Abstract

In this paper, a model for electromagnetic scattering of line structures is established based on high frequency approximation approach - ray tracing. This electromagnetic ray tracing (ERT) model gives the advantage of identifying each physical field that contributes to the total solution of the scattering phenomenon. Besides the geometrical optics field, different diffracted fields associated with the line structures are also discussed and formulated. A step by step addition of each electromagnetic field is given to elucidate the causes of a disturbance in the amplitude profile. The accuracy of the ERT model is also discussed by comparing with the reference finite difference time domain (FDTD) solution, which shows a promising result for a single polysilicon line structure with width of as narrow as 0.4 wavelength.

© 2008 Optical Society of America

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References

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  1. C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
    [CrossRef]
  2. S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
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    [CrossRef]
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    [CrossRef]
  9. C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
    [CrossRef]
  10. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics Publishing, Philadelphia, US, 2001).
    [CrossRef]

2007

C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
[CrossRef]

2005

S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
[CrossRef]

2003

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

1999

A. V. Osipov and A. N. Norris, "The Malyuzhinets theory for scattering from wedge boundaries: a review," Wave Motion 29, 313-340 (1999).
[CrossRef]

1995

1974

R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

1962

Foong, Y. M.

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Keller, J. B.

Khoh, A.

C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
[CrossRef]

S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
[CrossRef]

Koh, H. P.

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Koo, C. K.

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Moharam, M. G.

Norris, A. N.

A. V. Osipov and A. N. Norris, "The Malyuzhinets theory for scattering from wedge boundaries: a review," Wave Motion 29, 313-340 (1999).
[CrossRef]

Osipov, A. V.

A. V. Osipov and A. N. Norris, "The Malyuzhinets theory for scattering from wedge boundaries: a review," Wave Motion 29, 313-340 (1999).
[CrossRef]

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Pommet, D. A.

Siew, Y. K.

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Tan, C. B.

C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
[CrossRef]

S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
[CrossRef]

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Yeo, S. H.

C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
[CrossRef]

S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
[CrossRef]

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Electromagnetics

C. B. Tan, A. Khoh, and S. H. Yeo, "Higher Order Asymptotic Analysis of Impedance Wedge Using Uniform Theory of Diffraction," Electromagnetics 27, 23-39 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

S. H. Yeo, C. B. Tan, and A. Khoh, "Rigorous coupled wave analysis of front-end-of-line wafer alignment marks," J. Vac. Sci. Technol. B 23, 186-195 (2005).
[CrossRef]

Proc. IEEE

R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Proc. SPIE

C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, "Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process," Proc. SPIE 5038, 1211-1218 (2003).
[CrossRef]

Wave Motion

A. V. Osipov and A. N. Norris, "The Malyuzhinets theory for scattering from wedge boundaries: a review," Wave Motion 29, 313-340 (1999).
[CrossRef]

Other

EM Explorer, http://www.emexplorer.net.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics Publishing, Philadelphia, US, 2001).
[CrossRef]

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Figures (17)

Fig. 1.
Fig. 1.

Decomposition of a single line structure formed on substrate into individual model.

Fig. 2.
Fig. 2.

Decomposition of an isolated impedance double wedge into two impedance wedges. The notation, η ¯ denotes the normalized surface impedance for the wedges.

Fig. 3.
Fig. 3.

Geometry of a single impedance wedge with plane wave incidence.

Fig. 4.
Fig. 4.

Plane wave reflection from the bottom surface, SB

Fig. 5.
Fig. 5.

Diffracted rays are reflected from the substrate surface, SB.

Fig. 6.
Fig. 6.

Cut-off point for diffracted field emanating from edge at point B.

Fig. 7.
Fig. 7.

Isolated line structure formed on substrate.

Fig. 8.
Fig. 8.

Amplitude of the individual electromagnetic field that contributes to the total solution for the scattering of an isolated single polysilicon line structure. (a) Geometrical optics field (i) incident field, (ii) reflected field from the top surface of the line structure, ST and (iii) reflected field from the substrate surface, SB. (b) Diffracted field (i) direct diffracted field and (ii) diffracted-reflected field.

Fig. 9.
Fig. 9.

(a). Total geometrical optics field (b) Combination of total geometrical optics field and direct diffracted field (c) Total solution for the electromagnetic scattering of a polysilicon line structure formed on silicon substrate.

Fig. 10.
Fig. 10.

Amplitude and phase of the total field for an isolated single polysilicon line structure with width, l of (a) 8 µm and (b) 1 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.

Fig. 11.
Fig. 11.

