ELLIPSOMETRIC FORMULAS FOR AN INDEX PROFILE OF SMALL AMPLITUDE BUT ARBITRARY SHAPE

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ELLIPSOMETRIC FORMULAS FOR AN INDEX PROFILE OF SMALL AMPLITUDE BUT

ARBITRARY SHAPE

J. Charmet, P.-G. de Gennes

To cite this version:

J. Charmet, P.-G. de Gennes. ELLIPSOMETRIC FORMULAS FOR AN INDEX PROFILE OF

SMALL AMPLITUDE BUT ARBITRARY SHAPE. Journal de Physique Colloques, 1983, 44 (C10),

pp.C10-27-C10-29. �10.1051/jphyscol:19831004�. �jpa-00223453�

JOURNAL DE PHYSIQUE

Colloque CIO, supplbment a u n012, Tome 44, d k e m b r e 1983 page C10-27

ELLIPSOMETRIC FORMULAS FOR AN INDEX PROFILE OF SMALL AMPLITUDE BUT ARBITRARY SHAPE

J.C. Charmet and P.G. d e Gennes

EcoZe SupGrieure de Physique e t de Chimie IndustrrieZZes de Zu viZZe de Paris, (E.S.P.C.I.),

rue Vauquelin,

Paris Cedex 05, France

Resume - Le p l u s souvent l e s r e f l e c t a n c e s de couches non homogenes s o n t c a l - FdZG p a r r e s o l u t i o n numerique des e q u a t i o n s de flaxwel l . Pour c e r t a i n s

problPmes physiques c e t t e procedure ne permet pas b i e n de r e s o u d r e l e p r o b l e m e i n v e r s e , c ' e s t - 2 - d i r e de d e t e r m i n e r l e p r o f i l d ' i n d i c e n ( z ) a p a r t i r des mesures e l l i p s o m e t r i q u e s ( $ e t A ) . I c i nous c a l c u l o n s l e s r e f l e c t a n c e s e x p l i - c i t e m e n t pour n ' i m p o r t e q u e l l e forme de n ( z ) p a r une a p p r o x i m a t i o n de E r v a l a b l e a u p r e m i e r o r d r e en n ( z ) - no (013 n e s t 1 ' i n i c e u mi i e u ' i n c i - dence). Par c o n t r e , l l e f f e t de l a p a r o i r e f ~ d c h i s s a n t d e (endz = b) ^es: i n c o r - p o r e dans l e probleme non p e r t u r b @ .

e t

s o n t a i n s i exnrimes en f o n c t i o n de l a t r a n s f o r m e e de F o u r i e r complexe r ( 2 q )

r ' + i r " du p r o f i l (oil q e s t l a composante normale du v e c t e u r d ' o n d e i n c i d e n t ) . Pour des couches enaisses ( e

A/4n) c e c i d o i t p e r m e t t r e une r e c o n s t r u c t i o n complete du p r o f i l . Pour des couches minces ( e

~ / 4 n ) on d e t e r m i n e seulement l e s p r e m i e r s moments du p r o f i l d ' i n d i c e . Pour i l l u s t r e r ces techniques, nous d i s c u t o n s deux exemples q u i f o n t i n t e r v e n i r un p r o f i l d ' i n d i c e l e n t e ~ n e n t d e c r o i s s a n t : ( i ) e f f e t s de p a r o i s u r un melange b i n a i r e c r i t i q u e ; ( i i ) a d s o r p t i o n de polymeres f l e x i b l e s

a p a r t i r d ' u n bon s o l v a n t .

A b s t r a c t - The r e f l e c t a n c e o f non homogeneous l a y e r s i s u s u a l l y c a l c u l a t e d by numerical s o l u t i o n o f t h e flaxwell e q u a t i o n s . T h i s r e q u i r e s a s p e c i f i c model f o r t h e l a y e r s t r u c t u r e . ble a r e i n t e r e s t e d h e r e i n t h e i n v e r s e problem : t o f i n d t h e r e f r a c t i o n i n d e x p r o f i l e n ( z ) f r o m t h e e l l ip s o n e t r i c d a t a

and

We have c a l c u l a t e d t h e r e f l e c t a n c e s e x p l i c i t l ~ i n a 1 s t Born a p p r o x i m a t i o n

( i . e . t o f i r s t o r d e r i n n ( z ) - no w h e r e n o i s t h e i n d e x o f t h e p u r e l i q u i d ) . The e f f e c t o f t h e r e f l e c t i n g w a l l a t z

0 i s i n c o r p o r a t e d e x a c t l y . F i n a l l y we express

and

i n terms o f t h e complex F o u r i e r t r a n s f o r m r ( 2 q )

r ' + i r "

o f t h e p r o f i l e (where q i s t h e normal component o f t h e i n c i d e n t wave v e c t o r ) . F o r t h i c k d i f f u s e l a y e r s ( e

A/4a) t h i s s h o u l d a l l o w f o r a complete recon- s t r u c t i o n o f t h e p r o f i l e . F o r t h i n l a y e r s ( e

A/4n) what i s r e a l l y measured i s t h e moments ro and rl ( o f o r d e r 0 and 1 ) o f t h e i n d e x p r o f i l e . To i l l u s - t r a t e t h e s e methods, we d i s c u s s two s p e c i f i c examples, which a r e a s s o c i a t e d w i t h a s l o w l y d e c r e a s i n g i n d e x p r o f i l e : ( i ) w a l l e f f e c t s i n c r i t i c a l b i n a r y m i x t u r e s ; ( i i ) polymer a d s o r p t i o n from a good s o l v e n t .

