## Robust level-set-based inverse lithography |

Optics Express, Vol. 19, Issue 6, pp. 5511-5521 (2011)

http://dx.doi.org/10.1364/OE.19.005511

Acrobat PDF (717 KB)

### Abstract

Level-set based inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem, interpreted with a time-dependent model, and then solved as a partial differential equation with finite difference schemes. This paper focuses on developing level-set based ILT for partially coherent systems, and upon that an expectation-orient optimization framework weighting the cost function by random process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results demonstrating the benefits of defocus-aberration-aware level-set based ILT are presented.

© 2011 Optical Society of America

## 1. Introduction

*k*

_{1}regime and minimum design pitches to sub-100nm. Due to the wave nature of light, as dimensions approach sizes comparable to or smaller than the wavelength of the light used in the photolithography process, the bandlimited imaging system introduces undesirable distortions and artifacts. Along the way, resolution enhancement techniques (RET) [1, 2] are essential in optical lithography, which include modified illumination schemes and optical proximity correction (OPC). The latter predistorts the mask patterns such that printed patterns are as close to the desired shapes as possible. Rule-based OPC [3, 4], in which various geometries are treated by different empirical rules, and model-based OPC, which is more complex and involves the computation of a weighted sum of pre-simulated results for simple edges and corners that are stored in a library, are two main approaches to OPC. Moving beyond model-based OPC, inverse lithography technology (ILT) is becoming a strong candidate for 32nm and below low-

*k*

_{1}regime.

8. S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. **4**(9), 1252–1257 (1995). [PubMed]

10. 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**(19), 14746–14760 (2008). [PubMed]

13. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [PubMed]

27. E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express **17**(15), 12259–12268 (2009). [PubMed]

## 2. The Constrained level set time-dependent model formulation

### 2.1. Mathematical model for partially coherent systems

**x**denotes spatial coordinates (

*x,y*), and 𝒯 {·} maps the input intensity function

*U*(

**x**) to the output intensity function

*I*(

**x**). Due to the lowpass nature of the optical lithography imaging system,

*I*(

**x**) is typically a blurred version of

*U*(

**x**). Suppose the desired circuit pattern is

*I*

_{0}(

**x**). The objective of inverse lithography is to find a predistorted input intensity function

*Û*(

**x**) which minimizes its distance with the desired output, i.e., in which

*d*(·, ·) is an appropriately defined distance metric, such as the

*ℓ*

_{2}norm.

13. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [PubMed]

*a*being the steepness of the sigmoid and

*t*being the threshold.

_{r}35. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. **21**(15), 2770–2777 (1982). [PubMed]

36. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express **17**(7), 5783–5793 (2009). [PubMed]

**x**= (

*x,y*),

**x**

_{1}= (

*x*

_{1},

*y*

_{1}) and

**x**

_{2}= (

*x*

_{2},

*y*

_{2}).

*U*(

**x**) is the mask pattern,

*γ*(

**x**

_{1}–

**x**

_{2}) is the complex degree of coherence and

*H*(

**x**) represents the amplitude impulse response of the optical system, namely point spread function (PSF). The term

*γ*(

**x**

_{1}–

**x**

_{2}) is generally a complex number, whose magnitude represents the extent of optical interaction between two spatial locations

**x**

_{1}= (

*x*

_{1},

*y*

_{1}) and

**x**

_{2}= (

*x*

_{2},

*y*

_{2}) of the light source. It is the inverse Fourier transform of the image of the illumination shape Γ(

**x**) in the lens pupil. Common illumination sources are shown in Fig. 1, which include conventional circular, annular, and dipole, all introducing partial coherence into illumination.

*A*defined by

**x**∈ [

*γ*(

**x**) needed are those inside the square area

*A*defined by

_{γ}**x**∈ [−

*G,G*]. We can expand

*γ*(

**x**) as a 2-D Fourier series of periodicity 2

*G*in both the

*x*and

*y*directions, and therefore,

*γ*(

**x**) can be rewritten as and also where

*ω*

_{0}=

*π*/

*G*,

**m**= (

*m*,

_{x}*m*),

_{y}*m*and

_{x}*m*being integers within the range of [−

_{y}*D*,

*D*], and

**m**·

**x**=

*m*+

_{x}x*m*. Substituting (5) into (4), the light intensity on the wafer is given by where Combining the aerial image formation in (7) and the logarithmic sigmoid function in (3) describing resist effects, we have the image formation equation for a partially coherent imaging system as It is observed from (7) that the partially coherent image is equal to the superposition of coherent systems. In what follows, we will drop the arguments

_{y}y**x**whenever there is no ambiguity.

