Optimization: Mathematical Modeling

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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} + \nabla \cdot (\mathbf{u}C) = D\nabla^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<j}\beta_{ij}x_i x_j + \varepsilon

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:

\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}

Least squares estimator:

\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}

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:

y(\mathbf{x}) \sim \mathcal{GP}\big(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')\big)

Squared exponential (RBF) kernel:

k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left(-\frac{1}{2}\sum_{d=1}^{n}\frac{(x_d - x'_d)^2}{\ell_d^2}\right)

Where:

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

Matérn 5/2 kernel (alternative):

k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \left(1 + \sqrt{5}r + \frac{5r^2}{3}\right) \exp\left(-\sqrt{5}r\right)

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

Predictive mean:

\mu(\mathbf{x}_*) = \mathbf{k}_*^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{y}

Predictive variance:

\sigma^2(\mathbf{x}_*) = k(\mathbf{x}_*, \mathbf{x}_*) - \mathbf{k}_*^T(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{k}_*

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:

\mathbf{y} = f_{\theta}(\mathbf{x}) = \sigma_L\left(\mathbf{W}_L \cdots \sigma_1(\mathbf{W}_1\mathbf{x} + \mathbf{b}_1) \cdots + \mathbf{b}_L\right)

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):

\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}}

\mathcal{L}_{\text{data}} = \frac{1}{N}\sum_{i=1}^{N}\|y_i - f_\theta(\mathbf{x}_i)\|^2

\mathcal{L}_{\text{physics}} = \frac{1}{M}\sum_{j=1}^{M}\left\|\mathcal{F}[f_\theta](\mathbf{x}_j)\right\|^2

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

3. Optimization Formulations

3.1 Standard Constrained Optimization

General form:

\min_{\mathbf{x}} \quad f(\mathbf{x}) = \|\mathbf{y}(\mathbf{x}) - \mathbf{y}_{\text{target}}\|_2^2

Subject to:

\begin{aligned} \mathbf{y}_L &\leq \mathbf{y}(\mathbf{x}) \leq \mathbf{y}_U \quad &\text{(specification limits)} \\ \mathbf{x}_L &\leq \mathbf{x} \leq \mathbf{x}_U \quad &\text{(equipment limits)} \\ \mathbf{g}(\mathbf{x}) &\leq 0 \quad &\text{(process constraints)} \end{aligned}

Weighted multi-output objective:

f(\mathbf{x}) = \sum_{j=1}^{m} w_j \left(\frac{y_j(\mathbf{x}) - y_{j,\text{target}}}{\Delta_j}\right)^2

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

3.2 Multi-Objective Optimization

Vector optimization problem:

\min_{\mathbf{x}} \quad \mathbf{F}(\mathbf{x}) = \begin{bmatrix} f_1(\mathbf{x}) \\ f_2(\mathbf{x}) \\ \vdots \\ f_k(\mathbf{x}) \end{bmatrix}

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:

f_i(\mathbf{x}^{(1)}) \leq f_i(\mathbf{x}^{(2)}) \quad \forall i \quad \text{and} \quad \exists j: f_j(\mathbf{x}^{(1)}) < f_j(\mathbf{x}^{(2)})

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:

\min_{\mathbf{x}} \quad \mathbb{E}[f(\mathbf{x}, \boldsymbol{\xi})] + \kappa \cdot \sqrt{\text{Var}[f(\mathbf{x}, \boldsymbol{\xi})]}

Where $\kappa$ is the risk aversion parameter.

Uncertainty propagation (first-order Taylor):

\mathbb{E}[y] \approx y(\bar{\mathbf{x}})

\text{Var}[y] \approx \sum_{i=1}^{n}\left(\frac{\partial y}{\partial x_i}\right)^2 \sigma_{x_i}^2 + 2\sum_{i<j}\frac{\partial y}{\partial x_i}\frac{\partial y}{\partial x_j}\text{Cov}(x_i, x_j)

Sensitivity coefficient:

S_i = \left|\frac{\partial y}{\partial x_i}\right| \cdot \frac{\sigma_{x_i}}{\sigma_y}

Chance-constrained optimization:

\Pr\big(y(\mathbf{x}, \boldsymbol{\xi}) \in [\text{LSL}, \text{USL}]\big) \geq 1 - \alpha

For Gaussian outputs:

\mu - z_{\alpha/2}\sigma \geq \text{LSL} \quad \text{and} \quad \mu + z_{\alpha/2}\sigma \leq \text{USL}

