Abstract

A monochromatic waveguide model that can predict the optical microscope images of line objects with arbitrary edge geometry is presented. The lines may be patterned in thick layers, including multilayer structures with sloping, curved, and undercut edges; granular structures such as lines patterned in polysilicon; and asymmetric objects. The model is used to illustrate the effects of line edge structure on the optical image. Qualitative agreement with experimentally obtained optical image profiles is demonstrated. Application of the model to studying the effects of variations in layer thickness and edge geometry on linewidth measurements made at different stages of manufacturing integrated-circuit devices is discussed.

© 1988 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–532.
  2. L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.
  3. D. Nyyssonen, “Theory of optical edge detection and imaging of thick layers,” J. Opt. Soc. Am. 72, 1425–1436 (1982),
    [CrossRef]
  4. See Ref. 1, Sec. 13.5, pp. 633–664.
  5. P. W. Barber, D. Y. Wang, M. B. Long, “Scattering calculations using a microcomputer,” Appl. Opt. 20, 1121–1123 (1981).
    [CrossRef] [PubMed]
  6. J. P. Hugonin, R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,”J. Opt. Soc. Am. 71, 664–674 (1981).
    [CrossRef]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,”IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
    [CrossRef]
  9. J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansions used to describe the field diffracted by a grating,”J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  10. See Ref. 1, p. 453.
  11. D. Nyyssonen, “Optical linewidth measurement on patterned metal layers,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 65–70 (1984).
    [CrossRef]
  12. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  13. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  14. F. G. Kaspar, “Computation of light transmitted by a thick grating, for application to contact printing,”J. Opt. Soc. Am. 64, 1623–1630 (1974),
    [CrossRef]
  15. W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1966; Dover, New York, 1979).
  16. See Ref. 1, p. 612.
  17. See Ref. 1, p. 579.

1982 (1)

1981 (3)

1974 (1)

1973 (1)

1971 (1)

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,”IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

1966 (1)

Barber, P. W.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–532.

Burckhardt, C. B.

Cadilhac, M.

Hugonin, J. P.

Kaspar, F. G.

Long, M. B.

Magnus, W.

W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1966; Dover, New York, 1979).

Martin, L. C.

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.

Neureuther, A. R.

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,”IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

Nyyssonen, D.

D. Nyyssonen, “Theory of optical edge detection and imaging of thick layers,” J. Opt. Soc. Am. 72, 1425–1436 (1982),
[CrossRef]

D. Nyyssonen, “Optical linewidth measurement on patterned metal layers,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 65–70 (1984).
[CrossRef]

Petit, R.

Wang, D. Y.

Winkler, S.

W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1966; Dover, New York, 1979).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–532.

Zaki, K. A.

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,”IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

K. A. Zaki, A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,”IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

J. Opt. Soc. Am. (6)

Other (9)

W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1966; Dover, New York, 1979).

See Ref. 1, p. 612.

See Ref. 1, p. 579.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–532.

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.

See Ref. 1, Sec. 13.5, pp. 633–664.

See Ref. 1, p. 453.

D. Nyyssonen, “Optical linewidth measurement on patterned metal layers,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 65–70 (1984).
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

The geometry of most line objects encountered in integrated circuits may be approximated by low-order polynomials.

Fig. 2
Fig. 2

(a) Cross section of a hypothetical thick line object and (b) the corresponding multilayer representation.

Fig. 3
Fig. 3

Orientation of the line structure and incident fields in the linewidth measuring system. The y direction is out of the page and parallel to the length of the line patterns.

Fig. 4
Fig. 4

Theoretical image intensity profile of a 6.0-μm-wide line centered at zero, patterned in a 0.6-μm-thick SiO2 layer on silicon. λ = 530nm.

Fig. 5
Fig. 5

The effect of edge geometry on the calculated optical image profile. The edge geometry is superimposed upon the image profile for reference.

Fig. 6
Fig. 6

Theoretical image intensity profiles of a line patterned in a noisy dielectric layer on silicon. A different set of noise data is used for each curve.

Fig. 7
Fig. 7

Physical profile model of the asymmetric window structure in the resist used to compute the image profiles shown in Fig. 8. (Dimensions are given in micrometers.)

