Computational Scaling Research Articles

Block-based inverse lithography technology with adaptive level-set algorithm

Inverse lithography technology (ILT) is a key computational lithography approach aimed at inversely optimizing the photomask pattern to compensate for the image distortion in advanced optical lithography process. Traditional ILT algorithms, despite their capacity of significantly enhancing the image quality, bring challenges to the computational efficiency and mask manufacturability. To overcome those problems, this paper proposes a novel block-based ILT method driven by the level-set algorithm. This method leverages overlapped basis blocks with a level-set support area for mask representation, thus reducing the mask complexity. To circumvent the slow convergence rate dictated by the conventional Euler time step of the Courant-Friedrichs-Lewy condition, this research adopts the Barzilai-Borwein algorithm to update level set function using adaptive time step, which accelerates the optimization process. In addition, a testbed of digital lithography system is established to verify the proposed ILT method with a calibrated imaging model. It shows that the proposed method is superior over the widely-used and state-of-the-art ILT methods in terms of convergence speed and mask manufacturability.

Efficient curvilinear optical proximity correction using non-uniform B-spline curves.

The curvilinear mask has received much attention in recent years due to its better lithography imaging fidelity than the Manhattan mask. As a significant part of computational lithography techniques, the curvilinear OPC optimally designs the mask contour represented by parametric curves to generate a curvilinear mask structure. However, the current curvilinear OPC process is computationally intensive and contains redundant data. In this paper, a curvilinear OPC method using the non-uniform B-spline curve, together with a knot removal process, is proposed to improve the optimization efficiency and reduce the mask file size. The non-uniform B-spline curve is used to characterize curvilinear mask structure without a complex splicing process, which can effectively reduce the computation complexity. To our best knowledge, knot removal theory is for the first time applied to solve the redundant data problem in curvilinear OPC. Simulations and comparisons verify the superior optimization efficiency and data reduction (DRON) rate of the proposed method.

Plasmonic lithography fast imaging model based on least square fitting for periodic patterns

As a new and alternative lithography technology, plasmonic lithography can break through the diffraction limit of traditional lithography by exciting the surface plasmon polaritons to make the evanescent wave at the mask participate in imaging. Plasmonic lithography is capable of fabricating deep subwavelength structures for nanophotonics, metasurfaces, and various other applications, and it is expected to be applied to integrated circuit manufacturing. The photoresist aerial image distribution of different mask patterns can be calculated by establishing an imaging model, which is the basis for understanding and further optimizing imaging. Based on the idea of machine learning and least square fitting, a fast imaging model for plasmonic lithography is established, including a one-dimensional line/space periodic pattern and a two-dimensional square hole pattern, which can be used as a supplement to the previous model developed by Ding et al. [Opt. Express 31, 192 (2023)OPEXFF1094-408710.1364/OE.476825]. Compared with the rigorous numerical method, the fast imaging model can greatly improve the calculation speed with high accuracy. Under the same hardware conditions, the calculation speed of the 1D fast imaging model is improved by two orders of magnitude, and the 2D fast imaging model is improved by about 25 times, which creates conditions for the development of computational lithography technology.

Feature Vector Effectiveness Evaluation for Pattern Selection in Computational Lithography

Pattern selection is crucial for optimizing the calibration process of optical proximity correction (OPC) models in computational lithography. However, it remains a challenge to achieve a balance between representative coverage and computational efficiency. This work presents a comprehensive evaluation of the feature vectors’ (FVs’) effectiveness in pattern selection for OPC model calibration, leveraging key performance indicators (KPIs) based on Kullback–Leibler divergence and distance ranking. Through the construction of autoencoder-based FVs and fast Fourier transform (FFT)-based FVs, we compare their efficacy in capturing critical pattern features. Validation experimental results indicate that autoencoder-based FVs, particularly augmented with the lithography domain knowledge, outperform FFT-based counterparts in identifying anomalies and enhancing lithography model performance. These results also underscore the importance of adaptive pattern representation methods in calibrating the OPC model with evolving complexities.

