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ON THE ACCURACY OF ELLIPSOMETRIC
THICKNESS DETERMINATIONS FOR VERY THIN FILMS
D. Chandler-Horowitz, G. Candela
To cite this version:
D. Chandler-Horowitz, G. Candela. ON THE ACCURACY OF ELLIPSOMETRIC THICKNESS
DETERMINATIONS FOR VERY THIN FILMS. Journal de Physique Colloques, 1983, 44 (C10),
pp.C10-23-C10-26. �10.1051/jphyscol:19831003�. �jpa-00223452�
JOURNAL DE PHYSIQUE
Colloque C10, suppl&ment au n012, Tome 44, dkernbre 1983 page C 10- .23
ON THE ACCURACY OF ELLIPSOMETRIC THICKNESS DETERMINATIONS FOR VERY THIN FILMS
D . Chandler-Horowitz and G.A. Candela
Semiconductor Materials and Processes Division, NationaZ Bureau o f Standards, Washington, DC 20234, U.S. A .
%sum&. L ' i n c e r t i t u d e s u r l a mesure B l l i p s o m e t r i q u e d e I ' B p a i s s e u r d ' u n f i l m s u r un s u b s t r a t p e u t e t r e d 6 t . e r m i n Q e d 1 u n e f a s o n q u a n t i t a t i v e . Nous
a v o n s u t i l i s e l a s o l u t i o n d e s d i f f e r e n t i e l l e s d e l ' e p a i s s e u r du f i l m e t d e l ' i n d i c e d e r e f r a c t i o n . Les r e s u l t a t s d e s c a l c u l s p o u r l e s y s t e m e a i r - S i O -
S i s o n t p r d s e n t e s . C e t t e t h e o r i e a d t 6 u t i l i s e e p o u r c a l c u l e r l 1 i n c e r t i t u 3 e s u r l a v a l e u r d e 1 ' B p a i s s e u r . Nous a v o n s a u s s i c a l c u l d c e t t e e r r e u r quand l ' i n c e r t i t u d e s u r l ' i n d i c e d e r e f r a c t i o n e s t connue. Nous montrons q u e l l e s i n c e r t i t u d e s c o n t r i b u e n t l e p l u s B l a p r e c i s i o n t o t a l e s u r l a mesure d e 1 ' d p a i s s e u r .
A b s t r a c t . The u n c e r t a i n t y i n t h e e l l i p s o m e t r i c d e t e r m i n a t i o n of t h e t h i c k - n e s s o f a f i l m on a s u b s t r a t e c a n be found q u a n t i t a t i v e l y . We have u s e d t h e s o l u t i o n of t h e d i f f e r e n t i a l s of f i l m t h i c k n e s s and r e f r a c t i v e i n d e x . Re- s u l t s o f c a l c u l a t i o n s f o r t h e a i r - S i 0 2 - S i s y s t e m a r e p r e s e n t e d . mis t h e o r y h a s been used to c a l c u l a t e t h e u n c e r t a i n t y i n t h e v a l u e o f t h e t h i c k n e s s a s a f u n c t i o n of wavelength. W e h a v e a l s o c a l c u l a t e d t h i s u n c e r t a i n t y f o r known u n c e r t a i n t y i n t h e f i l m ' s r e f r a c t i v e i n d e x . We show which u n c e r t a i n - ties c o n t r i b u t e t h e most to t h e o v e r a l l a c c u r a c y of a t h i c k n e s s measurement.
I. INTRODUCTION
The need t o measure t h e t h i c k n e s s o f v e r y t h i n f i l m s , l e s s t h a n 50 nm t h i c k , and t o c h a r a c t e r i z e t h i n i n t e r l a y e r s p r o v i d e s t h e i n c e n t i v e to t a k e i n t o a c c o u n t a l l p o s s i - b l e u n c e r t a i n t i e s i n p a r a m e t e r s which c a n a f f e c t t h e a c c u r a c y o f a n e l l i p s o m e t r i c measurement. D e s p i t e t h e c o m p l i c a t e d n a t u r e o f t h e e l l i p s o m e t r i c e x p r e s s i o n s , t h e u n c e r t a i n t i e s i n t h e d e t e r m i n e d p a r a m e t e r s s u c h a s t h e t h i c k n e s s and r e f r a c t i v e i n d e x of a f i l m on a s u b s t r a t e c a n be c a l c u l a t e d . For a p r o p e r t r e a t m e n t o f t h e c a l c u l a t i o n , one must u s e t h e e x a c t s o l u t i o n o f t h e d i f f e r e n t i a l s of f i l m t h i c k n e s s and r e f r a c t i v e i n d e x . T h i s c a l c u l a t i o n depends on t h e f o l l o w i n g e x p e r i m e n t a l v a r i - a b l e s : t h e e l l i p s o m e t r i c a l l y measured a n g l e s A and $I, t h e a n g l e of i n c i d e n c e 0, t h e a m b i e n t and s u b s t r a t e i n d i c e s of r e f r a c t i o n , and the wavelength o f t h e l i g h t . I n t h i s p a p e r , t h e u n c e r t a i n t y i n t h e d e t e r m i n e d f i l m t h i c k n e s s f o r known u n c e r t a i n t i e s i n A , $I, and 0 i s c a l c u l a t e d a s a f u n c t i o n of w a v e l e n g t h f o r t h e Si02-Si system.
