Abstract

To resolve observed absorption anomalies in the visible region in very thin films on evaporated metals a model is proposed which assumes that: (1) absorption can be separated into x, y, and z components and each of the components decreases exponentially with distance from the film-metal interface and that (2) the absorption at a point in the film is given by the sum of products involving the relative electric-field components and the respective absorption coefficients in the coordinate directions. A procedure is presented for calculation of the relative electric field components for a given film, substrate, wavelength, and angle of incidence. The relative tangential electric field has a dominant influence on the absorption, particularly for very thin films. This conclusion is based on the observed correlation between the slopes of the measured absorption and the calculated tangential field. The substrate has an important influence. This is evident in differences in measured absorption and calculated fields comparing Au and Cr substrates.

© 1966 Optical Society of America

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References

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  1. S. A. Francis and A. H. Ellison, J. Opt. Soc. Am. 49, 131 (1959).
    [Crossref]
  2. F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963).
    [Crossref]
  3. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
    [Crossref]
  4. F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 61.
  5. N. M. Bashara and C. T. Doty, J. Appl. Phys. 35, 3498 (1964).
    [Crossref]
  6. F. Rouard and F. Bousquet, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), IV, p. 147.
  7. L. G. Schulz, J. Opt. Soc. Am. 44, 357 (1954); L. G. Schulz and F. R. Tangherlini, J. Opt. Soc. Am. 44, 363 (1954).
    [Crossref]
  8. A. C. Hall, J. Opt. Soc. Am. 55, 911 (1965).
  9. Contamination films of water on Cr and Au are discussed by McCrackin et al.3
  10. The variation may be partly due to a change of substrate properties caused by a difference of structure.
  11. D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965).
    [Crossref]
  12. The approximate equation for the thickness used is derived assuming that higher-order terms of the ratio (d/λ0) can be neglected. This assumption holds best for film thickness <50 Å.
  13. V. V. Andreeva, Corrosion 20, 35t. (1964).
    [Crossref]
  14. L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961).
    [Crossref]
  15. O. Wiener, Ann. Physik 40, 203 (1890).
    [Crossref]
  16. See the discussion by Hayfield and White17 on this point, p. 161.
  17. P. C. S. Hayfield and G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 157.
  18. We are uncertain of the homogeneity of a 22-Å film of polybutadiene.
  19. The changes of film absorption for the film on Au between 4900 and 5300 Å are not dealt with by our model nor is the rapid increase of absorption for films on both substrates as measurements extend into the ultraviolet. The changes of absorption between 4900 and 5300 Å are probably related to the large change of the optical properties of Au, Fig. 1. In contrast, the properties of Cr are considerably different, Fig. 2.
  20. Compare Figs. 1 and 2 with Fig. 12.
  21. The initial part of this section follows a presentation by Hayfield and White.17 Fry’s22 analysis of the field in a film sandwiched between air and a metal base was used by Francis and Ellison. Strachan23 has also studied this general problem.
  22. T. C. Fry, J. Opt. Soc. Am. 22, 307 (1932).
    [Crossref]
  23. C. Strachan, Proc. Cambridge Phil. Soc. 29, 116 (1933).
    [Crossref]
  24. The absorption and phase lag which occur in the traversals of the film are not shown explicitly in Fig. 15 but are included in the analysis.

1965 (2)

1964 (2)

V. V. Andreeva, Corrosion 20, 35t. (1964).
[Crossref]

N. M. Bashara and C. T. Doty, J. Appl. Phys. 35, 3498 (1964).
[Crossref]

1963 (2)

F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963).
[Crossref]

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

1961 (1)

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961).
[Crossref]

1959 (1)

1954 (1)

1933 (1)

C. Strachan, Proc. Cambridge Phil. Soc. 29, 116 (1933).
[Crossref]

1932 (1)

1890 (1)

O. Wiener, Ann. Physik 40, 203 (1890).
[Crossref]

Andreeva, V. V.

V. V. Andreeva, Corrosion 20, 35t. (1964).
[Crossref]

Bartell, L. S.

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961).
[Crossref]

Bashara, N. M.

D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965).
[Crossref]

N. M. Bashara and C. T. Doty, J. Appl. Phys. 35, 3498 (1964).
[Crossref]

Bousquet, F.

F. Rouard and F. Bousquet, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), IV, p. 147.

Churchill, D.

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961).
[Crossref]

Colson, J. P.

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 61.

Doty, C. T.

