Maxwell-Garnett theory extended for Cu-Pbl2 cermets

Jean-Marc Thériault and Germain Boivin

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 wave-length, 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.

I. Introduction The optical properties of small metal particles im­

bedded in a dielectric matrix have been of great interest since the work by Lord Rayleigh and J. C. Maxwell-Garnett.1,2 In recent years, this interest has rapidly grown owing to the applications of these materials to many fields of physics. The theoretical and practical interest is quite well evidenced in the AIP Conference Proceedings on Inhomogeneous Media.3 In its original form, the Maxwell-Garnett theory can approximately predict the value of the dielectric constant of a metal-dielectric mixture from the dielectric constants of the components and the volume fraction: in this theory, the metal particles are supposed to be spherical and small compared with the wavelength. To take into account the shape of the particles, Cohen et al.4 ex­tended the theory to include the ellipsoidal shapes. The size of the particle plays an important role in the optical constants of metal particles,5'6 and the reduction of the mean free path of the conduction electrons should be taken into account.4-7

When dealing with discontinuous media one of the major difficulties encountered is to define the local field acting on the particles. We refer the reader to an in­teresting comparative study by Grosse and Greffe8

where they analyze the numerous theories derived from various approaches for the local field.

In this paper, we propose a new method of determi­nation of optical constants of cermet materials using the

The authors are with Laval University, Physics Department, LROL, Quebec, Quebec G1K 7P4.

Received 17 September 1984. 0003-6935/84/244494-05$02.00/0. © 1984 Optical Society of America.

Maxwell-Garnett theory extended to ellipsoidal shape particles including a correction for the mean free path of the electrons. Starting from experimental values of bulk optical constants of a cermet,9,10 we were able to determine the shape factor and the size of the particles by a separate fit of the real and imaginary parts of the dielectric constant for a particular wavelength. Since these two parameters are of geometrical nature we used them to check the validity of this theory throughout the visible spectrum. Alternately, this method has been used to test the limit of validity of three effective me­dium theories.

II. Theory Knowledge of the real and imaginary parts of the

dielectric constant of a cermet suffices to determine the geometrical parameters of a particle, that is, the shape factor and the mean radius. For ellipsoidal particles, the modified Maxwell-Garnett formula can be written4

where Kc = K1c — iK2c - dielectric constant of the cermet,

Kd = K1d - iK2d = dielectric constant of the

dielectric, Kp = K1p — iK2p = dielectric constant of the

particle, q = volume fraction of metal, and ƒ = shape factor of the metal particles.

On the other hand, the dielectric constant of the metal particle can be derived from that of the bulk metal by the relation

where Km = K1m — iK2m is the dielectric constant of bulk metal.

4494 APPLIED OPTICS / Vol. 23, No. 2 4 / 1 5 December 1984

Table I. Experimental Dielectric Constants K= (n — ik)2 at λ = 632.8 nm

KF - [ω2pF]/[ω(ig — ω)] is the contribution of free

electrons in bulk metal:

where g = damping parameter, L = mean free path in the bulk metal, R = the radius of the particles, F = fraction of free electrons that is effective,

ωp = plasma frequency, and EF = Fermi energy.

These results are similar to those quoted in Ref. 7. So, once the dielectric constant Kc has been deter­

mined experimentally, it is possible to obtain a solution for f and R. From Eq. (1) the filling factor can be ex­pressed as

For a given volume fraction q, values for f and R are obtained by solving a pair of equations given by the real and imaginary parts of Eq. (6) [ƒ = fr - ifi] with the condition that ft = 0, since the shape factor is of geo­metrical nature and should be real.

III. Experimental In a recent paper11 we proposed new recording media

for optical data storage consisting of a double-layer (metal-cermet) antireflecting coating. To produce such a coating, consistent values of the optical constants of the cermet are necessary. In previous paper,12,13 we have shown that determination of optical constants by photometric method leads to values which vary with the thickness of the layer. In a subsequent paper,9 we conclusively elucidate this problem by using an inter-ferometric method of measurement of optical constants. This method was applied to Cu-PbI2 cermets, but it can be extended to any cermet material. In this paper we will use the experimental values as they correspond to those of bulk material as required by the hypothesis of the Maxwell-Garnett theory.