The u n c e r t a i n t y i n t h e S i 0 2 f i l m t h i c k n e s s i s a l s o c a l c u l a t e d f o r t h e s p e c i a l c a s e when t h e v a l u e o f t h e f i l m ' s r e f r a c t i v e i n d e x and i t s u n c e r t a i n t y is g i v e n .
2 . THEORY
We b e g i n by b r i e f l y summarizing t h e g e n e r a l u n c e r t a i n t y t h e o r y . l r The e l l i p s o - m e t r i c p a r a m e t e r s A and $I a r e d e f i n e d by t h e complex r e f l e c t a n c e r a t i o Rp/Rs:
Here Rp and R . a r e t h e r a t i o s of t h e r e f l e c t e d t o t h e i n c i d e n t e l e c t r i c f i e l d a m p l i t u d e s of t h e l i g h t f o r p o l a r i z a t i o n s p a r a l l e l p, and p e r p e n d i c u l a r s, t o t h e p l a n e o f i n c i d e n c e . The r e f l e c t a n c e r a t i o i s a f u n c t i o n p which, f o r t h e c a s e o f a n a m b i e n t - f i l m - s u b s t r a t e model o f t h e sample, depends o n t h e f o l l o w i n g p a r a m e t e r s : NO, t h e i n d e x o f r e f r a c t i o n o f t h e a m b i e n t ; N1, t h e i n d e x of r e f r a c t i o n o f t h e f i l m ;
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831003
C10-24 JOURNAL DE PHYSIQUE
N2, the i n d e x o f r e f r a c t i o n o f t h e s u b s t r a t e ; t, t h e t h i c k n e s s o f the f i l m ; 0, t h e a n g l e o f i n c i d e n c e ; a n d A, t h e w a v e l e n g t h o f t h e l i g h t .
From e q ( 1 ), t h e f o l l o w i n g r e a l f u n c t i o n s f o r A a n d J, are:
A = A (NO, N1, N2, t , 4, A ) ( 2
I n e l l i p s o m e t r y o n e m e a s u r e s t h e two e l l i p s o m e t r i c p a r a m e t e r s A a n d $ a n d t h e n u s e s c o m p u t a t i o n a l t e c h n i q u e s t o f i n d , f o r e x a m p l e , t h e two unknown p a r a m e t e r s , N1 and t.
A l t h o u g h t h e e q u a t i o n s c a n n o t be s o l v e d e x p l i c i t l y f o r t h e s e unknown p a r a m e t e r s , t h e e x p r e s s i o n s f o r t h e i r d i f f e r e n t i a l s c a n be o b t a i n e d e x p l i c i t l y . When t h e t o t a l d i f f e r e n t i a l s o f e q s ( 2 ) and ( 3 ) are t a k e n a n d it i s assumed t h a t a l l u n c e r t a i n t i e s a r e known e x c e p t t h o s e i n t h e f i l m t h i c k n e s s a n d r e f r a c t i v e i n d e x , 6 t a n d 6N1, a n d t h a t a l l t h e u n c e r t a i n t i e s a r e i n d e p e n d e n t random d e v i a t i o n s from t h e i r t r u e v a l u e s , t h e r o o t mean s q u a r e u n c e r t a i n t i e s i n t a n d N1 a r e g i v e n by:
and
H e r e , o n l y the c o n t r i b u t i o n s o f t h e u n c e r t a i n t i e s i n t h e i n s t r u m e n t a l p a r a m e t e r s , 6A, 6J,, a n d 64, a r e c o n s i d e r e d . The m a g n i t u d e s o f t h e c o e f f i c i e n t s o f e a c h o f t h e u n c e r t a i n t i e s a r e g i v e n i n T a b l e 1. The v a l u e s o f t h e p a r t i a l d e r i v a t i v e s o f A a n d J, w i t h respect t o Nl, t, a n d @ f o u n d i n t h e t a b l e c a n be o b t a i n e d as f o l l o w s .
