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Faraday effect studies of CdS and CdIn2S4 Gaglardi, Kenneth Joseph 1972

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YiVt FARADAY EFFECT STUDIES OF CdS AND Cdln^S, by KENNETH JOSEPH GAGLARDI B . S o . , U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 5 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE Ph.D. i n t h e Department of PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June, 1972 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study . I f u r t h e r agree t h a t pe rmiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT The F a r a d a y e f f e c t i s used t o s t u d y the p r o p e r t i e s o f donor and c o n d u c t i o n e l e c t r o n s i n CdS and C d l r ^ S ^ . The e x p e r i m e n t s on CdS g i v e a d d i t i o n a l i n f o r m a t i o n t o t h a t o b t a i n e d f r o m p r e v i o u s ESR measurements w h i c h showed p o s s i b l e ' i m p u r i t y b a n d i n g * a t c o n c e n t r a t i o n s much l e s s t h a n 2x10^/cc. The e l e c t r o n s a r e shown t o behave a t h i g h f r e q u e n c i e s as i f t h e y were f r e e and a t l o w f r e q u e n c i e s a s i f t h e y were l o c a l i z e d . The c o n c e n t r a t i o n dependence o f t h e c o n d u c t i o n band e l e c t r o n i c e f f e c t i v e mass was measured u s i n g the f r e e c a r r i e r F a r a d a y e f f e c t and found t o be c o n s t a n t t o w i t h i n t h e e x p e r i m e n t a l e r r o r o f 7% o v e r the c o n c e n t r a t i o n range 3xl0 17 t o 7.6xl0 1 8/cc. The u n c e r t a i n t y i s much l e s s t h a n p r e s e n t l y r e p o r t e d i n the l i t e r a t u r e but too l a r g e t o v e r i f y the i n c r e a s e of 2% o v e r the c o n c e n t r a t i o n range p r e d i c t e d by a k.p c a l c u l a t i o n . A l a r g e r F a r a d a y r o t a t i o n t h a n a n t i c i p a t e d was ob s e r v e d a t 2°K and a n e x p l a n a t i o n i s p r e s e n t e d i n terms of a d d i t i o n a l e l e c t r i c - d i p o l e m a g n e t o - a b s o r p t i o n e n t e r i n g t h r o u g h t h e s p i n - o r b i t i n t e r a c t i o n . T h i s i s i n t e r p r e t e d a s t h e o b s e r v a t i o n o f t h e s o - c a l l e d 'combined r e s o n a n c e ' e f f e c t i n CdS. The c o n d u c t i o n band e l e c t r o n i c e f f e c t i v e mass o f C d I n 2 S ^ was determined f r o m F a r a d a y e f f e c t measurements and found t o be m%i=0.17± .02. To o b t a i n t h i s v a l u e i t i i i "'was n e c e s s a r y t o measure t h e i n d e x o f r e f r a c t i o n i n t h e n e a r i n f r a r e d s p e c t r a l r e g i o n between 0 . 6 and 1.6yK. A v a l u e of 2.57 was found a t 1 . 6 / t . TABLE OF CONTENTS Page Abstract i i Table of Contents i v L i s t of Figures v i L i s t of Tables v i i Acknowledgements v i i i CHAPTER I INTRODUCTION AND PURPOSE 1 CHAPTER II THEORY 6 A-Faraday E f f e c t 6 1- Free Carrier Faraday E f f e c t 7 a-Degenerate S t a t i s t i c s 8 b-Non-degenerate S t a t i s t i c s 9 2- Interband Faraday E f f e c t 10 3- Combined Resonance E f f e c t 13 4- 0ther Factors 14 B - E l e c t r i c a l Properties 16 1- Conductivity 16 a-D.C. Conductivity 16 b-A.C. Conductivity 16 i - Conduction v i a the Group V e l o c i t y 16 i i - Conduction via Hopping 17 2- Conductivity Temperature Dependence 18 a-Carrier Concentration 18 b-Mobility 19 3- H a l l E f f e c t 20 CHAPTER I I I EXPERIMENTAL ARRANGEMENT 24 A - A p p a r a t u s f o r F a r a d a y E f f e c t Measurements 24 1- Source 24 2- Monochromator 26 3- I n i t i a l P o l a r i z e r 26 . 4 - C r y o s t a t ' 29 5- D e t e c t i o n System 29 6- Temperature D e t e r m i n a t i o n 33 7- Kagnet 33 8-Magnetic F i e l d D e t e r m i n a t i o n s 36 9- Sample P r e p a r a t i o n 36 V 10- A l i g n m e n t 37 11- Measurement of the F a r a d a y Angle 38 12- M i r r o r C o r r e c t i o n 38 B-Apparatus f o r E l e c t r i c a l Measurements 41 1- E l e c t r i c a l F r o b e 4-1 2- Temperature Measurements 1+1 3- Sample P r e p a r a t i o n 43 4- E l e c t r i c a l C o n t a c t s 46 5- R e s i s t i v i t y Measurement P r o c e d u r e 46 6- H a l l E f f e c t 48 C-Apparatus f o r I n d e x o f R e f r a c t i o n Measurements 50 CHAPTER IV RESULTS AND ANALYSIS $2 A-CdS 52 1- F a r a d a y E f f e c t R e s u l t s 52 2- D i s c u s