OPC Model Parameters and OPC Methodology

This document summarizes the model-based OPC framework, including its mathematical foundations and a clear comparison between Rule-based OPC and Model-based OPC, from a practical semiconductor manufacturing perspective.

1. OPC Model Parameters and Practical Usage

1.1 Overall OPC Modeling Structure

OPC models the transfer of mask geometry to wafer patterns using a combination of optical imaging, photoresist response, and process effects.

$$ \text{Mask} \;\xrightarrow{\text{Optics}}\; \text{Aerial Image} \;\xrightarrow{\text{Resist}}\; \text{Wafer Contour} $$

1.2 Optical Model – Hopkins Formulation

Under partially coherent illumination, the aerial image intensity on the wafer is described by the Hopkins equation:

$$ I(x,y) = \iint TCC(u_1,v_1,u_2,v_2) M(u_1,v_1) M^*(u_2,v_2) e^{i2\pi[(u_1-u_2)x+(v_1-v_2)y]} du_1 dv_1 du_2 dv_2 $$

M(u,v): Fourier spectrum of the mask
TCC: Transmission Cross Coefficient (NA, σ, illumination, aberrations)

In practical OPC tools, the TCC is decomposed using SVD,
allowing the Hopkins formulation to be approximated as a sum of
coherent imaging systems to reduce computational cost.

1.3 Resist Model – Mack Model

(1) Acid diffusion blur

$$ E_{\text{eff}} = E * G, \quad G(r)=\frac{1}{2\pi\sigma^2}e^{-r^2/2\sigma^2} $$

(2) Nonlinear resist response

$$ R(E_{\text{eff}}) = \frac{1}{1 + \left(\frac{E_0}{E_{\text{eff}}}\right)^\gamma} $$

E₀: Dose-to-clear threshold
γ: Resist contrast

In OPC, the wafer pattern edge is defined as the contour satisfying R = R_th.

1.4 OPC Model Calibration Flow

1. Test mask fabrication
2. Wafer exposure
3. CD-SEM measurement
4. Simulation vs. measurement comparison
5. Parameter optimization
6. Error minimization

2. Rule-based OPC vs. Model-based OPC

2.1 Conceptual Comparison

Category	Rule-based OPC	Model-based OPC
Foundation	Empirical rules	Physics-based models
Prediction capability	None	Wafer contour prediction
2D pattern handling	Limited	Strong
Computation cost	Low	High

2.2 Rule-based OPC

Lookup-table driven geometric corrections
Biasing, hammerheads, and serif insertion
Fast and robust for simple, repetitive patterns
Limited adaptability to process changes

2.3 Model-based OPC

The objective of model-based OPC is to minimize the wafer contour error:

$$ \min_M \| C_{\text{wafer}}(M) - C_{\text{target}} \|^2 $$

Inverse lithography problem
High accuracy for advanced nodes
Requires well-calibrated lithography models

2.4 Industry Standard: Hybrid OPC

Rule-based OPC  →  Coarse correction
Model-based OPC →  Fine tuning
Verification    →  Sign-off

In production environments, Rule-based and Model-based OPC are always combined
to achieve both high throughput and high accuracy.

3. Key Takeaways

Hopkins formulation models optical image formation
Mack model captures nonlinear photoresist behavior
Model-based OPC provides high accuracy through simulation
Rule-based OPC enables fast, empirical corrections

Modern OPC is inherently hybrid.

4. Inverse Lithography Technology (ILT)

4.1 What is ILT?

Inverse Lithography Technology (ILT) is an advanced computational lithography technique that formulates mask synthesis as a global optimization problem, rather than applying local geometric corrections.

While traditional OPC modifies mask shapes heuristically, ILT directly computes the optimal mask pattern that produces the desired wafer image.

OPC asks: “How should I tweak this edge?”

ILT asks: “What mask gives me the best wafer result?”

4.2 Mathematical Formulation of ILT

ILT solves an inverse problem by minimizing a cost function:

$$ \min_M \Big( \| C_{\text{wafer}}(M) - C_{\text{target}} \|^2 + \lambda \cdot R(M) \Big) $$

M: mask representation (often pixel-based)
C_wafer: simulated wafer contour
C_target: target wafer pattern
R(M): regularization term
λ: trade-off parameter

The regularization term enforces manufacturability, such as:

Minimum feature size
Mask complexity constraints
SRAF suppression

4.3 ILT vs. Model-based OPC (Key Difference)

Aspect	Model-based OPC	ILT
Optimization scope	Local (edge-based)	Global (mask-level)
Mask representation	Polygon edges	Pixel / level-set
Solution type	Heuristic iterative	Mathematical optimum
Mask complexity	Controlled	Very high

4.4 Typical ILT Workflow

1. Target wafer pattern definition
2. Initial mask guess (often binary)
3. Lithography simulation (Hopkins + resist)
4. Gradient-based optimization
5. Mask regularization
6. Verification and manufacturability check

4.5 Why ILT Produces Better Results

Simultaneous optimization of all features
Superior CD uniformity
Improved process window
Better handling of complex 2D patterns

4.6 Why ILT Is Not Fully Deployed in Production

Extremely high computational cost
Highly complex mask shapes
Mask write time explosion
Inspection and repair challenges

As a result, ILT is often applied selectively
(e.g., critical layers or hotspots),
rather than full-chip deployment.

4.7 Industry Practice: Hybrid OPC + ILT

Rule-based OPC     → Initial bias and simplification
Model-based OPC    → Fine contour correction
ILT                → Hotspot / critical region optimization
Verification       → Sign-off

4.8 Interview-ready Summary Statement

“Inverse Lithography Technology formulates mask synthesis as a global optimization problem using physics-based lithography models. While it offers superior pattern fidelity and process window, its computational and manufacturing costs limit its use to selected critical regions in production.”

5. Final Takeaway

Rule-based OPC: Fast, empirical, local
Model-based OPC: Accurate, physics-driven
ILT: Global optimization, highest fidelity

Modern computational lithography uses a hierarchical combination of all three approaches.

↩ Go to OPC / RET Technical Notes Index
↩ Go to Home – Tech Notes Park