Abstract

Existing models for optical properties of metallic particle films differ in their treatment of the mechanisms for particle interaction. For model comparison and improvement we fit the calculated optical spectra to obtained ones experimentally by variation of several structural parameters. The deviation of the model’s structure parameters from those obtained experimentally (by analysis of transmission electron microscope pictures) permits comparison of models. We find that for thermally treated silver island films in the mass thickness range 1–6 nm the particle interaction is described best by a quasi-static dipole–dipole interaction, neglecting retardation effects and mirror dipole effects.

© 1995 Optical Society of America

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References

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  1. J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 203, 385 (1904).
    [Crossref]
  2. J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 205, 237 (1906).
    [Crossref]
  3. E. David, Z. Phys. 114, 389 (1939).
    [Crossref]
  4. F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).
  5. K. Baba, R. Yamada, S. Nakao, and M. Miyagi, Appl. Opt. 32, 3137 (1993).
    [Crossref] [PubMed]
  6. H. Schopper, Z. Phys. 130, 565 (1951).
    [Crossref]
  7. S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 61, 62 (1971).
    [Crossref]
  8. T. Yamaguchi, H. Takahashi, and A. Sudoh, J. Opt. Soc. Am. 68, 1039 (1978).
    [Crossref]
  9. T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
    [Crossref]
  10. H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Wiley, New York, 1957).
  11. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [Crossref]
  13. U. Kreibig and C. Fragstein, Z. Phys. 224, 307 (1969).
    [Crossref]
  14. C. G. Granquvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
    [Crossref]
  15. C. G. Granquvist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
    [Crossref]
  16. T. Andersson and C. G. Granquvist, J. Appl. Phys. 48, 1673 (1977).
    [Crossref]
  17. S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
    [Crossref]
  18. J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, 1957).
  19. M. Meier, A. Wokaun, and P. F. Liao, J. Opt. Soc. Am. B 2, 931 (1985).
    [Crossref]
  20. H. Mayer, Physik dünner Schichten (Springer-Verlag, Berlin, 1950).
  21. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  22. D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1983).
  23. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  24. B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
    [Crossref]
  25. P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
    [Crossref]
  26. R. Ruppin, Surf. Sci. 127, 108 (1983).
    [Crossref]

1993 (1)

1987 (1)

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

1985 (1)

1983 (3)

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1983).

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[Crossref]

R. Ruppin, Surf. Sci. 127, 108 (1983).
[Crossref]

1978 (2)

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

T. Yamaguchi, H. Takahashi, and A. Sudoh, J. Opt. Soc. Am. 68, 1039 (1978).
[Crossref]

1977 (2)

C. G. Granquvist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
[Crossref]

T. Andersson and C. G. Granquvist, J. Appl. Phys. 48, 1673 (1977).
[Crossref]

1976 (1)

C. G. Granquvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[Crossref]

1974 (1)

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

1971 (1)

1969 (1)

U. Kreibig and C. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

1951 (1)

H. Schopper, Z. Phys. 130, 565 (1951).
[Crossref]

1939 (1)

E. David, Z. Phys. 114, 389 (1939).
[Crossref]

1906 (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 205, 237 (1906).
[Crossref]

1904 (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 203, 385 (1904).
[Crossref]

Aitchison, J.

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, 1957).

Anderson, V. E.

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

Andersson, T.

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

T. Andersson and C. G. Granquvist, J. Appl. Phys. 48, 1673 (1977).
[Crossref]

Aussenegg, F. R.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Baba, K.

Barber, P. W.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Brown, J. A. C.

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, 1957).

Brunner, H.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Buhrman, R. A.

C. G. Granquvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[Crossref]

Chang, R. K.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

David, E.

E. David, Z. Phys. 114, 389 (1939).
[Crossref]

Ferrell, T. L.

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Fragstein, C.

U. Kreibig and C. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

Granquvist, C. G.