s i o n o f 300°K R e s u l t s 52 a- A Z Component 55 b- A " 2 Component 57 3- D i s c u s s i o n of t h e 2°K R e s u l t s 59 a- A ~ Z Component 59 b- A 1 Component -Samples C2-A, C l and C3 60 c- A2" Component -Samples #2 .and C2-B 61 4 - R e s u l t s o f E l e c t r i c a l E x p e r i m e n t s 62 B - C d l n ^ S ^ 68 1- F a r a d a y E f f e c t R e s u l t s 68 2- I n d e x o f R e f r a c t i o n 68 3- S l e c t r i c a l R e s u l t s 70 4 - A n a l y s i s of R e s u l t s 70 CHAPTER V CONCLUSIONS 72 A - G e n e r a l C o n c l u s i o n s 72 B - S p e c i f i c C o n c l u s i o n s 73 APPENDIX ' 75 A-Chromic A c i d E t c h 75 B-Indium P l a t i n g S o l u t i o n 76 REFERENCES 77 LIST OF FIGURES Figure T i t l e Page I I I - l Block diagram of Faraday-experimental arrangement 25 III-2 Layout of the monochromator 27 III-3 Infrared spectrum of source 28 III-4 I n i t i a l p o l a r i z e r 30 III-5 Liquid helium cryostat 31 III-6 Schematic of Faraday e l e c t r o n i c s 32 III-7 Rotating p o l a r i z e r assembly 35 III-8 C a l i b r a t i n g apparatus 40 III-9 Sample holder for e l e c t r i c a l studies 42 111-10 Germanium thermometer el e c t r o n i c s 44 I I I - l l H a l l sample 45 I I I - 12 Schematic of H a l l e l e c t r o n i c s 47 IV- 1 Observed Faraday rotation 54 IV-2 Comparison with the r e s u l t s of Ebina et a l . 58 IV-3 Temperature dependent r e s i s t i v i t y data 63 IV-4 Temperature dependent H a l l constant 64 IV-5 Frequency dependent resistance 66 IV-6 Refractive index of CdIn~S, 69 v i i L I S T OF TABLES Ta b l e T i t l e Page IV-1 F a r a d a y R o t a t i o n a t 300°K 53 IV-2 F a r a d a y R o t a t i o n a t 2°K 53 IV-3 C o n d u c t i o n Band E f f e c t i v e Mass 55 IV-4 E x t r a R o t a t i o n a t 2°K 59 1T-5 R e s i s t i v i t y and F r e e C a r r i e r C o n c e n t r a t i o n 67 IV-6 R e s i s t i v i t y A c t i v a t i o n E n e r g i e s 67 IV-7 F a r a d a y R o t a t i o n a t 300° K 68 IV-8 CdIn~S, E l e c t r i c a l R e s u l t s 69 V l l l ACKN^V/LEDGI^SKTS , I w i s h t o t h a n k Dr. C F . S c h w e r d t f e g e r f o r h i s c o n t i n u o u s i n t e r e s t and h e l p f u l a d v i c e t h r o u g h o u t t h e s e i n v e s t i g a t i o n s and d u r i n g t h e t h e s i s p r e p a r a t i o n . The o t h e r members o f t h e s u p e r v i s o r y c o m m i t t e e , D r s . B a r r i e , B i c h a r d and P r o f e s s o r B u r g e s s gave h e l p f u l a d v i c e and s u g g e s t i o n s . T h e i r k i n d n e s s i s ackn o w l e d g e d w i t h t h a n k s , . My t h a n k s t o Mr. Dave Q u i r t who w i l l i n g l y p r e f o r m e d ESR measurements on each o f t h e CdS s a m p l e s . The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s was s u p p o r t e d by 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 Canada, g r a n t # A2228 and t h e D e f e n c e R e s e a r c h B o a r d o f Canada, g r a n t 7f DR3- 9 5 3 5 - 1 0 . The a u t h o r r e c e i v e d f i n a n c i a l a s s i s t a n c e t h r o u g h N a t i o n a l R e s e a r c h C o u n c i l S c h o l a r s h i p s and K i l l a m f e l l o w s h i p s , and w i s h e s t o e x t e n d h i s t h a n k s . I w i s h t o t h a n k my w i f e , E u n i c e f o r h e r p a t i e n c e and t r u s t . 1 CHAPTER I INTRODUCTION AND PURPOSE Fa r a d a y e f f e c t measurements i n s e m i c o n d u c t o r s have y i e l d e d i m p o r t a n t i n f o r m a t i o n about t h e i r band s t r u c t u r e and t h e e f f e c t i v e mass o f t h e m a j o r i t y . c a r r i e r s . S e v e r a l r e v i e w a r t i c l e s have appeared o v e r the p a s t few y e a r s w h i c h d e a l m a i n l y w i t h such measurements and t h e i m p o r t a n t i n f o r m a t i o n w h i c h may be c a l c u l a t e d from them. The m o t i v a t i o n f o r a p p l y i n g t h e F a r a d a y e f f e c t t o the p r e s e n t i n v e s t i g a t i o n stems from-'.the r e s u l t s of e l e c t r o n s p i n r e s o n a n c e (ESR) s t u d i e s o f donor e l e c t r o n s i n C d S 2 and CdIn 2S4 \ The ESR spectrum o f e l e c t r o n s i n s h a l l o w donor s t a t e s o f n-type CdS c o n s i s t s o f a s i n g l e a b s o r p t i o n l i n e w h i c h i s o b s e r v e d o n l y a t low t e m p e r a t u r e s . The g - v a l u e o f t h e l i n e i s indep e n d e n t o f t h e dopant and i s