Abstract

The Maxwell Garnett and Bruggeman effective medium theories are derived for the average dielectric permeability of heterogeneous materials from a unified theoretical approach. It starts by specifying two random unit cells which represent different microstructures. Requiring that these cells should not be detectable by electromagnetic radiation when embedded in an effective medium, we show from an extended optical theorem that the forward scattering amplitude must vanish. Setting the leading term in the expansion series of this quantity equal to zero yields the effective medium theories pertaining to the two microstructures. The remaining terms provide estimates of the accuracy of the approximations. This approach is then used in numerical computations for Co–Al2O3 cermets.

© 1981 Optical Society of America

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  1. R. Landauer, AIP Conf. Proc. 40, 2 (1978).
    [CrossRef]
  2. L. K. H. van Beek, Prog. Dielectr. 7, 69 (1967).
  3. W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
    [CrossRef]
  4. C. J. F. Böttcher, P. Bordewijk, Theory of Electric Polarization (Elsevier, Amsterdam, 1978), Vol. 2, p. 476.
  5. C. Grosse, J. -L. Greffe, J. Chim. Phys. 76, 305 (1979).
  6. A. J. Sievers, in Solar Energy Conversion: Solid State Physics Aspects, B. O. Seraphin, Ed. (Springer, Heidelberg, 1979), p. 57.
    [CrossRef]
  7. C. G. Granqvist, J. Phys. Paris, to be published.
  8. J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Philos. Trans. R. Soc. London 205, 237 (1906).
    [CrossRef]
  9. D. A. G. Bruggeman, Ann. Phys. Leipzig 24, 636 (1935).
    [CrossRef]
  10. D. Stroud, Phys. Rev. B: 12, 3368 (1975).
    [CrossRef]
  11. M. Hori, J. Math. Phys. 14, 514, 1942 (1973); J. Math. Phys. 16, 1772 (1975); J. Math. Phys. 18, 487 (1977); M. Hori, F. Yonezawa, J. Math. Phys. 15, 2177 (1974); J. Math. Phys. 16; 352, 365 (1975); J. Phys. C: 10, 229 (1977).
    [CrossRef]
  12. A unified derivation of the Maxwell Garnett and Bruggeman theories has long been known for the static case [Ref. 5; J. A. Reynolds, J. M. Hough, Proc. Phys. Soc. London 70, 769 (1957)]. Historically, this has been extended to optical properties by the quasi-static approximation which assumes that the particles are smaller than the wavelength by a large but unspecified factor.
  13. G. B. Smith, J. Phys. D: 10, L39 (1977).
    [CrossRef]
  14. G. B. Smith, Appl. Phys. Lett. 35, 668 (1979).
    [CrossRef]
  15. W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
    [CrossRef]
  16. W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
    [CrossRef]
  17. C. F. Bohren, D. P. Gilra, J. Colloid Interface Sci. 72, 215 (1979).
    [CrossRef]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  19. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  20. This structure has been phrased cermet topology in Ref. 16. This terminology is awkward since cermets cannot be assumed a priori to have a certain well-defined microstructure and since the structural problem is not a topological one in the mathematical sense.
  21. Experimental work which substantiates this point is found in H. G. Craighead, Ph.D. Thesis, Cornell U., 1980, unpublished; see also C. G. Granqvist, J. Appl. Phys. 50, 2916 (1979).
    [CrossRef]
  22. R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
    [CrossRef]
  23. D. Stroud, F. P. Pan, Phys. Rev. B: 17, 1602 (1978).
    [CrossRef]
  24. See p. 30 of Ref. 18, which gives results for particles in nonabsorbing media.
  25. G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
    [CrossRef]
  26. Smith (Ref. 14) used the same principal approach but considered only magnetic dipole contributions to δMG.
  27. D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
    [CrossRef]
  28. C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
    [CrossRef]
  29. L. Genzel, U. Kreibig, Z. Phys. B: 37, 93 (1980).
    [CrossRef]
  30. See p. 145 of Ref. 18 and arguments on p. 195 of Ref. 19, which state results for spheres in nonabsorbing media.
  31. C. G. Granqvist, O. Hunderi, Phys. Rev. B: 16, 3513 (1977).
    [CrossRef]
  32. U. Kreibig, Z. Phys. B: 31, 39 (1978).
    [CrossRef]
  33. Examples can be found in C. G. Granqvist, R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
    [CrossRef]
  34. U. Kreibig, J. Phys. Paris 38, C2-97 (1977); in Growth and Properties of Metal Clusters, J. Bourdon, Ed. (Elsevier, Amsterdam, 1980), p. 371.
    [CrossRef]
  35. H. G. Craighead, R. A. Buhrman, Appl. Phys. Lett. 31, 423 (1977); J. Vac. Sci. Technol. 15, 269 (1978).
    [CrossRef]
  36. G. A. Niklasson, C. G. Granqvist, J. Appl. Phys. 50, 5500 (1979).
    [CrossRef]
  37. L. P. Gorkov, G. M. Éliashberg, Zh. Eksp. Teor. Fiz. 48, 1407 (1965) [Sov. Phys. JETP 21, 940 (1965)].
  38. A. Kawabata, R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
    [CrossRef]
  39. L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
    [CrossRef]
  40. The assumption of small filling factor is inherent in some recent derivations of the Maxwell Garnett theory; see, for example, L. Genzel, in Festkörperprobleme, H. J. Queisser, Ed. (Vieweg, Braunschweig, 1974), Vol. 14, p. 183; S. Berthier, J. Lafait, J. Phys. Paris 40, 1093 (1979); S. Hayashi, N. Nakamori, H. Kanamori, J. Phys. Soc. Jpn. 46, 176 (1979).
    [CrossRef]
  41. R. C. McPhedran, D. R. McKenzie, Proc. R. Soc. London Ser. A: 359, 45 (1978); D. R. McKenzie, R. C. McPhedran, G. H. Derrick, Proc. R. Soc. London Ser. A: 362, 211 (1978).
    [CrossRef]
  42. S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
    [CrossRef]
  43. Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
    [CrossRef]
  44. G. A. Niklasson, C. G. Granqvist, to be published.
  45. Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
    [CrossRef]
  46. M. M. Kirillova, B. M. Charikov, Opt. Spektrosk. 17, 254 (1964) [Opt. Spectrosc. USSR 17, 134 (1964)].
  47. P. B. Johnson, R. W. Christy, Phys. Rev. B: 9, 5056 (1974).
    [CrossRef]
  48. L. Harris, J. Opt. Soc. Am. 45, 27 (1955); J. Opt. Soc. Am. 52, 223 (1962).
    [CrossRef]
  49. J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).
  50. G. A. Niklasson, unpublished.

