Introducing plasma processes into semiconductor manufacturing has enabled the fabrication of high-aspect-ratio micropatterns [1,2]. These advancements have paved the way for patterns with complex three-dimensional stacked structures, improving the performance of semiconductor equipment [3,4]. The complexity of equipment structures increases as equipment performance improves; moreover, improvements in process capabilities increase the number of process steps required to improve the device performance [5].

Some plasma processes have been developed to achieve high density and uniformity [6,7]. Achieving the requisite plasma density and uniformity for semiconductor processes involves extensive trial and error because of the complex interactions between process gases and electrons [8], coupled with the effects of process variables, such as power and pressure [9].

Data-driven machine learning (ML) methods have been widely adopted across various scientific fields [10,11]. ML serves as an effective method for tackling complex and computationally expensive realworld systems [12]. These ML-based optimization approaches have demonstrated their effectiveness as valuable tools for addressing intricate engineering problems, with potential applications in optimizing plasma processes [13].

Data from the simulations of Ar/O₂ inductively coupled plasma (ICP) discharge were used for model training before designing the deep learning (DL) model. The DL model was optimized through six selection criteria, being trained with power, pressure, and O₂ ratio as input data to predict the electron density, electron temperature, potential, and densities of Ar⁺, O²⁺, O⁻, and O⁺. The performance of the deep-learning model was validated by comparing its interpolation and extrapolation results against the ground truth. Interpolation predicts a constant pattern within the range of the trained data, while extrapolation extends predictions beyond that range [14].

This paper is structured to provide a comprehensive overview of the experimental details, encompassing the presentation of the DL model and optimization methodology. The results and discussion section delves into the detailed optimization results and validation of the DL model against the ground truth. The conclusion section summarizes the study.

2.1. Gaseous electronics conference (GEC) chamber structure

Figure 1 depicts an ICP chamber with a radius and height of 30 and 30 cm, respectively. Simulation results for the plasma discharge were obtained using COMSOL Multiphysics. The ICP chamber was subjected to the conditions listed in Table I to generate a dataset for prediction using the DL model. Subsequently, the electron density, electron temperature, potential, and densities of Ar⁺, O²⁺, O⁻, and O⁺ were calculated. The frequency value and flow rate were fixed at 13.56 MHz at 250 sccm, respectively. Moreover, the ranges of O₂ ratio, pressure, and power were set to 0.3–0.7 (ITI:0.1), 10–50 mTorr (ITI:10 mTorr), and 100–180 W (ITI:20 W) to assess the changes in plasma variables based on parameters mentioned above for constructing the DL model.

Table 1 . Chamber parameter condition.

Parameter	Value
Frequency (MHz)	13.56
Quality factor (Qf)	250
Power (W)	100 ~ 180
Pressure (mTorr)	10 ~ 50
O2 ratio	0.3 ~ 0.7

Figure 1. GEC ICP chamber structure: (a) 3D image and (b) 2D image.

2.2. Deep neural network (DNN) prediction mechanism

The prediction process follows a step-by-step procedure. This method can mitigate inaccurate predictions by sequentially predicting O₂ ratios, pressures, and power levels; furthermore, this approach demonstrates superior performance compared to existing DNN models [15,16]. A total of 25 data files were generated by maintaining the pressure and power at fixed values and organizing the files depending on the O₂ ratio. The optimal DNN model for the data was determined through a cross-validation process. The first prediction was based on the O₂ ratio, followed by additional predictions based on pressure. Finally, we interpolated and extrapolated the three variables by predicting them based on power. Accuracy was confirmed by comparing the predicted data with the original data.

2.3. DNN model optimization

The DNN models were cross-validated using six criteria: input data normalization, output data normalization, train type, train network, activation function, and hidden layer, which were compared to select the optimized DNN model. We compared the mean absolute error (MAE) of each dataset and selected the lowest case (Table II). The cases of the train network and hidden layer were not selected based on MAE. The accuracy of the data was low because of overfitting, stemming from excessively complex and large-scale neural networks [17,18]. Furthermore, we selected the most reasonable case while considering the time required for training. We aimed to minimize the underfitting and overfitting by optimizing the DNN model.

Table 2 . Details of each case.

