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Field-induced optical anisotropy in thin niobium oxide films Yee, Kai Kwan 1974

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FIELD-INDUCED OPTICAL ANISOTROPY IN THIN NIOBIUM OXIDE FILMS by KAI KWAN YEE B.A.Sc, Uni v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1974 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes is fo r f inanc ia l gain sha l l not be allowed without my written permission. Depa rtment The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date > c . 2 , I JILL , ABSTRACT An automated ellipsometer was used to study field-induced opti-cal anisotropy in anodic niobium oxide films. The oxide films were found to change from the optically isotropic state to the optically ani-sotropic state when an electric f i e l d was applied normal to the film surface. The anisotropic refractive indices of the oxide films decreased quadratically while the thickness of the films increased quadratically with the applied f i e l d . The quadratic electro-optic coefficients were determined. The changes i n refractive indices and i n thickness of the oxide films were found to be independent of time. Field recrystallization of the anodic -niobium oxide films was investigated using a scanning electron microscope. The results are ccirps^sd. Vt—till oss r^°po*v"t — — for ~ si? od.ic ~ t BJV "t 51 J^T" • oy^ Hp. f 7. T TT»C -1 rt t~.r» mi re-lished literature. i TABLE OF CONTENTS Page ABSTRACT , . . . . . . . . . . . . . i TABLE OF CONTENTS i i LIST OF ILLUSTRATIONS i v ACKNOWLEDGEMENT v i 1. INTRODUCTION . 1 2. THEORY OF ELECTRO-OPTIC EFFECT IN THIN OXIDE FILMS 3 3. PREVIOUS WORK 9 4. PRINCIPLES OF ELLIPSOMETRY 10 4.1 Ellipsometry f o r a Homogeneous Is o t r o p i c Substrate Covered by a Single-layer Homogeneous Is o t r o p i c Film. 10 4.2 Ellipsometry for Homogeneous U n i a x i a l Anisotropic Non-absorbing Films on I s o t r o p i c Substrates 12 5. EXPERIMENTAL ARRANGEMENTS 16 5.1 The Ellipsometer 16 5.1.1 System Components 16 5.1.2 The Balancing Routine 16 5.1.3 Ellipsometer Alignment . 19 5.2 Anodization C e l l 22 5.3 Window Erro r 24 5.3.1 E r r o r of Windows Before Mounting on the C e l l . 24 5.3.2 E r r o r of Windows A f t e r Mounting on the C e l l . . 28 5.4 Sample Preparation 28 6. RESULTS 30 6.1 O p t i c a l Anisotropy of Anodic Niobium Oxide Films. . . 30 6.1.1 Experimental Procedure 30 6.1.2 Results 31 6.1.3 Discussion 36 6.2 F i e l d and Time Dependence of E l e c t r o - o p t i c and E l e c -t r o s t r i c t i v e E f f e c t s 37 6.2.1 Experimental Procedure 37 6.2.2 Results 38 6.2.3 Discussion 42 6.3 R e p r o d u c i b i l i t y of Ellipsometer Measurements. . . . . 43 i i P a g e 7 . F I E L D R E C R Y S T A L L I Z A T I O N O F A N O D I C N I O B I U M O X I D E F I L M S . . . . 4 7 8 . C O N C L U S I O N S -53> B I B L I O G R A P H Y . 5 4 A P P E N D I X . 5 6 i i i LIST OF ILLUSTRATIONS Page Fi g . 2.1 I n d i c a t r i x before and a f t e r an e l e c t r i c f i e l d i s applied 7 F i g . 4.1 R e f l e c t i o n from a film-covered surface 10 F i g . 5.1 Schematic of the automated ellipsometer system 17 F i g . 5.2 Zero error i n p o l a r i z e r and analyzer scales 21 F i g . 5.3 Structure of the c e l l f o r i n s i t u ellipsometer measurements 23 F i g . 5.4 Poincare sphere representation of the e f f e c t of window error on the p o l a r i z a t i o n state of l i g h t 24 Fi g . 5.5 Window error i n A against r o t a t i o n of the window 27 F i g . 6.1 Lower portion of the A - i J i p l o t f o r increasing thickness of a niobium oxide f i l m up to two cycles 32 Fi g . 6.2 Upper portion of the A — p l o t 33 Fi g . 6.3 (a) Theoretical values of minimum A as functions of 3 and n 0 34 (b) T h e o r e t i c a l values of maximum A as functions of 8 and n Q 35 F i g . 6.4 The change i n r e f r a c t i v e indices n Q and n g against E . . 39 Fi g . 6.5 The change i n f i l m thickness against E 2 40 F i g . 6.6 The change i n ordinary r e f r a c t i v e index and i n thickness with time when the f i e l d was switched from zero to 1.78 x 10 6 v/cm 41 Fi g . 6.7 Interpolating a set of A-iJ> data 44 F i g . 6.8 Error (distance) between two experimental A - i p curves versus thickness of the oxide f i l m : (a) No applied f i e l d , (b) With applied f i e l d 45 i v P a g e F i g . 7 . 1 R e c r y s t a l l i z e d a r e a s o f a n o d i c n i o b i u m o x i d e f i l m 2 f o r m e d a t c o n s t a n t 0 . 2 m A / c m t o 1 5 0 v o l t s , a t 2 3 ° C i n 0 . 2 N H2SO4 4 9 F i g . 7 . 2 S a m e s a m p l e a s F i g . 7 . 1 . R e c r y s t a l l i z e d a f t e r t h e a m o r p h o u s f i l m w a s e t c h e d o f f w i t h H F . 5 0 F i g . 7 . 3 R e c r y s t a l l i z e d a r e a s o f n i o b i u m o x i d e f i l m a n o d i z e d a t 2 3 ° C , i n 0 . 2 N H ^ S O ^ , a t c o n s t a n t 0 . 2 m A / c m 2 t o 1 4 9 v o l t s , t h e n a t c o n s t a n t 1 2 0 v o l t s f o r 1 0 h o u r s 5 1 F i g . A . l L o c u s o f t h e e l e c t r i c v e c t o r f o r e l l i p t i c a l l y p o l a r i z e d l i g h t i n a p l a n e n o r m a l t o t h e d i r e c t i o n o f p r o p a g a t i o n . 5 6 F i g . A . 2 R e p r e s e n t a t i o n o f p o l a r i z e d l i g h t o n P o i n c a r e s p h e r e . . . 5 8 F i g . A . 3 R e p r e s e n t a t i o n o f p h a s e r e t a r d a t i o n o n P o i n c a r e s p h e r e . . 5 8 v A C K N O W L E D G E M E N T I w i s h t o e x p r e s s m y s i n c e r e g r a t i t u d e f o r t h e g u i d a n c e r e c e i v e d f r o m m y s u p e r v i s o r D r . L . Y o u n g t h r o u g h o u t t h e c o u r s e o f t h i s w o r k . I a l s o t h a n k D r . M . H o p p e r a n d M r . W . C o r n i s h f o r m a n y h e l p f u l d i s c u s s i o n s . I a m i n d e b t e d t o M e s s r s . J . S t u b e r , A . M a c K e n z i e , H . B l a c k , a n d A . L a c i f f o r t h e i r t e c h n i c a l a s s i s t a n c e , a n d t o M i s s S h e l a g h L u n d f o r h e r p a t i e n c e i n t y p i n g t h i s t h e s i s . F i n a l l y , t h e f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a i s g r a t e f u l l y a c k n o w l e d g e d . v i 1 1. INTRODUCTION It has been suggested"*" that films of Ta^O^ and Nb^ O^ - could be used as l i g h t modulators because of the dependence of t h e i r o p t i c a l 2 constants on e l e c t r i c f i e l d . Films of la^O^ and ^ 2 ^ 5 n a v e been found to be a promising material for use as o p t i c a l waveguides because they are easy to prepare, are chemically stable, and have high r e f r a c t i v e indices and large bandgaps. The e l e c t r o - o p t i c properties of Ta^O^ films have been investigated i n some d e t a i l " * " ' ^ ' ^ ' " ^ ^ . • The objective of the present work was to extend the knowledge i n the e l e c t r o - o p t i c properties of niobium oxide fi l m s which have not. been f u l l y . investigated'. In the published l i t e r a t u r e , t h i n Ta^O^ films have been taken as o p t i c a l l y i s o t r o p i c except by Cornish and Young^ who showed that such films became o p t i c a l l y anisotropic when an e l e c t r i c f i e l d was applied across the f i l m s . The work described i n t h i s thesis was to study t h e o r e t i c a l l y and experimentally the field-induced o p t i c a l anisotropy which i t was expected that anodic niobium oxide films would show. In s i t u ellipsometry was used i n t h i s study. Presented i n chapter 2 of t h i s thesis i s the theory of e l e c t r o -optic e f f e c t i n t h i n oxide fil m s . Chapter 3 describes the previous work on e l e c t r o - o p t i c e f f e c t i n films of T a ^ and Nb20 . The ellipsometry p r i n c i p l e s which have been applied i n t h i s work are presented i n chapter 4. Chapter 5 describes the general experimental arrangements: the automated ellipsometer, the c e l l , the windows, and the preparation of samples. Described also i n this chapter i s a method which was devised to investigate the birefringence of the c e l l windows, which i s a possible 2 source of error with i n s i t u ellipsometry. Presented i n chapter 6 are the experimental r e s u l t s on: (a) the field-induced o p t i c a l anisotropy i n anodic niobium oxide f i l m s , (b) quadratic e l e c t r o - o p t i c and e l e c t r o s t r i c t i v e e f f e c t s , and (c) the r e p r o d u c i b i l i t y of ellipsometer measurements. F i e l d r e c r y s t a l l i z a t i o n of the niobium oxide films was found to be a serious problem encountered i n anodization during t h i s work. The r e c r y s t a l l i z e d oxides were examined by using a scanning electron microscope. The res u l t s are discussed i n chapter 7. Conclusions are presented i n chapter 8. 