OPTICAL CONSTANTS OF LAYER STRUCTURES FROM ELLIPSOMETRIC DATA

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OPTICAL CONSTANTS OF LAYER STRUCTURES FROM ELLIPSOMETRIC DATA

S. Logothetidis, J. Spyridelis

To cite this version:

S. Logothetidis, J. Spyridelis. OPTICAL CONSTANTS OF LAYER STRUCTURES FROM ELLIPSOMETRIC DATA. Journal de Physique Colloques, 1983, 44 (C10), pp.C10-31-C10-34.

�10.1051/jphyscol:19831005�. �jpa-00223454�

JOURNAL DE PHYSIQUE

Colloque CIO, supplement au n012, T o m e

44, d k e m b r e 1983 page C

10-

3 1

OPTICAL

CONSTANTS

OF LAYER

STRUCTURES

FROM ELLIPSOMETRIC

DATA

S. Logothetidis and J. Spyridelis

Physics Laboratory, University of Thessaloniki, ThessaZoniki, Greece

~6sumi5 - I1 est possible de calculer lesconstantesoptiques nl,kl et

n,,,n,,

dl un cristal uniaxe, en utilisant des mesures ellipsom&triques sur un plan perpendiculaireB l'axe optique, avec l'aide d'un algorithme tr&s efficace.

Cette mkthode B 6t6 appliquGe au composB B-GaSe pour des longueurs d'onde de

3600

A

2 6800

A .

Abstract - With the help of an efficient algorithm it is possible to calculate the optical constants n~ . ^{kl and rill , kll} of an uniaxial crystal using ellipsome- tric measurements on a plane perpendicular to the optic axis (the cleavage

has been applied to B-GaSe compound in the spectral region

The determination of the optical constants in single-axis crystals, with the optic axis c normal to thc crystal surface, presents many difficulties. /1,2/. The method proposed by F. Meyer et a1

/ 3 / ,

according to which the optical constants of €-Case were calculated using ellipsometric data obtained from the crystal planes parallel and normal to the optic axis of the crystal, cannot be considered as a general one, due to thc fact that the layer crystals are extremely thin. N.K. Nangia et a1 /4/

havc rccently calculated, with satisfactory precision, the complex refractive index iiL normal to the optic axis c but not thc complex refractive index iill parallel to the optic axis c, from reflectivity measurements of normal and oblique incidence to a crystal plane mormal to the optic axis c of the layer compound Bi2 (Tel-,Sx)3.

In this paper we shall try to present a method for the determination of the optical constants of layer compounds, which is based on a combination of algorithmssuitablc for the solution of the non-linear least square problem of ellipsometric data.

I

-

GENERAL PRINCIPLESAW LEAST SQUARE PROBLEM IN ELLIPSOHETRY.

Ellipsometry studies the interaction between a polarized beam of light and an opti- cal medium

/ 5 / .

The change of the polarized state of the beam can be described by two independent parameters, i.e.

A

and I. In single-layer crystals, with their optic axis normal to the crystal surface, the Fresnel coefficiens will bc

1

-2

2

i

2

i.

=

[cosB-(nl - ^sin

⁸⁾

I / [cos8+(iif - sin 0) 1 (1)

2

5i

z

; _P .

^[

^iilql

^COSB

^{- (6; - sin}

^{0 )}

^{I i [Gl core}

⁺

^{(7: -sin}

⁸⁾

¹

( 2 )

where

0 represents

thc angle of incidence and fil =nl -ikl, 9,

^=n,,

- ^ikll are the prin- cipal complex refractive indices.

The Fresnel coefficients of relations

( 1 )

and

(2)

are related to the ellipsometric parameters A and

Y

as 6

= (? / > ) =

tanY.exp(jd).

P

s

For a certain wave-length

X

and for M measurements under Oi(i=l,?,..,M) different angles of incidence the following non-linear relations

will be derived: A: s

d(A,0.) and

I: = I

(A,Bi), where A=(al,a2,. . ,anlT represents the parameters vector. Due to the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831005

ClO-32

J O U R N A L

DE PHYSlQUE

e x p e r i m e n t a l e r r o r s a n d t o t h e n o n - l i n e a r i t y o f t h e s e r e l a t i o n s , f o r t h e d c t e r m i n a - t i o n o f t h e p a r a m e t e r s v e c t o r A , i t i s n e c e s s a r y t o a p p l y a n i t e r a t i o n mcthod t o t h e r r o r f u n c t i o n G(A) / 6 / .

