Semiconductor Manufacturing Process Metrology: Mathematical Modeling

Keywords: metrology, scatterometry, ellipsometry, x-ray reflectometry, inverse problems, optimization, statistical inference, mathematical modeling

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

$$

abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$

$$

abla \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:

$$ R(\mathbf{p}) = \int | abla \mathbf{p}| \, d\mathbf{x} $$

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:

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):

$$ \mathcal{L}_{\text{physics}} = \frac{1}{M} \sum_{j=1}^{M} \left\| abla \times \mathbf{E}_\theta - i\omega\mu\mathbf{H}_\theta \right\|^2 $$

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

8.2 Total Measurement Uncertainty (TMU)

$$ TMU^2 = TMU_{\text{precision}}^2 + TMU_{\text{accuracy}}^2 + TMU_{\text{sample}}^2 $$

Components:

• Precision: Repeatability and reproducibility
• Accuracy: Systematic offset from reference
• Sample: Variation in test structures

8.3 Bayesian Approaches

8.3.1 Posterior Inference

$$ P(\mathbf{p}|\mathbf{y}) \propto P(\mathbf{y}|\mathbf{p}) \cdot P(\mathbf{p}) $$

8.3.2 Sampling Methods

• Markov Chain Monte Carlo (MCMC):
• Metropolis-Hastings
• Hamiltonian Monte Carlo
• No-U-Turn Sampler (NUTS)

• Variational Inference:

$$ q^*(\mathbf{p}) = \arg\min_{q \in \mathcal{Q}} D_{KL}\left(q(\mathbf{p}) \| P(\mathbf{p}|\mathbf{y})\right) $$

9. Emerging Mathematical Challenges

9.1 Compressed Sensing for Spectroscopic Metrology

$$ \min_{\mathbf{x}} \|\mathbf{x}\|_1 \quad \text{subject to} \quad \mathbf{A}\mathbf{x} = \mathbf{y} $$

Restricted Isometry Property (RIP):

$$ (1-\delta_s)\|\mathbf{x}\|_2^2 \leq \|\mathbf{A}\mathbf{x}\|_2^2 \leq (1+\delta_s)\|\mathbf{x}\|_2^2 $$

9.2 Hybrid Metrology Data Fusion

Combine multiple measurement techniques (OCD + SEM + AFM):

$$ P(\mathbf{p}|\mathbf{y}_1, \mathbf{y}_2, \ldots) \propto P(\mathbf{p}) \prod_i P(\mathbf{y}_i|\mathbf{p}) $$

Weighted combination (Gaussian case):

$$ \hat{\mathbf{p}}_{\text{fused}} = \left(\sum_i \mathbf{\Sigma}_i^{-1}\right)^{-1} \sum_i \mathbf{\Sigma}_i^{-1} \hat{\mathbf{p}}_i $$

10. Summary: The Mathematical Ecosystem

-
                    ┌─────────────────────────────────────┐
                    │     PHYSICAL FORWARD MODELS         │
                    │  Maxwell, Schrödinger, Monte Carlo  │
                    └──────────────────┬──────────────────┘
                                       │
            ┌──────────────────────────┼──────────────────────────┐
            │                          │                          │
            ▼                          ▼                          ▼
    ┌───────────────┐        ┌─────────────────┐        ┌─────────────────┐
    │   INVERSE     │        │   STATISTICAL   │        │    MACHINE      │
    │   PROBLEMS    │        │   INFERENCE     │        │    LEARNING     │
    │               │        │                 │        │                 │
    │ Optimization  │        │ Bayesian UQ     │        │ Neural networks │
    │ Regulartic    │        │ MCMC            │        │ Surrogates      │
    │ Library search│        │ Information     │        │ PINNs           │
    └───────────────┘        └─────────────────┘        └─────────────────┘
            │                          │                          │
            └──────────────────────────┼──────────────────────────┘
                                       │
                    ┌──────────────────┴──────────────────┐
                    │        PROCESS CONTROL              │
                    │   Run-to-run, APC, SPC, VM          │
                    └─────────────────────────────────────┘

Key Equations

A.1 Inverse Problem

$$\hat{\mathbf{p}} = \arg\min_{\mathbf{p}} \left\| \mathbf{y} - \mathcal{F}(\mathbf{p}) \right\|^2 + \lambda R(\mathbf{p})$$

A.2 Ellipsometry

$$\rho = \tan(\Psi)e^{i\Delta}$$

A.3 XRR Parratt

$$R_j = \frac{r_{j,j+1} + R_{j+1}e^{2ik_{z,j+1}d_{j+1}}}{1 + r_{j,j+1}R_{j+1}e^{2ik_{z,j+1}d_{j+1}}}$$

A.4 Fisher Information

$$[\mathbf{I}]_{ij} = \mathbb{E}\left[\frac{\partial \ln L}{\partial p_i}\frac{\partial \ln L}{\partial p_j}\right]$$

A.5 Gaussian Process

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

A.6 EWMA Control

$$\hat{d}_t = \lambda y_{t-1} + (1-\lambda)\hat{d}_{t-1}$$

A.7 Bayesian Posterior

$$P(\mathbf{p}|\mathbf{y}) \propto P(\mathbf{y}|\mathbf{p}) \cdot P(\mathbf{p})$$

Notation Reference

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