3 2. THEORY OF ELECTRO-OPTIC EFFECT IN THIN OXIDE FILMS The e f f e c t of changing the r e f r a c t i v e index of a material by applying an e l e c t r i c f i e l d ( i n our case a dc f i e l d ) i s c a l l e d the e l e c -t r o - o p t i c e f f e c t . The r e f r a c t i v e index of a material as a function of the d i r e c t i o n of the light-wave normal i s s p e c i f i e d by the " o p t i c a l i n -d i c a t r i x " which i s the e l l i p s o i d defined by B ± j x ± X j = 1 ( i , j = 1,2,3) (2.1) where x^, X£ and x^ are the p r i n c i p a l axes of the r e l a t i v e d i e l e c t r i c impermeability tensor at o p t i c a l frequencies, where 3E° B i j = K o 9D° and K Q i s the p e r m i t t i v i t y of a vacuum, and the superscript o s i g n i f i e s o p t i c a l frequency f i e l d s . The change i n r e f r a c t i v e index of a material due to an applied f i e l d or other cause i s i n general represented by a change i n the shape, s i z e and o r i e n t a t i o n of the i n d i c a t r i x . The oxide films under study are believed to be amorphous and hence should be i s o t r o p i c with no e l e c t r i c f i e l d applied. I t s r e f r a c t i v e index, n^, can be represented by an i n d i c a t r i x which i s a sphere of the form B (x, 2 + x 2 + x 2) = 1 o J- z J where B = 1/n2, x,. i s defined along the d i r e c t i o n normal to the f i l m o n' 3 b surface, x^ and X2 are p a r a l l e l to the f i l m surface. When an e l e c t r i c f i e l d i s applied normal to the f i l m surface, the oxide f i l m has a r o t a t i o n a l symmetry about the axis i n x^ d i r e c t i o n . The i n d i c a t r i x should therefore become an e l l i p s o i d of r o t a t i o n about 4 the f i l m normal. I f we had f i e l d s of one frequency, say dc, instead of dc plus o p t i c a l , the e l e c t r i c displacement D could be expressed i n terms of the e l e c t r i c f i e l d as D. = a. . E. + 6. ., E .E +).,. „E,E,E, + ... ( i , j , k , £ = 1,2,3) (2.2) where a^_., B ^ j ^ a n d Y ^ j j ^ a r e tensor components. Let us suppose that the d i r e c t i o n of the e l e c t r i c f i e l d i s now reversed so that E., E, and J k E become -E., -E, and -E , r e s p e c t i v e l y . Since the oxide f i l m i s am-orphous, the reversed f i e l d "sees" an i d e n t i c a l oxide f i l m as o r i g i n a l l y so that i s expected to become -D^. Equation 2.2 becomes -D. = -a. .E. + 6. E.E. -y. .. .E.E, E„ + ... l i] ] ljk j k 'ljkJl j K H (i,j , k , J l = 1,2,3) (2.3) Equations 2.2 and 2.3 can hold simultaneously only i f the c o e f f i c i e n t s for the even power terms equal zero. The s t a t i c p e r m i t t i v i t y of the amorphous oxide f i l m i s defined 3D. K i j = a i r = a i j + Y i j k A E * + ••• ( i , j , k , £ = 1,2,3) (2.4) In our case, we are concerned with the p e r m i t t i v i t y at o p t i c a l f r e -quencies for r e l a t i v e l y low i n t e n s i t y l i g h t as a function of a r e l a t i v e l y large dc e l e c t r i c f i e l d . By analogy with the above dc c a s e ^ the per-m i t t i v i t y at o p t i c a l frequencies can be expressed as a power serie s i n the dc e l e c t r i c f i e l d : K l j = 3 i y = a i j + b i j k A E £ + • • • (l,j,k,£ = 1,2,3) (2.5) 5 where the superscript o (as before) stands f o r o p t i c a l frequencies, E k and E^ are the components of the dc e l e c t r i c f i e l d . Terms higher than power two are neglected. By contrast with non-centrosymmetric c r y s t a l s there i s no l i n e a r e l e c t r o - o p t i c e f f e c t . The c o e f f i c i e n t a ^ i s the o p t i c a l p e r m i t t i v i t y when the dc f i e l d i s zero. 8 9 Since i t i s considered ' that p o l a r i z a t i o n i s the more funda-mental p h y s i c a l v a r i a b l e i n e l e c t r o - o p t i c e f f e c t as compared to the e l e c t r i c f i e l d , the r e l a t i o n f o r the quadratic e l e c t r o - o p t i c e f f e c t i s conventionally expressed as B i j = B o + 8 i j k A P i > ° r A B i j " 8 i j k £ P k P £ (i,j,k,£ = 1,2,3) (2.6) where P i s the p o l a r i z a t i o n , and A B „ i s the change i n r e l a t i v e o p t i c a l impermeability when a f i e l d i s applied. 10 I t i s known that f o r thermodynamic reasons AB.. = AB.. f o r a l l P kP^. Hence 8ijk£ 8jik£ Since P^P^ = P £P k» we have 8 i j k i , 8ij£k The tensor relation 2.6' can therefore be reduced to the matrix form AB = g A m mn n (m,n = 1,2,...6) (2.7) where AB = m AB AB AB AB AB AB 11 22 33 23 13 12 6 A = n mn P P 2 3 P P r r 3 p P 1 2 and — 0 gmn " Sijk£ when n = 1,2,3 when n = 4,5,6 In the case considered here, the p o l a r i z a t i o n i s i n d i r e c -t i o n . Using the form of g for i s o t r o p i c material"^, equation 2.7 i s ° mn written out i n f u l l AB^ " S l l S12 g12 0 0 0 0 g12 P 3 2 AB 2 g12 g l l 812 0 0 0 0 812 P 3 AB 3 g12 812 g l l 0 0 0 P 2 3 811 P 3 2 (2.8) AB, 4 0 0 0 g44 0 ' 0 0 0 AB 5 0 0 0 0 g44 •o 0 0 AB, 6 0 0 0 0 0 844_ 0 0 where g ^ may be shown to depend on g ^ and g-j^' By d e f i n i t i o n B^, 344 and B^ correspond to the three p r i n c i p a l r e f r a c t i v e indices as B. = —2-l n. ( i = 1,2,3) (2.9) For our case.B^ = =j= B^, because of the symmetry. From equation 2.9 AB. = -(-^-r) An. ( i = 1,2,3) I Because the change i n r e f r a c t i v e index a f t e r a f i e l d i s applied i s small compared to the z e r o - f i e l d r e f r a c t i v e index n^, we may replace n^ by n n to a s u f f i c i e n t approximation. The changes i n r e f r a c t i v e indices are therefore n~ A n l = A n 2 = " 2 n v n g12 P3" (2.10) n A n 3 = " f " g l l P 3 2 Because An^ = A i ^ =f= An^, when a dc e l e c t r i c f i e l d i s applied normal to the f i l m surface, the oxide f i l m i s expected to be u n i a x i a l a n i s o t r o p i c with i t s o p t i c axis i n the d i r e c t i o n of the surface normal. The i n d i c a t r i x f o r the ani s o t r o p i c f i l m i s an e l l i p s o i d of r o t a t i o n about the surface normal, the sign of the e f f e c t s being, as i t turns out, as shown i n exaggerated manner i n F i g . 2.1. The ordinary r e f r a c t i v e index n F i g . 2.1 I n d i c a t r i x before and a f t e r an e l e c t r i c f i e l d i s applied. i n the x^x 2 plane, and the extraordinary r e f r a c t i v e index n £ i n x^ d i r e c -t i o n are expressed as n An = n - n = x— g, 0 P 0 2 o o n 2 °12 3 An = n - n = - n g,, P_ e e n —TJ— 11 3 (2.11) Since An Q and An g both have quadratic dependence on the applied f i e l d they should be r e l a t e d through the r e l a t i o n An = B An e • o (2.12) where B i s a constant independent of the applied f i e l d . The value of 6 also characterizes the changing of the f i l m from the i s o t r o p i c state to 8 the anisotropic state when a f i e l d i s applied. If 3 equals one, the f i l m remains i s o t r o p i c . 9 3. PREVIOUS WORK The e l e c t r i c field-induced modulation of i n t e n s i t y r e f l e x i o n i n anodic oxide films was discovered by Holden and Ullman^'^. The modulation was a t t r i b u t e d to the e l e c t r o s t r i c t i v e compression of the oxide f i l m s . 1 3 Frova and M i g l i o r a t o ' , analyzing the electromodulation of the interference spectrum of anodic T a 2 ° 5 f i l m s , suggested that the dominant e f f e c t was the e l e c t r o - o p t i c modulation o'ffethe r e f r a c t i v e index of the f i l m s , rather than e l e c t r o s t r i c t i o n . The experimental values of the quadratic e l e c t r o - o p t i c c o e f f i c i e n t s of Ta^O films were compared to those of oxygen-octahedra f e r r o e l e c t r i c c r y s t a l s . Ord, Hopper and Wang , applying ellipsometry to the problem, reportedhthattitheifdilmhthltcknessriinereasediMnea.rly'iwhereas' the r e f r a c t i v e index decreased l i n e a r l y on increasing the applied f i e l d . They also suggested that the only contribution to the change i n r e f r a c t i v e index with f i e l d was the change i n the bulk density of the f i l m ( i . e . the f i l m thickness ). Only a narrow range of e l e c t r i c f i e l d was used i n t h e i r studies. A l l the above authors took the oxide films as o p t i c a l l y i s o t r o p i c . Cornish and Young^ showed that anodic tantalum oxide films were o p t i c a l l y i s o t r o p i c in.the absence of an e l e c t r i c f i e l d , but became anisotropic when an e l e c t r i c f i e l d was applied normal to the f i l m surface. Also varying the applied f i e l d from zero to j u s t below the formation f i e l d , they showed that the changes i n anisotropic r e f r a c t i v e indices and thickness were both q u a d r a t i c a l l y field-dependent. They observed fast and slow components i n the time-dependence of the changes i n indices and thickness. 10 4. PRINCIPLES OF ELLIPSOMETRY 4.1 Ellipsometry for a Homogeneous Isotr o p i c Substrate Covered by a Single-layer Homogeneous Is o t r o p i c Film Ellipsometry i s a convenient method for the o p t i c a l i n v e s t i g a t i o n of t h i n films on r e f l e c t i n g surfaces immersed i n l i q u i d s , provided that the r e f r a c t i v e index of the sample i s s u f f i c i e n t l y d i f f e r e n t from that of the l i q u i d . The p r i n c i p l e of ellipsometry i s to measure the changes of state of p o l a r i z a t i o n when an e l l i p t i c a l l y p o l a r i z e d monochromatic l i g h t beam i s r e f l e c t e d by a surface, or a surface covered with a transparent f i l m . From these changes, the o p t i c a l constants of the system can be calculated. Ellipsometry for systems of homogeneous i o s t r o p i c l a y e r s , as shown i n F i g . 