Abstract

When polarized light is reflected from a film-covered surface, the variations in Δ and Ψ with the film thickness d are given by the exact equation of ellipsometry. Δ is the change in the difference between the phases of the parallel (p) and the perpendicular (s) components of the polarized light on reflection, and Ψ is the arc tangent of the factor by which the amplitude ratio of the p and s components changes on reflection. A first-order calculation of the exact equation is presented from which the two Archer equations for Ψ and Ψ (which are more accurate than those derived by Drude), relating their variations with the film thickness d, have been derived. Until now, both of these equations have been considered to be valid only in the thin-film region by most of the authors except Burge and Bennett who questioned the usefulness of the linear approximation for Ψ. It is shown that the Archer equation for Ψ is not valid in the thin-film region and a new generalized equation for Ψ is derived which is valid not only in the thin-film region (∼100 Å) but also in certain ranges of the thick-film region. Further, a generalized approximate equation for Δ has also been derived which holds in the thin-film as well as in certain ranges of the thick-film regions. Experimental data to check the validity of the generalized approximate equation for Δ in the thick-film region are presented.

© 1965 Optical Society of America

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References

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  1. P. Drude, Ann. Phys. Chem. 36, 865 (1889);L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935);O. S. Heavens, Optical Properties of Thin Solid Films, (Butterworths Scientific Publications Inc., London, 1955), pp. 55, 125;M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
    [Crossref]
  2. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 38 (1955).
  3. A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
    [Crossref]
  4. R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).In this paper there were a few misprints and Archer had neglected the term 2 αα1 sin2 φ0in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨin a recent paper (see Ref. 5).
    [Crossref]
  5. R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
    [Crossref]
  6. D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
    [Crossref]
  7. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
    [Crossref]
  8. C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).
    [Crossref]
  9. Rigorously speaking, there exists a film thickness dx= 2CΨ/(CΔ2−CΨ2) below which Eq. (24) is not a good approximation. Thus, the thickness range between zero and dx could be termed as a “very thin film region”; however, it is practically nonexistent for most of the cases. For the SiO2-Si case, as an example, dx≅0.066 Å. This was pointed out to the author by W. K. H. Panofsky.
  10. G. E. Moore, in Microelectronics, E. Keonjian, ed. (McGraw Hill Book Co., Inc., New York,1963), pp. 276–279;B. E. Deal, J. Electrochem. Soc. 110, 527 (1963);E. Duffek and D. Pilling, presented at the Electrochem. Soc. Meeting in San Francisco, California, 9–13 May 1965, Abstract No. 111;J. Andrus, U. S. Patent No. 3,122, 817, 3March1964.
    [Crossref]
  11. A. E. Lewis, J. Electrochem. Soc. 111, 1007 (1964).
    [Crossref]
  12. S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London1948).
  13. H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
    [Crossref]

1965 (1)

R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
[Crossref]

1964 (3)

A. E. Lewis, J. Electrochem. Soc. 111, 1007 (1964).
[Crossref]

H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
[Crossref]

D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
[Crossref]

1962 (1)

1957 (1)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).In this paper there were a few misprints and Archer had neglected the term 2 αα1 sin2 φ0in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨin a recent paper (see Ref. 5).
[Crossref]

1955 (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 38 (1955).

1949 (1)

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

1939 (1)

1889 (1)

P. Drude, Ann. Phys. Chem. 36, 865 (1889);L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935);O. S. Heavens, Optical Properties of Thin Solid Films, (Butterworths Scientific Publications Inc., London, 1955), pp. 55, 125;M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
[Crossref]

Archer, R. J.

R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
[Crossref]

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

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).In this paper there were a few misprints and Archer had neglected the term 2 αα1 sin2 φ0in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨin a recent paper (see Ref. 5).
[Crossref]

Bennett, H. E.

Burge, D. K.

Cooper, H. W.

H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
[Crossref]

Drude, P.

P. Drude, Ann. Phys. Chem. 36, 865 (1889);L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935);O. S. Heavens, Optical Properties of Thin Solid Films, (Butterworths Scientific Publications Inc., London, 1955), pp. 55, 125;M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
[Crossref]

Evitts, H. C.

H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
[Crossref]

Flaschen, S. S.

H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
[Crossref]

Gobeli, G. W.

R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
[Crossref]

Hanson, M.

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

Leberknight, C. E.

Lewis, A. E.

A. E. Lewis, J. Electrochem. Soc. 111, 1007 (1964).
[Crossref]

Lustman, B.

Moore, G. E.

