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Demonstrate The Performance Of Tensor-based Process Control Using An Example Semiconductor Manufacturing Process With Unstable Disturbances

Demonstrate The Performance Of Tensor-based Process Control Using An Example Semiconductor Manufacturing Process With Unstable Disturbances

Process Control

3 main points

✔️ The control of complex data is discussed using the motif of positional superposition control in semiconductor manufacturing processes, especially in lithography processes.
✔️ The curse of dimensionality is mitigated by building a high-dimensional process model and estimating the parameters of the model using a tensor-on-vector regression algorithm.
Based on tensor parameter estimation, design an EWMA controller for tensor data with theoretically guaranteed stability, monitor control residuals, and prevent significant drift due to uncontrollable high-dimensional disturbances.

Tensor-based process control and monitoring for semiconductor manufacturing with unstable disturbances
written by Yanrong LiJuan DuFugee TsungWei Jiang
[Submitted on 31 Jan 2024]
Comments: accepted by arXiv
  Machine Learning (stat.ML); Machine Learning (cs.LG); Image and Video Processing (eess.IV); Systems and Control (eess.SY)


The images used in this article are from the paper, the introductory slides, or were created based on them.


To cope with the complex data collected from sensors installed in manufacturing systems, we propose a new process control and monitoring method to reduce high-dimensional, image-based overlay errors in semiconductor manufacturing processes. The method mitigates the curse of dimensionality by building a high-dimensional process model and estimating the parameters of the model using a tensor-on-vector regression algorithm. Furthermore, based on the tensor parameter estimation, an EWMA controller is designed for the tensor data with theoretically guaranteed stability. Considering that the control recipe adjustment cannot compensate for all high-dimensional disturbances, the control residuals are monitored to prevent significant drift due to uncontrollable high-dimensional disturbances. Through extensive simulations and real-world case studies, we evaluate the performance of the EWMA controller in the parameter estimation algorithm and tensor space and verify the superiority of the proposed method over existing image-based feedback controllers.


The importance of overlay error detection and correction will be explored in depth, focusing on the semiconductor manufacturing process, particularly the lithography process. Lithography aims to minimize misalignment that occurs between adjacent material layers, which directly affects the performance of semiconductor devices. Reducing overlay errors is critical to ensuring high quality products.

To accurately identify overlay errors, error measurements at specific locations on the wafer are represented as images. This process is illustrated in Figure 1, where the error at each measurement point is represented by an arrow, the length and direction of which indicate the magnitude and direction of the error. This image-based representation provides a better visual understanding of the error and allows for precise correction.

Figure 1: Illustration of overlay errors on wafers

Control recipes to compensate for overlay errors include wafer position and orientation, lens height, and more. Adjusting these parameters compensates for errors and improves the accuracy of the manufacturing process. However, the high dimensionality and complexity of overlay errors make full compensation by these control recipes difficult. To address this challenge, the introduction of tensor-based process control and monitoring methods is proposed.

The core of the proposed approach is to efficiently handle high-dimensional image data collected from manufacturing processes and build tensor-based models for process control and monitoring. The framework of the approach, illustrated in Figure 2, includes estimation of high-dimensional parameters, design of an EWMA controller for tensor data, and a monitoring strategy based on control residuals. This improves process stability and enhances the tolerance of the manufacturing process to disturbances.

Figure 2. Tensor-based process control and monitoring

It details how the proposed tensor-based method is superior to traditional control methods and how it works as a solution to the challenges faced in working with high-dimensional data. Tensor-based methods enable more effective process control and monitoring, taking into account the multidimensionality and complexity of process data. This is expected to increase efficiency, improve quality, and reduce costs of semiconductor manufacturing processes. A detailed introduction is provided to provide an in-depth understanding of thecurrent challenges in the semiconductor manufacturing industry and the potential impact of the proposedtensor-based methodology.

Related Research

Existing process control and monitoring methods

Run-to-Run ( R2R) Control: In many complex manufacturing systems, including semiconductor manufacturing, R2R control is widely used to stabilize quality across a series of production tasks. Methods range from single input-single output (SISO) systems to multiple input-multiple output (MIMO) systems, but the focus here is primarily on univariate or multivariate data.

Process Monitoring: Various process monitoring methods have been developed for semiconductor manufacturing processes, including PCA (Principal Component Analysis)-based methods and on-line monitoring methods based on local cumulative sum statistics. While these methods can handle high dimensionality and non-Gaussianity of data, they have limitations in effectively handling more complex data structures such as image data.

Application of Tensor Analysis

Tensor-based regression: Tensor analysis is being applied to regression models to handle complex structures and high-dimensional data. This has led to the proposal of new tensor regression methods, for example, variable-on-tensor regression models in brain imaging data analysis and multiple tensor-on-tensor regression approaches for multiple input tensors of different order.

Tensor-based monitoring: Tensor-based monitoring methods are applied for anomaly detection and fault diagnosis. Image-based process monitoring methods, such as those based on low-rank tensor decomposition, are used to detect changes in manufacturing processes.


