metrology lab,metrology
Controlled environment for precise measurements.
923 technical terms and definitions
Controlled environment for precise measurements.
# Semiconductor Manufacturing Process Metrology: Science, Mathematics, and Modeling A comprehensive exploration of the physics, mathematics, and computational methods underlying nanoscale measurement in semiconductor fabrication. ## 1. The Fundamental Challenge Modern semiconductor manufacturing produces structures with critical dimensions of just a few nanometers. At leading-edge nodes (3nm, 2nm), we are measuring features only **10–20 atoms wide**. ### Key Requirements - **Sub-angstrom precision** in measurement - **Complex 3D architectures**: FinFETs, Gate-All-Around (GAA) transistors, 3D NAND (200+ layers) - **High throughput**: seconds per measurement in production - **Multi-parameter extraction**: distinguish dozens of correlated parameters ### Metrology Techniques Overview | Technique | Principle | Resolution | Throughput | |-----------|-----------|------------|------------| | Spectroscopic Ellipsometry (SE) | Polarization change | ~0.1 Å | High | | Optical CD (OCD/Scatterometry) | Diffraction analysis | ~0.1 nm | High | | CD-SEM | Electron imaging | ~1 nm | Medium | | CD-SAXS | X-ray scattering | ~0.1 nm | Low | | AFM | Probe scanning | ~0.1 nm | Low | | TEM | Electron transmission | Atomic | Very Low | ## 2. Physics Foundation ### 2.1 Maxwell's Equations At the heart of optical metrology lies the solution to Maxwell's equations: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ $$ \nabla \cdot \mathbf{D} = \rho $$ $$ \nabla \cdot \mathbf{B} = 0 $$ Where: - $\mathbf{E}$ = Electric field vector - $\mathbf{H}$ = Magnetic field vector - $\mathbf{D}$ = Electric displacement field - $\mathbf{B}$ = Magnetic flux density - $\mathbf{J}$ = Current density - $\rho$ = Charge density ### 2.2 Constitutive Relations For linear, isotropic media: $$ \mathbf{D} = \varepsilon_0 \varepsilon_r \mathbf{E} = \varepsilon_0 (1 + \chi_e) \mathbf{E} $$ $$ \mathbf{B} = \mu_0 \mu_r \mathbf{H} $$ The complex dielectric function: $$ \tilde{\varepsilon}(\omega) = \varepsilon_1(\omega) + i\varepsilon_2(\omega) = \tilde{n}^2 = (n + ik)^2 $$ Where: - $n$ = Refractive index - $k$ = Extinction coefficient ### 2.3 Fresnel Equations At an interface between media with refractive indices $\tilde{n}_1$ and $\tilde{n}_2$: **s-polarization (TE):** $$ r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t} $$ $$ t_s = \frac{2 n_1 \cos\theta_i}{n_1 \cos\theta_i + n_2 \cos\theta_t} $$ **p-polarization (TM):** $$ r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t} $$ $$ t_p = \frac{2 n_1 \cos\theta_i}{n_2 \cos\theta_i + n_1 \cos\theta_t} $$ With Snell's law: $$ n_1 \sin\theta_i = n_2 \sin\theta_t $$ ## 3. Mathematics of Inverse Problems ### 3.1 Problem Formulation Metrology is fundamentally an **inverse problem**: | Problem Type | Description | Well-Posed? | |--------------|-------------|-------------| | **Forward** | Structure parameters → Measured signal | Yes | | **Inverse** | Measured signal → Structure parameters | Often No | We seek parameters $\mathbf{p}$ that minimize the difference between model $M(\mathbf{p})$ and data $\mathbf{D}$: $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 $$ Or with weighted least squares: $$ \chi^2 = \sum_{k=1}^{N} \frac{\left( M_k(\mathbf{p}) - D_k \right)^2}{\sigma_k^2} $$ ### 3.2 Levenberg-Marquardt Algorithm The workhorse optimization algorithm interpolates between gradient descent and Gauss-Newton: $$ \left( \mathbf{J}^T \mathbf{J} + \lambda \mathbf{I} \right) \delta\mathbf{p} = \mathbf{J}^T \left( \mathbf{D} - M(\mathbf{p}) \right) $$ Where: - $\mathbf{J}$ = Jacobian matrix (sensitivity matrix) - $\lambda$ = Damping parameter - $\delta\mathbf{p}$ = Parameter update step The Jacobian elements: $$ J_{ij} = \frac{\partial M_i}{\partial p_j} $$ **Algorithm behavior:** - Large $\lambda$ → Gradient descent (robust, slow) - Small $\lambda$ → Gauss-Newton (fast near minimum) ### 3.3 Regularization Techniques For ill-posed problems, regularization is essential: **Tikhonov Regularization (L2):** $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} - \mathbf{p}_0 \right\|^2 $$ **LASSO Regularization (L1):** $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} \right\|_1 $$ **Bayesian Inference:** $$ P(\mathbf{p} | \mathbf{D}) = \frac{P(\mathbf{D} | \mathbf{p}) \cdot P(\mathbf{p})}{P(\mathbf{D})} $$ Where: - $P(\mathbf{p} | \mathbf{D})$ = Posterior probability - $P(\mathbf{D} | \mathbf{p})$ = Likelihood - $P(\mathbf{p})$ = Prior probability ## 4. Thin Film Optics ### 4.1 Ellipsometry Fundamentals Ellipsometry measures the change in polarization state upon reflection: $$ \rho = \tan(\Psi) \cdot e^{i\Delta} = \frac{r_p}{r_s} $$ Where: - $\Psi$ = Amplitude ratio angle - $\Delta$ = Phase difference - $r_p, r_s$ = Complex reflection coefficients ### 4.2 Transfer Matrix Method For multilayer stacks, the characteristic matrix for layer $j$: $$ \mathbf{M}_j = \begin{pmatrix} \cos\delta_j & \frac{i \sin\delta_j}{\eta_j} \\ i\eta_j \sin\delta_j & \cos\delta_j \end{pmatrix} $$ Where the phase thickness: $$ \delta_j = \frac{2\pi}{\lambda} \tilde{n}_j d_j \cos\theta_j $$ And the optical admittance: $$ \eta_j = \begin{cases} \tilde{n}_j \cos\theta_j & \text{(s-pol)} \\ \frac{\tilde{n}_j}{\cos\theta_j} & \text{(p-pol)} \end{cases} $$ **Total system matrix:** $$ \mathbf{M}_{total} = \mathbf{M}_1 \cdot \mathbf{M}_2 \cdot \ldots \cdot \mathbf{M}_N = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} $$ **Reflection coefficient:** $$ r = \frac{\eta_0 m_{11} + \eta_0 \eta_s m_{12} - m_{21} - \eta_s m_{22}}{\eta_0 m_{11} + \eta_0 \eta_s m_{12} + m_{21} + \eta_s m_{22}} $$ ### 4.3 Dispersion Models **Lorentz Oscillator Model:** $$ \varepsilon(\omega) = \varepsilon_\infty + \sum_j \frac{A_j}{\omega_j^2 - \omega^2 - i\gamma_j \omega} $$ **Tauc-Lorentz Model (for amorphous semiconductors):** $$ \varepsilon_2(E) = \begin{cases} \frac{A E_0 C (E - E_g)^2}{(E^2 - E_0^2)^2 + C^2 E^2} \cdot \frac{1}{E} & E > E_g \\ 0 & E \leq E_g \end{cases} $$ With $\varepsilon_1$ obtained via Kramers-Kronig relations: $$ \varepsilon_1(E) = \varepsilon_{1,\infty} + \frac{2}{\pi} \mathcal{P} \int_{E_g}^{\infty} \frac{\xi \varepsilon_2(\xi)}{\xi^2 - E^2} d\xi $$ ## 5. Scatterometry and RCWA ### 5.1 Rigorous Coupled-Wave Analysis For a grating with period $\Lambda$, electromagnetic fields are expanded in Fourier orders: $$ E(x,z) = \sum_{m=-M}^{M} E_m(z) \exp(i k_{xm} x) $$ Where the diffracted wave vectors: $$ k_{xm} = k_{x0} + \frac{2\pi m}{\Lambda} = k_0 \left( n_1 \sin\theta_i + \frac{m\lambda}{\Lambda} \right) $$ ### 5.2 Eigenvalue Problem In each layer, the field satisfies: $$ \frac{d^2 \mathbf{E}}{dz^2} = \mathbf{\Omega}^2 \mathbf{E} $$ Where $\mathbf{\Omega}^2$ is a matrix determined by the Fourier components of the permittivity: $$ \varepsilon(x) = \sum_n \varepsilon_n \exp\left( i \frac{2\pi n}{\Lambda} x \right) $$ The eigenvalue decomposition: $$ \mathbf{\Omega}^2 = \mathbf{W} \mathbf{\Lambda} \mathbf{W}^{-1} $$ Provides propagation constants (eigenvalues $\lambda_m$) and field profiles (eigenvectors in $\mathbf{W}$). ### 5.3 S-Matrix Formulation For numerical stability, use the scattering matrix formulation: $$ \begin{pmatrix} \mathbf{a}_1^- \\ \mathbf{a}_N^+ \end{pmatrix} = \mathbf{S} \begin{pmatrix} \mathbf{a}_1^+ \\ \mathbf{a}_N^- \end{pmatrix} $$ Where $\mathbf{a}^+$ and $\mathbf{a}^-$ represent forward and backward propagating waves. The S-matrix is built recursively: $$ \mathbf{S}_{1 \to j+1} = \mathbf{S}_{1 \to j} \star \mathbf{S}_{j,j+1} $$ Using the Redheffer star product $\star$. ## 6. Statistical Process Control ### 6.1 Control Charts **$\bar{X}$ Chart (Mean):** $$ UCL = \bar{\bar{X}} + A_2 \bar{R} $$ $$ LCL = \bar{\bar{X}} - A_2 \bar{R} $$ **R Chart (Range):** $$ UCL_R = D_4 \bar{R} $$ $$ LCL_R = D_3 \bar{R} $$ **EWMA (Exponentially Weighted Moving Average):** $$ Z_t = \lambda X_t + (1 - \lambda) Z_{t-1} $$ With control limits: $$ UCL = \mu_0 + L \sigma \sqrt{\frac{\lambda}{2 - \lambda} \left[ 1 - (1-\lambda)^{2t} \right]} $$ ### 6.2 Process Capability Indices **$C_p$ (Process Capability):** $$ C_p = \frac{USL - LSL}{6\sigma} $$ **$C_{pk}$ (Centered Process Capability):** $$ C_{pk} = \min \left( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right) $$ **$C_{pm}$ (Taguchi Capability):** $$ C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}} $$ Where: - $USL$ = Upper Specification Limit - $LSL$ = Lower Specification Limit - $T$ = Target value - $\mu$ = Process mean - $\sigma$ = Process standard deviation ### 6.3 Gauge R&R Analysis Total measurement variance decomposition: $$ \sigma^2_{total} = \sigma^2_{part} + \sigma^2_{gauge} $$ $$ \sigma^2_{gauge} = \sigma^2_{repeatability} + \sigma^2_{reproducibility} $$ **Precision-to-Tolerance Ratio:** $$ P/T = \frac{6 \sigma_{gauge}}{USL - LSL} \times 100\% $$ | P/T Ratio | Assessment | |-----------|------------| | < 10% | Excellent | | 10-30% | Acceptable | | > 30% | Unacceptable | ## 7. Uncertainty Quantification ### 7.1 Fisher Information Matrix The Fisher Information Matrix for parameter estimation: $$ F_{ij} = \sum_{k=1}^{N} \frac{1}{\sigma_k^2} \frac{\partial M_k}{\partial p_i} \frac{\partial M_k}{\partial p_j} $$ Or equivalently: $$ F_{ij} = -E \left[ \frac{\partial^2 \ln L}{\partial p_i \partial p_j} \right] $$ Where $L$ is the likelihood function. ### 7.2 Cramér-Rao Lower Bound The covariance matrix of any unbiased estimator is bounded: $$ \text{Cov}(\hat{\mathbf{p}}) \geq \mathbf{F}^{-1} $$ For a single parameter: $$ \text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)} $$ **Interpretation:** - Diagonal elements of $\mathbf{F}^{-1}$ give minimum variance for each parameter - Off-diagonal elements indicate parameter correlations - Large condition number of $\mathbf{F}$ indicates ill-conditioning ### 7.3 Correlation Coefficient $$ \rho_{ij} = \frac{F^{-1}_{ij}}{\sqrt{F^{-1}_{ii} F^{-1}_{jj}}} $$ | |$\rho$| | Interpretation | |--------|----------------| | < 0.3 | Weak correlation | | 0.3 – 0.7 | Moderate correlation | | > 0.7 | Strong correlation | | > 0.95 | Severe: consider fixing one parameter | ### 7.4 GUM Framework According to the Guide to the Expression of Uncertainty in Measurement: **Combined standard uncertainty:** $$ u_c^2(y) = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i) + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} u(x_i, x_j) $$ **Expanded uncertainty:** $$ U = k \cdot u_c(y) $$ Where $k$ is the coverage factor (typically $k=2$ for 95% confidence). ## 8. Machine Learning in Metrology ### 8.1 Neural Network Surrogate Models Replace expensive physics simulations with trained neural networks: $$ M_{NN}(\mathbf{p}; \mathbf{W}) \approx M_{physics}(\mathbf{p}) $$ **Training objective:** $$ \mathcal{L} = \frac{1}{N} \sum_{i=1}^{N} \left\| M_{NN}(\mathbf{p}_i) - M_{physics}(\mathbf{p}_i) \right\|^2 + \lambda \left\| \mathbf{W} \right\|^2 $$ **Speedup:** Typically $10^4$ – $10^6 \times$ faster than RCWA/FEM. ### 8.2 Physics-Informed Neural Networks (PINNs) Incorporate physical laws into the loss function: $$ \mathcal{L}_{total} = \mathcal{L}_{data} + \lambda_{physics} \mathcal{L}_{physics} $$ Where: $$ \mathcal{L}_{physics} = \left\| \nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} \right\|^2 + \ldots $$ ### 8.3 Gaussian Process Regression A non-parametric Bayesian approach: $$ f(\mathbf{x}) \sim \mathcal{GP}\left( m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}') \right) $$ **Common kernel (RBF/Squared Exponential):** $$ k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left( -\frac{\left\| \mathbf{x} - \mathbf{x}' \right\|^2}{2\ell^2} \right) $$ **Posterior prediction:** $$ \mu_* = \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{y} $$ $$ \sigma_*^2 = k_{**} - \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{k}_* $$ **Advantages:** - Provides uncertainty estimates naturally - Works well with limited training data - Interpretable hyperparameters ### 8.4 Virtual Metrology Predict wafer properties from equipment sensor data: $$ \hat{y} = f(FDC_1, FDC_2, \ldots, FDC_n) $$ Where $FDC_i$ are Fault Detection and Classification sensor readings. **Common approaches:** - Partial Least Squares (PLS) regression - Random Forests - Gradient Boosting (XGBoost, LightGBM) - Deep neural networks ## 9. Advanced Topics and Frontiers ### 9.1 3D Metrology Challenges Modern structures require 3D measurement: | Structure | Complexity | Key Challenge | |-----------|------------|---------------| | FinFET | Moderate | Fin height, sidewall angle | | GAA/Nanosheet | High | Sheet thickness, spacing | | 3D NAND | Very High | 200+ layers, bowing, tilt | | DRAM HAR | Extreme | 100:1 aspect ratio structures | ### 9.2 Hybrid Metrology Combining multiple techniques to break parameter correlations: $$ \chi^2_{total} = \sum_{techniques} w_t \chi^2_t $$ **Example combination:** - OCD for periodic structure parameters - Ellipsometry for film optical constants - XRR for density and interface roughness **Mathematical framework:** $$ \mathbf{F}_{hybrid} = \sum_t \mathbf{F}_t $$ Reduces off-diagonal elements, improving condition number. ### 9.3 Atomic-Scale Considerations At the 2nm node and beyond: **Line Edge Roughness (LER):** $$ \sigma_{LER} = \sqrt{\frac{1}{L} \int_0^L \left[ x(z) - \bar{x} \right]^2 dz} $$ **Power Spectral Density:** $$ PSD(f) = \frac{\sigma^2 \xi}{1 + (2\pi f \xi)^{2(1+H)}} $$ Where: - $\xi$ = Correlation length - $H$ = Hurst exponent (roughness character) **Quantum Effects:** - Tunneling through thin barriers - Discrete dopant effects - Wave function penetration ### 9.4 Model-Measurement Circularity A fundamental epistemological challenge: ``` - ┌──────────────┐ ┌──────────────┐ │ Physical │ ───► │ Measured │ │ Structure │ │ Signal │ └──────────────┘ └──────────────┘ ▲ │ │ ▼ │ ┌──────────────┐ │ │ Model │ └────────────◄─┤ Inversion │ └──────────────┘ ``` **Key questions:** - How do we validate models when "truth" requires modeling? - Reference metrology (TEM) also requires interpretation - What does it mean to "know" a dimension at atomic scale? ## Key Symbols and Notation | Symbol | Description | Units | |--------|-------------|-------| | $\lambda$ | Wavelength | nm | | $\theta$ | Angle of incidence | degrees | | $n$ | Refractive index | dimensionless | | $k$ | Extinction coefficient | dimensionless | | $d$ | Film thickness | nm | | $\Lambda$ | Grating period | nm | | $\Psi, \Delta$ | Ellipsometric angles | degrees | | $\sigma$ | Standard deviation | varies | | $\mathbf{J}$ | Jacobian matrix | varies | | $\mathbf{F}$ | Fisher Information Matrix | varies | ## Computational Complexity | Method | Complexity | Typical Time | |--------|------------|--------------| | Transfer Matrix | $O(N)$ | $\mu$s | | RCWA | $O(M^3 \cdot L)$ | ms – s | | FEM | $O(N^{1.5})$ | s – min | | FDTD | $O(N \cdot T)$ | s – min | | Monte Carlo (SEM) | $O(N_{electrons})$ | min – hr | | Neural Network (inference) | $O(1)$ | $\mu$s | Where: - $N$ = Number of layers / mesh elements - $M$ = Number of Fourier orders - $L$ = Number of layers - $T$ = Number of time steps
# Semiconductor Manufacturing Process Metrology: Mathematical Modeling
## 1. The Core Problem Structure
Semiconductor metrology faces a fundamental **inverse problem**: we make indirect measurements (optical spectra, scattered X-rays, electron signals) and must infer physical quantities (dimensions, compositions, defect states) that we cannot directly observe at the nanoscale.