4.1, has been described i n d e t a i l by many authors 11,12,13 so that only a b r i e f o u t l i n e need be given here. LIGHT SUBSTRATE A M B I E N C E n, -> 7 »" F I L M F i g . 4 . 1 R e f l e c t i o n from a film-covered surface The e l e c t r i c f i e l d vector of an incident plane wave can be resolved into component E , p a r a l l e l to the plane of incidence, and component E , normal to the plane of incidence. A system of an i s o t r o p i c substrate covered by an i s o t r o p i c f i l m r e f l e c t s p- and s-polarized l i g h t i n t o p- and s-polarized l i g h t r e s p e c t i v e l y . The t o t a l r e f l e c t i o n c o e f f i c i e n t s for p - l i g h t , R , and s - l i g h t , R , have the form P s " 2 1 6 r l 2 4 + r23 e R = = " -216 ( 4 - 1 } 1 + r 1 2 r 2 3 e where r ^ 2 a n < i r 2 ^ a r e t n e Fresnel r e f l e c t i o n c o e f f i c i e n t s for the ambience/ f i l m and film/substrate i n t e r f a c e r e s p e c t i v e l y , and 6 i s the quantity . 2ft d , 2 2 . 2 . . N l / 2 0 . <S = — ( n 2 - n s i n <)>].)) (4.2) where d=*= thickness of f i l m A = wavelength of l i g h t i n vacuum n^= r e f r a c t i v e index of ambience n 2= r e f r a c t i v e index of f i l m $1= angle of incidence. The Fresnel c o e f f i c i e n t s for p- and s - l i g h t are of the form n 2 cos $i c o s <t>2 "12 n 2 cos <\>i + n^ cos <t>2 ,P _ n^ cos 4* j - n 2_cos <J>2 "12 n^cos $i + n 2 cos <j>2 ( 4 . 3 ) s The changes of state of p o l a r i z a t i o n on r e f l e c t i o n are characterized by the r e l a t i v e amplitude reduction, tan ' i j i , and r e l a t i v e phase change, A , which are related to R and R through the "fundamental P s equation of ellipsometry" R tan ^ e = --f— .(4.4) K S An ellipsometer measures the quanti'ties^dand A. I f enough measurements are taken, or enough information about the substrate i s a v a i l a b l e , the 12 o p t i c a l constants of the f i l m can be calculated u t i l i z i n g equations 4.1, 4.2 and 4.3. 4.2 Ellipsometry for Homogeneous Uni a x i a l Anisotropic Nonabsorbing Films on I s o t r o p i c Substrates. The r e f l e c t i o n , r e f r a c t i o n and absorption of l i g h t by anisotropic 14 materials were f i r s t studied by Drude . The complex Fresnel equations for reflectance for u n i a x i a l anisotropic absorbing c r y s t a l s were derived 15 16 by Mosteller and Wooten . Schopper formulated the o p t i c a l properties 17 18 of b i r e f r i n g e n t f i l m s . Azzam and Bashara and De Smet studied ellipsometry f o r anisotropic materials withe t h e i r optic axes p a r a l l e l to 19 20 the material surface.. Both Den Engelsen, and Tomar and Srivastava studied ellipsometry for a u n i a x i a l anisotropic f i l m between two i s o t r o p i c media i n a three layer system, with the optic axis of the anisotropic f i l m normal to the f i l m surface. Here we w i l l r e s t r i c t ourselves to the case studied by Den Engelsen and by Tomar et a l . since i t applies to the system studied i n this work. As shown i n chapter 2, for a system of a u n i a x i a l l y anisotropic f i l m on an i s o t r o p i c substrate, with the optic axis normal to the f i l m surface, the r e f r a c t i v e index i s represented by an o p t i c a l i n d i c a t r i x which i s an e l l i p s o i d of rotation about the surface normal. The s - l i g h t and the p - l i g h t i n the f i l m respond to two d i f f e r e n t r e f r a c t i v e i n d i c e s , namely n and n res p e c t i v e l y . These two r e f r a c t i v e indices obey the s p equat ions <: ua t i ons 2 2 n = n (4.5) o s 2 • ' 2 A 2 2 , n sxn <f> n cos d> „ P 2 P - + P 2 P = 1 (4.6) n n e o 13 where and n Q are respectively the extraordinary and ordinary r e f r a c t i v e i n d i c e s , and <j> i s the r e f r a c t i o n angle for p - l i g h t . The d i r e c t i o n s of the wave normals obey Snell's law n, s i n <()i = n s i n cf) = n s i n <j> = n„ s i n <j>„ (4.7) 1 x p p o o 3 3 where n^ = r e f r a c t i v e index of ambience n^ = r e f r a c t i v e index of substrate <(>]_ = angle of incidence tf>o = r e f r a c t i o n angle f o r s - l i g h t cj>3 = r e f r a c t i o n angle i n substrate. The r e f l e c t i o n c o e f f i c i e n t s R and R are calculated i n the same s P wayf as fo.rtisotropic,:'fiilmseexcepttthatand andanc aredusedtinstead of,, the • ;s,p O p p i s o t r o p i c r e f r a c t i v e index. For the s - l i g h t r s + r|! exp (-2iB ) R = — — = (4.8) S 1 + <2 r23 e X P ( " 2 i § o > s s where r and r„. are the Fresnel c o e f f i c i e n t s for the s - l i g h t at the 12 23 ambience/film and film/substrate i n t e r f a c e respectively. The c o e f f i c i e n t s r l 2 a n d r23 a r e n. cos d)n - n cos d> cs = 1 1 o o 12 n n cos <h + n cos <b (4.9) o n cos <j> - n_ cos <b„ s o ^o 3 Y 3 23 n cos <b + n„ cos <f>„ o o 3 3 i s the phase difference f o r the ordinary wave 2ir d n cos <f) = 2ir d( n^ - n!? s i n ^ t f i i ) 6 = ? ° = : : ( 4 . 1 0 ) o A A where A i s the wavelength of the l i g h t i n vacuum, and d i s the thickness of theaanisotropic f i l m . For the p - l i g h t r P 1 ? + exp (-218 ) R = — — - — (4.11) P 1 + r!2 r23 *** ( - 2 i e e > where r?„ and r^ are the Fresnel c o e f f i c i e n t s 12 ^23 2 2 2 1/2 n n cos A T - n, ( n - n, s i h <h ) p = e o 1 1 e 1 1 r12 A ' / 2 2 • 2 A ^ / 2 n n cos <f>-i + n, ( n - n, s i n d>i ) e o 1 1 e 1 1 2 2 2 1/2 n. ( n - n, s i n <f>i ) - n n cos <|>Q p _ 3 e 1 1 e o _ r23 2 2 2 1/2 n„ ( nz - n, s i n <j>i ) + n n cos <j>q 3 e 1 e o (4.12) The phase difference 8 i s e 2 2 2 1/2 2iTd n cos i> 2i\d n (n - n, s i n <j>i) ^ O N •FT p rp = o e 1 1 (4.13) e A X n e The e l l i p s o m e t r i c angles and A are found from the usual expression R lA p tan e R s It i s seen from equation 4.5 that n g i s equal to the radius of the c i r c u l a r p r i n c i p a l section of the i n d i c a t r i x , and i s independent of the incident angle. The Fresnel r e f l e c t i o n c o e f f i c i e n t s for s - l i g h t at both interfaces are therefore analp.gous to the i s o t r o p i c case. But the Fresnel c o e f f i c i e n t s for p - l i g h t and the phase difference 8g are d i f f e r e n t from those f o r i s o t r o p i c media. In the case of i s o t r o p i c non-absorbing f i l m , the locus i n the A-ij; plane for increasing f i l m thickness follows a closed loop. However, for anisotropic non-absorbing f i l m s , the phase differences 8 and 8 are not the same. Since 8 and 8 vary with f i l m o e o e thickness d i f f e r e n t l y , the r e l a t i v e phase change A depends on thickness 15 t h r o u g h t w o d i f f e r e n t e l e m e n t s , e x p ( - 2 i 8 Q ) a n d e x p ( - 2 i 8 e ) . T h e r e f o r e , a s t h e f i l m t h i c k n e s s i n c r e a s e s , t h e A - i J j = c u r v e s p i r a l s t o w a r d s e i t h e r h i g h e r A v a l u e s o r l o w e r A v a l u e s , d e p e n d i n g o n t h e s i g n o f b i r e f r i n g e n c e . 16 5. EXPERIMENTAL ARRANGEMENTS 5.1 The Ellipsometer 5.1.1 System components The automated ellipsometer system used was a Rudolph (type 43603-200B) interfaced to a DEC PDP-8/E computer (Fig. 5.1). The l i g h t source was a He-Ne l a s e r (Spectra Physics model 133) o producing l i g h t of wavelength 6328 A. The p o l a r i z e r and analyzer were each driven by a stepping motor (IMC Magnetic Corporation #PIN 008-008) which produced a 0.01° r o t a t i o n per step. The azimuthal p o s i t i o n s of the p o l a r i z e r and analyzer were detected by Decitrak shaft encoders (TR 511-CW/D). The encoder unit outputs the angles as BCD to the computer. A RCA photomultiplier (RCA 8645 )tube with Kepco regulated voltage source) was used as a detector. I t s output voltage (error signal) was amplified by a v a r i a b l e gain a m p l i f i e r with an adjustable zero con-t r o l , and then monitored as one of the four analog inputs multiplexed i n the i n t e r f a c e to a s i n g l e A/D converter. For convenience of alignment the error s i g n a l was also displayed on a meter which was connected to the input of the A/D converter. A Tyco d i g i t a l voltmeter was also interfaced to the computer. Elapsed time was measured by a clock incorporated i n the i n t e r f a c e . The clock consisted of a divide-by-six counter which was triggered by a 60 Hz s i g n a l , giving a basic unit of 0.1 seconds. 5.1.2 The balancing routine The p r i n c i p l e of operation of the ellipsometer i s as follows. The l i g h t passing through the p b l a r i z e r becomes l i n e a r l y p o l a r i z e d at an angle to the plane of incidence. The quarter wave plate changes the MOTOR DRIVE CIRCUITS DVM OUTPUT. ANALOG -INPUTS ENCODER OUTPUT UNIT INTERFACE VARIA DL E GAIN AMPLIFIER PDP8E COMPUTER 111 III TELETYPE F i g . 5.1 Schematic of the automated ellipsometer system. 