1980 (3)

W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
[CrossRef]

L. Genzel, U. Kreibig, Z. Phys. B: 37, 93 (1980).
[CrossRef]

Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
[CrossRef]

1979 (5)

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, J. Appl. Phys. 50, 5500 (1979).
[CrossRef]

C. F. Bohren, D. P. Gilra, J. Colloid Interface Sci. 72, 215 (1979).
[CrossRef]

G. B. Smith, Appl. Phys. Lett. 35, 668 (1979).
[CrossRef]

C. Grosse, J. -L. Greffe, J. Chim. Phys. 76, 305 (1979).

1978 (6)

R. Landauer, AIP Conf. Proc. 40, 2 (1978).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
[CrossRef]

D. Stroud, F. P. Pan, Phys. Rev. B: 17, 1602 (1978).
[CrossRef]

U. Kreibig, Z. Phys. B: 31, 39 (1978).
[CrossRef]

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

R. C. McPhedran, D. R. McKenzie, Proc. R. Soc. London Ser. A: 359, 45 (1978); D. R. McKenzie, R. C. McPhedran, G. H. Derrick, Proc. R. Soc. London Ser. A: 362, 211 (1978).
[CrossRef]

1977 (4)

C. G. Granqvist, O. Hunderi, Phys. Rev. B: 16, 3513 (1977).
[CrossRef]

U. Kreibig, J. Phys. Paris 38, C2-97 (1977); in Growth and Properties of Metal Clusters, J. Bourdon, Ed. (Elsevier, Amsterdam, 1980), p. 371.
[CrossRef]

