427
14
Effective Medium Theories
Consider a composite of at least two nonmixable components: well-separated
nanoparticles statistically distributed in a nonabsorbing homogeneous matrix (see
Figure 14.1). Its optical response is determined both by the particles and by the
matrix material and is difficult to predict in general. However, if it is possible to
replace the inhomogeneous composite by a homogeneous material of one common
dielectric function εeff, the reflectance, transmittance, and absorbance of this
medium can be calculated as a linear response. For that purpose, a model for the
dielectric function εeff of this effective medium must be established as a function of
the particle properties and the surrounding matrix and the concentration of particles in the composite.
Figure 14.1 Scheme of the effective medium. The realistic composite is replaced by a
homogeneous effective medium.
The fundamental question is how to obtain this effective dielectric function εeff
in terms of the dielectric function ε of the nanoparticles and εM of the matrix
and of suitably chosen topology parameters. A considerable number of equations
for the effective dielectric function are available today based upon different
approximative models. The most important will be reviewed and discussed in this
chapter.
The most limiting assumption of all effective medium theories is the assumption
that scattering by the particles can be neglected, for which the nanoparticles must
be very small compared with the wavelength of light. Hence their application to
Optical Properties of Nanoparticle Systems: Mie and Beyond. Michael Quinten
Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-41043-9
428
14 Effective Medium Theories
Table 14.1 Values of the dielectric constant and the scattering cross-section of very small
particles in the Rayleigh limit and at the same wavelength λ = 514 nm.
Material
Dielectric constant ε
σ sca ∝
Ag
GaAs
Si
Si3N4
SiO2
−10.3 + i0.205
17.66 + i3.207
17.89 + i0.525
4.15 + i0
2.13 + i0
1.853
0.725
0.72
0.263
0.075
ε - εM
ε + 2ε M
2
composites with transparent, purely scattering particles is questionable. However,
also for strongly absorbing particles the question may arise of whether they can
be applied because those particles usually scatter the light more strongly than
transparent particles. This can be recognized from Table 14.1, where the scattering
cross-sections of small spherical particles are calculated in the Rayleigh approximation for a fixed size 2R and at wavelength λ = 514 nm. Compared with a silica
sphere, the silver sphere of same size scatters the light about 25 times more
strongly. This is the result for a silver sphere off resonance. In resonance the scattering cross-section of the silver sphere is additionally increased by a factor of 3–4
due to the SPP resonance. As a consequence of this stronger scattering, the failure
of effective medium models must become obvious earlier for metal nanoparticles
than for dielectric nanoparticles.
A further restricting assumption is that the particles must be well separated.
Electromagnetic particle–particle interactions are usually omitted.
The existing effective medium models essentially differ in the way in which an
average is calculated from the dielectric functions ε and εM of the embedded nanoparticles and the matrix, respectively.
The first effective medium concept goes back to Newton (see [821]), later modified by Beer [822], Gladstone and Dale [823], Landau and Lifschitz [824], and
Looyenga [825]. It is based on simply averaging certain powers of the dielectric
functions of the two mixed components, weighted with the filling factor f :
ε eff = fε + (1 − f )ε M
1
1
Newton
1
2
ε eff
= fε 2 + (1 − f )ε M2 Beer, Gladstone and Dale
1
1
(14.1)
(14.2)
1
3
ε eff
= fε 3 + (1 − f )ε M3 Landau and Lifschitz, Looyenga
(14.3)
By applying upper and lower bounds, Lichtenecker [821] obtained the logarithmic
mixture rule, which was later used in careful investigations of effective medium
theories by Niklasson et al. [826]:
log (ε eff ) = f log (ε ) + (1 − f ) log (ε M )
(14.4)
14 Effective Medium Theories
The simplest approach to an effective medium that explicitly considers also the
shape of the particles stems from J. C. Maxwell Garnett1) [827] in 1904 for spherical
inclusions:
ε eff = ε M
ε eff − ε M
ε − εM
(ε + 2ε M ) + 2 f (ε − ε M )
= f
or
ε eff + 2ε M
ε + 2ε M
(ε + 2ε M ) − f (ε − ε M )
(14.5)
Using the abbreviation
Λ=
ε − εM
ε + 2ε M
(14.6)
the Maxwell-Garnett equation can be rewritten as
ε eff = ε M
1 + 2 fΛ
1 − fΛ
(14.7)
The advantage of this equation becomes obvious in the following. For spherical
particles that are small compared with the wavelength, the dipole polarizability α
is (see also Section 4.7)
α = 3Vp
ε − εM
= 3Vp Λ
ε + 2ε M
(14.8)
This relation between Λ and α now allows the Maxwell-Garnett equation to be
extended to nonspherical particles. Expressions for α for ellipsoids, spheroids, and
cubes were given in Chapter 9.
