MorphOPC: Advancing Mask Optimization with Multi-scale Hierarchical Morphological Learning

Technical Summary of MorphOPC: Advancing Mask Optimization with Multi-scale Hierarchical Morphological Learning

Problem Statement
As semiconductor feature sizes shrink to the nanometer scale, accurately transferring circuit patterns from photomasks to silicon wafers becomes increasingly difficult due to optical diffraction and process variations. Optical Proximity Correction (OPC) is the standard technique used to compensate for these distortions. While traditional model-based OPC and Inverse Lithography Techniques (ILT) offer high accuracy, they are computationally prohibitive for full-chip applications due to the need for repeated forward-backward simulations and iterative optimization loops.

Recent machine learning (ML) approaches, such as GAN-OPC, DAMO, and Neural-ILT, have emerged as fast surrogates to accelerate mask generation. However, these generative models typically rely on statistical learning of the layout-to-mask mapping. They often fail to capture the explicit geometric reasoning required for accurate mask optimization. For instance, specific geometric configurations (e.g., tip-to-tip patterns requiring hammerhead extensions, or L-shaped corners needing serif corrections) demand analytical understanding of local geometric context rather than just statistical pattern matching. Existing ML models often produce suboptimal masks because they lack an interpretable framework for modeling these shape-dependent transformations.

Methodology
To address the limitations of purely statistical generative models, the authors propose MorphOPC, a novel approach that formulates mask generation as a sequence of learned morphological operations.

• Morphological Abstraction: The core insight is that OPC corrections can be viewed as displacing edge segments inward or outward, which corresponds directly to mathematical morphology operations: erosion (shrinking features) and dilation (expanding features). Instead of manually designing structuring elements (SEs), MorphOPC learns adaptive, non-flat SEs and the coefficients for these operations directly from convolutional feature maps.
• Neural Morphological Modules: The architecture introduces a MorphBasic block that replaces standard convolutional aggregation with learnable max-plus and min-plus algebraic operations. These blocks utilize per-channel learnable structuring surfaces ($w_c$) and bias terms ($\beta_c$) to perform dilation and erosion. A learnable gating mechanism dynamically balances the contribution of dilation and erosion for each channel.
• Multi-scale Hierarchical Learning: Recognizing that lithography requires modeling both local geometric primitives (edges, corners) and non-local pattern interactions, the authors introduce a MultiScaleMorph module. This module splits feature channels into groups ($s$) and processes them through a hierarchical residual framework. Each group employs morphological blocks with distinct kernel sizes (e.g., 3x3 and 5x5), allowing the network to capture features with varying receptive field sizes.
• Architecture: MorphOPC integrates these modules into a standard encoder-decoder structure. The encoder extracts hierarchical features, which are refined by the multi-scale morphological modules. The decoder reconstructs the high-resolution mask, fusing the geometrically refined features via skip connections.
• Training Strategy: The model employs a two-stage training process. First, it is pre-trained to minimize the Mean Squared Error (MSE) between the predicted mask and the ground-truth mask. Subsequently, it is fine-tuned using a GAN-style paradigm that incorporates a lithography simulator. The generator is optimized using a composite loss that includes mask reconstruction error, printing error (difference between simulated wafer and target), and an adversarial objective to enhance realism.

Key Contributions

1. Formulation of Mask Generation as Morphology: The paper formulates the OPC problem as a composition of learned dilation and erosion operations applied to local layout patterns, providing an interpretable framework for geometric transformations.
2. MorphOPC Architecture: The authors propose a multi-scale hierarchical architecture that integrates differentiable morphological modules into an encoder-decoder framework, enabling the network to learn adaptive structuring elements and context-dependent operation combinations.
3. Performance and Generalization: Extensive experiments demonstrate that MorphOPC achieves state-of-the-art (SOTA) performance across multiple benchmarks (GAN-OPC, LithoBench, MaskOpt) and exhibits superior generalization to unseen, complex design patterns compared to existing learning-based OPC methods.

Experimental Results
The authors evaluated MorphOPC against SOTA learning-based baselines (GAN-OPC, DAMO, Neural-ILT, CFNO) and optimization-based methods (MultiILT, CurvyILT) across several datasets:

• Benchmark Performance: On the GAN-OPC, LithoBench (MetalSet and ViaSet), and MaskOpt datasets, MorphOPC consistently achieved the lowest $\ell_2$ error (printing fidelity) and competitive Edge Placement Error (EPE) and Process Variation Band (PVB). For example, on the MaskOpt EBOPC task, MorphOPC reduced $\ell_2$ error by 18.7% compared to GAN-OPC and 8.5% compared to DAMO.
• Generalization: In tests on the ICCAD 2013 benchmark (unseen layouts), MorphOPC demonstrated robust generalization, achieving the lowest $\ell_2$ error and Shot count while maintaining competitive PVB, outperforming baselines that often suffered from high error rates on complex patterns.
• Ablation Studies: Removing the morphological modules or the multi-scale hierarchy resulted in significant performance degradation, confirming the necessity of both geometric reasoning and multi-scale feature extraction.
• Efficiency: While maintaining superior mask quality, MorphOPC retains the speed advantage of ML methods, operating in the order of $10^{-2}$ to $10^{-1}$ seconds, significantly faster than optimization-based OPCs (which take seconds). Furthermore, when used to initialize optimization-based OPCs, MorphOPC provided the most effective starting point, yielding the best final printing quality.

Significance
The paper claims that MorphOPC represents a significant advancement by bridging the gap between the speed of machine learning and the geometric rigor required for accurate mask optimization. By embedding differentiable morphological operations that align with fundamental lithographic behaviors (expansion and contraction), the model ensures that mask optimization is not only computationally efficient but also explainable. The results suggest that MorphOPC has strong potential as a scalable, high-fidelity surrogate for traditional OPC methods in large-scale industrial design flows, effectively balancing printing fidelity with manufacturability (low shot count).