Uncertainty quantification and parameter optimization of plasma etching process using heteroscedastic Gaussian process

TL;DR

The paper addresses uncertainty quantification and robust design for plasma etching in semiconductor manufacturing by combining a heteroscedastic Gaussian process (hetGP) surrogate with reliability-based design optimization (RBDO). It introduces a two-stage hetGP to separately model mean thickness and input-dependent noise, enabling explicit decomposition of epistemic and aleatoric uncertainties, and couples this with an augmented probability of failure $ar{P}_{f}$ within a double-loop Monte Carlo optimization to produce designs that minimize thickness variability while meeting reliability targets. The main contributions include (i) a practical hetGP framework validated on experimental etching data across nine wafer locations, (ii) a rigorous uncertainty decomposition and reliability analysis, and (iii) a robust design procedure (RBRDO-AE) that accounts for surrogate-model uncertainty, yielding more reliable process recipes. The findings demonstrate improved predictive accuracy and process reliability, with a demonstrated trade-off between robustness and variance, and the approach is positioned as generalizable to other semiconductor processes such as photolithography.

Abstract

This study presents a comprehensive framework for uncertainty quantification (UQ) and design optimization of plasma etching in semiconductor manufacturing. The framework is demonstrated using experimental measurements of etched depth collected at nine wafer locations under various plasma conditions. A heteroscedastic Gaussian process (hetGP) surrogate model is employed to capture the complex uncertainty structure in the data, enabling distinct quantification of (a) spatial variability across the wafer and (b) process-related uncertainty arising from variations in chamber pressure, gas flow rate, and RF power. Epistemic uncertainty due to sparse data is further quantified and incorporated into a reliability-based design optimization (RBDO) scheme. The proposed method identifies optimal process parameters that minimize spatial variability of etch depth while maintaining reliability under both aleatory and epistemic uncertainties. The results demonstrate that this framework effectively integrates data-driven surrogate modeling with robust optimization, enhancing predictive accuracy and process reliability. Moreover, the proposed approach is generalizable to other semiconductor processes, such as photolithography, where performance is highly sensitive to multifaceted uncertainties.

Figures (12)

• Figure 1: Illustration of the hetGP-based RBDO framework.
• Figure 2: A schematic of plasma etching process
• Figure 3: Experimental setup of the plasma-assisted etching process and the fabricated samples.
• Figure 4: Visualization of the training data used to construct the surrogate model. (a) Sample points in the three-dimensional process parameter space. (b) Measured thickness showing the mean and standard deviation at each sample point.
• Figure 5: Relationship between three input parameters and spatial variability in wafer