R e f l e c t a n c e measurements can g i v e us a r a t h e r d e t a i l e d i n f o r m a t i o n on t h e s t r u c t u r e o f d i f f u s e i n t e r f a c e s , even when t h e t h i c k n e s s e o f t h e i n t e r f a c e i s somewhat s m a l l ~ r t h a n t h e o p t i c a l wavelength

: we can probe c o n v e n i e n t l y v a l u e s o f e

X/4n 500 8.

F o r o b l i q u e i n c i d e n c e , one can d e f i n e two conplex r e f l e c t a n c e c o e f f i c i e n t s , Rs and R c o r r e s p o n d i n g t o t h e two p o l a r i s a t i o n s t a t e s "s" and "p".

P The q u a n t i t y o f e x p e r i -

mental i n t e r e s t i s t h e r a t i o : R /P

t a n 9 eiA. F o r s i m p l e s i t u a t i o n s , such as a

w a l l covered b y a s l a b o f t h i c k n e s s e and c o n s t a n t r e f r a c t i v e i n d e x nl, we have r e l a - t i v e l y s i m p l e f o r m u l a s f o r Rs and R /1,2/. There a r e a number of cases, however, where t h i s model i s n o t s a t i s f a c t o r y . Lie s h a l l q u o t e two examples o f c u r r e n t i n t e r e s t P a ) a c r i t i c a l b i n a r y l i q u i d m i x t u r e , n e a r a w a l l , shows a p r o f i l e o f a c o n c e n t r a t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831004

JOURNAL DE PHYSIQUE

(

( v e r s u s d i s t a n c e t o t h e w a l l , z ) o f t h e f o r m /3,4/ : @ ( z ) - pi..) = ( c o n ~ t . ) ( a / z ) ' / ~ where B and

a r e c r i t i c a l exponents

0.5 i n t h r e e dimensions) and a i s a mole- c u l a r s i z e ;

b ) an adsorbed l a y e r o f f l e x i b l e polymers ( i n a good .

s o l v e n t ) i s e x p e c t e d t o show a c o n c e n t r a t i o n p r o f i l e /5,6/ : $ ( z )

const.(a/z)"'. I n b o t h cases ( a ) and ( b ) , $ ( z ) i s a s l o w l y d e c r e a s i n g f u n c t i o n o f z, and i t i s n o t a t a l l p e r m i s s i b l e t o r e p l a c e t h e p r o f i l e by a s l a b o f f i n i t e t h i c k n e s s e and c o n s t a n t i n d e x nl : i f we d i d t h i s , t h e a p p a r e n t t h i c k n e s s e would become a f u n c t i o n o f t h e e x p e r i m e n t a l c o n d i t i o n s (wavelength, a n g l e o f i n c i d e n c e ) .

Another i m p o r t a n t f e a t u r e o f cases (a) and ( b ) i s t h a t t h e r e i s a c u t t o f f zmax t o t h e p r o f i l e . I n case ( a ) , zmax i s t h e c o r r e l a t i o n l e n g t h E and i s f i n i t e when we a r e n o t e x a c t l y a t t h e c r i t i c a l p o i n t . I n case ( b ) , zmax i s t h e c o i l s i z e / 6 / . One o f t h e q u e s t i o n s t o be c l a r i f i e d i s t h e i n f l u e n c e o f t h i s c u t o f f on

and

T h i s has l e d us t o search f o r a g e n e r a l f o r m u l a t i o n where

and

would be w r i t t e n e x p l i c i t 1 as f u n c t i o n a l s o f t h e p r o f i l e . I t i s n o t t o o h a r d t o r e a l i s e t h i s p r o g r a n

7 i - d w e n t e o c a l p e r t u r b a t i o n $ ( z ) - p(m) i s s m a l l . G!e d e r i v e t h e r e f l e c t a n c e a m p l i - tudes t o f i r s t o r d e r i n t h i s l o c a l p e r t u r b a t i o n , by a procedure r e m i n i s c e n t o f t h e Born a p p r o x i m a t i o n i'n quantum mechanics / 7 / . The problem i s n o t e n t i r e l y c l a s s i c a l , however, because we i n c o r p o r a t e i n t h e 0 o r d e r wave f u n c t i o n s t h e e f f e c t of t h e r e f l e c t i n g w a l l ( a t z = 0 ) .

