Abstract

It is shown that the unknown thicknesses of any two transparent layers in an arbitrary multilayer system from a single ellipsometric measurement can be found by the solution of an eighth-degree real polynomial. The method gives directly all the possible physical solutions, which are computed from the real roots of the polynomial. The coefficients of the polynomial are determined by the angle of incidence, the refractive indices of all the phases, and the thicknesses of the other layers. The method is used on a simulated system of air/silicon nitride/silicon oxide/silicon.

© 1996 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.
  2. S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
    [CrossRef]
  3. R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).
  4. R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
    [CrossRef] [PubMed]
  5. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  6. J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
    [CrossRef] [PubMed]
  7. J.-P. Drolet, S. C. Russev, M. I. Boyanov, R. M. Leblanc, “Polynomial inversion of the single transparent layer problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284–3292 (1994).
    [CrossRef]
  8. S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
    [CrossRef]
  9. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  10. D. Charlot, A. Maruani, “Ellipsometric data processing: an efficient method and an analysis of the relative errors,” Appl. Opt. 24, 3368–3373 (1985).
    [CrossRef] [PubMed]

1994 (2)

1993 (1)

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[CrossRef]

1991 (2)

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
[CrossRef] [PubMed]

1990 (1)

1985 (1)

1983 (1)

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

Azzam, R. M. A.

R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
[CrossRef] [PubMed]

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

Bosch, S.

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[CrossRef]

Boyanov, M. I.

Charlot, D.

Drolet, J.-P.

Georgieva, D. D.

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Leblanc, R. M.

Lekner, J.

Maruani, A.

Russev, S. C.

J.-P. Drolet, S. C. Russev, M. I. Boyanov, R. M. Leblanc, “Polynomial inversion of the single transparent layer problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284–3292 (1994).
[CrossRef]

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

Appl. Opt. (3)

J. Mod. Opt. (1)

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. (Paris) (1)

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

Surf. Sci. (1)

S. Bosch, “Double layer ellipsometry: an efficient numerical method for data analysis,” Surf. Sci. 289, 411–417 (1993).
[CrossRef]

Other (2)

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 4.

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Figures (8)

Fig. 1
Fig. 1

Multilayer system structure with labeling of layers and definition of electric fields for the scattering matrix method.

Fig. 2
Fig. 2

Scatter of values of ψ and Δ for which the simulated system air/silicon nitride/silicon oxide/silicon has two solutions. For values outside this domain the system has no solutions.

Fig. 3
Fig. 3

Solid curves show the behavior of the solutions for d1 with angle of incidence for the system with d1 = 122 nm and d2 = 104 nm. The system value is stable and can be discriminated from the other solution with a second measurement at another angle of incidence. The other curves are the solutions when a systematic error of +0.02° (dotted) and −0.02° (dotted–dashed) are introduced in ψ before the inversion.

Fig. 4
Fig. 4

Solutions and ψ error curves for d2 corresponding to Fig. 3. Systematic errors of +0.02° (dotted) and −0.02° (dotted–dashed) are introduced in ψ before the inversion.

Fig. 5
Fig. 5

Solutions and Δ error curves for d1. Systematic errors of +0.02° (dotted) and −0.02° (dotted–dashed) are introduced in Δ before the inversion.

Fig. 6
Fig. 6

Same as Fig. 5, but for d2.

Fig. 7
Fig. 7

Solutions and ϕ error curves for d1. Systematic errors of +0.1° (dotted) and −0.1° (dotted–dashed) are introduced in ϕ before the inversion.

Fig. 8
Fig. 8

Same as Fig. 7, but for d2.

Equations (32)