In the Bruggeman ansatz [828], the dielectric constant is given as
ε eff = E ± E 2 +
εε M
2
(14.9)
with
E=
ε M (2 − 3 f ) − ε (1 − 3 f )
4
(14.10)
Usually, the positive square root is used. However, if the imaginary part of εeff
becomes negative, the negative square root has to be used to obtain the correct
dielectric constant.
Bergman [829–831] developed a generalized effective medium model. The starting
point of the Bergman theory is the expression for the orientation averaged polarizability of one nanoparticle of variable shape:
C i (ε − ε M )
ε
i =1 M + Gi (ε − ε M )
N
α = Vp ∑
(14.11)
1) His approach is well known as the Maxwell-Garnett model, although Maxwell is in fact the
third Christian name of James Clerk Maxwell Garnett.
429
430
14 Effective Medium Theories
The next step is to introduce the variables t = εM/(εM − ε) and gi = Ci/εM. On going
formally to a macroscopic sample with large numbers of arbitrarily irregular inclusions, the number of modes N will increase sufficiently to extend N formally to
infinity. Then, gi becomes a continuously variable quantity denoted β between the
boundaries of zero and unity. The summation over i can be replaced by integration
over β and we formally obtain
1

g (β ) 
ε eff = ε M 1 − f ∫
dβ 
t−β 

0
(14.12)
where the function g(β ) is called the spectral density, which reflects the topology of
the sample.
This is the Bergman equation (or Bergman theorem) [829–831]. Here, εeff is defined
from the assumption that the volume average of the energy density W of the
electric field in a capacitor filled with inhomogeneous material is invariant if the
inhomogeneous sample is replaced by a fictitious homogeneous effective medium
of εeff. In this concept, the volume fraction f just represents a scaling factor without
a direct influence on topology. The dielectric properties of particle and matrix
materials enclosed merely in the variable t occur separately from the topology in
the Bergman equation in a transparent but formal way.
The explicit consideration of structural effects and electrodynamic interaction was
introduced in effective medium approaches for the first time by Lamb et al. [832]. They
used a multiple-scattering approach that yields an effective propagation wavevector.
Thus lowest-order corrections of the filling factor and weak scattering are included.
Felderhof and Jones [833–836] introduced the electrodynamic influences of
neighboring particles in statistically inhomogeneous samples as a correction of
the Maxwell-Garnett equation:
3f


ε eff = ε M  1 +
 1 − 3t − f 
(14.13)
with respect to two-cluster pair correlations by a correction term C(t):
3f


ε eff = ε M 1 +

 1 − 3t − f − C (t ) 
(14.14)
where the variable t is defined as t = εM/(εM − ε). C(t) was computed for particle
pairs from the direct dipolar coupling. The result is
pair
ε eff
=
( 4 πNα )2VT  ∞


2 (ε − ε M ) r 2
 ∫ dr ⋅ r 
3
3
3
3 
3
2
2
ε
ε
r
R
r
R
−
+
+
(
)
(
)
M

 2R

∞
2



2 (ε − ε M ) r
+ ∫ dr ⋅ r 

3
r
R 3 ) + ε M ( 2r 3 − R 3 )  
ε
+
(

2R
(14.15)
where r is center-to-center distance of the nanoparticles, R the nanoparticle radius,
α the single particle polarizability, N the number of particles in the sample volume,
and VT the single particle volume.