Table I shows the experimental dielectric constant K = (n — ik)2 for Cu-PbI2 cermets of various concen­trations and that of lead iodide. To solve Eq. (6), we used a graphical method. The dielectric constant of copper is calculated using Eq. (6) and the physical constants of copper quoted in Table II. Then the val­ues of Kp, Kd , and Kc are inserted in Eq. (2) to obtain the value f = fr — ifi . Table III shows these values for R ranging from 0.8 to 96.6 nm. The value fi = 0 is de­termined by interpolation of the values of R and fr for a given concentration. Figure 1 shows the variation of ƒ and R with the concentration. In this figure, we give results for two different dielectric constants of copper quoted in literature, one published by Schultz and Tangherlini16 and the other published by Johnson and Christy.17 It can be seen that, according to how we choose one value or the other, the results are different for R but little different for ƒ. It means that the di-

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Table II. Physical Constants of Bulk Copper

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)

Fig. 1. Radius R and shape factor ƒ 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

electric constant of a particle is the same if, for example, we choose a radius R = 1.8 nm with the dielectric con­stant of Johnson and Christy or a radius R = 1.4 nm with that of Schultz and Tangherlini for a volume fraction q = 0.06.

This discrepancy in the values of the dielectric con­stant is most probably due to the different conditions of preparation of the films. On the other hand, one can see the effect on R of an increase in the volume fraction q. This result is similar to that reported by Abeles et αl.18 for Ni-SiO2, Pt-SiO2, and Au-Al2O3 cermets. Their results obtained with an electron microscope show a linear increase in the radius R with the concentration q with a mean radius of 0.5 nm for low concentration. For Cu-PbI2 cermets we also obtain a linear variation for R while the calculated mean radius is ~1.5 nm. For the shape factor, it can be seen from Table III that the mean value f = 0.41 indicates that the particles are el­lipsoids, which means that the minor axis/major axis ratio is equal to ~½. This result is to be compared with those obtained by Cohen et al.4 They found that for the Ag-SiO2 cermet the metal particles are of ellipsoidal shape (b/a = 1/1.7); that is, their shape factor lays be­tween the sphere and the cylinder with ƒ = 0.4. For the cermet we also note that the shape factor decreases as the volume fraction increases. This means that for a higher volume fraction the particles tend to be more and more spherical; the conclusion seems quite natural, since for a higher evaporation rate the impinging atoms have greater mobility.

This method has also been applied to the Bruggeman effective medium theory19-20 and to the Hanai-Bruggeman theory.21 The effective medium theory extended to ellipsoids states that20

for which the shape factor is

Using the values of KcKpKd and q we found that no value of R can make ƒi = 0.

For the Hanai-Bruggeman theory, we have8

which leads to the following expressions for the filling factor:

These two equations are derived from the real and imaginary parts of Eq. (9). In this case, the method is slightly modified. For various values of R, f is calcu­lated by means of Eq. (10), and inserted in Eq. (11), which has to be satisfied for the particular known value (1 — q)2. We again found that no value of R could sat­isfy the above equations. We conclude that for this particular type of mixture these two mentioned theories are inappropriate to explain the experimentally deter­mined optical constants.

IV. Spectral Properties of Dielectric Constant of Cermets

As mentioned previously, the shape factor ƒ is a purely geometrical quantity, and once determined at a particular wavelength the same value should be used to calculate the optical constants of a particular cermet throughout the visible spectrum. Figures 2 and 3 show the experimental values of the real and imaginary parts of the dielectric constant for a volume fraction q = 0.04. The experimental values (curve E) have been measured on a 26.0-nm thick cermet film. We have shown in previous work12,13 that bulk optical constants of a cer­met material can be obtained from measurements car­ried out on a relatively thin film. In this case the sur­face effect has little influence.9

Figures 2 and 3 show the influence of parameters R and f on the dielectric constant at various wavelengths. Curves 1 and 2 correspond to the mean free path in bulk material (R = 96.6 nm) for spherical and ellipsoidal shapes, respectively, while curves 3 and 4 correspond to a restricted mean free path (R = 1.5 nm) for spherical and ellipsoidal shapes. Agreement is particularly good for curve 4, which has been calculated with the correct size and shape particle. Figure 4 shows an excellent agreement between calculated and measured values of the real and imaginary part of the dielectric constant for a volume fraction q = 0.08.

