Abstract

A deeper understanding of imaging behavior is needed with the widespread adoption of optical proximity correction in advanced lithography processes. To gain insight into the printing behavior of different mask pattern configurations, we derive edge-based and vertex-based image models by combining concepts contained in the geometrical theory of diffraction and Hopkins’s image model. The models are scalar models and apply to planar, perfectly conducting mask objects.

© 2004 Optical Society of America

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References

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  1. “ITRS Roadmap,” at http://public.itrs.net/Files/2002Update-Litho.pdf .
  2. J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
    [CrossRef]
  3. N. Cobb, A. Zakhor, E. Miloslavsky, “Mathematical and CAD framework for proximity correction,” in Optical Microlithography, G. E. Fuller, ed., Proc. SPIE2726, 208–221 (1996).
    [CrossRef]
  4. J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
    [CrossRef]
  5. F. M. Schellenberg, “Resolution enhancement with OPC/PSM,” Future Fab Intl., 9(2000).
  6. B. E. A. Saleh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
    [CrossRef]
  7. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  8. H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).
  9. J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik 80, 171–183 (1988).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  11. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Institute of Electrical Engineers, London, 1986).
  12. SOLID-C, a software product (release 5.6.2) (SIGMA-C GmbH, http://www.sigma-c.com ).
  13. V. A. Borovikov, B. Ye. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, London, 1994).
  14. R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 2000).
  15. A. Khoh, “Image formation using geometrical theory of diffraction and its applications to lithography,” Ph.D. dissertation (National University of Singapore, Singapore, 2003).
  16. A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
    [CrossRef]

2000

F. M. Schellenberg, “Resolution enhancement with OPC/PSM,” Future Fab Intl., 9(2000).

1997

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

1988

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik 80, 171–183 (1988).

1981

B. E. A. Saleh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

1977

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

1962

Borovikov, V. A.

V. A. Borovikov, B. Ye. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, London, 1994).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 2000).

Caldwell, R.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Chen, J. F.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Choi, B.-I.

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Cobb, N.

N. Cobb, A. Zakhor, E. Miloslavsky, “Mathematical and CAD framework for proximity correction,” in Optical Microlithography, G. E. Fuller, ed., Proc. SPIE2726, 208–221 (1996).
[CrossRef]

Flagello, D.

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Hopkins, H. H.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Institute of Electrical Engineers, London, 1986).

Keller, J. B.

Khoh, A.

A. Khoh, “Image formation using geometrical theory of diffraction and its applications to lithography,” Ph.D. dissertation (National University of Singapore, Singapore, 2003).

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Kinber, B. Ye.

V. A. Borovikov, B. Ye. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, London, 1994).

Komrska, J.

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik 80, 171–183 (1988).

Laidig, T.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Liebchen, A.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Miloslavsky, E.

N. Cobb, A. Zakhor, E. Miloslavsky, “Mathematical and CAD framework for proximity correction,” in Optical Microlithography, G. E. Fuller, ed., Proc. SPIE2726, 208–221 (1996).
[CrossRef]

Milster, T.

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Nakagawa, K. H.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

Samudra, G. S.

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Schellenberg, F. M.

F. M. Schellenberg, “Resolution enhancement with OPC/PSM,” Future Fab Intl., 9(2000).

Wampler, K. E.

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

Wu, Y.-H.

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

Zakhor, A.

N. Cobb, A. Zakhor, E. Miloslavsky, “Mathematical and CAD framework for proximity correction,” in Optical Microlithography, G. E. Fuller, ed., Proc. SPIE2726, 208–221 (1996).
[CrossRef]

Future Fab Intl.

F. M. Schellenberg, “Resolution enhancement with OPC/PSM,” Future Fab Intl., 9(2000).

J. Opt. Soc. Am.

J. Vac. Sci. Technol. B

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, “Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars,” J. Vac. Sci. Technol. B 15, 2426–2433 (1997).
[CrossRef]

Opt. Eng.

