Abstract

Aerial image through focus in the presence of aberrations and electromagnetic edge effects modeled by adding ±π/2 phase at pattern edges is expanded by a quadratic equation with respect to focus. The quadratic equation is expressed by four coefficients that are adequately independent of both mask layout and the variations in the optical setting in projection printing, thus saving the computation cost of the quadratic fit for each individual layout edge position in a new mask pattern or variation from a nominal optical setting. The error of this method is less than 1% for any typical integrated circuit features. This accuracy holds when the defocus is less than one Rayleigh unit (0.5λ/NA2, where λ is a wavelength and NA is the numerical aperture) and the root mean square of the existing aberration is less than 0.02λ, which encompasses current lithography practice. More importantly, the method is a foundation for future first-cut accurate algebraic imaging models that have sufficient speed for assessing the desired or undesired changes in the through-focus images of millions of features as the optical system conditions change. These optical system changes occur naturally across the image field, and aberration levels are even programmed in tuning modern tooling to compensate for electromagnetic mask edge effects.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010 (4)

M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. dissertation (University of California, Berkeley, 2010), Chap. 3.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Am. A 27, 1311–1321(2010).
[CrossRef]

K. Yamazoe, “Fast fine-pixel aerial image calculation in partially coherent imaging by matrix representation of modified Hopkins equation,” Appl. Opt. 49, 3909–3915 (2010).
[CrossRef] [PubMed]

2009 (1)

C. C. Hu, Modern Semiconductor Devices for Integrated Circuits (Prentice-Hall, 2009), p. 284.

2008 (1)

2007 (3)

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Microlithogr., Microfabr., Microsyst. 6, 031004 (2007).
[CrossRef]

C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley & Sons, 2007), Chap. 3.
[CrossRef]

L. T.-N. Wang and A. R. Neureuther, “Lateral interactions between standard cells using pattern matching,” Proc. SPIE 6730, 673010 (2007).
[CrossRef]

2006 (2)

J. Tirapu-Azpiroz and E. Yablonovitch, “Incorporating mask topography edge diffraction in photolithography simulations,” J. Opt. Soc. Am. A 23, 821–828 (2006).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “Fast lithography simulation under focus variations for OPC and layout optimizations,” Proc. SPIE 6156, 615618 (2006).
[CrossRef]

2005 (1)

A. K. Wong, Optical Imaging in Projection Microlithography, Vol.  TT66, SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.
[CrossRef]

2002 (3)

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr., Microfabr., Microsyst. 1, 253–269(2002).
[CrossRef]

D. Fuard, M. Besacier, and P. Schiavone, “Assessment of different simplified resist models,” Proc. SPIE 4691, 1266–1277(2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2002), Chap. 10.

2001 (3)

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

A. K. Wong, Resolution Enhancement Techniques in Optical Microlithography, Vol.  TT47, SPIE Tutorial Texts in Optical Engineering (SPIE, 2001), Chap. 2.
[CrossRef]

H. J. Levinson, Principles of Lithography, 1st ed. (SPIE, 2001), Chap. 2.

1998 (1)

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California, Berkeley, 1998), Chap. 4.

1994 (1)

B. J. Lin, “The exposure-defocus forest,” Jpn. J. Appl. Phys. 33, 6756–6764 (1994).
[CrossRef]

1970 (1)

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Opt. Acta 17, 337–347 (1970).
[CrossRef]

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

Adam, K.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr., Microfabr., Microsyst. 1, 253–269(2002).
[CrossRef]

Barakat, R.

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Opt. Acta 17, 337–347 (1970).
[CrossRef]

Baselmans, J.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Besacier, M.

D. Fuard, M. Besacier, and P. Schiavone, “Assessment of different simplified resist models,” Proc. SPIE 4691, 1266–1277(2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2002), Chap. 10.

Cobb, N. B.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California, Berkeley, 1998), Chap. 4.

Conley, W.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Cummings, K.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Flagello, D.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Foster, J.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Fuard, D.

D. Fuard, M. Besacier, and P. Schiavone, “Assessment of different simplified resist models,” Proc. SPIE 4691, 1266–1277(2002).
[CrossRef]

Garza, C. M.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Hamatani, M.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

Hu, C. C.

C. C. Hu, Modern Semiconductor Devices for Integrated Circuits (Prentice-Hall, 2009), p. 284.

Iriuchijima, Y.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Ishikawa, J.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Kohno, H.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Kosugi, J.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Levinson, H. J.

