Abstract

In deep ultraviolet lithography simulations, conventional application of Kirchhoff’s boundary conditions on the mask surface provides the so-called “thin-mask” approximation of the object field. Current subwavelength lithographic operation, however, places a serious limitation on this approximation, which fails to account for the topographical, or “thick-mask,” effects. In this paper, a new simulation model is proposed that is theoretically founded on the well-established physical theory of diffraction. This model relies on the key result that diffraction effects can be interpreted as an intrinsic edge property, and modeled with just two fixed parameters: width and transmission coefficient of a locally determined boundary layer applied to each chrome edge. The proposed model accurately accounts for thick-mask effects of the fields on the mask, greatly improving the accuracy of aerial image simulations in photolithography, while maintaining a reasonable computational cost.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).
  2. H. J. Levinson, Principles of Lithography (SPIE, 2001).
  3. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with phase shifting-mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982).
  4. C. Pierrat and A. Wong, "The MEF revisited: Low k1 effects versus mask topography effects," in Optical Microlithography XVI, A.Yen, ed., Proc. SPIE 5040, 193-202 (2003).
  5. M. S. Yeung and E. Barouch, "Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157 nm optical lithography," IEEE Electron Device Lett. 21, 433-435 (2000).
    [CrossRef]
  6. P. Y. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (Foreign Technology Division, Air Force Systems Command, 1971).
  7. K. Adam and A. R. Neureuther, "Simplified models for edge transitions in rigorous mask modeling," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4346, 331-344 (2001).
  8. A. Khoh, G. S. Samudra, W. Yihong, T. Milster, and B.-I. Choi, "Image formation by use of the geometrical theory of diffraction," J. Opt. Soc. Am. A 21, 959-967 (2004).
    [CrossRef]
  9. J. Tirapu-Azpiroz, "Analysis and modeling of photomask near-fields in subwavelength deep ultraviolet lithography," Ph.D. dissertation (University of California at Los Angeles, 2004).
  10. C. T. Tai, "Direct integration of field equations," Electromagn. Waves : Prog. Electromagn. Res. 28, 339-359 (2000).
    [CrossRef]
  11. P. Y. Ufimtsev, "Rubinowicz and the modern theory of diffracted rays," Electromagnetics 15, 547-565 (1995).
    [CrossRef]
  12. P. Y. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
    [CrossRef]
  13. J. Tirapu-Azpiroz and E. Yablonovitch, "Modeling of near-field effects in sub-wavelength deep ultraviolet lithography," in Future Trends of Microelectronics 2003, S.Luryi, J.Xu, and A.Zaslavsky, eds. (Wiley-IEEE, 2004), pp. 80-92.
  14. A. K. Wong and A. R. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Trans. Electron Devices 41, 895-902 (1994).
    [CrossRef]
  15. J. Tirapu-Azpiroz and E. Yablonovitch, "Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography," Proc. SPIE 5377, 1528-1535 (2004).
    [CrossRef]
  16. T. V. Pistor, A. R. Neureuther, and R. J. Socha, "Modeling oblique incidence effects in photomasks," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4000, 228-237 (2000).

2004

A. Khoh, G. S. Samudra, W. Yihong, T. Milster, and B.-I. Choi, "Image formation by use of the geometrical theory of diffraction," J. Opt. Soc. Am. A 21, 959-967 (2004).
[CrossRef]

J. Tirapu-Azpiroz and E. Yablonovitch, "Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography," Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

2000

M. S. Yeung and E. Barouch, "Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157 nm optical lithography," IEEE Electron Device Lett. 21, 433-435 (2000).
[CrossRef]

C. T. Tai, "Direct integration of field equations," Electromagn. Waves : Prog. Electromagn. Res. 28, 339-359 (2000).
[CrossRef]

1995

P. Y. Ufimtsev, "Rubinowicz and the modern theory of diffracted rays," Electromagnetics 15, 547-565 (1995).
[CrossRef]

1994

A. K. Wong and A. R. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

1991

P. Y. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

Adam, K.

K. Adam and A. R. Neureuther, "Simplified models for edge transitions in rigorous mask modeling," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4346, 331-344 (2001).

