Optical Proximity Correction (OPC): Mathematical Modeling

1. The Physical Problem

When projecting mask patterns onto a silicon wafer using light (typically 193nm DUV or 13.5nm EUV), several phenomena distort the image:

• Diffraction: Light bending around features near or below the wavelength
• Interference: Constructive/destructive wave interactions
• Optical aberrations: Lens imperfections
• Resist effects: Photochemical behavior during exposure and development
• Etch loading: Pattern-density-dependent etch rates

OPC pre-distorts the mask so that after all these effects, the printed pattern matches the design intent.

Key Parameters

2. Hopkins Imaging Model

The foundational mathematical framework for partially coherent lithographic imaging comes from Hopkins' theory (1953).

Aerial Image Intensity

The aerial image intensity at position $\mathbf{r} = (x, y)$ is given by:

$$ I(\mathbf{r}) = \iiint\!\!\!\iint TCC(\mathbf{f}_1, \mathbf{f}_2) \cdot M(\mathbf{f}_1) \cdot M^*(\mathbf{f}_2) \cdot e^{2\pi i (\mathbf{f}_1 - \mathbf{f}_2) \cdot \mathbf{r}} \, d\mathbf{f}_1 \, d\mathbf{f}_2 $$

Where:

• $M(\mathbf{f})$ — Fourier transform of the mask transmission function
• $M^*(\mathbf{f})$ — Complex conjugate of $M(\mathbf{f})$
• $TCC$ — Transmission Cross Coefficient
• $\mathbf{f} = (f_x, f_y)$ — Spatial frequency coordinates

Transmission Cross Coefficient (TCC)

The TCC encodes the optical system characteristics:

$$ TCC(\mathbf{f}_1, \mathbf{f}_2) = \iint J(\mathbf{f}) \cdot H(\mathbf{f} + \mathbf{f}_1) \cdot H^*(\mathbf{f} + \mathbf{f}_2) \, d\mathbf{f} $$

Where:

• $J(\mathbf{f})$ — Source (illumination) intensity distribution (mutual intensity at mask)
• $H(\mathbf{f})$ — Pupil function of the projection lens
• $H^*(\mathbf{f})$ — Complex conjugate of pupil function

Pupil Function

For an ideal circular aperture:

$$ H(\mathbf{f}) = \begin{cases} 1 & \text{if } |\mathbf{f}| \leq \frac{NA}{\lambda} \\ 0 & \text{otherwise} \end{cases} $$

With aberrations included:

$$ H(\mathbf{f}) = P(\mathbf{f}) \cdot e^{i \cdot W(\mathbf{f})} $$

Where $W(\mathbf{f})$ is the wavefront aberration function (Zernike polynomial expansion).

3. SOCS Decomposition

Sum of Coherent Systems

To make computation tractable, the TCC (a Hermitian matrix when discretized) is decomposed via eigenvalue decomposition:

$$ TCC(\mathbf{f}_1, \mathbf{f}_2) = \sum_{n=1}^{N} \lambda_n \cdot \phi_n(\mathbf{f}_1) \cdot \phi_n^*(\mathbf{f}_2) $$

Where:

• $\lambda_n$ — Eigenvalues (sorted in descending order)
• $\phi_n(\mathbf{f})$ — Eigenvectors (orthonormal kernels)

Image Computation

This allows the image to be computed as a sum of coherent images:

$$ I(\mathbf{r}) = \sum_{n=1}^{N} \lambda_n \left| \mathcal{F}^{-1}\{\phi_n \cdot M\} \right|^2 $$

Or equivalently:

$$ I(\mathbf{r}) = \sum_{n=1}^{N} \lambda_n \left| I_n(\mathbf{r}) \right|^2 $$

Where each coherent image is:

$$ I_n(\mathbf{r}) = \mathcal{F}^{-1}\{\phi_n(\mathbf{f}) \cdot M(\mathbf{f})\} $$

Practical Considerations

• Eigenvalue decay: $\lambda_n$ decay rapidly; typically only 10–50 terms needed
• Speedup: Converts one $O(N^4)$ partially coherent calculation into $\sim$20 $O(N^2 \log N)$ FFT operations
• Accuracy: Trade-off between number of terms and simulation accuracy

