Simulations of immersion lithography

Bai, Min; Lei, Junjiang; Zhang, Lin; Shiely, James P.

doi:10.1117/12.601497

Cited by 6 publications

(8 citation statements)

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“…[9][10][11][12]), in which the kernel functions are quite restrictive and are required to have compact supports in general. In the following we study the Hopkins equation in the spatial domain, which allows a larger class of kernel functions in applications.…”

Section: Extending Hopkins Equationmentioning

confidence: 99%

“…Particularly, a polarized light source can be decomposed and calculated in this form, if we properly choose for s i 's the x-, y-components of the vector field defined by a polarized source and choose for p i ; q i the components of the transfer matrix of an imaging system (see [11] for the definition). It is important to point out that a Hermitian kernel given in this summation form is positive definite if it represents the extended Hopkins equation in vector form.…”

Section: Extending Hopkins Equationmentioning

confidence: 99%

“…The Hopkins equation is a general mathematical description for a microscope or an optical exposure system in applications, which has widely been adopted by the micro optical lithography community (e.g. see [8][9][10][11]). In the Hopkins equation, function p is the spatial representation of a pupil transfer function defined in the frequency domain, which may include optical effects of an aperture stop, lens aberration and thin films effects, while function s is the spatial representation of an illumination source defined in the frequency domain.…”

Section: Hopkins Equationmentioning

confidence: 99%

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Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

Lei¹,

Wen²,

Ai³

et al. 2009

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In this article, we derive an analytical approximation to compute the aerial images on general geometric domains. Under the framework in Hilbert space, we develop a general approximation model for aerial image intensity, which is particularly valid for the computation of the aerial image with polarized illumination source. Our mathematical result is a generalization of classical Mercer's theorem provided that the integral kernel is Hilbert-Schmidt. By applying Synopsys's simulation tool Progen, we demonstrate the effectiveness of our proposed approximation with polarized illumination sources (65 nm and below). Since our approach setting is quite general, the proposed approximation in this article is suitable for modification in various applications.

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Section: Extending Hopkins Equationmentioning

confidence: 99%

Section: Extending Hopkins Equationmentioning

confidence: 99%

Section: Hopkins Equationmentioning

confidence: 99%

See 1 more Smart Citation

Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

Lei¹,

Wen²,

Ai³

et al. 2009

Applicable Analysis

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“…Not only are Hopkins's kernels semi-positive definite operators in a Hilbert space, but such linear combinations as representations of vector forms of the Hopkins equation are as well. The extension of the Hopkins equation to the vector case is usually done in the frequency domain (e.g., [7], [3], [6], [1] where p 's are functions defined in the frequency domain, (x, y, z) in the sub-indexes mean the x, y, z components of an electric field in the exit pupil, and (X, Y) mean the x, y components of an electric field in the entrance pupil. Here, we neglected the z-component of a field in the entrance pupil because of the reduction factor four of a projection system commonly used nowadays.…”

Section: Vector Modelingmentioning

confidence: 99%

A study for polarized illumination effects in photo resist

et al. 2005

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INTRODUCTIONUsing a polarized illumination source is a promising RET technique for improvement of wafer printability for features of 65 nm and below. Polarization effects could be considered in several different stages of lithography modeling and simulation. For example, light propagation in thin films, wave superstition and interference in the thin film stack, and mask-induced polarization all deserve special attention and delicate treatment because TE and TM waves have different behaviors through these stages. In this paper we consider effects of polarized illumination in photo resist, using the Kirchhoff approximation for masks. We discuss some theoretical aspects of our vector modeling methods and show an example of simulation for polarized illumination effects.

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“…To model these hyper-NA systems, current state-of-the-art OPC modeling engines are already capable of modeling thin-film energy coupling, vector diffraction, polarization illumination, and immersion, imaging. [1][2][3] However, current OPC simulators * do not consider the loss of spatial frequency content due to pupil apodization or pellicle film effects. Both of these effects cause a loss of critical high-spatial-frequency information in the imaging process.…”

Section: Introductionmentioning

confidence: 98%

Novel apodization and pellicle optical models for accurate optical proximity correction modeling at 45 and <inline-formula><math altimg="none" display="inline" overflow="scroll"><mrow><mn>32</mn><mspace width="0.3em" /><mi>nm</mi></mrow></math></inline-formula>

Zhang

Lucas

Adrichem³

et al. 2007

J. Micro/Nanolith. MEMS MOEMS

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An accurate optical model is the foundation of an accurate optical proximity correction ͑OPC͒ model, which has always been the key for successful implementation of model-based OPC. As critical dimension ͑CD͒ control requirements become severe at the 45-and 32-nm device generations, OPC model accuracy and hence optical model accuracy requirements become more stringent. In previous generations, certain optical effects could be safely ignored. For example, the transmission attenuation particularly at high spatial frequencies caused by lens apodization effects and organic pellicle films was ignored or not accurately modeled in conventional OPC simulators. These effects are now playing a more important role in OPC modeling as technology scales down. Our simulations indicate these effects can cause CD modeling errors of 5 nm or larger, at the 45-nm technology node and beyond. Therefore, they must be accurately modeled in OPC modeling. In our OPC modeling methodology, we propose two novel low-pass-filter models to capture the frequency-dependent transmission attenuation due to lens apodization and to pellicle films. These parameterized novel lowpass-filter models ensure that lens apodization and pellicle-film-induced transmission attenuation can be appropriately account for through regression during the experimental OPC model calibration stage in the case where no measured transmission data are available, thus enabling physics-centric OPC model building with considerably higher accuracy. We can then avoid overfitting the OPC model, which could cause instability in the OPC correction stage. The validity and efficiency of the proposed novel models are also verified using an industry-standard lithography simulator as well as an experimental OPC model calibration at the 45-nm technology node.

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Simulations of immersion lithography

Cited by 6 publications

References 0 publications

Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

A study for polarized illumination effects in photo resist

Novel apodization and pellicle optical models for accurate optical proximity correction modeling at 45 and <inline-formula><math altimg="none" display="inline" overflow="scroll"><mrow><mn>32</mn><mspace width="0.3em" /><mi>nm</mi></mrow></math></inline-formula>

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