## Computation lithography: virtual reality and virtual virtuality

Optics Express, Vol. 17, Issue 15, pp. 12259-12268 (2009)

http://dx.doi.org/10.1364/OE.17.012259

Acrobat PDF (297 KB)

### Abstract

Computation lithography is enabled by a combination of physical understanding, mathematical abstraction, and implementation simplification. An application in the virtual world of computation lithography can be a virtual reality or a virtual virtuality depending on its engineering sensible-ness and technical feasibility. Examples under consideration include design-for-manufacturability and inverse lithography.

© 2009 Optical Society of America

## 1. Introduction

## 2. Ingredients of computation lithography

## 2.1. Specifications

## 2.2. Physical understanding

*L*

_{1}. In conjunction with

*L*

_{1}, the lens

*L*

_{2}forms an image of the source in the pupil. This lens

*L*

_{2}performs a Fourier transform of the object such that the field across the pupil plane is the spectrum of the object. Low-spatial-frequency components pass closer to the center of the pupil, and higher-frequency components are nearer to the periphery of the pupil. The highest frequencies are cut off by the pupil. The lens

*L*

_{3}forms the image by combining the filtered frequency components.

## 2.3. Mathematical abstraction

*x, y*) is given by [27]:

*x,y*) can be interpreted as the weighted sum (

*w*) of contributions from neighboring objects

*O*(

*x*′

_{0};

*y*′

_{0}) and

*O*(

*x*″

_{o};

*y*″

_{o}).

## 2.4. Implementation simplification

*φ*and corresponding eigenvalues λ

_{k}_{k}[21], such that

*w*(

*x*′

_{0},

*y*′

_{o};

*x*″

_{o},

*y*″

_{o})=∑

^{∞}

_{k=1}

*λ*(

_{k}φ_{k}*x*′

_{o},

*y*′

_{o})φ

*k*∗(

*x*″o,

*y*″o):

*dominant eigenvectors:*

**K***N*is the total number of rectangles, the superscript (

*n*) indexes the rectangular shapes, and

*t*

^{(n)}

_{fg}represents the transmittance of the

*n*

^{th}shape [22, 23]. Substituting the above expression into Eq. (2) results in

*ψ*(

_{k}*x,y*) are independent of the object, they can be precomputed and stored. Image calculation hence becomes an addition operation, improving computation efficiency tremendously.

## 3. Virtual reality and virtual virtuality

## 3.1. Physical DFM

*reference*and the second

*trial*. We can define the hotspot matching rate as

*𝓟*

_{missing}=1-

*𝓟*

_{matching}:

*σ*) of the hotspot threshold for both simulators, it is computed that

*𝓟*

_{matching}=73.4% and

*𝓟*

_{missing}=

*𝓟*

_{extra}=26.6% respectively [31]. In addition, if we want to increase the matching rate by raising the trial threshold, the tradeoff is a significantly higher extra rate [31]. Thus, more stringent accuracy specifications than such theoretical bounds are unrealistic.

## 3.2. Electrical DFM

## 3.3. Inverse lithography

36. A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. **16**, 774–788 (2007).
[PubMed]

*t*represents the threshold) [36

_{r}36. A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. **16**, 774–788 (2007).
[PubMed]

*O*(

*x,y*) is the mask transmittance,

*O*

_{opt}(

*x,y*) is the optimal mask,

*Î*(

*x,y*) is the desired circuit pattern, and

*H*(

*x,y*) is the optical transfer function (OTF) of the imaging system. Eq. (4) can be considered an

*unconstrained*optimization. Fig. 2 shows an example of this computation. The target pattern is given in (a), and the unconstrained mask design is given in (b). The output aerial image is given in (c). By thresholding this image, we can obtain a binary circuit pattern that is very close to the target pattern. However, this design has a problem if the imaging parameter is not accurate. Consider, in particular, that the imaging system has 160 nm defocus. As shown in (d), the aerial image has the two rectangles merged together, and no thresholding can bring the pattern close to the original design.

*β*is a Gaussian distributed focus error. We can append the quadratic phase factor exp{-

*jπ*(

*f*

^{2}+

*g*

^{2})

*β*} to the Fourier transform of

*H*(

*x;y*), and optimize instead

*𝓔*denotes expectation over the variable

_{β}*β*. We call this a mask design regularized for robustness against focus variation. Note that the regularization is achieved implicitly rather than with an explicit penalty term, as in a typical image reconstruction and restoration algorithm [38]. Fig. 3 shows the mask design for the same pattern as before, but using this new methodology. We can see in (b) that the resulting mask design is very different from that in Fig. 2. It is, in fact, closer to what we normally expect with OPC. In (c), the output aerial image at nominal focus value is good in the sense that proper thresholding would result in a pattern as designed. Furthermore, it has sharper edges, making it a more robust design. In particular, as shown in (d), the design is still useful even at 160 nm defocus, as we can still separate the two rectangles by thresholding at around 0.5.

