ACCURACY IN THE DETERMINATION OF THIN FILMS OPTICAL CONSTANTS BY THE ATR METHOD

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ACCURACY IN THE DETERMINATION OF THIN FILMS OPTICAL CONSTANTS BY THE ATR

METHOD

G. Hincelin

To cite this version:

G. Hincelin. ACCURACY IN THE DETERMINATION OF THIN FILMS OPTICAL CONSTANTS BY THE ATR METHOD. Journal de Physique Colloques, 1983, 44 (C10), pp.C10-109-C10-112.

�10.1051/jphyscol:19831023�. �jpa-00223480�

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JOURNAL DE PHYSIQUE

Colloque CIO, supplément au n°12, Tome M, décembre 1983 page C10-I09

ACCURACY IN THE DETERMINATION OF THIN FILMS OPTICAL CONSTANTS BY THE ATR METHOD

G. Hincelin

CNAM, Laboratoire de Physique du Vide et des Composants Electroniques, 292, rue Saint-Martin, 75141 Paris Cedex OS, France

Résumé : On montre que l'existence d'erreurs systématiques importantes provenant des imperfections du faisceau d'analyse permettent d'expliquer la décroissance anormale de l'épaisseur calculée en fonction de la longueur d'onde dans le proche infrarouge.

Abstract : In this paper we present evidence that important systematic errors, owing to the light beam imperfections, account for the anomalous decrease of the calculated values of the film thickness as a function of the wavelength in the near infrared.

In the Kretschmann and Raether configuration /1//2/ (prism/thin metal film/vacuum), a Surface Plasma Wave (SPW) can be excited at the metal/vacuum interface, by matching the incoming wavevector component parallel to the surface Kx and the wavevector Kq

of the SPW : P

(1) n and cp are the prism refraction index and incidence angle respectively.

At a fixed frequency to, the experimental reflectance R' = f(cp) varies readily as a function of tp, and a pronounced minimum occurs at cp = cp . The optical model is ap- proximated by a perfectly plane and homogeneous film of thickness D, with a local die lectric constant £ (u), inserted between two semi infinite dielectric media of real permittivities n (to) and £„((*>) = 1 respectively.

The dielectric constant e, (to) and the thickness D of the film can be determined for each frequency io by a fitting of the theoretical R (cp) curve, computed with the help of the classical Fresnel's formula. The optical layout and measurement methods have been described in a previous publication /3/.

I - DISTORTIONS OF THE RESONANCE CURVES.

The theoretical resonance curve R (9) is calculated with the hypothesis of a perfectly parallel and monochromatic light beam, whereas the real experimental beam has finite angular aperture and spectral width.

The angular distribution of incident radiation flux as a function of cp, can be re- presented by a "rectangular" function F (<P) of half angular aperture Oo - The densi- ty of spectral luminance D(^) is assumed to have a triangulary shape, its half width at half maximum being denoted AX.

Denoting by V Q , ^0 the mean values of the incidence angle and central wavelength respectively, we assume that the experimental reflectance R1 is given by the following

convolution : "

(2) At a fixed wavelength ^0, and for a given set of the three parameters (e ,e.,D) the curve R' = f(cp0) may be numerically computed for different values of cp0. The distortions in the resonance curve, can be expressed in terms of the difference 6R(cp0) between R p ^ o ) and Rp(cp0) at each incidence angle cp0.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831023

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C10-110 JOURNAL DE PHYSIQUE

I n o r d e r t o c h a r a c t e r i z e t h e s e d i s t o r t i o n s i n a more s i m p l i f i e d way, we u s e a n ap- p r o x i m a t e r e l a t i o n f o r t h e t h e o r e t i c a l r e f l e c t a n c e i n t h e v i c i n i t y o f K O . t h e com- p l e x w a v e v e c t o r o f t h e SPW a t t h e boundary between t h e two h a l f s p a c e s E ~ / E ~ . For t h e plasma f i l m o f s u f f i c i e n t t h i c k n e s s D , t h e w a v e v e c t o r is a p p r o x i m a t e d by KSo = K O

+

AK /1//4/ and f i n a l l y :

J ~ ( K , ) . J ~ ( A K ) R ( K ) = l - 4.