A f t e r some c a l c u l a t i o n we can w r i t e t h e e l l i p s o m e t r i c r a t i o t o f i r s t o r d e r i n 6n as : Rp/% = t a n +eiA = r /r ( 1 + I A P ( - 2 9 )

i e i ( 2 q ) ) where :

P s

- rs and r a r e t h e r e f l e c t a n c e s f o r t h e w a l l i n d i r e c t c o n t a c t w i t h a h a l f space o f i n d e x n ( w i t h o u t d i f f u s e l a y e r ) ; P

- fi and B a r e two known f u n c t i o n s of q and o f t h e u n p e r t u r b e d r e f l e c t a n c e s

:

-1 -1 -1 2 -1

A(q)

a ( q ) ( r p - ^rs - b ( q ) rp a ( q )

no(w/c) q B(q) = a ( q ) ( r p - r s ) - b ( q ) rp b ( g )

2 n i l q

- and r ( k ) t h e F o u r i e r t r a n s f o r m o f t h e i n d e x p r o f i l e

Our d i s c u s s i o n o f what i s r e a l l y measured i n an e l l i p s o m e t r i c experiment i s r e s t r i c t e d t o d i f f u s e l a y e r s where t h e r e f r a c t i v e i n d e x i s n o t t o o d i f f e r e n t f r o m t h e b u l k v a l u e . B u t w i t h i n t h e s e l i m i t s we can d i s c u s s a r b i t r a r y shapes f o r t h e i n d e x p r o f i l e . Two l i m i t i n g cases a r e o f p a r t i c u l a r i n t e r e s t :

( i ) t h i c k l a y e r s where o n e c a n r e c o n s t r u c t t h e p r o f i l e f r o m t h e d a t a b y s t a n d a r d F o u r i e r t r a n s f o r m procedures. (Note t h a t t h e Kramers K r o n i g r e l a t i o n s g i v e us some e x t r a i n f o r m a t i o n ) ;

( i i ) t h i n l a y e r s ( t h i c k n e s s zmax below X/4n where

i s t h e o p t i c a l wavelenqth) where one may expand R /R i n powers o f (471 Z,,~/A). The d a t a g i v e o n l y two i n f o r m a t i o n s :

P s

t h e f i r s t moments ro and rl o f t h e p r o f i l e . !t!e r e c o v e r h e r e a s e t o f r e l a t i o n s between r e f l e c t a n c e s and moments which was f i r s t c o n s t r u c t e d by Lekner /a/.

For t h i s second case, i t i s sometimes i n s t r u c t i v e t o d e f i n e an a p p a r e n t t h i c k n e s s ea f o r t h e l a y e r : ea

2rl/ro. F o r a homooeneous s l a b o f t h i c k n e s s e, we r e t u r n t o ea

e e x a c t l y ) . When we a n a l y s e t h e moment s t r u c t u r e f o r t h e examples o f d i f f u s e l a y e r s

a ) f o r a c r i t i c a l b i n a r y f l u i d , where t h e p r o f i l e decreases s l o w l y 1 i ke z - l / * , t h e (0)

moment i s very dependent on t h e c u t o f f F, ( t h e c o r r e l a t i o n l e n t a h ) ro * and t h e apparent t h i c k n e s s i s of o r d e r 6 : i n t h i s c a s e ea i s a r a t h e r good imaae o f t h e o v e r a l l s i z e ;

b) f o r f l e x i b l e polymer a d s o r p t i o n , t h e p r o f i l e i s s l i g h t l y s t e e p e r ( z q 4 I 3 ) and t h i s l e a d s t o very d i f f e r e n t conclusions. The ( 0 ) voment is u n s e n s i t i v e t o t h e c u t o f f RF ( t h e c o i l s i z e ) , and t h e e f f e c t i v e t h i c k n e s s e a i s not proportional t o RF, but t o a weaker power e,

Thus t h i s second example shows very v i v i d l y tlie p i t 2 f a l l s of a l l models based on homogeneous s l a b s .

References

/ 1 / - AZZAF,l R . , BASHARA N.P., "Ellipsometry and p o l a r i s e d l i g h t " , North Holland, h s t e r d a m (1977).

/2/ - ^{BORN I;., WOLF} E . , " P r i n c i p l e s of o p t i c s " , Pergamon Press (1975).

/3/ - FISHER M . , de GENNES P . G . , C . R . Acad. S c i . ( P a r i s ) 8287 (1978) 207.

AU-YANG H . , FISHER It., Phys. Rev. R21 (1980) 3956. -

AU-YANG H . , FISHER M., Physica - l O l A (1980) 255.

/4/ - FRANK C . , SCHNATTERLEY S . , Phys. Rev. L e t t . 46 (1362) 762.

/5/ - de GENNES P.G., "Scaling concepts i n polymer physics", Cornell U. Press, Ithaca (NY) (1979).

/6/ - de GENNES P.G., ~ ~ a c r o m o l e c u l e s 2 (1981) 1637.

/7/ - F1ORSE P.%., FESHEACH C., "Methods of t h e o r e t i c a l physics", flc Graw H i l l , ch 9.3 (1053).

/8/ - LEKNER J . , Physica - 112A (1982) 544.