Abstract

The Maxwell-Garnett theory extended to include the shape factor and the size of the metal particles embedded in a dielectric matrix is proposed to explain the observed optical constants of a Cu–PbI2 cermet material. Both the shape factor and size of the particles are obtained by a separate fit of the real and imaginary parts of the measured dielectric constant. These geometrical parameters, once determined for a particular wavelength, can be used to derive the optical constants throughout the visible spectrum. Fairly good agreement is observed for a volume fraction ranging from 0 to 12%. Alternatively, the method derived in this paper can serve as a test for various theories proposed to predict the optical properties of composite materials.

© 1984 Optical Society of America

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References

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  1. Lord Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1892).
    [CrossRef]
  2. J. C. Maxwell-Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 205, 237 (1906).
    [CrossRef]
  3. Electrical Transport and Optical Properties of Inhomogeneous Media, AIP Conf. Proc.40 (1978).
  4. R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
    [CrossRef]
  5. R. H. Doremus, “Optical Properties of Thin Metallic Films in Island Form,” J. Appl. Phys. 37, 2775 (1966).
    [CrossRef]
  6. J. P. Marton, “Optical Properties of Islands Films of Metal at Long Wavelengths,” J. Appl. Phys. 40, 5383 (1969).
    [CrossRef]
  7. P. H. Lissberger, R. G. Nelson, “Optical Properties of Thin Film Au-MgF2 Cermets,” Thin Solid Films 21, 159 (1974).
    [CrossRef]
  8. C. Grosse, J. L. Greffe, “Permittivité statique des émulsions,” J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).
  9. G. Boivin, J. M. Thériault, “Influence of Surfaces Effects in the Determination of the Optical Constants of Cu-PbI2 Cermets Films,” Appl. Opt. 23, 4245 (1984).
    [CrossRef] [PubMed]
  10. J. M. Thériault, Ph.D. Thesis, U. Laval (1983).
  11. G. Boivin, J. M. Thériault, “Caractéristiques optiques de la photodécomposition de cermets et d’antireflets formés de Cu-PbI2,” Can. J. Phys. 62, 811 (1984).
    [CrossRef]
  12. J. M. Thériault, M.Sc. Thesis, U. Laval (1979).
  13. J. M. Thériault, G. Boivin, “Cermets de Cu-PI2 en couches minces: fabrication et détermination des constantes optiques,” Can. J. Phys. 61, 612 (1983).
    [CrossRef]
  14. A. V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967).
  15. C. Kittel, Introduction à l’étude de I’état solide (Dunod, Paris, 1972).
  16. L. G. Schultz, F. R. Tangherlini, “Optical Constants of Silver, Gold, Copper, and Aluminum. I. The Absorption Coefficient k,” J. Opt. Soc. Am. 44, 357 (1954).
    [CrossRef]
  17. P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  18. B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
    [CrossRef]
  19. D. M. Wood, N. W. Ashcroft, “Effective Medium Theory of Optical Properties of Small Particle Composites,” Philos. Mag. 35, 269 (1977).
    [CrossRef]
  20. S. Berthier, J. Lafait, “Black Chromium Coatings: Experimental and Calculated Optical Properties using Inhomogeneous Medium Theories,” J. Phys. 40, 1093 (1979).
    [CrossRef]
  21. T. Hanoi, “Theory of the Dielectric Dispersion due to the Interfacial Polarisation and its Application to Emulsions,” Kolloid-Z. 171, 23 (1960).
    [CrossRef]

1984 (2)

G. Boivin, J. M. Thériault, “Influence of Surfaces Effects in the Determination of the Optical Constants of Cu-PbI2 Cermets Films,” Appl. Opt. 23, 4245 (1984).
[CrossRef] [PubMed]

G. Boivin, J. M. Thériault, “Caractéristiques optiques de la photodécomposition de cermets et d’antireflets formés de Cu-PbI2,” Can. J. Phys. 62, 811 (1984).
[CrossRef]

1983 (1)

J. M. Thériault, G. Boivin, “Cermets de Cu-PI2 en couches minces: fabrication et détermination des constantes optiques,” Can. J. Phys. 61, 612 (1983).
[CrossRef]

1979 (2)

C. Grosse, J. L. Greffe, “Permittivité statique des émulsions,” J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).

S. Berthier, J. Lafait, “Black Chromium Coatings: Experimental and Calculated Optical Properties using Inhomogeneous Medium Theories,” J. Phys. 40, 1093 (1979).
[CrossRef]

1977 (1)

D. M. Wood, N. W. Ashcroft, “Effective Medium Theory of Optical Properties of Small Particle Composites,” Philos. Mag. 35, 269 (1977).
[CrossRef]

1975 (1)

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

1974 (1)

P. H. Lissberger, R. G. Nelson, “Optical Properties of Thin Film Au-MgF2 Cermets,” Thin Solid Films 21, 159 (1974).
[CrossRef]

1973 (1)

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370 (1972).
[CrossRef]

1969 (1)

J. P. Marton, “Optical Properties of Islands Films of Metal at Long Wavelengths,” J. Appl. Phys. 40, 5383 (1969).
[CrossRef]

1966 (1)

R. H. Doremus, “Optical Properties of Thin Metallic Films in Island Form,” J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

1960 (1)

T. Hanoi, “Theory of the Dielectric Dispersion due to the Interfacial Polarisation and its Application to Emulsions,” Kolloid-Z. 171, 23 (1960).
[CrossRef]

1954 (1)

1906 (1)

J. C. Maxwell-Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

1892 (1)

Lord Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1892).
[CrossRef]

Abeles, B.