t h e same a s t h a t measured f o r the c o n d u c t i o n e l e c t r o n s t h r o u g h o p t i c a l Zeeman e f f e c t s t u d i e s ^ The absence o f h y p e r f i n e s t r u c t u r e has 5 been e x p l a i n e d by c o n s i d e r i n g t h a t t h e e l e c t r o n s a r e i n a band o f d e l o c a l i z e d s t a t e s r e s u l t i n g i n m o t i o n a l a v e r a g i n g o f t h e h y p e r f i n e i n t e r a c t i o n s . T h i s would be c o n s i s t e n t w i t h t h e e x p l a n a t i o n g i v e n f o r the s i n g l e ESR l i n e o b s e r v e d i n m o d e r a t e l y doped n-type s i l i c o n e x c e p t f o r the f a c t 2 t h a t i t o c c u r s a t a l o w e r donor c o n c e n t r a t i o n , f o r example c h l o r i n e a t o m s / c c ^ Upon i n c r e a s i n g the c o n c e n t r a t i o n t h e i n t e n s i t y o f the ESR l i n e f i r s t i n c r e a s e s and t h e n d e c r e a s e s , f i n a l l y d i s a p p e a r i n g a t N s 2 x l 0 l 8 / c c f r e e c a r r i e r s . T h i s has been i n t e r p r e t e d as a m e r g i n g of t h e i m p u r i t y s t a t e s w i t h t h e c o n d u c t i o n band where s p i n r e l a x a t i o n e f f e c t s broaden the r e s o n a n c e t o t h e p o i n t o f u n o b s e r v a b i l i t y . I n a l l c a s e s t h e number of uncompensated s h a l l o w donors quoted as b e i n g p r e s e n t i n t h e sample was based on a room t e m p e r a t u r e H a l l e f f e c t measurement and was a p p r o x i m a t e l y an o r d e r o f magnitude g r e a t e r t h a n t h a t c a l c u l a t e d f r o m t h e i n t e n s i t y of the ESR s i g n a l . The s u g g e s t i o n of d e l o c a l i z e d s t a t e s o r t h a t an ' i m p u r i t y band' i s formed i s s u p p o r t e d by the work of M o t t and Twose (196l)^who p r e d i c t e d t h a t t h e o nset of b a n d i n g s h o u l d o c c u r a t a donor c o n c e n t r a t i o n such t h a t t h e a v e r a g e i n t e r d o n o r s e p a r a t i o n i s a p p r o x i m a t e l y t h r e e e f f e c t i v e Bohr r a d i i . F o r CdS, t h i s c o n c e n t r a t i o n i s ~-2.5xl0 ^ / c c . Tviott^ has argued t h a t a t 0°K t h e o n s e t of b a n d i n g should be sharp, i . e . t h e a c t i v a t i o n energy s h o u l d f a l l d i s c o n t i n u o u s l y t o z e r o as t h e donor c o n c e n t r a t i o n i s i n c r e a s e d . The p r e s e n t work on CdS was u n d e r t a k e n t o d e t e r m i n e t h e c h a r a c t e r i s t i c s o f e l e c t r o n s i n t h e donor l e v e l s f o r c o n c e n t r a t i o n s of d o n o r s w h i c h a r e > 2 . 5 x l o l 7 / c c . The c h a r a c t e r i s t i c s c o n s i d e r e d were t h e e l e c t r o n i c e f f e c t i v e mass and the mechanism of c o n d u c t i o n . The s t u d y was made 3 a s a f u n c t i o n o f c o n c e n t r a t i o n between 3 x l O ^ - 7 / c c and 7.6xlcA* V c c. Room t e m p e r a t u r e H a l l e f f e c t measurements were made t o d e t e r m i n e t h e . c o n c e n t r a t i o n s . The n e a r i n f r a r e d F a r a d a y e f f e c t was chosen t o measure t h e e f f e c t i v e mass f o r the f o l l o w i n g r e a s o n s : a) i t measures a bare mass f o r the f r e q u e n c i e s used, b) t h e measurements can be performed w i t h e q u a l f a c i l i t y a t room and l i q u i d h e l i u m t e m p e r a t u r e s , c ) t h e mass measured i s t h a t of e l e c t r o n s near t h e F e r m i l e v e l and d) the equipment was r e a d i l y a s s e m b l e d . P r e v i o u s . F a r a d a y measurements on CdS were made by B a l k a n s k i and H o p f i e l d (1962 )^ a t room t e m p e r a t u r e f o r 'pure' and g a l l i u m doped (N= 6 x 1 0 - ^ / 0 0) samples o n l y ; the l a t t e r i n d i c a t i n g a f r e e c a r r i e r e f f e c t i v e mass of (0.20±.01)]%. to ^ Other o p t i c a l d e t e r m i n a t i o n s of m t o d a t e have y i e l d e d v a l u e s c l o s e t o 0.20m„, but the measurements a t d i f f e r e n t c o n c e n t r a t i o n s have shown more t h a n a 3 5 % s c a t t e r . The t h e o r e t i c a l k.p f o r m u l a t i o n o f Cardona (1961) p r e d i c t s a 2cJo i n c r e a s e i n m* o v e r the c o n c e n t r a t i o n range 3x10^-7 t o 8x10-^ / c c . S i n c e the F a r a d a y measurements y i e l d m* a t the F e r m i l e v e l f o r de g e n e r a t e m a t e r