### 1.1 Mathematical Formulation
The general measurement model:
$$
\mathbf{y} = \mathcal{F}(\mathbf{p}) + \boldsymbol{\epsilon}
$$
**Variable Definitions:**
- $\mathbf{y}$ — measured signal vector (spectrum, image intensity, scattered amplitude)
- $\mathbf{p}$ — physical parameters of interest (CD, thickness, sidewall angle, composition)
- $\mathcal{F}$ — forward model operator (physics of measurement process)
- $\boldsymbol{\epsilon}$ — noise/uncertainty term
### 1.2 Key Mathematical Challenges
- **Nonlinearity:** $\mathcal{F}$ is typically highly nonlinear
- **Computational cost:** Forward model evaluation is expensive
- **Ill-posedness:** Inverse may be non-unique or unstable
- **High dimensionality:** Many parameters from limited measurements
## 2. Optical Critical Dimension (OCD) / Scatterometry
This is the most mathematically intensive metrology technique in high-volume manufacturing.
### 2.1 Forward Problem: Electromagnetic Scattering
For periodic structures (gratings, arrays), solve Maxwell's equations with Floquet-Bloch boundary conditions.
#### 2.1.1 Maxwell's Equations
$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
$$
$$
\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
$$
#### 2.1.2 Rigorous Coupled Wave Analysis (RCWA)
**Field Expansion in Fourier Series:**
The electric field in layer $j$ with grating vector $\mathbf{K}$:
$$
\mathbf{E}(\mathbf{r}) = \sum_{n=-N}^{N} \mathbf{E}_n^{(j)} \exp\left(i(\mathbf{k}_n \cdot \mathbf{r})\right)
$$
where the diffraction wave vectors are:
$$
\mathbf{k}_n = \mathbf{k}_0 + n\mathbf{K}
$$
**Key Properties:**
- Converts PDEs to eigenvalue problem
- Matches boundary conditions at layer interfaces
- Computational complexity: $O(N^3)$ where $N$ = number of Fourier orders
### 2.2 Inverse Problem: Parameter Extraction
Given measured spectra $R(\lambda, \theta)$, find best-fit parameters $\mathbf{p}$.
#### 2.2.1 Optimization Formulation
$$
\hat{\mathbf{p}} = \arg\min_{\mathbf{p}} \left\| \mathbf{y}_{\text{meas}} - \mathcal{F}(\mathbf{p}) \right\|^2 + \lambda R(\mathbf{p})
$$
**Regularization Options:**
- **Tikhonov regularization:**
$$
R(\mathbf{p}) = \left\| \mathbf{p} - \mathbf{p}_0 \right\|^2
$$
- **Sparsity-promoting (L1):**
$$
R(\mathbf{p}) = \left\| \mathbf{p} \right\|_1
$$
- **Total variation:**
$$
R(\mathbf{p}) = \int |\nabla \mathbf{p}| \, d\mathbf{x}
$$
#### 2.2.2 Library-Based Approach
1. **Precomputation:** Generate forward model on dense parameter grid
2. **Storage:** Build library with millions of entries
3. **Search:** Find best match using regression methods
**Regression Methods:**
- Polynomial regression — fast but limited accuracy
- Neural networks — handle nonlinearity well
- Gaussian process regression — provides uncertainty estimates
### 2.3 Parameter Correlations and Uncertainty
#### 2.3.1 Fisher Information Matrix
$$
[\mathbf{I}(\mathbf{p})]_{ij} = \mathbb{E}\left[\frac{\partial \ln L}{\partial p_i}\frac{\partial \ln L}{\partial p_j}\right]
$$
#### 2.3.2 Cramér-Rao Lower Bound
$$
\text{Var}(\hat{p}_i) \geq \left[\mathbf{I}^{-1}\right]_{ii}
$$
**Physical Interpretation:** Strong correlations (e.g., height vs. sidewall angle) manifest as near-singular information matrices—a fundamental limit on independent resolution.
## 3. Thin Film Metrology: Ellipsometry
### 3.1 Physical Model
Ellipsometry measures polarization state change upon reflection:
$$
\rho = \frac{r_p}{r_s} = \tan(\Psi)\exp(i\Delta)
$$
**Variables:**
- $r_p$ — p-polarized reflection coefficient
- $r_s$ — s-polarized reflection coefficient
- $\Psi$ — amplitude ratio angle
- $\Delta$ — phase difference
### 3.2 Transfer Matrix Formalism
For multilayer stacks:
$$
\mathbf{M} = \prod_{j=1}^{N} \mathbf{M}_j = \prod_{j=1}^{N} \begin{pmatrix} \cos\delta_j & \dfrac{i\sin\delta_j}{\eta_j} \\[10pt] i\eta_j\sin\delta_j & \cos\delta_j \end{pmatrix}
$$
where the phase thickness is:
$$
\delta_j = \frac{2\pi}{\lambda} n_j d_j \cos(\theta_j)
$$
**Parameters:**
- $n_j$ — refractive index of layer $j$
- $d_j$ — thickness of layer $j$
- $\theta_j$ — angle of propagation in layer $j$
- $\eta_j$ — optical admittance
### 3.3 Dispersion Models
#### 3.3.1 Cauchy Model (Transparent Materials)
$$
n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}
$$
#### 3.3.2 Sellmeier Equation
$$
n^2(\lambda) = 1 + \sum_{i} \frac{B_i \lambda^2}{\lambda^2 - C_i}
$$
#### 3.3.3 Tauc-Lorentz Model (Amorphous Semiconductors)
$$
\varepsilon_2(E) = \begin{cases}
\dfrac{A E_0 C (E - E_g)^2}{(E^2 - E_0^2)^2 + C^2 E^2} \cdot \dfrac{1}{E} & E > E_g \\[10pt]
0 & E \leq E_g
\end{cases}
$$
with $\varepsilon_1$ derived via Kramers-Kronig relations:
$$
\varepsilon_1(E) = \varepsilon_{1\infty} + \frac{2}{\pi} \mathcal{P} \int_0^\infty \frac{\xi \varepsilon_2(\xi)}{\xi^2 - E^2} d\xi
$$
#### 3.3.4 Drude Model (Metals/Conductors)
$$
\varepsilon(\omega) = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2 + i\gamma\omega}
$$
**Parameters:**
- $\omega_p$ — plasma frequency
- $\gamma$ — damping coefficient
- $\varepsilon_\infty$ — high-frequency dielectric constant
## 4. X-ray Metrology Mathematics
### 4.1 X-ray Reflectivity (XRR)
#### 4.1.1 Parratt Recursion Formula
For specular reflection at grazing incidence:
$$
R_j = \frac{r_{j,j+1} + R_{j+1}\exp(2ik_{z,j+1}d_{j+1})}{1 + r_{j,j+1}R_{j+1}\exp(2ik_{z,j+1}d_{j+1})}
$$
where $r_{j,j+1}$ is the Fresnel coefficient at interface $j$.
#### 4.1.2 Roughness Correction (Névot-Croce Factor)
$$
r'_{j,j+1} = r_{j,j+1} \exp\left(-2k_{z,j}k_{z,j+1}\sigma_j^2\right)
$$
**Parameters:**
- $k_{z,j}$ — perpendicular wave vector component in layer $j$
- $\sigma_j$ — RMS roughness at interface $j$
### 4.2 CD-SAXS (Critical Dimension Small Angle X-ray Scattering)
#### 4.2.1 Scattering Intensity
For transmission scattering from 3D nanostructures:
$$
I(\mathbf{q}) = \left|\tilde{\rho}(\mathbf{q})\right|^2 = \left|\int \Delta\rho(\mathbf{r})\exp(-i\mathbf{q}\cdot\mathbf{r})d^3\mathbf{r}\right|^2
$$
#### 4.2.2 Form Factor for Simple Shapes
**Rectangular parallelepiped:**
$$
F(\mathbf{q}) = V \cdot \text{sinc}\left(\frac{q_x a}{2}\right) \cdot \text{sinc}\left(\frac{q_y b}{2}\right) \cdot \text{sinc}\left(\frac{q_z c}{2}\right)
$$
**Cylinder:**
$$
F(\mathbf{q}) = 2\pi R^2 L \cdot \frac{J_1(q_\perp R)}{q_\perp R} \cdot \text{sinc}\left(\frac{q_z L}{2}\right)
$$
where $J_1$ is the first-order Bessel function.
