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:

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

2.1.2 Rigorous Coupled Wave Analysis (RCWA)

Field Expansion in Fourier Series:

The electric field in layer $j$ with grating vector $\mathbf{K}$:

where the diffraction wave vectors are:

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

Regularization Options:

• Tikhonov regularization:

• Sparsity-promoting (L1):

• Total variation:

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

2.3.2 Cramér-Rao Lower Bound

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:

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:

where the phase thickness is:

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)

3.3.2 Sellmeier Equation

3.3.3 Tauc-Lorentz Model (Amorphous Semiconductors)

with $\varepsilon_1$ derived via Kramers-Kronig relations:

3.3.4 Drude Model (Metals/Conductors)

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:

where $r_{j,j+1}$ is the Fresnel coefficient at interface $j$.

4.1.2 Roughness Correction (Névot-Croce Factor)

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:

4.2.2 Form Factor for Simple Shapes

Rectangular parallelepiped:

Cylinder:

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:

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:

5.1.2 Gaussian Process Regression

Common Kernel Functions:

• Squared Exponential (RBF):

• Matérn 5/2:

5.2 Run-to-Run Control

5.2.1 EWMA Controller

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)

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

First few Zernike polynomials:

5.3.2 Gaussian Random Fields

For spatially correlated residuals:

Common correlation functions:

• Exponential:

• Gaussian:

6. Overlay Metrology Mathematics

6.1 Higher-Order Correction Models

Overlay error as polynomial expansion:

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

Minimizes the volume of the confidence ellipsoid for parameter estimates.

6.2.2 Information-Theoretic Approach

Maximize expected information gain:

7. Machine Learning Integration

7.1 Physics-Informed Neural Networks (PINNs)

Combine data fitting with physical constraints:

Components:

• Data loss:

• Physics loss (example: Maxwell residual):

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:

• Instance segmentation — Precise defect boundary delineation

8. Uncertainty Quantification

8.1 GUM Framework (Guide to Uncertainty in Measurement)

Combined standard uncertainty:

8.2 Total Measurement Uncertainty (TMU)

Components:

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

8.3 Bayesian Approaches

8.3.1 Posterior Inference

8.3.2 Sampling Methods

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

• Variational Inference:

9. Emerging Mathematical Challenges

9.1 Compressed Sensing for Spectroscopic Metrology

Restricted Isometry Property (RIP):

9.2 Hybrid Metrology Data Fusion

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

Weighted combination (Gaussian case):

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

A.2 Ellipsometry

A.3 XRR Parratt

A.4 Fisher Information

A.5 Gaussian Process

A.6 EWMA Control

A.7 Bayesian Posterior

Notation Reference

Source: ChipFoundryServices — Search this topic — Ask CFSGPT