## Impacts of cost functions on inverse lithography patterning |

Optics Express, Vol. 18, Issue 22, pp. 23331-23342 (2010)

http://dx.doi.org/10.1364/OE.18.023331

Acrobat PDF (1433 KB)

### Abstract

For advanced CMOS processes, inverse lithography promises better patterning fidelity than conventional mask correction techniques due to a more complete exploration of the solution space. However, the success of inverse lithography relies highly on customized cost functions whose design and know-how have rarely been discussed. In this paper, we investigate the impacts of various objective functions and their superposition for inverse lithography patterning using a generic gradient descent approach. We investigate the most commonly used objective functions, which are the resist and aerial images, and also present a derivation for the aerial image contrast. We then discuss the resulting pattern fidelity and final mask characteristics for simple layouts with a single isolated contact and two nested contacts. We show that a cost function composed of a dominant resist-image component and a minor aerial-image or image-contrast component can achieve a good mask correction and contour targets when using inverse lithography patterning.

© 2010 OSA

## 1. Introduction

3. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE **6283**, 62830X (2006). [CrossRef]

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). [CrossRef] [PubMed]

4. 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). [CrossRef] [PubMed]

7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. **5**, 043002 (2006). [CrossRef]

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). [CrossRef] [PubMed]

11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE **7140**, 714014 (2008). [CrossRef]

7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. **5**, 043002 (2006). [CrossRef]

9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**(23), 15066–15079 (2007). [CrossRef] [PubMed]

12. X. Ma and G. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A **25**(12), 2960–2970 (2008). [CrossRef]

14. 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). [CrossRef] [PubMed]

## 2. Methodology

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

21. 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). [CrossRef] [PubMed]

*E*and

*I*represent the electrical field and image intensity respectively.

*q*

^{th}optical kernel, ⊗ denotes the convolution calculation, and

**the mask function.**

*o**λ*is the eigenvalue of the

_{q}*q*

^{th}kernel with

*n*kernels in total.

22. W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE **5377**, 1536–1543 (2001). [CrossRef]

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

9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**(23), 15066–15079 (2007). [CrossRef] [PubMed]

*a*describes the sensitivity of the photoresist reacting with the light, which depicts the slope of sidewall profiles, and

*t*is the parameter of the constant threshold level. Here, the value is set and normalized to 0.5.

_{r}*DI*performs the operation that calculates the difference in the row direction, and

*ID*

^{T}computes the column difference.

*I*(r) is the target image as a function of the pixel index

_{t}*r*.

*r*is arranged in the radial direction from the geometric center of a drawn pattern, where

*r*is the index of the boundary pixels. The generation of the pixel index is adopted from a wavefront expansion technique previously developed by us [11

_{0}11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE **7140**, 714014 (2008). [CrossRef]

11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE **7140**, 714014 (2008). [CrossRef]

*b*and

*c*are constants to adjust for the target aerial image profile. Here, we choose a target image that is unity within the drawn and zero outside, corresponding to α = 1,

*b*= -∞ and

*c*= ∞. Since the designed target is normalized to unity at the maximal, the optical model used for simulations must be calibrated with experimental values. Once the target aerial image is defined, the target resist image is obtained by the corresponding sigmoid transformation and the target image contrast is the differential operation of the target aerial image. As a result, the three objective functions that evaluate the differences between the desired and the calculated profiles can be expressed as in Eq. (8), Eq. (9) and Eq. (10), respectively:

*F*,

_{I}*F*and

_{R}*F*represent the costs for the aerial image, resist image, and aerial image contrast, respectively. The norm

_{C}*F*by assigning three weighting coefficients,

*F*,

_{I}*F*, and

_{R}*F*in Eq. (11), represent normalized objective functions.

_{C}**that minimizes a given constraint and therefore can be expressed as in Eq. (12):where**

*ô***is a mask function. A gradient-search method used to calculate**

*o***is explained next. Since the gradient operation calculates the derivatives of the mask, the discrete binary mask described by**

*ô***, must be first parameterized by a continuous variable**

*o***in order to obtain an analyzable form. Here, a sinusoidal transformation is employed to convert a binary drawn mask in to a continuous grey-level mask [8**

*θ*8. A. Poonawala and P. Milanfar, “Mask design for optical microlithography--an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [CrossRef] [PubMed]

**varying between zero and**

*θ***. Moreover, Eq. (13) can also be extended to a phase-shift mask (PSM) intuitively by multiplying a complex phase term [9**

*π*9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**(23), 15066–15079 (2007). [CrossRef] [PubMed]

7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. **5**, 043002 (2006). [CrossRef]

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

**15**(23), 15066–15079 (2007). [CrossRef] [PubMed]

**16**(3), 774–788 (2007). [CrossRef] [PubMed]

## 3. Results and discussion

*Rayleigh*criterion:

*R*= 0.61

*λ*/NA ~170nm, while for the pair, both the spacing and dimensions are below the limit. Both layouts are represented in 256 × 256 pixel tiles with a pixel dimension of 10 nm. An annular illumination source is employed with σ

_{in}= 0.4 and σ

_{out}= 0.7. The partially coherent illumination model contains eight kernels, where

*λ*= 193nm,

*NA*= 0.7. Moreover, the constant threshold for the aerial image intensity is set and normalized to 0.5. Therefore the threshold parameter

*t*, required for the resist image transformation in Eq. (3) is set to 0.5, while

_{r}*a*is chosen to be 90 to represent a conventional resist profile. We discuss the impact that the previously defined cost function components in various superposition configurations have on the resulting characteristics of the corrected grey-level masks, the contours and the aerial image intensities.

*γ*) = (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively. The corresponding aerial images and contours of the single large contact are shown in Fig. 2(d)–2(f). As shown in Fig. 2(a), the cost function based on only the aerial-image component results in mask features concentrated on the drawn pattern, while in Fig. 2(b) the correction shows very large serifs, and in Fig. 2(c) concentric rings. The mask features very much reveal the nature of each objective function as discussed below.

_{I}, γ_{R}, γ_{C}*γ*) = (1, 1, 0), (1, 0, 1), (0, 1, 1), and (1, 1, 1), respectively. When dealing with combined objective functions, we find that while the resulting mask features have the footprint from both components, one component always appears dominant to the others. So, in Fig. 3(a), 3(b), and 3(d) the aerial image dominates in the cost function, while in Fig. 3(c), the aerial image contrast dominates the resist image. The resulting contours and aerial image intensities are thus also dictated by the dominant objective function, as shown in Fig. 3(e)–3(h). However, since all the linear combinations of the cost function components can result in the exposure of photoresist for the large isolated contact, it is not conclusive as to what combination is best for inverse lithography patterning.

_{I}, γ_{R}, γ_{C}## 4. Conclusion

## Acknowledgments

## References and links

1. | D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE |

2. | C. Hung, B. Zhang, E. Guo, L. Pang, Y. Liu, K. Wang, and G. Dai, “Pushing the lithography limit: Applying inverse lithography technology (ILT) at the 65nm generation,” Proc. SPIE |

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

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

5. | K. Nashold and B. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A |

6. | B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. |

7. | Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. |

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

9. | X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express |

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. | J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE |

12. | X. Ma and G. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A |

13. | X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express |

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

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

16. | J. W. Goodman, |

17. | A. K. Wong, |

18. | N. B. Cobb, |

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

20. | J. S. Leon, |

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

22. | W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE |

23. | M. Minoux, |

24. | E. Hecht, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.3960) Imaging systems : Microlithography

(110.1758) Imaging systems : Computational imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: August 23, 2010

Revised Manuscript: October 11, 2010

Manuscript Accepted: October 15, 2010

Published: October 20, 2010

**Citation**

Jue-Chin Yu and Peichen Yu, "Impacts of cost functions on inverse lithography patterning," Opt. Express **18**, 23331-23342 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23331

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### References

- D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE 6154, 534–542 (2006).
- C. Hung, B. Zhang, E. Guo, L. Pang, Y. Liu, K. Wang, and G. Dai, “Pushing the lithography limit: Applying inverse lithography technology (ILT) at the 65nm generation,” Proc. SPIE 6154, 61541M (2006). [CrossRef]
- L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE 6283, 62830X (2006). [CrossRef]
- 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). [CrossRef] [PubMed]
- K. Nashold and B. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A 2(5), 635–643 (1985). [CrossRef]
- B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
- Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5, 043002 (2006). [CrossRef]
- A. Poonawala and P. Milanfar, “Mask design for optical microlithography--an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007). [CrossRef] [PubMed]
- X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE 7140, 714014 (2008). [CrossRef]
- X. Ma and G. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008). [CrossRef]
- X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express 16(24), 20126–20141 (2008). [CrossRef] [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). [CrossRef] [PubMed]
- M. Born, and E. Wolf, Principles of Optics, 7th(expanded) ed. (Cambridge University Press, 1999).
- J. W. Goodman, Statistical Optics (John Wiley and Sons, 1985).
- A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).
- N. B. Cobb, Fast optical and process proximity correction algorithms for integrated circuit manufacturing (University of California at Berkeley, Berkely, California, 1998).
- E. Y. Lam and A. K. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009). [CrossRef] [PubMed]
- J. S. Leon, Linear Algebra with applications, 6th ed. (Prentice-Hall, 2002).
- 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). [CrossRef] [PubMed]
- W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE 5377, 1536–1543 (2001). [CrossRef]
- M. Minoux, Mathematical programming theory and algorithms (John Wiley and Sons, Chichester, 1986).
- E. Hecht, Optics, 4thed (Addison Wesley, San Francisco, 2002).

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