Feb 2, 1994 - d'une réduction de l'amsotropie et rapproche les fréquences de résonance de plasmon de surface vers celle de la sphère. Par ailleurs, dans le.
Phys.
J.
France
I.
(1994)
4
303-318
1994.
FEBRUARY
PAGE
303
Classification
Physics
Absn.acts
78.20
78.65
Anisotropic Serge
medium
effective
theories
Be«hier
d'optique
Laboratoire
75252
(Received
7
June
(*),
Solides
des
05,
Cedex
Paris
Université
Pierre
Curie,
Marie
et
80, 4 place
Boite
Jussieu,
France
1993,
revised
propriétés
théories
des
the
2D
The
optical classical
theory
metallic
the
substrate
a
this
the
m
tant
properties 3D
Yamaguchi
of
inclusions
paper, it is inclusions
des
milieu
of the
Abstract.
framework
du
effectif,
October1993)
22
inhomogènes étudiées dans le amsotropes sont Gamett et (3D) (théories de Maxwell dimensions (2D) (théorie de Yamaguchi et ai. ). L'anisotropie du milieu de Bruggeman) qu'à deux effectif peut l'alignement d'inclusions sphériques dans un système à deux ou soit de provenir non distribution trois dimensions, de la même des plane du système 2D, particules soit pour sphériques. Nous ici que ce milieu effectif déformation fictive des induit montrons anisotrope une réduction inclusions d'une de l'amsotropie et rapproche les fréquences de qui va dans le sens de plasmon de surface celle de la sphère. Par ailleurs, résonance dans le cas de la théorie de vers modifie cela la valeur du de optique. Ces théories, Bruggeman, seuil percolation bien toujours très utilisées, qu'anciennes, particulier pour prédire l'absorption optique des sont en présenté ici doit donc impérativement être pris en compte. composites. L'effet Les
Résumé.
cadre
optiques
accepted
1993,
October
4
2D
shown
m
this
medium
The
origm
systems,
or
ai.
et
trie 3D
In
dimensions
trois
anisotropic
of
effective
configuration. that
milieux
à
bath
mhomogeneous
theories of the
the
of
amsotropy
cases,
it
Ieads
to
surrounding amsotropic medium anisotropy and shifts the the
an
the
anisotropic a
the
Bruggeman, and nonspherical shape of (even if sphencal) on and
inclusions
mduces
within
studied
are
Gamett either
is
of the
distribution
media
Maxwell
effective
fictitious
medium.
In
deformation
of
wavelengths toward the sphere plasmon In the affects of the field theory of Bruggeman, also the it resonance. case mean percolation value. Although threshold of these theories qmte old, they are still some are now especially for the predictions of the absorption of the extensively used, media. composite Therefore the effect presented here for the first time should be taken mto account.
the
which
reduces
resonance
Introduction.
1.
Heterogeneous applications, (*)
Unité
constitutes
matter m
associée
which
au
granular CNRS
n
swce
films
781.
trie and
seventies
an
nanocerrnets,
important i
e.
field
inclusions
of of
basic
research
metallic
and
pa«icles
JOURNAL
304
(dielectric
growing selective
function
e~) of
nanometric
importance,
in
size
by
stimulated
PHYSIQUE
DE
dielectric
a
their
I
(dielectric
host
applications
numerous
2
N°
in
e~), take
function field
the
a
spectrally
of
coating.
electromagnetic prope«ies of inhomogeneous works materials have their on study of three dimensional (3D) inhomogeneous media the beginning of the at m (Maxwell Gamett [1, 2], Bruggeman [31. Ail XXth .). these thermes, extensively century studied, first consider spherical isotropic inclusions and lead to an isotropic effective medium characterised effective dielectric function (DF) e~. The derivations of the by a scalar Maxwell theories for ellipsoidal inclusions aligned along the same axis j have Gamett and Bruggeman for time [4-6] been developed long and the lime of the at two net most a came same as dimensional (2D) theories (Yamaguchi et ai. [7-9], Bedeaux and Vlieger [10-12]). Ail these depolarization factors along the main axes of the ellipsoid related to the approaches introduce interactions ellipsoid shape in the 3D case [13], to the shape and between neighbouring particles and with the substrate in the 2D case (one then speaks of effective de polarization Theoretical
origin
the
factors).