Multi-scale analysis of urban forests and socioeconomic patterns in a desert city, Phoenix, Arizona

Understanding the relationship between various socioeconomic factors and urban forest structure is essential for directing resources to ensure equitable distribution of green space. Through a case study of a desert city, i.e., Phoenix, AZ, this study provides a novel application of Multiscale Geographically Weighted Regression (MGWR) in which we explore the spatially variable relationships between a wide array of socioeconomic indicators and urban forest attributes. Through the computation of various scales of influence for different explanatory variables, MGWR enhances our analysis’s precision and stresses the association between socioeconomic status and urban forest structure at local and regional scales. Our results indicate that although there has been a pattern of green inequality where minority and low-income communities have less access to urban forests, education levels were mostly insignificant based on the MGWR results. In some instances, higher incomes are negatively correlated with tree canopy coverage. Additionally, the stem density model outperformed the canopy coverage model in terms of prediction accuracy. This research adds a new dimension to urban forestry literature and emphasizes the value of customized urban planning strategies and the environmental justice implications of urban forestry, particularly in arid environments.

Symmetry-preserving modeling for lithographic imaging.

In computational imaging and lithography, it has been a challenge for a numerical model to faithfully preserve symmetries in the physical imaging system. In this Letter, we present a project-to-symmetry-subspace (PTSS) method to prevent symmetry loss during the iterative generation of optical kernels. Essentially, PTSS is to project iterative vectors onto a predefined symmetric subspace when decomposing the transmission cross coefficient (TCC). Simulation results demonstrate the PTSS-generation of a truncated set of optical kernels that are substantially free of symmetry error, regardless of the order of truncation.

High-fidelity source mask optimization for suppressing line-end shortening.

Source mask optimization (SMO) is a widely used computational lithography technique for compensating lithographic distortion. However, line-end shortening is still a key factor that cannot be easily corrected and affects the image fidelity of lithography at advanced nodes. This paper proposes a source mask optimization method that suppresses line-end shortening and improves lithography fidelity. An adaptive hybrid weight method is employed to increase the weights of the line end during the optimization, which adaptively updates the weights in each iteration according to the edge placement error (EPE). A cost function containing a penalty term based on the normalized image log slope (NILS) is established to ensure the fidelity of the overall feature when paying more attention to the line-end region. The scope of this penalty term is regulated by widening and extending the split contour to further reduce the line-end shortening. Simulation results show that the proposed method can effectively suppress the line-end shortening and improve the lithography fidelity compared with the traditional SMO method.

Investigating the Efficiency of Using U-Net, Erf-Net and DeepLabV3 Architectures in Inverse Lithography-based 90-nm Photomask Generation

The paper deals with the inverse problem of computational lithography. We turn to deep neural network algorithms to compute photomask topologies. The chief goal of the research is to understand how efficient the neural net architectures such as U-net, Erf-Net and Deep Lab v.3, as well as built-in Calibre Workbench algorithms, can be in tackling inverse lithography problems. Specially generated and marked data sets are used to train the artificial neural nets. Calibre EDA software is used to generate haphazard patterns for a 90 nm transistor gate mask. The accuracy and speed parameters are used for the comparison. The edge placement error (EPE) and intersection over union (IOU) are used as metrics. The use of the neural nets allows two orders of magnitude reduction of the mask computation time, with accuracy keeping to 92% for the IOU metric.

Three-dimensional plasmonic lithography imaging modeling based on the RCWA algorithm for computational lithography.

This paper reminds the principle and characteristics of plasmonic lithography, and points out the importance of establishing a fast and high precision plasmonic lithography imaging model and developing computational lithography. According to the characteristics of plasmonic lithography, the rigorous coupled-wave analysis (RCWA) algorithm is a very suitable alternative algorithm. In this paper, a three-dimensional plasmonic lithography model based on RCWA algorithm is established for computational lithography requirements. This model improves the existing RCWA algorithm, that is, deduces the formula for calculating the light field inside the structure and proposes the integration, storage and invocation of the scattering matrix to improve the computation speed. Finally, the results are compared with commercial software for the two typical patterns. The results show that the two calculation results are very close, with the root mean square error (RMSE) less than 0.04 (V/m)2. In addition, the calculation speed can be increased by more than 2 times in the first calculation, and by about 8 times by integrating, storing and invoking the scattering matrix, which creates conditions for the development of plasmonic computational lithography.

EUV mask model based on modified Born series.

