Abstract

The thin oxide film present on most metals and semiconductors is difficult to detect and, if uncorrected for, can cause serious errors in the ellipsometric determination of optical constants. The reflectance at normal incidence calculated from these incorrect optical constants will, in general, be higher than that measured experimentally for a film-covered sample. A comparison of the approximate, linear, thin-film relations of Drude and Archer with the exact equations shows that the Drude relations are more inaccurate than are usually reported; in some cases serious errors result for films as thin as 10 Å. Archer’s relations are more accurate than Drude’s in most cases. However, if either of these sets of relations is used to correct measured ellipsometric values to the no-film condition, considerable errors in determining the optical constants may occur even if the film thickness is known. The variation in the ellipsometric parameters ψ and Δ with angle of incidence cannot be used to determine the presence of a thin surface film, since calculations show that the variation of these parameters with angle of incidence for a film-covered surface is practically identical with that for a fictitious film-free surface with appropriate optical constants. Also, the requirement that Rs2=Rp at 45° angle of incidence is only marginally useful in detecting a surface film.

© 1964 Optical Society of America

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References

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  1. P. H. Berning, G. Hass, and R. P. Madden, J. Opt. Soc. Am. 50, 586 (1960).
    [Crossref]
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications Ltd., London, 1955), p. 125.
  3. R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
    [Crossref]
  4. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs, Skrifter 1, 1 (1955).
  5. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
    [Crossref]
  6. A. Rothen, Rev. Sci. Instr. 16, 26 (1945);Rev. Sci. Instr. 28, 283 (1957).
    [Crossref]
  7. McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
    [Crossref]
  8. P. Theroux and R. C. Plumb, J. Opt. Soc. Am. 53, 1451 (1963).
  9. J. A. Berning and P. H. Berning, J. Opt. Soc. Am. 50, 813 (1960).
    [Crossref]
  10. The formulas as given in A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 315, are incorrect.The correct expressions are given by L. Tronstad (Ref. 11) with the additional correction that both expressions should be multiplied by n0 as noted by A. B. Winterbottom (Ref. 12). Unfortunately, typographical errors are also present in Winterbottom’s published equations. The equations here have been slightly rearranged for greater clarity.
  11. L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
    [Crossref]
  12. See Ref. 4, p. 40.
  13. R. J. Archer (private communication);the equations have been slightly rearranged for greater clarity.
  14. J. R. Partington, An Advanced Treatise on Physical Chemistry (John Wiley & Sons, Inc., New York, 1953), Vol. IV, p. 506. The notation used by Partington differs from that used in this paper.
  15. A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
    [Crossref]
  16. I. N. Shklyarevskii and N. A. Nosulenko, Opt. i Spectro-skopiya 12, 769 (1962)[English transl.: Opt. Spectry. 12, 435 (1962)].
  17. See Ref. 4, p. 42.
  18. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 617. The notation used by Born and Wolf differs from that used in this paper.
  19. L. G. Schulz and F. R. Tangherlini, J. Opt. Soc. Am. 44, 362 (1954).
    [Crossref]
  20. R. J. Archer, Phys. Rev. 110, 354 (1958).
    [Crossref]
  21. M. A. Barrett, Norges Tekniske Hogskole, Trondheim, Norway (private communication).

1963 (2)

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

P. Theroux and R. C. Plumb, J. Opt. Soc. Am. 53, 1451 (1963).

1962 (2)

R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
[Crossref]

I. N. Shklyarevskii and N. A. Nosulenko, Opt. i Spectro-skopiya 12, 769 (1962)[English transl.: Opt. Spectry. 12, 435 (1962)].

1960 (2)

1958 (1)

R. J. Archer, Phys. Rev. 110, 354 (1958).
[Crossref]

1957 (1)

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

1955 (1)

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

1954 (1)

1949 (1)

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

1945 (1)

A. Rothen, Rev. Sci. Instr. 16, 26 (1945);Rev. Sci. Instr. 28, 283 (1957).
[Crossref]

1935 (1)

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[Crossref]

Archer, R. J.

