Abstract

Measurements and calculations using elliptically polarized light to determine the thickness and index of refraction of barium stearate films on a variety of substrates are described. Attention is focused especially upon problems associated with the study of very thin films, that is, films which are less than 100 Å thick.

Measurements of the optical constants of metal substrates are presented. An apparent dependence of the index of refraction n and extinction coefficient κ and independence of the absorption coefficient k upon the angle of incidence is observed and is discussed.

Several effects, including accuracy and sensitivity, of the angle of incidence in studying film growth are noted. It is shown that there exists a characteristic angle of incidence at which the growth of a film produces a negligible change in the ellipticity, and it is shown how advantage may be taken of this to facilitate measurements of rapidly changing surfaces.

It is demonstrated that extraneous films are satisfactorily accounted for in measuring thin films by means of the Drude equations through their effects upon the apparent optical constants of the substrate. Advantage may be taken of this fact to extend the thickness range over which the Drude linear equations are applicable by a computational procedure.

The problem of the anomalous index of refraction of very thin films as determined by the Drude equations is considered. It is concluded that the anomalies revealed by the Drude equations are associated with the system under study and are not caused by any inherent limitation in the Drude equations. It appears that at the interface between a dielectric layer and metal system there is an absorbing region which produces the anomaly in the refractive index calculated by the Drude equations and for which explicit allowance must be made in order to make accurate measurements of the thickness of very thin films.

© 1963 Optical Society of America

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References

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  1. A. B. Winterbottom, Norske Videnskabers Selskab 45, 1–149 (1955).
  2. A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).
  3. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955).
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London, 1959).
  5. Reference 2, p. 288 et seq.
  6. Reference 3, p. 181 et seq.
  7. Reference 2, p. 292.
  8. Reference 1, p. 121.
  9. A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
    [Crossref]
  10. J. A. Faucher, G. M. McManus, and H. T. Trurnit, J. Opt. Soc. Am. 48, 51–54 (1958).
    [Crossref] [PubMed]
  11. J. B. Bateman and M. W. Harris, Ann. N.Y. Acad. Sci. 53, 1964 (1951).
    [Crossref]
  12. A. Vasicek, Czech. J. Phys. 4, 204–220 (1954).
    [Crossref]
  13. Reference 1, p. 128 et seq.
  14. Technique due to W. Gray.
  15. R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960).
    [Crossref]
  16. K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
    [Crossref]
  17. F. P. Mertens and R. C. Plumb (to be published).
  18. J. W. Swaine and R. C. Plumb, J. Appl. Phys. 33, 2378–2382 (1962).
    [Crossref]
  19. K. Blodgett and I. Langmuir, Phys. Rev. 51, 964–982 (1937).
    [Crossref]

1962 (1)

J. W. Swaine and R. C. Plumb, J. Appl. Phys. 33, 2378–2382 (1962).
[Crossref]

1960 (1)

1958 (1)

1955 (1)

A. B. Winterbottom, Norske Videnskabers Selskab 45, 1–149 (1955).

1954 (1)

A. Vasicek, Czech. J. Phys. 4, 204–220 (1954).
[Crossref]

1951 (1)

J. B. Bateman and M. W. Harris, Ann. N.Y. Acad. Sci. 53, 1964 (1951).
[Crossref]

1945 (1)

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

1937 (1)

K. Blodgett and I. Langmuir, Phys. Rev. 51, 964–982 (1937).
[Crossref]

1935 (1)

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

Bateman, J. B.

J. B. Bateman and M. W. Harris, Ann. N.Y. Acad. Sci. 53, 1964 (1951).
[Crossref]

Blodgett, K.

K. Blodgett and I. Langmuir, Phys. Rev. 51, 964–982 (1937).
[Crossref]

Blodgett, K. B.

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London, 1959).

Faucher, J. A.

Harris, M. W.

J. B. Bateman and M. W. Harris, Ann. N.Y. Acad. Sci. 53, 1964 (1951).
[Crossref]

Heavens, O. S.

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

Langmuir, I.

K. Blodgett and I. Langmuir, Phys. Rev. 51, 964–982 (1937).
[Crossref]

McManus, G. M.

Mertens, F. P.

F. P. Mertens and R. C. Plumb (to be published).

Plumb, R. C.

J. W. Swaine and R. C. Plumb, J. Appl. Phys. 33, 2378–2382 (1962).
[Crossref]

R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960).
[Crossref]

F. P. Mertens and R. C. Plumb (to be published).

Rothen, A.

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

Swaine, J. W.

J. W. Swaine and R. C. Plumb, J. Appl. Phys. 33, 2378–2382 (1962).
[Crossref]

Trurnit, H. T.

Vasicek, A.

A. Vasicek, Czech. J. Phys. 4, 204–220 (1954).
[Crossref]

A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).

Winterbottom, A. B.

A. B. Winterbottom, Norske Videnskabers Selskab 45, 1–149 (1955).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London, 1959).

