ELLIPSOMETRY OF UNSUPPORTED AND EMBEDDED THIN FILMS

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ELLIPSOMETRY OF UNSUPPORTED AND EMBEDDED THIN FILMS

R. Azzam

To cite this version:

R. Azzam. ELLIPSOMETRY OF UNSUPPORTED AND EMBEDDED THIN FILMS. Journal de

Physique Colloques, 1983, 44 (C10), pp.C10-67-C10-70. �10.1051/jphyscol:19831013�. �jpa-00223470�

ELLIPSOMETRY OF UNSUPPORTED AND EMBEDDED THIN

FILMS

R.M.A. Azzam

Department of EZectrical Engineer~ing, llniversity o f New Orleans, Lakefront, New Orzeans, Louisiana 70748, U.S.A.

~ 6 s u m 6

-

Nous d b c r i v o n s i c i une methode $ x p l i c i t e n o u v e l l e de d e t e r m i n a t i o n s i m u l t a n h e de l ' i n d i c e complexe e t de l ' e p a i s s e u r des couches minces absorbantes sans s u p p o r t p a r e l l i p s o m e t r i e de r e f l e x i o n e t t r a n s m i s s i o n . C e t t e t e c h n i q u e a

6 t 6

a p p l i q u 6 e au cas d ' u n e lame d ' o r .

A b s t r a c t

-

A t e c h n i q u e f o r t h e simultaneous and e x p l i c i t d e t e r m i n a t i o n o f -- t h e complex r e f r a c t i v e index and t h i c k n e s s o f an unbacked o r embedded t h i n f i l m by combined r e f l e c t i o n and t r a n s m i s s i o n e l l i p s o m e t r y a t any a n g l e o f i n c i d e n c e i s p r e s e n t e d and a p p l i e d t o an u l t r a t h i n g o l d f o i l .

R e c e n t l y we have shown t h a t i t i s p o s s i b l e t o determine, ~ . i m p l q a ~ d d L k e a Y q , b o t h t h e complex r e f r a c t i v e i n d e x and t h i c k n e s s o f a b s o r b i n g , unsupported (unbacked) o r embedded, t h i n f i l m s b y combined r e f l e c t i o n and t r a n s m i s s i o n e l l i p s o m e t r y a t 45'ang1e o f i n c i d e n c e /I/. To o u r knowledge, t h i s i s t h e o n l y case t h a t p e r m i t s e x p f i c i t ( n o n - i t e r a t i v e ) i n v e r s i o n t o o b t a i n a l l t h r e e parameters o f t h e f i l m , w h i c h i s assumed t o be i n c o n t a c t on b o t h s i d e s w i t h t h e same t r a n s p a r e n t medium, o r by d i f f e r e n t media o f t h e bane r e f r a c t i v e index. T h i s paper has two o b j e c t i v e s . F i r s t , we have found, t o o u r own s u r p r i s e , t h a t t h e t e c h n i q u e remains as s i m p l e a t ailg a n g l e o f o b l i q u e i n c i d e n c e $ as i t i s a t 45: T h i s i m p o r t a n t g e n e r a l i - z a t i o n i s discussed. Second, we p r e s e n t r e s u l t s o f t h e s u c c e s s f u l a p p l i c a t i o n of t h e method t o u l t r a t h i n g o l d f o i l s .

THEORY --

F i g u r e 1 shows t h e r e f l e c t i o n and t r a n s m i s s i o n o f c o l l i m a t e d monochromatic t o t a l l y p o l a r i z e d l i g h t b y a p l a n e - p a r a l l e l t h i n f i l m o f t h i c k n e s s d (medium 1 ) surrounded b y a t r a n s p a r e n t medium 3. Both media a r e assumed t o be homogeneous and o p t i c a l l y i s o t r o p i c . $ i s an a r b i t r a r y a n g l e o f i n c i d e n c e and p and s r e p r e s e n t t h e l i n e a r p o l a r i z a t i o n s p a r a l l e l and p e r p e n d i c u l a r t o t h e p l a n e o f i n c i d e n c e , r e s p e c t i v e l y . The e l l i p s o m e t r i c r a t i o s o f p and s complex r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s a r e g i v e n b y /I/:

~t = (1-r;) ( 1 - r g X ) / ( l - r g ) (1-r; X), (2)

X = exp { - j 4 n ( d / ~ ) S , I , ( 3 )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831013

C 10-68 JOURNAL DE PHYSIQUE

F i g . 1 R e f l e c t i o n and transmission o f l i g h t by an unsupported t h i n f i l m .

r and r s a r e t h e 0-1 (ambient-film] i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t s , N, i s the cgmplex r e f r a c t i v e index of t h e fi m and No i s t h e r e a l r e f r a c t i v e index o f t h e surrounding medium a t t h e wavelength X o f the i n c i d e n t l i g h t . From Eqs. ( 1 ) and (2), we f i n d t h a t t h e r a t i o

i s independent o f X, hence o f f i l m thickness. I t can a l s o be r e a d i l y v e r i f i e d /I/

t h a t a i s determined o n l y by t h e r e f l e c t e d and t r a n s m i t t e d p o l a r i z a t i o n states, independently o f t h e i n c i d e n t p o l a r i z a t i o n , provided t h a t t h e l a t t e r has nonzero p and s components.

With No known, Eq. ( 5 ) has N1 as i t s o n l y unknown. To s i m p l i f y t h e s o l u t i o n o f Eq.

( 5 ) f o r N,

,

we had o r i g i n a l l y suggested /1/ t h a t $ be chosen a t 45', so t h a t 4belbs c o n d i t i o n /2/ ( r p = r s 2 ) c o u l d be used. Later, we discovered t h a t e x a c t l y the same s i m p l i c i t y i s a t t a i n e d a t every angle o f o b l i q u e incidence. A t a n y $ , Abelbs c o n d i t i o n i s replaced by a more general r e l a t i o n s h i p /3/:

When Eq. ( 6 ) i s s u b s t i t u t e d i n t o Eq. ( 5 ) , a q u a r t i c equation i n rs i s obtained w i t h c o e f f i c i e n t s determined by a a n d $ . F o r t u i t u o u s l y , t h e q u a r t i c equation i s r e d u c i b l e t o t h e f o l l o w i n g q u a d r a t i c equation:

where

= [2cos21$+a~l+cos~2$)1/(1

+

a c o s & ) . O f the two r o o t s o f Eq. (7),

rs = 4 [u

+

^{( u 2}

-

4141, ( 9 )

t h a t f o r which

lrS 1

< 1 i s selected. Next N1 i s obtained from rs by /3/:

1 - r ²

N, = [ s i n 7 $ + c ~ s f ( ~ +

14.

(10)

To determine t h e thickness d, Eq. (1) ( o r , e q u i v a l e n t l y , Eq. ( 2 ) ) i s solved f o r X : X = ( l / r p r s ) ( v r r s - r p ) / ( c r r p

-

r s ) . (11)

Two values f o r d can be obtained from complex X . To see t h i s , Eq. ( 3 ) i s r e w r i t t e n as :

X = exp (-d/Da) exp (-jZnd/D ),

P ⁽¹²⁾

where

and a s e r i e s o f thicknesses,

a r e obtained, where m i s an i n t e g e r . I d e a l l y , o n l y one value o f dp, a t a c e r t a i n m, must agree w i t h da.

APPLICATION TO ULTRATHIN GOLD FOIL

S e l f - s u p p o r t i n g , l i g h t - t r a n s m i t t i n g , t h i n f i l m s o f g o l d have been prepared and ex- t e n s i v e l y characterized, using transmission i n t e r f e r o m e t r y , by Casset / 4 / , and have r e c e i v e d prominent a t t e n t i o n r e c e n t l y /5/. Here we present t h e f i r s t e l l i p s o m e t r i c study o f such f i l m s u s i n g t h e simple and novel method we have j u s t described.

Two-zone n u l l e l l i p s o m e t r y /6/ was performed using He-Ne l a s e r l i g h t ( ~ = 6 3 2 8 1 ) r e - f l e c t e d and t r a n s m i t t e d by a non-opaque and unbacked g o l d f o i l k i n d l y supplied by Gary Reeves o f Los Alamos National Laboratory. The f i l m was mounted on a brass 0 r i n g o f about 5m i n s i d e diameter. The measured r e f l e c t i o n and transmission e l l i p s o m e t r i c parameters

(I&.,

A,) and (

k,

dt) o f the f i l m a r e given i n Table 1 a t t h r e e angles on incidence: 45, 60, 70"Recall t h a t t h e complex r a t i o o f p and s r e f l e c t i o n o r transmission c o e f f i c i e n t s i s given by pk=tan lk, e x p ( j ~ ~ ) where k = r o r t . ) We have a l s o measured t h e i n t e n s i t y t r a n s m i t t a n c e o f t h e f i l m a t normal incidence and found i t t o be 0.03866 ( i .e. = 4%).

