hierarchical all-reduce, distributed training
Multi-level aggregation.
hierarchical attention, transformer
Multi-level attention structure.
hierarchical context, llm architecture
Multi-level context organization.
hierarchical fusion, multimodal ai
Multi-level fusion strategy.
hierarchical planning, ai agents
Hierarchical planning operates at multiple abstraction levels from high-level goals to low-level actions.
hierarchical pooling, graph neural networks
Hierarchical pooling creates multi-resolution graph representations through successive coarsening operations.
high availability (ha),high availability,ha,reliability
System remains operational despite failures.
high dimensional optimization, bayesian optimization, gaussian process, response surface, doe, design of experiments, pareto optimization, robust optimization, surrogate modeling, tcad, run to run control
# 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:**
$$
y = \beta_0 + \sum_{i=1}^{n}\beta_i x_i + \sum_{i=1}^{n}\beta_{ii}x_i^2 + \sum_{i 50$ parameters) | PCA, PLS, sparse regression (LASSO), feature selection |
| Small datasets (limited wafer runs) | Bayesian methods, transfer learning, multi-fidelity modeling |
| Nonlinearity | GPs, neural networks, tree ensembles (RF, XGBoost) |
| Equipment-to-equipment variation | Mixed-effects models, hierarchical Bayesian models |
| Drift over time | Adaptive/recursive estimation, change-point detection, Kalman filtering |
| Multiple correlated responses | Multi-task learning, co-kriging, multivariate GP |
| Missing data | EM algorithm, multiple imputation, probabilistic PCA |
## 6. Dimensionality Reduction
### 6.1 Principal Component Analysis (PCA)
**Objective:**
$$
\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
```
┌────────────────────────────────────────────────────────────────────────────────────┐
│ RECIPE OPTIMIZATION FRAMEWORK │
├────────────────────────────────────────────────────────────────────────────────────┤
│ │
│ RECIPE PARAMETERS PROCESS MODEL │
│ ───────────────── ───────────── │
│ x₁: Temperature (°C) ───► ┌───────────────┐ │
│ x₂: Pressure (mTorr) ───► │ │ │
│ x₃: Gas flow 1 (sccm) ───► │ y = f(x;θ) │ ───► y₁: Thickness (nm) │
│ x₄: Gas flow 2 (sccm) ───► │ │ ───► y₂: Uniformity (%) │
│ x₅: RF power (W) ───► │ + ε │ ───► y₃: CD (nm) │
│ x₆: Time (s) ───► └───────────────┘ ───► y₄: Defects (#/cm²) │
│ ▲ │
│ │ │
│ Uncertainty ξ │
│ │
├────────────────────────────────────────────────────────────────────────────────────┤
│ OPTIMIZATION PROBLEM: │
│ │
│ min Σⱼ wⱼ(E[yⱼ] - yⱼ,target)² + λ·Var[y] │
│ x │
│ │
│ subject to: │
│ y_L ≤ E[y] ≤ y_U (specification limits) │
│ Pr(y ∈ spec) ≥ 0.9973 (Cpk ≥ 1.0) │
│ x_L ≤ x ≤ x_U (equipment limits) │
│ g(x) ≤ 0 (process constraints) │
│ │
└────────────────────────────────────────────────────────────────────────────────────┘
```
## 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
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