process optimization

**Optimization: Mathematical Modeling** 1. Context A recipe is a vector of controllable parameters: $$ \mathbf{x} = \begin{bmatrix} T \\ P \\ Q_1 \\ Q_2 \\ \vdots \\ t \\ P_{\text{RF}} \end{bmatrix} \in \mathbb{R}^n $$ 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: $$ \frac{\partial C}{\partial t} + abla \cdot (\mathbf{u}C) = D abla^2 C + R(C, T) $$ Where: - $C$ = Species concentration - $\mathbf{u}$ = Velocity field - $D$ = Diffusion coefficient - $R(C, T)$ = Reaction rate Surface reaction kinetics (Arrhenius form): $$ k_s = A \exp\left(-\frac{E_a}{RT}\right) $$ Where: - $A$ = Pre-exponential factor - $E_a$ = Activation energy - $R$ = Gas constant - $T$ = Temperature Deposition rate (transport-limited regime): $$ r = \frac{k_s C_s}{1 + \frac{k_s}{h_g}} $$ 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: $$ y = \beta_0 + \sum_{i=1}^{n}\beta_i x_i + \sum_{i=1}^{n}\beta_{ii}x_i^2 + \sum_{i 50$ parameters) | PCA, PLS, sparse regression (LASSO), feature selection | | Small datasets (limited wafer runs) | Bayesian methods, transfer learning, multi-fidelity modeling | | Nonlinearity | GPs, neural networks, tree ensembles (RF, XGBoost) | | Equipment-to-equipment variation | Mixed-effects models, hierarchical Bayesian models | | Drift over time | Adaptive/recursive estimation, change-point detection, Kalman filtering | | Multiple correlated responses | Multi-task learning, co-kriging, multivariate GP | | Missing data | EM algorithm, multiple imputation, probabilistic PCA | 6. Dimensionality Reduction 6.1 Principal Component Analysis (PCA) Objective: $$ \max_{\mathbf{w}} \quad \mathbf{w}^T\mathbf{S}\mathbf{w} \quad \text{s.t.} \quad \|\mathbf{w}\|_2 = 1 $$ Where $\mathbf{S}$ is the sample covariance matrix. Solution: Eigenvectors of $\mathbf{S}$ $$ \mathbf{S} = \mathbf{W}\boldsymbol{\Lambda}\mathbf{W}^T $$ Reduced representation: $$ \mathbf{z} = \mathbf{W}_k^T(\mathbf{x} - \bar{\mathbf{x}}) $$ Where $\mathbf{W}_k$ contains the top $k$ eigenvectors. 6.2 Partial Least Squares (PLS) Objective: Maximize covariance between $\mathbf{X}$ and $\mathbf{Y}$ $$ \max_{\mathbf{w}, \mathbf{c}} \quad \text{Cov}(\mathbf{Xw}, \mathbf{Yc}) \quad \text{s.t.} \quad \|\mathbf{w}\|=\|\mathbf{c}\|=1 $$ 7. Multi-Fidelity Optimization Combine cheap simulations with expensive experiments: Auto-regressive model (Kennedy-O'Hagan): $$ y_{\text{HF}}(\mathbf{x}) = \rho \cdot y_{\text{LF}}(\mathbf{x}) + \delta(\mathbf{x}) $$ Where: - $y_{\text{HF}}$ = High-fidelity (experimental) response - $y_{\text{LF}}$ = Low-fidelity (simulation) response - $\rho$ = Scaling factor - $\delta(\mathbf{x}) \sim \mathcal{GP}$ = Discrepancy function Multi-fidelity GP: $$ \begin{bmatrix} \mathbf{y}_{\text{LF}} \\ \mathbf{y}_{\text{HF}} \end{bmatrix} \sim \mathcal{N}\left(\mathbf{0}, \begin{bmatrix} \mathbf{K}_{\text{LL}} & \rho\mathbf{K}_{\text{LH}} \\ \rho\mathbf{K}_{\text{HL}} & \rho^2\mathbf{K}_{\text{LL}} + \mathbf{K}_{\delta} \end{bmatrix}\right) $$ 8. Transfer Learning Domain adaptation for tool-to-tool transfer: $$ y_{\text{target}}(\mathbf{x}) = y_{\text{source}}(\mathbf{x}) + \Delta(\mathbf{x}) $$ Offset model (simple): $$ \Delta(\mathbf{x}) = c_0 \quad \text{(constant offset)} $$ Linear adaptation: $$ \Delta(\mathbf{x}) = \mathbf{c}^T\mathbf{x} + c_0 $$ GP adaptation: $$ \Delta(\mathbf{x}) \sim \mathcal{GP}(0, k_\Delta) $$ 9. Complete Optimization Framework ```svg ``` 10. Equations: Process Modeling | Model Type | Equation | |:-----------|:---------| | Linear regression | $y = \mathbf{X}\boldsymbol{\beta} + \varepsilon$ | | Quadratic RSM | $y = \beta_0 + \sum_i \beta_i x_i + \sum_i \beta_{ii}x_i^2 + \sum_{i