H. J. Levinson, Principles of Lithography, 1st ed. (SPIE, 2001), Chap. 2.

Lin, B. J.

B. J. Lin, “The exposure-defocus forest,” Jpn. J. Appl. Phys. 33, 6756–6764 (1994).
[CrossRef]

Mack, C. A.

C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley & Sons, 2007), Chap. 3.
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “Fast lithography simulation under focus variations for OPC and layout optimizations,” Proc. SPIE 6156, 615618 (2006).
[CrossRef]

Miller, M.

M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. dissertation (University of California, Berkeley, 2010), Chap. 3.

Neureuther, A. R.

L. T.-N. Wang and A. R. Neureuther, “Lateral interactions between standard cells using pattern matching,” Proc. SPIE 6730, 673010 (2007).
[CrossRef]

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr., Microfabr., Microsyst. 1, 253–269(2002).
[CrossRef]

Pan, D. Z.

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Microlithogr., Microfabr., Microsyst. 6, 031004 (2007).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “Fast lithography simulation under focus variations for OPC and layout optimizations,” Proc. SPIE 6156, 615618 (2006).
[CrossRef]

Roman, B.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Schiavone, P.

D. Fuard, M. Besacier, and P. Schiavone, “Assessment of different simplified resist models,” Proc. SPIE 4691, 1266–1277(2002).
[CrossRef]

Schippers, M.

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

Shi, S. X.

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Microlithogr., Microfabr., Microsyst. 6, 031004 (2007).
[CrossRef]

Shibazaki, Y.

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

Tirapu-Azpiroz, J.

Wang, L. T.-N.

L. T.-N. Wang and A. R. Neureuther, “Lateral interactions between standard cells using pattern matching,” Proc. SPIE 6730, 673010 (2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2002), Chap. 10.

Wong, A. K.

A. K. Wong, Optical Imaging in Projection Microlithography, Vol.  TT66, SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.
[CrossRef]

A. K. Wong, Resolution Enhancement Techniques in Optical Microlithography, Vol.  TT47, SPIE Tutorial Texts in Optical Engineering (SPIE, 2001), Chap. 2.
[CrossRef]

Yablonovitch, E.

Yamazoe, K.

Yu, P.

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Microlithogr., Microfabr., Microsyst. 6, 031004 (2007).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “Fast lithography simulation under focus variations for OPC and layout optimizations,” Proc. SPIE 6156, 615618 (2006).
[CrossRef]

Appl. Opt. (1)

J. Microlithogr., Microfabr., Microsyst. (2)

P. Yu, S. X. Shi, and D. Z. Pan, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Microlithogr., Microfabr., Microsyst. 6, 031004 (2007).
[CrossRef]

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlithogr., Microfabr., Microsyst. 1, 253–269(2002).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

B. J. Lin, “The exposure-defocus forest,” Jpn. J. Appl. Phys. 33, 6756–6764 (1994).
[CrossRef]

Opt. Acta (1)

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Opt. Acta 17, 337–347 (1970).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

Proc. SPIE (5)

H. Kohno, Y. Shibazaki, J. Ishikawa, J. Kosugi, Y. Iriuchijima, and M. Hamatani, “Latest performance of immersion scanner S620D with the Streamlign platform for the double patterning generation,” Proc. SPIE 7640, 76401O (2010).
[CrossRef]

C. M. Garza, W. Conley, B. Roman, M. Schippers, J. Foster, J. Baselmans, K. Cummings, and D. Flagello, “Ring test aberration determination and device lithography correlation,” Proc. SPIE 4346, 36–44 (2001).
[CrossRef]

L. T.-N. Wang and A. R. Neureuther, “Lateral interactions between standard cells using pattern matching,” Proc. SPIE 6730, 673010 (2007).
[CrossRef]

D. Fuard, M. Besacier, and P. Schiavone, “Assessment of different simplified resist models,” Proc. SPIE 4691, 1266–1277(2002).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “Fast lithography simulation under focus variations for OPC and layout optimizations,” Proc. SPIE 6156, 615618 (2006).
[CrossRef]

Other (8)

M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. dissertation (University of California, Berkeley, 2010), Chap. 3.

A. K. Wong, Optical Imaging in Projection Microlithography, Vol.  TT66, SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.
[CrossRef]

C. C. Hu, Modern Semiconductor Devices for Integrated Circuits (Prentice-Hall, 2009), p. 284.

A. K. Wong, Resolution Enhancement Techniques in Optical Microlithography, Vol.  TT47, SPIE Tutorial Texts in Optical Engineering (SPIE, 2001), Chap. 2.
[CrossRef]

H. J. Levinson, Principles of Lithography, 1st ed. (SPIE, 2001), Chap. 2.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2002), Chap. 10.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California, Berkeley, 1998), Chap. 4.