This study proposes a methodology for effectively controlling and monitoring high-dimensional image-based overlay errors in semiconductor manufacturing processes, particularly lithography processes, using a tensor-based approach. The methodology introduces a new framework for handling high-dimensional process data and provides specific computational procedures and implementation details.

Problem description and formulation

A major challenge in the lithography process is minimizing quality variation, which manifests itself in the form of overlay error. These errors are caused by subtle misalignments that occur between multiple layers on a wafer. The authors employed a multi-dimensional tensor data structure to represent these overlay errors. The representation of the model in tensor form is defined in Equation (1), which allows the misalignment of each layer to be captured as a multidimensional array.

B is a tensor parameter.

Parameter Estimation

The tensor-on-vector regression algorithm is used to estimate the parameters B of the process model. This estimation is an extension of the basic least squares method to tensor space and is shown in Equation (2) assuming that B exists in a lower dimension to eliminate the curse of dimensionality. In addition, an estimation method using sparse learning is introduced to improve the accuracy and computational efficiency of parameter estimation. Algorithm 1 shows the detailed procedure of this estimation method and clearly explains the computational process at each step.

Figure 4 shows a comparison of different algorithms.

Figure 4. PEE of tensor parameter estimates by different algorithms

Online control scheme

The proposed control scheme applies an exponentially weighted moving average (EWMA) controller to tensor data. The stability of the method is theoretically guaranteed, and the proof is developed based on equation (10). An example of the application of the control scheme to actual process data is visually illustrated in Figure 1, which shows the improvement in the overlay error before and after the control.

Residual monitoring

A monitoring technique using tensor-based control charts has also been developed. This method evaluates in real time whether the process is within the planned control range by analyzing control residuals. The specific monitoring procedure is described in Algorithm 2 and includes methods for setting various control limits. Table 1 provides a summary of MAE values for varying correlation values between disturbance and control recipe. Stability conditions for the EWMA controller in tensor space were identified when controlling image-based quality variables.

Table 1: Summary of MAE with different parameter estimators

Case study

The proposed tensor-based process control and monitoring method is applied to actual semiconductor manufacturing processes and its effectiveness is analyzed in detail through specific case studies.

Comparison of different process controllers

The proposed EWMA controller is compared to a conventional image-based feedback controller. Specifically, simulation experiments were conducted using the same data set to compare the control performance of both controllers. In Tables 2 and 3, the statistical performance measures (mean error, standard deviation, etc.) for each controller are presented, confirming the overall superior performance of the proposed EWMA controller.

Table 2. performance based on IMA disturbances

Table 3. performance based on ARIMA disturbances

Performance analysis of monitoring methods

To evaluate the effectiveness of the proposed tensor-based monitoring approach, the impact of different monitoring settings is analyzed for four cases using data collected from actual manufacturing lines. The analysis specifically evaluated the ability to detect anomalies in the manufacturing process at an early stage.

Case A: Average shift

CaseB: Distributed shift

Case C: Gradual drift by IMA process

Case D: Gradual drift by ARIMA process

Figure 5 (a). Graph of 𝑇2 and 𝑄 chart for Case A

Figure 5 (b). Graph of 𝑇2 and 𝑄 chart for Case B

Figure 5 (c). EWMA chart for Case C

Figure 5 (d). EWMA chart for case D

Through this case study, it is shown how effectively the proposed tensor-based process control and monitoring method works in a real manufacturing environment. A detailed analysis based on actual data further supports the validity and practicality of the methodology and provides valuable insights into future improvements and potential applications .


With the proliferation of sensor technology, quality measurements are no longer limited to univariate or multivariate variables, but now include image-based quality variables. It is not limited to univariate or multivariate variables and includes image-based quality variables. This study focused on the design of a tensor-based process control and monitoring scheme for lithographic processes with image-based quality variables. This study focuses on the design of tensor-based process control and monitoring schemes for lithographic processes with image-based quality variables. Limited Number of Input Control Variables Due to the limited number of input control variables, we classify disturbances into two types. These two types of disturbances increase the complexity of high-dimensional tensor parameter estimation and make identifiable parameter estimation and process controller design very complex. To improve the accuracy of parameter estimators for process control, existing least-squares estimation algorithms in tensor space are improved by sparse learning to improve their limits, especially when the control recipe and the first type of disturbance in the offline data set are highly correlated. The EWMA control scheme is then extended to tensor space to compensate for the first type of disturbance and introduce different monitoring methods for the second type of disturbance. Importantly, the properties of the parameter estimation algorithm and the stability of the EWMA controller are guaranteed. Large-scale simulations and case studies for validation are proposed to illustrate the efficiency of the proposed parameter estimation, process control, and monitoring methods. By comparison with existing image-based process controls, we found that the EWMA control is superior in predicting and compensating for autocorrelated unstable disturbances. Finally, T2, Q, and EWMA charts were also employed to efficiently monitor the variability of a second, different type of disturbance arising from the manufacturing environment, establishing a comprehensive process control and monitoring methodology.

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友安 昌幸 (Masayuki Tomoyasu) avatar
JDLA G certificate 2020#2, E certificate2021#1 Japan Society of Data Scientists, DS Certificate Japan Society for Innovation Fusion, DX Certification Expert Amiko Consulting LLC, CEO

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