For t h e s o l u t i o n o f t h e p r o b l e m i t i s n e c e s s a r y t o know t h e s e n s i t i v i t y o f G(A) i n i t s minimum r e g i o n a s w e l l a s t h e mode o f i t s c h a n g e f o r c e r t a i n d c f i n i t c v a l u e s o f p a r a m e t e r s ak / 7 / . Such a n a n a l y s i s c a n b e made w i t h t h e a i d o f a q u a d r a t i c form o f t h e T a y l o r e x p a n s i o n o f t h e f u n c t i o n G(A) a r o u n d t h e r e g i o n o f t h e b e s t v a l u e A , i . e .

w h e r e I i . .(A) r e p r e s e n t s t h e i j c l e m e n t o f H e s s i a n m a t r i x / 6 / .

1J

I n f i g . (1) t h e c r o s s - s e c t i o n s G ( a k ) a r e p r e s e n t e d when a l l t h e p a r a m e t e r s e x c e p t ak a r e f i x e d a t A*. C o n t o u r s o f G(A) combined w i t h t h e c r o s s - s c c t i o n s G(a ) c o m p l e t e thc p i c t u r e o f i t s b e h a v i o u r i n a n e x t e n d e d a r e a a r o u n d t h e b e s t v a l u e A * . k ~ h e in f o r m a - t i o n o b t a i n e d i n t h i s way f o r t h e b c h a v i o u r o f f u n c t i o n G(A) m a i n l y i n t h e a r e a o f t h e b e s t v a l u c o f G(A*) c a n b c u s e d i n o r d e r t o c h o o s e t h e b c s t a l g o r i t h m f o r a c e r t a i n o b j e c t i v e f u n c t i o n (;(A).

I n t h e c a s e o f a l a y e r compound w i t h o p t i c a l c o n s t a n t s a k ( n L , k l , n l , k l l ) we o b t a i n t h e c u r v e s o f f i g . (1). We o b s e r v c a s t r o n g d c p c n d e n c e o f f u n c t l o n ~ ( i ) on p a r a m e t e r s

5

and kl

.

We a l s o n o t i c e t h a t c u r v e s G ( n l ) , G ( k l l ) h a v e a q u a d r a t i c form a r o u n d A* . T h i s means t h a t b y a n y method, Gauss-Newton o r m o d i f i e d Gauss-Newton / 8 / , t h e c o n v e r g e n c e w i l l b e f a s t a n d t h e e v a l u a t i o n o f nL and kll p r e c i s c T h e f l a t n e s s o f c u r v e s G(nll) and G ( k l l ) a s shown i n f i g . ( I ) a r o u n d t h e mlnlmum o f G(A) s u g g c s t s a weak d e p e n d e n c e o n t h o s e p a r a m e t e r s . 'The parameters c o n v e r g e n c e w i l l b e s l o w s i n c e i t d e p e n d s on t h e v a l u e o f e l e m e n t s H i i / 6 / .

'The programme we a p p l i e d f o r t h e d e t e r m i n a t i o n o f t h e parameters (nl

,

kL

.rill

,kll ) was t h e MINIUT / 9 / . I t c o n s i s t s o f t h r e e d i f f e r e n t m i n i m i z a t i o n m e t h o d s w h i c h c a n b c a p p l i e d q u i t e i n d e p e n d e n t l y and by which wc c a n overcome t h e main d i f f i c u l t i e s f o r t h e d e t e r m i n a t i o n o f t h e p a r a m e t e r s (nl ,kL ,nil , k l l ) i . e . t h e i n i t i a l v a l u e s o f t h e p a - r a m e t e r s , t h e weak d e p e n d e n c e o f t h e e l l i p s o m e t r i c a n g l e s A and

+'

/ 1 , 2 / o n t h e p a r a - m e t e r s 11 ,kli and f i n a l l y t h e "bad'' b e h a v i o u r o f t h e f u n c t i o n G(ak) w h e r e a =TI , k k . Each o n c o f t h e t h r c e m i n i m i z a t i o n m e t h o d s we u s e d f o l l o w s a c e r t a i n p r o c c $ u r e . T e f i r s t o n e which i s a Monte C a r l o /9/ method i s a p p l i e d f o r t h e i n i t i a l m ' n ' m i z a t i o n o f t h e f u n c t i o n , s o t h a t t h e r e q u i r e m e n t f o r t h e c l o s e s t i n i t i a l v a l u c A t o f o f t h e v e c t o r p a r a m e t e r A t o t h e A* i s n o t n c c e s s a r y , e s p e c i a l l y f o r t h e m o s t p r o b l e m a t i c p a r a m e t e r s

rill

and kll. T h i s mcthod l i m i t s t h e p a r a m e t e r v e c t o r A t o t h e c l o s e s t p o s - s i b l e a r e a a r o u n d A