The
effective
fundamental
two
DF
distinct
anisotropic (and itself is structurally
be
can
film
from
deduced reasons
:
2
m
thermes
3
dimensions
depolarization
the
anisotropic,
is
these and
factor
if
even
is
~
tensor)
a
of
made
shape
the
of
the
dimensions
two
m
spherical
tensonal,
and
amsotropic
now
because
inclusions
for
inclusions
because
the
(Fig. I).
E~
'
Ell
a
c
, j
b j
3D
~
a)
~i
~
~/ r
Ejj
2D
b)
Fig.
In
l.
The
bath
effective
neglect immersed
ongms
cases,
DF the
in
~~, fact an
each each
that
of
anisotropy
the
theory one
the
anisotropic
allows
of
the
effective
determmation
the
independently from modelling cells unit medium.
It
is
the
the the atm
medium
of the
three In
others.
inclusion
of this
3D
m
paper
and
mean
to
values
words,
other
and
configuratiÔns.
2D
it's
present
of the these
environment
improved
tensorial
approaches are
now
effective
N°
ANISOTROPIC
2
theories
medium exact
revisit,
we
Gamett
in
section
immersed
in
effect on the for the anisotropic depolarization The tensor. depolarization is outlined in section 2. In section 3, tensor historical and still extensively used 3D theories of Maxwell the original 2D theory of Yamaguchi. main The results are
accourt
the
and
4.
depolarization
the
tensor.
medium
effective
effective
(EMT) it is
theories
inhomogeneous
the
that
so
305
THEORIES
effective
context,
classical
most
interact
to
Bruggeman,
effective
The
In
order of the
this
in
and
summansed
2.
in
formulation
MEDIUM
EFFECTIVE
media
[14,
medium
that
assumed
modelled
be
can
particles
the
single
as
coated
do
not
inclusions
15] (Fig. 2).
Ce Ea A~
Eb Ab
Fig.
A
2.
m
unit
shape,
same
find
we
between
relation
The
and
e~
deterrnining the (and generally static) electric theories
field.
if
If
both
Materai
matenals
a
a
directly
are
immersed
is
and
b
have
immersed
the
in
the
functions
One
of
the
constituents
inclusion
equations
obtains
then
of
coated
the
of
the
e~
and
submitted
when
following
e~
is
to
a
for
forrn
then local
3D
:
Eb
~e +
p is the curer
(2)
reduce
to
the
2D
the
~b)
~~
Se)
Aa (Sa
+
p
=
classical
theory
of
of
Maxwell
"
we
p)
+
Eb ~
Î
~
~~
~~
0
~e)
depolarization
the
(2)
BR,
=
factors
of the
that equations to be identical, so and Bruggeman. Gamett
~
FÎ
E~(F ~
+
p)
Eb(Î
+
~
~
+
~~~
are
,
inner
(1) and
have
f~(Fjj l
and A~
assumed
(1)
MG
~b)
Ab (Eh
Se +
expressions Yamaguchi,
Se
~~
p)
(l
E~
+'Îa~~a
Eb
volume filling factor, A~ respectively, generally
menai
ellipsoids,
and
e~
E~
2~b(Ee
+
~~ p
For
theories.
medium.
Gamett,
Maxwell
dielectric
the
E~
where
the
theory.
polarizability
by
obtained
of
static
the quasi effective
in
in
expression
Bruggeman
the
grues
medium
immersed
is
classical
the
it
effective
the
inclusion
coated
medium,
effective
modellmg
cell
b, the
material
+
~~
Î~(Î
p)
Fjj
Î ~
E~(Î
~3~
~
~~~
p)
F ~
JOURNAL
306
PHYSIQUE
DE
I
N° 2
respectively the non degenerated and the degenerated values of the that substrate the effect of the F(Ajj, A~ ) taking into effective account tensor of depolanzation Up ail detailed the will below. be tensor components to any now, (A~, A~, Ajj and Ai depend only on the shape of the ellipsoid. For a general ellipsoid (axis a, b, given by [16, 17] : factors c), the depolarization are Fi
where
F,j are depolarization and
Î~
~"
((q
f(q)
with
~ A~
~ =
Arctg
(e
e
e~
for
and
~
2
degenerate
the
c~))
well
integral
The
axis
)
the
When
c,
be
cannot
e
forrn,
closed
degenerates
into
a
(prolate)
=
,
~~~
fi
with
in
evaluated
general ellipsoid
fi e
~Î
~' ~~
have
we
with
e
~'
respectively
(oblate)
=
values
variations
known
~'~
[18, 19].
Aa, The
~"
f(q
+
rotation
with
~
~~j) (Log
=
+
available
are
spheroid
oblate
or
(q
b~)(q
+
tabulations
extensive
but
profane
a~)(q
+
=
Î~)
~Î~
of A
with
b
the
Ab
A~ )/2
i ~
ratio
axis
=
(7)
are
shown
figure
m
3.
Aa=(1-Ac)/2
A~
3
Fig.
3.
Geometrical
These
expressions
coordinate
system
problem
treated
inclusions
purely has
a
in
3D
academic
direct
factors
here)
6
is
spheroid.
by solving
assumption that the isotropic. In fact, this
medium,
on
and :
the
the
shape position,
the
9
7
cla
ratio
above
of the width
Laplace equation
the
the
problem
influence
a
deterrnined
are
with
for
4 axis
externat is
never
calculation
elhpsoid, and
the
(the
medium
the
except
case,
requires and then
amplitude
some
the
of the
0 in
ha =
effective for
an
elhpsoidal
medium
sphencal
discussion.
This
m
the
isotropic is
not
a
depolanzation factor value, surface plasmon modes m
N°
EFFECTIVE
ANISOTROPIC
2
metallic
the
percolation equal to A.
Moreover,
inclusions. concentration It is
then
in
which
p~,
the
307
Bruggeman,
of
anisotropy
of the
exactly depolarization
the
on
critical
the
transition,
non-metal
metal
influence
the
theory
field
mean
deterrnines
evaluate
to
necessary
the
THEORIES
MEDIUM
is
factor. Ail The
previous problem is
tensorial
DF
where
=V.D=
p
the
where
here
=
system
to
a
new
equation
holds
potential
far
m
from
that,
if
is
system
except
define
we
the
field
extemal
the
system,
that
e~
the
is of
axes
coordinate
~~~~~
consider
anisotropic direction
~'
~~~'
* ~
2
~
j~ o
the holds
We
of
an
tensorial
DF
+
a'2)3/2 (q 2
(s~ )~
+
immersed
z'
xK
be,
~~~~ ,
notice
that
in
the
coordinate
new
>
~~YY~~~' -1/2
CEzz
> =
changes to
which
In
the
becomes
into
another
inclusion
of
an
ellipsoid isotropic DF
extemal
electric
coordinate
new
system
with
different
s~ field is
immersed
applied
axes.
in
in
above,
defined
=
an
the
the
:
+
c'~)i'~
=
°~
~~ (q(ei
)xx +
a2)3'2 (q(si )~
+
b2)12 (q(ei)~~
(i1) +
c
~
~~~
expressions for the other mean values A~ and A~. For inclusions in an amsotropic depolarization depends both on ils shape and on the anisotropy of the outer tensor b'~=a'~, The integral expressions have analytic solutions (II) when ie.. a~/(s~)~. Except for unprobable specific cases, this occurs for spheroids (b a) in a uniaxial anisotropic (s~)~). Whatever medium the inclusion shape ((s~)~~ the factor Ai can be complex effective depolanzation under of the wntten one =
=
may
(s~~)-1/2
z
=
,
the
medium.
b~/(s~ )~~
~'~
l/2
~-
ellipsoidal s~,
:
:
~
~
b'2)i'2 (q
o
similar
(9)
already
can
into
principal axes (a). and equation (5)
abc
medium,
(8)
~~ (q
=
and
~P,
ôz
z'
by
Y ~~YY~~
"
apparently
problem
with
of
one
equation ~'~'~'
~
inclusion the
dielectric of
Laplace
the
=
system
homogeneous. ellipsoid change
also
the
ellipsoidal
now
D and
,
C
the
validity of displacement
the
ôy
ôx
Eo s/)~
~~
that
electric
region. by a Laplace's
free
defined
axes
=
~
We
involve
the
charge
medium
e~~
=
=
> so
not
a
~2 ~2 ~2 (s~~+s~~~+e~.~j
=-
such
if
new
~' the
to
s-V-Va
~Po so
does
related
p is
m
along the principal axes of the tensor e. are (isotropic medium), p 0 does involve not s~~ obtained by changmg from a cartesian coordinate solution be a can of e, such coordinate that the Laplace system parallel to the principal axes extemal field along the z axis, the the outer medium [2 Il. For a homogeneous the inclusion boundary is given by :
coordinate see
0.
ha
(s.V~P)=-
an
Laplace equation anisotropic extemal
charges
electric
charge density
-V.