14.1 Theoretical Results for Dielectric Nanoparticle Composites
All these results for εeff can be used to calculate the reflectance, transmittance,
and absorbance of the composite using Maxwell’s relation:
n + iκ = ε1 + iε 2
(14.16)
between the dielectric function and the refractive index, Fresnel’s equations for
reflectance and transmittance, and
γ a (λ ) =
4π
κ
λ
(14.17)
for the absorption constant.
Extensions of existing effective medium theories have regularly been made by
different authors. Mainly the Maxwell-Garnett and the Bruggeman equations have
been extended to obtain better agreement between experiment and theory. We will
come back to these extensions later in Section 14.3. Here, we will treat only the first
and most prominent extension by Gans and Happel outlined in the information box.
14.1
Theoretical Results for Dielectric Nanoparticle Composites
The similarity of effective medium models with the molar refraction used to calculate the refractive index of mixed glasses makes them attractive for calculation
of the refractive index of dielectric nanoparticle–dielectric matrix composites. In
Table 14.2 we give results obtained for Al2O3, TiO2, ZrO2, and SiO2 nanoparticles
embedded in the polymer poly(methyl methacrylate) (PMMA) using the MaxwellGarnett equation. The idea is to increase or to lower the refractive index of PMMA
by oxide nanoparticle inclusions.
Actually, the inclusion of the highly refracting Al2O3, TiO2, and ZrO2 nanoparticles results in an increased refractive index. Vice versa, the inclusion of SiO2
Table 14.2
Refractive indices of PMMA with several oxide nanoparticle inclusions.
Material
PMMA
Al2O3 in PMMA
TiO2 in PMMA
ZrO2 in PMMA
SiO2 in PMMA
f
0.01
0.10
0.01
0.10
0.01
0.10
0.01
0.10
n
486 nm
587 nm
656 nm
1.4977
1.5005
1.5246
1.5094
1.6152
1.5042
1.5629
1.4974
1.4943
1.4918
1.4945
1.5185
1.5026
1.6015
1.4982
1.5564
1.4915
1.4884
1.4895
1.4919
1.5159
1.4998
1.5958
1.4956
1.5531
1.4889
1.4859
431
432
14 Effective Medium Theories
INFO – Mie Theory-Based Extension of Maxwell-Garnett Theory
An extension of Maxwell-Garnett’s equation to the full expressions for the electric partial modes an and the magnetic partial modes bn according to Mie’s
theory instead of the simple approximative Λ was made by Gans and Happel
[837] in 1909 and later by Doyle [838] and Ruppin [839].
Gans and Happel obtained the following effective dielectric function (for the
electric field) and the effective magnetic permeability (for the magnetic field):
ε eff = εm
GH
1 + 2 fΛmagn
1 + 2 fΛ GH
el
; µeff = µm
GH
GH
1 − fΛ el
1 − fΛmagn
(14.18)
with
Λ GH
el =
3i
3i
2 πR
GH
εM
= 3 b1 with x =
(a1 − a2 ) ; Λmagn
λ
2x 3
2x
(14.19)
The derivation of εeff and µeff by Gans and Happel is correct only up to a certain
point. Then, the authors made the erroneous assumption that the electric
partial modes an represent only the electric field and the magnetic partial modes
bn represent only the magnetic field. This assumption is wrong! As we have
already shown in Section 5.1, the electric or transverse magnetic modes an stand
for both (!) E™ and H™. Vice versa, the magnetic or transverse electric modes
bn stand for both ETE and HTE. Hence it is incorrect to assign the an only to the
electric field and the bn only to the magnetic field.
In a correct derivation, only εeff can be deduced, but polarization dependent:
TM,TE
ε eff
= εm
1 + 2 fΛ TM,TE
1 − fΛ TM,TE
(14.20)
with
Λ TM =
3i 
100
3i
2 πR
εM
3a1 + 2 a2  ; Λ TE = 3 b1 with x =

λ
x
2x 3 
2x
(14.21)
An effective magnetic permeability cannot be deduced. Hence the use of Λ GH
magn
may lead to confusion, since according to Equation 14.18 for nonmagnetic
materials with ε, εM, and µ = µM = 1, a magnetization M of the effective medium
may result for large enough filling factors. For instance, for glass nanoparticles
(ε = 2.25) in air (εM = 1) and a filling factor f = 0.3, the effective permeability
would amount to µeff ≈ 1.001, and for silver nanoparticles in air it would even
amount to µeff ≈ 1.1, both at wavelength λ = 633 nm.