R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
[Crossref]

R. J. Archer, Phys. Rev. 110, 354 (1958).
[Crossref]

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

R. J. Archer (private communication);the equations have been slightly rearranged for greater clarity.

Barrett, M. A.

M. A. Barrett, Norges Tekniske Hogskole, Trondheim, Norway (private communication).

Berning, J. A.

Berning, P. H.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 617. The notation used by Born and Wolf differs from that used in this paper.

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 (Butter-worths Scientific Publications Ltd., London, 1955), p. 125.

Madden, R. P.

McCrackin,

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Nosulenko, N. A.

I. N. Shklyarevskii and N. A. Nosulenko, Opt. i Spectro-skopiya 12, 769 (1962)[English transl.: Opt. Spectry. 12, 435 (1962)].

Partington, J. R.

J. R. Partington, An Advanced Treatise on Physical Chemistry (John Wiley & Sons, Inc., New York, 1953), Vol. IV, p. 506. The notation used by Partington differs from that used in this paper.

Passaglia,

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Plumb, R. C.

Rothen, A.

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

A. Rothen, Rev. Sci. Instr. 16, 26 (1945);Rev. Sci. Instr. 28, 283 (1957).
[Crossref]

Schulz, L. G.

Shklyarevskii, I. N.

I. N. Shklyarevskii and N. A. Nosulenko, Opt. i Spectro-skopiya 12, 769 (1962)[English transl.: Opt. Spectry. 12, 435 (1962)].

Steinberg,

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Stromberg,

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Tangherlini, F. R.

Theroux, P.

Tronstad, L.

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[Crossref]

Vasicek, A.

The formulas as given in A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 315, are incorrect.The correct expressions are given by L. Tronstad (Ref. 11) with the additional correction that both expressions should be multiplied by n0 as noted by A. B. Winterbottom (Ref. 12). Unfortunately, typographical errors are also present in Winterbottom’s published equations. The equations here have been slightly rearranged for greater clarity.

Winterbottom, A. B.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 617. The notation used by Born and Wolf differs from that used in this paper.

J. Electrochem. Soc. (1)

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

J. Opt. Soc. Am. (5)

J. Res. Natl. Bur. Std. (U. S.) (1)

McCrackin, Passaglia, Stromberg, and Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Kgl. Norske Videnskab. Selskabs, Skrifter (1)

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

Opt. i Spectro-skopiya (1)

I. N. Shklyarevskii and N. A. Nosulenko, Opt. i Spectro-skopiya 12, 769 (1962)[English transl.: Opt. Spectry. 12, 435 (1962)].

Phys. Rev. (1)

R. J. Archer, Phys. Rev. 110, 354 (1958).
[Crossref]

Rev. Sci. Instr. (2)

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

A. Rothen, Rev. Sci. Instr. 16, 26 (1945);Rev. Sci. Instr. 28, 283 (1957).
[Crossref]

Trans. Faraday Soc. (1)

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[Crossref]

Other (8)

See Ref. 4, p. 40.

R. J. Archer (private communication);the equations have been slightly rearranged for greater clarity.

J. R. Partington, An Advanced Treatise on Physical Chemistry (John Wiley & Sons, Inc., New York, 1953), Vol. IV, p. 506. The notation used by Partington differs from that used in this paper.

See Ref. 4, p. 42.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 617. The notation used by Born and Wolf differs from that used in this paper.

The formulas as given in A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 315, are incorrect.The correct expressions are given by L. Tronstad (Ref. 11) with the additional correction that both expressions should be multiplied by n0 as noted by A. B. Winterbottom (Ref. 12). Unfortunately, typographical errors are also present in Winterbottom’s published equations. The equations here have been slightly rearranged for greater clarity.

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worths Scientific Publications Ltd., London, 1955), p. 125.

M. A. Barrett, Norges Tekniske Hogskole, Trondheim, Norway (private communication).

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Figures (4)

F. 1
F. 1

Exact (——) and approximate (Drude — · — · —, Archer — — —) values for ψ ψ ¯ for (a) aluminum oxide (n1 = 1.77) on aluminum (n = 0.786−5.47j) and for (b) silicon dioxide (n1 = 1.45) on silicon (n = 4.050−0.028 j). Both are for 70° incidence and λ5460 A.