P x 2 ( 3 )

e[K

]12

+ [Jm(Ko) + J m ( ~ K ) ]

l k x

-

^Sp

The r e f l e c t a n c e h a s a d i p a t :

K: = Re(K, + AK)

The s h a p e o f t h e r e s o n a n c e c u r v e i s deduced from t h e d i p c o o r d i n a t e s

(K,,

m R,) and t h e h a l f w i d t h a t h a l f h e i g h t AK+ g i v e n by :

A t a p a r t i c u l a r optimum t h i c k n e s s DoDt, f o r which J,(AK) = Jm(K,) t h e minimum r e - f l e c t a n c e R r e a c h e s a minimum e q u a l t o z e r o . I f 6R i s s m a l l enough, i t i s o b v i o u s t h a t t h e r e s o n a n c e d i p i s n o t a p p r e c i a b l y s h i f t e d by t h e beam i m p e r f e c t i o n s

(6K: = 0 ) . W i t h i n t h e s e a p p r o x i m a t i o n s , t h e d i s t o r t i o n s c a n be c h a r a c t e r i z e d by means o f two p a r a m e t e r s o n l y :

-

~R(K:) t h e d i s c r e p a n c y between t h e e x p e r i m e n t a l and t h e t h e o r e t i c a l minimum.

- t h e b r o d e n i n g 6K ( o r 6K/AK% i n r e l a t i v e v a l u e ) .

I n t h i s way, t h e p r e v i o u s p a r a m e t e r s c a n b e e a s i l y c a l c u l a t e d w i t h t h e h e l p o f t h e r e l a t i o n ( 2 ) .

T y p i c a l v a r i a t i o n s o f 6K/AKk a n d ~R(K:)

,

f o r a n a n g u l a r a p e r t u r e a, = 0.014 d e g r e e and a s p e c t r a l w i d t h Ah = 1 mm,are shown i n f i g u r e s 1 a a n d 1 b f o r s i l v e r l a y e r s of d i f f e r e n t t h i c k n e s s e s .

Fig. 1 a a n d 1 b : V a r i a t i o n s of 8K/AK+ a n d ~R(K:) v e r s u s t h e l i g h t w a v e l e n g t h f o r s i l v e r f i l m s o f t h i c k n e s s e s : a = 60 nm; b = 50 nm; c = 40 nm.

(- c o n t r i b u t i o n o f t h e a n g u l a r a p e r t u r e a , = 0.014 d e g r e e ;

---

c o n t r i b u t i o n o f t h e s p e c t r a l w i d t h Ah = 1 nm).

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I n o r d e r t o compare t h e r e l a t i v e i n f l u e n c e o f a , a n d Ah, t h e f u l l l i n e c u r v e s show t h e c o n t r i b u t i o n o f a , o n l y ( s u p p o s e d l y AXZO), w h e r e a s t h e d a s h e d l i n e c u r v e s show t h e c o n t r i b u t i o n o f Ah.

The limits o f t h e measurement u n c e r t a i n t i e s a r e a l s o r e p r e s e n t e d AR < 0.002;

AK/AK+ < 0 . 0 0 5 .

The d f s t o r t i o n s coming from a , i n c r e a s e r e a d i l y a s a f u n c t i o n o f t h e w a v e l e n g t h : t h e y a r e n e g l i g i b l e compared t o t h e measurement u n c e r t a i n t i e s a s l o n g a s A< 0.5 urn b u t become p r e p o n d e r a n t i n t h e n e a r i n f r a r e d . On t h e o t h e r hand, t h e d i s t o r t i o n s i n p r o v e n a n c e o f Ah n e v e r e x c e e d t h e l i m i t s o f t h e measurement u n c e r t a i n t i e s . I 1 - SYSTEMATIC ERRORS IN THE IR DOMAIN.