i a l s , t h e s e r e s u l t s p r e s e n t a check t o the c o n c e n t r a t i o n dependence of the e f f e c t i v e mass and hence t o t h e p o s s i b l e n o n - p a r a b o l i c i t y o f t h e c o n d u c t i o n band. F o r t h e c o n d u c t i o n mechanism, two p o s s i b i l i t i e s were c o n s i d e r e d : a) normal band c o n d u c t i o n v i a the d r i f t v e l o c i t y 4 o f t h e e l e c t r o n s and b) a h o p p i n g c o n d u c t i o n . The f i r s t i s f r e q u e n c y i n d e p e n d e n t f o r f r e q u e n c i e s w h i c h a r e low compared t o the phonon and o t h e r s c a t t e r i n g f r e q u e n c i e s . H opping c o n d u c t i o n may be f r e q u e n c y dependent a t low f r e q u e n c i e s i f the h o p p i n g t i m e s a r e s u f f i c i e n t l y l o n g . it P o l l a k and G e b a l l e (1961) o b s e r v e d a f r e q u e n c y dependent c o n d u c t i v i t y f r o m 10 t o 105 Hz i n s i l i c o n w h i c h t h e y a t t r i b u t e d t o h o p p i n g . The p r e s e n c e of c o n d u c t i o n v i a h o p p i n g i m p l i e s t h a t t he e l e c t r o n s a r e l o c a l i z e d f o r t i m e s w h i c h a r e a t l e a s t comparable t o the h o p p i n g t i m e s . F r e q u e n c y dependent r e s i s t i v i t y measurements o v e r the range 10 t o 10^ Hz were made a t low t e m p e r a t u r e s on t h e CdS samples t o i d e n t i f y t h e c o n d u c t i o n mechanism. The i n t e r e s t i n C d l ^ S ^ r e s u l t e d from the r e c e n t ESR 3 work o f R. K e r r i n w h i c h f u r t h e r a n a l y s i s of the r e s u l t s r e q u i r e d the knowledge o f t h e c o n d u c t i o n band e f f e c t i v e mass and the s p i n - o r b i t s p l i t t i n g o f t h e v a l e n c e band. A 1 3 r e c e n t work by Endo e t a l . (1970) r e p o r t e d a mass o f 0.21110 deduced from Seebeck c o e f f i c i e n t d a t a but t h e y gave n e i t h e r t h e d e t a i l s o f the measurements n o r an e s t i m a t e o f t h e e r r o r . U s i n g t h a t v a l u e of t h e e f f e c t i v e mass and t h e m a g n e t o r e s i s t a n c e a n i s o t r o p y w h i c h t h e y o b s e r v e d , Endo e t a l . (1970) made f o u r s u g g e s t i o n s f o r t h e t r a n s v e r s e and l o n g i t u d i n a l e f f e c t i v e masses. S i n o e the Fa r a d a y e f f e c t y i e l d s a d i f f e r e n t a v e r a g e o f l o n g i t u d i n a l and t r a n s v e r s e masses, i t s h o u l d d e t e r m i n e w h i c h o f t h e s u g g e s t i o n s of 5 13 Endo et a l . i s c o r r e c t . '3 The mass r e p o r t e d by Endo e t a l . would, be t h a t o f an e l e c t r o n i n e q u i l i b r i u m w i t h the phonon f i e l d o r a p o l a r o n mass but t h e s i z e of t h i s c o r r e c t i o n s h o u l d be s m a l l e r f o r the l e s s p o l a r C d l n g S ^ t h a n f o r CdS where i t i s —10%. N e g l e c t i n g t h i s c o r r e c t i o n and u s i n g the l o n g i t u d i n a l 13 and t r a n s v e r s e masses suggested by Endo et a l . one p r e d i c t s a F a r a d a y e f f e c t i v e mass i n the range 0.06 t o 0.09m0. 6 CHAPTER I I  THEORY A-F a r a d a y E f f e c t The F a r a d a y e f f e c t i s the r o t a t i o n of the p l a n e o f p o l a r i z a t i o n o f p l a n e p o l a r i z e d l i g h t p r o p a g a t i n g t h r o u g h a medium i n t h e d i r e c t i o n o f an a p p l i e d m a g n e t i c f i e l d . F o r low f i e l d s , t h i s r o t a t i o n i s d i r e c t l y p r o p o r t i o n a l t o the m a g n e t i c f i e l d s t r e n g t h . A s i m p l e e x p l a n a t i o n o f t h i s e f f e c t i s o b t a i n e d by c o n s i d e r i n g t h e p l a n e p o l a r i z e d wave as b e i n g composed o f r i g h t and l e f t c i r c u l a r l y p o l a r i z e d components w h i c h p r o p a g a t e a t d i f f e r e n t phase v e l o c i t i e s , c / n + and c / n . r e s p e c t i v e l y , where n + / n _ a r e t h e i n d i c e s o f r e f r a c t i o n o f r i g h t / l e f t c i r o u l a r l y p o l a r i z e d l i g h t and c i s the speed of l i g h t i n vacuo. The r e s u l t i s a r o t a t i o n o f the p l a n e o f p o l a r i z a t i o n by an amount 0 g i v e n by • 9 = T T ( < L - " a + ) (II.1) where to i s t h e a n g u l a r f r e q u e n c y and i i s the d i s t a n c e t r a v e r s e d . E q u a t i o n II.1 d e f i n e s t h e sense o f the r o t a t i o n a s p o s i t i v e f o r c l o c k w i s e r o t a t i o n s about the m a g n e t i c f i e l d d i r e c t i o n a s viewed by one l o o k i n g a l o n g the f i e l d . A t f r e q u e n c i e s where the a b s o r p t i o n i s n e g l i g i b l e and 7 f o r propagation of l i g h t a long an a x i s of a t l e a s t t h r e e -f o l d symmetry, the r o t a t i o n may be expressed i n terms of the c o n d u c t i v i t y tensor <TjkL - (^fL+i <^ as 0 - °** 1 ( I I . 