4. OPC Problem Formulation

Forward Problem

Given mask $M(\mathbf{r})$, predict wafer pattern $W(\mathbf{r})$:

$$ M \xrightarrow{\text{optics}} I(\mathbf{r}) \xrightarrow{\text{resist}} R(\mathbf{r}) \xrightarrow{\text{etch}} W(\mathbf{r}) $$

Mathematical chain:

1. Optical Model: $I = \mathcal{O}(M)$ — Hopkins/SOCS imaging 2. Resist Model: $R = \mathcal{R}(I)$ — Threshold or convolution model 3. Etch Model: $W = \mathcal{E}(R)$ — Etch bias and loading

Inverse Problem (OPC)

Given target pattern $T(\mathbf{r})$, find mask $M(\mathbf{r})$ such that:

$$ W(M) \approx T $$

This is fundamentally ill-posed:

• Non-unique: Many masks could produce similar results
• Nonlinear: The imaging equation is quadratic in mask transmission
• Constrained: Mask must be manufacturable

5. Edge Placement Error Minimization

Objective Function

The standard OPC objective minimizes Edge Placement Error (EPE):

$$ \min_M \mathcal{L}(M) = \sum_{i=1}^{N_{\text{edges}}} w_i \cdot \text{EPE}_i^2 $$

Where:

$$ \text{EPE}_i = x_i^{\text{printed}} - x_i^{\text{target}} $$

• $x_i^{\text{printed}}$ — Actual edge position after lithography
• $x_i^{\text{target}}$ — Desired edge position from design
• $w_i$ — Weight for edge $i$ (can prioritize critical features)

Constraints

Subject to mask manufacturability:

• Minimum feature size: $\text{CD}_{\text{mask}} \geq \text{CD}_{\min}$
• Minimum spacing: $\text{Space}_{\text{mask}} \geq \text{Space}_{\min}$
• Maximum jog: Limit on edge fragmentation complexity
• MEEF constraint: Mask Error Enhancement Factor within spec

Iterative Edge-Based OPC Algorithm

The classic algorithm moves mask edges iteratively:

$$ \Delta x^{(n+1)} = \Delta x^{(n)} - \alpha \cdot \text{EPE}^{(n)} $$

Where:

• $\Delta x$ — Edge movement from original position
• $\alpha$ — Damping factor (typically 0.3–0.8)
• $n$ — Iteration number

Convergence criterion:

$$ \max_i |\text{EPE}_i| < \epsilon \quad \text{or} \quad n > n_{\max} $$

Gradient Computation

Using the chain rule:

$$ \frac{\partial \text{EPE}}{\partial m} = \frac{\partial \text{EPE}}{\partial I} \cdot \frac{\partial I}{\partial m} $$

Where $m$ represents mask parameters (edge positions, segment lengths).

At a contour position where $I = I_{th}$:

$$ \frac{\partial x_{\text{edge}}}{\partial m} = -\frac{1}{| abla I|} \cdot \frac{\partial I}{\partial m} $$

The image log-slope (ILS) is a key metric:

$$ \text{ILS} = \frac{1}{I} \left| \frac{\partial I}{\partial x} \right|_{I = I_{th}} $$

Higher ILS → better process latitude, lower EPE sensitivity.

6. Resist Modeling

Threshold Model (Simplest)

The resist develops where intensity exceeds threshold:

$$ R(\mathbf{r}) = \begin{cases} 1 & \text{if } I(\mathbf{r}) > I_{th} \\ 0 & \text{otherwise} \end{cases} $$

The printed contour is the $I_{th}$ isoline.

Variable Threshold Resist (VTR)

The threshold varies with local context:

$$ I_{th}(\mathbf{r}) = I_{th,0} + \beta_1 \cdot \bar{I}_{\text{local}} + \beta_2 \cdot abla^2 I + \beta_3 \cdot ( abla I)^2 + \ldots $$

Where:

• $I_{th,0}$ — Base threshold
• $\bar{I}_{\text{local}}$ — Local average intensity (density effect)
• $

abla^2 I$ — Laplacian (curvature effect)

• $\beta_i$ — Fitted coefficients

Compact Phenomenological Models

For OPC speed, empirical models are used instead of physics-based resist simulation:

$$ R(\mathbf{r}) = \sum_{j=1}^{N_k} w_j \cdot \left( K_j \otimes g_j(I) \right) $$

Where:

• $K_j$ — Convolution kernels (typically Gaussians):

$$K_j(\mathbf{r}) = \frac{1}{2\pi\sigma_j^2} \exp\left( -\frac{|\mathbf{r}|^2}{2\sigma_j^2} \right)$$

