Abstract

It is shown that for a uniform transparent layer over a substrate the layer dielectric constant satisfies a fifth-degree polynomial. The problem of extracting the layer index and thickness from the ellipsometric measurement is then reduced to finding the roots of this polynomial. The coefficients of this polynomial are determined by the angle of incidence, the real incident-medium index, the complex substrate index, and the measured complex ellipsometric ratio ρ. This approach to the problem gives directly all the possible physical solutions without the need for initial guesses or ranges. Special cases are examined. Numerical analysis and error analysis are provided for the case of a silicon oxide layer over silicon.

© 1994 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. Information available on request from S. C. Russev and M. I. Boyanov at the address on the title page of the present paper.
  4. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  5. R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
    [CrossRef]
  6. R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).
  7. 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]
  8. E. E. Dagman, “Analytical solution of the inverse ellipsometry problem in the modeling of a single-layer reflecting system,” Opt. Spectrosc. (USSR) 62, 500–503 (1987).
  9. R. M. A. Azzam, B. E. Perilloux, “Constraint on the optical constants of a film-substrate system for operation as an external-reflection retarder at a given angle of incidence,” Appl. Opt. 24, 1171–1179 (1985).
    [CrossRef] [PubMed]
  10. R. M. A. Azzam, B. E. Perilloux, “Equalization of the complex reflection coefficients for the parallel and perpendicular polarizations of an absorbing substrate coated by a transparent thin film,” Opt. Acta 32, 767–777 (1985).
    [CrossRef]
  11. V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
    [CrossRef]
  12. F. K. Urban, “Ellipsometry algorithm for absorbing films,” Appl. Opt. 32, 2339–2344 (1993).
    [CrossRef]
  13. Y. Yoriume, “Method for numerical inversion of the ellipsometry equation for transparent films,” J. Opt. Soc. Am. 73, 888–891 (1983).
    [CrossRef]
  14. O. Hunderi, “New method for accurate determination of optical constants,” Appl. Opt. 11, 1572–1578 (1972).
    [CrossRef] [PubMed]
  15. A. R. Reinberg, “Ellipsometer data analysis with a small programmable desk calculator,” Appl. Opt. 11, 1273–1274 (1972).
    [CrossRef] [PubMed]
  16. T. Yamaguchi, H. Takahashi, “Ellipsometric method for separate measurement of nand dof a transparent film,” Appl. Opt. 14, 2010–2015 (1975).
    [CrossRef] [PubMed]
  17. F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
    [CrossRef]
  18. M. C. Dorf, J. Lekner, “Reflection and transmission ellipsometry of a uniform layer,” J. Opt. Soc. Am. A 4, 2096–2100 (1987).
    [CrossRef]

1993 (1)

1992 (1)

V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
[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)

1987 (2)

E. E. Dagman, “Analytical solution of the inverse ellipsometry problem in the modeling of a single-layer reflecting system,” Opt. Spectrosc. (USSR) 62, 500–503 (1987).

M. C. Dorf, J. Lekner, “Reflection and transmission ellipsometry of a uniform layer,” J. Opt. Soc. Am. A 4, 2096–2100 (1987).
[CrossRef]

1985 (2)

R. M. A. Azzam, B. E. Perilloux, “Constraint on the optical constants of a film-substrate system for operation as an external-reflection retarder at a given angle of incidence,” Appl. Opt. 24, 1171–1179 (1985).
[CrossRef] [PubMed]

R. M. A. Azzam, B. E. Perilloux, “Equalization of the complex reflection coefficients for the parallel and perpendicular polarizations of an absorbing substrate coated by a transparent thin film,” Opt. Acta 32, 767–777 (1985).
[CrossRef]

1983 (3)

1975 (1)

1972 (2)

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

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

Dagman, E. E.

E. E. Dagman, “Analytical solution of the inverse ellipsometry problem in the modeling of a single-layer reflecting system,” Opt. Spectrosc. (USSR) 62, 500–503 (1987).

Dorf, M. C.

Georgieva, D. D.

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

Hunderi, O.