H. G. Craighead, R. A. Buhrman, Appl. Phys. Lett. 31, 423 (1977); J. Vac. Sci. Technol. 15, 269 (1978).
[CrossRef]

G. B. Smith, J. Phys. D: 10, L39 (1977).
[CrossRef]

1976 (2)

Examples can be found in C. G. Granqvist, R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[CrossRef]

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

1975 (3)

L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
[CrossRef]

D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
[CrossRef]

D. Stroud, Phys. Rev. B: 12, 3368 (1975).
[CrossRef]

1974 (2)

R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, Phys. Rev. B: 9, 5056 (1974).
[CrossRef]

1973 (2)

M. Hori, J. Math. Phys. 14, 514, 1942 (1973); J. Math. Phys. 16, 1772 (1975); J. Math. Phys. 18, 487 (1977); M. Hori, F. Yonezawa, J. Math. Phys. 15, 2177 (1974); J. Math. Phys. 16; 352, 365 (1975); J. Phys. C: 10, 229 (1977).
[CrossRef]

W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
[CrossRef]

1967 (1)

L. K. H. van Beek, Prog. Dielectr. 7, 69 (1967).

1966 (1)

A. Kawabata, R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

1965 (1)

L. P. Gorkov, G. M. Éliashberg, Zh. Eksp. Teor. Fiz. 48, 1407 (1965) [Sov. Phys. JETP 21, 940 (1965)].

1964 (2)

M. M. Kirillova, B. M. Charikov, Opt. Spektrosk. 17, 254 (1964) [Opt. Spectrosc. USSR 17, 134 (1964)].

J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).

1957 (1)

A unified derivation of the Maxwell Garnett and Bruggeman theories has long been known for the static case [Ref. 5; J. A. Reynolds, J. M. Hough, Proc. Phys. Soc. London 70, 769 (1957)]. Historically, this has been extended to optical properties by the quasi-static approximation which assumes that the particles are smaller than the wavelength by a large but unspecified factor.

1955 (1)

1935 (1)

D. A. G. Bruggeman, Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

1904 (1)

J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Andersson, Å.

Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
[CrossRef]

Andersson, T.

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

Ashcroft, N. W.

W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
[CrossRef]

Blossey, D. F.

W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. P. Gilra, J. Colloid Interface Sci. 72, 215 (1979).
[CrossRef]

Bordewijk, P.

C. J. F. Böttcher, P. Bordewijk, Theory of Electric Polarization (Elsevier, Amsterdam, 1978), Vol. 2, p. 476.

Böttcher, C. J. F.

C. J. F. Böttcher, P. Bordewijk, Theory of Electric Polarization (Elsevier, Amsterdam, 1978), Vol. 2, p. 476.

Bruggeman, D. A. G.

D. A. G. Bruggeman, Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Buhrman, R. A.

H. G. Craighead, R. A. Buhrman, Appl. Phys. Lett. 31, 423 (1977); J. Vac. Sci. Technol. 15, 269 (1978).
[CrossRef]

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

Examples can be found in C. G. Granqvist, R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[CrossRef]

D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
[CrossRef]

Charikov, B. M.

M. M. Kirillova, B. M. Charikov, Opt. Spektrosk. 17, 254 (1964) [Opt. Spectrosc. USSR 17, 134 (1964)].

Christy, R. W.

P. B. Johnson, R. W. Christy, Phys. Rev. B: 9, 5056 (1974).
[CrossRef]

Cox, J. T.

J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).

Craighead, H. G.

H. G. Craighead, R. A. Buhrman, Appl. Phys. Lett. 31, 423 (1977); J. Vac. Sci. Technol. 15, 269 (1978).
[CrossRef]

Experimental work which substantiates this point is found in H. G. Craighead, Ph.D. Thesis, Cornell U., 1980, unpublished; see also C. G. Granqvist, J. Appl. Phys. 50, 2916 (1979).
[CrossRef]

Éliashberg, G. M.

L. P. Gorkov, G. M. Éliashberg, Zh. Eksp. Teor. Fiz. 48, 1407 (1965) [Sov. Phys. JETP 21, 940 (1965)].

Elliott, R. J.

R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
[CrossRef]

Garnett, J. C. M.

J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Genzel, L.