Ann. N.Y. Acad. Sci. (1)

J. B. Bateman and M. W. Harris, Ann. N.Y. Acad. Sci. 53, 1964 (1951).
[Crossref]

Czech. J. Phys. (1)

A. Vasicek, Czech. J. Phys. 4, 204–220 (1954).
[Crossref]

J. Am. Chem. Soc. (1)

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

J. Appl. Phys. (1)

J. W. Swaine and R. C. Plumb, J. Appl. Phys. 33, 2378–2382 (1962).
[Crossref]

J. Opt. Soc. Am. (2)

Norske Videnskabers Selskab (1)

A. B. Winterbottom, Norske Videnskabers Selskab 45, 1–149 (1955).

Phys. Rev. (1)

K. Blodgett and I. Langmuir, Phys. Rev. 51, 964–982 (1937).
[Crossref]

Rev. Sci. Instr. (1)

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

Other (10)

F. P. Mertens and R. C. Plumb (to be published).

Reference 1, p. 128 et seq.

Technique due to W. Gray.

A. Vasicek, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960).

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

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London, 1959).

Reference 2, p. 288 et seq.

Reference 3, p. 181 et seq.

Reference 2, p. 292.

Reference 1, p. 121.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of polarizing spectrometer.

Fig. 2
Fig. 2

Curves showing method of obtaining analyzer and compensator readings for extinction from photoelectric intensity measurements.

Fig. 3
Fig. 3

The variation of the apparent optical constants of gold and of gold covered by a 100-Å-thick barium stearate film with angle of incidence at 5461 Å.

Fig. 4
Fig. 4

Curves showing the phase difference Δ − Δ0 caused by various thicknesses of barium stearate on copper as a function of the angle of incidence.

Fig. 5
Fig. 5

The thickness (d1) and refractive index (n1) of a four-layer barium stearate film on gold as measured at several angles of incidence and computed by the Drude linear approximation.

Fig. 6
Fig. 6

The ellipticity change γ − γ0 produced by one monolayer of barium stearate as a function of the angle of incidence for gold and copper substrates.

Fig. 7
Fig. 7

Quarter-wave plate readings versus analyzer readings for the oxidation of aluminum measured at several angles of incidence.

Fig. 8
Fig. 8

The thickness of barium stearate films on gold as measured at an angle of incidence of 65.57° and calculated by the standard Drude linear approximation method and by applying the Drude equations in a step-by-step method.

Fig. 9
Fig. 9

Observed and calculated tanΨ vs Δ for several thicknesses of barium stearate film: ◑ observed; x calculated, assuming n = 1.5, n = 3.0, and nκ = 0; ○, □, ●, △ calculated for n = 1.5 and nκ = 0.3, 0.6, 1.1, and 1.3, respectively. Thicknesses in angstroms marked on curves.

Fig. 10
Fig. 10

Observed dependence of absorption coefficient on film thickness assuming that entire change in reflectivity is caused by absorption within the film.

Tables (1)

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Table I Drude optical constants on Al(oxidized) at 5461 Å.

Equations (8)

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n 2 k 2 = n 2 ( 1 κ 2 ) = sin 2 θ i [ 1 + tan 2 θ i ( cos 2 2 Ψ sin 2 2 Ψ sin 2 Δ ) ( 1 + sin 2 Ψ cos Δ ) 2 ] , 2 n k = 2 n 2 κ = sin 2 θ i tan 2 θ i sin 4 Ψ sin Δ / ( 1 + sin 2 Ψ cos Δ ) 2 ,
R ˆ p / R ˆ s = tan Ψ e i Δ ,
R ˆ = [ r ˆ 12 + r ˆ 23 exp ( 2 i δ ) ] / [ 1 + r ˆ 12 r ˆ 23 exp ( 2 i δ ) ] , δ = 2 π n ˆ 2 cos ϕ 2 d / λ 0 ,
r ˆ k l p = ( n ˆ 1 cos ϕ k n ˆ k cos ϕ 1 ) / ( n ˆ 1 cos ϕ k + n ˆ k cos ϕ 1 ) , r ˆ k 1 s = ( n ˆ k cos ϕ k n ˆ 1 cos ϕ 1 ) / ( n ˆ k cos ϕ k + n ˆ 1 cos ϕ 1 ) .
a = 1 κ 2 2 n 2 2 ( 1 + κ 2 2 ) 2 , a = 2 κ 2 n 2 2 ( 1 + κ 2 2 ) 2 ; j = ( Δ 0 Δ ) cos 2 ϕ 0 a , n 1 = 1 cos ϕ 0 [ 2 ( Ψ 0 Ψ ) a j sin 2 Ψ 0 + 1 ] 1 2 ; d 1 = j λ 4 π [ ( cos 2 ϕ 0 a ) 2 + a 2 ] 4 cos ϕ 0 sin 2 ϕ 0 [ 1 ( 1 / n 2 1 ) ] .
Ψ s / p = 1 / 2 cos 1 [ cos 2 γ cos 2 x ( 1 0.110 tan 2 x tan 2 γ ) ] , Δ p s = cos 1 [ cos 2 γ cos 2 x ( tan 2 x + 0.110 tan 2 γ ) / sin 2 Ψ ] ,
R = | ( n ˆ 1 ) / ( n ˆ + 1 ) | 2 .
ln ( R 0 / R ) = 4 π k 2 d / λ .