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

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References

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  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 6154, 61541M (2006).
    [CrossRef]
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    [CrossRef]
  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]
  5. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  18. N. B. Cobb, Fast optical and process proximity correction algorithms for integrated circuit manufacturing (University of California at Berkeley, Berkely, California, 1998).
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    [CrossRef] [PubMed]
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    [CrossRef]
  23. M. Minoux, Mathematical programming theory and algorithms (John Wiley and Sons, Chichester, 1986).
  24. E. Hecht, Optics, 4thed (Addison Wesley, San Francisco, 2002).

2009 (2)

2008 (4)

2007 (2)

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]

2006 (4)

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

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]

2001 (1)

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]

1995 (1)

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]

1985 (1)

1982 (1)

1981 (1)

B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).

Abrams, D.

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]

Abrams, D. S.

D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE 6154, 534–542 (2006).

Arce, G.

Arce, G. R.

Chan, S. H.

Chao, H.-Y.

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]

Chen, J.

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]

Dai, G.

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]

De Leone, R.

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]

Granik, Y.

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

Guo, E.

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]

Huang, C.

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]

Huang, W.

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]

Hung, C.

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]

Ku, Y.

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]

Kuo, C.

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]

Lam, E. Y.

Lin, B.

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]

Lin, C.

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]

Lin, J.

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]

Liu, R.

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]

Liu, Y.

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]

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]

Ma, X.

Milanfar, P.

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]

Nashold, K.

Pang, L.

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]

Poonawala, A.

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]

Rabbani, M.

Saleh, B.

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).

Saleh, B. E. A.

Sayegh, S.

B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).

Sherif, S.

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]

Wang, K.

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]

Wong, A. K.

Wong, A. K. K.

Yu, J.-C.

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]

Yu, P.

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]

Zhang, B.

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]

Appl. Opt. (1)

IEEE Trans. Image Process. (2)

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]

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]

J. Microlith. Microfab. Microsyst. (1)

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

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).

Opt. Express (5)

Proc. SPIE (5)

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]

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]

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]

Other (7)

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).

J. S. Leon, Linear Algebra with applications, 6th ed. (Prentice-Hall, 2002).

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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Figures (6)

Fig. 1
Fig. 1

Steepest descent algorithm. Three parameters γR , γI and γC are used to bias the relative cost of aerial image, resist image and image contrast.

Fig. 2
Fig. 2

The inverse results of a large isolated contact evaluated by every cost function component individually where (a) (γI, γR, γC ) = (1, 0, 0), (b) (γI, γR, γC ) = (0, 1, 0) and (c) (γI, γR, γC ) = (0, 0, 1). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.

Fig. 3
Fig. 3

The inverse results of a large isolated contact as with different combination of cost functions components where (a) (γI, γR, γC ) = (1, 1, 0), (b) (γI, γR, γC ) = (1, 0, 1), (c) (γI, γR, γC ) = (0, 1, 1) and (d) (γI, γR, γC ) = (1, 1, 1). The corresponding contours and aerial images of (a)-(d) are shown in (e)-(h) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.

Fig. 4
Fig. 4

The inverse results of two close contacts evaluated by every cost function component individually where (a) (γI, γR, γC ) = (1, 0, 0), (b) (γI, γR, γC ) = (0, 1, 0) and (c) (γI, γR, γC ) = (0, 0, 1). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.

Fig. 5
Fig. 5

The inverse results of two close contacts with different combination of cost functions components where (a) (γI, γR, γC ) = (1, 1, 0), (b) (γI, γR, γC ) = (1, 0, 1), (c) (γI, γR, γC ) = (0, 1, 1) and (d) (γR, γI, γC ) = (1, 1, 1). The corresponding contours and aerial images of (a)-(d) are shown in (e)-(h) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.

Fig. 6
Fig. 6

The inverse results of two close contacts with different combinations of cost function components where (a) (γI, γR, γC ) = (1, 0, 32.8947), (b) (γI, γR, γC ) = (0.0041, 1, 0) and (c) (γI, γR, γC ) = (0, 1, 0.0447). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.

Equations (17)

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E q ( x , y ) = φ q ( x , y ) o ( x , y ) ,
I ( x , y ) = q = 1 n λ q | E q ( x , y ) | 2 ,
T ( I ) = 1 1 + e a ( I t r ) .
T I = I x x ^ + I y y ^ ,
T I = ( D I ) x ^ + ( I D T ) y ^ ,
D = [ 1 1 O 1 1 O 1 1 1 1 ] N × N ,
I t ( r ) = { α 1 + e b ( r r 0 ) , G e o m e t r i c C e n t e r r B o u n d a r y , e c ( r r 0 ) , r B o u n d a r y ,
F I = I t I 2 ,
F R = T ( I t ) T ( I ) 2 ,
F C = x ( I t I a ) 2 + y ( I t I a ) 2 ,
F = γ I F I + γ R F R + γ C F C .
o ^ = argmin o ( x , y ) F ( o ( x , y ) ) ,
o = 1 + cos ( θ ) 2 .
F = γ I F I + γ R F R + γ C F C .
F I ( θ ) = ( q = 1 Q [ 2 ( I t I ) 2 E q ] φ q f l i p ) ( sin ( θ ) 2 ) ,
F R ( θ ) = ( q Q [ 2 ( T ( I t ) T ( I ) ) a ( 1 T ( I ) ) T ( I ) 2 E q ] φ q f l i p ) ( sin ( θ ) 2 ) ,
F C ( θ ) = ( q = 1 Q [ ( 2 D T ( D ( I t I ) ) 2 ( ( I t I ) D T ) D ) 2 E q ] φ q f l i p ) ( sin ( θ ) 2 ) ,

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