• $g_j(I)$ — Nonlinear functions: $I$, $I^2$, $\log(I)$, $\sqrt{I}$, etc.
• $w_j$ — Fitted weights
• $\otimes$ — Convolution operator

Physical Interpretation

Model Calibration

Parameters are fitted to wafer measurements:

$$ \min_{\theta} \sum_{k=1}^{N_{\text{test}}} \left( \text{CD}_k^{\text{measured}} - \text{CD}_k^{\text{model}}(\theta) \right)^2 + \lambda \|\theta\|^2 $$

Where:

• $\theta = \{w_j, \sigma_j, \beta_i, \ldots\}$ — Model parameters
• $\lambda \|\theta\|^2$ — Regularization term
• Test structures: Lines, spaces, contacts, line-ends at various pitches/densities

7. Inverse Lithography Technology

Full Optimization Formulation

ILT treats the mask as a continuous optimization variable (pixelated):

$$ \min_{M} \mathcal{L}(M) = \| W(M) - T \|^2 + \lambda \cdot \mathcal{R}(M) $$

Where:

• $W(M)$ — Predicted wafer pattern
• $T$ — Target pattern
• $\mathcal{R}(M)$ — Regularization for manufacturability
• $\lambda$ — Regularization weight

Cost Function Components

Pattern Fidelity Term:

$$ \mathcal{L}_{\text{fidelity}} = \int \left( W(\mathbf{r}) - T(\mathbf{r}) \right)^2 d\mathbf{r} $$

Or in discrete form:

$$ \mathcal{L}_{\text{fidelity}} = \sum_{\mathbf{r} \in \text{grid}} \left( W(\mathbf{r}) - T(\mathbf{r}) \right)^2 $$

Regularization Terms

Total Variation (promotes piecewise constant, sharp edges):

$$ \mathcal{R}_{TV}(M) = \int | abla M| \, d\mathbf{r} = \int \sqrt{\left(\frac{\partial M}{\partial x}\right)^2 + \left(\frac{\partial M}{\partial y}\right)^2} \, d\mathbf{r} $$

Curvature Penalty (promotes smooth contours):

$$ \mathcal{R}_{\kappa}(M) = \oint_{\partial M} \kappa^2 \, ds $$

Where $\kappa$ is the local curvature of the mask boundary.

Minimum Feature Size (MRC - Mask Rule Check):

$$ \mathcal{R}_{MRC}(M) = \sum_{\text{violations}} \text{penalty}(\text{violation severity}) $$

Sigmoid Regularization (push mask toward binary):

$$ \mathcal{R}_{\text{binary}}(M) = \int M(1-M) \, d\mathbf{r} $$

Level Set Formulation

Represent the mask boundary implicitly via level set function $\phi(\mathbf{r})$:

• Inside chrome: $\phi(\mathbf{r}) < 0$
• Outside chrome: $\phi(\mathbf{r}) > 0$
• Boundary: $\phi(\mathbf{r}) = 0$

Evolution equation:

$$ \frac{\partial \phi}{\partial t} = -v \cdot | abla \phi| $$

Where velocity $v$ is derived from the cost function gradient:

$$ v = -\frac{\delta \mathcal{L}}{\delta \phi} $$

Advantages:

• Naturally handles topological changes (features splitting/merging)
• Implicit curvature regularization available
• Well-studied numerical methods

Optimization Algorithms

Since the problem is non-convex, various methods are used:

1. Gradient Descent with Momentum:

$$ M^{(n+1)} = M^{(n)} - \eta abla_M \mathcal{L} + \mu \left( M^{(n)} - M^{(n-1)} \right) $$

2. Conjugate Gradient:

$$ d^{(n+1)} = - abla \mathcal{L}^{(n+1)} + \beta^{(n)} d^{(n)} $$

3. Adam Optimizer:

$$ m_t = \beta_1 m_{t-1} + (1-\beta_1) g_t $$ $$ v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2 $$ $$ M_{t+1} = M_t - \eta \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} $$

4. Genetic Algorithms (for discrete/combinatorial aspects)

5. Simulated Annealing (for escaping local minima)

8. Source-Mask Optimization

Joint Optimization

SMO optimizes both illumination source $S$ and mask $M$ simultaneously:

$$ \min_{S, M} \sum_{j \in \text{PW}} w_j \cdot \| W(S, M, \text{condition}_j) - T \|^2 $$

Source Parameterization

Pixelated Source:

$$ S = \{s_{ij}\} \quad \text{where } s_{ij} \in [0, 1] $$

Each pixel in the pupil plane is a free variable.