that, Nevertheless,
can
of
absence
volume
the
on
more
the
The
the validity of complicated for
based
are
somewhat
s
equation [20]. potential ~P by
We
results
308
JOURNAL
equivalent
forrns
=
~*
2 e*
log
~
i
~*
~
l
e
2
e* =
be
can
values
s~
s
)~~
e*
or
c~
(e
e*~
~ (s~
s~ )~~ b~
(
~*
=
( s~
)~~
Arctg
*
e *
) (obl. )
this
(12)
b~
(13) c~
)~,~
the
flattens
inclusion
and
theories,
~
or
of the complex eccentricity e* that the amsotropy expression (1.e. the depolarization factor) along the direction increases elongates it along the other. If we effective back the to tum now frequencies toward the sphere will lead to a shift of the plasmon mode from
seen
ellipsoidal
the
i
(pro. )
e *
~
j(
with
It
2
N°
I
:
1_
Ai
PHYSIQUE
DE
of
high
medium resonance
frequency. 3.
effective
The
medium
revisited.
theories
aligned along the direction ellipsoidal inclusions in a 3D system, same or ellipsoidal partiales with the rotational axis perpendicular to the films, deposited substrate (2D system), the effective dielectric inclusion have, by and the ellipsoidal tensor on a formalism applies. The unit cells parallel and the above which allow the nature, axes immersed in an anisotropic determination of s~ are medium of DF s~ which modifies their now eccentricity and thus the effective value s~ itself : in the anisotropic all the EMT apparent case, become self-consistent. We will point ouf this anisotropic effect with in the frame work of now the commonly used 3D theories (the Maxwell Gamett self-consistent theory and the most theory of Bruggeman) and 2D theory of Yamaguchi. the These theories will be briefly
If
consider
we
spherical
and
outlined. 3,1
THE
MAXWELL
GENERALISED
expression
MGT,
the
m
(1) s~
l
e~.
GARNETT
becomes
a
If
THEORY.
self
tensonal
l
s~.
+A*(e~
s~
l)
e~.
p
=
anisotropic
the
expression
s~.
of
(14)
s~.1)
eigenvalues of A*, the deterrnined where by equations (12) and (13), depend implicit system of non-independent equations is numerically solved using a forced of e~ (fl and distinct rotational axis of procedure. The two the components 1to shown in figure 4, and compared to the classical prediction of the MGT. are inclusion
Taking s~ but,
as
sphere, example
into
3.2
axis
from
p~.
ratio FIELD
consider a long meaning
Each We
can
model
to
effect the
of the
two
inclusions
dielectric
sigmficant modification 0.3, (Au-AI~O~, p reduced by more than a
of
the
THEORY
the time of A
OF
BRUGGEMAN.
particular ago
that
=
factor
the
More
modification
distinguish
two
situations.
different value
We
whether of A
have
convergence the
ellipsoid)
(Matrix
and
host.)
the
of the the an
2.
signification of the static percolation
may be quite of the effective
metallic
a
This
e~.
not
toward
cla
=
m
on
deeply affect the absolute value plasmon mode frequency of shape of the inclusion. In apparent 0.5), the new position corresponds to
does
amsotropy
absorption peaks
here
we
physical
theory.
a
MEAN
if
demonstrated
the
shifts
presented
THE
the
account
expected, indicating
effective
theory,
permuted
be
can
effect
form
the
l
+A*(e~-
l
s~.
introduce
we
consistent
may
effects
drastic
depolanzation threshold the
expected
factor
A.
It
with
has
this been
p~ is exactly equal to A and concentration far p is close or
deeply
mentioned
are
that,
affect for
predictions of this spheroidal inclusions,
the
N°
EFFECTIVE
ANISOTROPIC
2
degenerated
the can
of A*
values
deterrnine
two
and
for
and B,
A
regions, anisotropic the In
factor
directions.
other
regimes
These
effective
the
the
in
medium
is
typical
are
similar
very
qualitatively equivalent. the example given in the previous section, equal to 0.527 along the axis of rotation
between metallic
so
that
we
(1.e.
p~jj direction
of
represented by regions C and D Bruggeman theory. In the other predicted by the MG theory and
is
of the those
to
p~~ in the
and
effect
0.3,
concentration
p metallic
rotation,
This
by equation (7)
one
concentration
Ajj), leading to an and dielectric example, Ajj, the A-p diagram of figure 5. on Ai
non-degenerated to the regions falls where p extremely anisotropic medium, related
are
309
THEORIES
MEDIUM
the
effective
medium
is
equal along the
axis
an
and
ratio
0.236 dielectric
m
=
along
other
the
two
axes
to
0.5 other
two
a
depolarization
axes,
parallel
direction
a
gives
so
to
the
that
for
axis
a
of
(region C).
Maxwell 13 p
=
theory
Gamett
cla
o.3
=
o.5
ii
~
Component 9
7
II
Component t
3
~
l
-1
-3
0.2
o.4
o.6
0.8
1.2
1.4
Wavelength
1.6
1.8
~m)
a)
Fig.
4.
modelled not
Real
by
affected
the bu
(a)
and
classical the
imagmary (---)
absorption
(b)
part
of
the
effective
generahsed (~) peaks are shifted toward
and
Maxwell the
dielectric
Garnett
plasmon
function of Au-AI~O~ cermet theory. The absolute values are mode frequency of the sphere.
JOURNAL
310
PHYSIQUE
DE
N°
I
2
16
Maxwell P
14
"
theory
Gamett
cla
0.3
0.5
=
£
Component
/
Î
Î
g
1 '
1
1
',
6
,~
1
II
Compcnent
J
1
'
'
"
4
' " "
' "
(eV)
Energy
Fig.
4b.
~
à
Region
menai-like
A:
'Z
( c
)
g l-
u
~~~~°~ ~~
1/2
1/3
II:
Dielectric
1:
Metal-like
D
Region
B:
-like ~
Fig. 5. Region
A-p
The A
configuration.
diagram
correspond to Regions C and
and B
Depolarisation
showing D
the
media
the are
four modelled
specific
different
by
of the
the
regions Maxwell
Bruggeman
factors
by
modelled
theory
approach.
in
the
the
Bruggeman metalhc
and
theory. dielectric
2
N°
ANISOTROPIC
The
equation
tensorial
for e~.
~
Figures
6
different
present
the
l
e~
e~ +
A](~~.
the
classical
characteristic
theory
BR
regions.
e~)
l
~~
is
e~ +
expected,
l
e~.
~'~
of
influence
the
e~
e~)
Bruggeman anisotropic
the
of
the
Bruggeman
3
~~~~
~'
Af(e~,
predictions
modified
and As
~
31
THEORIES
MEDIUM
EFFECTIVE
p=0.3
theory effect
m
is
two
quite
theory cla=0.S
Component
II
2
,,--,,,,[[~
~~~Î
modif.
~ia~~i~ai
,,
~~~~------
-----~~~~~
i
part
3
2
1
Wavelength
p m)
a)
Fig.
la,
6.
medium
is
b)
theory along (Fig. 6b).
The
dielectric-like
of the
Bruggeman axis
of
in
rotation
directions other two it implies a together with significant modification of Bruggeman m the dielectric configuration. plasmon frequency.
shift the We
anisotropic
configuration. The effective (Fig. fia), conducting along the of the percolation threshold value with the frequency dielectric function (Au-Al~03). (c) The Theory of find mode the around the sphere contraction agam the
of the
extreme
oblate
inclusions
312
JOURNAL
Bruggeman
PHYSIQUE
DE
N°
I
,'
theory
~
ol~~n~~
2
'
~,"' ~~~~~~~~ '
/ /
/
j
c
'
/
~
'
~
,£
' '
~
G
' '
Q
'
z
'
+
'
g
'
'
Î
' '
~
t
' ,
Ul
/ ' ' '
,
' '
Classical
Modified
-1
Real
2
Wavelength
Fig.
~ m)
6b.
already mentioned, for a concentration 0.3 and an axis ratio cla 0.5, the p region C, the component perpendicular to the rotational metal-fike and axis c is dielectric fike along the axis of rotation. The anisotropy elongates the oblate ellipsoid, reduces depolarization factor m the direction of c (Ai now the fluctuates between 0.4 and 0.44, these values correspond to an effective ratio cla A* in the other directions 0.8) an increases two (0.28 ~Af~ 0.29). The ellipsoid tends to a sphere, the of A*, values and the mean so percolation threshold in this theory, become closer to p, so that the conductivity of the metallic decreases and the polarizability of the dielectric increases (Figs. 6a, b). In this components one connection, it is worth noting that the percolation threshold depends frequency. Far now on the A* from region, decreases with involving gradual increase of the the resonance a w, polarizability m the infrared region. As a this modified theory deterrnme cannot consequence, critical for the conductivity t and the polarizability s of anisotropic exponents more inhomogeneous media, than when the anisotropic effect is neglected, which leads to the static sensitive.