Mask model is a critical part of computational lithography (CL). Owing to the significant 3D mask effects, it is challenging to accurately and efficiently calculate the near field of extreme ultraviolet (EUV) masks with complex patterns. Therefore, a method based on the modified Born series (MBS) was introduced for EUV mask modeling. With comparable accuracy, the MBS method was two orders of magnitude faster than the finite-difference time-domain method for the investigated examples. Furthermore, the time required for MBS was further reduced when the mask pattern was slightly changed. The proposed method shows great potential for constructing an accurate 3D mask model in EUV CL with high efficiency.

Layout pattern analysis and coverage evaluation in computational lithography.

In advanced semiconductor technology nodes, the model accuracy of optical proximity correction (OPC) is the key for integrated circuit (IC) chip mask tape out, yield ramp up, and product time-to-market. An accurate model means a small prediction error for the full chip layout. As the full chip layout usually has large pattern variety, an optimal pattern set with good coverage is desired during the model calibration process. Currently, no existing solutions can provide the effective metrics to evaluate the coverage sufficiency of the selected pattern set before a real mask tape out, which may potentially cause higher re-tape out cost and product time-to-market delay due to the multiple rounds of model calibration. In this paper, we construct the metrics to evaluate the pattern coverage before any metrology data is obtained. The metrics are based on either the pattern's intrinsic, numerical feature representation, or its potential model simulation behavior. Experimental results show a positive correlation between these metrics and lithographic model accuracy. An incremental selection method is also proposed based on the pattern simulation error. It reduces up to 53% of the model's verification error range. These pattern coverage evaluation methods can improve the efficiency of OPC model building, and are, in turn, beneficial to the whole OPC recipe development process.

Emulation of deep-ultraviolet lithography using rapid-prototyping, electron-beam lithography for silicon photonics design.

We demonstrate a method to emulate the optical performance of silicon photonic devices fabricated using advanced deep-ultraviolet lithography (DUV) processes on a rapid-prototyping electron-beam lithography process. The method is enabled by a computational lithography predictive model generated by processing SEM image data of the DUV lithography process. We experimentally demonstrate the emulation method's accuracy on integrated silicon Bragg grating waveguides and grating-based, add-drop filter devices, two devices that are particularly susceptible to DUV lithography effects. The emulation method allows silicon photonic device and system designers to experimentally observe the effects of DUV lithography on device performance in a low-cost, rapid-prototyping, electron-beam lithography process to enable a first-time-right design flow.

Degenerate scales for thin elastic plates with Dirichlet boundary conditions

A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We extend several results about degenerate scales previously found for the Laplace and Lamé equation to the biharmonic equation in 2 dimensions. The degenerate scales are obtained from a \(4\times 4\) discriminant matrix whose shape is provided for different kinds of domain symmetry. We show that degenerate scales can be obtained from a minimization problem. Then, we compare the degenerate scales of two boundaries, one included within the other. For smooth star-shaped curves, we show that there are only two degenerate scales, give sufficient conditions for not being at a degenerate scale and produce bounds to the degenerate scales. For symmetric smooth simply connected curves there are also only two degenerate scales. For these symmetric cases, the use of complex variables allows us to go further and to link the problem of biharmonic equation to the one of plane elasticity and to give information, including bounds and exact values of degenerate scales, for many cases, while until now only very few ones were known. Our results have some consequences for the biharmonic outer radius defined by Pólya and Szegö. The numerical computation of degenerate scales by using boundary elements confirms the theoretical results.

Attenuated phase shift masks: an interview with Andreas Erdmann

JM3 Associate Editor Emily Gallagher, a principal researcher at imec, interviews Andreas Erdmann, head of the Fraunhofer IISB Computational Lithography and Optics Group. With Hazem Mesilhy and Peter Evanschitzky, Erdmann is the lead author of “Attenuated phase shift masks: a wild card resolution enhancement for extreme ultraviolet lithography?,” a review paper in the April-June 2022 issue of the Journal of Micro/Nanopatterning, Materials, and Metrology.

Plasmonic lithography fast imaging model based on the decomposition machine learning method.