Fig. 8
Fig. 8

Comparison of experimental and theoretical image intensity profiles of a window in photoresist on silicon. The window has either (a) vertical edge walls or (b) sloping edges. The experimental profiles are shown by the solid line, and the theoretical profiles are shown by the dashed line. The photoresist is assumed to have a thickness of 0.94 μm, and the geometry shown in Fig. 7 was used to model the window in the case of sloping edges.

Fig. 9
Fig. 9

Cross sections of the shapes used to model the polysilicon patterning stage of making a metal-on-oxide semiconductor transistor. The two shapes are (a) patterned resist on unetched polysilicon and (b) etched polysilicon with the resist removed. (Dimensions are given in micrometers.)

Fig. 10
Fig. 10

Theoretical image profiles of a resist line on polysilicon [structure shown in Fig. 9 (a)] for polysilicon thicknesses of (a) 0.5, (b) 0.6, and (c) 0.7 μm. A vertical edge is assumed.

Fig. 11
Fig. 11

Theoretical image profiles of a polysilicon line [structure shown in Fig. 9 (b)] for oxide thicknesses of (a) 85, (b) 105, and (c) 125 nm. A vertical edge and a polysilicon thickness of 6 μm are assumed.

Fig. 12
Fig. 12

Theoretical image profiles of a polysilicon line [structure shown in Fig. 9 (b)] for a range of edge curvatures defined by Eq. (26). Oxide thickness is 105 nm; polysilicon thickness is 0.6 μm.

Equations (44)

