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Infrared Absorption of Nanocermets Close to the Percolation Threshold
Serge Berthier, J. Peiro
To cite this version:
Serge Berthier, J. Peiro. Infrared Absorption of Nanocermets Close to the Percolation Threshold.
Journal de Physique III, EDP Sciences, 1997, 7 (3), pp.537-547. �10.1051/jp3:1997141�. �jpa-00249599�
J Phys III FFance 7 (1997) 537-547 MARCH 1997, PAGE 537
Infrared Absorption of Nanocermets Close to the Percolation
Threshold
S. Berthier (*) and J. Peiro
Laboratoire d'optique des Sohdes (**), Universit4 Pierre et Marie Curie, 4 place Jussieu,
75252 Paris Cedex 05, France
(Received 6 &1ay1996, accepted 22 October 1996)
PACS.78.20.-e Optical properties of bulk materials and thin films
Abstract. We present a two dimensional simulation of the optical absorption of gold granu-
lar films, using a real space renormalization procedure. This numerical effective medium theory takes into account the actual morphology of the films. The results are compared both with
experimental measurements and with other theoretical predictions asserting that the optical properties around the percolation threshold cannot be described by an effective dielectric func- tion. Nevertheless, we obtain good agreement between the simulations and the experimental data, whatever the morphology and the concentration It is concluded that the IR absorp-
tion can be attributed to the classical surface plasmon modes, generally situated iii the visible
spectral range for spheroidal inclusions, broadened here towards higher wavelengths due to the formation of fractal clusters of various sizes.
1. Introduction
Very thin metal films are known to be discontinuous and present a phase transition between
an insulating behaviour and a conducting one for a critical concentration pc, the so-called per- colation threshold. Several previous studies have shown that the optical properties of granular
metal present a very special behaviour in this phase transition region, namely:
a frequency independence of the optical properties in the infrared region at the optical percolation threshold;
a maximum of the optical absorption at and near the percolation threshold.
Initially observed on granular gold films 11,2j, this holds true for granular silver films [3j and, with differeIit quantitative value, for thin cermet Au-AI? 03 films [4j. These experimental
results give rise to various theoretical explanations. The "general aggregate theory" [5] and the scaling theory iii assert the fundamental impossibility for the effective medium theories to model these anomalous behaviours. The argument is that around the percolation threshold,
the quasi-static assumption is not valid and that the clusters are explored by the incident radiation on a length scale L(Lu much smaller than the correlation length (, so that the notion of
(*) Author for correspondence (**) UA CNRS D0781
© Les (ditions de Physique 1997
Dielectric Function (DF) is meaningless In both works, the contribution of the plasmon surface modes are neglected. By contrast, it has been shown that the Effective-Medium Theories
(EMT) account for these special behaviours [6-10j- Even more, the phase transition is shown to be correctly modelized, with critical exponents in good agreement with the theoretical and
experimental predictions, when the actual morphology is precisely taken into account This is the case with the Franck and Lobb algorithm [10] applied to the determination of the effective DF by Brouers et al. [9] and the real space renormalization procedure presented here ill,12].
These effective medium theories give a primordial part to the surface plasmon modes rather than to the length scale dependencies.
In this paper, we first present our experimental measurements on the optical and morpholog-
ical properties of gold granular films. The set of samples covers a broad range of concentrations around the percolation threshold and some of them have been annealed to modified their ge-
ometrical aspect. This allows an evaluation of the influence of the morphology on the optical properties in the percolation range. In Section 3 we outline the renormalization approach from
a theoretical point of view and Section 4 contains the comparison between the theory and the data Section 5 summarizes the results.
2. Experiments
The granular gold films are obtained by thermal evaporation onto optically polished borosilicate
glass, under high vacuum (mid- 10~~ torr), at room temperature. The gold deposited mass,
or mass thickness, is determined by a quartz crystal oscillator inside the evaporation chamber.
The film resistance, and so the critical mass thickness de where the percolation occurs, is
continuously measured between two parallel gold stripes. Just after deposition, half of the films
are annealed under vacuum at 98 °C during 24 hours, which deeply affect their geometrical, optical and electrical properties.