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

T. Andersson and C. G. Granquvist, J. Appl. Phys. 48, 1673 (1977).
[Crossref]

C. G. Granquvist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
[Crossref]

C. G. Granquvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hunderi, O.

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

C. G. Granquvist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Kinbara, A.

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 61, 62 (1971).
[Crossref]

Kreibig, U.

U. Kreibig and C. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

Leitner, A.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Liao, P. F.

Lobmaier, C.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Mantovani, J. G.

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

Marquardt, D. W.

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1983).

Massoudi, H.

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[Crossref]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 205, 237 (1906).
[Crossref]

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 203, 385 (1904).
[Crossref]

Mayer, H.

H. Mayer, Physik dünner Schichten (Springer-Verlag, Berlin, 1950).

Meier, M.

Miyagi, M.

Nakao, S.

Norrman, S.

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

Pittner, F.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Ruppin, R.

R. Ruppin, Surf. Sci. 127, 108 (1983).
[Crossref]

Russell, B. K.

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

Schalkhammer, T.

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

Schopper, H.

H. Schopper, Z. Phys. 130, 565 (1951).
[Crossref]

Sudoh, A.

Takahashi, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Wiley, New York, 1957).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Warmack, R. W.

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

Wokaun, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yamada, R.

Yamaguchi, T.

Yoshida, S.

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 61, 62 (1971).
[Crossref]

Appl. Opt. (1)

J. Appl. Phys. (2)

C. G. Granquvist and R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[Crossref]

T. Andersson and C. G. Granquvist, J. Appl. Phys. 48, 1673 (1977).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

J. Soc. Indust. Appl. Math. (1)

D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1983).

Phil. Trans. R. Soc. London Ser. A (2)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 203, 385 (1904).
[Crossref]

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. London Ser. A 205, 237 (1906).
[Crossref]

Phys. Rev. B (5)

S. Norrman, T. Andersson, C. G. Granquvist, and O. Hunderi, Phys. Rev. B 18, 674 (1978).
[Crossref]

C. G. Granquvist and O. Hunderi, Phys. Rev. B 16, 3513 (1977).
[Crossref]

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

B. K. Russell, J. G. Mantovani, V. E. Anderson, R. W. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 2151 (1987);P. Royer, J. P. Goudonnet, R. J. Warmack, and T. L. Ferrell, Phys. Rev. B 35, 3753 (1987).
[Crossref]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. B 27, 7251 (1983).
[Crossref]

Surf. Sci. (1)

R. Ruppin, Surf. Sci. 127, 108 (1983).
[Crossref]

Thin Solid Films (1)

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

Z. Phys. (3)

E. David, Z. Phys. 114, 389 (1939).
[Crossref]

U. Kreibig and C. Fragstein, Z. Phys. 224, 307 (1969).
[Crossref]

H. Schopper, Z. Phys. 130, 565 (1951).
[Crossref]

Other (7)

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, 1957).

F. R. Aussenegg, H. Brunner, A. Leitner, C. Lobmaier, T. Schalkhammer, and F. Pittner, “The metal island coated swelling polymer over mirror system (MICSPOMS): a new principle for measuring ionic strength,” Sensors Actuators (to be published).

H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Wiley, New York, 1957).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

H. Mayer, Physik dünner Schichten (Springer-Verlag, Berlin, 1950).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (7)

Fig. 1
Fig. 1

Parts of TEM’s taken from tempered (left) and untempered (right) silver island films of evaporated mass thickness 3 (top) and 6 (bottom) nm.

Fig. 2
Fig. 2

Size distribution of the island diameter x for a 3-nm mass thickness film obtained from evaluation of TEM pictures. A log-normal distribution function was fitted through the central points of the size classes.

Fig. 3
Fig. 3

Visualization of Eqs. (20) (see text). Median of the axial diameter x ¯ versus 2 q / π n parameter.