C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley & Sons, 2007), Chap. 3.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Example of optical setting. The gray circles in both figures correspond to the pupil edge. (a) Illumination. Each pixel shows a mutually incoherent point source. (b) Randomly generated wavefront aberration formed by Zernike polynomials up to thirty-sixth term, which has 0.01 λ RMS.

Fig. 2
Fig. 2

Test layout. Patterns are placed on a clear background. Black features show 0.06% intensity transmittance with π phase shift. Gray features have 100% transmittance but add ± π / 2 phase to approximate the mask edge effects. The gray features have a 4 nm width with a π / 2 phase for the X direction and an 8 nm width with a π / 2 phase for the Y direction. The dotted line shows the enlarged view of a portion of the test layout.

Fig. 3
Fig. 3

Simulated image obtained by the source integration algorithm.

Fig. 4
Fig. 4

Error of Eq. (2) for the test simulation in Subsection 3B.

Fig. 5
Fig. 5

Error analysis of Eq. (2) by the condition in Subsection 3C. The error bar shows ± σ at each defocus plane, where σ is the standard deviation at each defocus value. (a) Test aberration is 0.01 λ RMS. (b) Test aberration is 0.02 λ RMS.

Fig. 6
Fig. 6

Error of the conventional quadratic fit.

Fig. 7
Fig. 7

Test layout to check the process window accuracy. The pitch of the line and space pattern is 100 nm . The black and gray features have the same definition as in Fig. 2. The scripts A, B, and C show the identification for each bar.

Fig. 8
Fig. 8

Plots of mean Δ CDs at different exposure doses for each bar in Fig. 7. Test aberration is 0.01 λ RMS. (a) The mean CD difference of bar A. (b) The mean CD difference of bar B. (c) The mean CD difference of bar C.

Fig. 9
Fig. 9

Error analysis of Eq. (14) by the conditions in Subsection 3C. The error bar shows ± σ at each defocus plane, where σ is the standard deviation at each defocus value. Test aberrations are (a)  0.01 λ and (b)  0.02 λ RMS.

Equations (14)

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

I ( x , y , 0 ) = a ^ | T | a ^ ,
I ( x , y , z ) a ^ | T | a ^ z 2 + a ^ | T | a ^ z + a ^ | T | a ^ = Q ( x , y ) z 2 + L ( x , y ) z + I ( x , y , 0 ) ,
Δ I i ( x , y ) = I ( x , y , 0 ) I ( x , y , δ i ) ,
Δ I i ( x , y ) = Q ( x , y ) δ i 2 L ( x , y ) δ i .
Q ( x , y ) = α 1 Δ I 1 ( x , y ) + β 1 Δ I 2 ( x , y ) ,
L ( x , y ) = α 2 Δ I 1 ( x , y ) + β 2 Δ I 2 ( x , y ) ,
α 1 = δ 2 δ 1 δ 2 ( δ 2 δ 1 ) ,
β 1 = δ 1 δ 1 δ 2 ( δ 2 δ 1 ) ,
α 2 = δ 2 2 δ 1 δ 2 ( δ 2 δ 1 ) ,
β 2 = δ 1 2 δ 1 δ 2 ( δ 2 δ 1 ) .
I ( x , y , z ) = [ 6.0575 × 10 1 Δ I 1 ( x , y ) 6.0575 × 10 1 Δ I 2 ( x , y ) ] z 2 + [ 5.5034 × 10 1 Δ I 1 ( x , y ) + 5.5034 × 10 1 Δ I 2 ( x , y ) ] z + I ( x , y , 0 ) .
e ( z ) = max [ I ( x , y , z ) I ( x , y , z ) ] min [ I ( x , y , z ) I ( x , y , z ) ] max [ I ( x , y , z ) ] min [ I ( x , y , z ) ] × 100 ,
Δ CD ( w , z ) = 1 100 i = 1 100 [ CD i ( w , z ) CD i ( w , z ) ] .
I ( x , y , z ) = i = 1 4 [ α i Δ I 1 ( x , y ) + β i Δ I 2 ( x , y ) + γ i Δ I 3 ( x , y ) + ε i Δ I 4 ( x , y ) ] z ( 5 i ) + I ( x , y , 0 ) .

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