Krisdhasima, V.

V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
[CrossRef]

Lekner, J.

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

McGuire, J.

V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
[CrossRef]

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

Perilloux, B. E.

R. M. A. Azzam, B. E. Perilloux, “Equalization of the complex reflection coefficients for the parallel and perpendicular polarizations of an absorbing substrate coated by a transparent thin film,” Opt. Acta 32, 767–777 (1985).
[CrossRef]

R. M. A. Azzam, B. E. Perilloux, “Constraint on the optical constants of a film-substrate system for operation as an external-reflection retarder at a given angle of incidence,” Appl. Opt. 24, 1171–1179 (1985).
[CrossRef] [PubMed]

Reinberg, A. R.

Russev, S. C.

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

Sproull, R.

V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
[CrossRef]

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

Takahashi, H.

Urban, F. K.

Yamaguchi, T.

Yoriume, Y.

Appl. Opt. (6)

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. (2)

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).

J. Res. Nat. Bur. Stand. Sect. A (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,”J. Res. Nat. Bur. Stand. Sect. A 67, 363–377 (1963).
[CrossRef]

Opt. Acta (1)

R. M. A. Azzam, B. E. Perilloux, “Equalization of the complex reflection coefficients for the parallel and perpendicular polarizations of an absorbing substrate coated by a transparent thin film,” Opt. Acta 32, 767–777 (1985).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

E. E. Dagman, “Analytical solution of the inverse ellipsometry problem in the modeling of a single-layer reflecting system,” Opt. Spectrosc. (USSR) 62, 500–503 (1987).

Surf. Interface Anal. (1)

V. Krisdhasima, J. McGuire, R. Sproull, “A one-film-model ellipsometry program for the simultaneous calculation of protein film thickness and refractive index,” Surf. Interface Anal. 18, 453–456 (1992).
[CrossRef]

Other (2)

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

Information available on request from S. C. Russev and M. I. Boyanov at the address on the title page of the present paper.

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Figures (3)

Fig. 1
Fig. 1

Domain of possible solutions for the layer index and thickness when the polynomial is solved with experimental errors of ±0.05° in Ψ and Δ.

Fig. 2
Fig. 2

Scatter in index values produced from the roots of the polynomial when random errors are introduced in the simulated ellipsometric data. The errors are uniformly distributed within ±0.1° in Ψ and ±0.2° in Δ.

Fig. 3
Fig. 3

Scatter in thickness values related to the indices of Fig. 2.

Tables (2)

Tables Icon

Table 1 Comparative Results for Different Inversion Methods: Solution for Silicon Oxide Layer over Silicon Substratea

Tables Icon

Table 2 Five Polynomial Solutions for the Same Measurement

Equations (53)