As
=
medium
lies in
m
~
=
ANISOTROPIC
N° 2
EFFECTIVE
MEDIUM
313
THEORIES
' " '
/
c
,£
/
/
c
'
/
OE
/
u
'Z
/
/
«
3
Î
p
classical~/ /
.~
j
~~~~
cla
0.1
=
0.S
=
À
component
/'
~~~~,
theory
Bruggeman
,
/
modif.
Î
'
'
,,-,,
,
j
,
,,
,
/
'
'
',
/
',
/ /
' "
1 1
',
j i
/
/ l
Imaginary
1
part
,, -_,,
clos.
,,
',,
~~~jf
j
,, '-
2
1
~m)
Wavelength
Fig.
6c.
mean was
field
exponents
shown
follow
to
s
t
=
[22].
=
a
(The
function
Drude
plasma frequency w~~. In the Bruggeman dependent and independent terras, whose laws
:
P =
~
concentration
~
and
cla
no
w)~
and
w
=
w~
d
the
present but
=
p
=
one
with
solutions s
situation
same
O.l).
an
equation
=
t
=
frequency ~p-p~)/p~ with A depends as on dependent.)
longer suitable wavelength are
0.5
*'
p
The
dielectric effective
plasma
the
by p*
defined
procedure is decomposition Figures 6c, ratio
*~
e~ p
inclusions
dielectric
effective
with
effective
as
that
can
lead
1,
p~
=A
w
descnbed
medium
(region B). We observe the shift of the absorption peaks modification sphere corresponding to the apparent JO~R~AL
DP
PHYSIQUE
T
4,
N' 2
FEBRUARY
1994
is
the
with
terras
the
dielectric-hke toward
of the
wavelength scaling
of
the
function
p* in
is
m
of the reduced
the
theory.
the
theory
MG
effective
an
the
Bruggeman
the
all
and
dielectric ones.
region
infrared
the
deterrnination
metallic m
and
m
e~ is the
the
=
of the
the
to
where
of
function
polanzability P~ distinguish then
every mode
This
Bruggeman (same
axis
direction
the plasmon frequency shape of the spheroid, but m >2
314
DE
JOURNAL
PHYSIQUE
I
N° 2
Bruggeman p=0.1
theory cla=0.S
Component
II
modified
/ part
Real
~
~
classical Ù-S
modified
imaginary
part
i
~m)
Wavelength
Fig.
6c'.
contrast
3.3
THE
classical
namely
of
which
THEORY
theories
theory
takes
As
a
the
inclusion,
on
of
the
has
Yamaguchi et ai. given inclusion,
are
the
actual
~~ =
is
~p
deeply affected, p~)/p~ (with p~
due
the
to
apparent
A*). =
effects have to be expected from the principle the theory of Maxwell Gamett, same as on ai. and the theory of Bedeaux and Vlieger. We will focus which be analytically solved whereas the second can one, morphology of the film, needs an image and does not No
based
Yamaguchi et first approach
account
general it
YAMAGUCHI.
of
p*
concentration
which
into
Lorentz
OF
value
absolute
the
reduced
2D the
present
the
2D
attention
our
MGT,
the
to
modification
spectacular
the
formulation.
approach is no longer suitable in 2D be exactly calculated for a given
to
to
account
the
local
for
field
the
is
substrate then
the
effect
deterrnine distribution
by using
superposition
of
the the
the of
local
mirror
appfied
polansing
field
spheroids image field,
on
a
expedient. the
the
substrate.
field
For of
a
the
ANISOTROPIC
N° 2
image-dipole inclusions, forrn
and
of
of
field
and
presents
image couples. The dipoles (interspacing d). Even appreciable anisotropy considering of the image dipole (Fig. 7).
an
orientations
the
array
part)
(Real
assumed
are
spherical particular
for the
(pm)
Wavelength
2.5
1.5
1
0.5
315
and
square
a
THEORIES
MEDIUM
dipole
other
the
nodes
medium
effective
dipolar
of ail
the
onio
the
the
field
the
distributed
be
to
EFFECTIVE
Îd
~
~
à ~
~
é
Î
É
)
1 II
~
~
ÎÙ
t
M%
à O
C
£
~
~
Il
O
~ ~
O
'Q
~ ôQ
) (
# ~
$ ~i
5
o
~
©
~
.Î
À
Î ~
~ o
3
1
(Imaginary part)
Fig.