The room temperature transmittance T and reflectance R under normal incidence were measured using a Varian 5 spectrometer, in the range 0.35 ~Jm 3 ~Jm. The absorptance of the film A
= I R T is deduced from them. The data are presented in Figure I for the two sets of films (annealed and non annealed). We can clearly distinguish the insulator like samples (decreasing reflectance as the wavelength increases) and the metal like ones (high and
increasing reflectance, low and decreasing transmittance as the wavelength increases). Between these relatively dilute or concentrated samples, i. e. near the percolation threshold, we observe the so-called anomalous behaviours: the optical properties of the films are nearly independent of frequency, and the absorption is high, of about 35%. The volume fraction dependence of
the absorptance for ~
= 2.2 pm (this wavelength corresponds to the first observation of this phenomenon by Gadenne in Ref. ill is presented in Figure 2- As already observed, the near
infrared optical absorption is maximum for a filling factor slightly higher than the percolation
threshold pc.
The surface coverage parameter p is determined by Transmission Electron Microscopy ITEM)
and compared u.ith the corresponding mass-thickness d of the film. A linear relation between p and d is observed The TEM micrographs of the samples are scanned and digitalized so
that image treatment can be performed. Figure 3 shows the TEM micrographs of two sets of samples. It confirms the previous optical analysis: samples A are below the percolation threshold pc, saInples B are very close to pc and samples C well above it. The concentrations and mass tiicknesses of the presented samples are listed in Table I. We also indicate the reduced
concentrations p* = (p pc)/pc and the reduced mass-thickness d*
= id de)/dc which are
generally used in the percolation literature, the fractal dimension Df of the main clusters and
the resistance per square R D of the films The morphology has a direct influence on the
N°3 INFRARED ADSORPTION OF NANOCER&IETS.. 539
1
p<pc
o.5
A
R 0
0 0.5 1 1.5 2 2.5 3
Wavelength (wicrometer)
1
fl~pc
T
o.5
A
R 0
0 O-S I 1.5 2 2.5 3
Wavelength (micrometer) 1
p>pc
o-s T
A
0
0 O-S I 1.5 2 2.5 3
Wavelength (micrometer)
Fig. I Reflectance R, transmittance T and absorptance A of the annealed (~) and non-annealed (- -) granular gold films ~ersus wavelength in the visible and near infrared range
~'
~'0.3 1
= 2.2 pm
, ~
~ '
~ ' O
' '
W /~ ',
' 0 2 / ' ~
~
/ ,
1
,'
,
~ ' ,
"
0.I P
,
, ,
,' ,,'
-~Y 'P~~
,P~~~
0.3 0.4 O-S 0.6 0.7 0.8
Metal concentration p
Fig 2. Plots of the experimental absorptances A of the annealed (- -) and non-annealed (~)
films ~ersus the metal concentration p IA = 2.2 ~Jm)
Table1. Mean geometrical and electrical charactenstic3 of the samples.
d d* p Df RD
cxJ
annealed B 69.9 0.013 0.52 0.04 1.83 + 0.01 194
annealed C 107.4 0.56 0.67 0.34 1.88 + 0.01 7.2
non annealed A 0.52 cxJ
non annealed B 60.1 0-019 0.63 0-016 1.90 + 0.01 231
non annealed C 90.4 0.53 0.72 0-16 1.93 + 0-01 II
percolation threshold value. For the annealed films, pc
m~ 0-5 and for the non annealed ones, pc m~
0 62.
3. Theory
To simulate the optical properties of 2D granular metal films, we consider a square lattice,
in fact a real digitised TEM micrograph or a computer simulated lattice, occupied by metal sites of dielectric function em(Lu) with a probability p and sites of dielectric function eo with a
probability ii p). In the case treated here, the noble metal DF is given by a Drude function:
~j
~~~~~ ~
ld(ld +1/71'
where P, Lop and r are the electronic polarization, the plasma frequency and the relaxation time respectively. For the gold inclusions considered in this work, P
= 6.5, hLup = 9-2 eV and
h/r is corrected from the finite size effects following the classical relation hIT
= Afro + vF/r.