2 ) 2. n. c where n i s the average of n. and n + and i s u s u a l l y taken to be the zero f i e l d r e f r a c t i v e index. The Faraday r o t a t i o n r e s u l t s from the magneto-d i s p e r s i o n which i s c a u s a l l y r e l a t e d t o the magneto-a b s o r p t i o n . The Faraday r o t a t i o n i s l a b e l l e d by the magneto-a b s o r p t i o n mechanism on which i t depends. For the present work there are s e v e r a l mechanisms which c o n t r i b u t e to the measured r o t a t i o n . They are: 1) f r e e c a r r i e r Faraday r o t a t i o n which r e s u l t s from magneto-absorption between Landau l e v e l s i n a s i n g l e band. 2) interband Faraday r o t a t i o n which r e s u l t s from magneto-a b s o r p t i o n between two bands. This i n c l u d e s the Faraday r o t a t i o n due to donor l e v e l to conduction band magneto-a b s o r p t i o n . 3) Faraday r o t a t i o n due t o combined resonance magneto-a b s o r p t i o n . 4) other f a c t o r s a f f e c t i n g the Faraday r o t a t i o n . 1-Free C a r r i e r Faraday E f f e c t The f r e e c a r r i e r Faraday r o t a t i o n has been t r e a t e d 14-by M i t c h e l l (1955) among others using a c l a s s i c a l Prude-8 Z e n e r a p p r o a c h y i e l d i n g a . F a r a d a y r o t a t i o n g i v e n b y JL H ~ ^ a c ^ ( I I , 3 ) w h e r e N i s t h e f r e e c a r r i e r c o n c e n t r a t i o n , q i s t h e m a g n i t u d e o f t h e e l e c t r o n i c c h a r g e , H i s t h e e x t e r n a l m a g n e t i c f i e l d , a n d m* p i s a n a v e r a g e e l e c t r o n i c e f f e c t i v e m a s s . The m a s s i n e q u a t i o n I I . 3 d e p e n d s o n t h e e n e r g y -momentum r e l a t i o n and t h e a p p r o p i a t e s t a t i s t i c s and c a n be c a l c u l a t e d e x p l i c i t l y b y u s i n g t h e B o l t z m a n n t r a n s p o r t e q u a t i o n m o d i f i e d f o r t h e p r e s e n c e o f a n e x t e r n a l m a g n e t i c v f i e l d t o c a l c u l a t e t h e c o n d u c t i v i t y t e n s o r c o mponent < ^ 1 5 A g e n e r a l e x p r e s s i o n h a s b e e n g i v e n b y S t e p h e n a n d L i d i a r d 16 ( 1 9 5 8 ) b a s e d o n t h e w o r k o f A b e l e s a n d M e i b o o m ( 1 9 5 4 ) * F o r l o w f i e l d s ( " c ^ l ; w c = — i s t h e c y c l o t r o n f r e q u e n c y ) , tn*c h i g h f r e q u e n c i e s ' (wa>1) a n d f o r a s i n g l e e n e r g y s u r f a o e w i t h t h e z - a x i s b e i n g a n a x i s o f a t l e a s t t h r e e f o l d s y m m e t r y , t h e y o b t a i n e d w h i c h u p o n c o m p a r i s o n w i t h I I . 3 y i e l d s ( I I . 4 ) J£ ( I I . 5 ) , 3 >wx->v3 ik K ^ E q u a t i o n I I . 5 i n d i c a t e s t h a t t h e m a s s m e a s u r e d b y t h e F a r a d a y e f f e c t i s a n a v e r a g e o v e r a l l e l e c t r o n s w e i g h t e d b y o r e f f e c t i v e l y a n a v e r a g e o f t h o s e e l e c t r o n s w i t h i n - 2 k T o f t h e F e r m i l e v e l E p . a - D e g e n e r a t e S t a t i s t i c s F o r d e g e n e r a t e s t a t i s t i c s ( E F » k T ) , t h e m e a s u r e d m a s s 9 i s e s s e n t i a l l y the mass .at t h e F e r m i l e v e l and f o r t h i s case case the f r e e c a r r i e r F a r a d a y e f f e c t can be used t o map out the E ( k ) r e l a t i o n s h i p . F o r s p h e r i c a l energy s u r f a c e s though not n e c e s s a r i l y p a r a b o l i c , and f o r a F e r m i l e v e l E y i n the band such t h a t 3% -^ i s non-zero o n l y a t Ey, the F a r a d a y r o t a t i o n i s g i v e n by 3 1 i£ z (II.6) 2, t/3 L J *-where kt. = (3'«N) .. Comparison o f I I . 3 and I I . 6 g i v e s ^ = "TV (4c) (1I-7) I f t h e band i s p a r a b o l i c , mj i s e q u i v a l e n t t o 1 1 >3Ef - c o n s t a n t ( I I . 