Barouch, E.

M. S. Yeung and E. Barouch, "Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157 nm optical lithography," IEEE Electron Device Lett. 21, 433-435 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

Choi, B.-I.

Khoh, A.

Levenson, M. D.

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

Levinson, H. J.

H. J. Levinson, Principles of Lithography (SPIE, 2001).

Milster, T.

Neureuther, A. R.

A. K. Wong and A. R. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

T. V. Pistor, A. R. Neureuther, and R. J. Socha, "Modeling oblique incidence effects in photomasks," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4000, 228-237 (2000).

K. Adam and A. R. Neureuther, "Simplified models for edge transitions in rigorous mask modeling," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4346, 331-344 (2001).

Pierrat, C.

C. Pierrat and A. Wong, "The MEF revisited: Low k1 effects versus mask topography effects," in Optical Microlithography XVI, A.Yen, ed., Proc. SPIE 5040, 193-202 (2003).

Pistor, T. V.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, "Modeling oblique incidence effects in photomasks," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4000, 228-237 (2000).

Samudra, G. S.

Simpson, R. A.

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

Socha, R. J.

T. V. Pistor, A. R. Neureuther, and R. J. Socha, "Modeling oblique incidence effects in photomasks," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4000, 228-237 (2000).

Tai, C. T.

C. T. Tai, "Direct integration of field equations," Electromagn. Waves : Prog. Electromagn. Res. 28, 339-359 (2000).
[CrossRef]

Tirapu-Azpiroz, J.

J. Tirapu-Azpiroz and E. Yablonovitch, "Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography," Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

J. Tirapu-Azpiroz and E. Yablonovitch, "Modeling of near-field effects in sub-wavelength deep ultraviolet lithography," in Future Trends of Microelectronics 2003, S.Luryi, J.Xu, and A.Zaslavsky, eds. (Wiley-IEEE, 2004), pp. 80-92.

J. Tirapu-Azpiroz, "Analysis and modeling of photomask near-fields in subwavelength deep ultraviolet lithography," Ph.D. dissertation (University of California at Los Angeles, 2004).

Ufimtsev, P. Y.

P. Y. Ufimtsev, "Rubinowicz and the modern theory of diffracted rays," Electromagnetics 15, 547-565 (1995).
[CrossRef]

P. Y. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

P. Y. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (Foreign Technology Division, Air Force Systems Command, 1971).

Viswanathan, N. S.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

Wong, A.

C. Pierrat and A. Wong, "The MEF revisited: Low k1 effects versus mask topography effects," in Optical Microlithography XVI, A.Yen, ed., Proc. SPIE 5040, 193-202 (2003).

Wong, A. K.

A. K. Wong and A. R. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

Yablonovitch, E.

J. Tirapu-Azpiroz and E. Yablonovitch, "Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography," Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

J. Tirapu-Azpiroz and E. Yablonovitch, "Modeling of near-field effects in sub-wavelength deep ultraviolet lithography," in Future Trends of Microelectronics 2003, S.Luryi, J.Xu, and A.Zaslavsky, eds. (Wiley-IEEE, 2004), pp. 80-92.

Yeung, M. S.

M. S. Yeung and E. Barouch, "Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157 nm optical lithography," IEEE Electron Device Lett. 21, 433-435 (2000).
[CrossRef]

Yihong, W.

Electromagn. Waves

C. T. Tai, "Direct integration of field equations," Electromagn. Waves : Prog. Electromagn. Res. 28, 339-359 (2000).
[CrossRef]

Electromagnetics

P. Y. Ufimtsev, "Rubinowicz and the modern theory of diffracted rays," Electromagnetics 15, 547-565 (1995).
[CrossRef]

P. Y. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

IEEE Electron Device Lett.

M. S. Yeung and E. Barouch, "Limitation of the Kirchhoff boundary conditions for aerial image simulation in 157 nm optical lithography," IEEE Electron Device Lett. 21, 433-435 (2000).
[CrossRef]

IEEE Trans. Electron Devices

A. K. Wong and A. R. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Trans. Electron Devices 41, 895-902 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Proc. SPIE

J. Tirapu-Azpiroz and E. Yablonovitch, "Fast evaluation of photomask near-fields in sub-wavelength 193 nm lithography," Proc. SPIE 5377, 1528-1535 (2004).
[CrossRef]

Other

T. V. Pistor, A. R. Neureuther, and R. J. Socha, "Modeling oblique incidence effects in photomasks," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4000, 228-237 (2000).

