt level, the Taylor–Proudman vortex disappeared and in its place, a large buoyancy vortex formed inside the melt as shown in Fig. 6(c and d). Under these conditions, most oxygen released from the melt was transferred into melt free surface, which is another reason for the decrease in oxygen concentration as a function of crystal length.

Fig. 6.

The melt flow stream for crystal lengths (a) 175 mm, (b) 350 mm, (c) 525 mm, and (d) 700 mm.

After investigating the oxygen distribution during the CZ process, a new adjustment for the oxygen level could be thoroughly evaluated. To evaluate the new adjustment in equation (12), the experimental investigations were conducted. In this situation, single silicon crystal growth was produced under the same conditions with crucible rotation speeds 1, 2.5, 5, 7.5, and 10 rpm. The FTIR technique was used to calculate the oxygen concentrations in these crystals in the axial directions at ingot centreline. The software developed for this study enables the analysis of FTIR absorbance or transmittance data to calculate the concentration of oxygen interstitials (Oi) using three distinct algorithms: the standard test method (STM), the short baseline technique (SBT), and the curved baseline technique (CBT) based on ASTM F1188.2 standards. The outcomes of the simulation and experiment are shown in Fig. 7. The simulation results are also depicted for the original equation (8) to allow a comprehensive comparison for oxygen content at the ingot centreline. The simulated oxygen concentration in the silicon melt and crystal as a function of crucible angular speed resulting from the simulation is also shown in Fig. 8.

Fig. 7.

The oxygen concentration as a function of crucible speed.

Fig. 8.

The oxygen distribution in the melt and silicon ingot at the end of the CZ silicon crystal growth. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

The results reveal that when equation (8) is applied without our adjustment term, increasing the crucible speed from 1 to 2.5 rpm caused the oxygen concentration to decrease. Under these conditions, only the melt flow pattern affected the distribution of oxygen when the oxygen was simply assumed to dissolve into the melt due to the crucible's temperature. Under these conditions, the decrease in oxygen concentration was due to melt flow pattern. As shown in Fig. 7, this oxygen concentration decline was not observed in experimental data. On the other hands, when equation (12) is used, due to considering the angular speed of crucible, the oxygen concentration near the crucible wall rises. Although a greater barrier was formed between the crucible wall and crystal front in this situation due to the melt flow pattern, the oxygen content of the ingot increased. This finding shows more compatibility with experimental observations.

Furthermore, equation (8) shows little change in oxygen concentration as the crucible speed increases from 2.5 to 10 rpm, but our experimental data show a greater change in oxygen concentration. When considering the melt flow changed as the crucible speed was increased from 2.5 to 10 rpm based only on equation (8), the crucible temperature cannot have sufficiently affected this process, and the final oxygen concentration would have remained the same. On the other hand, when the crucible angular speed was considered in the calculation of the final oxygen concentration approach to experimental observation, it affected he final oxygen concentration. Statistical analysis revealed that simulation results obtained from proposed model, generally yielded slightly lower mean values and exhibited higher variability compared to experimental data. For instance, at 1 rpm, the mean radial oxygen concentration was 15.71 ppma (simulation) with a standard deviation of 0.51 ppma, while for experimental data, it was 15.64 ppma with a standard deviation of 0.42 ppma. Similarly, at 2.5 rpm, the mean radial oxygen concentration was 16.60 ppma (simulation) with a standard deviation of 0.52 ppma, whereas for experimental data, it was 17.05 ppma with a standard deviation of 0.39 ppma.

It is worth mentioning that from the results depicted in Fig. 8, it can be inferred that the oxygen concentration exhibits a slight concave trend towards the melt. This suggests that in the radial direction, the oxygen concentration is higher towards the center of the wafer. This radial decrease in oxygen concentration is attributed to the evaporation of oxygen at the free surface adjacent to the side section of the ingot. Additionally, during the process, segregated oxygen tends to accumulate near the solid-liquid interface, leading to an increase in oxygen concentration at the center of the ingot.

4. Conclusion

This study demonstrates that the crucible speed does have a significant role in determining the oxygen concentration near the crucible wall and in the silicon ingot. When using the existing equations without considering crucible angular speed, an increase in crucible speed from 1 to 2.5 rpm resulted in a decrease in oxygen concentration. However, experimental observations indicate that the oxygen content actually increased as the crucible speed. increased. By including the FR term in modified equation (12), the simulation results align more closely with the experimental data, emphasizing the importance of considering crucible–melt interactions. By considering the effects of crucible-melt friction and crucible quality, we have developed a modified approach to simulate and predict the oxygen distribution during CZ crystal growth. The proposed modifications yield more accurate results compared to the existing equations and are in good agreement with experimental observations. This study contributes to the ongoing efforts to control crystal quality and optimize CZ crystal growth processes for producing high-quality monocrystalline silicon for semiconductor applications.

Towards the conclusion, it is imperative to underscore that the interaction between the molten substance and the crucible is incorporated empirically through coefficients labeled as “FR". This coefficient's value is greatly influenced by the type and quality of the crucible. For diverse types of quartz crucibles, this empirical parameter must be determined through a comparison between simulation and experimental results once more. Future research endeavors could delve deeper into studying the interaction between the molten substance and the crucible surface with greater precision. Furthermore, investigating the impact of coating crucible quartz with various materials could offer insights into controlling the infusion of oxygen into the molten substance.

Funding

This work was supported by the National Science and Technology Council of Taiwan under [grant numbers NSTC: 112-2222-E−167 -001 -MY3].

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

CRediT authorship contribution statement

Amir Reza Ansari Dezfoli: Writing – review & editing, Software, Resources, Project administration, Investigation, Funding acquisition, Formal analysis, Conceptualization. Zary Adabavazeh: Writing – original draft, Investigation, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.