18 l i n e a r l y p olarized l i g h t to e l l i p t i c a l l y p o l a r i z e d l i g h t . The p o l a r i z a -t i o n of the l i g h t i s further changed by r e f l e c t i o n from the sample. I f the p o l a r i z e r and the quarter wave plate are set such that the l i g h t r e f l e c t e d from the sample becomes l i n e a r l y p o l a r i z e d , the l i g h t can be extinguished by the analyzer. Under t h i s condition the ellipsometer i s said to be "balanced". Balancing the ellipsometer i s automated by using the PDP-8/E computer. The computer program begins by i n i t i a l i n g various pointers, and asking f o r date and s p e c i f i c a t i o n of time i n t e r v a l between successive balancings. A heading i s then printed out and the balancing routine i s started. The balancing routine i s based on the p r i n c i p l e that the l i g h t i n t e n s i t y detected by the photomultiplier v a r i e s symmetrically about the ex t i n c t i o n p o s i t i o n of the p o l a r i z e r or the analyzer f o r small excursion 21'. from the e x t i n c t i o n p o s i t i o n . The p o l a r i z e r i s balanced f i r s t since i t s e x t i n c t i o n s e t t i n g i s independent of the exact s e t t i n g of the anal-yzer f o r small deviations of the l a t t e r . The computer program f i r s t deter-mines which way the motor should be stepped to minimize the error s i g n a l . I t steps the motor i n that d i r e c t i o n and takes error s i g n a l readings a f t e r each step u n t i l a set of 16 readings are taken and summed. The stepping of the motor i s continued u n t i l the error s i g n a l goes through the minimum and begins to increase again. The t h i r d sum back from the minimum i s stored. The program then calculates a second sum of 16 error s i g n a l readings. As the stepping continues t h i s sum i s updated a f t e r each step so that i t contains only the 16 most recent readings. When the second sum equals the f i r s t sum on the other side of the minimum, 19 the balance p o s i t i o n which correspond to the mid-point between these two equal sums can be located, and the p o l a r i z e r i s stepped to that p o s i t i o n . The analyzer i s then adjusted by the same routine, followed by another balancing of the p o l a r i z e r . A f t e r the p o l a r i z e r and the analyzer have been balanced, the net number of steps each motor takes i s compared with the change i n po-s i t i o n of the respective shaft encoder. This checks for any er r o r due to gear backlash or to the motor missing steps. The program then reads the encoder settings of the p o l a r i z e r and analyzer, the step e r r o r s , the clock time, and the d i g i t a l voltmeter. The readings are stored i n a b u f f e r area i n the computer memory to wait for output on the teletype which i s operated on an in t e r r u p t b a s i s . The buffer area consists of 1664 locations and i s arranged i n a c i r c u l a r manner. The time required f or each balancing cycle i s about 2 seconds, but p r i n t i n g out the data requires 5 to 6 seconds. 5.1.3 Ellipsometer alignment Alignment of the ellipsometer followed the method devised by 22 Aspnes and Studna . In t h i s alignment procedure the symmetry of the ellipsometer was used to proyide the information needed f o r i t s own alignment. The accuracy of the alignment was l i m i t e d p r i m a r i l y by that of the ellipsometer i t s e l f . The alignment method i s outlined below, (a) Determination of zero error i n angle of incidence scale With'.the compensator removed, and the p o l a r i z e r and analyzer arms i n straight-through p o s i t i o n , the azimuth angles of both p o l a r i z e r and analyzer were set at 90°. The analyzer arm was rotated about the straight-through p o s i t i o n i n 0.01° steps u n t i l the peak error s i g n a l was 20 found. From the scale reading f o r the peak s i g n a l p o s i t i o n , the angle of incidence scale error was determined to be -0.04°. (b) Determination of zero error i n p o l a r i z e r and analyzer azimuthal angles With the compensator removed, the angle of incidence set at 62.77°, and both the p o l a r i z e r and the analyzer set at 90° azimuthal angles, an o p t i c a l l y f l a t quartz plate was mounted as a beam r e f l e c t o r . The p o s i t i o n of the quartz plate was adjusted u n t i l a maximum s i g n a l was •detected by the detector. Then-the'analyzer was balanced near A = 0° for d i f f e r e n t set values, of P-about 90° to obtain a s t r a i g h t l i n e plot of P versus A. In the same procedure the p o l a r i z e r was balanced near P = 0 for d i f f e r e n t set values of A about 90°. - The r e s u l t s are shown i n F i g . 5.2. The i n t e r s e c t i o n of the two s t r a i g h t l i n e s i n F i g . 5.2 gave the error of the P and A scales. This error was corrected f o r by f i r s t s e t t i n g the P and A to the i n t e r s e c t i o n values, and then adjusting the shaft encoders so that the output of one encoder read 0.00° while the other read 90.00°. (c) Setting the quarter wave plate azimuth angle at 315° In tracking the growth of films the ellipsometer readings were taken with the quarter wave plate set at 315°. The s e t t i n g of the QWP was as follows. With the p o l a r i z e r and the analyzer i n straight-through p o s i t i o n and set at 315° and 45° re s p e c t i v e l y i n azimuth angle, the QWP was i n s t a l l e d . The azimuth angle of the QWP was set around 315°, and then f i n e adjustment was made u n t i l a minimum erro r s i g n a l was obtained. In t h i s condition the f a s t axis of the QWP was p a r a l l e l to the p o l a r i z e r transmission a x i s , which was at 315° to the plane of incidence. ' I I I i L I I _ l I —I 1 i -.4 .5 .6 .7 .8 .9 0 .7 .2 .3 .4 .5 .6 90 P/deg. F i g . 5.2 Zero error i n p o l a r i z e r and analyzer s c a l e s . x - x - x balancing p o l a r i z e r with analyzer stationary near 90°. o - o - o balancing analyzer with p o l a r i z e r s t a t i o n a r y near 90°. 22 (d) Determination of the transmission r a t i o , T , and the phase retarda-t i o n , A , of the quarter wave p l a t e . An inconel s l i d e with a mirror surface was used as a sample and was c a r e f u l l y aligned. The angle of incidence was set at 62.77°. Two-zone readings were then taken. For perfect alignment of the sample and with an i d e a l quarter wave pla t e , there would be no diffe r e n c e between readings of two d i f f e r e n t zones* To minimize the zone difference the phase s h i f t produced by the compensator was a l t e r e d by adjusting the micrometer on the compensator. The f i n a l two-zone readings were taken. Using 23 McCrackin's program the transmission r a t i o , T c, and the phase retarda-t i o n , A(i, were calculated to be 0.997 and 90.00° r e s p e c t i v e l y . 5.2 Anodization C e l l The anodization c e l l used i n our studies had to be compatible with i n s i t u ellipsometer measurements. I t had to meet the requirements that i t be r e s i s t a n t to chemicals, have small window e r r o r , and be easy to adjust i n the alignment procedure. With these factors i n mind, the c e l l was designed as shown i n F i g . 5.3. The c e l l was made of a t e f l o n cylinder with a bottom piece threaded i n . The angle between the two windows was 125.54°, which made the angle of incidence 62.77°. A more d e t a i l e d d e s c r i p t i o n of the win-dows w i l l be given i n the next section. The c e l l was supported by a brass base which was f i t t e d to the standard sample holder f o r the e l l i p -someter. There were two sets of screws on the base, one f o r adjusting the t i l t of the c e l l and the other one for locking the p o s i t i o n once the desired t i l t i n g had been made. The c e l l could be t i l t e d and rotated without disturbing the alignment of the sample. TEFLON CYLINDER WINDOW •1_ f- H BASE •A NOT TO SCALE L _ 23 SCREWS SHAFT F i g . 5.3 Structure of the c e l l f o r i n s i t u ellipsometer measurements. 24 5 . 3 W i n d o w E r r o r T h e b i r e f r i n g e n c e o f t h e w i n d o w s i s a p o s s i b l e s o u r c e o f e r r o r i n i n s i t u e l l i p s o m e t e r m e a s u r e m e n t s . T o m i n i m i z e t h e w i n d o w e r r o r t w o % - i n c h t h i c k , f u s e d s i l i c a w i n d o w s ( 1 - i n c h i n d i a m e t e r , 1 / 2 0 w a v e i n f l a t -n e s s o n b o t h s i d e s , a n d 1 s e c . p a r a l l e l i s m ) s u p p l i e d b y O r i e l C o r p o r a t i o n w e r e u s e d . T h e w i n d o w s w e r e t e s t e d b e f o r e a n d a f t e r t h e y w e r e m o u n t e d o n t h e c e l l . 5 . 3 . 1 E r r o r o f w i n d o w s b e f o r e m o u n t i n g o n t h e c e l l T h e w i n d o w s , p l a c e d b e t w e e n t h e q u a r t e r w a v e p l a t e a n d t h e a n a l y z e r , w e r e t e s t e d i n d i v i d u a l l y b y t h e e l l i p s o m e t e r i n a s t r a i g h t -t h r o u g h a r r a n g e m e n t . T h e e f f e c t o f t h e b i r e f r i n g e n c e o f t h e w i n d o w s o n t h e p o l a r i z a t i o n s t a t e o f t h e l i g h t b e a m c a n b e b e s t v i s u a l i z e d u s i n g t h e P o i n c a r e s p h e r e ( s e e a p p e n d i x ) s h o w n i n F i g . 