A STATISTICAL THEORY FOR THE DIELECTRIC PROPERTIES OF THIN ISLAND FILMS, APPLICATION AND COMPARISON WITH EXPERIMENTAL RESULTS

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A STATISTICAL THEORY FOR THE DIELECTRIC PROPERTIES OF THIN ISLAND FILMS,

APPLICATION AND COMPARISON WITH EXPERIMENTAL RESULTS

D. Bedeaux, J. Vlieger

To cite this version:

D. Bedeaux, J. Vlieger. A STATISTICAL THEORY FOR THE DIELECTRIC PROPERTIES OF THIN ISLAND FILMS, APPLICATION AND COMPARISON WITH EXPERIMENTAL RESULTS.

Journal de Physique Colloques, 1983, 44 (C10), pp.C10-379-C10-382. �10.1051/jphyscol:19831077�.

�jpa-00223535�

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J O U R N A L D E PHYSIQUE

Colloque CIO, supplkrnent a u n012, Tome 44, dkcernbre 1983 page Cl0-379

A STATISTICAL THEORY FOR THE DIELECTRIC PROPERTIES OF THIN ISLAND FILMS, APPLICATION AND COMPARISON WITH EXPERIMENTAL RESULTS

D. Bedeaux and J . v l i e g e r r

I n s t i t u t t f o r T e o r e t i s k Fysikk, U n i v e r s i t e t e t i Trondheim, Norges l'ekniske Hbgskole, N 7034 Trondheim-NTH, Norway

r ~ n s t i t u u t - ~ o r e n t z voor T h e o r e t i s c h e Natuurkunde, R i j k s u n i v e r s i t e i t t e Leiden,Nieuwsteeg 18, 2 3 1 1 SB Leiden, The Netherlands

Rdsurn6

-

A p a r t i r de photographies de microscopic Q l e c t r o n i q u e de f i l m s minces e t d i s c o n t i n u s d ' o r , Q t u d i e s expgrimentalement p a r Norman e t c o l l a - b o r a t e u r s , on c a l c u l e l a t r a n s m i t t i v i t c B l ' a i d e d ' u n e t h s o r i e s t a t i s t i q u e pr6sent6e pr6cZdement. Les r 6 s u l t a t s s o n t en a s s e z bon accord avec l e s v a l e u r s expgrimentales.

A b s t r a c t

-

Using t h e e l e c t r o n micrographs of t h i n d i s c o n t i n u o u s g o l d f i l m s , s t u d i e d e x p e r i m e n t a l l y by Norrman e t a l . , we c a l c u l a t e d t h e t r a n s m i t t a n c e on the b a s i s of a s t a t i s t i c a l theory given i n a p r e v i o u s paper. The r e s u l t s agree r a t h e r w e l l w i t h t h e experimental v a l u e s .

The s t a t i s t i c a l theory1) f o r t h e d i e l e c t r i c s u s c e p t i b i l i t i e s of a t h i n i s l a n d f i l m i s used t o c a l c u l a t e t h e t r a n s m i t t a n c e of normally i n c i d e n t l i g h t through gold f i l m s on a g l a s s s u b s t r a t e 2 ) . Taking l o c a l f i e l d s i n t o account, t h e d i e l e c t r i c s u s c e p t i - b i l i t y p a r a l l e l t o t h e s u b s t r a t e i s found t o have t h e f o l l o w i n g g e n e r a l form

where cis i s t h e average p o l a r i z a b i l i t y of t h e i s l a n d s p a r a l l e l t o t h e s u r f a c e of t h e s u b s t r a t e p e r u n i t of s u r f a c e a r e a . Furthermore, ~ 1 , has t h e d i m e n s i o n a l i t y of an i n v e r s e l e n g t h and accounts f o r l o c a l f i e l d e f f e c t s p a r a l l e l t o t h e s u r f a c e . An e x p l i c i t e x p r e s s i o n f o r lcll w i l l be given below. The t r a n s m i t t a n c e of normally in- c i d e n t l i g h t , d i v i d e d by t h e F r e s n e l v a l u e , i s given i n terms of y by3)

where A i s t h e wavelength of t h e i n c i d e n t l i g h t , c t h e v e l o c i t y of li n t h e r e f r a c t i v e index of t h e s u b s t r a t e . For t h e f i l m under c o n s i d e r a t i o n g y t t ~ d i >