Process capability index:

C_{pk} = \min\left(\frac{\text{USL} - \mu}{3\sigma}, \frac{\mu - \text{LSL}}{3\sigma}\right) \geq 1.33

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):

\text{EI}(\mathbf{x}) = \mathbb{E}\big[\max(y^* - y(\mathbf{x}), 0)\big]

Closed form (for GP):

\text{EI}(\mathbf{x}) = (y^* - \mu(\mathbf{x}))\Phi(z) + \sigma(\mathbf{x})\phi(z)

Where:

z = \frac{y^* - \mu(\mathbf{x})}{\sigma(\mathbf{x})}

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

Upper Confidence Bound (UCB):

\text{UCB}(\mathbf{x}) = \mu(\mathbf{x}) + \kappa \sigma(\mathbf{x})

Where $\kappa$ balances exploration vs. exploitation.

Probability of Improvement (PI):

\text{PI}(\mathbf{x}) = \Phi\left(\frac{y^* - \mu(\mathbf{x})}{\sigma(\mathbf{x})}\right)

4. Run-to-Run (R2R) Control

4.1 EWMA Controller

Exponentially Weighted Moving Average prediction:

\hat{y}_{k+1} = \lambda y_k + (1-\lambda)\hat{y}_k

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

Recipe update:

x_{k+1} = x_k + G^{-1}(y_{\text{target}} - \hat{y}_{k+1})

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

4.2 Model Predictive Control (MPC)

Optimization problem:

\min_{\{u_k, \ldots, u_{k+N-1}\}} \sum_{j=0}^{N-1} \left[\|y_{k+j} - y_{\text{target}}\|_{\mathbf{Q}}^2 + \|\Delta u_{k+j}\|_{\mathbf{R}}^2\right]

Subject to:

\begin{aligned} \mathbf{x}_{k+j+1} &= \mathbf{A}\mathbf{x}_{k+j} + \mathbf{B}u_{k+j} \quad &\text{(state dynamics)} \\ y_{k+j} &= \mathbf{C}\mathbf{x}_{k+j} \quad &\text{(output equation)} \\ u_{\min} &\leq u_{k+j} \leq u_{\max} \quad &\text{(input constraints)} \end{aligned}

Where:

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

5. Mathematical Challenges

Challenge	Mathematical Approach
High dimensionality ($n > 50$ 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

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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
Gaussian Process	$y(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x},\mathbf{x}'))$
Neural Network	$y = \sigma_L(\mathbf{W}_L\cdots\sigma_1(\mathbf{W}_1\mathbf{x}+\mathbf{b}_1)+\mathbf{b}_L)$

Optimization

Formulation	Equation
Least squares	$\min_\mathbf{x} \	\mathbf{y}(\mathbf{x}) - \mathbf{y}_{\text{target}}\	_2^2$
Robust	$\min_\mathbf{x} \mathbb{E}[f] + \kappa\sqrt{\text{Var}[f]}$
Chance-constrained	$\Pr(y \in \text{spec}) \geq 1-\alpha$
Expected Improvement	$\text{EI}(\mathbf{x}) = (y^*-\mu)\Phi(z) + \sigma\phi(z)$

Run-to-Run Control

Controller	Equation
EWMA	$\hat{y}_{k+1} = \lambda y_k + (1-\lambda)\hat{y}_k$
Recipe update	$x_{k+1} = x_k + G^{-1}(y_{\text{target}} - \hat{y}_{k+1})$

Notation:

Symbol	Description
$\mathbf{x}$	Recipe parameter vector
$\mathbf{y}$	Process output vector
$n$	Number of input parameters
$m$	Number of outputs
$\boldsymbol{\beta}$	Regression coefficients
$\mathcal{GP}$	Gaussian Process
$k(\cdot,\cdot)$	Kernel/covariance function
$\mu(\mathbf{x})$	Predictive mean
$\sigma^2(\mathbf{x})$	Predictive variance
$\mathbb{E}[\cdot]$	Expectation operator
$\text{Var}[\cdot]$	Variance operator
$\Phi(\cdot)$	Standard normal CDF
$\phi(\cdot)$	Standard normal PDF
$\boldsymbol{\xi}$	Random disturbance/uncertainty
$G$	Process gain matrix
LSL, USL	Lower/Upper Specification Limits
$C_{pk}$	Process capability index

process optimizationrecipe optimizationresponse surface methodologyrsmgaussian processbayesian optimizationrun to run controlr2rrobust optimizationmulti-objective optimization

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