Abstract

We present an efficient method for calculating the reflectivity of three-dimensional gratings on multilayer films based on a finite-element, Green’s function approach. Our method scales as NlogN, where N is the number of plane waves used in the expansion. Therefore, it is much more efficient than the commonly adopted rigorous-coupled-wave analysis (RCWA), which scales as N3. We demonstrate the effectiveness of this method by applying it to a two-dimensional periodic array of contact holes on a multilayer film. We find that our Green’s function approach is about one order of magnitude faster than the RCWA approach when applied to typical contact holes considered in industry. For most cases, this method is efficient enough for application as a real-time, critical-dimension metrology tool.

© 2006 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, 'Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,' J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  2. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, 'Stable implementation of the rigorous coupled-wave analysis of surface-relief gratings: enhanced transmittance matrix approach,' J. Opt. Soc. Am. A 12, 1077-1085 (1995).
    [CrossRef]
  3. L. Li, 'Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,' J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  4. L. Li, 'Use of Fourier series in the analysis of discontinuous structures,' J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  5. L. Li, 'Note on the S-matrix propagation algorithm,' J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]
  6. P. Lalanne and G. M. Morris, 'Highly improved convergence of the coupled wave method for TM polarization,' J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  7. G. Granet and B. Guizal, 'Really efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,' J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  8. P. Lalanne, 'Improved formulation of the coupled-wave method for two-dimensional gratings,' J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  9. G. Granet, 'Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,' J. Opt. Soc. Am. A 16, 2510-2516 (1999).
    [CrossRef]
  10. E. Popov and M. Neviére, 'Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,' J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  11. See for example, Fayyazuddin and Riazuddin, Quantum Mechanics, (World Scientific, 1990), p. 368.
  12. M. Paulus and O. J. F. Martin, 'Green's tensor technique for scattering in two-dimensional stratified media,' Phys. Rev. E 63, 066615 (2001).
    [CrossRef]
  13. Note that the choice of the initial guess is rather arbitrary. It has been noted that there is no difference in terms of convergence speed whether the estimated initial guess jor the random vector generated by the QMR routine is used.
  14. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (National Bureau of Standards, Applied Mathematics Series 55, 1972), p. 496.

2003 (1)

2001 (1)

M. Paulus and O. J. F. Martin, 'Green's tensor technique for scattering in two-dimensional stratified media,' Phys. Rev. E 63, 066615 (2001).
[CrossRef]

2000 (1)

1999 (1)

1997 (1)

1996 (4)

1995 (2)

Gaylord, T. K.

Granet, G.

Grann, E. B.

Guizal, B.

Lalanne, P.

Li, L.

Martin, O. J.

M. Paulus and O. J. F. Martin, 'Green's tensor technique for scattering in two-dimensional stratified media,' Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Neviére, M.

Paulus, M.

M. Paulus and O. J. F. Martin, 'Green's tensor technique for scattering in two-dimensional stratified media,' Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Pommet, D. A.

Popov, E.

J. Opt. Soc. Am. A (10)

G. Granet, 'Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,' J. Opt. Soc. Am. A 16, 2510-2516 (1999).
[CrossRef]

P. Lalanne, 'Improved formulation of the coupled-wave method for two-dimensional gratings,' J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

P. Lalanne and G. M. Morris, 'Highly improved convergence of the coupled wave method for TM polarization,' J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet and B. Guizal, 'Really efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,' J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

L. Li, 'Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,' J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

L. Li, 'Use of Fourier series in the analysis of discontinuous structures,' J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, 'Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,' J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, 'Stable implementation of the rigorous coupled-wave analysis of surface-relief gratings: enhanced transmittance matrix approach,' J. Opt. Soc. Am. A 12, 1077-1085 (1995).
[CrossRef]

E. Popov and M. Neviére, 'Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,' J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

L. Li, 'Note on the S-matrix propagation algorithm,' J. Opt. Soc. Am. A 20, 655-660 (2003).
[CrossRef]

Phys. Rev. E (1)

M. Paulus and O. J. F. Martin, 'Green's tensor technique for scattering in two-dimensional stratified media,' Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Other (3)

Note that the choice of the initial guess is rather arbitrary. It has been noted that there is no difference in terms of convergence speed whether the estimated initial guess jor the random vector generated by the QMR routine is used.

Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, M.Abramowitz and I.A.Stegun, eds. (National Bureau of Standards, Applied Mathematics Series 55, 1972), p. 496.

See for example, Fayyazuddin and Riazuddin, Quantum Mechanics, (World Scientific, 1990), p. 368.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram for ellipsometric measurement of a 2D periodic array of contact holes.

Fig. 2
Fig. 2

Complex reflection amplitude r versus wavelength of the incident light calculated by the Green’s function method for TE and TM modes. The real and imaginary parts are marked by Re r and Im r , respectively. Solid circles, k-space version; solid curves: r-space version.

Fig. 3
Fig. 3

Difference between reflection amplitude calculated by the GF approach ( r ) and RCWA ( r r c w a ) versus wavelength of the incident light for TE and TM modes: solid (open) triangles, real (imaginary) part for k-space version; solid (open) circles, real (imaginary) part for r-space version with uniform-grid sampling. Solid curves, r-space version with high-order basis expansion.