,

r e d u c i n g t h e v a l u e o f t h e f u n c t i o n G(A) t o a g r e a t e x t e n t . T h e s e c o n d m e t h o d , t h e o n e o f N e l d e r and Mead / 6 / , i s a s a f e a n d q u i t c r a p i d method when we a r e f a r f r o m t h e minlmum a n d c a n b e a p p l i e d f o r c o n v e r g c n c e t o t h e e x a c t minimum.

I n t h e c a s e o f o u r f u n c t i o n G ( A ) t h i s method g i v e s v e r y good v a l u e s t o t h e p a r a m c t c r v e c t o r A a n d c o n s e q u e n t l y a v c r y s m a l l v a l u e t o t h e f u n c t i o n G(A), n e c e s s a r y f o r t h e good e s t i m a t i o n o f a l l t h e p a r a m e t e r s . T h e t h i r d m e t h o d , t h c method o f F l e t c h e r / l o / , i s r a p i d a r o u n d a minimum o r a n e a r l y q u a d r a t i c a r e a a l t h o u g h s l o w e r i n t h c c a s e o f a "badly1' b e h a v c d f u n c t i o n / h / . T h i s method g i v e s v e r y good r e s u l t s f o r a l l t h e p a - r a m e t e r s i f t h e v a l u e o f G(A) i s v c r y s m a l l when t h c f i r s t i t e r a t i o n c y c l e b e g i n s and t h u s t h e p a r a m e t e r v c c t o r A i s v e r y c l o s e t o A * . T h i s c a n b e a c h i e v e d by t h e two f o r m e r m i n i m i z a t i o n m e t h o d s .

I I

-

EXPERIMENTAL PROCEDURE AND RESIII.I'S.

For t h e e l l i p s o m e t r i c m e a s u r c m c n t s a 8 - C a s e c r y s t a l o f 6 x 6 x 0 , 2 mm h a s b e e n u s e d . T h e back s i d e o f . t h e c r y s t a l was made r o u g h a n d b l a c k e n e d t o p r e v e n t u n d e s i r e d r e f l e c t i o n . E l l i p s o m e t r i c m e a s u r e m e n t s w e r e p c r f o r m c d u s i n g s t a n d a r d t e c h n i q u e s o n a G a c r t n e r L-119X e l l i p s o m e t e r i n t h e PCSA c o n f i g u r a t i o n . A s t h e m e a s u r e m e n t s had t o b e rclcascld by a s many e r r o r s a s p o s s i b l e , i t was n c c e s s a r y t o d c v e l o p s u i t a b l e p r o c e d u r e s f o r t h e a l i g n e m e n t a n d c a l i b r a t i o n o f t h e e l l i p s o m e t e r / 1 1 , 1 2 / . A s p e c i a l a t t e n t i o n was p a i d i n t h e t u n i n g o f f t h e B a b i n e t - S o l e i l c o m p e n s a t o r f o r e x a c t q u a r t e r wave r e t a r - d a t i o n / 1 3 / . T h u s , t h e r e a d i n g s w e r e t a k e n i n f o u r z o n e s t o c o r r e c t a l l d e v i a t o n s and e r r o r s . F o r m e a s u r e m e n t s i n t h e 3600 - 6 8 0 0

R

w a v e l e n g t h r e g i o n t h e e l l i p s o m e t e r was e q u i p p e d w i t h a H a l o g e n lamp and a s c t o f i n t e r f e r e n c e f i l t e r s , o f FWHM b e t w e e n 20 t o 5 0

8 .

'The a n g l e s o f i n c i d e n c e i n t h e e l l i p s o m c t r i c measurements, c o v e r c d t h e r a n g e

F i g . 1 - C r o s s s c c t i o n s o f t h e e r r o r f u n c t i o n . F i g . 2

-

Thc c o r r e c t c d o p t i c a l con- (H) f o r n p a r a m e t e r , (*) f o r kl parame- s t a n t s nl ,kL and

rill

^, kll o f B-GaSc a t t c r and

(A)

^{f o r}

^rill

p a r a m c t e r and

(-1

f o r kII. room t c m p e r a t l l r e .

from 40' t o 7 0 ~ . I J s u a l l y f i v e measurements wcrc made f o r each wavelength

171e f o u r o p t i c a l c o n s t a n t s f o r each w a v e l e n t h can b e d e t e r m i n e d from t h e f i v e measu- r e d e l l i p s o m e t r i c a n g l e s (Ai "iJi), c o r r e s p o n d i n g t o t h e d i f f e r e n t a n g l e s of incidence

ei.