## 5. Statistical Process Control Mathematics
### 5.1 Virtual Metrology
Predict wafer properties from tool sensor data without direct measurement:
$$
y = f(\mathbf{x}) + \varepsilon
$$
#### 5.1.1 Partial Least Squares (PLS)
Handles high-dimensional, correlated inputs:
1. Find latent variables: $\mathbf{T} = \mathbf{X}\mathbf{W}$
2. Maximize covariance with $y$
3. Model: $y = \mathbf{T}\mathbf{Q} + e$
**Optimization objective:**
$$
\max_{\mathbf{w}} \text{Cov}(\mathbf{X}\mathbf{w}, y)^2 \quad \text{subject to} \quad \|\mathbf{w}\| = 1
$$
#### 5.1.2 Gaussian Process Regression
$$
y(\mathbf{x}) \sim \mathcal{GP}\left(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')\right)
$$
**Common Kernel Functions:**
- **Squared Exponential (RBF):**
$$
k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left(-\frac{\|\mathbf{x} - \mathbf{x}'\|^2}{2\ell^2}\right)
$$
- **Matérn 5/2:**
$$
k(r) = \sigma_f^2 \left(1 + \frac{\sqrt{5}r}{\ell} + \frac{5r^2}{3\ell^2}\right) \exp\left(-\frac{\sqrt{5}r}{\ell}\right)
$$
### 5.2 Run-to-Run Control
#### 5.2.1 EWMA Controller
$$
\hat{d}_t = \lambda y_{t-1} + (1-\lambda)\hat{d}_{t-1}
$$
$$
x_t = x_{\text{nom}} - \frac{\hat{d}_t}{\hat{\beta}}
$$
**Parameters:**
- $\lambda$ — smoothing factor (typically 0.2–0.4)
- $\hat{\beta}$ — estimated process gain
- $x_{\text{nom}}$ — nominal recipe setting
#### 5.2.2 Model Predictive Control (MPC)
$$
\min_{\mathbf{u}} \sum_{k=0}^{N} \left\| y_{t+k} - y_{\text{target}} \right\|_Q^2 + \left\| \Delta u_{t+k} \right\|_R^2
$$
subject to:
- Process dynamics: $\mathbf{x}_{t+1} = \mathbf{A}\mathbf{x}_t + \mathbf{B}\mathbf{u}_t$
- Output equation: $y_t = \mathbf{C}\mathbf{x}_t$
- Constraints: $\mathbf{u}_{\min} \leq \mathbf{u}_t \leq \mathbf{u}_{\max}$
### 5.3 Wafer-Level Spatial Modeling
#### 5.3.1 Zernike Polynomial Decomposition
$$
W(r,\theta) = \sum_{n=0}^{N} \sum_{m=-n}^{n} a_{nm} Z_n^m(r,\theta)
$$
**First few Zernike polynomials:**
| Index | Name | Formula |
|-------|------|---------|
| $Z_0^0$ | Piston | $1$ |
| $Z_1^{-1}$ | Tilt Y | $2r\sin\theta$ |
| $Z_1^1$ | Tilt X | $2r\cos\theta$ |
| $Z_2^0$ | Defocus | $\sqrt{3}(2r^2-1)$ |
| $Z_2^{-2}$ | Astigmatism | $\sqrt{6}r^2\sin2\theta$ |
| $Z_2^2$ | Astigmatism | $\sqrt{6}r^2\cos2\theta$ |
#### 5.3.2 Gaussian Random Fields
For spatially correlated residuals:
$$
\text{Cov}\left(W(\mathbf{s}_1), W(\mathbf{s}_2)\right) = \sigma^2 \rho\left(\|\mathbf{s}_1 - \mathbf{s}_2\|; \phi\right)
$$
**Common correlation functions:**
- **Exponential:**
$$
\rho(h) = \exp\left(-\frac{h}{\phi}\right)
$$
- **Gaussian:**
$$
\rho(h) = \exp\left(-\frac{h^2}{\phi^2}\right)
$$
## 6. Overlay Metrology Mathematics
### 6.1 Higher-Order Correction Models
Overlay error as polynomial expansion:
$$
\delta x = T_x + M_x \cdot x + R_x \cdot y + \sum_{i+j \leq n} c_{ij}^x x^i y^j
$$
$$
\delta y = T_y + M_y \cdot y + R_y \cdot x + \sum_{i+j \leq n} c_{ij}^y x^i y^j
$$
**Physical interpretation of linear terms:**
- $T_x, T_y$ — Translation
- $M_x, M_y$ — Magnification
- $R_x, R_y$ — Rotation
### 6.2 Sampling Strategy Optimization
#### 6.2.1 D-Optimal Design
$$
\mathbf{s}^* = \arg\max_{\mathbf{s}} \det\left(\mathbf{X}_s^T \mathbf{X}_s\right)
$$
Minimizes the volume of the confidence ellipsoid for parameter estimates.
#### 6.2.2 Information-Theoretic Approach
Maximize expected information gain:
$$
I(\mathbf{s}) = H(\mathbf{p}) - \mathbb{E}_{\mathbf{y}}\left[H(\mathbf{p}|\mathbf{y})\right]
$$
## 7. Machine Learning Integration
### 7.1 Physics-Informed Neural Networks (PINNs)
Combine data fitting with physical constraints:
$$
\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}}
$$
**Components:**
- **Data loss:**
$$
\mathcal{L}_{\text{data}} = \frac{1}{N} \sum_{i=1}^{N} \left\| y_i - f_\theta(\mathbf{x}_i) \right\|^2
$$
- **Physics loss (example: Maxwell residual):**
$$
\mathcal{L}_{\text{physics}} = \frac{1}{M} \sum_{j=1}^{M} \left\| \nabla \times \mathbf{E}_\theta - i\omega\mu\mathbf{H}_\theta \right\|^2
$$
### 7.2 Neural Network Surrogates
**Architecture for forward model approximation:**
- **Input:** Geometric parameters $\mathbf{p} \in \mathbb{R}^d$
- **Hidden layers:** Multiple fully-connected layers with ReLU/GELU activation
- **Output:** Simulated spectrum $\mathbf{y} \in \mathbb{R}^m$
**Speedup:** $10^4$ – $10^6\times$ over rigorous simulation
### 7.3 Deep Learning for Defect Detection
**Methods:**
- **CNNs** — Classification and localization
- **Autoencoders** — Anomaly detection via reconstruction error:
$$
\text{Score}(\mathbf{x}) = \left\| \mathbf{x} - D(E(\mathbf{x})) \right\|^2
$$
- **Instance segmentation** — Precise defect boundary delineation
## 8. Uncertainty Quantification
### 8.1 GUM Framework (Guide to Uncertainty in Measurement)
Combined standard uncertainty:
$$
u_c^2(y) = \sum_{i} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i) + 2\sum_{i
# Semiconductor Manufacturing Process Metrology: Science, Mathematics, and Modeling A comprehensive exploration of the physics, mathematics, and computational methods underlying nanoscale measurement in semiconductor fabrication. ## 1. The Fundamental Challenge Modern semiconductor manufacturing produces structures with critical dimensions of just a few nanometers. At leading-edge nodes (3nm, 2nm), we are measuring features only **10–20 atoms wide**. ### Key Requirements - **Sub-angstrom precision** in measurement - **Complex 3D architectures**: FinFETs, Gate-All-Around (GAA) transistors, 3D NAND (200+ layers) - **High throughput**: seconds per measurement in production - **Multi-parameter extraction**: distinguish dozens of correlated parameters ### Metrology Techniques Overview | Technique | Principle | Resolution | Throughput | |-----------|-----------|------------|------------| | Spectroscopic Ellipsometry (SE) | Polarization change | ~0.1 Å | High | | Optical CD (OCD/Scatterometry) | Diffraction analysis | ~0.1 nm | High | | CD-SEM | Electron imaging | ~1 nm | Medium | | CD-SAXS | X-ray scattering | ~0.1 nm | Low | | AFM | Probe scanning | ~0.1 nm | Low | | TEM | Electron transmission | Atomic | Very Low | ## 2. Physics Foundation ### 2.1 Maxwell's Equations At the heart of optical metrology lies the solution to Maxwell's equations: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ $$ \nabla \cdot \mathbf{D} = \rho $$ $$ \nabla \cdot \mathbf{B} = 0 $$ Where: - $\mathbf{E}$ = Electric field vector - $\mathbf{H}$ = Magnetic field vector - $\mathbf{D}$ = Electric displacement field - $\mathbf{B}$ = Magnetic flux density - $\mathbf{J}$ = Current density - $\rho$ = Charge density ### 2.2 Constitutive Relations For linear, isotropic media: $$ \mathbf{D} = \varepsilon_0 \varepsilon_r \mathbf{E} = \varepsilon_0 (1 + \chi_e) \mathbf{E} $$ $$ \mathbf{B} = \mu_0 \mu_r \mathbf{H} $$ The complex dielectric function: $$ \tilde{\varepsilon}(\omega) = \varepsilon_1(\omega) + i\varepsilon_2(\omega) = \tilde{n}^2 = (n + ik)^2 $$ Where: - $n$ = Refractive index - $k$ = Extinction coefficient ### 2.3 Fresnel Equations At an interface between media with refractive indices $\tilde{n}_1$ and $\tilde{n}_2$: **s-polarization (TE):** $$ r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t} $$ $$ t_s = \frac{2 n_1 \cos\theta_i}{n_1 \cos\theta_i + n_2 \cos\theta_t} $$ **p-polarization (TM):** $$ r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t} $$ $$ t_p = \frac{2 n_1 \cos\theta_i}{n_2 \cos\theta_i + n_1 \cos\theta_t} $$ With Snell's law: $$ n_1 \sin\theta_i = n_2 \sin\theta_t $$ ## 3. Mathematics of Inverse Problems ### 3.1 Problem Formulation Metrology is fundamentally an **inverse problem**: | Problem Type | Description | Well-Posed? | |--------------|-------------|-------------| | **Forward** | Structure parameters → Measured signal | Yes | | **Inverse** | Measured signal → Structure parameters | Often No | We seek parameters $\mathbf{p}$ that minimize the difference between model $M(\mathbf{p})$ and data $\mathbf{D}$: $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 $$ Or with weighted least squares: $$ \chi^2 = \sum_{k=1}^{N} \frac{\left( M_k(\mathbf{p}) - D_k \right)^2}{\sigma_k^2} $$ ### 3.2 Levenberg-Marquardt Algorithm The workhorse optimization algorithm interpolates between gradient descent and Gauss-Newton: $$ \left( \mathbf{J}^T \mathbf{J} + \lambda \mathbf{I} \right) \delta\mathbf{p} = \mathbf{J}^T \left( \mathbf{D} - M(\mathbf{p}) \right) $$ Where: - $\mathbf{J}$ = Jacobian matrix (sensitivity matrix) - $\lambda$ = Damping parameter - $\delta\mathbf{p}$ = Parameter update step The Jacobian elements: $$ J_{ij} = \frac{\partial M_i}{\partial p_j} $$ **Algorithm behavior:** - Large $\lambda$ → Gradient descent (robust, slow) - Small $\lambda$ → Gauss-Newton (fast near minimum) ### 3.3 Regularization Techniques For ill-posed problems, regularization is essential: **Tikhonov Regularization (L2):** $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} - \mathbf{p}_0 \right\|^2 $$ **LASSO Regularization (L1):** $$ \min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} \right\|_1 $$ **Bayesian Inference:** $$ P(\mathbf{p} | \mathbf{D}) = \frac{P(\mathbf{D} | \mathbf{p}) \cdot P(\mathbf{p})}{P(\mathbf{D})} $$ Where: - $P(\mathbf{p} | \mathbf{D})$ = Posterior probability - $P(\mathbf{D} | \mathbf{p})$ = Likelihood - $P(\mathbf{p})$ = Prior probability ## 4. Thin Film Optics ### 4.1 Ellipsometry Fundamentals Ellipsometry measures the change in polarization state upon reflection: $$ \rho = \tan(\Psi) \cdot e^{i\Delta} = \frac{r_p}{r_s} $$ Where: - $\Psi$ = Amplitude ratio angle - $\Delta$ = Phase difference - $r_p, r_s$ = Complex reflection coefficients ### 4.2 Transfer Matrix Method For multilayer stacks, the characteristic matrix for layer $j$: $$ \mathbf{M}_j = \begin{pmatrix} \cos\delta_j & \frac{i \sin\delta_j}{\eta_j} \\ i\eta_j \sin\delta_j & \cos\delta_j \end{pmatrix} $$ Where the phase thickness: $$ \delta_j = \frac{2\pi}{\lambda} \tilde{n}_j d_j \cos\theta_j $$ And the optical admittance: $$ \eta_j = \begin{cases} \tilde{n}_j \cos\theta_j & \text{(s-pol)} \\ \frac{\tilde{n}_j}{\cos\theta_j} & \text{(p-pol)} \end{cases} $$ **Total system matrix:** $$ \mathbf{M}_{total} = \mathbf{M}_1 \cdot \mathbf{M}_2 \cdot \ldots \cdot \mathbf{M}_N = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} $$ **Reflection coefficient:** $$ r = \frac{\eta_0 m_{11} + \eta_0 \eta_s m_{12} - m_{21} - \eta_s m_{22}}{\eta_0 m_{11} + \eta_0 \eta_s m_{12} + m_{21} + \eta_s m_{22}} $$ ### 4.3 Dispersion Models **Lorentz Oscillator Model:** $$ \varepsilon(\omega) = \varepsilon_\infty + \sum_j \frac{A_j}{\omega_j^2 - \omega^2 - i\gamma_j \omega} $$ **Tauc-Lorentz Model (for amorphous semiconductors):** $$ \varepsilon_2(E) = \begin{cases} \frac{A E_0 C (E - E_g)^2}{(E^2 - E_0^2)^2 + C^2 E^2} \cdot \frac{1}{E} & E > E_g \\ 0 & E \leq E_g \end{cases} $$ With $\varepsilon_1$ obtained via Kramers-Kronig relations: $$ \varepsilon_1(E) = \varepsilon_{1,\infty} + \frac{2}{\pi} \mathcal{P} \int_{E_g}^{\infty} \frac{\xi \varepsilon_2(\xi)}{\xi^2 - E^2} d\xi $$ ## 5. Scatterometry and RCWA ### 5.1 Rigorous Coupled-Wave Analysis For a grating with period $\Lambda$, electromagnetic fields are expanded in Fourier orders: $$ E(x,z) = \sum_{m=-M}^{M} E_m(z) \exp(i k_{xm} x) $$ Where the diffracted wave vectors: $$ k_{xm} = k_{x0} + \frac{2\pi m}{\Lambda} = k_0 \left( n_1 \sin\theta_i + \frac{m\lambda}{\Lambda} \right) $$ ### 5.2 Eigenvalue Problem In each layer, the field satisfies: $$ \frac{d^2 \mathbf{E}}{dz^2} = \mathbf{\Omega}^2 \mathbf{E} $$ Where $\mathbf{\Omega}^2$ is a matrix determined by the Fourier components of the permittivity: $$ \varepsilon(x) = \sum_n \varepsilon_n \exp\left( i \frac{2\pi n}{\Lambda} x \right) $$ The eigenvalue decomposition: $$ \mathbf{\Omega}^2 = \mathbf{W} \mathbf{\Lambda} \mathbf{W}^{-1} $$ Provides propagation constants (eigenvalues $\lambda_m$) and field profiles (eigenvectors in $\mathbf{W}$). ### 5.3 S-Matrix Formulation For numerical stability, use the scattering matrix formulation: $$ \begin{pmatrix} \mathbf{a}_1^- \\ \mathbf{a}_N^+ \end{pmatrix} = \mathbf{S} \begin{pmatrix} \mathbf{a}_1^+ \\ \mathbf{a}_N^- \end{pmatrix} $$ Where $\mathbf{a}^+$ and $\mathbf{a}^-$ represent forward and backward propagating waves. The S-matrix is built recursively: $$ \mathbf{S}_{1 \to j+1} = \mathbf{S}_{1 \to j} \star \mathbf{S}_{j,j+1} $$ Using the Redheffer star product $\star$. ## 6. Statistical Process Control ### 6.1 Control Charts **$\bar{X}$ Chart (Mean):** $$ UCL = \bar{\bar{X}} + A_2 \bar{R} $$ $$ LCL = \bar{\bar{X}} - A_2 \bar{R} $$ **R Chart (Range):** $$ UCL_R = D_4 \bar{R} $$ $$ LCL_R = D_3 \bar{R} $$ **EWMA (Exponentially Weighted Moving Average):** $$ Z_t = \lambda X_t + (1 - \lambda) Z_{t-1} $$ With control limits: $$ UCL = \mu_0 + L \sigma \sqrt{\frac{\lambda}{2 - \lambda} \left[ 1 - (1-\lambda)^{2t} \right]} $$ ### 6.2 Process Capability Indices **$C_p$ (Process Capability):** $$ C_p = \frac{USL - LSL}{6\sigma} $$ **$C_{pk}$ (Centered Process Capability):** $$ C_{pk} = \min \left( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right) $$ **$C_{pm}$ (Taguchi Capability):** $$ C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}} $$ Where: - $USL$ = Upper Specification Limit - $LSL$ = Lower Specification Limit - $T$ = Target value - $\mu$ = Process mean - $\sigma$ = Process standard deviation ### 6.3 Gauge R&R Analysis Total measurement variance decomposition: $$ \sigma^2_{total} = \sigma^2_{part} + \sigma^2_{gauge} $$ $$ \sigma^2_{gauge} = \sigma^2_{repeatability} + \sigma^2_{reproducibility} $$ **Precision-to-Tolerance Ratio:** $$ P/T = \frac{6 \sigma_{gauge}}{USL - LSL} \times 100\% $$ | P/T Ratio | Assessment | |-----------|------------| | < 10% | Excellent | | 10-30% | Acceptable | | > 30% | Unacceptable | ## 7. Uncertainty Quantification ### 7.1 Fisher Information Matrix The Fisher Information Matrix for parameter estimation: $$ F_{ij} = \sum_{k=1}^{N} \frac{1}{\sigma_k^2} \frac{\partial M_k}{\partial p_i} \frac{\partial M_k}{\partial p_j} $$ Or equivalently: $$ F_{ij} = -E \left[ \frac{\partial^2 \ln L}{\partial p_i \partial p_j} \right] $$ Where $L$ is the likelihood function. ### 7.2 Cramér-Rao Lower Bound The covariance matrix of any unbiased estimator is bounded: $$ \text{Cov}(\hat{\mathbf{p}}) \geq \mathbf{F}^{-1} $$ For a single parameter: $$ \text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)} $$ **Interpretation:** - Diagonal elements of $\mathbf{F}^{-1}$ give minimum variance for each parameter - Off-diagonal elements indicate parameter correlations - Large condition number of $\mathbf{F}$ indicates ill-conditioning ### 7.3 Correlation Coefficient $$ \rho_{ij} = \frac{F^{-1}_{ij}}{\sqrt{F^{-1}_{ii} F^{-1}_{jj}}} $$ | |$\rho$| | Interpretation | |--------|----------------| | < 0.3 | Weak correlation | | 0.3 – 0.7 | Moderate correlation | | > 0.7 | Strong correlation | | > 0.95 | Severe: consider fixing one parameter | ### 7.4 GUM Framework According to the Guide to the Expression of Uncertainty in Measurement: **Combined standard uncertainty:** $$ u_c^2(y) = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i) + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} u(x_i, x_j) $$ **Expanded uncertainty:** $$ U = k \cdot u_c(y) $$ Where $k$ is the coverage factor (typically $k=2$ for 95% confidence). ## 8. Machine Learning in Metrology ### 8.1 Neural Network Surrogate Models Replace expensive physics simulations with trained neural networks: $$ M_{NN}(\mathbf{p}; \mathbf{W}) \approx M_{physics}(\mathbf{p}) $$ **Training objective:** $$ \mathcal{L} = \frac{1}{N} \sum_{i=1}^{N} \left\| M_{NN}(\mathbf{p}_i) - M_{physics}(\mathbf{p}_i) \right\|^2 + \lambda \left\| \mathbf{W} \right\|^2 $$ **Speedup:** Typically $10^4$ – $10^6 \times$ faster than RCWA/FEM. ### 8.2 Physics-Informed Neural Networks (PINNs) Incorporate physical laws into the loss function: $$ \mathcal{L}_{total} = \mathcal{L}_{data} + \lambda_{physics} \mathcal{L}_{physics} $$ Where: $$ \mathcal{L}_{physics} = \left\| \nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} \right\|^2 + \ldots $$ ### 8.3 Gaussian Process Regression A non-parametric Bayesian approach: $$ f(\mathbf{x}) \sim \mathcal{GP}\left( m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}') \right) $$ **Common kernel (RBF/Squared Exponential):** $$ k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left( -\frac{\left\| \mathbf{x} - \mathbf{x}' \right\|^2}{2\ell^2} \right) $$ **Posterior prediction:** $$ \mu_* = \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{y} $$ $$ \sigma_*^2 = k_{**} - \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{k}_* $$ **Advantages:** - Provides uncertainty estimates naturally - Works well with limited training data - Interpretable hyperparameters ### 8.4 Virtual Metrology Predict wafer properties from equipment sensor data: $$ \hat{y} = f(FDC_1, FDC_2, \ldots, FDC_n) $$ Where $FDC_i$ are Fault Detection and Classification sensor readings. **Common approaches:** - Partial Least Squares (PLS) regression - Random Forests - Gradient Boosting (XGBoost, LightGBM) - Deep neural networks ## 9. Advanced Topics and Frontiers ### 9.1 3D Metrology Challenges Modern structures require 3D measurement: | Structure | Complexity | Key Challenge | |-----------|------------|---------------| | FinFET | Moderate | Fin height, sidewall angle | | GAA/Nanosheet | High | Sheet thickness, spacing | | 3D NAND | Very High | 200+ layers, bowing, tilt | | DRAM HAR | Extreme | 100:1 aspect ratio structures | ### 9.2 Hybrid Metrology Combining multiple techniques to break parameter correlations: $$ \chi^2_{total} = \sum_{techniques} w_t \chi^2_t $$ **Example combination:** - OCD for periodic structure parameters - Ellipsometry for film optical constants - XRR for density and interface roughness **Mathematical framework:** $$ \mathbf{F}_{hybrid} = \sum_t \mathbf{F}_t $$ Reduces off-diagonal elements, improving condition number. ### 9.3 Atomic-Scale Considerations At the 2nm node and beyond: **Line Edge Roughness (LER):** $$ \sigma_{LER} = \sqrt{\frac{1}{L} \int_0^L \left[ x(z) - \bar{x} \right]^2 dz} $$ **Power Spectral Density:** $$ PSD(f) = \frac{\sigma^2 \xi}{1 + (2\pi f \xi)^{2(1+H)}} $$ Where: - $\xi$ = Correlation length - $H$ = Hurst exponent (roughness character) **Quantum Effects:** - Tunneling through thin barriers - Discrete dopant effects - Wave function penetration ### 9.4 Model-Measurement Circularity A fundamental epistemological challenge: ``` - ┌──────────────┐ ┌──────────────┐ │ Physical │ ───► │ Measured │ │ Structure │ │ Signal │ └──────────────┘ └──────────────┘ ▲ │ │ ▼ │ ┌──────────────┐ │ │ Model │ └────────────◄─┤ Inversion │ └──────────────┘ ``` **Key questions:** - How do we validate models when "truth" requires modeling? - Reference metrology (TEM) also requires interpretation - What does it mean to "know" a dimension at atomic scale? ## Key Symbols and Notation | Symbol | Description | Units | |--------|-------------|-------| | $\lambda$ | Wavelength | nm | | $\theta$ | Angle of incidence | degrees | | $n$ | Refractive index | dimensionless | | $k$ | Extinction coefficient | dimensionless | | $d$ | Film thickness | nm | | $\Lambda$ | Grating period | nm | | $\Psi, \Delta$ | Ellipsometric angles | degrees | | $\sigma$ | Standard deviation | varies | | $\mathbf{J}$ | Jacobian matrix | varies | | $\mathbf{F}$ | Fisher Information Matrix | varies | ## Computational Complexity | Method | Complexity | Typical Time | |--------|------------|--------------| | Transfer Matrix | $O(N)$ | $\mu$s | | RCWA | $O(M^3 \cdot L)$ | ms – s | | FEM | $O(N^{1.5})$ | s – min | | FDTD | $O(N \cdot T)$ | s – min | | Monte Carlo (SEM) | $O(N_{electrons})$ | min – hr | | Neural Network (inference) | $O(1)$ | $\mu$s | Where: - $N$ = Number of layers / mesh elements - $M$ = Number of Fourier orders - $L$ = Number of layers - $T$ = Number of time steps
Very fine pitch BGA.
Small breaks in intended continuous features.
Small unwanted connections between features.
Small solder bumps for 3D interconnect.
PL with microscale resolution.
High spatial resolution XRF.
Precision length measurement tool.
Surface roughness at micron scale.
Image electrical properties at nanoscale.
Non-contact lifetime measurement.
Quantify crystal orientation differences.
Use predictive models for process control.
Fit geometric model to optical data.
EMC absorbing water.
Protect from moisture.
Protect from humidity.
Classification of moisture sensitivity.