Plasmonic lithography can make the evanescent wave at the mask be resonantly amplified by exciting surface plasmon polaritons (SPPs) and participate in imaging, which breaks through the diffraction limit in conventional lithography. It provides a reliable technical way for the study of low-cost, large-area and efficient nanolithography technology. This paper introduces the characteristics of plasmonic lithography, the similarities and the differences with traditional DUV projection lithography. By comparing and analyzing the already existed fast imaging model of mask diffraction near-field (DNF) of DUV/EUV lithography, this paper introduces the decomposition machine learning method of mask diffraction near-field into the fast imaging of plasmonic lithography. A fast imaging model of plasmonic lithography for arbitrary two-dimensional pattern is proposed for the first time. This model enables fast imaging of the input binary 0&1 matrix of the mask directly to the light intensity distribution of photoresist image (PRI). The illumination method employs the normal incidence with x polarization, the normal incidence with y polarization and the quadrupole illumination with TM polarization respectively. The error and the operating efficiency between this fast imaging model and the rigorous electromagnetic model is compared. The test results show that compared with the rigorous electromagnetic computation model, the fast imaging model can greatly improve the calculation efficiency and maintain high accuracy at the same time, which provides great conditions for the development of computational lithography such as SMO/OPC for plasmonic lithography technology.

Can optical proximity correction solution be learned? The learning limit and a general learning framework

BackgroundOptical proximity correction (OPC) is an indispensable technology that has been propelling the advancement of computational lithography technology. To tightly control edge placement error (EPE) and maintain lithography process window, the demands on OPC computational resources and OPC turnaround time are growing rapidly with alarming acceleration. To tame the trend, machine learning technologies have been explored; however, an in-depth discussion on OPC solution learning limit is still lacking.AimWe aim to present an in-depth discussion on OPC solution learning limit and propose a general machine learning OPC framework that can be extended to curvilinear mask OPC technology.ApproachIn this study, we first investigate the machine learning OPC learning limit by examining noninverse lithography technology (non-ILT) OPC solution space characteristics inherited from edge segmentation and control point setting rules and then propose a general machine learning OPC framework that can take full advantage of deep convolution neural network (DCNN) learning capability while being able to preserve mask data high resolution.ResultsWith this machine learning OPC framework, we have achieved models with average absolute model error <1 nm for 14-nm metal layer. With single GPU, the average time for machine learning OPC models to produce results of 3840 nm × 3840 nm area is 8.74 ms for single channel input model and 12.65 ms for six channels input model.ConclusionsFor non-ILT OPC solution, there is an intrinsic learning limit inherited from edge segmentation rules. Machine learning OPC models should be content with learning low order OPC solutions. This intrinsic learning limit of non-ILT OPC solution may diminish for ILT OPC solution when the constraint on degrees of freedom of OPC solution is lifted. The machine learning OPC framework we proposed is general and extendable to curvilinear OPC technology.

Machine learning optical proximity correction with generative adversarial networks

BackgroundAlgorithmic breakthroughs in machine learning (ML) have allowed increasingly more applications developed for computational lithography, gradually shifting focus from hotspot detection to inverse lithography and optical proximity correction (OPC). We proposed a pixelated mask synthesis method utilizing deep-learning techniques, to generate after-development-inspection (ADI) contour and mask feature generation.AimConventional OPC correction consists of two parts, the simulation model which predicts the expected contour signal, and the correction script that modifies the actual layout. With practicality in mind, we collected modeling wafer data from scratch, then implemented ML models to reproduce conventional OPC actions, mask to contour prediction, and design to mask correction.ApproachTwo generative adversarial networks (GANs) were constructed, a pix2pix model was first trained to learn the correspondences between mask image and paired ADI contour image collected on wafer. The second model is embedded into machine learning mask correction (ML-OPC) framework, output mask is optimized through minimizing pixel difference between design target and simulated contour.ResultsTwo different magnification SEM image datasets were collected and studied, with the higher magnification showing better simulator pixel accuracy. Supervised training of the correction model provided a quick prototype mask synthesis generator, and combination of unsupervised training allowed mask pattern synthetization from any given design layout.ConclusionsThe experimental results demonstrated that our ML-OPC framework was able to mimic conventional OPC model in producing exquisite mask patterns and contours. This ML-OPC framework could be implemented across full chip layout.

Informatics-based computational lithography for phase-shifting mask optimization.