Parametric Source:

• Annular: $(R_{\text{inner}}, R_{\text{outer}})$
• Quadrupole: $(R, \theta, \sigma)$
• Freeform: Spline or Zernike coefficients

Alternating Optimization

Algorithm:

Initialize: S⁰, M⁰
for k = 1 to max_iter:
    # Step 1: Fix S, optimize M (standard OPC)
    M^k = argmin_M L(S^(k-1), M)
    
    # Step 2: Fix M, optimize S
    S^k = argmin_S L(S, M^k)
    
    # Check convergence
    if |L^k - L^(k-1)| < tolerance:
        break

Note: Step 2 is often convex in $S$ when $M$ is fixed (linear in source pixels for intensity-based metrics).

Mathematical Form for Source Optimization

When mask is fixed, the image is linear in source:

$$ I(\mathbf{r}; S) = \sum_{ij} s_{ij} \cdot I_{ij}(\mathbf{r}) $$

Where $I_{ij}$ is the image contribution from source pixel $(i,j)$.

This makes source optimization a quadratic program (convex if cost is convex in $I$).

9. Process Window Optimization

Multi-Condition Optimization

Real manufacturing has variations. Robust OPC optimizes across a process window (PW):

$$ \min_M \sum_{j \in \text{PW}} w_j \cdot \mathcal{L}(M, \text{condition}_j) $$

Process Window Dimensions

Worst-Case (Minimax) Formulation

$$ \min_M \max_{j \in \text{PW}} \text{EPE}_j(M) $$

This is more conservative but ensures robustness.

Soft Constraints via Barrier Functions

$$ \mathcal{L}_{PW}(M) = \sum_j w_j \cdot \text{EPE}_j^2 + \mu \sum_j \sum_i \max(0, |\text{EPE}_{ij}| - \text{spec})^2 $$

Process Window Metrics

Common Process Window (CPW):

$$ \text{CPW} = \text{Focus Range} \times \text{Dose Range} $$

Where all specs are simultaneously met.

Exposure Latitude (EL):

$$ \text{EL} = \frac{\Delta \text{Dose}}{\text{Dose}_{\text{nom}}} \times 100\% $$

Depth of Focus (DOF):

$$ \text{DOF} = \text{Focus range where } |\text{EPE}| < \text{spec} $$

10. Stochastic Effects (EUV)

At EUV wavelengths (13.5 nm), photon counts are low and shot noise becomes significant.

Photon Statistics

Number of photons per pixel follows Poisson distribution:

$$ P(n | \bar{n}) = \frac{\bar{n}^n e^{-\bar{n}}}{n!} $$

Where:

$$ \bar{n} = \frac{E \cdot A \cdot \eta}{\frac{hc}{\lambda}} $$

• $E$ — Exposure dose (mJ/cm²)
• $A$ — Pixel area
• $\eta$ — Quantum efficiency
• $\frac{hc}{\lambda}$ — Photon energy

Signal-to-Noise Ratio

$$ \text{SNR} = \frac{\bar{n}}{\sqrt{\bar{n}}} = \sqrt{\bar{n}} $$

For reliable imaging, need $\text{SNR} > 5$, requiring $\bar{n} > 25$ photons/pixel.

Line Edge Roughness (LER)

Random edge fluctuations characterized by:

• 3σ LER: $3 \times \text{standard deviation of edge position}$
• Correlation length $\xi$: Spatial extent of roughness

Power Spectral Density:

$$ \text{PSD}(f) = \frac{2\sigma^2 \xi}{1 + (2\pi f \xi)^{2\alpha}} $$

Where $\alpha$ is the roughness exponent (typically 0.5–1.0).