Reducing the data o f Table 1 gave the r e s u l t s o f Table 2. I n t h i s t a b l e , n and k are t h e r e f r a c t i v e index and e x t i n c t i o n c o e f f i c i e n t o f t h e f i l m (N1 = n

-

j k ) , as determined by Eqs. ( 8 ) - ( l o ) , where a = pt/pr. da i s the f i l m thickness as obtained from Eq. (14). Equation (15) f a i l e d t o supply any reasonable value o f f i l m thickness; i t r e t u r n e d o n l y unacceptably l a r g e values o f d t h a t i n d i c a t e t h a t t h e f i l m i s opaque when i t i s not. T h i s discrepancy between !hewy and ex- periment can be explained as f o l l o w s . For t h e present f i l m ,

1x1,

as determined by Eq. ( l l ) , i s l e s s than 0.02 a t a l l angles o f incidence. When t h e absolute value o f a complex q u a n t i t y i s small, t h e u n c e r t a i n t y o f i t s argument becomes l a r g e . (Of course, i n t h e l i m i t

1x1

⁼ 0, a r g X i s indeterminate.) Because dp i s obtained from a r g X, i t , too, becomes u n c e r t a i n . E q u i v a l e n t l y , t h e c a l c u l a t e d phase-thickness parameter Dp i s too l a r g e (> l0000A a t a l l angles) t o be u s e f u l as a s e n s i b l e " y a r d s t i c k " w i t h which t o measure thickness o f he o r e r o f a few hundred A. By comparison, Da ( = 1741 a t 450 1691 a t 60'and 165k a t 78) i s o f t h e c o r r e c t s i z e t o s u i t t h e purpose f i lm-thickness determination. (Remember a1 so t h a t we are assuming an idealized mod& o f a homogeneous and i s o t r o p i c f i l m w i t h p a r a l l e l - p l a n e boundaries. )

Because the i n t e n s i t y transmittance a t normal incidence To i s a s e n s i t i v e f u n c t i o n o f f i l m thickness, we decided t o improve upon our f i l m - t h i c k n e s s determination by changing d from da t o minimize the d i f f e r e n c e , (Tom - To

1,

between t h e measured, Tqm = 0.03866, and c a l c u l a t e d , To

,

values o f To. The r e s u ? t i n g thicknesses are s ~ m p l y denoted by d i n Table 2. F n t e r e s t i n g l y and s i g n i f i c a n t l y , t h e degree of f i t t o e l l i p s o m e t r i c data, represented by t h e sum o f squares,

d i d n o t change appreciably. This lends considerable c r e d i b i l i t y t o t h e newly determined f i l m thickness d. I n Eq. (16), the s u b s c r i p t s m and c r e f e r t o

C 10-70 JOURNAL DE PHYSIQUE

Table 1. E l l i p s o m e t r i c Parameters ( i n deg.) o f an U l t r a t h i n Au F o i l

$ '4- Ar

2

A+

45O 43.432 153.474 59.321 -25.116

60' 42.455 127.560 69.620 -49.070

70' 41.952 101.318 76.246 -73.448

Table 2. F i l m C h a r a c t e r i s t i c s and Degree-of-Fit Parameters

$ n k da

(&-T(!)

SSr)(deg.')

I

Tom-T&

45' 0.311 -2.800 71 3 642 0.429 0.25X10- ⁴

60' 0.314 -2.846 777 629 0.364 0 . 3 6 ~ 1 0 - ⁴

70' 0.252 -2.899

4

654 616 0.303 l . l X 1 0 -

measured and c a l c u l a t e d values, r e s p e c t i v e l y . Table 2 l i s t s the r e s i d u a l ITom-Tocl and SSQ. Averaging t h e r e s u l t s i n Table 2, we o b t a i n t h e f o l l o w i n g f i l m parameters:

For comparison, we have determined ( u s i n g two-zone n u l l e l l i p s o m e t r y ) n and k o f a M c k ( o aque) g o l d f i l m deposited on glass; we obtained n = 0.333, k = -2.974 a t h = 63289. Because t h e o p t i c a l p r o p e r t i e s o f evaporated f i l m s depend on c o n d i t i o n s o f t h e i r p r e p a r a t i o n , t h e d i f f e r e n c e s between n and k o f t h e f o i l and those o f the t h i c k f i l m on glass are considered reasonable and n o t unexpected.

ACKNOWLEDGEMENT

I am g r a t e f u l t o Gary Reeves o f Los Alamos N a t i o n a l Laboratory f o r p r o v i d i n g me w i t h t h e unbacked g o l d f i l m . T h i s work was supported by t h e N a t i o n a l Science Foundation g r a n t DMR-8018417.

REFERENCES

1. AZZAM, R.M.A.

,

J. Opt. Soc. Am. 73 (1983) i n press.

2. ABELES, F., C. R. Acad. Sci

.

230 7 1 9 5 0 ) 1942.

3. AZZAM, R.M.A.

,

J. Opt. Soc.

Am.

69 (1979) 1007.

4. CASSET, J., J. Opt. Soc. Am.

69

n 9 7 9 ) 725, and references t h e r e i n .

5. NEWS ITEMS, I n d u s t . Res. Develop.

25

^{( 3 )} (1983) 43; Mech. Eng.

105

( 5 ) (1983) 71.

6. AZZAM, R.M.A. and BASHARA, N.M., E l l i p s o m e t r y and P o l a r i z e d L i g h t (North- Holland, Amsterdam, 1977).