Abstract

The use of exact reflection theory to interpret data allows the application of ellipsometry to the determination of the thicknesses and optical constants of surface films without the thickness limitations of approximate theory. Ellipsometer measurements as a function of the thicknesses and optical constants of a variety of different films on silicon substantiate the predictions of exact theory and yield the properties of the films.

© 1962 Optical Society of America

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References

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  1. P. Drude, Wied. Ann. 43, 126 (1891).
    [Crossref]
  2. L. Tronstad, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1931).
  3. F. A. Lucy, J. Chem. Phys. 16, 167 (1948).
    [Crossref]
  4. R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
    [Crossref]
  5. C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).
    [Crossref]
  6. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1955).
  7. A. B. Winterbottom, J. Opt. Soc. Am. 38, 1074 (1948).
    [Crossref]
  8. L. Tronstad and T. Hoverstad, Trans. Faraday Soc. 30, 349 (1934).
    [Crossref]
  9. A. Vasicek, Czechoslov. J. Phys. 8, 296 (1958).
    [Crossref]
  10. A. Vasicek, J. Opt. Soc. Am. 37, 145 (1947).
    [Crossref] [PubMed]
  11. A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
    [Crossref]
  12. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, Ltd., London, 1955).
  13. Figure 1 is not reproduced on a large enough scale nor is it sufficiently comprehensive in refractive index for the practical evaluation of ellipsometer data. On request, the author will supply tabulations of Δ and ψ as a function of the thicknesses and the indices of refraction of films on silicon. The tables give Δ and ψ to three decimal places for 28 values for the index of refraction as a function of δ in 2.5° increments from 0° to 180°.
  14. There are two sets of settings which cause extinction. For one set the analyzer angle A is always in the fourth quadrant. For the other set A is in the first quadrant. It is the former set for which the Eqs. (8), (9), and (10) are valid.
  15. A. B. Winterbottom, J. Sci. Instr. 14, 203 (1937).
    [Crossref]
  16. R. W. Ditchburn, J. Opt. Soc. Am. 45, 743 (1955).
    [Crossref]
  17. These limits for n1 cover any likely dielectric film. If the film is absorbing the conclusions deduced here and below are valid for k as large as 0.055. Thus the possibility that the film might be silicon hydride [see the discussion concerning the data of Fig. 6] is recognized.
  18. W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).
    [Crossref]
  19. J. R. Ligenza, J. Electrochem. Soc. (to be published).
  20. E. L. Jordan, J. Electrochem. Soc. 108, 478 (1961).
    [Crossref]
  21. R. J. Archer, J. Phys. Chem. Solids 14, 104 (1960).
    [Crossref]
  22. F. H. Constable, Proc. Roy. Soc. (London) A115, 570 (1927).
  23. R. B. Sosman, The Properties of Silica (Chemical Catalog Company, Inc., New York, 1927).
  24. J. R. Ligenza and W. G. Spitzer (private communication).
  25. A. J. Moulson and J. P. Roberts, Trans. Faraday Soc. 57, 1208 (1961).
    [Crossref]
  26. D. Kahng, Ph.D. dissertation, Ohio State University, (1959).
  27. J. R. Ligenza and W. G. Spitzer, J. Phys. Chem. Solids 14, 131 (1960).
    [Crossref]
  28. G. Hass and C. D. Salzberg, J. Opt. Soc. Am. 44, 181 (1954).
    [Crossref]
  29. G. Hass, J. B. Ramsey, and R. Thun, J. Opt. Soc. Am. 48, 324 (1958).
    [Crossref]

1961 (2)

E. L. Jordan, J. Electrochem. Soc. 108, 478 (1961).
[Crossref]

A. J. Moulson and J. P. Roberts, Trans. Faraday Soc. 57, 1208 (1961).
[Crossref]

1960 (2)

J. R. Ligenza and W. G. Spitzer, J. Phys. Chem. Solids 14, 131 (1960).
[Crossref]

R. J. Archer, J. Phys. Chem. Solids 14, 104 (1960).
[Crossref]

1958 (2)

1957 (1)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[Crossref]

1955 (3)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1955).

R. W. Ditchburn, J. Opt. Soc. Am. 45, 743 (1955).
[Crossref]

W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).
[Crossref]

1954 (1)

1949 (1)

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

1948 (2)

1947 (1)

1939 (1)

1937 (1)

A. B. Winterbottom, J. Sci. Instr. 14, 203 (1937).
[Crossref]

1934 (1)

L. Tronstad and T. Hoverstad, Trans. Faraday Soc. 30, 349 (1934).
[Crossref]

1931 (1)

L. Tronstad, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1931).

1927 (1)

F. H. Constable, Proc. Roy. Soc. (London) A115, 570 (1927).

1891 (1)

P. Drude, Wied. Ann. 43, 126 (1891).
[Crossref]

Archer, R. J.