Stochastic Defect Probability

Probability of a stochastic failure (missing contact, bridging):

$$ P_{\text{fail}} = 1 - \prod_{\text{features}} (1 - p_i) $$

For rare events, approximately:

$$ P_{\text{fail}} \approx \sum_i p_i $$

Stochastic-Aware OPC Objective

$$ \min_M \mathbb{E}[\text{EPE}^2] + \lambda_1 \cdot \text{Var}(\text{EPE}) + \lambda_2 \cdot P_{\text{fail}} $$

Monte Carlo Simulation

For stochastic modeling:

1. Sample photon arrival: $n_{ij} \sim \text{Poisson}(\bar{n}_{ij})$ 2. Simulate acid generation: Proportional to absorbed photons 3. Simulate diffusion: Random walk or stochastic PDE 4. Simulate development: Threshold with noise 5. Repeat $N$ times, compute statistics

11. Machine Learning Approaches

Neural Network Forward Models

Train networks to approximate expensive simulations:

$$ \hat{I} = f_\theta(M) \approx I_{\text{optical}}(M) $$

Architectures:

• CNN: Convolutional neural networks for local pattern effects
• U-Net: Encoder-decoder for image-to-image translation
• GAN: Generative adversarial networks for realistic image generation

Training:

$$ \min_\theta \sum_{k} \| f_\theta(M_k) - I_k^{\text{simulation}} \|^2 $$

End-to-End ILT with Deep Learning

Directly predict corrected masks:

$$ \hat{M}_{\text{OPC}} = G_\theta(T) $$

Training data: Pairs $(T, M_{\text{optimal}})$ from conventional ILT.

Loss function:

$$ \mathcal{L} = \| W(G_\theta(T)) - T \|^2 + \lambda \| G_\theta(T) - M_{\text{ref}} \|^2 $$

Hybrid Approaches

Combine ML speed with physics accuracy:

1. ML Initialization: $M^{(0)} = G_\theta(T)$ 2. Physics Refinement: Run conventional OPC starting from $M^{(0)}$

Benefits:

• Faster convergence (good starting point)
• Physics ensures accuracy
• ML handles global pattern context

Neural Network Architectures for OPC

12. Computational Complexity

Scale of Full-Chip OPC

• Features per chip: $10^9 - 10^{10}$
• Evaluation points: $\sim 10^{12}$ (multiple points per feature)
• Iterations: 10–50 per feature
• Optical simulations: $O(N \log N)$ per FFT

Complexity Analysis

Single feature OPC:

$$ T_{\text{feature}} = O(N_{\text{iter}} \times N_{\text{SOCS}} \times N_{\text{grid}} \log N_{\text{grid}}) $$

Full chip:

$$ T_{\text{chip}} = O(N_{\text{features}} \times T_{\text{feature}}) $$

Result: Hours to days on large compute clusters.

Acceleration Strategies

Hierarchical Processing:

• Identify repeated cells (memory arrays, standard cells)
• Compute OPC once, reuse for identical instances
• Speedup: $10\times - 100\times$ for regular designs

GPU Parallelization:

• FFTs parallelize well on GPUs
• Convolutions map to tensor operations
• Multiple features processed simultaneously
• Speedup: $10\times - 50\times$

Approximate Models:

• Kernel-based: Pre-compute influence functions
• Variable resolution: Fine grid only near edges
• Neural surrogates: Replace simulation with inference

Domain Decomposition:

• Divide chip into tiles
• Process tiles in parallel
• Handle tile boundaries with overlap or iteration

13. Mathematical Toolkit Summary

Equations Quick Reference

Hopkins Imaging

$$ I(\mathbf{r}) = \iiint\!\!\!\iint TCC(\mathbf{f}_1, \mathbf{f}_2) \cdot M(\mathbf{f}_1) \cdot M^*(\mathbf{f}_2) \cdot e^{2\pi i (\mathbf{f}_1 - \mathbf{f}_2) \cdot \mathbf{r}} \, d\mathbf{f}_1 \, d\mathbf{f}_2 $$

SOCS Image

$$ I(\mathbf{r}) = \sum_{n=1}^{N} \lambda_n \left| \mathcal{F}^{-1}\{\phi_n \cdot M\} \right|^2 $$

EPE Minimization

$$ \min_M \sum_{i} w_i \left( x_i^{\text{printed}} - x_i^{\text{target}} \right)^2 $$

ILT Cost Function

$$ \min_{M} \| W(M) - T \|^2 + \lambda \cdot \mathcal{R}(M) $$

Level Set Evolution

$$ \frac{\partial \phi}{\partial t} = -v \cdot | abla \phi| $$

Poisson Photon Statistics

$$ P(n | \bar{n}) = \frac{\bar{n}^n e^{-\bar{n}}}{n!} $$