G. E. Moore, in Microelectronics, E. Keonjian, ed. (McGraw Hill Book Co., Inc., New York,1963), pp. 276–279;B. E. Deal, J. Electrochem. Soc. 110, 527 (1963);E. Duffek and D. Pilling, presented at the Electrochem. Soc. Meeting in San Francisco, California, 9–13 May 1965, Abstract No. 111;J. Andrus, U. S. Patent No. 3,122, 817, 3March1964.
[Crossref]

Rothen, A.

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

Tolansky, S.

S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London1948).

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 38 (1955).

Ann. Phys. Chem. (1)

P. Drude, Ann. Phys. Chem. 36, 865 (1889);L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935);O. S. Heavens, Optical Properties of Thin Solid Films, (Butterworths Scientific Publications Inc., London, 1955), pp. 55, 125;M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
[Crossref]

J. Electrochem. Soc. (3)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).In this paper there were a few misprints and Archer had neglected the term 2 αα1 sin2 φ0in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨin a recent paper (see Ref. 5).
[Crossref]

A. E. Lewis, J. Electrochem. Soc. 111, 1007 (1964).
[Crossref]

H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. Chem. Solids (1)

R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
[Crossref]

Kgl. Norske Videnskab. Selskabs Skrifter (1)

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 38 (1955).

Rev. Sci. Instr. (1)

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

Other (3)

Rigorously speaking, there exists a film thickness dx= 2CΨ/(CΔ2−CΨ2) below which Eq. (24) is not a good approximation. Thus, the thickness range between zero and dx could be termed as a “very thin film region”; however, it is practically nonexistent for most of the cases. For the SiO2-Si case, as an example, dx≅0.066 Å. This was pointed out to the author by W. K. H. Panofsky.

G. E. Moore, in Microelectronics, E. Keonjian, ed. (McGraw Hill Book Co., Inc., New York,1963), pp. 276–279;B. E. Deal, J. Electrochem. Soc. 110, 527 (1963);E. Duffek and D. Pilling, presented at the Electrochem. Soc. Meeting in San Francisco, California, 9–13 May 1965, Abstract No. 111;J. Andrus, U. S. Patent No. 3,122, 817, 3March1964.
[Crossref]

S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London1948).

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Figures (4)

F. 1
F. 1

Curve showing the variation of Ψ and Δ with film thickness d as given by the exact Eq. (7). The substrate is Si (n2=4.051, k2=0.027) and the film is SiO2 (n1 = 1.46). The angle of incidence φ0 is 70° and the wavelength of light λ is 5461 Å. The relationship between the film thickness d and the phase difference δ introduced by the film is given by Eq. (8). The film thickness is marked off in terms of δ on the curve. The point O on the curve represents the ( Ψ ¯ , Δ ¯ ) point characteristic of the film-free substrate. For Si, Ψ ¯ = 11.77 ° and Δ ¯ = 179.05 °. As the film starts growing on the substrate, the (Ψ,Δ) point starts moving away from O on the curve in the counterclockwise direction. If the refractive index n1 of the film remains constant, the (Ψ,Δ) point follows the curve and meets the point O again when the film thickness is such that its corresponding δ=π. This is defined as the completion of the zeroth order of film thickness. When the film thickness increases further, the (Ψ,Δ) point retraces the curve and crosses the point O at the completion of each order of the film thickness. For a film thickness in the mth order, the point O is crossed m times. The regions OA and OB marked off on the curve define the thin-film region and the supplement to thin-film region, respectively. Approximate equations are derived in the text using a first-order calculation in the exact Eq. (7) which relate the variations in Ψ and Δ with the film thickness d.

F. 2
F. 2

Plot of the difference between the film thicknesses (SiO2 on Si) calculated from Archer’s Eq. 1 [CΔ from Eq. (13)] and the exact Eq. (7) vs the film thickness d. The difference between the film thicknesses calculated from Eq. (22) and the exact Eq. (7) is also plotted. The optical constants and other parameters are the same as in Fig. 1.

F. 3
F. 3

Plot of the difference between the film thicknesses (SiO2 on Si) calculated from Archer’s Eq. (2) [CΨ from Eq. (14)] and the exact Eq. (7) vs the film thickness d. The difference between the film thicknesses calculated from Eq. (25) and the exact Eq. (7) is also plotted. Note that this difference is much less when Eq. (25) is used as compared with that obtained with Eq. (2). The optical constants and other parameters are the same as in Fig. 1.

F. 4
F. 4

Plot of ( Δ ¯ Δ ) and ( Ψ Ψ ¯ ) vs film thickness d as given by exact Eq. (7). Note that for thin films, say up to 100 Å ( Δ ¯ Δ ), curve can be fitted by a straight line whereas a linear approximation for ( Ψ Ψ ¯ ) curve is a poor approximation. A quadratic relationship between ( Ψ Ψ ¯ ) and d is a better approximation.