B. E. A. Saleh, “Reduction of errors of microphotographic reproductions by optimal corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
[CrossRef]

Optik

J. Komrska, “Algebraic expressions of shape amplitudes of polygons and polyhedra,” Optik 80, 171–183 (1988).

Photograph. Sci. Eng.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Institute of Electrical Engineers, London, 1986).

SOLID-C, a software product (release 5.6.2) (SIGMA-C GmbH, http://www.sigma-c.com ).

V. A. Borovikov, B. Ye. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, London, 1994).

R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 2000).

A. Khoh, “Image formation using geometrical theory of diffraction and its applications to lithography,” Ph.D. dissertation (National University of Singapore, Singapore, 2003).

A. Khoh, D. Flagello, T. Milster, B.-I. Choi, G. S. Samudra, Y.-H. Wu, “Extending a GTD-based image formation technique to EUV lithography,” in Emerging Lithographic Technologies, R. L. Engelstad, ed., Proc. SPIE5037, 682–689 (2003).
[CrossRef]

“ITRS Roadmap,” at http://public.itrs.net/Files/2002Update-Litho.pdf .

J. F. Chen, T. Laidig, K. E. Wampler, R. Caldwell, K. H. Nakagawa, A. Liebchen, “A practical technology path to sub-0.10 micron process generations via enhanced optical lithography,” in Photomask Technology, F. Abboud, B. J. Grenon, eds., Proc. SPIE3873, 995–1016 (1999).
[CrossRef]

N. Cobb, A. Zakhor, E. Miloslavsky, “Mathematical and CAD framework for proximity correction,” in Optical Microlithography, G. E. Fuller, ed., Proc. SPIE2726, 208–221 (1996).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Image formation as a series of Fourier transforms.

Fig. 2
Fig. 2

Image formation of a 1D space.

Fig. 3
Fig. 3

Schematic showing how the angles ϕ and ϕ0 are measured.

Fig. 4
Fig. 4

Geometrical and edge contributions to the disturbance in the Fraunhoffer region, assuming a space width of 10 μm. The Fraunhoffer region is taken at 106λ away.

Fig. 5
Fig. 5

Comparing the fields calculated with Eq. (9) (approximate) and exact GTD ray tracing.

Fig. 6
Fig. 6

DER. NA=0.7; coherent illumination.

Fig. 7
Fig. 7

Reconstructing the image disturbance of a 1D space by use of the GTD.

Fig. 8
Fig. 8

Irradiance profile of a space of width 0.5 μm at (a) coherent illumination and (b) oblique illumination: σx=0.9, NA=0.7.

Fig. 9
Fig. 9

Footprints of geometrical and edge contributions in direction cosine space.

Fig. 10
Fig. 10

Definitions of l’s and n’s of a vertex.

Fig. 11
Fig. 11

(a) Magnitude and (b) phase of the VDER. NA=0.7, σ=0.

Fig. 12
Fig. 12

(a) Irradiance contours of a T pattern obtained by using (a) the vertex-based image model, and (b) SOLID-C. NA=0.7, σ=0.7.

Fig. 13
Fig. 13

Irradiance profiles along (a) cutlines x=-0.1 μm, x=0.1 μm, and x=0.3 μm and (b) cutlines y=-0.2 μm, y=0 μm, and y=0.4 μm.

Fig. 14
Fig. 14

Irradiance contours of a right-angled triangle obtained by using (a) the edge-based model and (b) SOLID-C. NA=0.7; oblique illumination: σx=0.6, σy=0.3.

Fig. 15
Fig. 15

Irradiance profiles along (a) cutlines x=-0.2 μm, x=0 μm, and x=0.2 μm and (b) cutlines y=-0.2 μm, y=0 μm, and y=0.2 μm.

Fig. 16
Fig. 16

(a) Magnitude and (b) phase of the two-dimensional DER. NA=0.7; coherent illumination.

