Regression Analysis

Semiconductor fabrication involves hundreds of sequential process steps, each governed by dozens of parameters. Regression analysis serves critical functions:

• Process Modeling: Understanding relationships between inputs and quality outputs
• Virtual Metrology: Predicting measurements from real-time sensor data
• Run-to-Run Control: Adaptive process adjustment
• Yield Optimization: Maximizing device performance and throughput
• Fault Detection: Identifying and diagnosing process excursions

Core Mathematical Framework

Ordinary Least Squares (OLS)

The foundational linear regression model:

Variable Definitions:

• $\mathbf{y}$ — $n \times 1$ response vector (e.g., film thickness, etch rate, yield)
• $\mathbf{X}$ — $n \times (k+1)$ design matrix of process parameters
• $\boldsymbol{\beta}$ — $(k+1) \times 1$ coefficient vector
• $\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \sigma^2\mathbf{I})$ — error term

OLS Estimator:

Variance-Covariance Matrix of Estimator:

Unbiased Variance Estimate:

Response Surface Methodology (RSM)

Critical for semiconductor process optimization, RSM uses second-order polynomial models.

Second-Order Model

Matrix Form:

Where:

• $\mathbf{b}$ — vector of first-order coefficients
• $\mathbf{B}$ — symmetric matrix of second-order coefficients

Stationary Point Analysis

Stationary Point (Optimum):

Nature Determination:

• All eigenvalues of $\mathbf{B}$ negative → Maximum
• All eigenvalues of $\mathbf{B}$ positive → Minimum
• Mixed signs → Saddle point

Canonical Analysis

Where $\lambda_i$ are eigenvalues and $w_i$ are canonical variables.

Regularized Regression Methods

Semiconductor data often exhibits multicollinearity and high dimensionality.

Ridge Regression (L2 Penalty)

Objective Function:

Closed-Form Solution:

Properties:

• Shrinks coefficients toward zero
• Does not perform variable selection
• Handles multicollinearity effectively

LASSO (L1 Penalty)

Objective Function:

Properties:

• Performs automatic variable selection
• Sets some coefficients exactly to zero
• Crucial for identifying which process parameters matter

Elastic Net

Objective Function:

Alternative Parameterization:

Where $\alpha \in [0,1]$ controls the mix between L1 and L2 penalties.

Partial Least Squares (PLS) Regression

The most important technique for semiconductor process modeling.

Why PLS?

• Handles high dimensionality ($p > n$)
• Addresses multicollinearity
• Captures latent variable structures
• Simultaneously models X and Y relationships

NIPALS Algorithm

1. Initialize: $\mathbf{u} = \mathbf{y}$

2. X-weight: $$\mathbf{w} = \frac{\mathbf{X}^\top\mathbf{u}}{\|\mathbf{X}^\top\mathbf{u}\|}$$

3. X-score: $$\mathbf{t} = \mathbf{X}\mathbf{w}$$

4. Y-loading: $$q = \frac{\mathbf{y}^\top\mathbf{t}}{\mathbf{t}^\top\mathbf{t}}$$

5. Y-score update: $$\mathbf{u} = \frac{\mathbf{y}q}{q^2}$$

6. Iterate until convergence

7. Deflate X and Y, extract next component

Model Structure

Where:

• $\mathbf{T}$ — score matrix (latent variables)
• $\mathbf{P}$ — X-loadings
• $\mathbf{Q}$ — Y-loadings
• $\mathbf{E}, \mathbf{F}$ — residuals

Spatial Regression for Wafer Maps

Wafer-level variation exhibits spatial patterns requiring specialized models.

Zernike Polynomial Decomposition

General Form:

Standard Zernike Polynomials (first few terms):

Orthogonality Property:

Gaussian Process Regression (Kriging)

Prior Distribution:

Common Kernel Functions:

Squared Exponential (RBF):

Matérn Kernel:

Where $K_ u$ is the modified Bessel function of the second kind.

Posterior Predictive Mean:

Posterior Predictive Variance:

Mixed Effects Models

Semiconductor data has hierarchical structure (wafers within lots, lots within tools).

General Model

Random Effects Distribution:

• $b_i^{(\text{tool})} \sim N(0, \sigma_{\text{tool}}^2)$
• $b_{ij}^{(\text{lot})} \sim N(0, \sigma_{\text{lot}}^2)$
• $\varepsilon_{ijk} \sim N(0, \sigma^2)$

Matrix Notation

Where:

• $\mathbf{b} \sim N(\mathbf{0}, \mathbf{G})$
• $\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \mathbf{R})$
• $\text{Var}(\mathbf{y}) = \mathbf{V} = \mathbf{Z}\mathbf{G}\mathbf{Z}^\top + \mathbf{R}$

REML Estimation

Restricted Log-Likelihood:

Where $\mathbf{r} = \mathbf{y} - \mathbf{X}\hat{\boldsymbol{\beta}}$.

