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Mean field theory and scaling laws for the optical properties of inhomogeneous media
Serge Berthier, K. Driss-Khodja, J. Lafait
To cite this version:
Serge Berthier, K. Driss-Khodja, J. Lafait. Mean field theory and scaling laws for the op- tical properties of inhomogeneous media. Journal de Physique, 1987, 48 (4), pp.601-610.
�10.1051/jphys:01987004804060100�. �jpa-00210475�
601
Mean field
theory
andscaling
laws for theoptical properties
ofinhomogeneous
mediaS. Berthier (*), K. Driss-Khodja (x) and J. Lafait
Laboratoire d’Optique des Solides (+), Université Pierre et Marie Curie, 4, place Jussieu,
75252 Paris Cedex 05, France
(Reçu le 22 juillet 1986, accepti le 25 novembre 1986)
Résumé. 2014 La théorie de Bruggeman de la fonction diélectrique optique des milieux inhomogènes est une
théorie de champ moyen. Au voisinage du seuil de percolation, ces théories prédisent des variations des
grandeurs physiques étudiées, en fonction de la concentration, selon des lois de puissance de la forme
(q 2014 qc)-03B4, où 03B4 est un exposant critique et qc la concentration critique de percolation. Ces lois n’ont jamais été
mises en évidence analytiquement pour la fonction diélectrique optique. Nous présentons donc une
formulation nouvelle de la théorie de Bruggeman où les lois de puissance apparaissent explicitement avec les exposants caractéristiques d’une théorie de champ moyen : s = 1 pour la polarisation et t = 1 pour la conduction. Or, des calculs de Monte-Carlo sur des réseaux aléatoires, ainsi que nos mesures sur de nombreux
systèmes, métal/diélectrique (Au-MgO, Pt-Al2O3, Fe-Al2O3) et celles effectuées par ailleurs sur Co-Al2O3 et Ag-KCl, qui sont présentées en détail ici, confirment que pour un système à trois dimensions,
s = 0,73 et t = 1 ,94. Les theories de champ moyen sont par conséquent inadéquates au voisinage du seuil de
percolation. Nous proposons donc d’introduire les valeurs exactes des exposants critiques dans notre nouvelle
formulation de la théorie de Bruggeman, obtenant ainsi une théorie valable à toute concentration. Cette nouvelle approche est testée avec succès sur tous les matériaux cités précédemment. Les écarts observés sur la
partie imaginaire de la fonction diélectrique peuvent être justifiés avec des arguments purement expérimen-
taux, qui cependant n’expliquent pas tout. C’est le principe même des théories de milieu effectif qui doit être
remis en cause à la percolation. Par ailleurs, nous proposons une méthode simple et originale pour déterminer la valeur expérimentale du seuil de percolation optique, qui est un paramètre essentiel de cette approche.
Abstract. 2014 The Bruggeman theory for the optical effective dielectric function of inhomogeneous media is a
mean field theory. However, the characteristic behaviour of such theories, namely the power law relation between the effective dielectric quantities and the filling factor q, of the form (q 2014 qc)-03B4, where 03B4 is a critical
exponent depending of the physical quantity under consideration (s = 1 for polarization and t = 1 for
conductivity), was never analitically demonstrated. In fact, these values are wrong, as confirmed by Monte-
Carlo calculations predicting the values s = 0.73 and t = 1.94 and by our experimental results on Au-MgO, Pt-Al2O3, Fe-Al2O3, as well as by many other measurements on Co-Al2O3 and Ag-KCl, for example. We
confirm that these values are frequency independent far from the dielectric anomaly and we present a new formulation of the Bruggeman theory in which the different contributions to the dielectric function appear
explicitly with their own exponents. A new formulation is then proposed by replacing the exponents by their proper values. One shows, on comparing to experimental results, that, in contrast to the BR theory, our approach is valid close to the percolation critical fraction. The agreement between theory and experiments is
rather satisfactory, except for some discrepancies on the imaginary part of the dielectric function. This problem
can be solved using experimental arguments but the validity of any approach based on an effective medium
theory has to be reconsidered near the percolation. We also propose an original but simple way to determine the experimental percolation threshold which is the most important parameter in this approach.
J. Physique 48 (1987) 601-610 AVRIL 1987,
Classification
Physics Abstracts
78.20
(+ ) U.A. au CNRS n° 040-781.
(*) Also : Universitd de Corse, BP 24, 20250 Corte.
(X ) Also: U.R. en Physique du Solide, Universite d’Oran-es-Sdnia, Algerie.