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ˆ n ( x ) = { η ˆ n 2 ( W n 2 + Δ n ) < x < ( W n 2 + Δ n ) 1 between the lines ,
2 E + ˆ μ k 0 2 E + grad E ˆ · grad ˆ = 0 ,
2 H + ˆ μ k 0 2 H + grad ˆ ˆ × curl H = 0 ,
2 E y + k 0 2 ˆ E y = 0 ( TE mode )
2 H y 1 ˆ x H y x + k 0 2 ˆ H y = 0 ( TM mode ) .
2 X ( x ) x 2 + k 0 2 ˆ ( x ) X ( x ) + α 2 X ( x ) = 0 ,
2 Z z 2 = α 2 Z ( z ) .
ˆ n ( x ) = q E q , n exp ( 2 π i q x / P ) ,
E q , n = { ( W n / P ) η ˆ n 2 + ( 1 W n / P ) η ˆ 0 2 , q = 0 ( η ˆ n 2 η ˆ 0 2 ) q π exp ( 2 π i Δ n q / P ) sin ( q π W n / P ) , q 0
2 X ( x ) x 2 + k 0 2 [ q E q , n exp ( 2 π i q x / P ) ] X ( x ) + α 2 X ( x ) = 0.
E y n ( x , z ) = n [ A m , n exp ( α m , n z ) + A m , n exp ( α m , n z ) ] × j B j , m , n exp ( 2 π i j x / P ) ,
H x n ( x , z ) = m [ A m , n ( α m , n i k 0 ) exp ( α m , n z ) A m , n ( α m , n i k 0 ) × exp ( α m , n z ) ] j B j , m , n exp ( 2 π i j x / P ) .
E y I = E 0 I exp ( i k 0 z ) ,
H x I = E 0 I exp ( i k 0 z ) ;
E y R ( x , z ) = j E j R exp { i k 0 [ ( λ j P ) x + K j R z ] } ,
H x R ( x , z ) = j K j R E j R exp { i k 0 [ ( λ j P ) x + K j R z ] } ,
E y T ( x , z ) = j E j T exp { i k 0 [ ( λ j P ) x + K j T z ] } ,
H x T ( x , z ) = j K j T E j T exp { i k 0 [ ( λ j P ) x + K j T z ] } ,
E 0 I δ j , 0 + E j R = m [ A m , 1 + A m , 1 ] B j , m , 1
E 0 I δ j , 0 + K j R E j R = m [ A m , 1 ( α m , 1 i k 0 ) A m , 1 ( α m , 1 i k 0 ) ] B j , m , 1 ,
δ j , 0 = { 1 , j = 0 0 , j 0 .
m [ A m , n exp ( α m , n Z n ) + A m , n exp ( α m , n Z n ) ] B j , m , n = m [ A m , n + 1 exp ( α m , n + 1 Z n ) + A m , m , 1 exp ( α m , n + 1 Z n ) ] B j , m , n + 1 ,
m [ A m , n ( α m , n i k 0 ) exp ( α m , n Z n ) A m , n ( α m , n i k 0 ) exp ( α m , n Z n ) ] × B j , m , n = m [ A m , n + 1 ( α m , n + 1 i k 0 ) exp ( α m , n + 1 Z n ) A m , n + 1 × ( α m , n + 1 i k 0 ) exp ( α m , n + 1 Z n ) ] B j , m , n + 1 .
m [ A m , N exp ( α m , N T ) + A m , N exp ( α m , N T ) ] B j , m , N = E j T exp ( i k 0 K j T T ) ,
m [ A m , N ( α m , N i k 0 ) exp ( α m , N T ) A m , N ( α m , N i k 0 ) exp ( α m , N T ) ] × B j , m , N = K j T E j T exp ( i k 0 K j T T ) .
[ E y I R H x I R ] = B 1 ( 0 ) · [ A 1 A 1 ] ,
B n ( Z n ) · [ A n A n ] = B n + 1 ( Z n ) · [ A n + 1 A n + 1 ] ,
B N ( T ) · [ A N A N ] = [ E y T H x T ] ,
B n ( Z n ) = [ B n 11 ( Z n ) B n 12 ( Z n ) B n 21 ( Z n ) B n 22 ( Z n ) ]
B j , m , n 11 ( Z n ) = exp ( α m , n Z n ) B j , m , n , B j , m , n 12 ( Z n ) = exp ( α m , n Z n ) B j , m , n , B j , m , n 21 ( Z n ) = ( α m , n i k 0 ) exp ( α m , n Z n ) B j , m , n , B j , m , n 22 ( Z n ) = ( α m , n i k 0 ) exp ( α m , n Z n ) B j , m , n .
( A 1 A 1 ) = [ B 1 ( Z 1 ) ] 1 · [ B 2 ( Z 1 ) ] · [ B 2 ( Z 2 ) ] 1 [ B N 1 ( Z N 1 ) ] 1 · [ B N ( Z N 1 ) ] · [ A N A N ] = B · [ A N A N ] ,
[ A N A N ] = [ B N ( Z N 1 ) ] 1 · [ B N 1 ( Z N 1 ) ] · [ B N 1 ( Z N 2 ) ] 1 [ B 2 ( Z 1 ) ] 1 · [ B 1 ( Z 1 ) ] · [ A 1 A 1 ] = B 1 · [ A 1 A 1 ] .
( E x I R H y I R ) = [ B 1 ( 0 ) ] · [ A 1 A 1 ] ,
[ B 11 B 12 B 21 B 22 ] · ( A 1 A 1 ) = ( E x T H y T ) ,
[ B 11 B 12 B 21 B 22 ] = [ B N ( T ) ] · B 1 .
[ D 11 D 12 D 21 D 22 ] · ( A 1 A 1 ) = ( R 0 ) ,
D j , m 11 = [ K j R ( α m , 1 i k 0 ) ] B j , m , 1 , D j , m 12 = [ K j R + ( α m , 1 i k 0 ) ] B j , m , 1 , D j , m 21 = [ K j T B 11 + B 21 ] , D j , m 22 = [ K j T B 12 + B 22 ] ,
R j = [ E 0 I ( 1 + K j R ) δ j , 0 ]
E j R = m [ A m , 1 + A m , 1 ] B j , m , 1 E 0 I δ j , 0 .
t ( x ) = j E j R exp ( i k 0 λ j x / P ) .
W ( z ) = j = 0 J X j ( z Z j ) j ,
ˆ n ( x ) = ˆ n ( x ) + j = 1 J C j δ ( x X j ) ,
W ( z ) = X 0 + X 2 · z 2
D · B n = α 2 I · B n

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