N. M. Bashara and C. T. Doty, J. Appl. Phys. 35, 3498 (1964).
[Crossref]

Ellison, A. H.

Francis, S. A.

Fry, T. C.

Hall, A. C.

Hayfield, P. C. S.

P. C. S. Hayfield and G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 157.

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 61.

Mertens, F. P.

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

Peterson, D. W.

Plumb, R. C.

Rouard, F.

F. Rouard and F. Bousquet, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), IV, p. 147.

Schulz, L. G.

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

Strachan, C.

C. Strachan, Proc. Cambridge Phil. Soc. 29, 116 (1933).
[Crossref]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

Theroux, P.

White, G. W. T.

P. C. S. Hayfield and G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 157.

Wiener, O.

O. Wiener, Ann. Physik 40, 203 (1890).
[Crossref]

Ann. Physik (1)

O. Wiener, Ann. Physik 40, 203 (1890).
[Crossref]

Corrosion (1)

V. V. Andreeva, Corrosion 20, 35t. (1964).
[Crossref]

J. Appl. Phys. (1)

N. M. Bashara and C. T. Doty, J. Appl. Phys. 35, 3498 (1964).
[Crossref]

J. Opt. Soc. Am. (6)

J. Phys. Chem. (1)

L. S. Bartell and D. Churchill, J. Phys. Chem. 65, 2242 (1961).
[Crossref]

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

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. 67A, 363 (1963).
[Crossref]

Proc. Cambridge Phil. Soc. (1)

C. Strachan, Proc. Cambridge Phil. Soc. 29, 116 (1933).
[Crossref]

Other (12)

The absorption and phase lag which occur in the traversals of the film are not shown explicitly in Fig. 15 but are included in the analysis.

F. L. McCrackin and J. P. Colson, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 61.

Contamination films of water on Cr and Au are discussed by McCrackin et al.3

The variation may be partly due to a change of substrate properties caused by a difference of structure.

The approximate equation for the thickness used is derived assuming that higher-order terms of the ratio (d/λ0) can be neglected. This assumption holds best for film thickness <50 Å.

F. Rouard and F. Bousquet, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1965), IV, p. 147.

See the discussion by Hayfield and White17 on this point, p. 161.

P. C. S. Hayfield and G. W. T. White, in Ellipsometry in the Measurement of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, and J. Kruger, Eds. (Natl. Bur. Std. Miscl. Publ., No. 256, Washington, D. C., 1964), p. 157.

We are uncertain of the homogeneity of a 22-Å film of polybutadiene.

The changes of film absorption for the film on Au between 4900 and 5300 Å are not dealt with by our model nor is the rapid increase of absorption for films on both substrates as measurements extend into the ultraviolet. The changes of absorption between 4900 and 5300 Å are probably related to the large change of the optical properties of Au, Fig. 1. In contrast, the properties of Cr are considerably different, Fig. 2.

Compare Figs. 1 and 2 with Fig. 12.

The initial part of this section follows a presentation by Hayfield and White.17 Fry’s22 analysis of the field in a film sandwiched between air and a metal base was used by Francis and Ellison. Strachan23 has also studied this general problem.

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Figures (16)

Fig. 1
Fig. 1

The optical properties of the vacuum-evaporated Au film used showing refractive index n and absorption coefficient k.

Fig. 2
Fig. 2

The optical properties of the vacuum-evaporated Cr film used showing refractive index n and absorption coefficient k.

Fig. 3
Fig. 3

The Δ and ψ angles for the curve-fitting of the polybutadiene film on Au. The solid line is the theoretical curve for n=1.57 and k=0.054. The experimental points 1 through 5 are for increasing film thickness. The point marked 0 is the bare substrate. The measuring wavelength is 4000 Å.

Fig. 4
Fig. 4

The wavelength dependence of the measured absorption coefficient kav of polybutadiene films (22, 55, and 136 Å) on a vacuum-evaporated, opaque Au substrate. The films were successively deposited on the same substrate. Incident angle −75°.

Fig. 5
Fig. 5

Shown are values of Δ vs ψ for 4000-Å light at an angle of incidence of 80°. The solid line is an experimental curve for polybutadiene on Cr. Dashed curves are assumed values of refractive index n of 1.57 and varying absorption coefficient k which are shown on the curves. The assumed film thickness d1 is shown also.

Fig. 6
Fig. 6

A continuation of Fig. 5 for thicker films.