1. Introduction.
The optical properties of metallic granular com- pounds are mainly characterised by : 1) an optical
cross over, i.e. a transition between a dielectric-like
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004804060100
and a metal-like optical behaviour and 2) the so-
called dielectric anomaly or resonant absorption
observed in the visible range of the spectrum, due to the excitation of a surface plasmon mode of the conduction electrons in the metallic grains.
The two main properties are now well established
experimentally and a brief confrontation with the effective medium theories shows that the-non-self- consistent theories (Maxwell Garnett [1, 2] and
derived models) are inadequate because they do not predict the optical cross over. In contrast, as shown
elsewhere [3] the self consistent theories of Brug-
geman [4] and Ping Sheng [5] account for both an optical cross over and a resonant absorption.
In section 2, we outline the Bruggeman theory,
with particular attention to the non metal-metal transition. The well-known general predictions and
demonstration [6-8] of the theory will not be re- peated and we will start the analysis directly from
the formulation. The equation is then solved as- suming that, in the I.R. spectrum, the dielectric function (D.F.) of the effective medium follows a
Drude formula. The analytic expression of the
effective dielectric function can then be expanded
into terms with a power-law dependence, related to
the d.c. conductivity and the dielectric polarization.
The validity of this expression is then discussed and
the analysis of each term is carried out in the
theoritical case of frequency independent dielectric
material and free electron metal inclusions.
We develop in section 3 our phenomenological
model by replacing the exponents of the BR theory by the critical exponents of the static conductivity
and the static dielectric function deduced from Monte-Carlo simulations. Section 4 is devoted to
experimental considerations. We finally show in
section 5 the agreement of this model with our
experimental results on Au-MgO, Pt-AI203, Fe-AI203 as well as those of Niklasson and Grangvist
for Co-A1203 [9, 10] and give our interpretation of
the discrepancies observed between the imaginary
part of the dielectric function calculated with our
model and the experimental results.
2. The Bruggeman theory.
2.1 THE BRUGGEMAN FORMULATION. - All the effective medium theories of the optical D.F., in the quasi-static approximation, can be derived from the Clausius-Mossotti relation based on the local field
theory. In the case of the Bruggeman mean-field theory, where the effective medium is the embedding
medium of each type of inclusion, it follows :
where
£e = £1 e + i E2 e is the effective dielectric function
(D.F.) of the medium,
Ea(Eb) is the D.F. function of the two types of inclusions,
q is the volume fraction of inclusions a, and L is the depolarization factor of identical ellipsoidal inclusions, supposed to have the same orientation
(the axis of the ellipsoids is parallel to the electric
field). The physical meaning of L has been discussed in detail elsewhere [11, 12].
The equivalent quadratic equation is :
whose analytic solutions are established in [13]. The
real and imaginary parts of these solutions follow two regimes :
- In the far infrared an optical cross over is
clearly observed : at low concentration El e is con-
stant and positive and e2e close to zero, whereas at
high concentration Cle is becoming negative and
both
I sl e
I and e2 e strongly increasing with the wavelength.- In the visible and near infrared a broad struc- ture is observed, attributed to the excitation of a
surface mode of the conduction electrons in the metallic grains.
The shape and position of this absorption are strongly dependent of the nature of the metal (noble
or transition), but when the metal concentration tends to L, the peak increases and moves towards infinite wavelengths.
Thus, even if not clearly pointed out, the transition
between the two regimes seems to occur around a
critical metal concentration close to L. Our first aim in this paper is to separate out analytically these two
characteristic behaviours. As it can be seen from the above considerations, this separation is not strictly possible close to the critical concentration, the whole spectral range being there concerned by the resonant absorption. We first have to determine in which concentration and wavelength ranges this absorption
can be neglected.
Equation (2) can be recast in terms of the ratio
R =
Eel Em
of the effective medium to metal inclusiondielectric functions :
(where E. is the metal D.F. and Ei is the matrix
(insulator) D.F.) whose solutions, real and imaginary
parts, are represented in figure 1 as a function of the
metal fraction around the percolation threshold for
different wavelengths in the infrared region. As can
603
Fig. 1. - Real (a) and Imaginary (b) parts of the normalized effective dielectric permeability R = Eel Em of
a free-electron metal (dielectric permeability Em)-dielec-
tric composite as a function of metal volume fraction around the percolation threshold (0.3 q 0.38) in the
infrared region (25 A 50 um).
be seen on this figure, the cermet to metal D.F. ratio is wavelength independent in a broad concentration range beyond the percolation. Thus, in this range, if the metal D.F. follows a Drude law, the cermet D.F.
can be modelized by a Drude function too :
(where Pe and wpe are effective wavelength indepen-
dent parameters) except around q = L where a wavelength dependence appears. In the far infrared,
e2e can be neglected and the contribution to E1 e is essentially wavelength independent. The effec-
tive dielectric function reduces to a constant term.