V. Conclusion In this paper, we studied the influence of the shape

factor and the size of particles on the dielectric constant

4496 APPLIED OPTICS / Vol. 23, No. 24 / 15 December 1984

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 ƒ. Fig. 4. Measured dielectric constant Kc = K1c — iK2c 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.

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.

of a metal-dielectric cermet material. We found that up to a concentration of 12% in volume, which repre­sents an interesting range of variation of optical con­stants for practical applications, the Maxwell-Garnett theory, extended to include the shape of the particle and the restriction of the mean free path of the conduction electrons, represents an adequate formulation of the physical situation. The fact that the shape factor represents a geometrical parameter allows one to de­termine both the shape factor f and the radius R of the

particles by a separate fit of the real and imaginary parts of the dielectric constant. These parameters serve to calculate the dielectric constant of the cermet throughout the visible spectrum, and the results show an excellent agreement with the experimental values. Alternately, the method can serve as a test for various proposed theories of composite materials.

The authors wish to thank the Quebec Department of Education and NSERC of Canada for financial as­sistance.

References 1. Lord Rayleigh, "On the Influence of Obstacles Arranged in

Rectangular Order upon the Properties of a Medium," Philos. Mag. 34, 481 (1892).

2. J. C. Maxwell-Garnett, "Colours in Metal Glasses and in Metallic Films," Philos. Trans. R. Soc. London 205, 237 (1906).

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

4. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, "Optical Properties of Granular Silver and Gold Films," Phys. Rev. B 8, 3689 (1973).

5. R. H. Doremus, "Optical Properties of Thin Metallic Films in Island Form," J. Appl. Phys. 37, 2775 (1966).

6. J. P. Marton, "Optical Properties of Islands Films of Metal at Long Wavelengths," J. Appl. Phys. 40, 5383 (1969).

7. P. H. Lissberger and R. G. Nelson, "Optical Properties of Thin Film Au-MgF2 Cermets," Thin Solid Films 21, 159 (1974).

8. C. Grosse and J. L. Greffe, "Permittivité statique des emulsions," J. Chim. Phys. Phys. Chim. Biol. 76, 305 (1979).

9. G. Boivin and J. M. Thériault, "Influence of Surfaces Effects in the Determination of the Optical Constants of Cu-PbI2 Cermets Films," Appl. Opt. 23, 4245 (1984).

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

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11. G. Boivin et J. M. Thériault, "Caractéristiques optiques de la photodécomposition de cermets et d'antireflets formes de Cu-PbI2," Can. J. Phys. 62, 811 (1984).

12. J. M. Thériault, M.Sc. Thesis, U. Laval (1979). 13. J. M. Thériault et G. Boivin, "Cermets de Cu-PI2 en couches

minces: fabrication et determination des constantes optiques," Can. J. Phys. 61, 612 (1983).

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

15. C. Kittel, Introduction à I'êtude de I'état solide (Dunod, Paris, 1972).

16. L. G. Schultz and F. R. Tangherlini, "Optical Constants of Silver, Gold, Copper, and Aluminum. I. The Absorption Coefficient k," J. Opt. Soc. Am. 44, 357 (1954).

17. P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370 (1972).

18. B. Abeles, P. Sheng, M. D. Coutts, and Y. Arie, "Structural and Electrical Properties of Granular Metal Films," Adv. Phys. 24, 407 (1975).

19. D. M. Wood and N. W. Ashcroft, "Effective Medium Theory of Optical Properties of Small Particle Composites," Philos. Mag. 35, 269 (1977).

20. S. Berthier and J. Lafait, "Black Chromium Coatings: Experi­mental and Calculated Optical Properties using Inhomogeneous Medium Theories," J. Phys. 40, 1093 (1979).

21. T. Hanoi, "Theory of the Dielectric Dispersion due to the Inter-facial Polarisation and its Application to Emulsions," Kolloid-Z. 171, 23 (1960).

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