Category	Case	Details
Input normalization	1	None
	2	Z-score normalize
Output normalization	1	None
	2	Log scale normalize
Train type	1	Trainbr
	2	Trainbfg
	3	Trainlm
Train network	1	Cascadeforwardnet
	2	Fitnet
	3	Feedforwardnet
Activation function	1	None
	2	ReLU
	3	Tanh
	4	Sigmoid
Hidden layer	1	(n[1] = 20, n[2] = 20, n[3] = 20)
	2	(n[1] = 10, n[2] = 40, n[3] = 10)
	3	(n[1] = 20, n[2] = 20, n[3] = 20, n[4] = 20)

2.4. Final selection

Input and output data normalizations were achieved through Zscore normalization and log normalization, respectively. Bayesian regulation (Trainbr) and feedforward net were train type and train network, respectively. The activation function was the rectified linear unit (ReLU), and the hidden layer was set as (n^[1] = 20, n^[2] = 20, n^[3] = 20) a selection deemed optimal for the DNN model. Applying Z-score and log-scale normalization serves the dual purpose of expediting the training and enhancing the generalization of the DNN [19]. Bayesian regulation, compared to standard backpropagation networks, offers increased potency and can minimize lengthy cross-validation [20]. The feedforward network is notable for its efficiency in terms of the number of configured units and its capacity to generalize to patterns not included in the training dataset [21]. Notably, the ReLU DNN model with sufficient layers can reproduce any linear finite element function [22]. Predictions were made for seven outputs, namely, electron density, electron temperature, potential, and densities of Ar⁺, O²⁺, O⁻, and O⁺, based on three input variables–O₂, pressure, and power. We trained the network with 20 neurons in each of the three layers.

3.1. Interpolation

We opted for specific variable values: O₂ = 0.4, p0 = 35 mTorr, and Pw = 150 W based on data from COMSOL (Multiphysics v. 6.1. COMSOL AB, Stockholm, Sweden). Interpolation predicts data within the bounds of the existing data. Hence, the optimized DNN model with O₂ = 0.4, p0 = 35 mTorr, and Pw = 150 W was predicted in two steps, by the sequential prediction of the pressure and power using the prepared data for O₂ = 0.4, p0 = 10–50 mTorr, and Pw = 100–180 W. We could predict p0 = 35 mTorr because our data included values for p0 = 10–50 mTorr. Subsequently, we obtained data for O₂ = 0.4, p0 = 35 mTorr, and Pw = 100–180 W, which enabled us to predict the value for Pw = 150 W, culminating in the prediction for O₂ = 0.4, p0 = 35 mTorr, and Pw = 150 W.

Figure 2(a) compares the data predicted by the DNN model in Step 2 with the ground truth from COMSOL. Seven variables were compared: electron density, electron temperature, electron potential, Ar⁺ ion density, O²⁺ ion density, O⁻ ion density, and O⁺ ion density. The average MAE was exceedingly low (approximately 1 %), even considering error accumulation from Step 1 and Step 2.

Figure 2. (a) O₂ = 0.4, p0 = 35 mTorr, and Pw = 150 W Step 2 comparison. (b) Original data and predicted data comparison in the middle of the chamber.

Figure 2(b) compares the original and DNN-predicted density data at the center of the chamber. The solid lines and diamond symbols represent the original data and predicted values, respectively. Consequently, the two lines were approximately identical. The selected optimized DNN model predicted the interpolation effectively.

3.2. Extrapolation

We selected the following variable values to verify the extrapolation performance: O₂ = 0.2, p0 = 60 mTorr, and Pw = 200 W. The extrapolation predicted data that fell outside the existing data range, demonstrating the process of using prepared data with O₂ = 0.3–0.7, p0 = 10–50 mTorr, and Pw = 100–180 W for data prediction with O₂ = 0.2, p0 = 60 mTorr, and Pw = 200 W. All three variables extended beyond the range of the prepared data set. We made a step-by-step prediction with the O₂ ratio, followed by pressure and power. Step 1 involved predicting 25 datasets corresponding to O₂ = 0.2, p0 = 10–50 mTorr, and Pw = 100–180 W. The predicted data for O₂ = 0.2, p0 = 20 mTorr, and Pw = 120 W were compared with the original data to determine the performance of the trained model. Figure 3(a) compares the data predicted by the DNN model in Step 1 and the ground truth of COMSOL. The average MAE was determined to be 1.65 %.

Figure 3. (a) O₂ = 0.2, p0 = 20 mTorr, and Pw = 120 W Step 1 comparison. (b) Comparing the average of original data and predicted data.

Figure 3(b) compares the predicted values of O₂ = 0.2, p0 = 10–50 mTorr, and Pw = 120 W with the chamber-wide average values obtained from the original data. The accuracy of the overall data predicted in Step 1 was compared with original data. The following four variables were compared: electron, O²⁺ density, electronic potential, and electric temperature. The solid lines and symbols (diamond) represent the original and DNN-predicted data, respectively.