Fig. 7
Fig. 7

The wavelength dependence of the measured absorption coefficient kav of polybutadiene films (50 and 300 Å) on a vacuum-evaporated, opaque Cr substrate. Films were successively deposited on the same substrate. Incident angle—80°.

Fig. 8
Fig. 8

The refractive index for the films of Fig. 4 (solid lines), compared with results of approximate relations (dashed lines).

Fig. 9
Fig. 9

The thickness dependence of the measured absorption coefficient kav of polybutadiene films. The curves with steeper slope are on an Au substrate, the others on a Cr substrate. Measurement wavelengths are shown. Incident angle: Au—75°, Cr—80°.

Fig. 10
Fig. 10

The absorption of a thick polybutadiene film (>1000 Å) on tin oxide. 1—transmittance of tin oxide on glass, 2—polybutadiene over the tin oxide, 3—the relative decrease in transmittance. (Similar results are obtained for a thick polybutadiene film on partly transparent Au).

Fig. 11
Fig. 11

The postulated variation of film absorption with distance from the film–metal substrate interface l. The bulk absorption is kb. The jth component (j=x, y, z, or xy) of the absorption at the film–metal interface is k0j.

Fig. 12
Fig. 12

Relative normal electric field strength g2 in a film at the film–substrate interface. The variation of g2 with wavelength for an assumed film of n=1.57, k=0.10 on an Au substrate and on a Cr substrate is shown. A film thickness of 20 Å is assumed.

Fig. 13
Fig. 13

Relative tangential electric field strength, h2=(1−g2), in a film at the film–substrate interface. The variation of h2 with wavelength for an assumed film of n=1.57, k=0.10 on an Au substrate and on a Cr substrate is shown. A film thickness of 20 Å is assumed.

Fig. 14
Fig. 14

Reflectance and transmittance at a single interface.

Fig. 15
Fig. 15

Multiple reflections within a film. Light of unit amplitude is incident at A. The cumulative effect of each reflection and transmission is shown as it would enter into the summation process. The absorption and phase lag which occur in the traversals of the film are not shown explicitly but are included in the analysis.

Fig. 16
Fig. 16

Contributions to the electric field at point H in a thin film due to multiple reflections. Changes due to the Fresnel reflectance and transmittance, as shown in Fig. 15 occur for each individual reflection and transmission.

Tables (3)

Tables Icon

Table I A comparison of the optical properties of Au at 5461 Å.

Tables Icon

Table II The relative vertical field for a film of a given refractive index. The film absorption coefficient, film thickness, wavelength, and substrate are varied. Au substrate—a, b, and c. Cr substrate—d and e. The substrate properties are shown in Figs. 1 and 2.

Tables Icon

Table III Comparison of the relative vertical field at the film–substrate interface using Schulz’s (Ref. 7) data for an Au substrate and slight variations in substrate properties. The properties of the assumed dielectric film are n=1.57, k=0.100. Group a—original data, b—all substrate values increased 5%, c—all substrate values decreased 5%.

Equations (43)