F i r s t , o n e t a k e s t h e p a r t i a l d e r i v a t i v e o f e q ( 1 ) w i t h r e s p e c t t o t h e f i l m t h i c k n e s s t a n d o b t a i n s :
By e q u a t i n g b o t h t h e real a n d t h e i m a g i n a r y p a r t s o f e q (61, o n e g e t s two e q u a t i o n s which c a n be s o l v e d f o r b o t h a $ / a t a n d a A / a t :
One g e t s s i m i l a r e q u a t i o n s f o r t h e p a r t i a l d e r i v a t i v e s o f A a n d J, w i t h r e s p e c t t o e i t h e r N1 o r 4 by s i m p l y i n t e r c h a n g i n g e i t h e r N l o r 4 w i t h t i n e q s ( 7 ) and (8).
The e x p r e s s i o n s f o r t h e p a r t i a l d e r i v a t i v e of p w i t h r e s p e c t t o t, N1, or $ d e p e n d o n t h e p a r t i c u l a r f i l m - s u b s t r a t e model c h o s e n . T h e s e e x p r e s s i o n s f o r the a m b i e n t - f i l m s u b s t r a t e model a r e s t r a i g h t f o r w a r d b u t a r e n o t g i v e n h e r e b e c a u s e o f t h e i r l e n g t h . T h e s e e x p r e s s i o n s a r e u s e d t o c a l c u l a t e t h e m a g n i t u d e o f K and o f t h e c o e f f i c i e n t s A t h r o u g h F.
For t h e s p e c i a l c a s e w h e r e t h e f i l m ' s r e f r a c t i v e i n d e x a n d i t s u n c e r t a i n t y a r e assumed known, o n e o b t a i n s e q u a t i o n s i n a s i m i l a r manner t o t h a t u s e d t o o b t a i n e q s
( 4 ) a n d ( 5 ) . The f i l m t h i c k n e s s u n c e r t a i n t y is g i v e n by
B e c a u s e e q s ( 9 ) a n d ( 1 0 ) g i v e two i n d e p e n d e n t d e t e r m i n a t i o n s o f t h e f i l m t h i c k n e s s u n c e r t a i n t y , the s m a l l e r o f t h e two v a l u e s c a n be u s e d . The c o e f f i c i e n t s P t h r o u g h U a r e a l s o g i v e n i n T a b l e 1.
3. RESULTS
W e f i r s t u s e t h e a b o v e t h e o r y t o c a l c u l a t e t h e e f f e c t o f w a v e l e n g t h on t h e t h i c k n e s s u n c e r t a i n t y . For t h e c a s e of a t h i n s i l i c o n d i o x i d e f i l m on a s i l i c o n s u b s t r a t e , e q ( 4 ) g i v e s t h e n u m e r i c a l r e s u l t s shown i n F i g u r e 1 f o r t h e f i l m t h i c k n e s s u n c e r t a i n t y i n p e r c e n t a s a f u n c t i o n o f w a v e l e n g t h u n d e r t h e f o l l o w i n g c o n d i t i o n s : 6 t and 6N1 a r e assumed t o be unknown and t h e u n c e r t a i n t i e s i n A , 9 , a n d 4 a r e e q u a l t o 0.01 d e g
( d a s h e d c u r v e s ) a n d 0.001 d e g ( s o l i d c u r v e s ) . P a i r s o f c u r v e s f o r t h r e e d i f f e r e n t
o x i d e t h i c k n e s s e s , 1.0, 10.0, and 50.0 nm, a r e shown. W e h a v e a l s o u s e d t h e t h e o r y t o c a l c u l a t e t h e t h i c k n e s s u n c e r t a i n t y a s a f u n c t i o n o f t h e u n c e r t a i n t y i n N1.