5 . 4 . F i g . 5.4 P o i n c a r e s p h e r e r e p r e s e n t a t i o n o f t h e e f f e c t o f w i n d o w e r r o r o n t h e p o l a r i z a t i o n . s t a t e o f l i g h t . 25 The window i s considered to be a wave plate with a small r e l a -t i v e phase retardation 6, and i t s f a s t and slow axes p a r a l l e l to the sur-face. Consider f i r s t the case when there i s no window and a l l the com-ponents of the ellipsometer are i d e a l . When the l i g h t i s extinguished by the analyzer, the p o l a r i z e r should be at the same azimuth angle as the quarter wave plate,•represented by Q on the Poincare sphere. When the window i s placed between the quarter wave plate and the analyzer, and at an azimuth angle <|>w, i t i s represented by W on the Poincare sphere. Under a balanced condition the p o l a r i z e r i s set to P. When the l i g h t goes through the quarter wave plate the point on the sphere representing the p o l a r i z a t i o n of the l i g h t beam i s rotated about the OQ axis by 90° from P to P^. The phase retardation of the window rotates P^ about the 0W axis by an angle 6 to T?2> which l i e s on'the equator, so that the l i g h t i s again l i n e a r l y p olarized and i s extinguished by the analyzer. Since the r o t a t i o n about 0W i s fi x e d by the angle 6, the distance P-^  t r a v e l s to P2» or i n other words the difference between P and Q, depends on the po-s i t i o n of W. When W coincides with or i s 180° away from Q, P should be the same as Q, as i n the case of no^window. As W i s rotated away from Q, the difference between P and Q which represents window err o r increases. The maximum differ e n c e occurs when W i s 90° away from Q, i . e . the fa s t axis of the window i s 45° away from that of the quarter wave pl a t e . When W i s on the other side of the sphere, t h i s d i f f e r e n c e should be of oppo-s i t e sign. Therefore when the window i s rotated about i t s surface normal, i t i s expected that the difference between the ellipsometer angle A (which equals the sum of readings (P, + P,) f o r zones 1 and 3 when Q = -45°) f o r 2 6 the case with window and that f o r the case without window w i l l vary i n a sin u s o i d a l form. Two cycles of such v a r i a t i o n are expected for one com-plete r o t a t i o n of the window because W makes two rotations on the Poin-care sphere. Using the p r i n c i p l e explained above, the v a r i a t i o n of window error with the r o t a t i o n of the window was examined. The window was mounted on a holder which allowed r o t a t i o n of the window. The angle of r o t a t i o n , r e l a t i v e to some a r b i t r a r y d i r e c t i o n , could be read on an attached c i r c u l a r d i a l with degrees marked. The window was rotated about the surface normal and f o r every 15 degrees of r o t a t i o n two-zone e l l i p -someter readings were taken. Care had to be taken that f o r every mea-surement the window surface was normal to the beam. The ellipsometer readings were compared to those' obtained when no Window was placed i n the ellipsometer arrangement, and the differences were the errors due to the window. No difference was observed i n Tp. The errors i n A for d i f -ferent rotations were as shown i n F i g . 5.5. F i g . 5.5 agrees very w e l l with the p r i n c i p l e discussed above. The error i n A v a r i e s s i n u s o i d a l l y with the r o t a t i o n angle. There are two cycles of v a r i a t i o n f o r one complete r o t a t i o n of the window. From F i g . 5.5 one can determine the d i r e c t i o n of the fast and slow axes of the window. The important conclusion from these window error measurements i s that the q u a l i t y of the window was considered very good. The maximum error i n A was ±0.04°. No error i n I|J was observed, implying the absorp-tions along the f a s t and slow axes of the window were e i t h e r the same or the difference could cause a change of l e s s than 0.01° i n F i g . 5.5 Window error i n A against r o t a t i o n of the window. A w i s the A value with window. A Q i s the A value with no window. Rotation angle i s r e l a t i v e to an a r b i t r a r y d i r e c t i o n . 28 5.3.2 Error of windows a f t e r mounting on the c e l l When the windows were mounted on the c e l l , more errors could be introduced by the stress exerted on them. To determine the window 24 error a method devised by McCrackin was used. An inconel s l i d e with a mirror surface was again used as a sample i n the ellipsometer measurements. The angle of incidence was. set at 62.77°. Two-zone ellipsometer readings were taken with the i n -conel s l i d e i n the c e l l and out of the c e l l . The diffe r e n c e between the readings for two cases gave the window error. For a l l experiments done i n t h i s work, the error i n A ranged from 0.02° to 0.06°. The er r o r i n was always within ±0.01° which was also the adjustment step of the analyzer and p o l a r i z e r . 5.4 Sample Preparation The samples used i n t h i s work were cut from a s i n g l e c r y s t a l niobium rod supplied by M a t e r i a l Research Corporation. The samples were cut i n the shape of a c i r c u l a r disk, %" i n diameter, 0.1" t h i c k , and with the surfaces oriented i n the (110) d i r e c t i o n . A heavy tantalum wire spot-welded at the back of the sample acted as an e l e c t r i c a l con-nection as w e l l as the support to the sample by f i t t i n g into a sample holder. The lower h a l f of the tantalum wire, which was to be immersed i n the e l e c t r o l y t e during the anodization of the niobium sample, was insulated beforehand by anodizing a thick oxide f i l m on i t . This was done with the sample masked by non-conductive Apiezon grease. Since the samples were used i n o p t i c a l studies, a f l a t r e f l e c t i n g surface on each sample was required. The surface was prepared by mech-25 a n i c a l followed by chemical p o l i s h i n g 29 One side of the sample was abraded on 0/0, 2/0, 3/0 and 4/0 emery papers i n that sequence. The sample was then immersed i n a chem-i c a l p o l i s h i n g s o l u t i o n which consisted of 1:1:1 by volume of 85% phos-phoric a c i d , 48% HF and 70% HNO^, f r e s h l y mixed i n that order. A gentle a g i t a t i o n by moving the sample about i n the sol u t i o n was required. The time of immersion required to obtain a good surface ranged from 5 to 20 minutes, depending on the mechanical condition of the surface. Upon removal from the s o l u t i o n , the sample was rinsed immediately i n d i s t i l l e d water to avoid uneven attack on the surface. The sample was then dipped i n HF for 5 seconds, followed by a d i s t i l l e d water r i n s e . 30 6 . R E S U L T S 6.1 O p t i c a l Anisotropy of Anodic Niobium Oxide Films 6.1.1 Experimental procedure The ellipsometry p r i n c i p l e s described i n chapter 4 were applied to study the o p t i c a l anisotrophy of the niobium oxide f i l m s . A f i l m growing under constant f i e l d was expected to be i n the anisotropic state. It was therefore expected that the ellipsometer angles measured during the growing process would s p i r a l e i t h e r upwards or downwards i n the A-^ plane. The experimental procedure was as follows. The polished niobium sample was aligned on the ellipsometer at an angle of incidence of 62.77°. The c e l l was positioned and aligned so that both windows were normal to the laser beam. The window erro r was checked beforehand by the method described i n section 5.3.2. The e l e c t r o -l y t e was 0.2N H2SO4. I t : w a s s t i r r e d and maintained at 25.0 ± 0.1°C using a temperature c o n t r o l l e r . A p l a t i n i z e d platinum electrode was used as the cathode. The sample was anodized at constant current of 0.11 mA/cm2 to 92 v o l t s (2240 A i n thickness). The thickness of the f i l m was so chosen for two reasons. (a) The ellipsometer angles A and i> at t h i s thickness were i n the region most s e n s i t i v e to the f i l m thickness. This was required i f the sample was to be used i n studies described i n section 6.2. (b) The f i l m was i n second cycle of the A - ijj curve while 92 v o l t s was below the voltage which would cause the oxide f i l m to r e c r y s t a l l i z e , as w i l l be d i s -cussed i n chapter 7. 3 1 The anodization process was p e r i o d i c a l l y interrupted by switching o f f the f i e l d ( i . e . s h o r t - c i r c u i t i n g the constant current supply). Contin-uous one-zone ellipsometer readings were taken during the growth process and during the periods with the f i e l d removed. Before and a f t e r the o v e r a l l anodization process,two-zone ellipsometer readings were taken. Based on these two-zone readings, corrections to the one-zone readings were made. Errors due to the windows and the quarter wave pl a t e were corrected f o r . The experimental meansurements were analyzed by p l o t t i n g them on a large graph paper and f i t t i n g them to t h e o r e t i c a l curves computed by programmes for IBM 3 6 0 / 6 7 . 