Doyle’s result is included in Λ™ since he only considered the TM dipole mode
a1. A conceptual inconsequence of this Mie-based extension is, however, that
this more general particle polarizability makes sense only if the higher multipoles
are essentially excited, that is, particle sizes are beyond the Rayleigh limit, automatically admitting the possibility of light scattering.
14.2 Theoretical Results for Metal Nanoparticle Composites
nanoparticles with a refractive index lower than that of PMMA also lowers the
refractive index of the composite. The effects of the inclusions depend on the
volume fraction f.
The Maxwell-Garnett approach seems to be helpful here to predict the refractive
index of dielectric–dielectric composites. Nevertheless, we want to point to the fact
that these nanoparticles still scatter the light. A well-suited measure of the influence of the scattering is the haze of such composites introduced in Section 2.3. If
Al2O3, TiO2, or ZrO2 nanoparticles with 2R = 10 nm or larger are suspended in a
PMMA plate of 1 mm thickness, the mean haze becomes intolerable for filling
factors f ≥ 0.01, according to the standard ASTM D 1003-97: Test Method for Haze
and Luminous Transmittance of Transparent Plastics.
Another application of effective medium approaches for dielectric nanoparticle
composites is the prediction of optical properties of transparent dielectric particles
with absorbing inclusions. This is an important problem in atmospheric science,
astronomy, and optical particle sizing. Hence theories have been developed to
calculate the scattering by particles composed of more than one distinct refractive
index. For examples, we refer to [840–842].
14.2
Theoretical Results for Metal Nanoparticle Composites
The original work of Garnett dealt with metal nanoparticle inclusions in a transparent medium to explain in a simple way the optical properties of such systems,
like colloidal suspensions or colored glasses. They were published even before
Gustav Mie developed his theory on light scattering by spherical particles. Later,
Bruggeman recognized that the Maxwell-Garnett approach is incomplete in the
sense that percolation, that is, conductive paths through the sample, at higher
volume fractions is not included. He developed a new approach for εeff (see Equation 14.9) that includes percolation.
If the filling factor f is less than 0.001, the results for εeff of the most prominent
Maxwell-Garnett and Bruggeman effective medium models and the mixing rule
of Looyenga do not differ. This changes drastically for higher volume fractions.
Figure 14.2 shows a comparison of these three models for Ag inclusions ( f = 0.1)
in a transparent matrix with a constant refractive index nM = 1.5 (approximately
valid for various crown glasses).
For the Maxwell-Garnett model, the refractive index neff of this composite
exhibits features that indicate the presence of a harmonic oscillator with a resonance wavelength of λ = 435 nm, which is close to the wavelength position of Ag
spheres with 2R = 2 nm in a medium with nM = 1.5 (λ = 421 nm). In consequence,
when using the corresponding κeff in calculation for the absorption constant
γa(λ) = 4πκeff(λ)/λ of a planar homogeneous medium, almost the same absorption
peak can be observed as in the extinction by single isolated Ag nanoparticles, but
shifted to longer wavelengths (red shift). This can be recognized from Figure 14.3,
where the absorption coefficient calculated with the above κeff is calculated. This
433
434
14 Effective Medium Theories
Figure 14.2 Comparison of the refractive index neff + iκeff from the Maxwell-Garnett, Brugge-
man and Looyenga models for Ag inclusions ( f = 0.1) in a transparent matrix with a constant
refractive index nM = 1.5.
Figure 14.3 Comparison of the absorption coefficient resulting from κeff from the Maxwell-
Garnett, Bruggeman and Looyenga models for Ag inclusions ( f = 0.1) in a transparent matrix
with a constant refractive index nM = 1.5.
