(371y) Gray-Box Model for Predicting Crystal Radius and Crystal Growth Rate of 300 mm Czochralski Single-Crystal Silicon Ingot Production Process

Nowadays, AI and IoT have become widespread, and demand for semiconductors has increased; therefore, reduction of costs and improvement of production efficiency are required in producing a single crystal silicon ingot, which is a key material of semiconductors.

A 300 mm diameter single crystal silicon ingot is mainly produced by the Czochralski (CZ) method. To keep high product quality, we need to precisely control the crystal radius and the crystal growth rate by manipulating the heater input, the crystal pulling rate, and the crucible rise rate. The CZ process has the following characteristics that make the control of quality difficult. The amount of the melt and the position of the crucible change with crystal growth, and the heat transfer characteristic changes with the positional relationship between the heater and the crucible. In addition, it takes a long time for heat transfer from the heater to the melt. Although PID control has been widely used because it is easy to implement, it has a limited control performance due to the above characteristics. Thus, expecting that model predictive control (MPC) would be useful to realize precise control, we constructed the gray-box model of an industrial 300 mm CZ process in our past research [Kato et al. 2019]. The gray-box model consists of two parts: the hydrodynamic-geometrical model which calculates the crystal radius, and the heat transfer model which calculates the crystal growth rate.

In the hydrodynamic-geometrical model, the variable related to the height of the meniscus part is estimated by the method proposed by Seto et al. [Seto et al. 2016]. The height of the meniscus part is calculated by the approximated solution of the Euler-Laplace equation [Ferguson 1912] in the hydrodynamic-geometrical model of the first-principle model [Zheng et al. 2018]. Although the Euler-Laplace equation assumes the steady state, this assumption may not be satisfied since the crucible position and the crystal growth rate change during the pulling process. To cope with this problem, Seto et al. proposed the method to calculate the value of ψ₁, which is related to the height of the meniscus part, by the statistical model [Seto et al. 2016]. The hydrodynamic-geometrical model needs the initial values of the crystal slope angle and the statistical model to calculate ψ₁. On the other hand, the heat transfer model requires the initial values of the heater temperature and the melt temperature [Zheng et al. 2018]. These variables are not measured in the industrial CZ process.

When the initial values were determined by minimizing the error of the controlled variables between the measured values and the calculated values using the gray-box model just before the prediction period, the prediction accuracy at the end of the pulling process was lower than the other period. We proposed the method to calculate the initial values by the statistical model, which was built with the data of past two batches, and the method could reduce the prediction error on average by 24% in comparison with the conventional method using only the data of the current batch. The prediction error, however, was sometimes larger than the tolerance.

In our gray box model, the heater is divided into three parts to describe the thermal environment which changes with the volume of the melt. The temperature of the middle part of the heater is assumed to be the same as that of the bottom part. When the proposed method in the previous paragraph was applied, both the temperature at the upper part of the heater and that at the lower part sometimes changed drastically during the fitting period, in which the initial values were determined. In the CZ process, the temperatures are essential variables related to product quality. Since the temperatures would not change rapidly, we assumed that this rapid change of the temperature decreased the prediction accuracy. To further improve the prediction accuracy, the difference between the temperatures at the upper part and the lower part of the heater was included in the objective function to determine the initial values. We calculated the initial values to minimize the proposed objective function and could improve the prediction accuracy of the whole pulling process. It is assumed that the fitting period used for parameter estimation also affects the prediction accuracy because the heat transfer between the heater and the melt takes a long time in the CZ process. We compared the prediction accuracy of the multiple periods for parameter estimation to clarify the effect of the fitting period. By using the model proposed in this research for MPC, the precise control would be realized in the pulling process.

References

S. Kato, H. Yoshioka, S. Kim, M. Kano, T. Fujiwara, M. Mizuta, and S. Hasebe. Model predictive control of Czochralski process producing 300 mm single crystal silicon ingot. The 8th International Symposium on Design, Operation and Control of Chemical Processes (PSE Asia 2019), Bangkok, Thailand, 2019.

T. Seto, S. Kim, M. Kano, T. Fujiwara, M. Mizuta, and S. Hasebe. Gray-box modeling of 300 mm Czochralski single-crystal Si production process, 2016 AIChE Annual Meeting, San Francisco, US, 2016.

A. Ferguson. LXXXIX. On the shape of the capillary surface formed by the external contact of a liquid with a cylinder of large radius, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 24, No. 144, pp. 837-844, 1912.

Z. Zheng, T. Seto, S. Kim, M. Kano, T. Fujiwara, M. Mizuta, S. Hasebe. A first-principle model of 300 mm Czochralski single-crystal Si production process for predicting crystal radius and crystal growth rate, Journal of Crystal Growth, Vol. 492, No. 15, pp. 105-113, 2018.