6 . 1 . 2 Results The experimental measurements of A and $ with the f i e l d on and off are shown i n Figs. 6 . 1 and 6 . 2 . F i g . 6 . 1 shows the lower portion of the A - i> p l o t up to two cycles, while F i g . 6 . 2 shows the upper p o r t i o n . These figures show that when the f i e l d i s on, the A - ty. curve s p i r a l s upwards. But once the f i e l d i s switched o f f , the A - # points on the two cycles f a l l on the same curve, and these experimental points f i t a t h e o r e t i c a l curve for a s i n g l e - l a y e r i s o t r o p i c f i l m with the f i l m r e f r a c t i v e index, n n , equal to 2 . 3 2 9 , and the index of the niobium substrate, n3, equal to 3 . 0 1 - i 3 . 5 5 5 . The experimental points obtained with the f i e l d on were f i t t e d to t h e o r e t i c a l curve f o r a homogeneous anisotropic f i l m i n the following way. The c h a r a c t e r i s t i c s of a i - I curve for an anisotropic f i l m i s the s p i r a l l i n g along the A-axis. The difference i n the values of minimum A or maximum A between each cycle depends on the constant (3, which i s defined 35 40 45 50 55 V / deg Fig. 6.1 Lower portion of the A-f plot for increasing thickness of a niobium oxide film up to two cycles. Solid points are experimental measurements for zero f i e l d : •, f i r s t cycle; w X , second cycle. Blank points are measurements with f i e l d on: o, f i r s t cycle; *° A, second cycle. ( ) is a theoretical curve for isotropic film. ( ) is a theoretical curve for anisotropic film* 246 y /deg Fig. 6.2 Upper portion of the A-fplot. Key as for Fig. 6.1. 34 2.30. 2.31 2.32 F i g . 6.3(a) T h e o r e t i c a l values of minimum A as functions of 3 and n D . ( ) i s f o r the f i r s t c y c l e . ( ) i s f o r the second c y c l e . Shaded areas are experimental values. 35 F i g . 6.3(b) T h e o r e t i c a l values of maximum A as functions of B and n Q . Key as f o r F i g . 6.3(a). 36 by equation 2.12, and the ordinary r e f r a c t i v e index n Q . The t h e o r e t i c a l values of minimum A and maximum A as functions of 3 and n Q have been computed and shown i n F i g . 6.3. The values of and n^ were based on that found from f i t t i n g the i s o t r o p i c curve. The experimental values of the minimum A's and maximum A are shown within a shaded area which repre-sents the estimated experimental e r r o r of ±0.05°. From F i g . 6.3,3 and n Q were estimated at such values that the f i t t i n g s of the three experimental values were compromised. By varying 3 and n Q about the estimated values, a curve which best f i t t e d the experimental points was obtained and i s shown i n Figs. 6.1 and 6.2. Points A, B and C i n F i g . 6.3, which a l l l i e within the l i m i t of experimental e r r o r , are values obtained from the chosen t h e o r e t i c a l curve shown i n Figs. 6.1 and 6.2. F i g . 6.3 therefore also gives some i n d i c a t i o n on how close the experimental points were f i t t e d to the t h e o r e t i c a l curve. The anisotropic o p t i c a l constants of the f i l m determined from curve f i t t i n g were as follows: 3 = 1.6 n D = n s = 2.3095 n e = 2.2978 n p = 2.3064 n n = 2.329 n 3 = 3.01 - i 3.555 A l l symbols above are as defined i n section 4.2. 6.1.3 Discussion The experimental r e s u l t s show that applying a f i e l d normal to the niobium oxide f i l m caused the f i l m to change from the o p t i c a l l y 37 i s o t r o p i c state to the o p t i c a l l y anisotropic s t a t e . This e f f e c t i s s i m i l a r to that reported for anodic Ta2C>5 films by Cornish and Young^. Fig . 4.3 shows that i f n Q i s varied a l i t t l e i n curve f i t t i n g , a reasonably good f i t can s t i l l be obtained by varying 3 from 1.55 to 1.65. The value 3 = 1.6 ± 0.05 for niobium oxide films happens to be the same as reported for Ta20s f i l m s ^ . This presumably r e f l e c t s the s i m i l a r i t y of these two oxides. The o p t i c a l anisotropy of the oxide f i l m with f i e l d applied could have been demonstrated better by three or more cycles of the A - \p curve. Unfortunately the oxide f i l m tended to r e c r y s t a l l i z e at high voltage, as w i l l be discussed i n chapter 7. Due to t h i s problem the oxide f i l m was grown to cover only one and a h a l f cycles of the A - ip curve. 6.2 F i e l d and Time Dependence of E l e c t r o - o p t i c and E l e c t r o s t r i c t i v e  E f f e c t s 6.2.1 Experimental procedure This experiment was designed to examine the f i e l d and time dependence of the anisotropic r e f r a c t i v e indices and the thickness of films of anodic niobium oxide. The same oxide f i l m grown i n the experi-ment described i n section 6.1 was used i n this study. The sample was l e f t i n the c e l l i n the same ellipsometer set up. A d-c voltage was applied across the f i l m and was monitored by a d i g i t a l voltmeter and recorded by the computer. The conduction current was monitored by a d i g i t a l ammeter. The voltage was varied i n 5-volt steps from zero to 60 v o l t s (2/3 of the formation f i e l d ) . The ellipsometer was balanced continuously during the time the f i e l d was on, and the time i n t e r v a l between two 38 successive balancings was approximately two seconds. In t h i s way the time dependences of An Q and Ad f o r every applied f i e l d were simultaneously examined. Between every two voltage steps, the f i e l d was switched o f f to check i f there had been any growth of the f i l m , i Corrections for ellipsometer errors were made to the measurements i n the same way as discussed i n section 6.1.1. The analysis of the data was based on the values of B, n^ and n3 obtained i n the experiment 6.1. The changes i n ordinary r e f r a c t i v e index, An Q, and i n thickness, Ad, were determined independently by f i t t i n g the experimental A - values to the contours of constant no and the contours of constant d i n the A - ^ domain. 6.2.2 Results With no f i e l d applied, the oxide f i l m had been found i n section 6.1 to have an i s o t r o p i c r e f r a c t i v e index, n , equal to 2.329, and thick-o ness equal to 2239.1 A. Fig . 6.4 shows An Q and An g as a function of E 2 . The experimental points shown are for A i ^ only since An g was determined from the equation 2.12. The quadratic f i e l d dependence of Ad i s shown i n F i g . 6.5. The experimental errors were estimated to be ±0.15 x 10~5 for An cand ±0.2 A for Ad. From the r e s u l t s shown i n F i g . 6.4 the quadratic e l e c t r o - o p t i c c o e f f i c i e n t s gn and g j 2 were computed using equations 2.11 and the r e l a t i o n P = e 0(K - 1)E where the r e l a t i v e d i e l e c t r i c constant, K, of anodic Nb205 films was taken 26 as 41.4 . The values of g i 2 and 8l 1 were determined to be 0.065 m^/coul 2 and 0.105 m^/coul 2 r e s p e c t i v e l y . 39 Fig. 6.5 The change in film thickness against E . 2.01 o i yl } I I l,l i i i I * t I { I f I j I I 0 10 20 30 40 SECONDS (a) 7.5 2 10 "73 0.51 0 l-,4 70 20 SECONDS (b) 30 40 F i g . 6.6 The change i n ordinary r e f r a c t i v e index and i n thickness with time when the f i e l d was switched from zero to 1.78 x 10 6 V/cm. 42 No time dependence of An Q and Ad was detected i n t h i s experiment. The changes i n n Q and d followed immediately the change i n applied f i e l d . F i g . 6.6 shows An Q and Ad as a function of time when the applied f i e l d was switched from zero to 1.78. x 10 6 V/cm. The deviations of the experimental values from the average values were within the estimated ellipsometer er r o r which i s represented by the error bars. 6.2.3 Discussion The r e s u l t s of this experiment show that the changes i n the ordinary and the extraordinary r e f r a c t i v e index, and the change i n t h i c k -ness of the niobium oxide films vary q u a d r a t i c a l l y with the applied e l e c -t r i c f i e l d . These r e s u l t s are s i m i l a r to those reported f o r films of tantalum oxide by Cornish and Young^. The observed quadratic e l e c t r o - o p t i c c o e f f i c i e n t s gi\ and g i 2 found i n t h i s experiment are i n the same order of magnitude as those reported f o r oxygen-octahedra f e r r o e l e c t r i c c r y s t a l s i n references 27 and 28, and for KTN c r y s t a l s by Chen et a l . From the experimental r e s u l t s for Ta 205 films reported by Cornish et a l . ^ , g 1 2 a n ^ §11 were estimated as 0.055 mVcoul 2 and 0.088 mVcoul 2 r e s p e c t i v e l y . These values are close to the r e s u l t s obtained i n t h i s experiment. However, the quadratic e l e c t r o -• opt i c c o e f f i c i e n t s f o r Ta 20s films. reported by Frova et al.'