vF is the Fermi velocity and To the bulk relaxation time IAfro
= 0.06 eV)- r is the mean free
N°3 INFRARED ADSORPTION OF NANOCERMETS. 541
Non annealed gold film Annealed gold films
ol,
~n
I
'
~
~ ,
Ill '
$
Fig 3. Digitised TEM micrographs of the two samples sets of gold granular films, as deposited on
the left, after 24 hours annealing at 98 °C on the right Metal is
in white
Step n ; L = 16 x 16 Step n+I : L = 8 X 8
1
Step n+3 : L = 2 x 2 Step n + 2 : L = 4 x 4
Fig. 4 Sequence of 3 renormalization transformations The size of the lattice, expressed m blocks number L, converges to unity as the homogenisation progresses
path of the electrons in the fractal clusters and is assumed, m this case, to be equal to the average width of the conducting arms. As determined from image treatment, r is of the order of 25 nm for the non annealed films and 125 nm for the annealed ones. To determine the effective DF of such lattices, the real space renormalization procedure consists m a repeated application of a classical effective medium theory (here the Maxwell Garnett theory modified by Cohen et al. [13]) to a small number of sites (blocks of four sites in our case). Each block
is then considered as a single super-site which becomes itself a part of a new super-block, and
so on. The final result of this sequence of transformations is to reduce the lattice to a single
site which has the same dielectric function than the entire lattice In much more details, the process, illustrated
m Figure 4, is the following. The initial lattice of size L (512 x 512 for our TEM micrographs) is divided into blocks of four sites each (2 x 2)<
N°3 INFRARED ADSORPTION OF NANOCERMETS.. 543
The local DF en of each block is determined using the Maxwell Gamett EMT in the insulating
or conducting configuration according to a conduction criterion:
~n 61(c) ~c(i) t§(c)
§(c) ~ 2i(fn §(c)) ~61(c) ~ ~(fc(1) fi(c))
ei is the dielectric function of the insulating phase and ec the DF of the conducting one Equal respectively to co and em at the first renormalization step, they are different in each block in the following steps. Following Kadanoff's method, the new lattice of size L/2 so constituted is
divided one more time into blocks of four sites and the process is repeated until L
= I. This
sequence of transformations, usually noted R2, allows to go from a description of the film at a
length scale a equal to the size of the initial pixel, to a description of the same system at the
length scales 2a, 4a..
The first aim of this approach is to take into account the actual morphology of the films,
with their local fluctuations and density variations. The results of this theory, when applied to random lattices or real structures, have been presented in details in previous papers ill,12,14].
It gives a good description of the optical transition, without any free parameters, both in 2D
and 3D cases.
The Franck and Lobb algorithm mentioned before deals with a square lattice occupied by bonds of various conductances- deduced from a real TEM micrograph by a classical site-bond
transformation It consists in a repeated application of series~parallel and star~triangle trans- formations, until a single bond is obtained that has the same effective conductance as the entire lattice. This procedure is very similar to the renormalization approach used in this work and leads, as can be seen latter, to the same conclusions.
4. Numerical Results
The calculated optical properties R, T and -4 of the granular Au films near the percolation threshold, where the data include the substrate contribution, are shown in Figure 5. They have been determined from the calculated effective DF with the exact Abeles equations [15]. The films thicknesses used in the calculation are the mass thicknesses divided by the concentration p. They give a good account of the experimental results, both in the visible and the infrared
range, especially near the percolation threshold where £&IT
are supposed to fail. The frequency independence in the IR range at the percolation is accurately reproduced.
The absorptance A = I R T of the two sets of films versits the reduced concentration is shown in Figure 6 for ~
= 2-2 ~m together with the theoretical predictions Two remarks arise from this figure. ii) Despite of very different morphologies,the absorption appears to be similar for the two sets offilms, and so independent ofthe morphology when plotted versits the reduced concentration p*. As already pointed out, the morphology affects the percolation threshold value but not the optical properties m it vicinity. iii) Over a wide range of concentrations, the renormalization theory provides a good description of the so-called anomalous absorption.