L. Genzel, U. Kreibig, Z. Phys. B: 37, 93 (1980).
[CrossRef]

L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
[CrossRef]

The assumption of small filling factor is inherent in some recent derivations of the Maxwell Garnett theory; see, for example, L. Genzel, in Festkörperprobleme, H. J. Queisser, Ed. (Vieweg, Braunschweig, 1974), Vol. 14, p. 183; S. Berthier, J. Lafait, J. Phys. Paris 40, 1093 (1979); S. Hayashi, N. Nakamori, H. Kanamori, J. Phys. Soc. Jpn. 46, 176 (1979).
[CrossRef]

Gilra, D. P.

C. F. Bohren, D. P. Gilra, J. Colloid Interface Sci. 72, 215 (1979).
[CrossRef]

Gorkov, L. P.

L. P. Gorkov, G. M. Éliashberg, Zh. Eksp. Teor. Fiz. 48, 1407 (1965) [Sov. Phys. JETP 21, 940 (1965)].

Granqvist, C. G.

Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, J. Appl. Phys. 50, 5500 (1979).
[CrossRef]

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

C. G. Granqvist, O. Hunderi, Phys. Rev. B: 16, 3513 (1977).
[CrossRef]

Examples can be found in C. G. Granqvist, R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[CrossRef]

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

C. G. Granqvist, J. Phys. Paris, to be published.

G. A. Niklasson, C. G. Granqvist, to be published.

Greffe, J. -L.

C. Grosse, J. -L. Greffe, J. Chim. Phys. 76, 305 (1979).

Grosse, C.

C. Grosse, J. -L. Greffe, J. Chim. Phys. 76, 305 (1979).

Harris, L.

Hass, H.

J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).

Hayashi, S.

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

Hirono, J.

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

Hori, M.

M. Hori, J. Math. Phys. 14, 514, 1942 (1973); J. Math. Phys. 16, 1772 (1975); J. Math. Phys. 18, 487 (1977); M. Hori, F. Yonezawa, J. Math. Phys. 15, 2177 (1974); J. Math. Phys. 16; 352, 365 (1975); J. Phys. C: 10, 229 (1977).
[CrossRef]

Hough, J. M.

A unified derivation of the Maxwell Garnett and Bruggeman theories has long been known for the static case [Ref. 5; J. A. Reynolds, J. M. Hough, Proc. Phys. Soc. London 70, 769 (1957)]. Historically, this has been extended to optical properties by the quasi-static approximation which assumes that the particles are smaller than the wavelength by a large but unspecified factor.

Hunderi, O.

Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
[CrossRef]

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

C. G. Granqvist, O. Hunderi, Phys. Rev. B: 16, 3513 (1977).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, Phys. Rev. B: 9, 5056 (1974).
[CrossRef]

Kanamori, H.

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

Kawabata, A.

A. Kawabata, R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kirillova, M. M.

M. M. Kirillova, B. M. Charikov, Opt. Spektrosk. 17, 254 (1964) [Opt. Spectrosc. USSR 17, 134 (1964)].

Kreibig, U.

L. Genzel, U. Kreibig, Z. Phys. B: 37, 93 (1980).
[CrossRef]

U. Kreibig, Z. Phys. B: 31, 39 (1978).
[CrossRef]

U. Kreibig, J. Phys. Paris 38, C2-97 (1977); in Growth and Properties of Metal Clusters, J. Bourdon, Ed. (Elsevier, Amsterdam, 1980), p. 371.
[CrossRef]

L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
[CrossRef]

Krumhansl, J. A.

R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
[CrossRef]

Kubo, R.

A. Kawabata, R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Lamb, W.

W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
[CrossRef]

Landauer, R.

R. Landauer, AIP Conf. Proc. 40, 2 (1978).
[CrossRef]

Leath, P. L.

R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
[CrossRef]

Martin, T. P.

L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
[CrossRef]

McKenzie, D. R.

R. C. McPhedran, D. R. McKenzie, Proc. R. Soc. London Ser. A: 359, 45 (1978); D. R. McKenzie, R. C. McPhedran, G. H. Derrick, Proc. R. Soc. London Ser. A: 362, 211 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, D. R. McKenzie, Proc. R. Soc. London Ser. A: 359, 45 (1978); D. R. McKenzie, R. C. McPhedran, G. H. Derrick, Proc. R. Soc. London Ser. A: 362, 211 (1978).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

Niklasson, G. A.