7.
Prediction The
substrate.
When whereas
the
trie
field
they
(Fig. 8). second
As
of the
amsotropy
one.
of
arises
from
parallel
in the
are a
is
theory
to
the
substrate
anisotropy
ai.
et
spherical gold
for
inclusions
deposited
enta
a
glass
configuration.
2D
film,
trie
direction
same
consequence, At least, trie
Yamaguchi the
(eV)
Energy
image-dipole are in opposite and applied field is perpendicular to the
dipole
when
the
effect may
is be
aise
in
the
first
induced
by
trie
shape
case
columnar
where
the
rotational
[24],
axis
of the
of
the
is
parallel
case
and
of trie
substrate
increased
in
inclusions.
In
[23] or with oblique depolarization is tensor structures components none from equation (11). If this deterrnined degenerated and they have to be numerically axis is (parallel to the film) degenerate, the tensor is perpendicular to the substrate, two mean values (6) et (7) and the anisotropic effect can be studied in a determined by the analytical expressions 3D theories (Fig. lb). similar the above way as
the
spheroid
minimized
directions
of
the
to
the
substrate
effective
DE
JOURNAL
316
PHYSIQUE
I
N° 2
1-
~
E
~
lÉ
~ P
l'
,-
,,
~
jt
1 '
Fig.
8.
The
The
influence
of
image-dipole
the
depolarization
effective
'j
pf
tensor,
different
is
for
the
by
determined
as
,
"-----~
directions
two
Yamaguchi
et
of
ai,
applied
the
has
field.
following
the
forrn
Fjj
~
~
=Ajj24
F =
~
A~ are perpendicular
the
and
Ajj
ratio
and
rotational
Î
h
terra
Î is
where
p~ ~
~/
For
Î.
result
of trie
with
inclusions
second
term
=
is trie
2
p~
0.716
F~+ Fa
Fs+
depolarization of the geometric (Eqs. (6) and (7)), d~ is the mass the distance between the dipole
adsorbate The
in
(16)
summation
over
Fa 2 d
F~ +
+
24~/
=
axis.
and
=
third
1/h,
7~
substrate
the
to
values
mean
y2
~~,
-0.716
p~ + p~
2
A~
~~~
~~
e
~~~~
d~ ~
d-.
Fa tensor
in
thickness
directions of the
parallel
film,
y the
and axis
image-dipole and h trie trie rotational axis perpendicular to trie substrate, trie dipole-image contribution and the represents image-dipole). This third ail trie couples (dipole, and
trie
preponderant in the perpendicular It can case. by modification of trie shape two terras apparent an seen are will neglect the of trie second This of the inclusion. In the following, variations terra. we approximation is valid for oblate spheroids (cla < 1). In finis case trie second terra, representing affected effective factor (only 5 fé in trie parallel case), is net greatly less than 10 ni of trie total modification of A (see Fig. 3) and we avoid trie questionable inversion of equations (6). by a substituting A* in equations (16) are shown in figure 9. Trie modifications introduced when Only trie parallel component is appreciably affected by trie amsotropic effect, considenng trie contributes
terra
be
that
relative
4.
only
amount
less
than
trie
first
of A
in
10 fb
to
Fjj
but
of F
the
effective
can
be
affected
depolarization
factor
F.
Conclusion.
initially developed for sphencal inclusions, historical effective medium theories, have extended ellipsoidal shapes. Except for the case of randomly oriented subsequently to deterrnined is anisotropic and the depolarization factors elhpsoids, the effective medium thus immersed in anisotropic medium, bath ellipsoidal inclusions, depend of the of the an now effective dielectric As ail the shape of the inclusion and of the tensor a consequence, e~. be numerically solved. This been in the theories self and has done become bave to consistent and trie effective of the spheroids particular cases where the dielectnc tensor present the axes analysis 2D is 3D but reduces symmetries. This always the systems to case in our same of inclusions perpendicular substrate. with the rotational axis the the The to systems main eccentricity of the spheroid and thus, the the result is that the anisotropy tends to minimize Most
been
EFFECTIVE
ANISOTROPIC
N° 2
MEDIUM
THEORIES
317
Yamaguchi ÎÎ Component
II (ta
the
S (TO
rot.
Component -L (to
the
subs.)