Equations (48)

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

Uˆ=U^GO+(U^d,1i+U^d,1r)+(U^d,2i+U^d,2r),
UˆU^GO+U^d,1+U^d,2,
U^d(r, ϕ, ϕ0)=U0D exp(-jkr)r,
D=-sgn(a)rK-(kr|a|),
a=2cos[(ϕ-ϕ0)/2].
sgn(x)=1forx0-1otherwise.
K-(x)=1π F-(x)exp[+j(x2+π/4)],
F-(x)=xexp(-jt2)dt.
Uˆ(L)=Dˆ(L)[exp(-jkLx0)-exp(+jkLx0)],
U(x)=-NA+NAD^(L)exp[-jkL(x+x0)]dL--NA+NAD^(L)exp[-jkL(x-x0)]dL,
DER(x)=-NA+NAD^(L)exp(-jkLx)dL,
U(x)=DER(x+x0)+[-DER(x-x0)].
U(x)=DER(x)  (x),
(x)=eN±δ(x-xe),
U(x)=DERi(x)  (x),
DERi(x)=AiD^(L)exp(-jkLx)dL.
I(x)=|U(x)|2/I0.
I(x)=i=1m|Ui(x)|2/(mI0),
Uˆ(L, M)=U^GO(L, M)+e=14U^edge,e(L, M)+v=14U^vertex,v(L, M),
Uˆ(L, M)=v=14[U^vertex,v(L, M)].
U^vertex,v(L, M)=U0Φ exp(-jkR)R,
Φ=1k2(q-qi)nv+1[(q-qi)nv+1]2+[(q-qi)lv+1]2×1(q-qi)lv+1-(q-qi)nv[(q-qi)nv]2+[(q-qi)lv]2×1(q-qi)lv,
Φ=1k2sin(β-α)×D^edge[(L-Li)cos α+(M-Mi)cos α]×D^edge[(L-Li)cos β+(M-Mi)cos β],
D^edge(L)-1/L.
U^vertex,v(L, M)=D^v(L-Li, M-Mi).
U^vertex,v(L, M)=D^v(L-Li, M-Mi)×exp{jk[(L-Li)xv+(M-Mi)yv]}.
 
U(x, y)=v=14AD^v(L-MLi, M-MMi)×exp{+jk[(L-MLi)xv+(M-MMi)yv]}×exp[-jk(Lx+My)]dLdM,
U(x, y)=v=14VDERv(x-xv, y-yv),
VDERv(x, y)=AiD^v(L, M)×exp[-jk(Lx+My)]dLdM.
Ai(L, M)
=1for(L+MLi)2+(M+MMi)2NA20otherwise.
U(x, y)=v=1NVDERv(x-xv, y-yv).
VDERv(x, y)=-VDERv+1(x, y).
U(x, y)=VDER(x, y)  (x, y),
(x, y)=v=1N±δ[x-xv, y-yv],
D^v(L, M)1|q|2qnvqlv-qnv+1qlv+1.
U(x, y)=v=1NAi1|q|2qnvqlv-qnv+1qlv+1×exp[-jkq(r-rv)]dLdM.
U(x, y)=Aiv=1N1|q|2qnvqlvexp(-jkqr)×[exp(+jkqrv)-exp(+jkqrv-1)]dLdM.
Rv=(rv+rv-1)/2;Lv=rv-rv-1,
U(x, y)=v=1NULv,
ULv=jkAi1|q|2q nvLvsinc(qLv/λ)×exp[-jkq(r-Rv)]dLdM.
qnv|q|2=-L(L)2+(M)2;
qLv=MLv.
ULv(x, y)=Ai-L(L)2+(M)2Lvsinc(MLv/λ)×exp[-jk(Lx+My)]dLdM.
ULv(x, y)=DER(x, y)  [rect(y/Lv)δ(x)],
DER(x, y)=Ai-L(L)2+(M)2×exp[-jk(Lx+My)]dLdM,
rect(x)=1if|x|120otherwise.

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