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Arie, Y.

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

Ashcroft, N. W.

D. M. Wood, N. W. Ashcroft, “Effective Medium Theory of Optical Properties of Small Particle Composites,” Philos. Mag. 35, 269 (1977).
[CrossRef]

Berthier, S.

S. Berthier, J. Lafait, “Black Chromium Coatings: Experimental and Calculated Optical Properties using Inhomogeneous Medium Theories,” J. Phys. 40, 1093 (1979).
[CrossRef]

Boivin, G.

G. Boivin, J. M. Thériault, “Influence of Surfaces Effects in the Determination of the Optical Constants of Cu-PbI2 Cermets Films,” Appl. Opt. 23, 4245 (1984).
[CrossRef] [PubMed]

G. Boivin, J. M. Thériault, “Caractéristiques optiques de la photodécomposition de cermets et d’antireflets formés de Cu-PbI2,” Can. J. Phys. 62, 811 (1984).
[CrossRef]

J. M. Thériault, G. Boivin, “Cermets de Cu-PI2 en couches minces: fabrication et détermination des constantes optiques,” Can. J. Phys. 61, 612 (1983).
[CrossRef]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Cody, G. D.

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Cohen, R. W.

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Coutts, M. D.

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Doremus, R. H.

R. H. Doremus, “Optical Properties of Thin Metallic Films in Island Form,” J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

Greffe, J. L.

C. Grosse, J. L. Greffe, “Permittivité statique des émulsions,” J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).

Grosse, C.

C. Grosse, J. L. Greffe, “Permittivité statique des émulsions,” J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).

Hanoi, T.

T. Hanoi, “Theory of the Dielectric Dispersion due to the Interfacial Polarisation and its Application to Emulsions,” Kolloid-Z. 171, 23 (1960).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Kittel, C.

C. Kittel, Introduction à l’étude de I’état solide (Dunod, Paris, 1972).

Lafait, J.

S. Berthier, J. Lafait, “Black Chromium Coatings: Experimental and Calculated Optical Properties using Inhomogeneous Medium Theories,” J. Phys. 40, 1093 (1979).
[CrossRef]

Lissberger, P. H.

P. H. Lissberger, R. G. Nelson, “Optical Properties of Thin Film Au-MgF2 Cermets,” Thin Solid Films 21, 159 (1974).
[CrossRef]

Marton, J. P.

J. P. Marton, “Optical Properties of Islands Films of Metal at Long Wavelengths,” J. Appl. Phys. 40, 5383 (1969).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Nelson, R. G.

P. H. Lissberger, R. G. Nelson, “Optical Properties of Thin Film Au-MgF2 Cermets,” Thin Solid Films 21, 159 (1974).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1892).
[CrossRef]

Schultz, L. G.

Sheng, P.

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

Sokolov, A. V.

A. V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967).

Tangherlini, F. R.

Thériault, J. M.

G. Boivin, J. M. Thériault, “Influence of Surfaces Effects in the Determination of the Optical Constants of Cu-PbI2 Cermets Films,” Appl. Opt. 23, 4245 (1984).
[CrossRef] [PubMed]

G. Boivin, J. M. Thériault, “Caractéristiques optiques de la photodécomposition de cermets et d’antireflets formés de Cu-PbI2,” Can. J. Phys. 62, 811 (1984).
[CrossRef]

J. M. Thériault, G. Boivin, “Cermets de Cu-PI2 en couches minces: fabrication et détermination des constantes optiques,” Can. J. Phys. 61, 612 (1983).
[CrossRef]

J. M. Thériault, M.Sc. Thesis, U. Laval (1979).

J. M. Thériault, Ph.D. Thesis, U. Laval (1983).

Wood, D. M.

D. M. Wood, N. W. Ashcroft, “Effective Medium Theory of Optical Properties of Small Particle Composites,” Philos. Mag. 35, 269 (1977).
[CrossRef]

Adv. Phys. (1)

B. Abeles, P. Sheng, M. D. Coutts, Y. Arie, “Structural and Electrical Properties of Granular Metal Films,” Adv. Phys. 24, 407 (1975).
[CrossRef]

Appl. Opt. (1)

Can. J. Phys. (2)

G. Boivin, J. M. Thériault, “Caractéristiques optiques de la photodécomposition de cermets et d’antireflets formés de Cu-PbI2,” Can. J. Phys. 62, 811 (1984).
[CrossRef]