J. Tirapu-Azpiroz and E. Yablonovitch, "Modeling of near-field effects in sub-wavelength deep ultraviolet lithography," in Future Trends of Microelectronics 2003, S.Luryi, J.Xu, and A.Zaslavsky, eds. (Wiley-IEEE, 2004), pp. 80-92.

J. Tirapu-Azpiroz, "Analysis and modeling of photomask near-fields in subwavelength deep ultraviolet lithography," Ph.D. dissertation (University of California at Los Angeles, 2004).

P. Y. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (Foreign Technology Division, Air Force Systems Command, 1971).

K. Adam and A. R. Neureuther, "Simplified models for edge transitions in rigorous mask modeling," in Optical Microlithography XIV, C.J.Progler, ed., Proc. SPIE 4346, 331-344 (2001).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

H. J. Levinson, Principles of Lithography (SPIE, 2001).

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

C. Pierrat and A. Wong, "The MEF revisited: Low k1 effects versus mask topography effects," in Optical Microlithography XVI, A.Yen, ed., Proc. SPIE 5040, 193-202 (2003).

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Figures (14)

Fig. 1
Fig. 1

(a) Geometry of a half-plane edge tangential to each point of the aperture edge. The integration strips on the metal surfaces along the diffraction cone are responsible for the edge waves diffracted by the object boundary. (b) Through Babinet’s principle the integration strips are taken on the edge complementary aperture.

Fig. 2
Fig. 2

Application of PTD to a rectangular aperture on perfectly electrically conducting plate and imaging through a 4 × , NA = 0.85 optical system. Relative error on the field amplitude due to edge diffraction is measured at the peak of the image.

Fig. 3
Fig. 3

Reticle cross sectional profile of typical alternating-phase-shifting masks.

Fig. 4
Fig. 4

Error measurement on aerial image field amplitude due to the thin-mask approximation relative to the rigorously evaluated mask field.

Fig. 5
Fig. 5

(a) Log-log plot of the relative error in the real component of the electric field on the wafer produced by the thin-mask approximation, as compared with the rigorously evaluated EM field, versus the harmonic mean of the opening height and width. (b) Log-log plot of the relative error in the imaginary component as a function of opening height (opening size in the direction perpendicular to polarization).

Fig. 6
Fig. 6

(a) Real component of the BL model. (b) Imaginary component of the BL model. (c) Final BL model as the superposition of both real and imaginary parts.

Fig. 7
Fig. 7

Comparison between the aerial field components produced by rigorously evaluated EM TEMPEST field solutions of the object field (solid curves) and both the corresponding thin-mask approximation (dashed curves) and our BL model (dashed–dotted curves), of a 1.6 λ , 180°-phase-shift square mask opening. (a) Intensity and phase of the field component along the polarization direction (x axis); (b) intensity and phase of the field coupled to the component along the optical axis (z axis).

Fig. 8
Fig. 8

Rms error in the intensity distribution, integrated over the focal plane and at two out-of-focus positions, of the approximated images relative to the rigorous EM fields for unpolarized, partially coherent illumination at 193 nm , NA = 0.85 , σ = 0.6 . (a) 180°-phase-shift openings; (b) clear openings.

Fig. 9
Fig. 9

(a) Kirchhoff scalar approximation (thin-mask model) of the field on the mask plane of a 78 nm (as measured at the wafer plane), half-pitch array of alternating 180°-shifter and clear line openings, with vertically polarized electric field at 193 nm . (b) Sketch of the actual object field obtained by rigorous electromagnetic FDTD TEMPEST simulation on the same mask. (c) BL model for the same mask features and illumination conditions. A cross-sectional view of the mask field for each case along cut A is displayed at the bottom for clarity.