R. J. Archer, J. Phys. Chem. Solids 14, 104 (1960).
[Crossref]

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[Crossref]

Constable, F. H.

F. H. Constable, Proc. Roy. Soc. (London) A115, 570 (1927).

Dash, W. C.

W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).
[Crossref]

Ditchburn, R. W.

Drude, P.

P. Drude, Wied. Ann. 43, 126 (1891).
[Crossref]

Hanson, M.

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

Hass, G.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, Ltd., London, 1955).

Hoverstad, T.

L. Tronstad and T. Hoverstad, Trans. Faraday Soc. 30, 349 (1934).
[Crossref]

Jordan, E. L.

E. L. Jordan, J. Electrochem. Soc. 108, 478 (1961).
[Crossref]

Kahng, D.

D. Kahng, Ph.D. dissertation, Ohio State University, (1959).

Leberknight, C. E.

Ligenza, J. R.

J. R. Ligenza and W. G. Spitzer, J. Phys. Chem. Solids 14, 131 (1960).
[Crossref]

J. R. Ligenza, J. Electrochem. Soc. (to be published).

J. R. Ligenza and W. G. Spitzer (private communication).

Lucy, F. A.

F. A. Lucy, J. Chem. Phys. 16, 167 (1948).
[Crossref]

Lustman, B.

Moulson, A. J.

A. J. Moulson and J. P. Roberts, Trans. Faraday Soc. 57, 1208 (1961).
[Crossref]

Newman, R.

W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).
[Crossref]

Ramsey, J. B.

Roberts, J. P.

A. J. Moulson and J. P. Roberts, Trans. Faraday Soc. 57, 1208 (1961).
[Crossref]

Rothen, A.

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

Salzberg, C. D.

Sosman, R. B.

R. B. Sosman, The Properties of Silica (Chemical Catalog Company, Inc., New York, 1927).

Spitzer, W. G.

J. R. Ligenza and W. G. Spitzer, J. Phys. Chem. Solids 14, 131 (1960).
[Crossref]

J. R. Ligenza and W. G. Spitzer (private communication).

Thun, R.

Tronstad, L.

L. Tronstad and T. Hoverstad, Trans. Faraday Soc. 30, 349 (1934).
[Crossref]

L. Tronstad, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1931).

Vasicek, A.

A. Vasicek, Czechoslov. J. Phys. 8, 296 (1958).
[Crossref]

A. Vasicek, J. Opt. Soc. Am. 37, 145 (1947).
[Crossref] [PubMed]

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1955).

A. B. Winterbottom, J. Opt. Soc. Am. 38, 1074 (1948).
[Crossref]

A. B. Winterbottom, J. Sci. Instr. 14, 203 (1937).
[Crossref]

Czechoslov. J. Phys. (1)

A. Vasicek, Czechoslov. J. Phys. 8, 296 (1958).
[Crossref]

J. Chem. Phys. (1)

F. A. Lucy, J. Chem. Phys. 16, 167 (1948).
[Crossref]

J. Electrochem. Soc. (2)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[Crossref]

E. L. Jordan, J. Electrochem. Soc. 108, 478 (1961).
[Crossref]

J. Opt. Soc. Am. (6)

J. Phys. Chem. Solids (2)

R. J. Archer, J. Phys. Chem. Solids 14, 104 (1960).
[Crossref]

J. R. Ligenza and W. G. Spitzer, J. Phys. Chem. Solids 14, 131 (1960).
[Crossref]

J. Sci. Instr. (1)

A. B. Winterbottom, J. Sci. Instr. 14, 203 (1937).
[Crossref]

Kgl. Norske Videnskab. Selskabs, Skrifter (2)

L. Tronstad, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1931).

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs, Skrifter 1(1955).

Phys. Rev. (1)

W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).
[Crossref]

Proc. Roy. Soc. (London) (1)

F. H. Constable, Proc. Roy. Soc. (London) A115, 570 (1927).

Rev. Sci. Instr. (1)

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

Trans. Faraday Soc. (2)

L. Tronstad and T. Hoverstad, Trans. Faraday Soc. 30, 349 (1934).
[Crossref]

A. J. Moulson and J. P. Roberts, Trans. Faraday Soc. 57, 1208 (1961).
[Crossref]

Wied. Ann. (1)

P. Drude, Wied. Ann. 43, 126 (1891).
[Crossref]

Other (8)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, Ltd., London, 1955).