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ρ = ( tan ψ ) exp ( i Δ ) = R p R s .
( E 0 + E 0 - ) = S ( E k + 1 + E k + 1 - ) ,
S = I 01 B 1 I 12 B 2 I k , k + 1 ,
I j , j + 1 = 1 t j , j + 1 [ 1 r j , j + 1 r j , j + 1 1 ] ,
B j = [ exp ( i β j ) 0 0 exp ( - i β j ) ] ,
β j = 2 π d j n j ( cos ϕ j ) / λ
R p = S 21 p S 11 p ,             R s = S 21 s S 11 s .
ρ = S 21 p S 11 s S 11 p S 21 s .
S = S B j S ,
S p = 1 X j X l T p [ 1 0 0 X j ] U p [ 1 0 0 X l ] V p ,
X j = exp ( - 2 i β j ) ,             X l = exp ( - 2 i β l ) ,
R p = X l t 21 p u 12 p v 21 p + t 21 p u 11 p v 11 p + ( X l t 22 p u 22 p v 21 p + t 22 p u 21 p v 11 p ) X j X l t 11 p u 12 p v 21 p + t 11 p u 11 p v 11 p + ( X l t 12 p u 22 p v 21 p + t 12 p u 21 p v 11 p ) X j ,
a X j 2 + b X j + c = 0.
( a a - c c ) 2 - ( - b a + c b ) ( - a b + b c ) = 0 ,
X j = - b a + c b a a - c c .
( - c a + a c ) 2 + ( b a - a b ) ( - c b + b c ) = 0 ,
X j = - c a + a c b a - a b .
ρ = ρ 0 + δ 1 d 1 + δ 2 d 2 , δ 1 = d 1 ρ ,             δ 2 = d 2 ρ ,
[ Re ( δ 1 ) Re ( δ 2 ) Im ( δ 1 ) Im ( δ 2 ) ] ( d 1 d 2 ) = [ Re ( ρ - ρ 0 ) Im ( ρ - ρ 0 ) ] .
Re ( δ 1 ) Im ( δ 2 ) - Im ( δ 1 ) Re ( δ 2 ) = 0.
Im ( δ 1 ) Re ( δ 1 ) = Im ( δ 2 ) Re ( δ 2 ) ,
ρ - ρ 0 = δ 2 d 2 + δ 1 d 1 .
a = a 2 X l 2 + a 1 X l + a 0 , b = b 2 X l 2 + b 1 X l + b 0 , c = c 2 X l 2 + c 1 X l + c 0 .
a 0 = α u 21 p u 21 s v 11 p v 11 s , a 1 = α ( u 21 p u 22 s v 11 p v 21 s + u 22 p u 21 s v 21 p v 11 s ) , a 2 = α u 22 p u 22 s v 21 p v 21 s , b 0 = ( β 1 u 11 p u 21 s + β 2 u 21 p u 11 s ) v 11 p v 11 s , b 1 = ( β 1 u 11 p u 22 s + β 2 u 21 p u 12 s ) v 11 p v 21 s + ( β 1 u 12 p u 21 s + β 2 u 22 p u 11 s ) v 21 p v 21 s , b 2 = ( β 1 u 12 p u 22 s + β 2 u 22 p u 12 s ) v 21 p v 21 s , c 0 = χ u 11 p u 11 s v 11 p v 11 s , c 1 = χ ( u 11 p u 12 s v 11 p v 21 s + u 12 p u 11 s v 21 p v 11 s ) , c 2 = χ u 12 p u 12 s v 21 p v 21 s ,
α = - t 22 p t 12 s + t 12 p t 22 s ρ , β 1 = - t 21 p t 12 s + t 11 p t 22 s ρ , β 2 = - t 22 p t 11 s + t 12 p t 21 s ρ , χ = - t 21 p t 11 s + t 11 p t 21 s ρ .
a a - c c = f 0 X l 4 + f 1 X l 3 + f 2 X l 2 + f 1 X l + f 0 X l 2 , - b a + c b = g 4 X l 4 + g 3 X l 3 + g 2 X l 2 + g 1 X l + g 0 X l 2 , - a b + b c = g 0 X l 4 + g 1 X l 3 + g 2 X l 2 + g 3 X l + g 4 X l 2 ,
f 0 = - c 0 c 2 + a 0 a 2 , f 1 = a 1 a 2 + a 0 a 1 - c 1 c 2 - c 0 c 1 , f 2 = a 2 a 2 + a 1 a 1 + a 0 a 0 - c 2 c 2 - c 1 c 1 - c 0 c 0 , g 0 = c 0 b 2 - b 0 a 2 , g 1 = - b 1 a 2 - b 0 a 1 + c 1 b 2 + c 0 b 1 , g 2 = - b 2 a 2 - b 1 a 1 - b 0 a 0 + c 2 b 2 + c 1 b 1 + c 0 b 0 , g 3 = - b 2 a 1 - b 1 a 0 + c 2 b 1 + c 1 b 0 , g 4 = - b 2 a 0 + c 2 b 0 .
h 0 X l 8 + h 1 X l 7 + h 2 X l 6 + h 3 X l 5 + h 4 X l 4 + h 3 X l 3 + h 2 X l 2 + h 1 X l + h 0 = 0 ,
h 0 = - g 0 g 4 + f 0 2 , h 1 = 2 f 1 f 0 - g 1 g 4 - g 0 g 3 , h 2 = f 1 2 + 2 f 2 f 0 - g 2 g 4 - g 1 g 3 - g 0 g 2 , h 3 = 2 f 2 f 1 + 2 f 0 f 1 - g 3 g 4 - g 2 g 3 - g 1 g 2 - g 0 g 1 , h 4 = f 2 2 + 2 f 1 f 1 + 2 f 0 f 0 - g 4 g 4 - g 3 g 3 - g 2 g 2 - g 1 g 1 - g 0 g 0 .
x l = i ( - X l γ + 1 ) X l γ + 1             or             X l = - x l + i γ ( x l + i )
( j 0 j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 ) = [ 1 1 1 1 1 1 1 1 1 - 8 i - 6 i - 4 i - 2 i 0 2 i 4 i 6 i 8 i - 28 - 14 - 4 2 4 2 - 4 - 14 - 28 56 i 14 i - 4 i - 6 i 0 6 i 4 i - 14 i - 56 i 70 0 - 10 0 6 0 - 10 0 70 - 56 i 14 i 4 i - 6 i 0 6 i - 4 i - 14 i 56 i - 28 14 - 4 - 2 4 - 2 - 4 14 - 28 8 i - 6 i 4 i - 2 i 0 2 i - 4 i 6 i - 8 i 1 - 1 1 - 1 1 - 1 1 - 1 1 ] ( h 0 γ 4 h 1 γ 3 h 2 γ 2 h 3 γ h 4 h 3 γ h 2 ( γ ) 2 h 1 ( γ ) 3 h 0 ( γ ) 4 ) .
d l = 1 4 i λ ln ( X l ) π n l cos ( ϕ l ) + 1 2 m λ n l cos ( ϕ l ) ,

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