Fig. 4
Fig. 4

CPU time required for three versions of the GF approach and RCWA: solid triangles, k-space version; solid (open) circles, r-space version with uniform-grid sampling (high-order basis expansion). Open squares, RCWA.

Equations (76)

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Ψ ( r ) = Ψ 0 ( r ) + G ( r , r ) V ( r ) Ψ ( r ) d r ,
r = r 0 + d r e i k 0 x x + i k 0 y y δ ( z ) G ( r , r ) V ( r ) Ψ ( r ) d r ,
× × E = k 2 E ,
ϵ E = 0 ,
H ( r ) × E ( r ) .
× × G k 2 G = G 2 G k 2 G = δ ( r r ) ,
G ( r r ) = n e i k n ( ρ ρ ) G n ( z , z ) .
[ ( i k n + z ̂ z ) ( i k n + z ̂ z ) G n + ( k n 2 z 2 ) G n ] k 2 G n = I δ ( z z ) ,
k 2 ( i k n + z ̂ z ) G n = ( i k n + z ̂ z ) δ ( z z ) ,
( q n 2 z 2 ) G n = [ I + ( i k n + z ̂ z ) ( i k n + z ̂ z ) k 2 ] δ ( z z ) ,
x ̂ = cos ϕ n x ̂ sin ϕ n y ̂ ,
y ̂ = sin ϕ n x ̂ + cos ϕ n y ̂ ,
[ G ] = [ G x x 0 G x z 0 G y y 0 G z x 0 G z z ]
G x x , G x z , G y y
ϵ a ( z ) q n 2 z G x x , ϵ a ( z ) q n 2 z G x z , z G y y ,
G y y = ( e q n z + e q n z R n ) f n ( z ) , for z > z ,
G y y = ( e q n z + e q n z r ¯ n ) g n ( z ) , for z < z ,
( f n g n ) = 1 2 q n ( z ) ( e Q z + r ¯ u ( R e Q z + e Q z ) u ( R e Q z + e Q z ) ) ,
R l = e Q l d l ( M l + M l + R l + 1 ) T l e Q l d l ,
T l ( M l + + M l R l + 1 ) 1 ,
r ¯ l + 1 = ( m + m + e Q l d l r ¯ l e Q l d l ) t l ,
t l ( m + + m e Q l d l r ¯ l e Q l d l ) 1 ,
G y y = 1 2 q n e q n z z + e q n z R n f n ( z ) + e q n z r ¯ n g n ( z ) .
G x x = ( e q n z + e q n z R ̃ n ) f ̃ n ( z ) , for z > z ,
G x x = ( e q n z + e q n z r ̃ n ) g ̃ n ( z ) , for z < z ,
( f ̃ n g ̃ n ) = q n 2 k 2 ( e Q z + r ̃ u ̃ ( R ̃ e Q z + e Q z ) u ̃ ( R ̃ e Q z + e Q z ) ) ,
G x x = q n 2 k 2 e q n z z + e q n z R ̃ n f ̃ n ( z ) + e q n z r ̃ n g ̃ n ( z ) .
G x x = ( e q n z + e q n z R ̃ n ) f ¯ n ( z ) , for z > z ,
G x x = ( e q n z + e q n z r ̃ n ) g ¯ n ( z ) , for z < z ,
( f ¯ n g ¯ n ) = i k n 2 k 2 ( e Q z + r ̃ u ̃ ( R ̃ e Q z e Q z ) u ̃ ( R ̃ e Q z e Q z ) ) .
G x z = i k n 2 k 2 sgn ( z z ) e q n z z + e q n z R ̃ n f ¯ n ( z ) g ¯ n ( z ) + e q n z r ̃ n = i k n q n 2 z G x x .
f l + 1 = ( M l + + M l R l + 1 ) 1 e Q l d l f l T l e Q l d l f l
g l = e Q l d l ( m + + m e Q l d l r ¯ l e Q l d l ) 1 g l + 1 e Q l d l t l g l + 1
G z x = i k n q n 2 z G x x ,
G z z = 1 q n 2 [ i k n z G x z + δ ( z z ) ] = k n 2 q n 4 z z G x x + 1 q n 2 δ ( z z ) .
G z x = i k n 2 k 2 sgn ( z z ) e q n z z + i k n q n [ e q n z R ̃ n f ̃ n ( z ) + e q n z r ̃ n g ̃ n ( z ) ] ,
G z z = k n 2 2 q n k 2 e q n z z + i k n q n [ e q n z R ̃ n f ¯ n ( z ) + e q n z r ̃ n g ¯ n ( z ) ] 1 k 2 δ ( z z ) .