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ρ = ( X rp 12 + rp 01 ) ( X rs 12 rs 01 + 1 ) ( X rp 12 rp 01 + 1 ) ( X rs 12 + rs 01 ) ,
X = exp [ - 4 π i ( cos ϕ 1 ) d n 1 λ ] ,
rp 01 = - n 0 cos ϕ 1 + n 1 cos ϕ n 0 cos ϕ 1 + n 1 cos ϕ ,
rp 12 = - n 1 cos ϕ 2 + n 2 cos ϕ 1 n 1 cos ϕ 2 + n 2 cos ϕ 1 ,
rs 01 = - n 1 cos ϕ 1 + n 0 cos ϕ n 1 cos ϕ 1 + n 0 cos ϕ ,
rs 12 = - n 2 cos ϕ 2 + n 1 cos ϕ 1 n 2 cos ϕ 2 + n 1 cos ϕ 1 ,
cos ϕ j = [ n j 2 - n 0 2 ( sin ϕ ) 2 ] 1 / 2 n j ,
a X 2 + b X + c = 0 ,
a = ( - rs 01 + rp 01 ρ ) rp 12 rs 12 ,
b = rp 12 - rp 01 rs 12 rs 01 + ρ ( rs 12 + rp 12 rp 01 rs 01 ) ,
c = - rp 01 + rs 01 ρ .
c X 2 + b X + a = 0.
X = a a - c c - a b + b c .
( - a b + b c ) ( - a b + b c ) - ( a a - c c ) 2 = 0 ,
- a 4 + a 2 b 2 - c 4 + 2 a 2 c 2 + b 2 c 2 - a c ( b ) 2 - b 2 a c = 0.
t = n 1 cos ϕ 1 n 0 cos ϕ ,
x = - tan ϕ tan ϕ 2 ,
y = n 2 cos ϕ 2 n 0 cos ϕ ,
z = ρ - 1 ρ + 1 .
rp 01 = ( t - 1 ) ( t - x y ) ( t + 1 ) ( t + x y ) ,
rp 12 = - ( t + x ) ( t + y ) ( t - x ) ( t - y ) ,
rs 01 = - t - 1 t + 1 ,
rs 12 = t + y t - y ,
ρ = z - 1 z + 1 ,
a = - ( t + 1 ) ( t - 1 ) ( t + x ) ( t + y ) 2 ( t + x z y ) C ,
b = 2 ( t + y ) ( t - y ) [ t 4 + ( 2 x y + x 2 z y + 2 x z + 1 ) t 2 + x 2 z y ] C ,
c = - ( t + 1 ) ( t - 1 ) ( t - x ) ( t - y ) 2 ( t - x z y ) C ,
C = ρ + 1 ( t + 1 ) 2 ( t - x ) ( t - y ) 2 ( t + x y ) ,
( - b 1 a 1 + c 1 b 1 ) ( - b 1 a 1 + c 1 b 1 ) - ( a 1 a 1 - c 1 c 1 ) 2 ( t 2 - 1 ) 2 = 0 ,
a 1 = - ( t + x ) ( t + y ) 2 ( t + x z y ) ,
b 1 = 2 ( t + y ) ( t - y ) [ t 4 + ( 2 x y + x 2 z y + 2 x z + 1 ) t 2 + x 2 z y ] ,
c 1 = - ( t - x ) ( t - y ) 2 ( t - x z y ) .
a 1 = - t 4 - α 3 t 3 - α 2 t 2 - α 1 t - α 0 ,
b 1 = 2 ( t 6 + α 4 t 4 + α 5 t 2 - α 0 ) ,
c 1 = - t 4 + α 3 t 3 - α 2 t 2 + α 1 t - α 0 ,
α 0 = x 2 y 3 z , α 1 = x y 2 + x y 3 z + 2 x 2 y 2 z , α 2 = y 2 + 2 x y + 2 x y 2 z + x 2 y z , α 3 = x + 2 y + x y z , α 4 = - y 2 + 2 x y + 2 x z + x 2 y z + 1 , α 5 = - 2 x y 3 - y 2 - x 2 y 3 z - 2 x y 2 z + x 2 y z .
a 1 a 1 - c 1 c 1 = 2 ( g 6 t 6 + g 4 t 4 + g 2 t 2 + g 0 ) t .
g 0 = α 0 α 1 + α 1 α 0 , g 2 = α 0 α 3 + α 1 α 2 + α 2 α 1 + α 3 α 0 , g 4 = α 1 + α 2 α 3 + α 3 α 2 + α 1 , g 6 = α 3 + α 3 .