This can be written, below the percolation, using the
same formalism as above the percolation, with W pe = 0 (in fact, there is no conduction) :
This is the starting point of our work, which introduces the expansion of the Bruggeman theory
in the framework of the scaling laws.
2.2 THE BRUGGEMAN MODEL AND THE SCALING LAWS. - Our aim, in this section, is to express the effective D.F. analytically as a sum of physical
contributions : polarizability of the dielectric, of the metallic clusters and of the infinite cluster after
percolation. We have shown in section 2 that the
Bruggeman D.F. can be modelled in the infrared by
a Drude function on both sides of the percolation
threshold. Thus, we can calculate the Drude par- ameters by replacing Ee by a Drude formula in
equation (2) :
with
considering that the metal DF follows the Drude law :
with
This leads to an equation which has to be valid
whatever the frequency. One can thus distinguish in
the new formulation of equation (4) the wavelength dependent and independent terms which both have to be made equal to zero :
for the frequency-independent term and:
for the frequency dependent term (which also in-
cludes Pe, solution of Eq. (5)).
These two equations are compatible only if the
term :
is equal to zero.
As can be directly seen on the formulation, this
term cannot be neglected in the vicinity of q = L, confirming that the Drude representation is not adequate close to the percolation threshold, due to
the importance of the dielectric anomaly at this
critical point.
The expansion of the effective D.F. as deduced from equation (5) and (6) with the above restrictions is :
- below the percolation :
- above the percolation :
with:
A simple formulation of the terms a and j8 cannot be given, but they represent the contribution of the dielectric anomaly to the effective D.F. in the
infrared and will be estimated later.
It is clear on this double formulation that the effective polarizability Pe diverges for q = L and
that the conductivity appears for the same critical concentration. This is the first demonstration of the well-admitted result that the Bruggeman model predicts the percolation at a critical metal fraction q,, equal to the depolarization factor L.
A reduced volume fraction q * can be introduced
below and beyond qc = L :
Using this reduced scale q *, equations (7) and (8)
become :
In these two equations, the Ed term can be related to a static polarization and the Drude term can be related to a conductivity via an effective number of conduction electrons ne involved in the effective
plasma frequency:
On both sides of thd percolation threshold, the polarizability diverges following q * -1 and above
qc, the effective plasma frequency follows a q *1 law.
The two remaining terms in Pe (q > q c) represent the polarization of the infinite cluster and of the dielec- tric grains respectively and are proportional to their respective volume in the medium.
Considering the formal analogy of these formulas with the expansions of physical quantities near a phase transition in the framework of the scaling
laws [14], one can see that the Bruggeman model predicts a critical exponent equal to 1 for both the
conductivity and the dielectric constant, in agree- ment with the classical results of the mean field theories.
Figure 2 represents the effective dielectric polari-
zation as calculated with the Bruggeman model (BR) for free-electron metal inclusions in a dielec- tric, as a function of the reduced concentration,
Fig. 2. - Effective polarization Pe of a free-electron metal-dielectric composite as a function of the metal volume fraction, deduced from a fit of the Bruggeman
D.F. (BR) (formula 4) and from our approximation (BRA) (formula (8)) for comparison.
compared to the effective polarization deduced from equations (7) and (8) (BRA). An illustration of the differences between these two curves (terms a and B) due to the contribution of the resonant absorption
is given in figure 3.
In the f ar infrared the discrepancy is negligible except in a narrow volume fraction range around the critical threshold. This range and the discrepancy
increase with the frequency and reach their max-
imum in the spectral range of the resonance. This resonant part of the Bruggeman D.F. obtained by subtracting the Drude contribution defined in (11) to
se is represented in figure 4. Its extension to infinite
wavelength at the percolation is clearly pointed out.
In short, one can distinguish two parts in the effective dielectric function of a typical noble
metallinsulator composite, as calculated by the Bruggeman model :
- A resonant part A Ee which is attributed to the excitation of interfacial polarization modes in the metallic grains.