G. A. Niklasson, C. G. Granqvist, J. Appl. Phys. 50, 5500 (1979).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, to be published.

G. A. Niklasson, unpublished.

Norrman, S.

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

Pan, F. P.

D. Stroud, F. P. Pan, Phys. Rev. B: 17, 1602 (1978).
[CrossRef]

Ramsey, J. B.

J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).

Reynolds, J. A.

A unified derivation of the Maxwell Garnett and Bruggeman theories has long been known for the static case [Ref. 5; J. A. Reynolds, J. M. Hough, Proc. Phys. Soc. London 70, 769 (1957)]. Historically, this has been extended to optical properties by the quasi-static approximation which assumes that the particles are smaller than the wavelength by a large but unspecified factor.

Ruppin, R.

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

Sievers, A. J.

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
[CrossRef]

A. J. Sievers, in Solar Energy Conversion: Solid State Physics Aspects, B. O. Seraphin, Ed. (Springer, Heidelberg, 1979), p. 57.
[CrossRef]

Smith, G. B.

G. B. Smith, Appl. Phys. Lett. 35, 668 (1979).
[CrossRef]

G. B. Smith, J. Phys. D: 10, L39 (1977).
[CrossRef]

Stroud, D.

D. Stroud, F. P. Pan, Phys. Rev. B: 17, 1602 (1978).
[CrossRef]

D. Stroud, Phys. Rev. B: 12, 3368 (1975).
[CrossRef]

Tanner, D. B.

D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
[CrossRef]

Tinga, W. R.

W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
[CrossRef]

van Beek, L. K. H.

L. K. H. van Beek, Prog. Dielectr. 7, 69 (1967).

van de Hulst, H. C.

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

Voss, W. A. G.

W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
[CrossRef]

Wood, D. M.

W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
[CrossRef]

Wyns, J.

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

AIP Conf. Proc. (2)

R. Landauer, AIP Conf. Proc. 40, 2 (1978).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, AIP Conf. Proc. 40, 240 (1978).
[CrossRef]

Ann. Phys. Leipzig (2)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

D. A. G. Bruggeman, Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Appl. Phys. Lett. (2)

G. B. Smith, Appl. Phys. Lett. 35, 668 (1979).
[CrossRef]

H. G. Craighead, R. A. Buhrman, Appl. Phys. Lett. 31, 423 (1977); J. Vac. Sci. Technol. 15, 269 (1978).
[CrossRef]

J. Appl. Phys. (4)

G. A. Niklasson, C. G. Granqvist, J. Appl. Phys. 50, 5500 (1979).
[CrossRef]

Å. Andersson, O. Hunderi, C. G. Granqvist, J. Appl. Phys. 51, 754 (1980).
[CrossRef]

Examples can be found in C. G. Granqvist, R. A. Buhrman, J. Appl. Phys. 47, 2200 (1976).
[CrossRef]

W. R. Tinga, W. A. G. Voss, D. F. Blossey, J. Appl. Phys. 44, 3897 (1973).
[CrossRef]

J. Chim. Phys. (1)

C. Grosse, J. -L. Greffe, J. Chim. Phys. 76, 305 (1979).

J. Colloid Interface Sci. (1)

C. F. Bohren, D. P. Gilra, J. Colloid Interface Sci. 72, 215 (1979).
[CrossRef]

J. Math. Phys. (1)

M. Hori, J. Math. Phys. 14, 514, 1942 (1973); J. Math. Phys. 16, 1772 (1975); J. Math. Phys. 18, 487 (1977); M. Hori, F. Yonezawa, J. Math. Phys. 15, 2177 (1974); J. Math. Phys. 16; 352, 365 (1975); J. Phys. C: 10, 229 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. D (1)

G. B. Smith, J. Phys. D: 10, L39 (1977).
[CrossRef]

J. Phys. Paris (2)

U. Kreibig, J. Phys. Paris 38, C2-97 (1977); in Growth and Properties of Metal Clusters, J. Bourdon, Ed. (Elsevier, Amsterdam, 1980), p. 371.
[CrossRef]

