Abstract

Phase-shifting masks (PSM) are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. Recently, a set of gradient-based PSM optimization methods have been developed to solve for the inverse lithography problem under coherent illumination. Most practical lithography systems, however, use partially coherent illumination due to non-zero width and off-axis light sources, which introduce partial coherence factors that must be accounted for in the optimization of PSMs. This paper thus focuses on developing a framework for gradient-based PSM optimization methods which account for the inherent nonlinearities of partially coherent illumination. In particular, the singular value decomposition (SVD) is used to expand the partially coherent imaging equation by eigenfunctions into a sum of coherent systems (SOCS). The first order coherent approximation corresponding to the largest eigenvalue is used in the PSM optimization. In order to influence the solution patterns to have more desirable manufacturability properties and higher fidelity, a post-processing of the mask pattern based on the 2D discrete cosine transformation (DCT) is introduced. Furthermore, a photoresist tone reversing technique is exploited in the design of PSMs to project extremely sparse patterns.

© 2008 Optical Society of America

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References

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  1. A. K. Wong, Resolution enhancement techniques, 1 (SPIE Press, 2001).
    [CrossRef]
  2. S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).
  3. F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," in Proc. SPIE 5377, 1-20 (2004).
    [CrossRef]
  4. F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).
  5. L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
    [CrossRef]
  6. A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007).
    [CrossRef] [PubMed]
  7. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
    [CrossRef]
  8. Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
    [CrossRef]
  9. Y. C. Pati and T. Kailath, "Phase-shifting masks for microlithography: Automated design and mask requirements," J. Opt. Soc. Am. A 11, 2438-2452 (1994).
    [CrossRef]
  10. A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A non-linear optimization approach," in Proc. SPIE, 6154, 1159-1172 (San Jose, CA, 2006).
  11. X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," in Proc. SPIE (San Jose, CA, 2007).
  12. X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007).
    [CrossRef]
  13. B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
    [CrossRef]
  14. E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38 (1981).
    [CrossRef]
  15. P. S. Davids and S. B. Bollepalli, "Generalized inverse problem for partially coherent projection lithography," in Proc. SPIE (San Jose, CA, 2008).
    [CrossRef]
  16. X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," in Proc. SPIE (Taiwan, 2008).
    [CrossRef]
  17. X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008).
    [CrossRef]
  18. N. Cobb, "Fast optical and process proximity correction algorithms for integrated circuit manufacturing," Ph.D. thesis, University of California at Berkeley (1998).
  19. J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, "Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT)," in Simulation of Semiconductor Processes and Devices, 12 (2007).
    [CrossRef]
  20. 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 (1982).
    [CrossRef] [PubMed]
  21. M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).
  22. R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (John Wiley and Sons, New York, 1995).
  23. N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE,  2440, 313-327 (1995).
  24. C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
    [CrossRef]

2008 (1)

X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008).
[CrossRef]

2007 (2)

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007).
[CrossRef] [PubMed]

2001 (1)

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

1996 (1)

B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
[CrossRef]

1995 (1)

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE,  2440, 313-327 (1995).

1994 (1)

1992 (1)

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
[CrossRef]

1982 (2)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

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 (1982).
[CrossRef] [PubMed]

1981 (1)

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38 (1981).
[CrossRef]

Arce, G. R.

X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008).
[CrossRef]

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007).
[CrossRef]

Cobb, N.

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE,  2440, 313-327 (1995).

Dunham, T.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Kailath, T.

Lavin, M.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Leipold, W.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Liebmann, L.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Liu, Y.

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
[CrossRef]

Ma, X.

X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008).
[CrossRef]

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007).
[CrossRef]

Mansfield, S.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Milanfar, P.

A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007).
[CrossRef] [PubMed]

Pati, Y. C.

Poonawala, A.

A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007).
[CrossRef] [PubMed]

Rabbani, M.

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 (1982).
[CrossRef] [PubMed]

Rosen, J.

B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
[CrossRef]

Saleh, B. E. A.

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 (1982).
[CrossRef] [PubMed]

Salik, B.

B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
[CrossRef]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

Wolf, E.

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38 (1981).
[CrossRef]

Wong, A.

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

Yariv, A.

B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
[CrossRef]

Zakhor, A.

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE,  2440, 313-327 (1995).

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
[CrossRef]

Appl. Opt. (1)

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 (1982).
[CrossRef] [PubMed]

IBM J. Res. Dev. (1)

L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, "TCAD development for lithography resolution enhancement," IBM J. Res. Dev. 45, 651-665 (2001).
[CrossRef]

IEEE Trans. Electron Devices (1)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
[CrossRef]

IEEE Trans. Image Process. (1)

A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007).
[CrossRef] [PubMed]

IEEE Trans. Semicond. Manuf. (1)

Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992).
[CrossRef]

J. Opt. Soc. Am. A (3)

X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008).
[CrossRef]

B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996).
[CrossRef]

Y. C. Pati and T. Kailath, "Phase-shifting masks for microlithography: Automated design and mask requirements," J. Opt. Soc. Am. A 11, 2438-2452 (1994).
[CrossRef]

Opt. Commun. (1)

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38 (1981).
[CrossRef]

Opt. Express (1)

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007).
[CrossRef]

Proc. SPIE (1)

N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE,  2440, 313-327 (1995).

Other (13)

C. Vogel, Computational methods for inverse problems (SIAM Press, 2002).
[CrossRef]

M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).

R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (John Wiley and Sons, New York, 1995).

P. S. Davids and S. B. Bollepalli, "Generalized inverse problem for partially coherent projection lithography," in Proc. SPIE (San Jose, CA, 2008).
[CrossRef]

X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," in Proc. SPIE (Taiwan, 2008).
[CrossRef]

N. Cobb, "Fast optical and process proximity correction algorithms for integrated circuit manufacturing," Ph.D. thesis, University of California at Berkeley (1998).

J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, "Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT)," in Simulation of Semiconductor Processes and Devices, 12 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A non-linear optimization approach," in Proc. SPIE, 6154, 1159-1172 (San Jose, CA, 2006).

X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," in Proc. SPIE (San Jose, CA, 2007).

A. K. Wong, Resolution enhancement techniques, 1 (SPIE Press, 2001).
[CrossRef]

S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).

F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," in Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).

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Figures (12)

Fig. 1.
Fig. 1.

Optical lithography system with partially coherent illuminations

Fig. 2.
Fig. 2.

A partially coherent system represented by a SVD decomposition of a sum of coherent systems.

Fig. 3.
Fig. 3.

Eigenvalues α k of sum of coherent systems decomposition by SVD.

Fig. 4.
Fig. 4.

Fig. 4. (a) The amplitude of the first equivalent kernel corresponding to the largest eigen-value |ϕ1(x,y)|, (b) second equivalent kernel corresponding to the second largest eigenvalue |ϕ2(x,y)|.

Fig. 5.
Fig. 5.

Approximated forward process model.

Fig. 6.
Fig. 6.

Top row (input masks), left to right: desired pattern, optimized real-valued mask and optimized trinary mask. The bottom row illustrates the corresponding binary output patterns. White, grey and black represent 1, 0 and -1, respectively. σ=0.3.

Fig. 7.
Fig. 7.

Top row (input masks), left to right: desired pattern, optimized real-valued mask and optimized trinary mask. The bottom row illustrates the corresponding binary output patterns. White, grey and black represent 1, 0 and -1, respectively. σ=0.6.

Fig. 8.
Fig. 8.

The relationship between the number of maintained DCT low frequency components and the output pattern errors.

Fig. 9.
Fig. 9.

Left to right: output pattern when the desired pattern is inputted, post-processed trinary mask with the DCT post-processing maintaining 136 low frequency components, and the binary output pattern of the post-processed trinary mask. White, grey and black represent 1, 0 and -1, respectively.

Fig. 10.
Fig. 10.

Left to right: desired pattern, output pattern when the desired pattern is inputted, optimized trinary mask, and the output pattern of the optimized trinary mask. White, grey and black represent 1, 0 and -1, respectively. σ=0.3.

Fig. 11.
Fig. 11.

Left to right: Photoresist distribution, optimized trinary mask using the photoresist tone reversing method, and the output pattern of the optimized trinary mask. White and black represent positive and negative photoresist, respectively in the first figure. White, grey and black represent 1, 0 and -1, respectively in the second and the third figures. σ=0.3.

Fig. 12.
Fig. 12.

Left to right: Photoresist distribution, post-processed trinary mask using the DCT post-processing maintaining 276 low frequency components, and the binary output pattern of the post-processed trinary mask. White and black represent positive and negative photoresist, respectively in the first figure. White, grey and black represent 1, 0 and -1, respectively in the second and the third figures. σ=0.3.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

I ( r ) = + M ( r 1 ) M ( r 2 ) γ ( r 1 r 2 ) h * ( r r 1 ) h ( r r 2 ) d r 1 d r 2 ,
I ( x , y ) = + TCC ( f 1 , g 1 ; f 2 , g 2 ) M ˜ ( f 1 , g 1 ) M ˜ * ( f 2 , g 2 )
× exp { i 2 π [ ( f 1 f 2 ) x + ( g 1 g 2 ) y ] } d f 1 d g 1 d f 2 d g 2 ,
TCC ( f 1 , g 1 ; f 2 , g 2 ) = + γ ˜ ( f , g ) h ˜ ( f + f 1 , g + g 1 ) h ˜ * ( f + f 2 , g + g 2 ) d f d g ,
I ( m , n ) = s ˜ H A s ˜ , m , n = 1 , 2 , , N ,
s ˜ i = M ˜ ( p , q ) exp [ i 2 π ( pm + qn ) ] i = 1 , 2 , , N 2 ,
I ( m , n ) = k = 1 N α k s ˜ T V k 2 .
h ˜ k ( p , q ) = S 1 ( V k ) = ( V k , 1 V k , N + 1 V k , N ( N 1 ) + 1 V k , 2 V k , N + 2 V k , N ( N 1 ) + 2 V k , N V k , 2 N V k , N 2 ) ,
h k ( m , n ) = IFFT { h ˜ k ( p , q ) } , m , n = 1 , 2 , , N .
I ( m , n ) = k = 1 N 2 α k h k ( m , n ) M ( m , n ) 2 .
I ( m , n ) k = 1 i α k h k ( m , n ) M ( m , n ) 2 , i = 1 , 2 , , N 2 .
γ ˜ ( f , g ) = λ 2 π ( σ NA ) 2 circ ( λ f 2 + g 2 σ NA )
= { λ 2 π ( σ NA ) 2 for f 2 + g 2 σ NA λ 0 elsewhere ,
h ( r ) = h ( x , y ) = J 1 ( 2 π r NA λ ) 2 π r NA λ .
h ˜ ( f , g ) = λ 2 π ( NA ) 2 circ ( λ f 2 + g 2 NA ) = { λ 2 π ( NA ) 2 for f 2 + g 2 NA λ 0 elsewhere .
D = d ( Z ( m , n ) , Z * ( m , n ) ) = d ( T { M ( m , n ) } , Z * ( m , n ) )
M ̂ ( m , n ) = arg min M ( m , n ) N × N d ( T { M ( m , n ) } , Z * ( m , n ) ) .
sig ( x ) = 1 1 + exp [ a ( x t r ) ] ,
M ̂ = arg min M d ( sig { H 1 { m ̲ } 2 } , Z * ) .
z ̲ i = 1 1 + exp [ a i = 1 N 2 h 1 , ij m ̲ j 2 + a t r ] , i = 1 , N 2 ,
m ̲ ̂ = arg min m ̂ ̲ { F ( m ̲ ) } ,
F ( m ̲ ) = z ̲ * z ̲ 2 2 = i = 1 N 2 ( z ̲ i 2 z ̲ i ) 2 ,
( θ ̂ ̲ ) = arg min θ ̲ { F ( θ ̲ ) }
= arg min θ ̲ { i = 1 N 2 ( z i * 1 1 + exp [ a j = 1 N 2 h 1 , ij cos θ ̲ j 2 + a t r ] ) 2 } .
F ( θ ̲ ) = d ̲ θ ̲ = 2 a × sin θ ̲ { ( H 1 * H [ ( z ̲ * z ̲ ) z ̲ ( 1 ̲ z ̲ ) ( H 1 * m ̲ ) ] }
+ 2 a × sin θ ̲ { ( H 1 H [ ( z ̲ * z ̲ ) z ̲ ( 1 ̲ z ̲ ) ( H 1 m ̲ ) ] } ,
θ ̲ k + 1 = θ ̲ k s θ ̲ d ̲ θ ̲ k ,
E = i = 1 N 2 z ̲ i * z ̲ bi 2 = i = 1 N 2 z ̲ i * T { M tri } 2 .
m ̂ ̲ = arg min m ̂ ̲ { F ( m ̲ ) + γ R ( m ̲ ) } ,
r D ( m ̲ i ) = 4.5 m ̲ i 4 + m ̲ i 2 + 3.5 , i = 1 , , N 2 .
R = k λ NA = 41.5 nm .

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