# Mathematical Modeling for MOL (Middle of Line) in Semiconductor Manufacturing ## Overview of MOL MOL (Middle of Line) is the critical transition layer in semiconductor fabrication between **FEOL** (Front End of Line - transistor fabrication) and **BEOL** (Back End of Line - metal interconnects). ### MOL Components - **Source/Drain Contacts**: Metal connections to transistor terminals - **Gate Contacts**: Connections to the gate electrode - **Local Interconnects (LI)**: Short-range wiring between nearby devices - **Via0**: Vertical connections from contacts to Metal 1 layer - **Trench Silicide (TS)**: Low-resistance contact formation At advanced technology nodes ($\leq 7\text{nm}$), MOL becomes increasingly critical due to contact resistance dominance and scaling challenges. ## 1. Contact Resistance Modeling ### 1.1 Specific Contact Resistivity For metal-semiconductor interfaces, the specific contact resistivity $\rho_c$ determines performance. **Thermionic-Field Emission** (dominant mechanism for modern contacts): $$ \rho_c \propto \exp\left(\frac{4\pi\sqrt{\epsilon_s m^*}\,\phi_B}{h\sqrt{N_D}}\right) $$ Where: - $\phi_B$ = Schottky barrier height (eV) - $N_D$ = Doping concentration (cm⁻³) - $m^*$ = Effective mass - $\epsilon_s$ = Semiconductor permittivity - $h$ = Planck's constant ### 1.2 Transmission Line Model (TLM) $$ R_c = \frac{\rho_c}{L_T} \coth\left(\frac{L}{L_T}\right) $$ **Transfer length:** $$ L_T = \sqrt{\frac{\rho_c}{R_{sh}}} $$ Where: - $R_c$ = Contact resistance ($\Omega$) - $L_T$ = Transfer length - $R_{sh}$ = Sheet resistance ($\Omega/\square$) - $L$ = Contact length ### 1.3 Scaled Contact Resistance At nanoscale dimensions, total contact resistance includes multiple components: $$ R_{total} = \frac{\rho_c}{A_c} + R_{spreading} + R_{interface} $$ **Spreading resistance** (increasingly important at small scales): $$ R_{spreading} \approx \frac{\rho}{4r} $$ Where $r$ is the circular contact radius. ## 2. Contact Etch Modeling ### 2.1 Aspect Ratio Dependent Etching (ARDE) High-aspect-ratio contact holes suffer from transport limitations. **Knudsen Transport Model:** $$ \frac{ER(AR)}{ER_0} = \frac{1}{1 + \frac{3 \cdot AR}{8} \cdot \frac{1}{K_n}} $$ Where: - $AR$ = Aspect Ratio (depth/width) - $ER_0$ = Reference etch rate - $K_n$ = Knudsen number ### 2.2 Ion-Enhanced Etching Model $$ ER = Y_i \Gamma_i E_i^{1/2} + k_n \Gamma_n \cdot \theta $$ Where: - $Y_i$ = Ion sputtering yield - $\Gamma_i$ = Ion flux (ions/cm²·s) - $\Gamma_n$ = Neutral flux - $E_i$ = Ion energy (eV) - $\theta$ = Surface coverage fraction - $k_n$ = Neutral reaction rate constant ### 2.3 Profile Evolution (Level Set Method) $$ \frac{\partial \phi}{\partial t} + V_n \left|\nabla \phi\right| = 0 $$ Where: - $\phi$ = Level set function - $V_n$ = Local etch rate in normal direction ## 3. Metal Fill Modeling (W, Co, Ru) ### 3.1 CVD Kinetics **Tungsten CVD Reaction:** $$ \text{WF}_6 + 3\text{H}_2 \rightarrow \text{W} + 6\text{HF} $$ **Deposition Rate:** $$ \frac{dh}{dt} = \frac{k_s k_g C_{bulk}}{k_g + k_s} $$ Where: - $h$ = Film thickness - $k_s$ = Surface reaction rate constant - $k_g$ = Mass transport coefficient - $C_{bulk}$ = Bulk precursor concentration ### 3.2 Step Coverage Model $$ SC = \frac{t_{bottom}}{t_{sidewall}} = \frac{1}{1 + \beta \cdot AR} $$ **Sticking coefficient dependence:** $$ \beta = \frac{s}{2(1-s)} $$ Where: - $SC$ = Step coverage ratio - $s$ = Sticking coefficient - $AR$ = Aspect ratio ### 3.3 ALD Super-Conformal Fill **Growth Per Cycle (GPC):** $$ GPC = \frac{\theta \cdot N_{sites} \cdot M}{\rho \cdot N_A} $$ Where: - $\theta$ = Surface site coverage (0 to 1) - $N_{sites}$ = Surface site density (sites/cm²) - $M$ = Molecular weight (g/mol) - $\rho$ = Film density (g/cm³) - $N_A$ = Avogadro's number ## 4. CMP Modeling for MOL ### 4.1 Preston's Equation (Modified) $$ RR = K_p \cdot P \cdot V \cdot f(chemistry) $$ Where: - $RR$ = Removal rate (nm/min) - $K_p$ = Preston coefficient - $P$ = Applied pressure - $V$ = Relative velocity ### 4.2 Pattern-Dependent Model **Effective Pressure Distribution:** $$ P_{eff}(x,y) = \frac{P_{applied}}{\rho_{local}(x,y)} $$ Where $\rho_{local}$ = Local pattern density ### 4.3 Dishing and Erosion **Dishing (metal recessing):** $$ D = K_1 \cdot w^\alpha \cdot t^\beta $$ **Erosion (oxide loss):** $$ E = K_2 \cdot \rho^\gamma \cdot t^\delta $$ Where: - $w$ = Line width - $t$ = Polish time - $\rho$ = Pattern density - $\alpha, \beta, \gamma, \delta$ = Empirical exponents ## 5. Thermal Modeling ### 5.1 Heat Diffusion (Annealing) $$ \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q_{source} $$ Where: - $\rho$ = Material density - $c_p$ = Specific heat capacity - $k$ = Thermal conductivity - $Q_{source}$ = Heat generation rate ### 5.2 Effective Thermal Conductivity (Multilayer) $$ \frac{1}{k_{eff,\perp}} = \sum_i \frac{t_i}{k_i \cdot t_{total}} + \sum_{interfaces} \frac{R_{TBR}}{t_{total}} $$ Where: - $t_i$ = Thickness of layer $i$ - $k_i$ = Thermal conductivity of layer $i$ - $R_{TBR}$ = Thermal boundary resistance ## 6. Stress and Reliability Modeling ### 6.1 Film Stress $$ \sigma = \frac{E_f}{1-\nu_f}(\alpha_s - \alpha_f)\Delta T + \sigma_{intrinsic} + \sigma_{growth} $$ Where: - $E_f$ = Film Young's modulus - $\nu_f$ = Film Poisson's ratio - $\alpha_s, \alpha_f$ = Thermal expansion coefficients (substrate, film) - $\Delta T$ = Temperature change ### 6.2 Electromigration (Black's Equation) $$ MTTF = A \cdot j^{-n} \exp\left(\frac{E_a}{k_B T}\right) $$ Where: - $MTTF$ = Mean Time To Failure - $j$ = Current density (A/cm²) - $n$ = Current exponent ($\approx 1-2$) - $E_a$ = Activation energy ($\approx 0.7-0.9$ eV for W/Co) - $k_B$ = Boltzmann constant - $T$ = Temperature (K) ### 6.3 Stress-Induced Voiding $$ \frac{\partial C}{\partial t} = D\nabla^2 C - \frac{D\Omega}{k_B T}\nabla \cdot (C \nabla \sigma) $$ Where: - $C$ = Vacancy concentration - $D$ = Diffusion coefficient - $\Omega$ = Atomic volume - $\sigma$ = Hydrostatic stress ## 7. RC Delay Modeling ### 7.1 Total RC for MOL Path $$ \tau_{MOL} = (R_c + R_{via} + R_{LI}) \cdot C_{total} $$ **Contact resistance dominance at advanced nodes:** $$ R_c \gg R_{interconnect} \quad \text{(for local paths)} $$ ### 7.2 Capacitance Modeling $$ C = C_{plate} + C_{fringe} + C_{coupling} $$ **For high-AR cylindrical contacts:** $$ C \approx \frac{2\pi \epsilon_0 \epsilon_r h}{\ln(r_2/r_1)} $$ Where: - $h$ = Contact height - $r_1, r_2$ = Inner and outer radii - $\epsilon_r$ = Relative permittivity ## 8. Process Variability Modeling ### 8.1 Statistical Model (Response Surface) $$ CD = CD_{nominal} + \sum_i a_i \Delta P_i + \sum_{i,j} b_{ij} \Delta P_i \Delta P_j + \epsilon $$ Where: - $CD$ = Critical dimension - $\Delta P_i$ = Process parameter deviation - $a_i, b_{ij}$ = Sensitivity coefficients - $\epsilon$ = Random error ### 8.2 Monte Carlo for Variability $$ \sigma_{R_c}^2 = \left(\frac{\partial R_c}{\partial \rho_c}\right)^2 \sigma_{\rho_c}^2 + \left(\frac{\partial R_c}{\partial A}\right)^2 \sigma_A^2 + \ldots $$ ### 8.3 Process Capability Index $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ Where: - $USL$ = Upper specification limit - $LSL$ = Lower specification limit - $\mu$ = Process mean - $\sigma$ = Standard deviation ## 9. TCAD Simulation Framework ### 9.1 Coupled Equations Solved **Poisson's Equation:** $$ \nabla \cdot (\epsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-) $$ **Continuity Equations:** $$ \frac{\partial n}{\partial t} = \frac{1}{q}\nabla \cdot \vec{J_n} + G - R $$ $$ \frac{\partial p}{\partial t} = -\frac{1}{q}\nabla \cdot \vec{J_p} + G - R $$ **Current Equations (Drift-Diffusion):** $$ \vec{J_n} = qn\mu_n \vec{E} + qD_n \nabla n $$ $$ \vec{J_p} = qp\mu_p \vec{E} - qD_p \nabla p $$ Where: - $\psi$ = Electrostatic potential - $n, p$ = Electron and hole concentrations - $N_D^+, N_A^-$ = Ionized donor and acceptor concentrations - $G, R$ = Generation and recombination rates - $\mu_n, \mu_p$ = Electron and hole mobilities - $D_n, D_p$ = Diffusion coefficients ## 10. Machine Learning Integration ### 10.1 Virtual Metrology $$ y_{predicted} = f_{NN}(\vec{x}_{sensors}, \vec{x}_{recipe}) $$ Where $f_{NN}$ represents a trained neural network mapping sensor data and recipe parameters to metrology outputs. ### 10.2 Process Optimization (Gaussian Process) **Gaussian Process Regression:** $$ y(\vec{x}) \sim \mathcal{GP}(m(\vec{x}), k(\vec{x}, \vec{x'})) $$ Where: - $m(\vec{x})$ = Mean function - $k(\vec{x}, \vec{x'})$ = Covariance kernel (e.g., RBF, Matérn) Used for Bayesian optimization of MOL process recipes. ## 11. Lithography and OPC for MOL ### 11.1 Aerial Image Modeling $$ I(x,y) = \left|\sum_n \sum_m c_{nm} P_{nm} \exp\left(i 2\pi \frac{n x + m y}{\lambda/NA}\right)\right|^2 $$ Where: - $c_{nm}$ = Diffraction order coefficients - $P_{nm}$ = Pupil function - $\lambda$ = Wavelength - $NA$ = Numerical aperture ### 11.2 OPC Edge Placement Error Minimization $$ \min \sum_{i} w_i |EPE_i|^2 + \lambda R(mask) $$ Where: - $EPE_i$ = Edge placement error at evaluation point $i$ - $w_i$ = Weight factor - $R(mask)$ = Regularization term - $\lambda$ = Regularization parameter ## 12. Key MOL Mathematical Models | **Parameter** | **Model Type** | **Critical Equation** | |---------------|----------------|----------------------| | Contact Resistance | Physics-based | $R = \rho_c/A + R_{spreading}$ | | Etch Profile | Level-set / MC | $\frac{\partial \phi}{\partial t} + V_n|\nabla \phi| = 0$ | | Metal Fill | Kinetic + Transport | Sticking coefficient models | | CMP | Empirical + Physics | Modified Preston equation | | Reliability | Arrhenius-based | Black's equation | | Variability | Statistical | Monte Carlo + RSM | ## 13. Advanced Considerations (Sub-5nm Nodes) At sub-5nm technology nodes, additional physics must be incorporated: - **Quantum Tunneling Effects**: Direct tunneling through thin barriers - **Interface Scattering**: Grain boundary and surface scattering - **Size-Dependent Resistivity**: Fuchs-Sondheimer model $$ \frac{\rho}{\rho_0} = 1 + \frac{3\lambda}{8t}(1-p) + \frac{3\lambda}{2d}\frac{R}{1-R} $$ Where: - $\lambda$ = Electron mean free path - $t$ = Film thickness - $d$ = Grain size - $p$ = Surface specularity parameter - $R$ = Grain boundary reflection coefficient
Shape of final package.