When f i t t i n g t h e t h e o r e t i c a l a n d e g p e r i m e n t a l r e s g n a n c e c u r v e s , t h e d i s , t o r t i o n s b e i n g i g n o r e d , s y s t e m a t i c e r r o r s 6 e r ' 6Se. 1 and S D o c c u r i n t h e v a l u e s o f t h e p a r a m e t e r s . F o r s m a l l v a r i a t i o n s d e r , d e . , dD o f t h e s e p a r a m e t e r s , t h e c o r r e s p o n - d i n g v a r i a t i o n s i n t h e s h a p e o f t h e r e s o n a n c e c u r v e dKx, dRm and ~ ( A K m ) c a n b e

e x p r e s s e d a s f o l l o w s :

4

The A ( I , J ) c o e f f i c i e n t s o f t h e m a t r i x [ A ] o f t h e p a r t i a l d e r i v a t i v e s a r e numeri- c a l l y computed from r e l a t i o n ( 4 ) t o ( 6 ) . The s y s t e m a t i c e r r o r s a r e t h e n s o l u t i o n s o f t h e s y s t e m ( 7 1 , t h e r i g h t member b e i n g [6K! = 0 , ~R(K!), SKI

6Ser which i s r o u g h l y p r o p o r t i o n a l t o t h e a n g u l a r s h i f t o f t h e r e s o n a n c e d i p , c a n be I g n o r e d i n a f i r s t a p p r o x i m a t i o n .

I t i s o f i n t e r e s t t o d i s t i n g u i s h t h e p a r t i a l e r r o r s owing t o t h e b r o a d e n i n g 6K

K R R

(SKEi and 6 D) from t h e s e owing t o t h e ~ R ( K ; ) t e r m ( 6 ei and 6 D)

,

^{w i t h}

K R K R

6 ' ~ ~ = 6 e i + 6 E . and 6% = 6 D

+

6 D. We have computed t h e s y s t e m a t i c e r r o r s i n t h e c a s e o f s i l v e r l a y e r s o f t h i c k n e s s e s Dl = 44 nm and D2 = 60 nm r e s p e c t i v e l y , t h e former c o r r e s p o n d i n g t o D < D t h e l a t e r t o D > Dopt

.

K O P ~ ' R s

T i g . 2 shows t h e v a r i a t i o n s o f 6 E:, 6 E,, 6 6; ( u p p e r c u r v e s ) and gRD, 6 K ~ , 6 ' ~

I I I

( l o w e r c u r v e s ) v e r s u s t h e w a v e l e n g t h i n t h e s p e c t r a l domain 0.6 t o 0.9 ym.

F i g . 2 : S y s t e m a t i c e r r o r s SSc.

a n d 6 ' ~ ( f u l l l i n e ) , f o r s i l v e r f i l m s o f t h i c k n e s s Dl = 44 nm (DID ) a n d D2 = 60 nm

o p t

(D>Dopt) a s a f u n c t i o n o f A.

c u r v e s b : S y s t e m a t i c e r r o r s 6 S ~ i and 6 S ~ , i f t h e r e l a t i v e b r o a d e n i n g is i n c r e a s e d by 70%.

V e r t i c a l b a r s show t h e l i m i t s o f measurement u n c e r t a i n t i e s .

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ClO-1 12 JOURNAL DE PHYSIQUE

Whatever t h e w a v e l e n g t h , a b r o a d e n i n g o f t h e r e s o n a n c e l e a d s t o a p o s i t i v e e r r o r

K K

6 c; and a n e g a t i v e e r r o r 6 D.

.L

On t h e o t h e r hand, t h e minimum i n c r e a s e & R ( K ~ ) l e a d s t o Op o s i t e e r r o r s , a c c o r d i n g o p t

t o t h e c o n d i t i o n D < D o r D > Dopt; i . e . Both SRei, 6

rP

D a r e n e g a t i v e i n t h e c a s e D < D and p o s i t i v e i n t h e c a s e D > Dopt.

o p t A l s o shown a r e t h e u p p e r limits o f

i n t r i n s i c measurement u n c e r t a i n t i e s AE; a n d AD, c a l c u l a t e d i n t h e same way f o r a

