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Semiconductor Manufacturing Process Recipe Optimization: Mathematical Modeling

1. Problem Context

A semiconductor 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. Practical 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. Key Equations Summary

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

11. References and Further Reading

Textbooks

Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th ed.
Rasmussen, C.E. & Williams, C.K.I. (2006). Gaussian Processes for Machine Learning.
Boyd, S. & Vandenberghe, L. (2004). Convex Optimization.

Semiconductor-Specific

May, G.S. & Spanos, C.J. (2006). Fundamentals of Semiconductor Manufacturing and Process Control.
Edgar, T.F., Butler, S.W., et al. (2000). "Automatic Control in Microelectronics Manufacturing", Automatica.

Bayesian Optimization

Shahriari, B. et al. (2016). "Taking the Human Out of the Loop: A Review of Bayesian Optimization", Proceedings of the IEEE.
Frazier, P.I. (2018). "A Tutorial on Bayesian Optimization", arXiv:1807.02811.

Notation Reference

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

dimensional optimization highhigh-dimensional optimizationbayesian optimizationgaussian processdoe

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