G n = [ G 11 G 12 G 13 G 21 G 22 G 23 G 31 G 32 G 33 ] = [ k x n 2 k n 2 G x x + k y n 2 k n 2 G y y k x n k y n ( G x x G y y ) k x n k n G x z k x n k y n k n 2 ( G x x G y y ) k y n 2 k n 2 G x x + k x n 2 k n 2 G y y k y n k n G x z k x n k n G z x k y n k n G z x G z z ] .
Ψ ( z ) Ψ 0 ( z ) = d z G ( z , z ) V ( z ) Ψ ( z ) ,
ϵ = [ α P 1 + ( 1 α ) ϵ 3 0 0 0 β P 1 + ( 1 β ) ϵ 3 0 0 0 ϵ 3 ] ,
E ̃ x P 1 E x = K x 1 [ i z ( ϵ E z ) K y ( ϵ E y ) ] .
E y ( z ) E y 0 ( z ) = d z ( G 21 V ̃ E ̃ x + G 22 V E y + G 23 V E z ) ,
E z ( z ) E z 0 ( z ) = d z ( G 31 V ̃ E ̃ x + G 32 V E y + G 33 V E z ) ,
V ̃ ( 1 P ϵ a ) k 0 2 .
E y ( z ) E y 0 ( z ) = d z G 22 ( z , z ) V E y ,
H y E ̃ z 0 ( z ) = d z [ z G ¯ ( z , z ) V ̃ z H y + G ¯ ( z , z ) K x V ϵ 1 K x H y ] ,
G ¯ ( z , z ) = 1 2 q n e q n z z k 2 i k n q n [ e q n z R ̃ n f ¯ n ( z ) + e q n z r ̃ n g ¯ n ( z ) ] .
Ψ ( z ) = j Ψ ( z j ) B 0 ( z z j ) for m = 0 ,
Ψ ( z ) = j = 0 M Ψ ( z j + ) B 1 ( z z j + ) for m = 1 ,
B 0 ( z z j ) = { 1 for z j < z < z + j 0 otherwise } ,
B 1 ( z z j + ) = { ( z z j ) Δ z j for z j < z < z j + ( z j + 1 + z ) Δ z j + 1 for z j + < z < z j + 1 + 0 otherwise } .
z j z j + d z e Q z Ψ ( z ) = e Q z j W j Ψ ( z j )
z j z j + d z e Q z Ψ ( z ) = e Q z j + W j Ψ ( z j ) ,
W j ( 1 e Q Δ z j ) Q 1 .
z j z j + d z z j z j + d z e Q z z = W j 0 ,
W j 0 = 2 [ Δ z j ( 1 e Q Δ z j ) Q 1 ] Q 1 = 2 W j + Δ z j ,
z j z j + d z z j z j + d z sgn ( z z ) e Q z z = 0 .
A ̂ X O X G ¯ V X = X 0 ,
O n , n ( j , j ) = δ n , n d z B m ( z , j ) B m ( z , j ) ,
V n , n ( j , j ) = I δ j , j d r e i ( k n k n ) ρ B m ( z , j ) B m ( z , j ) V ( z ) ,
G ¯ n , n ( j , j ) = δ n , n d z B m ( z , j ) G n ( z , z ) B m ( z , j ) d z .
Ψ j ( ρ , ϕ ) = m , ν C j ( m , ν ) ρ m 1 e i ν ϕ S m 1 ,
S m 1 z j z j + d z 0 a j 0 2 π ρ m d ϕ e i sin ( ϕ + ϕ n ) k n ρ e i ν ϕ e ± q n z d ρ
= 2 π m e i ν ϕ n W j ( m , ν ) e ± q n z j ± ( 1 e q n Δ z j ) q n 1 ,
ϕ n = tan 1 ( k n x k n y ) ,
z j ± z j ± Δ z j 2 ,
W j ( m , ν ) = 0 a j d ρ ( ρ a j ) m J ν ( k n ρ ) .
e i ( ν μ ) ϕ n 0 a j d ρ ( ρ a j ) m 0 a j d ρ ( ρ R ) m J μ ( k n ρ ) J ν ( k n ρ ) × e q n z z = e i ( ν μ ) ϕ n W j ( m , μ ) W j ( m , ν ) W n 0 ( j ) ,
W n 0 ( j ) = 2 [ Δ s ( 1 e q n Δ s ) q n ] q n .
W ( ν + 1 , ν ) = k 1 J ν + 1 ( k a j ) ,
W ( ν , ν ) = 2 ν 1 ( k a j ) ν a j π Γ ( ν + 1 2 ) [ J ν ( k a j ) H ν 1 ( k a j ) J ν 1 ( k a j ) H ν ( k a j ) ] ,
H ν 1 ( x ) + H ν + 1 ( x ) = 2 ν x H ν ( x ) + ( x 2 ) ν π Γ ( ν + 3 2 ) .
J ν + 1 ( x ) + J ν 1 ( x ) = ( 2 ν x ) J ν ( x ) ,
W ( m + 1 , ν 1 ) = 2 ν k a j W ( m , ν ) W ( m + 1 , ν + 1 ) .
J ν + 1 ( x ) = J ν 1 ( x ) + 2 J ν ( x )
W ( m , ν + 1 ) = W ( m , ν 1 ) + 2 [ a j J ν ( k a j ) m W ( m 1 , ν ) ] ( k a j ) .

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