8 ) As w i l l be i n d i c a t e d i n c h a p t e r I V , the c o n d u c t i o n band of CdS i s o n l y s l i g h t l y n o n - p a r a b o l i c . b-Non-degenerate S t a t i s t i c s F o r t h i s case %F/kT«l and ^iz i s s i g n i f i c a n t l y d i f f e r e n t f r om z e r o over t h e c o n d u c t i o n band. Now f o r a n o n - p a r a b o l i c band, e l e c t r o n s a t d i f f e r e n t e n e r g i e s have d i f f e r e n t masses and t h e measured i s t h e n an a v e r a g e o v e r t h e p e r t i n e n t energy r a n g e . The i n t e g r a l s i n e q u a t i o n I I . 5 must be e v a l u a t e d and hence an e x p l i c i t E ( k ) r e l a t i o n must be known. The f i r s t terms f o r CdS a r e a s m a l l l i n e a r term and a s l i g h t l y a n i s o t r o p i c q u a d r a t i c term? F o r C d I n 2 S ^ , v e r y l i t t l e i s known about the band 10 13 s t r u c t u r e . Recent measurements i n d i c a t e d t h a t t h e r e a r e 3 o r 6 e l l i p s o i d s o f e n ergy a l o n g the [ l o b ] d i r e c t i o n s . A ssuming a q u a d r a t i c dependence of E on k i n the v i c i n i t y o f t h e minima^ the F a r a d a y r o t a t i o n f o r f r e e c a r r i e r s can be c a l c u l a t e d by g e n e r a l i z i n g the r e s u l t s o f A b e l e s and 1& 15 M.eiboom a s o u t l i n e d by Stephen and L i d i a r d . The e f f e c t i v e mass measured by t h e F a r a d a y e f f e c t f o r e l l i p s o i d s a l o n g t h e pLOo] d i r e c t i o n s i s t h e n g i v e n by z 1/2 (11.10) op 3 m.£ where m^ and m t a r e t h e l o n g i t u d i n a l and t r a n s v e r s e masses r e s p e c t i v e l y . F o r t h e f r e q u e n c i e s used and f o r c u b i c symmetry t h i s e x p r e s s i o n i s independent of the f i e l d d i r e c t i o n ' . 5 The f r e e c a r r i e r F a r a d a y r o t a t i o n i n a l l o f the above c a s e s v a r i e s a s to"2 o r \ z and i s g i v e n a n a l y t i c a l l y by (11,6) 2- Inter-band F a r a d a y E f f e c t The i n t e r b a n d F a r a d a y e f f e c t has been t r e a t e d by s e v e r a l a u t h o r s . The p r e s e n t s e c t i o n u s e s th e d e r i v a t i o n 18 o f Boswarva e t a l . , a s r e f o r m u l a t e d by B a l k a n s k i and A m z a l l a g (1968) i n terms of the c o n d u c t i v i t y t e n s o r . The t o t a l H a m i l t o n i a n f o r t h i s c a s e i s g i v e n by X = Z ( K ; + H i ^ ) (11.11) i t h e sum o v e r i i s o v e r a l l e l e c t r o n s and >/0j. i s g i v e n by t h e o n e - e l e c t r o n H a m i l t o n i a n i n t h e absence of t h e r a d i a t i o n f i e l d but w i t h the p r e s e n c e o f t h e s t a t i c 11 m a g n e t i c f i e l d : H 2 = T S C + ^ + (sxvu).t? * ^(s-B) (11.12) where r i = "j? + 3: A and CU*-) i s t h e p e r i o d i c p o t e n t i a l , — > —> S t h e s p i n o p e r a t o r , H the e x t e r n a l f i e l d w i t h A i t s v e c t o r p o t e n t i a l . )4int r e p r e s e n t s t h e i n t e r a c t i o n o f t h e r a d i a t i o n f i e l d as a p e r t u r b a t i o n H u t = 4 t ( H . 1 3 ) where ff' = -~.Ea exp C H ** t - J + c.c. i s t h e v e c t o r p o t e n t i a l of t h e r a d i a t i o n f i e l d , and v? = — + J t>.* f S x V U) i s a g e n e r a l i z e d v e l o c i t y o p e r a t o r . The c o n d u c t i v i t y t e n s o r i s t h e n c a l c u l a t e d from t h e mean v a l u e o f the c u r r e n t and f o r f r e q u e n c i e s f a r from r e s o n a n t a b s o r p t i o n i s g i v e n by (II.14 ) where k i s the sum o v e r o c c u p i e d s t a t e s and k' t h e sum ov e r u n o c c u p i e d s t a t e s w i t h " r u ^ b e i n g the energy d i f f e r e n c e between s t a t e s k and k'. v*^ i s t h e m a t r i x element o f the v e l o c i t y o p e r a t o r and i s g i v e n by v k t ' = < k ' l ? v | k> ( I 1 < I 5 ) where J i s a u n i t v e c t o r a l o n g the e l e c t r i c f i e l d of t h e r a d i a t i o n . Boswarva, Howard and L i d i a r d (1962) c a l c u l a t e d t h e h i g h and low f r e q u e n c y dependence o f the r o t a t i o n . A t h i g h 12 f r e q u e n c i e s t h e y o b t a i n e d y - — * - t (11.16 w h i c h i s t h e same e x p r e s s i o n as f o r f r e e e l e c t r o n s and g i v e s a A* dependence ((9<C A 2 ). ' T h i s l i m i t i s a p p r o p r i a t e f o r t h e r o t a t i o n due t o donor l e v e l t o c o n d u c t i o n band m a g n e t o - a b s o r p t i o n and i n t h e e f f e c t i v e mass t r e a t m e n t o f the donor l e v e l s , t he mass i n e q u a t i o n " ( I I . 