The c a l c u l a t i o n o f t h e f o u r o p t i c a l c o n s t a n t s ( n l , k l , n ~ l , k l l ) c o n s i s t s e s s e n t i a l l y o f successively minimizing t h c G(A) f u n c t i o n . An i n i t i a l v a l u e no and kf' f o r t h i s c a l c u - l a t i o n was computed from t h e e q u a t i o n iio = N o t a n B . ( l - 1 6 . s i , n B i / ( l + 6 i ) 2 ) 4 , w i t h o n t

i!

t a k i n g i n t o a c c o u n t t h e a n i s o t r o p y o f t h e m a t e r i a i . 'This i s t r u c f o r a f i r s t o r d e r a p p r o x i m a t i o n . A d i f f i c u l t y a p p e a r e d i n t h e i n i t i a l v a l u e s o f nf,kl;): 'So c a l c u l a t e t h o s e two, e s p e c i a l l y i n t h e r e g i o n o f w a v e l e n g t h s o v c r t h e absorption e d g c o r u n d e r t h e e n c r y y gap v a l u e , whcrc t h c v a l u e s o f k l , k l a r e much s m a l l c r t h a n t h e r e s p e c t i v e v a l u e s of t h e r e f r a c t i v e i n d i c e s , i t i s t o a p p l y t h e mcthod o f t r a n s m i t t i n g beam under v a r i o u s o b l i q u e a n g l e s o f incidence and m u l t i p l e reflections. From t h e

~llaxima and minima o f t r a n s m i t t a n c e Ts(B) and T (B), i t i s p o s s i b l e t o c a l c u l a t e t h e P v a l u e s o f t h e c o r l s t a n t s nl ,rill, t h e t h i c k n e s s e s o f t h e c r y s t a l s and t o e v a l u a t e kl a n d k l l .

S i n c e t h e R-GaSe i s t r a n s p a r e r l t f o r l i g h t n e a r X ~ 6 3 0 0

a

( f i g . 2 ) t h e d e v i a t i o n s of A from 180°, must b c due t o a n u n c o n t r o l l e d o v e r l a y e r , a l w a y s p r e s e n t i n a i r . A l s o w a t e r was found t o be a d s o r b e d p t l y s i c a l l y . Thus, a c l e a n s u r f a c e exposed t o a i r would t a k e up s e v e r a l monolayers o f w a t e r I n a few h o u r s / 3 / .

We c a l c u l a t e d t h e f o u r p a r a m e t e r s nl ,rill ,kl ,kll f o r each wavelerlgth t w i c e . Thc f i r s t t i m e wc supposed t h a t t h c GaSc s u r f a c e was f r e e o f t h a t o v e r l a y e r and s o wc a p p l i e d t h e minimizing method, g i v e n by t h c G(A) f u n c t i o n . I n t h a t c a s c t h c v e c t o r of t h e p a r a m c t e r s was A=(n1 ,kl

,?

, k l l ) T . The s e c o n d t i m e we supposed t h a t on t h e B-GaSc s u r f a c c was formcd an isotropic, homogeneous d i e l e c t r i c f i l m / 3 / , w i t h a r e f r a c t i o n

index N1 Then, i n t h e (;(A) f u n c t i o r l , t h e v e c t o r o f t h e p a r a m e t e r s was A=(nl ,kl , n ~ l , kll ,]il,d)', where d i s t h c t h i c k n e s s o f t h e l a y e r f o r which d

<A.

I n t h a t c a s e we r e - g a r d e d t h a t t h e kll p a r a m e t e r was u n e f t ' e c t e d f r o m t h e o v e r l a y e r and s o we r e d u c c d t h e dimension o f t h e v e c t o r A i n o r d e r t o rninimizc e a s i l y t h e f u n c t i o n G(A). A f t e r w a r d s we c o r r e c t c d t h e kll p a r a m e t e r which was n o t changed b e c a u s e o f t h e o v c r l a y c r .

