Process Optimization

Optimization: Mathematical Modeling

1. Context

A recipe is a vector of controllable parameters:

Where:

• $T$ = Temperature (°C or K)
• $P$ = Pressure (mTorr or Pa)
• $Q_i$ = Gas flow rates (sccm)
• $t$ = Process time (seconds)
• $P_{\text{RF}}$ = RF power (Watts)

Goal : Find optimal $\mathbf{x}$ such that output properties $\mathbf{y}$ meet specifications while accounting for variability.

2. Mathematical Modeling Approaches

2.1 Physics-Based (First-Principles) Models

Chemical Vapor Deposition (CVD) Example

Mass transport and reaction equation:

Where:

• $C$ = Species concentration
• $\mathbf{u}$ = Velocity field
• $D$ = Diffusion coefficient
• $R(C, T)$ = Reaction rate

Surface reaction kinetics (Arrhenius form):

Where:

• $A$ = Pre-exponential factor
• $E_a$ = Activation energy
• $R$ = Gas constant
• $T$ = Temperature

Deposition rate (transport-limited regime):

Where:

• $C_s$ = Surface concentration
• $h_g$ = Gas-phase mass transfer coefficient

Characteristics:

• Advantages : Extrapolates outside training data, physically interpretable
• Disadvantages : Computationally expensive, requires detailed mechanism knowledge

2.2 Empirical/Statistical Models (Response Surface Methodology)

Second-order polynomial model:

Where:

• $\beta_0$ = Intercept
• $\beta_i$ = Linear coefficients
• $\beta_{ii}$ = Quadratic coefficients
• $\beta_{ij}$ = Interaction coefficients
• $\varepsilon$ = Random error, $\varepsilon \sim \mathcal{N}(0, \sigma^2)$

Matrix form:

Least squares estimator:

Design of Experiments (DOE) Options:

• Central Composite Design (CCD)
• Box-Behnken Design
• D-optimal designs
• Fractional factorial designs

2.3 Gaussian Process (Kriging) Models

GP prior:

Squared exponential (RBF) kernel:

Where:

• $\sigma_f^2$ = Signal variance
• $\ell_d$ = Length scale for dimension $d$

Matérn 5/2 kernel (alternative):

Where $r = \sqrt{\sum_d \frac{(x_d - x'_d)^2}{\ell_d^2}}$

Predictive mean:

Predictive variance:

Where:

• $\mathbf{k}_* = [k(\mathbf{x}_*, \mathbf{x}_1), \ldots, k(\mathbf{x}_*, \mathbf{x}_n)]^T$
• $\mathbf{K}$ = Gram matrix with $K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$
• $\sigma_n^2$ = Noise variance

Key Advantage : Native uncertainty quantification—critical for expensive semiconductor experiments.

2.4 Neural Network Models

Feedforward neural network:

Where:

• $\mathbf{W}_l$ = Weight matrix for layer $l$
• $\mathbf{b}_l$ = Bias vector for layer $l$
• $\sigma_l$ = Activation function (ReLU, tanh, etc.)

Physics-Informed Neural Networks (PINNs):

Where $\mathcal{F}$ is the differential operator from governing equations.

3. Optimization Formulations

3.1 Standard Constrained Optimization

General form:

Subject to:

Weighted multi-output objective:

Where $\Delta_j$ is the tolerance for output $j$.

3.2 Multi-Objective Optimization

Vector optimization problem:

Example objectives:

• $f_1(\mathbf{x})$ = Deviation from target thickness: $|t - t_{\text{target}}|$
• $f_2(\mathbf{x})$ = Within-wafer non-uniformity (WIWNU): $\frac{\sigma}{\mu} \times 100\%$
• $f_3(\mathbf{x})$ = Cycle time: $T_{\text{process}}$
• $f_4(\mathbf{x})$ = Defect density: $D_0$

Pareto dominance:
$\mathbf{x}^{(1)}$ dominates $\mathbf{x}^{(2)}$ if:

Solution methods:

• Weighted sum : $\min \sum_{i=1}^{k} w_i f_i(\mathbf{x})$
• $\varepsilon$-constraint : $\min f_1(\mathbf{x})$ s.t. $f_i(\mathbf{x}) \leq \varepsilon_i$ for $i \geq 2$
• Evolutionary algorithms : NSGA-II, MOEA/D

3.3 Robust Optimization

Sources of uncertainty $\boldsymbol{\xi}$:

• Control noise (recipe execution variability)
• Equipment drift
• Incoming material variation
• Measurement noise

Mean-variance formulation:

Where $\kappa$ is the risk aversion parameter.

Uncertainty propagation (first-order Taylor):

Sensitivity coefficient:

Chance-constrained optimization:

For Gaussian outputs:

Process capability index:

3.4 Bayesian Optimization

Algorithm for expensive-to-evaluate functions:

1. Fit GP surrogate: $y(\mathbf{x}) \sim \mathcal{GP}(m, k)$
2. Optimize acquisition function: $\mathbf{x}_{\text{next}} = \arg\max_{\mathbf{x}} \alpha(\mathbf{x})$
3. Evaluate true function: $y_{\text{next}} = f(\mathbf{x}_{\text{next}}) + \varepsilon$
4. Update dataset: $\mathcal{D} \leftarrow \mathcal{D} \cup \{(\mathbf{x}_{\text{next}}, y_{\text{next}})\}$
5. Repeat until budget exhausted

Acquisition functions:

Expected Improvement (EI):

Closed form (for GP):

Where:

• $\Phi(\cdot)$ = Standard normal CDF
• $\phi(\cdot)$ = Standard normal PDF
• $y^*$ = Best observed value

Upper Confidence Bound (UCB):

Where $\kappa$ balances exploration vs. exploitation.

Probability of Improvement (PI):

4. Run-to-Run (R2R) Control

4.1 EWMA Controller

Exponentially Weighted Moving Average prediction:

Where $\lambda \in (0,1]$ is the smoothing factor.

Recipe update:

Where $G = \frac{\partial y}{\partial x}$ is the process gain matrix.

4.2 Model Predictive Control (MPC)

Optimization problem:

Subject to:

Where:

• $\mathbf{Q}$ = Output weighting matrix
• $\mathbf{R}$ = Control effort weighting matrix
• $N$ = Prediction horizon

5. Mathematical Challenges

6. Dimensionality Reduction

6.1 Principal Component Analysis (PCA)

Objective:

Where $\mathbf{S}$ is the sample covariance matrix.

Solution: Eigenvectors of $\mathbf{S}$

Reduced representation:

Where $\mathbf{W}_k$ contains the top $k$ eigenvectors.

6.2 Partial Least Squares (PLS)

Objective: Maximize covariance between $\mathbf{X}$ and $\mathbf{Y}$

7. Multi-Fidelity Optimization

Combine cheap simulations with expensive experiments:

Auto-regressive model (Kennedy-O'Hagan):

Where:

• $y_{\text{HF}}$ = High-fidelity (experimental) response
• $y_{\text{LF}}$ = Low-fidelity (simulation) response
• $\rho$ = Scaling factor
• $\delta(\mathbf{x}) \sim \mathcal{GP}$ = Discrepancy function

Multi-fidelity GP:

8. Transfer Learning

Domain adaptation for tool-to-tool transfer:

Offset model (simple):

Linear adaptation:

GP adaptation:

9. Complete Optimization Framework

10. Equations:

Process Modeling

Optimization

Run-to-Run Control

Notation: