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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



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



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

~



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



~' ~~

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



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



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



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



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



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



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

~

©

~



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

~



~ 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.,