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Information-Theoretic Causal Modelling of Semiconductor Process Dynamics
Daniel Sørensen, Giorgio Melchiorre, Sudip Bandyopadhyay, Sandip Halder, Roel Wuyts, Bappaditya Dey
TL;DR
Semiconductor process control needs models that capture dynamic, nonlinear relationships beyond current FDC approaches. This paper infers causal dependencies from raw equipment time series using information-theoretic stochastic dynamical modelling, with approximately 43% of the strongest inferred relationships judged known or plausible by a domain expert.
Problem
Current semiconductor FDC approaches rely on thresholds, rules, and manual investigation while offering limited dynamic modelling of time-dependent and nonlinear relationships.
Method
The framework estimates stochastic process dynamics directly from multivariate equipment time series and quantifies causal dependencies through entropy-transfer rates.
Results
Approximately 43% of the strongest inferred relationships were identified as known or plausible by a domain expert, alongside previously unrecognised interactions.
Takeaways & Limitations
The framework provides a systematic, interpretable basis for modelling semiconductor manufacturing dependencies and exploring previously unrecognised interactions.
Takeaways & Limitations
The reported proportions are feasibility indications rather than validated full-graph accuracy measures because they assess only strongly weighted edges using one expert and vary across tools.
Abstract
from arXiv · showhide
With the progress of the semiconductor industry toward increasingly complex compute devices and tighter process tolerances, advanced process control has become crucial. This work explores a novel framework to infer the underlying dynamics of semiconductor processes, directly from raw equipment log-file time-series data. By modelling the tool dynamics as a stochastic dynamical system comprising (a) a deterministic component and (b) a stochastic component, we estimate entropy transfer rates between variables through the Liang-Kleeman and Pires formalism. Preliminary results indicated that 7.5% of the inferred dependencies were known, 36.0% were plausible, 17.5% represented previously uncharacterised relationships, and 39.0% were inconsistent with established process knowledge. These findings demonstrate the framework's capability to uncover novel causal insights, while motivating further improvements to reduce inconsistent findings.
I. INTRODUCTION
Semiconductor manufacturing increasingly requires advanced process control as process flows grow more complex and tolerances tighten. This work proposes an information-theoretic framework that models manufacturing processes as stochastic dynamical systems directly from multivariate time-series data.
- Motivation: Increasingly complex process flows and tighter tolerances make advanced process control crucial for high-volume semiconductor production.Fault Detection and Classification systems use sensor arrays to produce multivariate time-series data for analysis.
- Approach: The proposed framework estimates stochastic dynamical-system governing equations directly from observed multivariate time-series data.It is designed to model semiconductor manufacturing processes using an information-theoretic approach.
- Approach: The model separates process dynamics into deterministic and stochastic components.The deterministic component uses wavelet basis functions, while the stochastic component uses Gaussian basis functions.
- Paper structure: The paper presents its formalism, results, discussion and limitations, and conclusions with future work across Sections II–V.Section II covers the formalism; Sections III and IV cover results and discussion; Section V concludes with future work.
II. METHODS · A. Information Theory foundation
The framework models a D-dimensional stochastic dynamical system using deterministic mechanics and Wiener-process noise, then quantifies information transfer through entropy rates. Liang–Kleeman and Pires formalism separates entropy change into transfer, self-generation, direct variable contributions, and synergetic effects.
- A. Information Theory foundation: A D-dimensional state vector X evolves according to a stochastic differential equation with deterministic vector-field components and noise-diffusion coefficients.The noise is represented by independent Gaussian Wiener processes, with diffusion coefficients collected in matrix B and diffusivities defined by G ≡ BBT.
- A. Information Theory foundation: Shannon entropy H_Xi is defined as the expected negative logarithm of variable Xi’s probability density.The entropy rate is based on the probability density function ρi and the expectation operator E[·].
- A. Information Theory foundation: A positive entropy-change rate indicates information loss, whereas a negative rate indicates information gain.This interpretation applies to the rate of entropy change associated with each variable.
- A. Information Theory foundation: The rate of entropy change decomposes into global entropy transfer from complementary variables and self-entropy generation.The complementary vector is X~i = X/{Xi}.
- A. Information Theory foundation: Pires et al. decompose global entropy transfer into direct contributions from individual variables and a synergetic term caused by non-null conditional covariances.The synergetic contribution arises from nonlinearities in Fi and gi,j.
- A. Information Theory foundation: Under product-separable Fi and gi,i, entropy-transfer components split into deterministic and stochastic terms.This decomposition requires the stated product-separability condition.
- A. Information Theory foundation: The resulting transfer components identify Xj as causal and Xi as consequential, measure information flow in nats per unit time, and are non-bidirectional.The contextual variables are components independent of both Xi and Xj.
B. RET numerical computation
RET terms are computed from raw multivariate tool-log time series by estimating deterministic and stochastic components with regression and basis-function decompositions. Numerical complexity is controlled through wavelet selection and interaction restrictions, while surrogate testing assesses statistical significance.