lands a r e i n r e a s o n a b l e approximation p r o l a t e s p h e r o i d s w i t h t h e l o n g a x i s p a r a l l e l t o t h e surf ace of t h e s u b s t r a t e . The p o l a r i z a b i l i t y of such a s p h e r o i d i s given by4)

where V and e a r e t h e volume and t h e e c c e n t r i c i t y of t h e spheroid. Furthermore, E i s t h e d i e l e c t r i c c o n s t a n t of t h e i s l a n d m a t e r i a l ( g o l d i n t h i s case) and

sm

i s t h e , a s y e t u n s p e c i f i e d , d i e l e c t r i c c o n s t a n t of a medium i n which t h e i s l a n d

i s thought t o be embedded, which s a t i s f i e s 1< ; E~ i s t h e d i e l e c t r i c son- s t a n t of t h e s u b s t r a t e ( g l a s s i n t h i s c a s e ) . The d e p o l a r i z a t i o n f a c - t o r s i n t h e d i r e c t i o n of t h e l o n g , k = l , and t h e s h o r t , k=2, a x i s p a r a l l e l t o t h e sur- f a c e of t h e s u b s t r a t e a r e given by4)

L ( I ) ( e ) = M e ) - B ( e ) and L ( 2 ) ( e ) =

4 -

^{; A}( e ) - B ( e )

-

³ ^E^{- 1}

A ( e ) =

4

e (1-eZ) [ i n

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^-2.1 ^{and B(e)}

^=I

24 {cs+l

)

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831077

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Fig.2- The p o l a r i z a b i l i t y d e n s i t y of t h e f i l m s u s i n g E m = $ ( I + & )

Fig. 1- High-resolution electronmicrographs of r e g i o n s of t h e f i l m s

B i s due t o t h e i n t e r a c t i o n w i t h t h e image d i p o l e and we n o t e t h a t t h e spheroid touches t h e s u b s t r a t e s u r f a c e . For d i f f e r e n t i s l a n d s e . and Vi w i l l be d i f f e - r e n t . The average p o l a r i z a b i l i t y p a r a l l e l t o t h e surfac; i s given by

The weight t h i c k n e s s tw i s i n p r i n c i p l e e q u a l t o

F

^Vi p e r u n i t of s u r f a c e a r e a . I n p r a c t i c e one f i n d s a somewhat lower v a l u e 2 ) due t o so-called " l o s t weight".

We used t h e experimental v a l u e s t =1.5, 1.7, 2.6, 3.0 and 3.6 nm i n our c a l c u l a - t i o n . Using a computer a n a ~ ~ s i s ~ ) ~ of t h e e l e c t r o n micrographs, f i g . 1, we obtained t h e average p o l a r i z a b i l i t i e s . I n f i g . 2 t h e s e a r e given u s i n g E rn = L The d e s c r i p t i o n u s i n g t h e d i e l e c t r i c s u s c e p t i b i l i t y y f o r t h e f i l m g i v e s t h e same

t r a n s m i t t a n c e a s a p l a n e - p a r a l l e l p l a t e i f one u s e s

f o r t h e d i e l e c t r i c c o n s t a n t , f i l l i n g f r a c t i o n and t h i c k n e s s of t h e p l a t e . I f one would u s e s p h e r i c a l i s l a n d s and n e g l e c t s image d i p o l e s e q s . ( 2 ) - ( 6 ) g i v e t h e o r i g i n a l Maxwell G a r n e t t formula'). T h i s shows t h e f a c t t h a t t h e o p t i c a l t h i c k n e s s i n t h e u s u a l d e s c r i p t i o n may be c a l c u l a t e d i n terms of t h e l o c a l f i e l d f a c t o r K I ~

.

^This

f a c t o r , and t h u s t h e o p t i c a l t h i c k n e s s , i s given i n our theory i n terms of t h e cor- r e l a t i o n f u n c t i o n gw(r) f o r t h e d i s t r i b u t i o n of i s l a n d mass along t h e s u r f a c e of t h e s u b s t r a t e by

Here d i s t h e average d i s t a n c e of t h e c e n t r e of t h e i s l a n d s t o t h e s u b s t r a t e . The

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D

w

0 Y 0

Fig.3- The mass d i s t r i b u t i o n c o r r e l a - t i o n f u n c t i o n s of t h e f i l m s

Fig.4- The t r a n s m i t t a n c e a s a f u n c t i o n of t h e wavelength f o r t h e f i l m s . The continuous curves g i v e t h e experimental v a l u e s . The d o t s g i v e t h e computed v a l u e s .

0 2 C 5 :o ^{1 5}

second term betweenthe square b r a c k e t s i s due t o t h e image d i p o l e s . I n f i g . 3 t h e c o r r e l a t i o n f u n c t i o n s , o b t a i n e d from t h e electronmicrographs

,

a r e given f o r t h e va- r i o u s f i l m s . The r e s u l t i n g t r a n s m i t t a n c e s a r e compared w i t h t h e experimental v a l u e s i n f i g . 4 , where we used ^E =

.

^{I n}o r d e r t o g a i n some i n s i g h t i n t o how t h e v a r i o u s parameters l i k e t h z e c c e n t r i c i t y , &,

,

t h e o p t i c a l t h i c k n e s s and t h e image charge a f f e c t t h e computed t r a n s m i t t a n c e T

,

we have v a r i e d a l l t h e s e q u a n t i t i e s . F i g . 5 shows t h e v a l u e s of T f o r 3 d i f f e r e n t choices of E, and i t i s apparent t h a t both t h e l o c a t i o n of the minimum and t h e amount of a b s o r p t i o n depend s t r o n g l y on E

.

For a l l f i l m s we f i n d t h a t & =

5

( l + & ) i s t h e b e s t choice. A t h e o r e t i c a l moti- m v a t i o n f o r t h i s choice may alEo be giGen'b) based on t h e o b s e r v a t i o n t h a t E

should approach E i n t h e f = l l i m i t . Fig.6 shows t h a t accounting f o r t h e e f f image d i p o l e s i n c r e a s e s t h e width of t h e minimum i n T

.

This improves t h e agree- ment w i t h t h e experimental shape. I n f i g . 7 t h e v a l u e of T i s compared w i t h a va- l u e obtained by s t r e t c h i n g a l l t h e i s l a n d s . This shows t h a t t h e l o c a t i o n of t h e m i - nimum i s a s e n s i t i v e f u n c t i o n of t h e shape of t h e i s l a n d s . C l e a r l y t h e v a l u e s of the e c c e n t r i c i t i e s found from t h e electronmicrographs and used i n t h e c a l c u l a t i o n l e a d t o a s a t i s f a c t o r y v a l u e of t h i s l o c a t i o n . I n f i g . 8 t h e v a l u e of T i s compared t o a v a l u e obtained by d e c r e a s i n g t

.

^{I t}i s i n t e r e s t i n g t o n o t e t h a t t h e d e c r e a s e of t 'yas an e f f e c t comparablPP~o an i n c r e a s e of &

.

This e x p l a i n s why Norrman e t al'lPt were a b l e t o f i t t h e i r d a t a u s i n g E =1.2, m r a t h e r than t h e v a l u e E =

$ ( 1+& ) = 1.67 we used, u s i n g topt a s an a d j u s v a b l e parameter. I t should be e z phasiged however, t h a t t h e s t a t i s t ~ c a l a n a l y s i s l e a d s t o a unique v a l u e of t SO

t h a t we o b t a i n our v a l u e s of T w i t h o u t f i t t i n g any parameter. OP t REFERENCES

/ I / V l i e g e r , J. and Bedeaux, D . , Thin S o l i d Films

69

(1980) 107 Bedeaux, D. and V l i e g e r , J . , Thin S o l i d Films,

102

(1983) 265

/ 2 / Norrman, S . , Andersson, T . , Granqvist, C.G. and Hunderi, O . , P h y s . R e v . U (1978)

131

Bedeaux, D. and V l i e g e r , J . , P h y s i c a x (1973) 55

1674

/ 4 / Yamaguchi, T . , Yoshida,S and Kinbara, A , Thin S o l i d Films

2

(1974) 173

/ 5 / Kwee, R.S.H., Kandidaatsverslag (T.H.Delft, t h e Netherlands, unpublished)

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F i g . 5 - T h e computed t r a n s m i t t a n c e f o r

E m = I

,

j ( l + ~ ~ ) and E~ f o r two f i l m s

F i g .6

-

The corriputed t r a n s m i t t a n c e in- c l u d i n g and n e g l e c t i n g image d i p o l e s f o r two f i l m s

Fig.7-The computed t r a n s m i t t a n c e f o r two Fig.8-The computed t r a n s m i t t a n c e f o r v a l u e s of t h e average e c c e n t r i c i t y f o r two v a l u e s of t h e o p t i c a l t h i c k n e s s

two f i l m s f o r two f i l m s