c 1 b 1 - a 1 b 1 = 2 ( g 6 t 8 + f 7 i t 7 + f 6 t 6 + f 5 i t 5 + f 4 t 4 + f 3 i t 3 + f 2 t 2 + f 1 i t - g 0 ) t ,
f 1 = ( - α 0 α 5 - α 0 α 2 + α 5 α 0 + α 2 α 0 ) i , f 2 = α 1 α 5 - α 0 α 3 + α 5 α 1 - α 3 α 0 , f 3 = ( - α 0 - α 2 α 5 - α 0 α 4 + α 5 α 2 + α 4 α 0 + α 0 ) i , f 4 = α 3 α 5 + α 1 α 4 + α 5 α 3 + α 4 α 1 , f 5 = ( α 5 - α 0 - α 5 - α 2 α 4 + α 4 α 2 + α 0 ) i , f 6 = α 1 + α 3 α 4 + α 4 α 3 + α 1 , f 7 = ( α 4 - α 2 - α 4 + α 2 ) i .
s = n 1 2 ( cos ϕ 1 ) 2 n 0 2 ( cos ϕ ) 2 or s = ( x y + 1 ) - x y
= s ( cos ϕ ) 2 + ( sin ϕ ) 2 ,
j 6 s 6 + j 5 s 5 + j 4 s 4 + j 3 s 3 + j 2 s 2 + j 1 s + j 0 = 0 ,
j 0 = f 1 2 + 2 g 0 2 - 2 f 2 g 0 - 2 g 2 g 0 , j 1 = f 2 2 + 2 f 3 f 1 - g 2 2 - g 0 2 - 2 f 4 g 0 - 2 g 4 g 0 + 4 g 2 g 0 , j 2 = f 3 2 + 2 f 4 f 2 + 2 f 5 f 1 + 2 g 2 2 - 2 g 4 g 2 - 2 f 6 g 0 - 2 g 6 g 0 + 4 g 4 g 0 - 2 g 2 g 0 , j 3 = f 4 2 + 2 f 5 f 3 + 2 f 6 f 2 + 2 f 7 f 1 - g 4 2 - g 2 2 - 2 g 6 g 2 + 4 g 4 g 2 + 2 g 6 g 0 - 2 g 4 g 0 , j 4 = f 5 2 + 2 f 6 f 4 + 2 f 7 f 3 + 2 f 2 g 6 + 2 g 4 2 - 2 g 6 g 4 + 4 g 6 g 2 - 2 g 4 g 2 - 2 g 6 g 0 , j 5 = f 6 2 + 2 f 7 f 5 - g 6 2 + 2 f 4 g 6 - g 4 2 + 4 g 6 g 4 - 2 g 6 g 2 , j 6 = f 7 2 + 2 g 6 2 + 2 f 6 g 6 - 2 g 6 g 4 .
j 6 x 6 y 6 - j 5 x 5 y 5 + j 4 x 4 y 4 - j 3 x 3 y 3 + j 2 x 2 y 2 - j 1 x y + j 0 ,
k 5 5 + k 4 4 + k 3 3 + k 2 2 + k 1 + k 0 = 0 ,
k 0 = j 1 - 6 j 6 x 5 y 5 + 5 j 5 x 4 y 4 - 4 j 4 x 3 y 3 + 3 j 3 x 2 y 2 - 2 j 2 x y , k 1 = ( j 2 + 15 j 6 x 4 y 4 - 10 j 5 x 3 y 3 + 6 j 4 x 2 y 2 - 3 j 3 x y ) ( x y + 1 ) , k 2 = ( j 3 - 20 j 6 x 3 y 3 + 10 j 5 x 2 y 2 - 4 j 4 x y ) ( x y + 1 ) 2 , k 3 = ( j 4 + 15 j 6 x 2 y 2 - 5 j 5 x y ) ( x y + 1 ) 3 , k 4 = ( j 5 - 6 j 6 x y ) ( x y + 1 ) 4 , k 5 = ( x y + 1 ) 5 j 6 .
d = ( 1 4 λ π ( cos ϕ 1 ) n 1 ) i ln X + 1 2 m λ ( cos ϕ 1 ) n 1 .
X = g 6 t 8 + f 6 t 6 + f 4 t 4 + f 2 t 2 - g 0 - i ( f 7 t 6 + f 5 t 4 + f 3 t 2 + f 1 ) t ( t 2 - 1 ) ( g 6 t 6 + g 4 t 4 + g 2 t 2 + g 0 ) .
X 2 a + X b + c = 0 ,
X = c a - c a b a - a b .
( c b - c b ) ( - b a + a b ) - ( c a - c a ) 2 = 0.
= ( z x + z x + 2 z z x + z + z + 2 ) x z x 2 + z x 2 - 2 z z x 2 - 3 z x - 3 z x + 2 x - 4

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