The ACC and PCC results for single polysilicon line structure on silicon substrate.

Fig. 12.
Fig. 12.

Amplitude and phase of the total field for an isolated single polysilicon line structure with subwavelength width, l of (a) 0.25 µm and (b) 0.125 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.

Fig. 13.
Fig. 13.

Amplitude and phase of the total field for two identical 2 µm polysilicon line structures on silicon substrate with spacing of (a) 8 µm and (b) 1 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.

Fig. 14.
Fig. 14.

The ACC and PCC results for two-line structures of polysilicon on silicon substrate. The spacing in between the 2 µm lines is varied from 8 µm to 1 µm. A subwavelength space width of 0.5 µm is also included.

Fig. 15.
Fig. 15.

Amplitude and phase of the total field for two identical 2 µm polysilicon line structures on silicon substrate with spacing of 0.5 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.

Fig. 16.
Fig. 16.

The ACC and PCC results for single and two-line (equal spacing) structures across different real refractive indexes. Each single line has width of 1 µm and the observation distance, y is 0.8 λ. The imaginary part of the refractive index is taken as -0.1j.

Fig. 17.
Fig. 17.

The (a) CPU computational time and (b) memory used for the amplitude results obtained from the ERT model for single and two-line structures.

Tables (1)

Tables Icon

Table 1. Type of field and its associated refractive index used in the ERT model

Equations (19)

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U z i = exp ( j k ρ cos [ ϕ ϕ o ] )
U z r 1 ( ρ , ϕ ) = R TE , S T exp ( j k ρ cos [ ϕ + ϕ o ] )
R TE , S T = sin ( ϕ ) n e 2 cos 2 ( ϕ ) sin ( ϕ ) + n e 2 cos 2 ( ϕ )
U z d = U i ( Q E ) D TE ( ϕ , ϕ o ) exp ( j k ρ ) ρ
U z d = D TE ( ϕ , ϕ o ) exp ( j k ρ ) ρ
D TE ( ϕ , ϕ o ) = exp ( j π 4 ) 2 p ¯ 2 π k { cot [ π + ( ϕ ϕ o ) 2 p ¯ ] ψ ( p ¯ π 2 π ϕ ) ψ ( p ¯ π 2 ϕ o ) F [ 2 k ρ cos 2 ϕ ϕ o 2 ]
+ cot [ π ( ϕ ϕ o ) 2 p ¯ ] ψ ( p ¯ π 2 + π ϕ ) ψ ( p ¯ π 2 ϕ o ) F [ 2 k ρ cos 2 ϕ ϕ o 2 ]
+ R TE , S T cot [ π ( ϕ + ϕ o ) 2 p ] ψ ( p ¯ π 2 + π ϕ ) ψ ( p ¯ π 2 + ϕ o ) F [ 2 k ρ cos 2 ϕ + ϕ o 2 ]
+ R TE , S V cot [ π + ( ϕ + ϕ o ) 2 p ¯ ] ψ ( p ¯ π 2 π ϕ ) ψ ( 3 p ¯ π 2 + ϕ o ) F [ 2 k ρ cos 2 2 p ¯ π ( ϕ + ϕ o ) 2 ] }
U z r 2 ( ρ , ϕ ) = R TE , S B U z i ( Q r ) exp ( j k ρ r )
  = R TE , S B exp ( j k ρ Qr cos [ ϕ Qr + ϕ o ] ) exp ( j k ρ r )
U z rd = R TE , S B D TE ( ϕ , ϕ o ) exp ( j k ( ρ i + ρ r ) ρ i + ρ r
( y B II 2 + 2 y B II h 2 ) x r 1 2 + 2 h 2 2 x B II x r 1 h 2 2 x B II = 0
R f = η o Y η o + Y
[ B C ] = [ cos δ 1 ( j sin δ 1 ) η 1 j η 1 sin δ 1 cos δ 1 ] [ 1 η 2 ]
Δ E Z = E Z ( ERT ) E Z ( FDTD )
Δ Phase = Phase ( ERT ) Phase ( FDTD )
ACC i = { E z i ( ERT ) E z i ( FDTD ) for E z i ( ERT ) < E z i ( FDTD ) E z i ( FDTD ) E z i ( ERT ) for E z i ( ERT ) > E z i ( FDTD )
PCC i = { Phase i ( ERT ) Phase i ( FDTD ) for Phase i ( ERT ) < Phase i ( FDTD ) Phase i ( FDTD ) Phase i ( ERT ) for Phase i ( ERT ) > Phase i ( FDTD )

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