Tables (1)

Tables Icon

Table I Thickness of SiO2 layers on Si calculated from experimentally measured Δ using the exact Eq. (7) and the approximate Eqs. (27) and (32). Alternate measurements of the thicknesses of the SiO2 layers by multiple-beam interferometer are also tabulated. For Si, Δ ¯ = 179.05 °.

Equations (35)

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Δ Δ ¯ = ( 180 / π ) C Δ d ,
Ψ Ψ ¯ = ( 180 / π ) 1 2 ( sin 2 Ψ ¯ ) C Ψ d ,
C Δ ( Drude ) = 4 π λ · cos φ 0 sin 2 φ 0 ( cos 2 φ 0 α ) ( 1 1 / n 1 2 ) ( cos 2 φ 0 α ) 2 + α 1 2 ,
C Ψ ( Drude ) = 4 π λ · cos φ 0 sin 2 φ 0 · α 1 ( 1 n 1 2 cos 2 φ 0 ) ( 1 1 / n 1 2 ) ( cos 2 φ 0 α ) 2 + α 1 2 .
n 2 2 k 2 2 = n 0 2 sin 2 φ 0 × [ 1 + tan 2 φ 0 ( cos 2 2 Ψ ¯ sin 2 Δ sin 2 2 Ψ ¯ ) ( 1 + sin 2 Ψ ¯ cos Δ ¯ ) 2 ] ,
2 n 2 k 2 = n 0 2 sin 2 φ 0 tan 2 φ 0 sin 4 Ψ ¯ sin Δ ¯ ( 1 + sin 2 Ψ ¯ cos Δ ¯ ) 2 ,
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 ,
d = C n 1 · δ ,
C n 1 = ( λ / 2 π ) ( n 1 2 n 0 2 sin 2 φ 0 ) 1 2 .
d = ( m π + x ) C n 1 .
δ = π δ π ,
( tan Ψ / tan Ψ ¯ ) e i ( Δ Δ ¯ ) = 1 + C Ψ d i C Δ d ,
C Δ = 4 π λ · n 0 cos φ 0 sin 2 φ 0 ( n 1 2 n 0 2 ) [ μ ( 1 / n 1 2 α ) α 1 ν ] ( μ 2 + ν 2 ) ,
C Ψ = 4 π λ · n 0 cos φ 0 sin 2 φ 0 ( n 1 2 n 0 2 ) [ ν ( 1 / n 1 2 α ) α 1 μ ] ( μ 2 + ν 2 ) ,
μ = cos 2 φ 0 n 0 2 α + n 0 4 ( α 2 α 1 2 ) sin 2 φ 0 ,
ν = n 0 2 α 1 n 0 4 · 2 α α 1 sin 2 φ 0 ,
α = ( n 2 2 k 2 2 ) / [ ( n 2 2 + k 2 2 ) 2 ] ,
α 1 = 2 n 2 k 2 / [ ( n 2 2 + k 2 2 ) 2 ] .
( tan Ψ / tan Ψ ¯ ) cos ( Δ Δ ¯ ) = 1 + C Ψ d ,
( tan Ψ / tan Ψ ¯ ) sin ( Δ Δ ¯ ) = C Δ d .
tan ( Δ Δ ¯ ) = C Δ d / ( 1 + C Ψ d ) .
tan ( Δ Δ ¯ ) = C Δ d .
Δ Δ ¯ ( 180 / π ) C Δ d .
( tan Ψ / tan Ψ ¯ ) 1 = C Ψ d .
Ψ Ψ ¯ ( 180 / π ) 1 2 ( sin 2 Ψ ¯ ) · C Ψ d .
( tan 2 Ψ / tan 2 Ψ ¯ ) 1 = C Δ 2 d 2 ,
Ψ Ψ ¯ = ( 45 / π ) sin 2 Ψ ¯ · C Δ 2 d 2 .
d = ( m π + δ 0 ) C n 1 ,
Δ Δ ¯ = ( 180 / π ) C Δ ( d m π C n 1 ) ,
Ψ Ψ ¯ = ( 45 / π ) sin 2 Ψ ¯ C Δ 2 ( d m π C n 1 ) 2 .
Δ Δ ¯ = ( 180 / π ) C Δ ( C n 1 π d )
Ψ Ψ ¯ = ( 45 / π ) sin 2 Ψ ¯ C Δ 2 ( C n 1 π d ) 2 .
δ π = ( m + 1 ) π δ
Δ Δ ¯ = ( 180 / π ) C Δ [ C n 1 ( m + 1 ) π d ] ,
Ψ Ψ ¯ = ( 45 / π ) sin 2 Ψ ¯ C Δ 2 [ C n 1 ( m + 1 ) π d ] 2 .