## 4. Remarks

*Journal of Micro/Nanolithography, MEMS, and MOEMS*with a special section on double-patterning lithography [41]).

## Acknowledgment

## References and links

1. | F. H. Dill, W. Hornberger, P. Hauge, and J. Shaw, “Characterization of positive photoresists,” IEEE Trans. Electron Devices |

2. | K. L. Konnerth and F. H. Dill, “In-situ measurement of dielectric thickness during etching or developing process,” IEEE Trans. Electron Devices |

3. | F. Dill, “The basis for lithographic modeling,” in |

4. | C. A. Mack
, “Thirty years of lithography simulation,” in |

5. | A. Neureuther, “If it moves, simulate it!” in |

6. | M. Yeung, “Modeling aerial images in two and three dimensions,” in |

7. | D. Nyyssonen and C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory,” J. Opt. Soc. Am. A |

8. | K. Lucas, C.-M. Yuan, and A. Strojwas, “A Rigorous and Practical Vector Model for Phase Shifting Masks in Optical Lithography,” in |

9. | T. Matsuzawa, A. Moniwa, N. Hasegawa, and H. Sunami, “Two-Dimensional Simulation of Photolithography on Reflective Stepped Substrate,” IEEE Trans. Comput.-Aided Des. Int. Cir. Sys. |

10. | H. P. Urbach and D. A. Bernard, “Modeling latent image formation in photolithography using the Helmholtz equation,” in |

11. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

12. | R. Guerrieri, K. H. Tadros, J. Gamelin, and A. Neureuther, “Massively parallel algorithms for scattering in optical lithography,” IEEE Trans. Comput.-Aided Des. Int. Cir. Sys. |

13. | J. A. Sethian, “Fast marching level set methods for three-dimensional photolithography development,” in |

14. | S. Osher and J. A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” J. Comput. Phys. |

15. | J. F. Chen, T. Laidig, K. Wampler, and R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B |

16. | O. Otto and R. Henderson, “Advances in process matching for rules-based optical proximity correction,” in |

17. | N. Cobb and A. Zakhor, “Experimental Results on Optical Proximity Correction with Variable Threshold Resist Model,” in |

18. | M. Rieger and J. Stirniman, “Mask fabrication rules for proximity corrected patterns,” in |

19. | T. Waas, H. Eisenmann, and H. Hartmann, “Proximity Correction for high CD-Accuracy and Process Tolerance,” in |

20. | H.-Y. Liu, L. Karklin, Y.-T. Wang, and Y. C. Pati, “Application of alternating phase-shifting masks to 140 nm Gate Patterning II: Mask design and manufacturing tolerances,” in |

21. | H. Gamo, “Matrix Treatment of Partial Coherence,” in |

22. | Y. C. Pati, A. A. Ghazanfarian, and R. F. Pease, “Exploiting Structure in Fast Aerial Image Computation for Integrated Circuit Patterns,” IEEE Trans. Semi. Manufactur. |

23. | A. E. Rosenbluth, G. Gallatin, R. Gordon, W. Hinsberg, J. Hoffnagle, F. Houle, K. Lai, A. Lvov, M. Sanchez, and N. Seong, “Fast calculation of images for high numerical aperture lithography,” in |

24. | M. Born and E. Wolf, |

25. | J. W. Goodman, |

26. | A. K. Wong, |

27. | A. K. Wong, |

28. | J. Kim and M. Fan, “Hotspot detection on post-OPC layout using full-chip simulation-based verification tool: a case study with aerial image simulation,” in |

29. | S. D. Shang, Y. Granik, N. B. Cobb, W. Maurer, Y. Cui, L. W. Liebmann, J. M. Oberschmidt, R. N. Singh, and B. R. Vampatella, “Failure prediction across process window for robust OPC,” in |

30. | H. Mashita, T. Kotani, F. Nakajima, H. Mukai, K. Sato, S. Tanaka, K. Hashimoto, and S. Inoue, “Tool-induced hotspot fixing flow for high volume products,” in |

31. | A. K. K. Wong and E. Y. Lam, “The Nebulous Hotspot and Algorithm Variability,” in |

32. | J. H. Huang, Z. H. Lui, M. C. Jeng, P. K. Ko, and C. Hu, “A Robust Physical and Predictive Model for Deep-Submicrometer MOS Circuit Simulation,” Master’s thesis, University of California, Berkeley (1993). Memorandum No. UCB/ERL M93/57. |