J. T. Cox, H. Hass, J. B. Ramsey, J. Phys. Paris 25, 250 (1964).

J. Phys. Soc. Jpn. (2)

S. Hayashi, J. Hirono, H. Kanamori, R. Ruppin, J. Phys. Soc. Jpn. 46, 1602 (1979).
[CrossRef]

A. Kawabata, R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Opt. Spektrosk. (1)

M. M. Kirillova, B. M. Charikov, Opt. Spektrosk. 17, 254 (1964) [Opt. Spectrosc. USSR 17, 134 (1964)].

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

J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Philos. Trans. R. Soc. London 205, 237 (1906).
[CrossRef]

Phys. Rev. B (7)

D. Stroud, Phys. Rev. B: 12, 3368 (1975).
[CrossRef]

C. G. Granqvist, O. Hunderi, Phys. Rev. B: 16, 3513 (1977).
[CrossRef]

D. B. Tanner, A. J. Sievers, R. A. Buhrman, Phys. Rev. B: 11, 1330 (1975).
[CrossRef]

W. Lamb, D. M. Wood, N. W. Ashcroft, Phys. Rev. B: 21, 2248 (1980).
[CrossRef]

D. Stroud, F. P. Pan, Phys. Rev. B: 17, 1602 (1978).
[CrossRef]

P. B. Johnson, R. W. Christy, Phys. Rev. B: 9, 5056 (1974).
[CrossRef]

Such effective depolarization factors have been used to explain anomalous absorption in discontinuous noble metal films for many years; see S. Norrman, T. Andersson, C. G. Granqvist, O. Hunderi, Phys. Rev. B: 18, 674 (1978) and reference therein.
[CrossRef]

Phys. Rev. Lett. (1)

C. G. Granqvist, R. A. Buhrman, J. Wyns, A. J. Sievers, Phys. Rev. Lett. 37, 625 (1976); C. G. Granqvist, Z. Phys. B: 30, 29 (1978).
[CrossRef]

Proc. Phys. Soc. London (1)

A unified derivation of the Maxwell Garnett and Bruggeman theories has long been known for the static case [Ref. 5; J. A. Reynolds, J. M. Hough, Proc. Phys. Soc. London 70, 769 (1957)]. Historically, this has been extended to optical properties by the quasi-static approximation which assumes that the particles are smaller than the wavelength by a large but unspecified factor.

Proc. R. Soc. London Ser. A (1)

R. C. McPhedran, D. R. McKenzie, Proc. R. Soc. London Ser. A: 359, 45 (1978); D. R. McKenzie, R. C. McPhedran, G. H. Derrick, Proc. R. Soc. London Ser. A: 362, 211 (1978).
[CrossRef]

Prog. Dielectr. (1)

L. K. H. van Beek, Prog. Dielectr. 7, 69 (1967).

Rev. Mod. Phys. (1)

R. J. Elliott, J. A. Krumhansl, P. L. Leath, Rev. Mod. Phys. 46, 465 (1974).
[CrossRef]

Z. Phys. B (3)

L. Genzel, U. Kreibig, Z. Phys. B: 37, 93 (1980).
[CrossRef]

U. Kreibig, Z. Phys. B: 31, 39 (1978).
[CrossRef]

L. Genzel, T. P. Martin, U. Kreibig, Z. Phys. B: 21, 339 (1975).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

L. P. Gorkov, G. M. Éliashberg, Zh. Eksp. Teor. Fiz. 48, 1407 (1965) [Sov. Phys. JETP 21, 940 (1965)].

Other (13)

The assumption of small filling factor is inherent in some recent derivations of the Maxwell Garnett theory; see, for example, L. Genzel, in Festkörperprobleme, H. J. Queisser, Ed. (Vieweg, Braunschweig, 1974), Vol. 14, p. 183; S. Berthier, J. Lafait, J. Phys. Paris 40, 1093 (1979); S. Hayashi, N. Nakamori, H. Kanamori, J. Phys. Soc. Jpn. 46, 176 (1979).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, to be published.

G. A. Niklasson, unpublished.

See p. 145 of Ref. 18 and arguments on p. 195 of Ref. 19, which state results for spheres in nonabsorbing media.

See p. 30 of Ref. 18, which gives results for particles in nonabsorbing media.