Particularly noteworthy is the fact that these results are obtained from real TEM micrographs
without any free parameters. The assertion that the optical properties ofinhomogeneous media
near the percolation cannot be described by- an effective dielectric function falls in-
5. Discussion and Conclusions
The previous calculations and the simulations performed by Brouers et al. in reference [9] with the classical EMT of Bruggeman [16] and the Franck and Lobb fast algorithm have confirm
1
Annealed, p ~ pc
T
~ 0
p '~'
ci
, ', ~
R
0
.5 2 2.5 3
Wavelength (micrometer)
Non annealed, p ~ pc
Q nI
~, J
,
"
T o-s
R
o
0 0.5 1 1.5 2 2.5 3
Wavelength (micrometer)
Fig. 5. Reflectance R, transmittance T and absorptance A of the granular films, as calculated by the renormalization process (- -) near the percolation threshold pc, compared with the experimental
results (~)
that effective medium theories are not limited to low or high metal concentrations but are in excellent agreement with the experimental results in the transition region. According to these
approaches and by contrast with the scaling models of references ill and [5], the absorption
N°3 INFRARED ADSORPTION OF NANOCERMETS.. 545
O Annealed film
~ Non annealed films
~ '
l --.-- Renor. '
« i
_
0.2
'
~ ,l'
,' l 2.2
v~
0.1
-0.2 -0.1 0 0.1 0.2
Reduced concentration (p p )
Fig. 6 Absorptance ofthe gold granular films (annealed and non annealed) ~ersw the reduced con- centration (p-pc). The absorption variations of the two sample sets are nearly equivalent Comparison
with the theoretical results (m), A = 2 2 ~Jm.
of noble-metal granular films is attributed both to the surface plasmon modes and to the percolation law above p~.
The effective DF calculated by the real space renormalization procedure can be described at any concentration by using an effective Drude formalism, in such a way that the critical behaviour shows up explicitly ill,17] The effective polarization term P~ and the effective conduction term Luj~ both exhibit a power law dependence on )p pc) (Fig- 7)
[Fe " P01fl~flcl~~
~dje " A~jp-pc)~.
The calculated s and t values is
= 0.75 + 0.05, t
= 2 2 + 0.I) are in rather good agreement with the theoretical values (sth
" 0.73, tth
= 1.94) [iii.
P~ represents the static interfacial polarization and deals with the finite size clusters whose number and relative concentration is maximum at pc and starts to decrease above pc. The optical resonant absorption due to the surface plasmon modes in these clusters follows a sim- ilar variation. On the other hand, the metallic absorption from the infinite cluster starts to
increase at pc. The sum of these two contributions is maximum for a concentration slightly higher than pc, as experimentally observed. This can be observed in Figure 8 where the usual
absorption coefficient e2e/~ is plotted versits the reduced concentration p*. The contribution from percolating conducting electrons has been subtracted to emphazise the contribution of the
resonant cluster modes. We also plot the Franck and Lobb algorithm predictions, as calculated
by Brouers et al. for comparison. Great similarities can be observed betu,een these two EMT and with the experimental data.
In conclusion, we have shown in this work and m the previous theoretical presentations of the renormalization theory [11,12,17] that effective medium theories, when related to the exact
morphology of the medium, can account to the special behaviour of granular thin films in the
near infrared range, at and near the percolation threshold. They confirm the primordial role
30
. I
3
~
j j
2° ~e ~ / '~°~e IO
$ ~
) /
i ,~' 4~
t ~ '? ,m'
m
~ g (
$ lo / ' ' 5 ~
~ ~p ~
~ ~
~ , ~
~ .,@' ' j
u ,, u
~ , ~
uw ,,
, ~
~
-l -0.5 0 0.5 1
Reduced concentration p* = (p p )/P~
Fig. 7 Renormalized effective plasma frequency hLupe(m) and polarisation Fe (.) of a free electrons metal-dielectric composite deduced from a Drude fit of the infrared dielectric function, ~ersus the reduced concentration.
16
,/
12 . Renor.
/~
. l'
~
Q L F ~C~ ~'
$ i Q ,I'
j ~
' ', O . .,/
I Q ' Q
~ i f§Q O
'q ' I '
uJ 4
,
0
-1 -0.5 0 0.5 1
Reduced concentration p* (p p~)/P~
Fig. 8 Near infrared optical absorption, calculated with the renormalization procedure (.) The contributions of the resonant modes and conducting electrons have been separated Comparison with the predictions of the FL algorithm lo)
of the surface-plasmon modes of the finite size clusters and give a plausible interpretation of the so-called anomalous absorption. As already pointed out, the real space renormalization
procedure as been extended to 3D square lattices. It would be now of great interest to explore
the optical properties of inhoniogeneous media in the percolation range as one goes from a 2D to a 3D regime. Optical measurements on cermet films near the percolation threshold should be revisited and new measurements on transition metals and semiconductors granular or cermet films performed, to confirm the influence of plasmon modes on the optical absorption.