33. | S. Banerjee, P. Elakkumanan, L. W. Liebmann, J. A. Culp, and M. Orshansky, “Electrically driven optical proximity correction,” in |

34. | L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” in |

35. | S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express |

36. | A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. |

37. | N. Jia, A. K. Wong, and E. Y. Lam, “Robust Photomask Design with Defocus Variation Using Inverse Synthesis,” in |

38. | E. Y. Lam and J. W. Goodman, “Iterative Statistical Approach to Blind Image Deconvolution,” J. Opt. Soc. Am. A |

39. | B. Yenikaya and A. Sezginer, “A rigorous method to determine printability of a target layout,” in |

40. | A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. F. Molless, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlithogr., Microfabr., Microsyst. |

41. | W. H. Arnold, “Guest Editorial: Special Section on Double-Patterning Lithography,” J. Micro/Nanolith. MEMS MOEMS |

**OCIS Codes**

(110.3960) Imaging systems : Microlithography

(110.5220) Imaging systems : Photolithography

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 3, 2009

Revised Manuscript: April 29, 2009

Manuscript Accepted: June 22, 2009

Published: July 6, 2009

**Citation**

Edmund Y. Lam and Alfred K. Wong, "Computation lithography: virtual reality and virtual virtuality," Opt. Express **17**, 12259-12268 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12259

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### References

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- K. L. Konnerth and F. H. Dill, "In-situ measurement of dielectric thickness during etching or developing process," IEEE Trans. Electron Devices ED-22(7), 452-456 (1975).
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- A. Neureuther, "If it moves, simulate it!" in Proc. SPIE, H. J. Levinson andM. V. Dusa, eds., vol. 6924, p. 692402 (2008).
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- S. Osher and J. A. Sethian, "Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations," J. Comput. Phys. 79, 12-49 (1988).
- J. F. Chen, T. Laidig, K. Wampler, and R. Caldwell, "Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars," J. Vac. Sci. Technol. B 15(6), 2426-2433 (1997).
- O. Otto and R. Henderson, "Advances in process matching for rules-based optical proximity correction," in Proc. SPIE, vol. 2884, pp. 425-434 (1996).
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- M. Rieger and J. Stirniman, "Mask fabrication rules for proximity corrected patterns," in Proc. SPIE, vol. 2884, pp. 323-332 (1996).
- T. Waas, H. Eisenmann, and H. Hartmann, "Proximity Correction for high CD-Accuracy and Process Tolerance," in Proc. Symposium on Nanocircuit Engineering (1994).
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- S. D. Shang, Y. Granik, N. B. Cobb, W. Maurer, Y. Cui, L. W. Liebmann, J. M. Oberschmidt, R. N. Singh, and B. R. Vampatella, "Failure prediction across process window for robust OPC," in Proc. SPIE, A. Yen, ed., vol. 5040, pp. 431-440 (2003).
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- A. K. K. Wong and E. Y. Lam, "The Nebulous Hotspot and Algorithm Variability," in Proc. SPIE, vol. 7275, p. 727509 (2009).
- J. H. Huang, Z. H. Lui, M. C. Jeng, P. K. Ko, and C. Hu, "A Robust Physical and Predictive Model for Deep- Submicrometer MOS Circuit Simulation," Master’s thesis, University of California, Berkeley (1993). Memorandum No. UCB/ERL M93/57.
- S. Banerjee, P. Elakkumanan, L. W. Liebmann, J. A. Culp, and M. Orshansky, "Electrically driven optical proximity correction," in Proc. SPIE, V. K. Singh and M. L. Rieger, eds., vol. 6925, p. 69251W (2008).
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- S. H. Chan, A. K. Wong, and E. Y. Lam, "Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography," Opt. Express 16, 14,746-14,760 (2008).
- A. Poonawala and P. Milanfar, "Mask design for optical microlithography—an inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007). [PubMed]
- N. Jia, A. K. Wong, and E. Y. Lam, "Robust Photomask Design with Defocus Variation Using Inverse Synthesis," in Proc. SPIE, vol. 7140, p. 71401W (2008).
- Q5. E. Y. Lam and J. W. Goodman, "Iterative Statistical Approach to Blind Image Deconvolution," J. Opt. Soc. Am. A 17(7), 1177-1184 (2000).
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- Q6. A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. F. Molless, R. N. Singh, and A. K. K. Wong, "Optimum mask and source patterns to print a given shape," J. Microlithogr., Microfabr., Microsyst. 1(1), 13-30 (2002).
- W. H. Arnold, "Guest Editorial: Special Section on Double-Patterning Lithography," J. Micro/Nanolith. MEMS MOEMS 8(1), 011,001 (2009).

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