Fig. 4
Fig. 4

Experimental spectra for tempered silver island films with mass thicknesses from 1 to 6 nm.

Fig. 5
Fig. 5

Example of the quality of fit obtained by model variant M1. The arrows indicate the model curve. (The shoulder around 350 nm is due to higher-order plasma oscillation modes and cannot be considered by the model.)

Fig. 6
Fig. 6

Distribution function of the particle axial ratio γ obtained from the fitted spectra of tempered films. Mass thickness as a parameter.

Fig. 7
Fig. 7

Measured transmission spectra of untempered silver island films. Mass thickness as a parameter.

Tables (2)

Tables Icon

Table 1 Fitted and Measured Structure Parameters of Tempered Silver Films of Different Mass Thicknesses Corresponding to Model Variants M1, M2, and M3 for Oblate and Prolate Spheroids

Tables Icon

Table 2 Fitted Parameters of Untempered Silver Films of Different Mass Thicknesses Corresponding to Model Variants M1, M2, and M3 for Oblate and Prolate Spheroids

Equations (27)

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

χ i ( γ ) = m 1 + L i m m , i = x , y , z ,
L x = g ( e ) 2 e 2 { π 2 arctan [ g ( e ) ] } g 2 e 2 ,
g ( e ) = ( 1 e 2 ) / e 2 ,
e 2 = 1 ( c 2 / a 2 ) .
L x = 1 e 2 e 2 ( 1 + 1 2 e ln 1 + e 1 e ) ,
e 2 = 1 ( b 2 / a 2 ) .
( i m ) E = q 1 g ( γ ¯ , σ γ ; γ ) χ i ( λ ; γ ) d γ 1 + 0 β 1 g ( γ ¯ , σ γ ; γ ) χ i ( λ ; γ ) d γ E 0 , i = x , y , z ,
r ¯ π 2 x ¯ q .
m 0 β , = C , d w r ¯ ,
C = 2 C = 0.716 ,
0 β = γ 2 24 m η 3 s m s + m 0.719 1 s + m d w r ¯ ,
0 β = 2 γ 2 24 m η 3 s m s + m + 0.719 1 m 2 s s + m d w r ¯ ,
0 β , x , y = γ 2 24 m η 3 s m s + m + C , x , y 2 s + m d w r ¯ ,
0 β , z = 2 γ 2 24 m η 3 s m s + m + C , z 1 m 2 s s + m d w r ¯ ,
C x = j Φ j [ ( 1 i r ¯ k r j ) ( 2 3 cos 2 θ j ) + ( r ¯ k r j ) 2 cos 2 θ j ] ,
C y = j Φ j [ ( 1 i r ¯ k r j ) ( 3 cos 2 θ j 1 ) + ( r ¯ k r j ) 2 sin 2 θ j ] ,
C z = j Φ j [ ( 1 i r ¯ k r j ) + ( r ¯ k r j ) 2 ] ,
Φ j = 1 4 π r j 3 exp [ i k ( r j y j sin Θ 0 ) ] .
R = | ρ | 2 = ρ 12 + ρ 23 + 2 ρ 12 ρ 23 cos 2 δ 1 ρ 12 2 ρ 23 2 + 2 ρ 12 ρ 23 cos 2 δ ,
T = | τ | 2 n 3 n 1 = n 3 n 1 τ 12 2 τ 23 2 1 + ρ 12 2 ρ 23 2 + 2 ρ 12 ρ 23 cos 2 δ ,
ρ i j = n i n j n i + n j , τ i j = 2 n i n i + n j ;
δ = 2 π λ 0 n 2 d opt ,
χ 2 = { [ y i y ( x i , p ) ] / σ i } 2 ,
σ fit 2 = Σ N [ y i y ( x i , p ) ] 2 / N ,
f ¯ = q / n , f ¯ = π x ¯ 2 / 4 .
d 0 2 c ¯ .
b ¯ ( nm )

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