Fig. 10
Fig. 10

Aerial image intensity results at the focal plane of a 78 nm (as measured at the wafer plane), half-pitch array of alternating 180°-shifter and clear mask lines with an unpolarized, σ = 0.5 , partially coherent illumination at 193 nm and NA = 0.85 . (a) Aerial image produced by the thin-mask approximation with an rms error of 50.97%. (b) Aerial image produced by the rigorously evaluated object field. (c) Aerial image produced by the BL model with an rms error of 4.63%.

Fig. 11
Fig. 11

Cross-sectional view along the cuts of Fig. 10: (a)–(c) cut A; (d)–(f) cut B, (g)–(i) cut C, (j)–(l) cut D. Figures on the left-hand side show the image at the focal plane; figures at the center and right-hand side display the effect of defocus at 0.2 μ m out-of-focus and 0.4 μ m out-of-focus, respectively.

Fig. 12
Fig. 12

Rms error integrated over the wafer plane of periodic lines with different pitch dimensions, modeled by either the thin-mask or the BL models.

Fig. 13
Fig. 13

(a) Kirchhoff scalar approximation (thin-mask model) of the field on the mask plane of a 79 nm (as measured at the wafer plane), half-pitch array of alternating 180°-shifter and clear corner openings, with vertically polarized electric field at 193 nm . (b) Sketch of the actual object field obtained by rigorous electromagnetic FDTD TEMPEST simulation on the same mask. (c) BL model for the same mask features and illumination conditions.

Fig. 14
Fig. 14

Aerial image intensity results at the focal plane of a 79 nm (as measured at the wafer plane), half-pitch array of alternating 180°-shifter and clear square corners with an unpolarized, σ = 0.4 , partially coherent illumination at 193 nm and NA = 0.85 . (a) Aerial image produced by the thin-mask approximation with an rms error of 56.78%. (b) Aerial image produced by the rigorously evaluated object field. (c) Aerial image produced by the BL model with an rms error of 3.28%.

Tables (1)

Tables Icon

Table 1 Width and Transmission Coefficients of the Boundary-Layer Model for Two Types of Reticle Cross Section in Typical Alternating-Phase-Shifting-Mask Geometry

Equations (17)

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E S = j ω μ A j 1 ω ϵ ( A ) × F ,
H S = j ω ϵ F j 1 ω μ ( F ) + × A ,
A = S J e i k r 4 π r d s ,
F = S M e i k r 4 π r d s ,
F FR = C 0 M FR e i k r 4 π r d τ d l .
E image = E image K + E image FR = E image K ( 1 + Δ E E ) ,
Δ E E = E image FR E image K j 2 k ( 2 h + 2 w ) h w = j 2 k 4 2 h w w + h = j 4 Δ d d effective .
Δ E E = E image FR E image K j ( 2 h + 2 w ) Δ d h w = j boundary layer area total area .
F FR = j C 0 Δ d M K e i k r 4 π r d τ d l ,
Amplitude deficit = Re { Δ E E } = 4 Δ d d = 4 Δ d ( 2 w h ) ( w + h ) .
Relative imaginary error = Im { Δ E E } = β 2 Δ d h ,
E FR = η r ̂ × H FR = η r ̂ × C 2 [ ( H inc 1 ̂ ) F ( l , r ̂ ) + η 1 ( E inc 1 ̂ ) G ( l , r ̂ ) ] e i k r 4 π r d l ,
E image FR ( x , y ) = η r ̂ × x ̂ { H o 2 h NA 2 f 2 + g 2 1 cos ( π NA g w ) sinc ( π NA f h ) e j k ( f x + g y ) d f d g E o η 2 w NA 2
× f 2 + g 2 1 cos ( π NA f h ) sinc ( π NA g w ) e j k ( f x + g y ) d f d g } .
E image FR ( x , y ) η ( r ̂ × x ̂ ) H o NA 2 J 1 ( k x 2 + y 2 ) k r ( 2 h + 2 w ) ,
E image FR E o NA 2 J 1 ( k x 2 + y 2 ) k r ( 2 h + 2 w ) .
E image K j k 2 E o h w NA 2 J 1 ( k x 2 + y 2 ) k r .

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