### 2.2. Time-dependent model formulation in partially coherent imaging

### 2.3. Aberration-aware statistical model

*ρ,θ*are the polar coordinates in the exit pupil function, and

*f*between the image plane and focal plane, can be expressed as

*H*

_{0}in this paper. Aberration terms are incorporated into the PSF by multiplying an exponential term with aberration function as power in the frequency domain, i.e. [37] where ℱ denotes Fourier transform.

*ℓ*

_{2}norm and ℰ denotes expectation. One should note this expectation-orient minimizing problem practically weights the cost function

*α*(

**x**,

*t*) is computed as One should notice that Eq. (17) takes the same form as Eq. (12) with a different computation of

*α*(

**x**,

*t*), therefore Eq. (17) is also a PDE which could be solved by the proposed finite difference schemes in [26]. The computation stops after a certain number of iterations or when the value of the cost function has decreased below a certain threshold value.

## 3. Numerical results

*λ*= 193nm,

*NA*= 1.35, resolution Δ

*x*= 10nm/pixel, steepness of the sigmoid function

*a*= 85, threshold

*t*= 0.3, and therefore the same PSF

_{r}*H*

_{0}(

**x**) with the same size as that of the target pattern which is 101 × 101 for various experiments. The optimization stops after 50 iterations. In this paper, we apply the proposed algorithms on binary masks. However, it should be noted the same framework can be readily applied to phase-shifting masks (PSMs) by applying different levels sets to corresponding phases in the PSMs. Figure. 2(a) shows the target pattern, and (b), (c) and (d) are the output patterns under circular, annular, and dipole source illuminations in Fig. 1(a), (b) and (c), respectively. Respective pattern errors are also given.

*H*(

**x**) should be computed by degrading

*H*

_{0}(

**x**) with aberrations as in Eq. (15). For defocus aberration, conventionally, the relationship between the defocus coefficient

*z*by paraxial approximation can be described as [39] therefore, defocus term is incorporated into the amplitude spread function

*H*(

**x**) by multiplying an exponential term with power, in which

*N*is the image size and (

*u, v*) and (

*m, n*) are the frequency coordinates and normalized frequency coordinates corresponding to spatial coordinates (

*x,y*) respectively, in the frequency domain of the nominal PSF

*H*

_{0}(

**x**). We first construct the optimal input mask pattern with defocus variation using the statistical method described in Section 2.3 and then compare the generated results with that of the optimal input mask pattern constructed under nominal conditions presented in

*U*(

**x**) of Fig. 3(a). The optimal input mask pattern with defocus variation is presented in Fig. 4(a). It should be noted that since we are not introducing large aberration parameters and small weights accompany large aberration parameters, big difference in mask patterns obtained under nominal condition and using the statistical method in Fig. 4(a) is not expected. Figure. 4(c) plots the performance of the statistical method versus optimization only under nominal conditions with a defocus range of (−90nm, 90nm). It is observed that while under nominal conditions, the optimal input mask pattern under nominal condition outperforms the input mask computed using the statistical method, which is not a surprise since the former is intended specifically for nominal conditions and the latter is not, the input mask computed with statistical variability accounted for improves pattern fidelity with focus variation, obtaining fewer pattern errors than the optimal input mask pattern computed under the nominal condition. This is because the expectation operation tends to compensate the distortion brought by different defocus conditions on the input pattern mask. In Fig. 4(b), the input mask computed with statistical identity of coma variation accounted for is presented. The Zernike polynomial for coma is given as

*H*(

**x**) by multiplying nominal PSF

*H*

_{0}(

**x**) with an exponential term with power,

## 4. Conclusion

## Acknowledgment

*Theory, Modeling, and Simulation of Emerging Electronics*.