(ta
rot.
axis)
axis)
II
15
Î~ '
,
j
j
'
', ,, ,,,
---___
10
5
5
' ' '
' "
',, ,,
'-'----
Wavelength
Fig.
9.
anisotropic
The
absolute
the
anisotropy
value
of p~ is
effect
0.5
3
~
-5
(p
in the
pm)
Wavelength
m)
theory
1.5
1
of
Yamaguchi
et
ai.
(Au-granular,
p
=
This affects absolute of s~, depending less the values more or positions of the absorption peaks toward the interrnediate position This anisotropic effect has been experimentally in the 2D observed resonance. and interpreted as a multiple-image effect introduced in the theory of Yamaguchi shifts
and
(this
itself.
trie
effect
leads
to
an
1). Only
0.3, cla =
affected.
enhancement
of
the
anisotropy
of
the
effective
theory, sphere configuration et a/. [25, 26] the
on
of trie
medium).
predicted from trie self field theory of Bruggeman, consistent are means factor established depolanzation this theory. It is weII that of the due to the particular sense in This strictly in this theory. result has been percolation threshold equal to A the static p~ is polarisation follow optical conductivity and for optical frequencies, where demonstrated p~)", A. We form cntical equal of lp with laws in the exponents to unity and p~ power 1/3). For this inclusion (A have shown that result only valid for sphencal here p~ is which is a function charactenzed by p~ /p~ ellipsoidal shapes, A* depends on the anisotropy ~ threshold depends on the frequency. of the frequency and the percolation now have to be solved effective medium thermes When apphed to media, the simple anisotropic restriction concentrations and long classical low numencally. In addition their to to formulations limited (quasi-static approximation), the classical wavelengths to quasiare sphencal inclusions. More
drastic
effects
=
=
i
=
JOURNAL
318
DE
PHYSIQUE
I
N° 2
References
[17] [18] [19] [20]
C., Philos. Trans. R. Soc. Land. 203 (1904) 385. C., Philos. Trans. R. Soc. Land. B 205 (1904) 237. Bruggeman D. A. G., Ann. Phys. (Leip2 ) 24 (1935) 636. Cohen R. W., Cody G. D., Coutts M. D., Abeles B., Phys. Rev. B 8 (1973) 3689. Bilboul R. R., J. Appl. Phys. (J. Phys. D) 2 (1969) 921. Hunded O., Phys. Rev. B t6 (1977) 1353 and 3513. Granqvist C. G., Yamaguchi T., Yoshida S., Kimbara A., J. Opt. Soc. Am. 6t (1972) 634. A., Thin Sol. Films t3 (1972) 261. Yamaguchi T., Yoshida S., Kimbara Yamaguchi T., Yoshida S., Kimbara A., J. Opt. Soc. Am. 64 (1974) 1563. Bedeaux D., Vlieger J., Physica 67 (1973) 55. Fiaiice D., Vlieger J., J. Phys. 73 (1974j 287. Bedeaux D., Vlieger J., Physica 82A (1976) 221. Bedeaux Jackson J. D., Classical electrodynamics (John Willey & Sons, New York, 1975). Niklasson G. A., Thesis, Chalmers University, Gôteborg (1982). W., Word Ashcroft N. W., Phys. Rev. B Lamb D. M., 21(1978) 2248. Huffman Bohren C. F., D. R., m Absorption and scattenng of light by small particles (John Willey & Sons, New York, 1983) p, 141. Berthier S., Milieux Composites : Optique (Polytechnica, 1994). Osbom J. A., Phys. Rev. B 67 (1945) 351. Stomer E-C-, Philos Mag. 36 (1945) 803. C. J. F., Bordewijk P., Theory of Bôttcher Electnc Polarization (Elsevier, Amsterdam, 1978)
[21] [22] [23] [24] [25] [26]
Physica 75 (1974) 146. Driss-Khodja K., Lafait J., J. Phys. France 48 (1987) 601. Baba K., Miyagi M., J. Opt. Sac. Am. A 8 (1991) 619. Smith G. B., Opt. Commun. 7t (1989) 279. Truong V. V., Bosi G., Yamaguchi T., J. Opt. Soc. Am. A 5 (1988) 1379. Niklasson Graighead H. G., Thin Solid Films 125 (185) 165. G. A.,
[1] [2] [3] [4] [5] [6] [7] [8] [9] [loi il Ii [12] [13]
[14] [15] [16]
Maxwell
Garnett
J.
Maxwell
Garnett
J.
p.
427.
Bordewijk Berthier
S.,
P.,