14.2 Theoretical Results for Metal Nanoparticle Composites
Figure 14.4 Refractive index neff + iκef of a transparent matrix with a constant refractive index
nM = 1.5 with Ag inclusions as a function of the volume fraction of inclusions according to the
Maxwell-Garnett model.
is not so for the other two models. The Bruggeman neff appears to be composed
of a series of close-lying harmonic oscillators with different oscillator strengths.
However, the curvatures of neff and κeff are unexpectedly strange and cannot really
be explained by a sum over harmonic oscillators. Completely unexpected is the
behavior of the Looyenga neff. Whereas the real part is almost constant, the imaginary part, and hence the absorption in the composite, increases continuously with
increasing wavelength. Correspondingly, the absorption coefficient is also modified for the Bruggeman model. For the Looyenga mixing rule, the absorption even
appears to be almost identical with the absorption coefficient of a silver film.
Next, we consider the influence of the volume fraction on the refractive index
of the effective medium. We only give results for the Maxwell-Garnett model in
Figure 14.4. The volume fraction varies and takes the values f = 0.001, 0.01, 0.1,
0.2, 0.3, 0.4, and 0.5.
Despite the logarithmic scale used, the changes in the real part of neff cannot be
resolved for f = 0.001. The imaginary part κeff clearly shows the behavior of a harmonic oscillator. The harmonic oscillator behavior is maintained up to volume
fractions of f = 0.1. For higher volume fractions, clear deviations can be observed that
cannot be assigned to any realistic model for dielectric functions or refractive indices.
This trend can also be observed in the absorption coefficient displayed in Figure 14.5.
Finally, we consider the effect of the particle shape on the refractive index of the
composite. Again, we use Ag inclusions in a matrix with constant refractive index
nM = 1.5. The filling factor is f = 0.1, for which the Maxwell-Garnett approach still
results in a harmonic oscillator behavior of the refractive index neff. In all cases
presented in Figure 14.6 the refractive index neff and the absorption index κeff
435
436
14 Effective Medium Theories
Figure 14.5 Absorption coefficient of a transparent matrix with a constant refractive index
nM = 1.5 with Ag inclusions as a function of the volume fraction of inclusions according to the
Maxwell-Garnett model.
Figure 14.6 Refractive index neff + iκeff of a transparent matrix with a constant refractive index
nM = 1.5 with Ag inclusions as a function of the shape of the inclusions. Calculations
according to the Maxwell-Garnett model for f = 0.1.
14.3 Experimental Examples
Figure 14.7 Absorption coefficient of a transparent matrix with a constant refractive index
nM = 1.5 with Ag inclusions as a function of the shape of the inclusions. Calculations
according to the Maxwell-Garnett model for f = 0.1.
exhibit features that are similar to those of a harmonic oscillator. However, the
curvature, especially at the lower wavelength side, do not really correspond to a
harmonic oscillator behavior. The position of the resonance wavelength depends on
the particle shape. For ellipsoids and spheroids, clearly two resonances can be
identified, according to the geometry factors (depolarization factors). These resonances can be recognized also in the absorption coefficient in Figure 14.7. Except
for the sphere, all other particles lead to peaks in the absorption coefficient with
a shoulder at short wavelengths and an abrupt decrease for longer wavelengths.
14.3
Experimental Examples
In this last section, we give an overview of several experiments where effective
medium models were applied to calculate optical properties of the composite. In most
examples the original approach (Maxwell-Garnett, Bruggeman) was not sufficient and
was extended to take into account shapes, aggregation, sizes and shape distribution.
Abelès and Gittleman [843] examined the optical properties of various metal–
dielectric and semiconductor–dielectric composite systems. The results are compared with the predictions of the Maxwell-Garnett effective medium model and
437
438
14 Effective Medium Theories
its extension to an ellipsoidal shape. Only the original Maxwell-Garnett theory
predicts the characteristic optical features of granular metals (here Ag). In the case
of the semiconductor–dielectric composites (here Si–SiC), the observed red shift
in the transverse optical phonon frequency in the far-infrared region is very small
and does not allow one to discriminate between the two models. In the case of the
Ge–Al2O3 films, both models are in good agreement with the experimental results
for the optical constants near the absorption edge.