- B. RET numerical computation: The unknown driving equations are estimated from multivariate tool-log time series by regressing numerical gradients for F_i and squared residuals for ĝ_i,i.This provides data-driven estimates of the deterministic and stochastic components needed for RET computation.
- B. RET numerical computation: 30 most relevant wavelet candidates are selected using Stepwise Selection by Orthogonalisation after constructing a wavelet frame for smooth and non-smooth derivatives.Ricker wavelets provide flexible fitting for non-smooth semiconductor-processing signals.
- B. RET numerical computation: deg = 4 and m ∈{3, 4, 5, 6} restrict wavelet products and scale indices to address combinatorial growth and high-dimensional estimation failure.The product terms grow with (MN)^D and tend toward 0 for high-dimensional systems with D > 6.
- B. RET numerical computation: 100 surrogate time-series are generated per signal using IAAFT, and RET is deemed statistically significant when p-value < α with α = 0.05.Surrogate RETs are recomputed using surrogate realisations of X_j.
C. Data description
The methodology was evaluated using anonymized log-file data from three process recipes run on two matched FAB tools, each with two simultaneously operating chamber sides. Although the tools were intended to be matched, engineers reported minor discrepancies in subsequent wafer properties.
- Dataset: Data comprised three process recipes run on two matched FAB tools, each with two simultaneously operating chamber sides.The experiments used process log-file data acquired from the equipment.
- Dataset: The Tool1/Side1/RecipeA dataset included Var 10, a low-noise nonsmooth signal, and Var 22, a smooth but noisy signal.These signals were used to illustrate estimation of F_i with wavelet neural-network regression of variable derivatives.
- Dataset: Confidentiality constraints anonymized tool models, tool types, recipes, and variable names as Tool1/Tool2, RecipeA/RecipeB/RecipeC, and Var_i.Here, i denotes an integer variable index.
III. RESULTS · A. Wavelet Regression
Wavelet regression used a 90/10 train/test split and showed stable generalisation across ten datasets. Representative results captured both non-smooth derivative peaks and noisy-signal behaviour without over-fitting.
- A. Wavelet Regression: 90/10 train/test split was used for the wavelet regression, following common practice in related literature.Regression quality was assessed using the median mean squared error across variables after computing each variable’s MSE over all instances.
- A. Wavelet Regression: Ten datasets exhibited comparable train and test errors, indicating stable generalisation.The comparison used the median of mean squared errors for each dataset.
- A. Wavelet Regression: Representative Tool1/Side1/RecipeA results captured peak-shaped derivatives in Var 10 during non-smooth transitions without over-fitting.The same results also reproduced the behaviour of the noisy signal in Var 22.
B. RET-based Causal Graphs
RET values were used to construct directed causal graphs from each dataset, yielding 2,742 inferred edges overall. Graph agreement across repeated recipe executions was limited, with most edges appearing in only one graph.
- Graph construction: Each dataset’s summed deterministic and stochastic RET values formed a square adjacency matrix for a weighted directed causal graph.Causal graph generation was fully unsupervised and dependencies were validated through a blind evaluation protocol.
- Graph construction: 230 edges per graph on average produced 2,742 edges across all datasets.These totals describe the inferred graph structures before the blind dependency evaluation.
- Cross-graph agreement: At most 10% of edges appeared in at least three graphs for the same recipe, whereas at least 64% appeared in only one graph.The analysis compared graphs generated for the same recipe across tools and sides, which were expected to share core dependencies.
IV. DISCUSSION AND LIMITATIONS
The framework’s performance is attributed to wavelet-based regression fitting both non-smooth peak-shaped derivatives and smooth noise-afflicted signals. However, SSO selection from a very large wavelet candidate set introduces variability, computational cost, and uncertainty that limit practical and interpretive confidence.
- Wavelet-based regression accurately fits both peak-shaped derivatives from non-smooth signals and smooth, noise-afflicted signals.
- Selecting S = 30 wavelets from > 20 million candidates causes variability in regression quality and disagreement between inferred graphs.The selection procedure, SSO, was identified as responsible for both effects.
- SSO becomes computationally expensive despite being presented as efficient, making further increases in S prohibitive for practical FAB application.
- Future research should develop derivative-fitting methods with wavelet-like expressive power that are less sensitive to minor dataset variations.
- The reported proportions reflect only the most strongly weighted edges assessed by one expert and should be treated as feasibility indications rather than validated full-graph accuracy.The high Not-Possible fraction co-occurs with cross-tool graph variability traced to SSO sensitivity.
V. CONCLUSION
The study uses an information-theoretic framework with wavelet basis functions to infer causal dependencies in semiconductor manufacturing processes, detecting known and plausible relationships while identifying unrecognised interactions.
- The framework models semiconductor manufacturing processes from an information theory perspective to infer causal dependencies between process parameters.
- 43% of the largest inferred relationships were validated by a domain expert as known or plausible dependencies.
- Wavelet basis functions enable the framework to identify previously unrecognised interactions for further exploration.