- A Drude contribution whose components fol-
605
Fig. 3. - Relative discrepancies between the predictions
of the Bruggeman theory (Pe (BR)) and our approxi-
mation (Pe ((BRA)) for three wavelengths in the infrared
region. When the wavelength increases, the resonant part of the effective dielectric permeability decreases and the agreement is better.
Fig. 4. - Real (a) and imaginary (b) parts of the resonant part 4Be of the BR effective dielectric permeability
Ee of the composite, the Drude contribution having been subtracted, as a function of wavelength, for different metal volume fractions beyond and below the percolation.
low scaling laws below and above the percolation threshold, with an exponent equal to 1 :
with with
These values do not agree with the critical exponents s and t deduced from Monte Carlo calculations on
random networks, but we can now propose a proper formulation by replacing the exponents 1 by their
true values.
3. New phenomenological approach of the effective dielectric function.
3.1 THEORY. - To our knowledge, no measure- ment of the effective polarization Pe has been
undertaken above percolation but, below the per-
colation, we believe that the results obtained at low
frequencies on Ag-KOl cermets by Tanner et al. [15]
are presently the best available data and we can
reasonably assume the symmetry of such a phenomenon.
These measurements have given the following
value :
This value, confirmed by many theoretical calcula- tions [16-20], will be retained for the whole concen-
tration range.
On the other hand, the critical exponent for the dc conductivity, deduced from numerical simulations
on random percolating networks, is well known [21].
Recent calculations give t = 1.94 ± 0.01 [22, 23]. We
have two reasons for rejecting this value :
i) its validity range is reduced to the immediate
vicinity of the percolation critical fraction, and ii) in this range the Drude behaviour is largely
dominated by the resonant absorption and can no longer give a good description of the effective DF.
A more general value [24-27], verified by some experiments [28, 29] and valid on our results in a
larger range around qc, is t = 1.72. In fact this value calculated for lattice percolation may not apply rigourously to continuum percolation for which relatively larger values have been recently ob-
tained [30].
We propose a model in which these s and t exponents are put in formulae (10) and (11) instead
of the exponents 1. These two equations become,
when neglecting the a and B terms :
with :
The percolation critical fraction q, is explicitly invol-
ved in these formulae as one of the main parameters.
It is well known that this parameter is the only one
which is not predicted by the scaling laws. In the next section we present an original method for its determination.
3.2 EXPERIMENTAL DETERMINATION OF THE PER- COLATION THRESHOLD. - In order to apply our
model (formulae (14) and (15)), the knowledge of
qc is needed for calculating q *. This value is deduced from the experimental D.F. values by using the following procedure :
1. A fit of the experimental D.F. is performed
below percolation using equation (14), with q * as
free parameter.
2. The values of q * are then plotted as a function
of the q experimental values. If the s exponent is correct, the q * variations are nearly linear. The q experimental value at which q * is vanishing (interse-
ction of the experimental line with the q axis) corresponds to the percolation critical volume fraction qc.
The results of this procedure are represented in figures 5 (a, b, c) for the following cermet films
studied in our group: Au-MgO, Pt-AI203, Fe- A1203. We have also plotted the fit of the exper-
Fig. 5. - Reduced filling fraction as calculated using a fit
of experimental results with our model, versus the exper- imental filling fraction.
imental D.F. of Co-A1203 by Granquist and Niklas-
son [10]. The values of q,, obtained with this method
are the followings :
Except for Au-MgO, where no comparison can be
made because all our experimental results were
obtained below the percolation, these qc values are
in excellent agreement with those deduced from the d.c. resistivity measurements [31] (Fig. 6). We are
thus confident in the qc value deduced for Au-MgO.
Our model can now be applied to the experimental
D.F. without any free parameter.
4. Experimental results.
The test of this new model of the optical D.F. of inhomogeneous media near the percolation as well
as that of the experimental determination of the critical volume fraction have been performed on
three cermets studied in our laboratory: Au-MgO, Pt-A1203, Fe-A1203, and also on Co-AI203 studied by Niklasson and Granqvist. The preparation condi-
tions and the characterization of these cermets have been presented in earlier publications [32, 9]. We
will only insist here on the results directly related to
the percolation.
4.1 d.c. CONDUCTIVITY AND qc DETERMINATION.
- A first, but not very accurate method was used to
determine the critical volume fraction : the plot of
the resistivity versus the metallic volume fraction
(Fig. 6). It gives the following critical concen-
trations :
Fig. 6. - d.c. electrical resistivity of Pt-AI203, Au-MgO
and Fe-AI203 cermet films versus the metal volume fraction q. The dashed zone represents the domain of the
optical cross over.