Granqvist and Hunderi [844] prepared composites containing Au nanoparticles
with 2R = 3–4 nm. For interpretation of the measured optical transmittance they
used the Maxwell-Garnett and Bruggeman models and an extension of effective
medium models by Hunderi. For low filling factors the models are indistinguishable. For higher filling factors, the authors additionally considered deviations from
the spherical shape, the size distribution, and dielectric coatings on the particles,
but without experimental evidence that they affect the optical properties significantly. The authors also considered dipole–dipole coupling, which is accounted
for by a set of effective depolarization factors, that is, they assumed well-defined
geometric configurations of spheres. With this extension, they obtained in excellent agreement with the experimental data.
Norrman et al. [845] continued the experiments of Granqvist and Hunderi on
gold particles in a dielectric matrix. For interpretation of the optical transmittance
spectra they used the Maxwell-Garnett formalism extended to prolate spheroids.
Interactions were taken into account according to Bedeaux and Vlieger [846]. They
achieved in excellent agreement with the experimental results.
An overview of cermets (metal–ceramic composites) with application of the
Maxwell-Garnett and Bruggeman approaches was given by Granqvist [847].
Optical properties of granular tin films were investigated by Truong et al. [848]
in the spectral range 0.22–1.0 µm. Reflectance was calculated using the MaxwellGarnett approach. An extension of this theory – the revised dipole theory that
includes frequency-dependent terms neglected in the static approximation – was
also used to characterize the granular films with a better description of the optical
behavior when the particle size exceeds 10 nm.
Electron beam coevaporated Al–Si composite films were produced by Niklasson
and Craighead [849] consisting of a mixture of aluminum and silicon particles.
The microstructure suggests the use of the Bruggeman theory to account for the
optical properties of the films, since the microgeometry underlying this theory is
that of a random mixture of spheres of the two components. Indeed, very good
agreement between the experimental reflectance and predictions by the Bruggeman theory was found for volume fractions less than 0.33. Furthermore, the
agreement was obtained without the introduction of fitting parameters.
The Maxwell-Garnett theory extended to include the shape factor and the size
of the metal particles embedded in a dielectric matrix was used by Thériault and
Boivin [850] to explain the observed optical constants of a Cu–PbI2 cermet material.
Fairly good agreement was observed for a volume fraction ranging from f = 0
to 0.12.
14.3 Experimental Examples
Experimental spectral transmission curves of four cermets, Cu–PbCl2, Cu–NaF,
Ag–PbCl2, and Ag–NaF, were compared with calculated curves obtained from the
generalized Maxwell- Garnett theory by Chandonnet and Boivin [851]. The results
showed that the microstructure of the metal particles in the dielectric matrix is
independent of the nature of the dielectric, as predicted by the generalized Maxwell-Garnett theory. However, measurements also indicated that the microstructure is the same for both copper and silver cermets, which cannot be explained
within the context of the Maxwell-Garnett theory.
Heilmann et al. [852–854] investigated plasma–polymer–silver composites prepared by simultaneous or alternating plasma polymerization and metal evaporation. Transmission spectra were recorded and interpreted with the Maxwell-Garnett
and Bruggeman models. Fairly good agreement was obtained for the twodimensional distribution of the particles using the Maxwell-Garnett approach,
whereas for three-dimensional particle distributions in the matrix the Bruggeman
approach yielded better results. A comparison with the Bergman effective medium
model was also made [854].
Leitner et al. [855] examined the optical properties of a metal island film close
to a smooth metal surface. The silver island film was positioned close to a silver
mirror with a quartz interlayer in between. The interlayer thickness could be
changed stepwise. A dramatic change in the optical reflection behavior of this
mirror system was observed depending on the thickness of the interlayer. At a low
interlayer thickness the reflectivity loss is high in a broad spectral band, so the
mirror appears to be nearly black. At greater distances the mirror becomes colored:
as the spacer layer thickness is increased, the color varies from bright blue to
yellow, orange, and violet. The phenomenon was analyzed in terms of an effective
medium theory by using TEM data and an electromagnetic model for the optical
constants of the metal island film. For island films with a sufficiently high absorbance, the spectra are characterized by two sharp minima where the reflectivity
drops to values below 10−3.
