Abstract

When polarized light is reflected from a film-covered surface, the variations in Δ and Ψ with the film thickness <i>d</i> 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 Δ 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.

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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).
  2. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 38 (1955).
  3. A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
  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 φ0 in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨ in a recent paper (see Ref. 5).
  5. R. J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965).
  6. D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
  7. R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).
  8. C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).
  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, 3 March 1964.
  11. A. E. Lewis, J. Electrochem. Soc. 111, 1007 (1964).
  12. S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London 1948).
  13. H. C. Evitts, H. W. Cooper, and S. S. Flaschen, J. Electrochem. Soc. 111, 688 (1964).

Archer, R. J.

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

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 φ0 in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨ in a recent paper (see Ref. 5).

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

Bennett, H. E.

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

Burge, D. K.

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

Cooper, H. W.

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

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).

Evitts, H. C.

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

Flaschen, S. S.

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

Gobeli, G. W.

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

Hanson, M.

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

Leberknight, C. E.

C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).

Lewis, A. E.

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

Lustman, B.

C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).

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, 3 March 1964.

Rothen, A.

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

Tolansky, S.

S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London 1948).

Winterbottom, A. B.

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

Other (13)

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).

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

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

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 φ0 in the expressions for CΔ and CΨ. The author pointed this out to Archer and he retained this term in the expressions for CΔ and CΨ in a recent paper (see Ref. 5).

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

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

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

C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).

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, 3 March 1964.

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

S. Tolansky, Multiple Beam Interferometry of Surfaces and Films (Oxford at the Clarendon Press, London 1948).

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

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