*' were about f i v e times smaller than our r e s u l t s . This difference can be explained as follows. Frova et a l . , who used a field-modulated reflectance technique i n the study, assumed that the change i n thickness of the f i l m was n e g l i g i b l e . The reflectance technique a c t u a l l y measures the o p t i c a l , thickness which i s the product nd. Since -An i s i n fact accompanied by +Ad when a f i e l d i s applied, and Ad i s not n e g l i g i b l e , the value of An, 43 and hence the quadratic e l e c t r o - o p t i c c o e f f i c i e n t , obtained by Frova et a l . i s expected to be underestimated. Slow exponential time-dependences of the changes i n r e f r a c t i v e indices and the change i n thickness were reported for tantalum oxide films by Cornish and Young?. These slow time-dependences were not observed i n niobium oxide films i n our experiment. The tantalum oxide films used by Cornish et a l . were fabricated i n a d i f f e r e n t way from ours by growing f i r s t at constant current to a designated voltage, and then growing at that constant voltage for a few hours. This d i f f e r e n t process might have caused t h e i r f i l m to behave d i f f e r e n t l y , although the mechanism i s not understood. 6.3 Reprod u c i b i l i t y of Ellipsometer Measurements This experiment was designed to check whether o p t i c a l anistropy of the oxide f i l m was observed under c o n t r o l l e d conditions, and whether the experimental r e s u l t s were reproducible. A polished niobium sample was anodized and measured by the ellipsometer i n the same way as described i n section 6.1.1. The anodization current density was 0.087 mA/cm2 and the e l e c t r o l y t e was thermostated at 23°C. The f i l m was anodized to cover only one cycle of the A - jjj curve to make sure there was no r e c r y s t a l l i z a t i o n of the oxide. The sample was then taken out and the oxide f i l m was etched o f f using a s o l u t i o n of HF saturated with NH^F. With t h i s sample mounted back on the ellipsometer, the same experiment was run under i d e n t i c a l conditions. Hence two A - ip curves for two oxide films grown under as near as possible i d e n t i c a l conditions were obtained. 44 A computer program was w r i t t e n to check f o r the closeness of the two A - ip- curves. The algorithm of the program was as follows; For a point P-^  i n one set of the A - if; data, the program searches f o r a point Q2 i n the second set of data which i s c l o s e s t to P^, as shown i n F i g . 6.7. A F i g . 6.7 Inte r p o l a t i n g a set of A—Tp data The data point Q^, immediately preceding Q 2, and point Q3, immediately f o l -lowing Q2, are taken i n the same set. A second order polynomial which passes through these three points i s constructed by Lagrangian polynomial i n t e r p o -l a t i o n and points A j and A3 are found. The mid-point A2 between A^ and A3 i s then computed. Among the points A±, A 2, A3 and Q2 the one at the shortest distance from P-^  i s chosen and that distance i s considered as the true distance between the point P^ and the second A - if; curve. A l l points i n the f i r s t data set are examined i n t h i s way and the distance (error) between the two A - ijj curves are thus computed. F i g . .6.8 shows the e r r o r between two experimental A - if; curves as a function of f i l m thickness. For both the cases when the f i e l d was on and when the f i e l d was o f f , the mean value of e r r o r was 0.075°. This .20 .751 Q: o ft: Ui .05 ± 400 800 THICKNESS /A 1200 1600 (a) .20] .75 .05} • • • 0 400 800 THICKNESS / A (b) 1200 1600 Fig. 6.8 Error (distance) between two experimental A-f curves versus thickness of the oxide film (a) No applied field [ (b) With applied field 46 error can be resolved i n t o an error of 0.05° i n both A and ip. The s e n s i t i v i t y of the ellipsometer i s ±0.01° i n P and A readings ( i . e . ±0.02° i n A and ±0.01° i n ty). However, due to errors i n alignment, windows and component imperfection, the experimental error i s not l i m i t e d by the s e n s i t i v i t y . Also i n t h i s experiment removing and re-mounting the sample might have introduced more err o r i n alignment and windows, and etching o f f the oxide f i l m could have caused s l i g h t damages to the surface of the niobium substrate which i n turn could cause a s l i g h t difference i n experimental r e s u l t s . Therefore i t i s reasonable to estimate the experimental error as 0.05° i n both A and readings. The mean difference between the two A - IJJ curves determined i n t h i s experiment i s within the l i m i t of estimated experimental e r r o r . This experiment establishes the r e p r o d u c i b i l i t y of the ellipsometer measurements. 47 7. FIELD RE CRY S TALLIZ AT ION OF ANODIC NIOBIUM OXIDE FILMS As already stated, the thickness to which the anodic niobium oxide films could be grown was l i m i t e d by r e c r y s t a l l i z a t i o n of the oxide films i n the anodization process. The r e c r y s t a l l i z e d areas appearing on the f i l m were examined by a scanning electron microscope. I t was found that " f i e l d r e c r y s t a l l i z a t i o n " of anodic niobium oxide films was i n many 29 ways s i m i l a r to that of anodic T a 2 0 5 films reported by Jackson and •T 30,31 Vermilyea In " f i e l d r e c r y s t a l l i z a t i o n " the o r i g i n a l amorphous f i l m i s not, i n f a c t , r e c r y s t a l l i z e d , except possibly where a nucleus of c r y s t a l l i n e oxide i s produced, but i s displaced by a c r y s t a l l i n e phase which grows by anodization. Two mechanisms have been suggested on how r e c r y s t a l l i z a t i o n 31 s t a r t s . In the f i r s t one a small portion of the e x i s t i n g amorphous phase i s transformed into c r y s t a l l i n e oxide at a nucleation s i t e . I t has been suggested that the nucleation s i t e s may be small inc l u s i o n s of impurity which happen to be at the metal/oxide i n t e r f a c e . Stresses produced by a difference i n density between the c r y s t a l l i n e oxide and the amorphous oxide produce cracks i n the amorphous f i l m . Once a crack i s produced the c r y s t a l l i n e oxide has contact with the e l e c t r o l y t e and thickens r a p i d l y by anodization. The amorphous f i l m i s pushed up off the metal at the cracks and the c r y s t a l l i n e area grows r a d i a l l y . In the second suggested mechanism a nucleation s i t e created underneath the amorphous f i l m has access of e l e c t r o l y t e through a narrow pore i n the amorphous f i l m . New c r y s t a l l i n e oxide grows at the base of 29 the pore by anodization. This mechanism i s supported by Jackson who has recently confirmed the existence of small pores at the r e c r y s t a l l i z e d areas. 48 In F i g . 7.1 the scanning electron micrographs show the rec r y s -t a l l i z e d areas of a niobium oxide f i l m anodized at constant OJ'2 mA/cm to 150 v o l t s . The amorphous f i l m at these areas was pushed up and cracked. In F i g . 7.1(b) a d i s t i n c t i v e growth of c r y s t a l l i n e oxide within the crack can be seen. The cracked amorphous f i l m was peeled back by the c r y s t a l l i n e growth, and s p l i t into s t r i p s which curled outwards. The r e c r y s t a l l i z e d areas did not d i s t r i b u t e evenly over the sample surface. On the surface of the niobium sample there were a number of p i t s which had been produced from the chemical p o l i s h i n g process. However, these p i t s seemed to have no apparent e f f e c t on r e c r y s t a l l i z a t i o n of the f i l m . The amorphous f i l m on the sample shown i n F i g . 7.1 was etched o f f with HF. F i g . 7.2 shows the surface of the sample a f t e r the etching. The c r y s t a l l i n e oxide had remained on the surface. Evidently, as with Ta205 films the c r y s t a l l i n e oxide dissolves very slowly i n HF as compared to the amorphous oxide. However, i t was i n t e r e s t i n g to f i n d that part of the peeled back f i l m had not dissolved i n HF (Fig. 7.2(b)). The same 29 r e s u l t has been observed on anodic tantalum oxide films by Jackson It was suggested that as the amorphous f i l m was peeled back by the growing c r y s t a l s , a thi n layer of c r y s t a l l i n e oxide at the in t e r f a c e became de-tached and remained i n intimate contact with the amorphous f i l m , thus peeling back with the amorphous f i l m . F i g . 7.3 shows the r e c r y s t a l l i z e d areas of a f i l m anodized at constant 0.2 mA/cm2 to 149 v o l t s , and then at constant 120 V f o r 10 hours. The r e c r y s t a l l i z e d areas had grown r a d i a l l y from the i n i t i a l stage as shown i n F i g . 7.1 into large polygonal areas. The amorphous f i l m was 49 (b) Fig. 7.1 R e c r y s t a l l i z e d areas of anodic niobium oxide 2 f i l m formed at constant 0.2mA/cm to 150 v o l t s , at 23 C i n 0.2N H 2S0 4. (a) 8000x (b) 9200x (a) Fi g . 7.2 Same sample as F i g . 7.1. R e c r y s t a l l i z e d areas a f t e r the amorphous f i l m was etched o f f with HF. (a) 2000x (b) 14000x 5 2 peeled back further as the c r y s t a l s grew. The c r a t e r - l i k e centre of each polygonal area was presumably the s i t e where r e c r y s t a l l i z a t i o n started. Why i t was shaped d i f f e r e n t l y from the re s t of the area, or whether i t represented a pore i n the amorphous f i l m , as found i n tantalum oxide f i l m 29 by Jackson , i s not known. I t may be seen i n F i g . 