1 6 ) i s t h e e f f e c t i v e mass a t the bottom o f t h e c o n d u c t i o n band. The low f r e q u e n c y l i m i t a p p l i e s t o t r a n s i t i o n s between v a l e n c e and c o n d u c t i o n bands and i s g i v e n by <*>kV W k (11.17 ) where t h e sums a r e t h e same as f o r e q u a t i o n (11.14). and t h e momentum m a t r i x e l e m e n t s a r e d e f i n e d a s Pk'k = * Vk w i t h V^k = <<k ± L Vk'k (H.18) F o r w a v e l e n g t h s a p p r o p r i a t e t o t h e p r e s e n t work t h e i n t e r b a n d r o t a t i o n v a r i e s a s X~2. The f i n a l r o t a t i o n i s t h u s d e s c r i b e d by 9 = ft A 1 + E> X -1 (11.19) 13 3- Combined Resonance E f f e c t F r e e e l e c t r o n s i n CdS a t k>0 a r e d o u b l y d e g e n e r a t e ( s p i n ) . The a p p l i c a t i o n of a magn e t i c f i e l d removes t h i s d egeneracy and causes t h e c o n d u c t i o n band s t a t e s t o the Landau l e v e l and m s i s t h e s p i n i n d e x . S p i n -o r b i t i n t e r a c t i o n c a u s e s a m i x i n g o f the s t a t e s so t h a t where <*• » ^ and s t a t e s f a r t h e r away a r e mixed t o a s t i l l s m a l l e r e x t e n t , and r e p r e s e n t t h e breakdown o f the Ln. = ±t s e l e c t i o n r u l e f o r c y c l o t r o n a b s o r p t i o n . A n a l o g o u s l y t h e c y c l o t r o n r e s o n a n c e (CR) e l e c t r i c d i p o l e t r a n s i t i o n s ( A K = ± 1 , A m 3 - o ) a t the same energy a s t h e m a g n e t i c d i p o l e s p i n f l i p t r a n s i t i o n s (&n.= o ,£1^5=11) may o c c u r . These a r e t h e so c a l l e d combined resonance (KR) t r a n s i t i o n s . Rashba and Sheka (1962) c a l c u l a t e d the v e l o c i t y m a t r i x e l e m e n t s f o r c y c l o t r o n resonance^and f o r combined resonance w h i c h depends on the e x i s t e n c e o f a l i n e a r term i n t h e c o n d u c t i o n band energy momentum r e l a t i o n . F o r l i g h t p r o p a g a t i n g a l o n g the c - a x i s t h e y a r e c o a l e s c e i n t o Landau l e v e l s j n j i r L ^ where n l a b e l s 19 Combined Resonance C y c l o t r o n Resonance (11.21) 14 w h e r e A i s t h e e n e r g y d e p t h o f t h e e x t r e m u m l o o p o f t h e c o n d u c t i o n b a n d a n d = f^l " 0 , 1 8 The r e l a t i v e a b s o r p t i o n i n t e n s i t i e s v a r y a s t h e s q u a r e o f t h e m a t r i x e l e m e n t a n d t h i s r a t i o i s g i v e n b y V K R - 4 & (11.23) -4 - s 2.° t a k i n g 1^0,^3x10 eV a n d A-5.6x10 eV y i e l d s C % £ " *° (H.24) o r t h e c o m b i n e d r e s o n a n c e a b s o r p t i o n i s ~5$> o f t h e c y c l o t r o n r e s o n a n c e a b s o r p t i o n . T h e F a r a d a y e f f e c t a s s o c i a t e d w i t h KR a b s o r p t i o n s h o u l d a l s o be ~5$ o f t h a t a s s o c i a t e d w i t h CR a b s o r p t i o n s i n c e t h e c o m b i n e d r e s o n a n c e a b s o r p t i o n i s e l e c t r i c d i p o l e a b s o r p t i o n a t t h e s p i n - f l i p e n e r g y . T h e F a r a d a y e f f e c t a s s o c i a t e d w i t h c o m b i n e d r e s o n a n c e s h o u l d a l s o be o f t h e same s i g n a s t h a t a s s o c i a t e d w i t h c y c l o t r o n r e s o n a n c e s i n c e t h e a b s o r p t i o n f o r b o t h e f f e c t s i s p r e d o m i n a t e l y o f t h e l e f t c i r c u l a r l y p o l a r i z e d c o m p o n e n t . S i n c e t h e t r a n s i t i o n f r e q u e n c y o f c o m b i n e d r e s o n a n c e e f f e c t s a r e i n t h e . m i c r o w a v e r e g i o n f o r the f i e l d s u s e d a n d s i n c e t h e F a r a d a y e f f e c t i s o b s e r v e d a t n e a r i n f r a r e d f r e q u e n c i e s ( to » cO^^ ) t h e r o t a t i o n s h o u l d v a r y a s A z . . 