The optical properties of discontinuous copper films on quartz substrates at
volume fractions ranging from 0.19 to 1.0 were investigated by DobierzewskaMozrzymas and Bieganski [856]. Three phases could be distinguished: (1) the
dielectric phase for f < 0.54, (2) the percolation phase for f = 0.54 ± 0.05, and (3)
the metallic phase for f > 0.54. A modified Maxwell-Garnett theory (shape factor,
size effect, surface oxide layer) was successfully used to interpret the experimental
data for films with low volume fraction ( f ≈ 0.19–0.26).
Nagendra and Lamb [857] prepared germanium–silver composite thin films
having different concentrations of Ag, ranging from 7 to 40%, by DC cosputtering
of Ge and Ag. The morphology was determined to be island-like, that is, Ag particles distributed in Ge. The optical constants of the composites showed a semiconductor behavior, even for 40% Ag in Ge. Comparison of the n and k data
with the Maxwell-Garnett and Bruggeman theories showed that both theories have
limited scope in predicting the optical properties of semiconductor–metal composite films in the infrared region.
439
440
14 Effective Medium Theories
Based on the research of Niklasson et al. [826], the Maxwell–Garnett and Bruggeman theories for two-component systems were directly extended to threecomponent systems by Luo [858]. Two theories were obtained, of which one had
already been obtained by Wood and Ashcroft [859] and Stroud and Pan [860] by
different or similar methods. The improved theories were employed in the calculation of the spectral reflectance of three samples of electrolytic colored anodic
aluminum oxide film, Au–Cu–Al2O3 and Ag–Cu–Al2O3, and the results were compared with the experimental data.
Dalacu and Martinu [353] examined composites of Au nanoparticles with varying
size in SiO2 in the wavelength range 300–850 nm. They produced gold particles
by plasma deposition in an SiO2 film. Analysis was performed using the MaxwellGarnett effective medium approach. From comparison of the measured and calculated spectra, they determined the optical constants of the Au nanoparticles with
an additional evaluation of the interband edge.
Kürbitz et al. [861] doped a melt of a commercially manufactured glass with
copper compounds to prepare copper nanoparticles in glass. The glass obtained
was opaque black at the usual thickness and looked dark red in bulbs of incandescent lamps due to the high absorption caused by the particles. The pseudo-optical
constants of this material were determined as a function of wavelength in the
range 350–700 nm by ellipsometric measurements. They could be reproduced very
well by a model that consisted of a roughness layer situated on a substrate of glass
containing spherical copper particles with a Gaussian size distribution with
2R = 6.5 nm and σ = 1.6 nm and a volume concentration of 2.4 × 10−3. For this
modeling, the dielectric function of the roughness layer was approximated by the
Bruggeman effective-medium theory and that of the copper-containing glass substrate was calculated on the basis of the theory of Gans and Happel. The results
were verified by TEM investigations.
Mandal et al. [862] prepared silver–silica nanocomposite thin films by a highpressure DC sputtering technique. Films deposited at lower substrate temperature
showed a narrow distribution of nanoparticles with nearly spherical shape. An
increase in substrate temperature resulted in films with a non-uniform size and
shape due to the agglomeration of the nanoparticles. This size and shape distribution affects the optical absorbance spectra and results in a broad and asymmetric
surface plasmon band. A shape distribution introduced in the Maxwell-Garnett or
Bruggeman effective medium theory was found to give a reasonable description
of the experimentally observed optical absorption spectra.
Takeda et al. [863] investigated the linear and nonlinear optical properties of Cu
nanoparticle composites with high Cu concentrations. Negative Cu ions of 60 keV
were implanted into amorphous SiO2 at a fixed flux and various fluences. Optical
absorption increased with the Cu volume fraction but the SPP peak was maximized around a fraction of f = 0.1 and attenuated beyond that fraction. The best
agreement with the absorption spectra was found with the Maxwell-Garnett model.