I

p e r f e c t beam ( A X = a. = 0 ) a n d a b s o l u t e u n c e r t a i n t i e s on t h e e x p e r i m e n t a l v a l u e s dq < 0 . 0 1 d e g , AR < 0 . 0 0 2 , AK/AK+ < 0.005

S e v e r a l p o i n t s s h o u l d b e commented upon :

a ) A p a r t i a l c o m p e n s a t i o n o f t h e s y s t e m a t i c e r r o r s o n e . ( f o r D < D ) a n d o p t

D ( f o r D > D ) is o b s e r v e d . I n b o t h c a s e s l$ci

1

^{o r}

I

^$D\ n e v e r e x c e e d Asi o r AD o p t

r e s p e c t i v e l y .

K R

b ) I f D < Doot t h e n e g a t i v e v a l u e s o f d D a n d d D l e a d t o a c a l c u l a t e d t h i c k n e s s

- 8

v a l u e i n f e r i o r t o t h e t r u e t h i c k n e s s o f t h e l a y e r . The s y s t e m a t i c e r r o r

l & S ~ I

grows as a f u n c t i o n o f t h e w a v e l e n g t h a n d becomes g r e a t e r t h a n AD, beyond X =0.75um

R K

C) I n t h e c a s e where D > Dopt, 6 ci a n d 6 ci a r e p o s i t i v e , t h e n 6 ' ~ ~ i n c r e a s e s r e a d i l y v e r s u s A, a n d i m p o r t a n t e r r o r s a r e o b t a i n e d on t h e v a l u e s o f ei.

A s a m a t t e r o f f a c t , numerous e x p e r i m e n t s h a v e shown t h a t t h e c a l c u l a t e d t h i c k - n e s s a l w a y s d e c r e a s e s i n t h e i r domain a s w e l l f o r D<Dont, a s f o r D >DoDt, and t h e p r e c e d i n g r e s u l t s d o n o t t o t a l Ty a c c o u n t f o r t h e o b s e r v e 2 a n o m a l i e s .

We w i l l e m p h a s i z e t h a t t h e r e l a t i o n ( 3 ) i s t o o a c r u d e a p p r o x i m a t i o n f o r t h e e x a c t s h a p e o f t h e r e s o n a n c e d i p . When u s i n g t h e F r e s n e l ' s f o r m u l a i n p l a c e o f t h e s i m - p l i f i e d r e l a t i o n ( 3 ) , t h e e x a c t r e l a t i v e b r o a d e n i n g i s a l w a y s g r e a t e r t h a n GK/AK?, w h i l e t h e d ~ ( k ; ) t e r m is i n good a c c o r d a n c e . I n c r e a s i n g a r b i t r a r i l y SK/AK by 70

+

X

i n o u r a p p r o x i m a t e model, we o b t a i n t h e c u r v e s b , i n b e t t e r a g r e e m e n t w i t h t h e ex- p e r i m e n t a l o b s e r v a t i o n s .

CONCLUSION.

I n t h e i r domain, i m p o r t a n t d i s t o r t i o n s i n t h e r e s o n a n c e c u r v e s a r i s e from t h e an- g u l a r apertures, w h i l e t h e i n f l u e n c e o f AX r e m a i n s n e g l i g i b l e . These d i s t o r t i o n s l n v o l v e a n e g a t i v e s y s t e m a t i c e r r o r 6 " ~ which a c c o u n t s f o r t h e anomalous d e c r e a s e o f t h e measured t h i c k n e s s a s a f u n c t i o n o f A . Moreover a n i m p o r t a n t p o s i t i v e s y s - t e m a t i c e r r o r &'E i s i n d u c e d when D > Dopt, w h i l e i t i s o f minor i m p o r t a n c e i n t h e case where D

:

^Dopt.

REFERENCES :

/1/ E . KRETSCHMANN, Z. P h y s i k , 241, (19711, 313.

/2/ E. KRETSCHMANN a n d H. RAETER, Z. N a t u r f o r s c h A 23 (1968) 2135.

/3/ G. HINCELIN, Phys. Rev. B 2 4 (1981) 787.

/4/ I . POCKRAND, S u r f . S c i . 7 2 (1978) 577.