F i g u r e 2 shows t h e c a l c u l a t e d o x i d e f i l m t h i c k n e s s u n c e r t a i n t y i n p e r c e n t a s a f u n c - t i o n o f t h i c k n e s s f o r a 632.8-nm w a v e l e n g t h a n d u n c e r t a i n t i e s i n A , JI, a n d 4 of 0.01 d e g . The s l o p i n g l i n e a t r i g h t i s f o r a c o n v e n t i o n a l e l l i p s o m e t r i c c a l c u l a t i o n ( e q ( 4 ) ) i n which b o t h 6 t a n d 6N1 a r e unknown. The h o r i z o n t a l l i n e s show t h e r e s u l t s o f e q ( 9 ) o r ( 1 0 ) f o r t h e c a s e when o n e h a s known u n c e r t a i n t i e s i n t h e v a l u e of t h e o x i d e i n d e x , N
=1.46. They a r e 6N1 = t 1 . 0 , f 0 . 1 , tO.O1, a n d fO.OO1. S i n c e t h e s e c a l c u l a t i o n s a r e more r e s t r i c t i v e , t h e v a l u e s o b t a i n e d a r e n o t p l o t t e d t o t h e r i g h t of t h e s l o p e d l i n e s i n c e t h e y would t h e n c o r r e s p o n d t o a l a r g e r u n c e r t a i n t y t h a n t h a t of t h e c o n v e n t i o n a l c a l c u l a t i o n . F i g u r e 3 shows t h e same case e x c e p t t h a t t h e u n c e r t a i n t y i n A , 9, a n d 4 i s r e d u c e d t o 0.001 deg.
4. DISCUSSION
T h i s method o f u n c e r t a i n t y c a l c u l a t i o n a l l o w s o n e , w i t h t h e a i d o f t h e c o m p u t e r , t o q u i c k l y g e n e r a t e t h e e x a c t d e p e n d e n c e of a n y c a l c u l a t e d e l l i p s o m e t r i c u n c e r t a i n t i e s i n t e r m s o f known e l l i p s o m e t r i c u n c e r t a i n t i e s e v e n f o r v e r y complex s y s t e m s . I n o r d e r to u s e this method, o n e n e e d s t h e e x p r e s s i o n s f o r t h e p a r t i a l d e r i v i t i v e s of p w i t h r e s p e c t t o a l l e l l i p s o m e t r i c p a r a m e t e r s t h a t c o n t r i b u t e t o t h e o v e r a l l u n c e r - t a i n t y . F i g u r e s 1 , 2, a n d 3 show some o f t h e r e s u l t s f o r t h e f i l m t h i c k n e s s u n c e r - t a i n t y f o r t h e a i r - s i l i c o n d i o x i d e - s i l i c o n s y s t e m . From F i g u r e 1 , one o b s e r v e s t h a t t h e r e i s a l a r g e d e c r e a s e i n t h i c k n e s s u n c e r t a i n t y f o r t h e t h i n n e s t f i l m s a s t h e w a v e l e n g t h d e c r e a s e s from 600 nm t o b e l o w 400 nm. T h i s s t e e p f a l l o f f is d u e t o t h e
l a r g e i n c r e a s e i n the i m a g i n a r y part o f t h e s i l i c o n s u b s t r a t e ' s r e f r a c t i v e i n d e x a t s m a l l e r w a v e l e n g t h s . From F i g u r e 1 and eq ( 4 ) , one a l s o o b s e r v e s t h a t t h e u n c e r - t a i n t y i n t h e f i l m t h i c k n e s s i s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e u n c e r t a i n t y i n t h e e l l i p s o m e t r i c a n g l e s A , (I, a n d 4. One c a n g a i n i n c r e a s e d a c c u r a c y i n t h e d e t e r m i n e d f i l m t h i c k n e s s by d e c r e a s i n g t h e u n c e r t a i n t y i n A, 9 , o r 4 o r by u s i n g s h o r t e r wave- l e n g t h s . For e x a m p l e , a t X = 300 nm a n d & A , 6 9 , 6 $ = 0.001 d e g , one c a n d e t e r m i n e a l l t h i c k n e s s e s g r e a t e r t h a n 1 nm and t h e i r r e f r a c t i v e i n d i c e s to b e t t e r t h a n 1 p e r c e n t a c c u r a c y . I n F i g u r e s 2 a n d 3 t h e t h i c k n e s s u n c e r t a i n t y is shown f o r known v a l u e s o f t h e u n c e r t a i n t y i n t h e r e f r a c t i v e i n d e x o f t h e f i l m . As t h e o x i d e t h i c k - n e s s d e c r e a s e s , we s e e t h a t a s s i g n i n g a known u n c e r t a i n t y t o t h e f i l m r e f r a c t i v e i n d e x becomes v e r y t e m p t i n g , s i n c e i t s t h i c k n e s s u n c e r t a i n t y i s l o w e r t h a n t h a t ob- t a i n e d by t h e c o n v e n t i o n a l c a l c u l a t i o n . However, v e r y t h i n f i l m s may n o t h a v e t h e same r e f r a c t i v e i n d e x as t h e i r b u l k o r t h i c k - f i l m v a l u e s b e c a u s e of i n t e r l a y e r e f - f e c t s . T h e r e f o r e , i n most c a s e s , t h e c o n v e n t i o n a l e l l i p s o m e t r i c c a l c u l a t i o n s h o u l d be u s e d b e c a u s e t h e u n c e r t a i n t y i n t h e f i l m i n d e x i s a s much a n i n d e p e n d e n t l y un- known q u a n t i t y a s is i t s t h i c k n e s s .