Figure 1 is not reproduced on a large enough scale nor is it sufficiently comprehensive in refractive index for the practical evaluation of ellipsometer data. On request, the author will supply tabulations of Δ and ψ as a function of the thicknesses and the indices of refraction of films on silicon. The tables give Δ and ψ to three decimal places for 28 values for the index of refraction as a function of δ in 2.5° increments from 0° to 180°.

There are two sets of settings which cause extinction. For one set the analyzer angle A is always in the fourth quadrant. For the other set A is in the first quadrant. It is the former set for which the Eqs. (8), (9), and (10) are valid.

J. R. Ligenza, J. Electrochem. Soc. (to be published).

These limits for n1 cover any likely dielectric film. If the film is absorbing the conclusions deduced here and below are valid for k as large as 0.055. Thus the possibility that the film might be silicon hydride [see the discussion concerning the data of Fig. 6] is recognized.

D. Kahng, Ph.D. dissertation, Ohio State University, (1959).

R. B. Sosman, The Properties of Silica (Chemical Catalog Company, Inc., New York, 1927).

J. R. Ligenza and W. G. Spitzer (private communication).

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Figures (6)

Fig. 1
Fig. 1

Δ and ψ as a function of the index of refraction and thickness of transparent films on silicon. Each curve corresponds to a film of fixed index of refraction with thickness increasing in the direction of the arrow. The thickness scales are marked off in 20° increments in δ. Thickness is given by 15.17 δ / ( n 1 2 - 0.8830 ) 1 2 Å. The underlined numbers are indices of refraction.

Fig. 2
Fig. 2

Δ and ψ as a function of k and thickness for absorbing films on silicon with indices of refraction 2.2–ik. The thickness scales are the magnitude of the real part of δ in degrees. Film thickness is given, approximately, by 7.63δ(real) Å.

Fig. 3
Fig. 3

Schematic representation of the ellipsometer.

Fig. 4
Fig. 4

Experimental results for silica films on silicon: (○) Silicon oxidized in oxygen at 920°C or 1100°C; (●) Silicon oxidized in steam at 50 or 100 atm pressure and 650°C; (□) Anodized silicon. Curves are calculated using the indicated values for the indices of refraction of the films.

Fig. 5
Fig. 5

Experimental results for evaporated silicon monoxide (○) and cerium oxide (□) films on silicon. Curves are calculated using the indicated values for the indices of refraction of the films.

Fig. 6
Fig. 6

Experimental results for stain films on silicon. Curve is calculated using the value 2.12–0.055i for the index of refraction of the film.

Tables (1)

Tables Icon

Table I Properties of silica films on silicon.

Equations (13)

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Δ = ( β p - β s ) reflected - ( β p - β s ) incident
ψ = arc tan [ ( A p A s ) reflected / ( A p A s ) incident ] .
tan ψ e i Δ = r 1 p + r 2 p e - 2 i δ 1 + r 1 p r 2 p e - 2 i δ · 1 + r 1 s r 2 s e - 2 i δ r 1 s + r 2 s e - 2 i δ .
δ = ( 360 / λ ) d ( n 1 2 - sin 2 φ ) 1 2 deg ,
r p = n a cos φ b - n b cos φ a n a cos φ b + n b cos φ a ,
r s = n a cos φ a - n b cos φ b n a cos φ a + n b cos φ b .
δ ( real ) = 360 d λ [ ( n 1 2 - k 1 2 - sin 2 φ ) 2 + 4 n 1 2 k 1 2 ] 1 / 4 × cos [ 1 2 tan - 1 ( - 2 n 1 k 1 n 1 2 - k 1 2 - sin 2 φ ) ] .
tan Δ = sin δ tan ( 90 ° - 2 P 0 ) ,
tan ψ = cot L tan ( - A 0 ) ,
cos 2 L = - cos δ cos 2 P 0 .
I sin 2 ( A - A 0 ) + sin 2 A 0 sin 2 A sin 2 ( P - P 0 ) .
n 2 = k 2 + sin 2 φ tan 2 φ ( cos 2 2 ψ ¯ - sin 2 2 ψ ¯ sin 2 Δ ¯ ) ( 1 + sin 2 ψ ¯ cos Δ ¯ ) 2 + sin 2 φ
k = sin 2 φ tan 2 φ sin 4 ψ ¯ sin Δ ¯ 2 n ( 1 + sin 2 ψ ¯ cos Δ ¯ ) 2 ,