&lt;title&gt;Exact computation of scalar 2D aerial imagery&lt;/title&gt;

Gordon, Ronald L.

doi:10.1117/12.475687

Cited by 9 publications

(4 citation statements)

References 6 publications

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“…The approach taken in the current study is to find closed-form solutions from the research literature, and to use these solutions as an absolute standard for benchmarking lithography simulators. This is similar to the approach taken by Brunner [2] and Gordon [3].…”

Section: Introductionsupporting

confidence: 54%

Methods for benchmarking photolithography simulators: part III

Smith

Graves

Byers

et al. 2005

Optical Microlithography XVIII

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In the past, most lithography simulators have used the thin-mask or Kirchhoff approximation to calculate the diffraction pattern for imaging calculations. This approximation has been very accurate for binary reticles, and rigorous solutions to the full Maxwell equations were only required for "exotic" technologies such as alternating phase-shift masks and chromeless phase lithography (CPL). For the future technology nodes, the thin-mask approximation may be insufficient even for binary reticles. This means that solution of the full Maxwell equations will be required for most, if not all, lithography simulations, and that these simulators must be robust and accurate, especially when used by someone who is not an expert in solving the Maxwell equations.In a previous series of papers, we proposed benchmarks for lithography simulators drawn from the optics literature for aerial image and optical film-stack calculations. We extend this work and present benchmarks here for Maxwell equation solvers. These benchmarks can be easily applied to any mask topography simulator.

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Section: Introductionsupporting

confidence: 54%

Methods for benchmarking photolithography simulators: part III

Smith

Graves

Byers

et al. 2005

Optical Microlithography XVIII

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“…(8), and it is equal to or larger than (1 þ r) in optical lithography with the partial coherence factor of the source being r; N ¼ (2 N A þ 1)N R is the total number of CSFs with angle factor n from ÀN A to N A and radial factor s from 1 to N R ; u k (aq, h) stands for the kth CSF; and p k,l are the expansion coefficients. (8), and it is equal to or larger than (1 þ r) in optical lithography with the partial coherence factor of the source being r; N ¼ (2 N A þ 1)N R is the total number of CSFs with angle factor n from ÀN A to N A and radial factor s from 1 to N R ; u k (aq, h) stands for the kth CSF; and p k,l are the expansion coefficients.…”

Section: Theorymentioning

confidence: 99%

Fast aerial image simulations for partially coherent systems by transmission cross coefficient decomposition with analytical kernels

Gong

Liu

et al. 2012

Journal of Vacuum Science &Amp; Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phen

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Aerial image simulation is one of the most critical components in the model-based optical proximity correction (OPC), which has become a necessary part of resolution enhancement techniques used to improve the performance of subwavelength optical lithography. In this paper, a fast aerial image simulation method is proposed for partially coherent systems by decomposing the transmission cross coefficient (TCC) into analytical kernels. The TCC matrix is projected onto a function space whose basis is analytical circle-sampling functions (CSFs) and converted into a much smaller projected matrix. By performing singular value decomposition (SVD) to the projected matrix, its eigenvectors together with the CSFs are used to generate a set of analytical TCC kernels. The proposed method avoids directly performing SVD to the large TCC matrix, making it much more runtime efficient than the conventional SVD method. Furthermore, the grid size of the kernels can be flexibly set to any desired value in aerial image simulations, which is not realizable with the conventional SVD method. The comparison of aerial image intensity errors and edge placement errors calculated by the proposed method and the conventional SVD method has confirmed the validity of the proposed method. An OPC example is also provided to further demonstrate its efficiency.

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“…Further, it has been shown that the number of TCCs varies as the square of the area of the periodic unit cell, 10 it is clear that for larger areas, the numerical method becomes unwieldy.…”

Section: Illustration Of the Theorymentioning

confidence: 99%

Lithographic image simulation for the 21st century with 19th-century tools

Gordon¹,

Rosenbluth

2004

SPIE Proceedings

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Simulation of lithographic processes in semiconductor manufacturing has gone from a crude learning tool 20 years ago to a critical part of yield enhancement strategy today. Although many disparate models, championed by equally disparate communities, exist to describe various photoresist development phenomena, these communities would all agree that the one piece of the simulation picture that can, and must, be computed accurately is the image intensity in the photoresist.The imaging of a photomask onto a thin-film stack is the only phenomenon in the lithographic process that is described fully by well-known, definitive physical laws. Although many approximations are made in the derivation of the Fourier transform relations between the mask object, the pupil, and the image, these and their impacts are well-understood and need little further investigation.The imaging process in optical lithography is modeled as a partially-coherent, Köhler illumination system. As Hopkins has shown, we can separate the computation into 2 pieces: one that takes information about the illumination source, the projection lens pupil, the resist stack, and the mask size or pitch, and the other that only needs the details of the mask structure. As the latter piece of the calculation can be expressed as a fast Fourier transform, it is the first piece that dominates.This piece involves computation of a potentially large number of complex quantities called Transmission Cross Coefficients (TCCs), which are correlations of the pupil function weighted with the illumination intensity distribution. The advantage of performing the image calculations this way is that the computation of these TCCs represents an up-front cost, not to be repeated if one is only interested in changing the mask features, which is the case in Model-Based Optical Proximity Correction (MBOPC).1 The down side, however, is that the number of these expensive double integrals that must be performed increases as the square of the mask unit cell area; this number can cause even the fastest computers to balk if one needs to study medium-or long-range effects. One can reduce this computational burden by approximating with a smaller area, but accuracy is a concern, especially when building a model that will purportedly represent a manufacturing process. This work will review the current methodologies used to simulate the intensity distribution in air above the resist and address the above problems. More to the point, a methodology has been developed to eliminate the expensive numerical integrations in the TCC calculations, as the resulting integrals in many cases of interest can be either evaluated analytically, or replaced by analytical functions accurate to within machine precision. With the burden of computing these numbers lightened, more accurate representations of the image field can be realized, and better overall models are then possible.

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<title>Exact computation of scalar 2D aerial imagery</title>

Cited by 9 publications

References 6 publications

Methods for benchmarking photolithography simulators: part III

Methods for benchmarking photolithography simulators: part III

Fast aerial image simulations for partially coherent systems by transmission cross coefficient decomposition with analytical kernels

Lithographic image simulation for the 21st century with 19th-century tools

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