[Paper Review] Stability analysis of semiconductor manufacturing process with EWMA run-to-run controllers

Bing Ai, David Shan‐Hill Wong+1 more|arXiv (Cornell University)|Oct 30, 2015

Advanced Control Systems Optimization26 references3 citations

TL;DR

This paper proposes EWMA-I and EWMA-II controllers for semiconductor manufacturing run-to-run processes to address stability under metrology delays. Using Routh-Hurwitz and Lyapunov's direct method, it establishes necessary and sufficient conditions for stochastic stability in both single- and mixed-product processes, providing analytical stability regions under fixed and stochastic delays.

ABSTRACT

In the semiconductor manufacturing batch processes, each step is a complicated physiochemical batch process; generally it is difficult to perform measurements online or carry out the measurement for each run, and hence there will be delays in the feedback of the system. The effect of the delay on the stability of the system is an important issue which needs to be understood. Based on the exponentially weighted moving average (EWMA) algorithm, we propose two kinds of controllers, EWMA-I and II controllers for single product process and mixed product process in semiconductor manufacturing in this paper. For the single product process, the stabilities of systems with both controllers which undergo different kinds of metrology delays are investigated. Necessary and sufficient conditions for the stochastic stability are established. Routh-Hurwitz criterion and Lyapunov's direct method are used to obtain the stability regions for the system with fixed metrology delay. By using Lyapunov's direct method, the stability region is established for the system with fixed sampling metrology and with stochastic metrology delay. We also extended the theorems of single product process to mixed product process. Based on the proposed theorems, some numerical examples are provided to illustrate the stability of the delay system.

Motivation & Objective

• To address the challenge of system instability caused by delayed feedback in semiconductor batch processes due to infrequent online measurements.
• To develop EWMA-based run-to-run controllers (EWMA-I and EWMA-II) tailored for single and mixed product semiconductor manufacturing environments.
• To establish rigorous mathematical conditions for stochastic stability under various delay scenarios, including fixed and stochastic metrology delays.
• To extend stability analysis results from single-product to mixed-product processes, enhancing practical applicability.
• To provide numerical examples demonstrating the validity and effectiveness of the derived stability regions.

Proposed method

• Proposes two EWMA-based run-to-run controllers—EWMA-I and EWMA-II—specifically designed for single-product and mixed-product semiconductor processes.
• Applies the Routh-Hurwitz criterion to derive stability regions for systems with fixed metrology delays in single-product processes.
• Employs Lyapunov’s direct method to establish stability regions under both fixed sampling and stochastic metrology delays.
• Extends the theoretical stability conditions from single-product to mixed-product processes by adapting the controller structure and analysis framework.
• Uses mathematical modeling of the process dynamics and delay effects to derive sufficient and necessary conditions for stochastic stability.
• Validates the theoretical findings through numerical examples illustrating the stability boundaries under different delay distributions.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for stochastic stability in a single-product semiconductor manufacturing process with EWMA run-to-run control under fixed metrology delay?
• RQ2 How do fixed and stochastic metrology delays affect the stability of EWMA-based run-to-run control systems in semiconductor processes?
• RQ3 What is the stability region for systems with stochastic metrology delay, and how can it be analytically determined using Lyapunov’s direct method?
• RQ4 How can the stability analysis for single-product processes be extended to mixed-product semiconductor manufacturing environments?
• RQ5 What are the practical implications of the derived stability regions in real-world semiconductor manufacturing with measurement delays?

Key findings

• Necessary and sufficient conditions for stochastic stability are established for single-product processes under fixed metrology delay using the Routh-Hurwitz criterion.
• Lyapunov’s direct method successfully derives the stability region for systems with both fixed sampling and stochastic metrology delays in single-product processes.
• The stability analysis framework is successfully extended from single-product to mixed-product processes, preserving analytical rigor.
• Numerical examples confirm the validity of the derived stability regions, demonstrating the controllers' ability to maintain system stability under varying delay conditions.
• The proposed EWMA-I and EWMA-II controllers ensure stable process operation even with delayed metrology feedback, which is critical in semiconductor manufacturing.
• The study provides a theoretical foundation for selecting controller parameters that guarantee stability in the presence of measurement delays.