Step 2 used the data predicted in Step 1 for O₂ = 0.2, p0 = 10– 50 mTorr, and Pw = 100–180 W were used to predict data for O₂ = 0.2, p0 = 60 mTorr, and Pw = 100−180 W. Figure 4(a) compares the chamber plot and MAE of the original and predicted data. Among the data for O₂ = 0.2, p0 = 60 mTorr, Pw = 100–180 W, the specific case of O₂ = 0.2, p0 = 60 mTorr, Pw = 120 W was validated. The image comparison revealed that the original and predicted data were approximately identical. The average MAE was approximately 4 %.

Figure 4. (a) O₂ = 0.2, p0 = 60 mTorr, and Pw = 120 W Step 2 comparison. (b) Comparing the average of original data and predicted data.

Figure 4(b) compares the average values of the original and predicted data for O₂ = 0.2, p0 = 60 mTorr, and Pw = 100−180 W. Four variables were compared: electrons, O²⁺ density, electronic potential, and electric temperature. The solid lines and symbols denote the original and predicted data, respectively. The overall correspondence between the two datasets was striking despite some errors. The error in Step 3 was small because the accumulated error in Step 2 was adjusted.

Step 3 involved predicting for O₂ = 0.2, p0 = 60 mTorr, and Pw = 100–180 W based on the prediction made in Step 2. The O₂ and pressure were kept constant. Power data ranged from 100 to 180 W. Therefore, the predicted value could be 200 W. Figure 5(a) compares the ground truth and predicted data. The measured average MAE in Step 3 was 5.26 %.

Figure 5. (a) O₂ = 0.2, p0 = 60 mTorr, and Pw = 200 W Step 1 comparison. (b) Comparing original data and predicted data in the middle of the chamber.

Figure 5(b) compares the densities of electrons, Ar⁺, and O²⁺ at the center of the chamber. The error between the original (solid line) and predicted data (symbol) was negligible. The MAE was larger than Step 2 owing to the accumulation of errors. However, the MAE of 5.26 % was significant. The MAE was lower than 10 % for all variables.

We predicted O₂ = 0.8, p0 = 60 mTorr, and Pw = 200 W to accurately verify the extrapolation performance of the optimized DNN model. Our objective was to demonstrate the feasibility of bidirectional extrapolation by predicting O₂ = 0.2 and O₂ = 0.8 data, leveraging the prepared data spanning O₂ = 0.3–0.7. We predicted the seven variables in three steps, following the order of O₂, pressure, and power. Notably, Step 1 and Step 2 are identical. Therefore, we omitted both and introduced only Step 3.

Figure 6(a) compares the ground truth and predicted data images by measuring the MAE values. The average MAE observed in Step 3, with O₂ = 0.8, p0 = 60 mTorr, and Pw = 200 W, was 5.61 %. Notably, an MAE of 8.44 % or less was predicted for all variables. Figure 6(b) compares the average values of the data at the center of the chamber. The original and predicted data were approximately identical. The selected optimized DNN model proved the possibility of two-way extrapolation.

Figure 6. (a) O₂ = 0.8, p0 = 60 mTorr, and Pw = 200 W Step 1 comparison. (b) Comparing original data and predicted data in the middle of the chamber.

This study proposes an efficient model for determining the optimal process conditions by combining a DL model with plasma simulation data. We eliminated complex equations and multiple models using a DL model based on data pattern learning to understand the relationship between the given process conditions and plasma states.

The DL model demonstrates fast inference capability with respect to the ground truth and provides a reliable correlation between the process conditions and plasma, verified by the experimentally predicted plasma uniformity. A step-by-step algorithm and optimization iterations can be applied to determine the optimal processing conditions for plasma uniformity owing to the fast inference capability of the DL model. This approach promises substantial reductions in the iterative trial and error experimentation typically associated with exploring optimal process conditions. Furthermore, the applicability of the model can be broadened by controlling the process variables and dataset of the plasma simulation, indicating that the model can be applied to different process conditions in ICP.

The combined model can be a powerful tool for narrowing process windows and efficiently controlling complex plasma processing by considering different process and hardware conditions.

This work was supported by the National Research Council of Science & Technology (NST) grant from the Korean Government (MNIST) (No. CRC-20-01-NFRI). We would like to thank Editage (www.editage.co.kr) for English language editing.

The authors declare no conflicts of interest.