J. M. Thériault, G. Boivin, “Cermets de Cu-PI2 en couches minces: fabrication et détermination des constantes optiques,” Can. J. Phys. 61, 612 (1983).
[CrossRef]

J. Appl. Phys. (2)

R. H. Doremus, “Optical Properties of Thin Metallic Films in Island Form,” J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

J. P. Marton, “Optical Properties of Islands Films of Metal at Long Wavelengths,” J. Appl. Phys. 40, 5383 (1969).
[CrossRef]

J. Chim. Phys. Phys. Chim. Biol. (1)

C. Grosse, J. L. Greffe, “Permittivité statique des émulsions,” J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).

J. Opt. Soc. Am. (1)

J. Phys. (1)

S. Berthier, J. Lafait, “Black Chromium Coatings: Experimental and Calculated Optical Properties using Inhomogeneous Medium Theories,” J. Phys. 40, 1093 (1979).
[CrossRef]

Kolloid-Z. (1)

T. Hanoi, “Theory of the Dielectric Dispersion due to the Interfacial Polarisation and its Application to Emulsions,” Kolloid-Z. 171, 23 (1960).
[CrossRef]

Philos. Mag. (2)

D. M. Wood, N. W. Ashcroft, “Effective Medium Theory of Optical Properties of Small Particle Composites,” Philos. Mag. 35, 269 (1977).
[CrossRef]

Lord Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1892).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Phys. Rev. B (2)

R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Thin Solid Films (1)

P. H. Lissberger, R. G. Nelson, “Optical Properties of Thin Film Au-MgF2 Cermets,” Thin Solid Films 21, 159 (1974).
[CrossRef]

Other (5)

Electrical Transport and Optical Properties of Inhomogeneous Media, AIP Conf. Proc.40 (1978).

J. M. Thériault, Ph.D. Thesis, U. Laval (1983).

A. V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967).

C. Kittel, Introduction à l’étude de I’état solide (Dunod, Paris, 1972).

J. M. Thériault, M.Sc. Thesis, U. Laval (1979).

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Figures (4)

Fig. 1
Fig. 1

Radius R and shape factor f of ellipsoidal particle deduced from the modified Maxwell-Garnett theory with two published values of bulk dielectric constant of copper: △, Schultz and Tangherlini16; and ×, Johnson and Christy.17

Fig. 2
Fig. 2

Experimental values of the real part of the dielectric constant of a Cu–PbI2 cermet of volume fraction q = 0.04 compared with values calculated by the modified Maxwell-Garnett theory for various values of R and f.

Fig. 3
Fig. 3

Experimental values of the imaginary part of the dielectric constant of a Cu–PbI2 cermet of volume fraction q = 0.04 compared with values calculated by the modified Maxwell-Garnett theory for various values of R and f.

Fig. 4
Fig. 4

Measured dielectric constant Kc = K1ciK2c of a Cu–PbI2 cermet of volume fraction q = 0.08 as a function of wavelength compared with values calculated by the Maxwell-Garnett theory for R = 2 nm and f = 0.41.

Tables (3)

Tables Icon

Table I Experimental Dielectric Constants K = (nik)2 at λ = 632.8 nm

Tables Icon

Table II Physical Constants of Bulk Copper

Tables Icon

Table III Values of fr and fi Calculated for R Ranging from 0.8 to 96.6 nm Using the Maxwell-Garnett Theory Extended to Ellipsoidal Particles for Three-Volume Fractions (λ = 632.8 nm)

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

K c - K d f K c + ( 1 - f ) K d = q K p - K d f K p + ( 1 - f ) K d ,
K p = K m - g ( L - R ) g L + i ω R K F ,
ω p 2 = n e 2 / ɛ 0 m ,
g = ɛ 0 ω 2 F / σ 0 ,
L = ( 2 E F / m g 2 ) 1 / 2 ,
f = K d [ q K p ( R ) - K c + ( 1 - q ) K d ] ( 1 - q ) ( K c - K d ) [ K p ( R ) - K d ] .
( 1 - f ) K c 2 + K c [ ( f - q ) K p + ( f + q - 1 ) K p ] - f K d K d = 0 ,
f = K c [ K c - q K p + ( q - 1 ) K d K p K c + K d K c - K p K d - K c 2 ] .
( K p - K c K p - K d ) [ K d K c ] f = ( 1 - q ) ,
f = tan - 1 [ ( K 1 c - K 1 p ) ( K 2 d - K 2 p ) - ( K 1 d - K 1 p ) ( K 2 c - K 2 p ) ( K 1 c - K 1 p ) ( K 1 d - K 1 p ) + ( K 2 c - K 2 p ) ( K 2 d - K 2 p ) ] tan - 1 [ K 1 c K 2 d - K 1 d K 2 c K 1 c K 1 d + K 2 c K 2 d ] ,
( 1 - q ) 2 = ( K 1 c - K 1 p ) 2 + ( K 2 c - K 2 p ) 2 ( K 1 d - K 1 p ) 2 + ( K 2 d - K 2 p ) 2 ( K 1 d 2 + K 2 d 2 K 1 c 2 + K 2 c 2 ) f .

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