F. 2
F. 2

Exact (——) and approximate (Drude — · — · —, Archer — — —) values for Δ Δ ¯ for the materials in Fig. 1.

F. 3
F. 3

Error in (a) ψ ψ ¯ and (b) Δ Δ ¯ caused by use of the approximate linear equations of Drude (— · — · —) and Archer (— — —). Values are for a silicon dioxide film (n1 = 1.45) on silicon (n = 4.050−0.028j) at 70° angle of incidence and λ5460 Å.

F. 4
F. 4

Rs2/Rp at 45° angle of incidence for an aluminum oxide film on aluminum (λ5460 Å).

Tables (2)

Tables Icon

Table I Pseudo-indices for silicon covered with a 100 Å silicon dioxide film at λ 5460 Å. The true index for silicon is 4.050–0.028j.

Tables Icon

Table II Comparison of reflectance and tanψ for silicon covered with a 100 Å silicon dioxide film calculated using the 70° pseudo-index (3.839–0.872j) and using the true values for silicon and silicon dioxide.

Equations (14)

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n s = n cos Θ ,
n p = n / cos Θ ,
Δ Δ ¯ = 720 ( t / λ ) n 0 cos φ sin 2 φ ( n 0 2 n 1 2 ) [ ( a 1 / n 1 2 ) ( cos 2 φ a n 0 2 ) a 2 n 0 2 ] ( cos 2 φ a n 0 2 ) 2 + a 2 n 0 4 ,
ψ ψ ¯ = 360 ( t / λ ) a n 0 sin 2 ψ ¯ cos φ sin 2 φ ( n 0 2 n 1 2 ) ( cos 2 φ n 1 2 / n 0 2 ) ( cos 2 φ a n 0 2 ) 2 + a 2 n 0 4 ,
Δ Δ ¯ = 720 ( t / λ ) cos φ sin 2 φ ( cos 2 φ a ) ( 1 1 / n 1 2 ) ( cos 2 φ a ) 2 + a 2 ,
ψ ψ ¯ = 360 ( t / λ ) a sin 2 ψ ¯ cos φ sin 2 φ ( 1 n 1 2 cos 2 φ ) ( 1 1 / n 1 2 ) ( cos 2 φ a ) 2 + a 2 .
Δ Δ ¯ = 720 ( t / λ ) cos φ sin 2 φ ( 1 n 1 2 ) { ( a 1 / n 1 2 ) [ ( cos 2 φ a + sin 2 φ ( a 2 a 2 ) ] a 2 } [ cos 2 φ a + sin 2 φ ( a 2 a 2 ) ] 2 + a 2 ,
ψ ψ ¯ = 360 ( t / λ ) a sin 2 ψ ¯ cos φ sin 2 φ ( 1 n 1 2 ) { ( a 1 / n 1 2 ) ( 1 2 a sin 2 φ ) + [ cos 2 φ a + sin 2 φ ( a 2 a 2 ) ] } / { [ cos 2 φ a + sin 2 φ ( a 2 a 2 ) ] 2 + a 2 } .
tan Δ ¯ = sin Q tan 2 P ,
cos 2 ψ ¯ = cos Q sin 2 P ,
tan 2 Q = 2 n k / ( n 2 k 2 sin 2 φ ) ,
tan P = [ n 4 + 2 n 2 k 2 + k 4 2 ( n 2 k 2 ) sin 2 φ + sin 4 φ ] 1 4 sin φ tan φ .
n 2 k 2 = tan 2 φ sin 2 φ ( cos 2 2 ψ ¯ sin 2 2 ψ ¯ sin 2 Δ ¯ ) ( 1 + sin 2 ψ ¯ cos Δ ¯ ) 2 + sin 2 φ ,
2 n k = tan 2 φ sin 2 φ sin 4 ψ ¯ sin Δ ¯ ( 1 + sin 2 ψ ¯ cos Δ ¯ ) 2 ,