W e h a v e d e s i g n e d a n d c o n s t r u c t e d a t NBS a f u l l y a u t o m a t e d s p e c t r o s c o p i c e l l i p s o m e t e r o f h i g h a c c u r a c y , s e e F i g u r e 4, t h a t i n c o r p o r a t e s t h e r e s u l t s o f t h i s p a p e r a n d p r i o r consideration^.^^^ The e l l i p s o m e t e r is d e s i g n e d w i t h a t h e t a - t w o - t h e t a
g o n i o m e t e r a n d o p t i c a l b e n c h s y s t e m s o t h a t v a r i o u s e l l i p s o m e t r i c methods o f o b t a i n - i n g t h e m e a s u r a b l e p a r a m e t e r s , A a n d 9, c a n be u s e d . T h r e e i m p o r t a n t t e c h n i q u e s b e i n g s t u d i e d a r e t h e c o n v e n t i o n a l n u l l method, t h e r o t a t i n g a n a l y z e r method, a n d t h e p r i n c i p a l a n g l e method. The g o n i o m e t e r s ' a n g l e s a r e a l l a c c u r a t e t o 0.001 deg.
The monochromator s c a n s t h e w a v e l e n g t h r a n g e o f 185 t o 2650 nm. I n a d d i t i o n , t h e r e a r e two f i x e d l a s e r w a v e l e n g t h s , 632.8 nm and 441.6 nm. The e l l i p s o m e t e r h a s a s a m p l e a l i g n m e n t s y s t e m which u t i l i z e s two q u a d r a n t d e t e c t o r s .
R e f e r e n c e s
1. 0. C h a n d l e r - H o r o w i t z , E l l i p s o m e t r i c A c c u r a c y a n d t h e P r i n c i p a l Angle of I n c i - d e n c e , Proc. Soc. P h o t o - O p t i c a l I n s t r u m . E n g r s . 342, I n t e g r a t e d C i r c u i t M e t r o l - ogy, 121-130 ( 1 9 8 2 ) .
2. D. C h a n d l e r - H o r o w i t z and G. A. C a n d e l a , P r i n c i p a l Angle S p e c t r o s c o p i c E l l i p s o m - e t r y U t i l i z i n g a R o t a t i n g A n a l y z e r , Appl. O p t i c s 2, 2972-2977 ( 1 9 8 2 ) .
3. R. M. A. Azzam and N. M. R a s h a r a , E l l i p s o m e t r y a n d P o l a r i z e d L i g h t ( N o r t h Hol-
l a n d , New York, 1 9 7 7 ) , Chap. 4.
JOURNAL DE PHYSIQUE
F i g . 1. T h i c k n e s s u n c e r t a i n t y a s a f u n c t i o n o f w a v e l e n g t h ; s e e t e x t f o r e x p l a n a t i o n .
OXIDE THICKNESS [nm]
F i g . 2. T h i c k n e s s u n c e r t a i n t y a s a f u n c t i o n o f o x i d e t h i c k n e s s f o r g i v e n u n c e r t a i n t i e s i n t h e r e f r a c t i v e i n d e x o f the o x i d e f i l m f o r t h e c a s e o f 614, 6A, 641 = 0.01 d e g . X = 632.8 nm.
F i g . 3. T h i c k n e s s u n c e r t a i n t y a s a f u n c t i o n o f o x i d e t h i c k n e s s f o r g i v e n u n c e r t a i n t i e s i n t h e r e f r a c t i v e i n d e x o f t h e o x i d e f i l m f o r t h e c a s e of 6 $ , 6 A , 6 @ = 0.001 d e g , X = 632.8 nm.
SPECTROSCOPIC ELLIPSOMETER
Caranwl lhlll E l h p r m t ~ r . M E R o t r t m 1 ~ h z . r E l l g r m t n . RAE
P , m p l l Ant* Rotatm Aluhzer E l g m t e r . PA-RAE