## References and links

1. | A. K.-K. Wong, |

2. | F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” |

3. | O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE |

4. | S. Shioiri and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE |

5. | L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE |

6. | Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE |

7. | Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE |

8. | S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. |

9. | Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A |

10. | 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 |

11. | S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing , pp. 1832–1835 (2008). |

12. | A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE |

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

14. | S. H. Chan, A. K. Wong, and E. Y. Lam, “Inverse synthesis of phase-shifting mask for optical lithography,” in OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007). |

15. | V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE |

16. | A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE |

17. | N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE |

18. | N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask design with defocus variation using inverse synthesis,” Proc. SPIE |

19. | S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. |

20. | J. A. Sethian and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” |

21. | S. Osher and N. Paragios, |

22. | F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. |

23. | S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. |

24. | A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. |

25. | L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” |

26. | Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express |

27. | E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express |

28. | E. Y. Lam and A. K. Wong, “Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/Nanolithogr. MEMS MOEMS |

29. | N. Jia and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. |

30. | Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. of SPIE |

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

32. | J. W. Goodman, |

33. | H. H. Hopkins, “On the diffraction theory of optical images,” Proc. of the Royal Soc. of London |

34. | A. K.-K. Wong, |

35. | B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. |

36. | X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express |

37. | B. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis, Groningen University (1942). |

38. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A |

39. | P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” |

**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: January 10, 2011

Revised Manuscript: February 17, 2011

Manuscript Accepted: February 18, 2011

Published: March 9, 2011

**Citation**

Yijiang Shen, Ningning Jia, Ngai Wong, and Edmund Y. Lam, "Robust level-set-based inverse lithography," Opt. Express **19**, 5511-5521 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5511

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### References

- A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, Bellingham, WA, 2001).
- F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
- O. W. Otto, J. G. Garofalo, K. K. Low, C.-M. Yuan, R. C. Henderson, C. Pierrat, R. L. Kostelak, S. Vaidya, and P. K. Vasudev, “Automated optical proximity correction: a rules-based approach,” Proc. SPIE 2197, 278–293 (1994).
- S. Shioiri, and H. Tanabe, “Fast optical proximity correction: analytical method,” Proc. SPIE 2440, 261–269 (1995).
- L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE 6607, 660739 (2007).
- Y. Liu, and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE 1264, 401–412 (1990).
- Y. Liu, and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE 1463, 382–399 (1991).
- S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. 4(9), 1252–1257 (1995). [PubMed]
- Y. C. Pati, and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11(9), 2438–2452 (1994).
- 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(19), 14746–14760 (2008). [PubMed]
- S. H. Chan, and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in Proceedings of IEEE International Conference on Image Processing, pp. 1832–1835 (2008).
- A. Poonawala, and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE 5674, 114–127 (2005).
- A. Poonawala, and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007). [PubMed]
- S. H. Chan, A. K. Wong, and E. Y. Lam, “Inverse synthesis of phase-shifting mask for optical lithography,” in OSA Topical Meeting in Signal Recovery and Synthesis, p. SMD3 (2007).
- V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
- A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” Proc. SPIE 6520, 65202Q (2007).
- N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” Proc. SPIE 7520, 752032 (2009).
- N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask design with defocus variation using inverse synthesis,” Proc. SPIE 7140, 71401W (2008).
- S. Osher, and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001).
- J. A. Sethian, and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. 10, 167–184 (1997).
- S. Osher, and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer Verlag New York, NJ, USA, 2003).
- F. Santosa, ““A level-set approach for inverse problems involving obstacles,” ESAIM Contr¨ole Optim,” Calc. Var. 1, 17–33 (1996).
- S. Osher, and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171(1), 272–288 (2001).
- A. Marquina, and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comput. 22, 387–405 (2000).
- L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” Proc. SPIE 6924, 69240T (2008).
- Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009).
- E. Y. Lam, and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009). [PubMed]
- E. Y. Lam, and A. K. Wong, ““Nebulous hotspot and algorithm variability in computation lithography,” J. Micro/ Nanolithogr,” MEMS MOEMS 9(3), 033002 (2010).
- N. Jia, and E. Y. Lam, “Machine learning for inverse lithography: Using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010).
- Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE 7748, 77481U (2010).
- M. Born, and E. Wolf, Principles of Optics (Pergamon Press Oxford, 1980).
- J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 2000).
- H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. 217A(1130), 408–432 (1953).
- A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, Bellingham, WA, 2005).
- B. E. A. Saleh, and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982). [PubMed]
- X. Ma, and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009). [PubMed]
- B. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis, Groningen University (1942).
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976).
- P. Dirksen, J. Braat, A. Janssen, and A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory,” Proc. SPIE 5754, 263 (2005).

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