Phase-shifting mask (PSM) is widely used in semiconductor fabrication to improve the resolution and image fidelity of optical lithography process. However, the theoretical image fidelity limit of the PSM lithography process is not fully understood. In this paper, the information transmission mechanism of PSM layout in optical lithography system is unveiled from the perspective of information theory. First, an information channel model is established to depict the information transfer of PSM layout in optical lithography system. Based on the law of information entropy, the optimal mutual information (OMI) of the PSM lithography process is derived, and the theoretical lower bound of the PSM image error is obtained. Finally, an informatics-based computational lithography approach is proposed to optimize the PSM, which achieves higher image fidelity compared to the traditional optimization algorithm. The benefit of PSM over the binary lithography mask is also discussed from the information theoretical aspects.

A scalability study of the Ice-sheet and Sea-level System Model (ISSM, version 4.18)

Abstract. Accurately modelling the contribution of Greenland and Antarctica to sea level rise requires solving partial differential equations at a high spatial resolution. In this paper, we discuss the scaling of the Ice-sheet and Sea-level System Model (ISSM) applied to the Greenland Ice Sheet with horizontal grid resolutions varying between 10 and 0.25 km. The model setup used as benchmark problem comprises a variety of modules with different levels of complexity and computational demands. The core builds the so-called stress balance module, which uses the higher-order approximation (or Blatter–Pattyn) of the Stokes equations, including free surface and ice-front evolution as well as thermodynamics in form of an enthalpy balance, and a mesh of linear prismatic finite elements, to compute the ice flow. We develop a detailed user-oriented, yet low-overhead, performance instrumentation tailored to the requirements of Earth system models and run scaling tests up to 6144 Message Passing Interface (MPI) processes. The results show that the computation of the Greenland model scales overall well up to 3072 MPI processes but is eventually slowed down by matrix assembly, the output handling and lower-dimensional problems that employ lower numbers of unknowns per MPI process. We also discuss improvements of the scaling and identify further improvements needed for climate research. The instrumented version of ISSM thus not only identifies potential performance bottlenecks that were not present at lower core counts but also provides the capability to continually monitor the performance of ISSM code basis. This is of long-term significance as the overall performance of ISSM model depends on the subtle interplay between algorithms, their implementation, underlying libraries, compilers, runtime systems and hardware characteristics, all of which are in a constant state of flux. We believe that future large-scale high-performance computing (HPC) systems will continue to employ the MPI-based programming paradigm on the road to exascale. Our scaling study pertains to a particular modelling setup available within ISSM and does not address accelerator techniques such as the use of vector units or GPUs. However, with 6144 MPI processes, we identified issues that need to be addressed in order to improve the ability of the ISSM code base to take advantage of upcoming systems that will require scaling to even higher numbers of MPI processes.

Structural validity of the Eating Disorder Examination-Questionnaire: A systematic review.

ObjectiveThe main aim was to perform a systematic literature review of studies investigating the factor structure of the Eating Disorder Examination‐Questionnaire (EDE‐Q), a widely used measure of eating pathology. Secondary aims were to summarize the quality of reporting of latent variable (factor) analyses in these studies and review support for different factor solutions.MethodLiterature was identified through Scopus, Medline, PsycInfo, and ProQuest databases published up to February 23, 2022 and outreach via an international listserv. All studies published in English reporting factor analysis of the EDE‐Q were included with few restrictions. Sixty studies including 63,389 participants met inclusion criteria.ResultsThe originally proposed four‐factor solution received little empirical support, although few alternative models have been robustly evaluated. Items assessing shape and weight concerns frequently coalesce in factor solutions, suggesting that these constructs are closely related. Investigations of brief versions of the EDE‐Q have produced more consistent findings, suggesting that these measures, particularly a seven‐item version, might be useful alternatives to the full version. Quality of studies was reasonable, with important methodological elements of factor analysis often reported.DiscussionThe findings are of relevance to practitioners and researchers, suggesting that the “original” factor structure of the EDE‐Q should be reconsidered and that use of a seven‐item version is to be encouraged.Public SignificanceSelf‐report questionnaires are widely used in the assessment of disordered eating. The current study found that there is little consensus about the structure of a common measure of eating psychopathology. There is more consistent support for a brief, seven‐item, version assessing dietary restraint, body dissatisfaction, and overvaluation of weight and shape.