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k x = k b + k 0 x e - l / d s ,
k y = k b + k 0 y e - l / d s ,
k z = k b + k 0 z e - l / d s .
k l = m 2 k x + p 2 k y + g 2 k z ,
m 2 = E 1 x 2 E 1 x 2 + E 1 y 2 + E 1 z 2 ,
p 2 = E 1 y 2 E 1 x 2 + E 1 y 2 + E 1 z 2 ,
g 2 = E 1 z 2 E 1 x 2 + E 1 y 2 + E 1 z 2 .
k l = k b + ( h 2 k 0 x y + g 2 k 0 z ) e - l / d s .
k av = 1 d 1 0 d 1 k l d l .
k av = k b + [ h 2 k 0 x y + g 2 k 0 z ] ( 1 - e - d 1 / d s ) d s / d 1 .
k av = k b + [ h 2 ( k 0 x y - k 0 z ) + k 0 z ] ( 1 - e - d 1 / d s ) d s / d 1
k av = k b + [ g 2 ( k 0 z - k 0 x y ) + k 0 x y ] ( 1 - e - d 1 / d s ) d s / d 1 .
m 2 = E 1 x / E i p 2 E 1 x / E i p 2 + E 1 y / E i s 2 + E 1 z / E i p 2 ,
p 2 = E 1 y / E i s 2 E 1 x / E i p 2 + E 1 y / E i s 2 + E 1 z / E i p 2 ,
g 2 = E 1 z / E i p 2 E 1 x / E i p 2 + E 1 y / E i s 2 + E 1 z / E i p 2 .
E r = r E i
E t = t E i ,
r p = N 2 cos ϕ 1 - N 1 cos ϕ 2 N 2 cos ϕ 1 + N 1 cos ϕ 2
r s = N 1 cos ϕ 1 - N 2 cos ϕ 2 N 1 cos ϕ 1 + N 2 cos ϕ 2 .
t p = 2 N 1 cos ϕ 1 N 2 cos ϕ 1 + N 1 cos ϕ 2
t s = 2 N 1 cos ϕ 1 N 1 cos ϕ 1 + N 2 cos ϕ 2 .
A = A 0 e - 2 π k 1 d 1 / λ 0 ,
N 1 sin ϕ 1 = N 2 sin ϕ 2 .
r = - r ,
t t = 1 - r 2 ,
δ 0 = ( 2 π / λ 0 ) N 1 d 1 cos ϕ 1 .
E 1 p = E i p [ e + i δ a ( t p 1 - t p 1 r p 2 r p 1 e - i 2 δ 0 + t p 1 r p 2 2 r p 1 2 e - i 4 δ 0 - ) ] .
E 1 p = E i p [ t p 1 e + i δ a / ( 1 + r p 1 r p 2 e - i 2 δ 0 ) ] × ( a x cos ϕ 1 - a z sin ϕ 1 ) ,
E 1 p = E i p [ e + i ( δ b - δ a ) ( t p 1 r p z e - i 2 δ 0 - t p 1 r p 2 2 r p 1 e - i 4 δ 0 + t p 1 r p 2 3 r p 1 2 e - i 6 δ 0 - ) ]
E 1 p = E i p [ t p 1 r p 2 e + i ( δ b - δ a - 2 δ 0 ) 1 + r p 1 r p 2 e - i 2 δ 0 ] × ( - a x cos ϕ 1 - a z sin ϕ 1 ) .
E 1 x = E i p [ t p 1 ( e + i δ a - r p 2 e + i ( δ b - δ a - 2 δ 0 ) ) 1 + r p 1 r p 2 e - i 2 δ 0 ] a x cos ϕ 1
E 1 z = - E i p [ t p 1 ( e + i δ a + r p 2 e + i ( δ b - δ a - 2 δ 0 ) ) 1 + r p 1 r p 2 e - i 2 δ 0 ] a z sin ϕ 1 .
E 1 y = E 1 s = E i s [ t s 1 ( e + i δ a + r s 2 e + i ( δ b - δ a - 2 δ 0 ) ) 1 + r s 1 r s 2 e - i 2 δ 0 ] a y .
δ a = ( 2 π N 1 z / λ 0 cos ϕ 1 ) = ( z δ 0 / d 1 cos 2 ϕ 1 )
δ b = ( 2 π / λ 0 ) 2 ( d a sin ϕ 1 ) sin ϕ 0 .
δ a = ( 2 π N 1 / λ 0 ) d a
δ b = ( 2 δ a / N 1 ) sin ϕ 1 sin ϕ 0 = 2 sin 2 ϕ 1 δ a = ( z / d 1 ) 2 tan 2 ϕ 1 δ 0 .
E 1 x = E i p [ t p 1 ( e + i δ 0 / cos 2 ϕ 1 ) ( 1 - r p 2 ) 1 + r p 1 r p 2 e - i 2 δ 0 ] a x cos ϕ 1
E 1 z = - E i p [ t p 1 ( e + i δ 0 / cos 2 ϕ 1 ) ( 1 + r p 2 ) 1 + r p 1 r p 2 e - 2 i δ 0 ] a z sin ϕ 1
E 1 y = E i s [ t s 1 ( e + i δ 0 / cos 2 ϕ 1 ) ( 1 + r s 2 ) 1 + r s 1 r s 2 e - i 2 δ 0 ] a y .
E 1 x = E i p [ t p 1 ( 1 - r p 2 e - i 2 δ 0 ) ( 1 + r p 1 r p 2 e - i 2 δ 0 ) ] a x cos ϕ 1
E 1 z = E i p [ t p 1 ( 1 + r p 2 e - i 2 δ 0 ) ( 1 + r p 1 r p 2 e - i 2 δ 0 ) ] a z sin ϕ 1
E 1 y = E i s [ t s 1 ( 1 + r s 2 e - i 2 δ 0 ) ( 1 + r s 1 r s 2 e - i 2 δ 0 ) ] a y .