Physics-Informed Regression Models

Arrhenius-Based Models (Thermal Processes)

Rate Equation:

Linearized Form (for regression):

Parameters:

• $k$ — rate constant
• $A$ — pre-exponential factor
• $E_a$ — activation energy (J/mol)
• $R$ — gas constant (8.314 J/mol·K)
• $T$ — absolute temperature (K)

Preston's Equation (CMP)

Basic Form:

Extended Model:

Where:

• MRR — material removal rate
• $K_p$ — Preston coefficient
• $P$ — applied pressure
• $V$ — relative velocity

Lithography Focus-Exposure Model

Variables:

• CD — critical dimension
• $E$ — exposure dose
• $F$ — focus offset

Bossung Curve: Plot of CD vs. focus at various exposure levels.

Virtual Metrology Mathematics

Predicting quality measurements from equipment sensor data in real-time.

Model Structure

Where $\mathbf{x}_{\text{FDC}}$ is Fault Detection and Classification sensor data.

EWMA Run-to-Run Control

Exponentially Weighted Moving Average:

Properties:

• $\lambda \in (0,1]$ — smoothing parameter
• Smaller $\lambda$ → more smoothing
• Larger $\lambda$ → faster response to changes

Kalman Filter Approach

State Equation:

Measurement Equation:

Update Equations:

Predict: $$ \hat{\mathbf{x}}_{k|k-1} = \mathbf{A}\hat{\mathbf{x}}_{k-1|k-1} $$

Update: $$ \mathbf{K}_k = \mathbf{P}_{k|k-1}\mathbf{H}^\top(\mathbf{H}\mathbf{P}_{k|k-1}\mathbf{H}^\top + R)^{-1} $$

Classification and Count Models

Logistic Regression (Binary Outcomes)

For pass/fail or defect/no-defect classification:

Model:

Logit Link:

Log-Likelihood:

Newton-Raphson Update:

Where $\mathbf{W} = \text{diag}(\pi_i(1-\pi_i))$.

Poisson Regression (Defect Counts)

Model:

Probability Mass Function:

Model Validation and Diagnostics

Goodness of Fit Metrics

Coefficient of Determination:

Adjusted R-Squared:

Root Mean Square Error:

Mean Absolute Error:

Cross-Validation

K-Fold CV Error:

Leave-One-Out CV:

Information Criteria

Akaike Information Criterion:

Bayesian Information Criterion:

Diagnostic Statistics

Variance Inflation Factor:

Where $R_j^2$ is the $R^2$ from regressing $x_j$ on all other predictors.

Rule of thumb: VIF > 10 indicates problematic multicollinearity.

Cook's Distance:

Leverage:

Where $\mathbf{H} = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top$ is the hat matrix.

Studentized Residuals:

Bayesian Regression

Provides full uncertainty quantification for risk-sensitive manufacturing decisions.

Bayesian Linear Regression

Prior:

Posterior:

Posterior Parameters:

Predictive Distribution

For conjugate priors, this is a Student-t distribution.

Credible Intervals

95% Credible Interval for $\beta_j$:

Design of Experiments (DOE)

Full Factorial Design

For $k$ factors at 2 levels:

Fractional Factorial Design

Resolution:

• Resolution III: Main effects aliased with 2-factor interactions
• Resolution IV: Main effects clear; 2FIs aliased with each other
• Resolution V: Main effects and 2FIs clear

Central Composite Design (CCD)

Components:

• $2^k$ factorial points
• $2k$ axial (star) points at distance $\alpha$
• $n_0$ center points

Rotatability Condition:

D-Optimal Design

Maximizes the determinant of the information matrix:

Equivalently, minimizes the generalized variance of $\hat{\boldsymbol{\beta}}$.

I-Optimal Design

Minimizes average prediction variance:

Reliability Analysis

Cox Proportional Hazards Model

Hazard Function:

Where:

• $h(t|\mathbf{x})$ — hazard at time $t$ given covariates $\mathbf{x}$
• $h_0(t)$ — baseline hazard
• $\boldsymbol{\beta}$ — regression coefficients

Partial Likelihood

Where $\mathcal{R}(t_i)$ is the risk set at time $t_i$.

Challenge-Method Mapping

Equations Quick Reference

Estimation

Prediction Interval

Confidence Interval for $\beta_j$

Process Capability

Reference