Smith (Ref. 14) used the same principal approach but considered only magnetic dipole contributions to δMG.

C. J. F. Böttcher, P. Bordewijk, Theory of Electric Polarization (Elsevier, Amsterdam, 1978), Vol. 2, p. 476.

A. J. Sievers, in Solar Energy Conversion: Solid State Physics Aspects, B. O. Seraphin, Ed. (Springer, Heidelberg, 1979), p. 57.
[CrossRef]

C. G. Granqvist, J. Phys. Paris, to be published.

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

This structure has been phrased cermet topology in Ref. 16. This terminology is awkward since cermets cannot be assumed a priori to have a certain well-defined microstructure and since the structural problem is not a topological one in the mathematical sense.

Experimental work which substantiates this point is found in H. G. Craighead, Ph.D. Thesis, Cornell U., 1980, unpublished; see also C. G. Granqvist, J. Appl. Phys. 50, 2916 (1979).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) and (b) depict two microstructures for heterogeneous two-phase media; (c) and (d) show the corresponding random unit cells used to derive the effective dielectric permeability within the Maxwell Garnett and Bruggeman theories.

Fig. 2
Fig. 2

Limiting radius versus filling factor for the Maxwell Garnett and Bruggeman effective medium theories as computed for a fixed wavelength. The theories are correct to a precision governed by δMG and δBr.

Fig. 3
Fig. 3

Limiting radius versus filling factor for the Maxwell Garnett and Bruggeman effective medium theories as computed for four different wavelengths. The theories are correct to a precision governed by a fixed magnitude of δMG and δBr.

Equations (13)

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C ext = 4 π Re [ S ( 0 ) / k 2 ] .
k = 2 π ɛ ¯ 1 / 2 / λ ,
S ( 0 ) = 0 ,
S c s ( 0 ) = 1 2 n = 1 ( 2 n + 1 ) ( α n + β n ) ,
S c s ( 0 ) = i ( k b ) 3 ( ɛ B - ɛ ¯ ) ( ɛ A + 2 ɛ B ) + f ( 2 ɛ B + ɛ ¯ ) ( ɛ A - ɛ B ) ( ɛ B + 2 ɛ ¯ ) ( ɛ A + 2 ɛ B ) + f ( 2 ɛ B - 2 ɛ ¯ ) ( ɛ A - ɛ B ) + O [ ( k b ) 5 ] .
f = ( a / b ) 3 ,
ɛ ¯ MG - ɛ B ɛ ¯ MG + 2 ɛ B = f ɛ A - ɛ B ɛ A + 2 ɛ B ,
ɛ ¯ MG = ɛ B ɛ A + 2 ɛ B + 2 f ( ɛ A - ɛ B ) ɛ A + 2 ɛ B - f ( ɛ A - ɛ B ) .
S s ( 0 ) = i ( k b ) 3 ɛ - ɛ ¯ ɛ + 2 ɛ ¯ + O [ ( k b ) 5 ] ,
f ɛ A - ɛ ¯ Br ɛ A + 2 ɛ ¯ Br + ( 1 - f ) ɛ B - ɛ ¯ Br ɛ B + 2 ɛ ¯ Br = 0 ,
( ɛ B - ɛ ¯ ) ( ɛ B + 2 ɛ B ) + f ( 2 ɛ B + ɛ ¯ ) ( ɛ A - ɛ B ) ( ɛ B - 2 ɛ ¯ ) ( ɛ A + 2 ɛ B ) + f ( 2 ɛ B - 2 ɛ ¯ ) ( ɛ A - ɛ B ) + δ MG ( ɛ A , ɛ B , ɛ ¯ , f , b , λ ) = 0.
ɛ ¯ = ɛ B [ ɛ A + 2 ɛ B + 2 f ( ɛ A - ɛ B ) ] ( 1 + δ MG ) [ ɛ A + 2 ɛ B - f ( ɛ A - ɛ B ) ] ( 1 - 2 δ MG ) ,
f ɛ A - ɛ ¯ ɛ A + 2 ɛ ¯ + ( 1 - f ) ɛ B - ɛ ¯ ɛ B + 2 ɛ ¯ + δ Br ( ɛ A , ɛ B , ɛ ¯ , f , b , λ ) = 0.

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