Tool holding cavities.
Time mold is closed.
Design of molding tool.
Excess compound at parting line.
Time mold is open.
Temperature of tool.
Underfill via transfer molding.
Polymer material for encapsulation.
Total time per package.
Control variables for molding.
Wafer used to check tool performance and cleanliness.
Wafers used to check tool status.
# Semiconductor Manufacturing Monte Carlo Simulation: The Mathematics
## 1. Introduction
### 1.1 Why Monte Carlo for Semiconductors?
Semiconductor manufacturing involves:
- **Nanometer-scale features** (3nm, 5nm nodes) where atomic-scale randomness matters
- **Hundreds of process steps**, each with inherent variability
- **High-dimensional parameter spaces** (100s–1000s of variables)
- **Rare event statistics** (yield prediction for 99%+ target yields)
Classical numerical methods fail due to the **curse of dimensionality**. Monte Carlo's key advantage:
$$
\text{Error} = O\left(\frac{1}{\sqrt{N}}\right) \quad \text{independent of dimensionality}
$$
### 1.2 Key Applications
- **Process variability modeling**: Understanding how variations in lithography, etching, doping affect device parameters
- **Yield prediction**: Estimating what percentage of chips will work
- **Circuit performance analysis**: Predicting speed, power consumption distributions
- **Design for manufacturability (DFM)**: Ensuring designs are robust to process variations
- **Statistical timing analysis**: Understanding timing margins
- **Device physics simulation**: Modeling carrier transport, quantum effects
## 2. Fundamental Monte Carlo Mathematics
### 2.1 Basic Monte Carlo Integration
To estimate an integral:
$$
I = \int_D f(x) \, p(x) \, dx
$$
The **Monte Carlo estimator**:
$$
\hat{I} = \frac{1}{N} \sum_{i=1}^{N} f(x_i), \quad x_i \sim p(x)
$$
**Error bound** (Central Limit Theorem):
$$
\text{Standard Error} = \frac{\sigma}{\sqrt{N}}
$$
where $\sigma$ is the standard deviation of $f(x)$.
### 2.2 Random Number Generation
#### 2.2.1 Linear Congruential Generator
$$
X_{n+1} = (aX_n + c) \mod m
$$
- **Parameters**: multiplier $a$, increment $c$, modulus $m$
- **Period**: at most $m$
#### 2.2.2 Box-Muller Transform (Uniform → Gaussian)
$$
Z_0 = \sqrt{-2 \ln U_1} \cos(2\pi U_2)
$$
$$
Z_1 = \sqrt{-2 \ln U_1} \sin(2\pi U_2)
$$
where:
- $U_1, U_2 \sim \text{Uniform}(0,1)$
- $Z_0, Z_1 \sim \mathcal{N}(0,1)$
#### 2.2.3 Inverse Transform Method
$$
X = F^{-1}(U)
$$
where $F$ is the CDF of the desired distribution and $U \sim \text{Uniform}(0,1)$.
### 2.3 Modern PRNGs for Parallel Computing
- **Mersenne Twister (MT19937)**: Period $2^{19937} - 1$
- **Xorshift**: Fast, good statistical properties
- **PCG (Permuted Congruential Generator)**: Statistically excellent
- **Counter-based (Philox, Threefry)**: Ideal for GPU parallelization
## 3. Process Variation Modeling
### 3.1 Parameter Decomposition
A device parameter $P$ is modeled as:
$$
P = P_{\text{nom}} + \Delta P_{\text{sys}}(x,y) + \Delta P_{\text{global}} + \Delta P_{\text{local}}
$$
where:
- **Systematic variation**:
$$
\Delta P_{\text{sys}}(x,y) = \sum_{i,j} a_{ij} x^i y^j
$$
(spatial polynomial)
- **Global variation** (wafer-to-wafer, lot-to-lot):
$$
\Delta P_{\text{global}} \sim \mathcal{N}(0, \sigma_g^2)
$$
- **Local variation** (device-to-device):
$$
\Delta P_{\text{local}} \sim \mathcal{N}(0, \sigma_l^2)
$$
### 3.2 Spatial Correlation Structure
Local variations often exhibit spatial correlation:
$$
\text{Cov}(\Delta P(\mathbf{r}_1), \Delta P(\mathbf{r}_2)) = \sigma^2 \cdot \rho(|\mathbf{r}_1 - \mathbf{r}_2|)
$$
#### Common Correlation Functions
| Model | Formula | Characteristics |
|-------|---------|-----------------|
| Exponential | $\rho(d) = e^{-d/\lambda}$ | Sharp near-field correlation |
| Gaussian | $\rho(d) = e^{-(d/\lambda)^2}$ | Smoother correlation decay |
| Matérn | $\rho(d) = \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{d}{\lambda}\right)^\nu K_\nu\left(\frac{d}{\lambda}\right)$ | Flexible smoothness parameter $\nu$ |
### 3.3 Generating Correlated Samples
Given covariance matrix $\mathbf{\Sigma}$, use **Cholesky decomposition**:
$$
\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^T
$$
Generate correlated samples:
$$
\mathbf{X} = \boldsymbol{\mu} + \mathbf{L}\mathbf{Z}
$$
where $\mathbf{Z}$ is a vector of independent standard normals.
### 3.4 Pelgrom's Mismatch Law
For transistor mismatch (critical for analog/SRAM):
$$
\sigma(\Delta V_{th}) = \frac{A_{VT}}{\sqrt{WL}}
$$
$$
\sigma\left(\frac{\Delta \beta}{\beta}\right) = \frac{A_\beta}{\sqrt{WL}}
$$
where:
- $A_{VT}$: Threshold voltage mismatch coefficient (typical: 1-5 mV·μm)
- $A_\beta$: Current factor mismatch coefficient (typical: 1-2 %·μm)
- $W$: Gate width
- $L$: Gate length
## 4. Statistical Static Timing Analysis (SSTA)
### 4.1 Gate Delay Model
$$
d = d_0 + \sum_i a_i \Delta P_i + \sum_i \beta_{ii} (\Delta P_i)^2 + \sum_{i
Measure oxide quality and interface.
MSL 1 through 6.
Complete polarization characterization.
Full polarization scatterometry.
Parallel electron beams for faster writing.
Parallel beams for faster mask writing.
Transfer multiple layers sequentially.
Use multiple imaging modes together.
Split design into multiple masks for multi-patterning.
Use multiple litho-etch cycles to achieve finer pitch than single exposure.
Share wafer among multiple customers to reduce prototyping cost.
Share mask costs across designs.
Withstand repeated heating.
Mechanically press pattern into resist.