7.3(b) that the surface of the recrys-t a l l i z e d area (and the underside of the peeled-back f i l m , which was : . ac t u a l l y a t h i n layer of cr y s t a l s ) has a fibrous nature. The f i b r e s are oriented r a d i a l l y . I t can also be seen from F i g . 7.2(b) that small f i b r e - or rod-shaped c r y s t a l s had attached themselves to the edge of the r e c r y s t a l l i z e d oxides. The fibrous nature of the r e c r y s t a l l i z e d surface 29 30 31 has also been observed on tantalum oxide f i l m s ' ' It was found that i n constant current (constant f i e l d ) anodiza-t i o n , r e c r y s t a l l i z a t i o n of oxide films seemed to depend on the voltage. Although a c r i t i c a l voltage has not beenddetermined i n t h i s i n v e s t i g a t i o n , 2 i t was found that a f i l m anodized at 0.2 mA/cm di d not show r e c r y s t a l -l i z a t i o n below 100 v o l t s . I t was found that the constant current/voltage anodization technique produced r e c r y s t a l l i z e d oxide even when the constant voltage was below the value at which r e c r y s t a l l i z a t i o n would not have occurred i n constant current anodization. R e c r y s t a l l i z a t i o n was found on an oxide 2 f i l m anodized at 0.2 mA/cm to 80 v o l t s , and then at constant 80 v o l t s for A hours. This suggests that the period a voltage i s applied across the f i l m may be one of the main factors which cause r e c r y s t a l l i z a t i o n . 53 8. CONCLUSIONS The field-induced o p t i c a l anisotropy i n niobium oxide films was studied using i n s i t u ellipsometry. F i e l d r e c r y s t a l l i z a t i o n of the oxide films was investigated using a scanning e l e c t r o n microscope. It was found that anodic niobium oxide films were o p t i c a l l y i s o -t r o p i c i n the absence of an e l e c t r i c f i e l d , but became anisotropic when an e l e c t r i c f i e l d was applied normal to the f i l m surface. The a n i s o t r o p i c r e f r a c t i v e indices of the oxide f i l m decreased q u a d r a t i c a l l y with the applied f i e l d , while the f i l m thickness increased q u a d r a t i c a l l y with the applied f i e l d . The r a t i o g of the change i n extraordinary r e f r a c t i v e index, Atile,. to the change i n ordinary r e f r a c t i v e index, An Q, was found to be 1.6 ± 0.05. This value i s the same as that reported f o r tantalum oxide f i l m s ^ . The quadratic e l e c t r o - o p t i c c o e f f i c i e n t s of the niobium oxide films were found to be i n the same order of magnitude as those reported 8 28 fo r oxygen-octahedra f e r r o e l e c t r i c c r y s t a l s and KTN c r y s t a l s ' No time dependences of e l e c t r o - o p t i c and e l e c t r o s t r i c t i v e e f -f e c t s were observed. Studies on f i e l d r e c r y s t a l l i z a t i o n of niobium oxide films showed that the c r y s t a l l i n e oxide was i n many ways s i l i m a r to that reported on 29 30 31 tantalum oxide films ' ' . The c r y s t a l l i n e oxide was found to consist of a number of small f i b r e - or rod-shape c r y s t a l s with t h e i r long axes oriented i n r a d i a l d i r e c t i o n . The c r y s t a l l i n e oxide dissolved very slowly i n HF as compared to the amorphous oxide. The occurrence of r e c r y s t a l l i z a t i o n was found to depend on voltage i n the constant current anodization, and depend on time i n the constant current/voltage anodization. 54 BIBLIOGRAPHY 1. A. Frova and P. M i g l i o r a t o , Appl. Phys. Letters 13, 328 (1968). 2. Y.C. Cheng and W.D. Westvood, J . E l e c t r o n i c Mat. 3_, 37 (1974). 3. A. Frova and P. M i g l i o r a t o , Appl. Phys. Letters 15, 406 (1969). 4. B.J. Holden and F.G. Ullman, J . Electrochem. Soc. 116, 280 (1969). 5. F.G. Ullman and B.J. Holden, B u l l . Am. Phys. Soc. 12, 1132 (1967). 6. J.L. Ord, M.A. Hopper and W.P. Wang, J. Electrochem. Soc. 119, 439 (1972). 7. W.D. Cornish and L. Young, Proc. R. Soc. Lond. A. 335, 39 (1973). 8. F.S. Chen, J.E. Geusic, S.K. Kurtz, J.G. Skinner and S.H. Wemple, J. Appl. Phys. 37, 388 (1966). 9. W. Kanzig i n " S o l i d State Physics", edited by F. Seitz and D. Turn-b u l l (Academic Press Inc., New York, 1957), Vol. 4, p. 89. 10. J.F. Nye, "Physical Properties of C r y s t a l s " , Oxford Press (1957). 11. F.L. McCrackin, E. Passaglia, R.R. Stromberg and H. Steinberg, J . Res. Nat. Bur. Stand. A. 67, 363 (1963). R a c 12. K.H.-Zaininger and A.G. Revesz, RCA Review 25 (1), 85 (1964). 13. R.J. Archer, "Manual on Ellipsometry", Gaertner S c i e n t i f i c Corp. (1968) 14. P. Drude, Wied. Ann. Phys. 32, 623 (1887). 15. L.P. Mosteller, J r . and F. Wooten, J . Opt. Soc. Am. 58, 511 (1968). 16. O.S. Heavens, "Thin Film Physics", Methuen, London, 1970, p. 88. 17. R.M.A. Azzam and N.M. Bashara, J . Opt. Soc. Am. 64, 128 (1974). 18. De Smet, J . Opt. Soc. Am. 6_4, 631 (1974) 19. D. Den Engelsen, J . Opt. Soc. Am. 61, 1460 (1971). 20. M. Tomar and V.K. Srivastava, Thin S o l i d Films 15, 207 (1973). 21. R.J. Archer, J . Opt. Soc. Am. 52, 970 (1962). 22. D.E. Aspnes and A.A.. Studna, App. Opt. 10, 1024 (1971). 55 23. F.L. McCrackin, "A Fortran Program f o r Analysis of Ellipsometer measurements", N.B.S. Technical Note 479 (1969). 24. F.L. McCrackin, J . Opt. Soc. Am. 60, 57 (1970). 25. M.L. Ki n t e r , I. Weissman and W.W. Stein, J . Appl. Phys. 41^ , 828 (1970). 26. L. Young, "Anodic Oxide Films", Academic Press, New York, 1961. 27. S.H. Wemple, M. Di Domenico, J r . , and I. Cambibel, Appl. Phys. Letters 12, 209 (1968). 28. M. Di Domenico, J r . and S.H. Wemple, J . Appl. Phys. 40, 720 (1969). 29. N.F. Jackson, J . Appl. Electrochem. 3_» 91 (1973). 30. D.A. Vermilyea, J . Electrochem. Soc. 102, 207 (1955). 31. D.A. Vermilyea, J . Electrochem. Soc. 104, 542 (1957). 32. W.A. S h u r c l i f f , "Polarized L i g h t " , HarvardcUniversity Press, 1962. 56 APPENDIX Poincare Sphere Representation of P o l a r i z a t i o n of Light For a l i g h t wave t r a v e l l i n g i n the z d i r e c t i o n of an a r b i t r a r y coordinate system, the e l e c t r i c vector i s given by E x = aj cos (x + &i) Ey = a 2 cos (x + <52) where T = w(t - z / v ) , and u and v are the angular frequency and l i n e a r v e l o c i t y of the l i g h t , r e s p e c t i v e l y . The p o l a r i z a t i o n of the l i g h t wave can be described by the amplitudes aj and a 2 , and the phase di f f e r e n c e 6 = S 2 - 6 j . This e l e c t r i c vector has an e l l i p t i c a l locus i n the plane normal to the d i r e c t i o n of propagation, as shown i n F i g . A . l . The e l l i p t i c a l locus i s characterized by the azimuth, cf>, and the e l l i p t i c i t y x» tan x = ±~ a where a and b are the semi-axes of the e l l i p s e . F i g . A . l Locus of the e l e c t r i c vector for e l l i p t i c a l l y p o l a r i z e d l i g h t i n a plane normal to the d i r e c t i o n of propagation. 57 Another representation of the state of p o l a r i z a t i o n of the l i g h t i s by the Stokes parameters 5 0 = a\ + a 2 51 - - a f 5 2 = 2a}a 2 cos 6 53 = 2&i&2 s i n 6 where SQ i s proportional to the i n t e n s i t y of the l i g h t wave. For ellipsometry the Stokes parameters can be normalized so that SQ = 1 because one i s not concerned with the i n t e n s i t y of l i g h t . I t can be ; shown-'--'-»32 that the normalized Stokes parameters are given by 5 0 = 1 51 = cos 2a = cos 2x cos 2(j> (A.l) 5 2 = s i n 2a cos6 = cos 2x s i n 2cf> 53 = s i n 2a sin6 = s i n 2x where a i s an a u x i l i a r y angle defined by A 2 tan a = — a l I t can be seen from equations A . l that SQ = 1, 2<j>, 2x are the sp h e r i c a l coordinates equivalent to the cartesian coordinates S j , S 2 and S 3. Thus the state of p o l a r i z a t i o n of a l i g h t wave can be represented by a point P on a sphere of unit radius, as shown i n F i g . A.2. This sphere i s c a l l e d the Poincare sphere. 58 Fig. A.2 Representation of polarized l i g h t on Poincare sphere. The Poincare sphere provides a convenient method for representing polarized l i g h t and for predicting how o p t i c a l elements w i l l change the pol a r i z a t i o n . From equations A.1 and Fig. A.2 i t can be seen that the equator of the Poincare sphere represents l i n e a r l y polarized l i g h t . The upper and lower poles, c^ and c 2 , represent c i r c u l a r l y polarized l i g h t . Other points on the sphere indicate e l l i p t i c a l p o l a r i z a t i o n . The Poincare sphere i s used to predict the eff e c t of a retarder 59 (also c a l l e d wave plate) on a beam of l i g h t i n the following method. The p o l a r i z a t i o n of a l i g h t beam incident on a retarder i s represented by P i n Fig. A.3. R represents a retarder whose phase retardation i s 6 and whose azimuth i s <j> . After the l i g h t beam has passed through the retarder, i t s p o l a r i z a t i o n i s obtained by ro t a t i n g P about the axis OR through an angle 6. The f i l i a l l o c a t i o n P' represents the p o l a r i z a t i o n of l i g h t emerging from the retarder. 

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