C10-34 JOURNAL

DE

PHYSIQUE

The c o r r e c t o p t i c a l c o n s t a n t s a r e c a l c u l a t e d by t h i s p r o c e d u r e a s shown i r ~ f i g . 2 . The d e v i a t i o n i n kl p a r a m e t e r was 0 . 0 7 a t s h o r t e r w a v e l e n g t h s a n d 0 . 0 5 a t l o n g e r wa- v e l e n g t h s ( a t t r a n s p a r e n t r e g i o n ) . A s i m i l a r c h a n g e was n o t i c e d a l s o f o r t h e nl p a r a - m e t e r . While f o r t h e p a r a m e t e r s

yl

and k l l . t h e r e s p e c t i v e d e v i a t i o n was w i t h i n t h e l i m i t o f e s t i m a t i o n e r r o r . F u l l d e t a i l s w i l l b e g i v e n e l s e w h e r e ; we i n d i c a t e h e r e o n l y t h e e s s e n t i a l r e s u l t s . The c a l c u l a t i o n of n k

y

,kll,,. was good even f o r any random e r r o r o f t h e c l l i p s o m e t r i c a n g l e s :'~6,:A6 k : l b . I h e m i n i m i z i n g method we used was good f o r M=3 i n c a s e we d i d n o t r e g a r d a t f i i n f i l m . The e r r o r i n t h e e l l i p s o m e - t r i c a n g l e

A?,

b e c a u s e o f a produccd o v e r l a y e r

&A?

i n c r e a s e s w i t h i n c r e a s i n g a n g l e of i n c i d c n c e ' a n d a p p r o a c h e s t h e p r i n c i p a l a n g l e o t ' i n c i d e n c e . A comparison o f t h e p r e s e n t nl,

rill

and kl ,kl, v a l u e s v e r s u s

X

w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s shows t h a t t h e p r e s e n t work a g r e e s with t h e d a t a o b t a i n e d by Meyer e t a 1 /3/, T o u l e t e l a1 /14/

and Akhundov e t a 1 / 1 5 / . D i f f e r e n c e s i n kl below t h e 3eV (-4200

a )

e n e r g y , between o u r r e s u l t s and t h o s e o f hleycr / 3 / may b e due t o t h e d i f f e r e n t growth method o f GaSe and t o t h e d i f f e r e n t t r e a t m e n t o f t h e s u r f a c e l a y e r .

111 - CONCLUSIONS.

The p r e s e n t work d e m o n s t r a t e s how c l l i p s o m e t r y , o n a s u r f a c e p e r p e n d i c u l a r t o t h e o p t i c a l a x i s c , can b e used t o d e t e r m i n e t h e o p t i c a l c o n s t a n t s o f a c r y s t a l w i t h u n i a x i a l symmetry, i n t h e case t h a t in the prvgrammc a s u i t a b l e and e f f i c i e n t a l g o r i t h m w i l l b e u s e d . By measuring on a s u r f a c e , s o u r c e s f o r s y s t e m a t i c e r r o r s might b e e a s i l y c o n t r o l l e d . To o b t a i n a c c e p t a b l e s o l u t i o n s , t h e r e q u i r e m e n t s on i n i t i a l e s t i - m a t i o n s o f t h e unknown p a r a m e t e r s

rill

and kll a r c i m p o r t a n t . A c c e p t a b l e s o l u t i o n s a r e o b t a i n e d even i f t h e m e a s u r i n g e r r o r s o f d and '4' a r c a s l a r g e a s f 0.1'.

We must be c a r e f u l i n t h e case t h a t an o v e r l a y e r i s formed on t h e s u r f a c c b e c a u s c o f a d s o r p t i o n . I n t h a t c a s e we s h o u l d a l s o t a k e i t i n t o c o n s i d e r a t i o n f o r t h e a c c u r a t e c a l c u l a t i o n o f t h e o p t i c a l p a r a m e t e r s . Our r e s u l t s f o r GaSe a r c i n good agreement w i t h p r e v i o u s l y p i l b l i s h e d d a t a .

Aknowledgmerlts

.

-

Wc wish t o t.hank Ur. K . blanolikas f o r t h i s h e l p f o r t h e i d e n t i f i c a t i o n o f t h e s t r u c t u - r e o f o u r compound w i t h t h a t o f 8 - C a s e . S . L . would l i k e t o e x p r e s s e s h i s i n d e b t e d n e s s t o t h e p u b l i c B e n e f i t F o u n d a t i o n "Al.EXANDER S . ONASSIS" f o r t h e award o f a f e l l o w s h i p . REFERENCES.

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