4-0ther F a c t o r s F o r c o m p l e t e n e s s two o t h e r m e c h a n i s m s s h o u l d be 15 mentioned a l t h o u g h t h e y c o n t r i b u t e n e g l i g i b l y t o the p r e s e n t i n v e s t i g a t i o n . at a) S p i n - o r b i t i n t e r a c t i o n : B e n n e t t and S t e r n (1965) i n c l u d e d t h e e f f e c t o f s p i n - o r b i t i n t e r a c t i o n on t h e v e l o c i t y m a t r i x e l e m e n t s . F o r f r e q u e n c i e s w h i c h a r e not c l o s e t o the t r a n s i t i o n f r e q u e n c y t h i s e f f e c t i n c r e a s e s •-'the F a r a d a y r o t a t i o n by E s 0 / S g where E s o i s the s p i n - o r b i t s p l i t t i n g o f the v a l e n c e band and E„ i s t h e band gap energy. T h i s c o n t r i b u t i o n t a k e s on a A" 2 dependence f o r t h e p r e s e n t measurements and i s i m p o s s i b l e t o s e p a r a t e from t h e nor m a l i n t e r b a n d r o t a t i o n . b) S p i n l e v e l p o p u l a t i o n e f f e c t s : M i t c h e l l , P a l i k and W a l l i s (1965) o b s e r v e d t h a t a f r e q u e n c y i n d e p e n d e n t n e g a t i v e r o t a t i o n o f ~10 deg/kG-cm was added t o t h e F a r a d a y r o t a t i o n o f PbS a t 100° K. The e f f e c t was found t o be p r o p o r t i o n a l t o 1/T. They e x p l a i n e d the r e s u l t by p r o p o s i n g t h a t t h e a b s o r p t i o n o f one c i r c u l a r l y p o l a r i z e d component o f t h e p l a n e p o l a r i z e d beam was s e l e c t i v e l y b l o c k e d by the e l e c t r o n p o p u l a t i o n d i f f e r e n c e o f t h e c o n d u c t i o n band Zeeman l e v e l s . From t h e i r a n a l y s i s , t h i s becomes s i g n i f i c a n t o n l y i f t h e s i g n s of t h e gy r o m a g n e t i c r a t i o s o f the v a l e n c e and c o n d u c t i o n bands a r e d i f f e r e n t . F o r CdS t h e y a r e o f the same s i g n and t h e C d I n 2 S ^ measurements were performed a t room t e m p e r a t u r e where t h e e f f e c t i s n e g l i g i b l e . B - E l e o t r i c a l P r o p e r t i e s 1 C o n d u c t i v i t y  a-D.C. C o n d u c t i v i t y An n-type s e m i c o n d u c t o r h a v i n g a c o n c e n t r a t i o n o f The i n v e r s e o f t h e c o n d u c t i v i t y »/<^  i s c a l l e d the r e s i s t i v i t y / . The m o b i l i t y depends upon t h e t e m p e r a t u r e and s c a t t e r i n g mechanism and i s g i v e n by where t i s t h e a v e r a g e r e l a x a t i o n t i m e and sT i s the e l e c t r o n i c e f f e c t i v e mass. b-A.C. C o n d u c t i v i t y i - C o n d u c t i o n v i a the Group V e l o c i t y The A.C. c o n d u c t i v i t y <Ktf) f o r n o r m a l band c o n d u c t i o n e q u a l s t h e D.C. c o n d u c t i v i t y a t low f r e q u e n c i e s f o r most m a t e r i a l s s i n c e t h e s c a t t e r i n g t i m e s a r e much s h o r t e r t h a n the p e r i o d o f the a p p l i e d v o l t a g e . The A.C. c o n d u c t i v i t y i s g i v e n by 'N e l e c t r o n s p e r u n i t volume o f ave r a g e m o b i l i t y ^ and cha r g e -q has a D.C. c o n d u c t i v i t y g i v e n by (11.25) (11.26) (11.27) A s t h e f r e q u e n c y to/zxi becomes of t h e o r d e r of t h e A.C. c o n d u c t i v i t y becomes complex - t h a t i s , t h e c u r r e n t 17 r e s p o n s e I s out o f phase w i t h t h e a p p l i e d v o l t a g e . i i - C o n d u c t i o n v i a Hopping I n compensated s e m i c o n d u c t o r s w i t h a s m a l l c o n c e n t r a t i o n o f donors the t h e r m a l energy a l o n e i s u s u a l l y i n s u f f i c i e n t a t low t e m p e r a t u r e s to e x c i t e c a r r i e r s t o o b t a i n a s t e a d y s t a t e c u r r e n t upon a p p l i c a t i o n o f a D.C. f i e l d . A change o f p o t e n t i a l however i s a f f e c t e d i m p l y i n g new e q u i l i b r i u m c o n d i t i o n s ^ . e . a p o l a r i z a t i o n of t h e e l e c t r o n d i s t r i b u t i o n . The p o l a r i z a t i o n i s o b t a i n e d by e l e c t r o n s s h i f t i n g from one donor s i t e t o a n o t h e r and has been termed h o p p i n g . The f i r s t s u g g e s t i o n o f a c o n d u c t i o n mechanism o f t h i s t y pe was made by Hung and G l e i s s m a n (1950) 12 P o l l a k and G e b a l l e (1961) mada A.C. c o n d u c t i v i t y s t u d i e s on n - t y p e s i l i c o n t o o b t a i n i n f o r m a t i o n about t h e c h a r a c t e r i s t i c h o p p i n g t i m e s . They f o u n d t h a t the c o n d u c t i v i t y i n c r e a s e d a s a f u n c t i o n o f f r e q u e n c y I n t h e i r samples a c c o r d i n g t o t h e r e l a t i o n 0 8 K.c. - r ~ - c o n s t a n t x to * (11.28 ) and t h a t <rflrf. was p r o p o r t i o n a l t o t h e p r o d u c t o f donor and a c c e p t o r c o n c e n t r a t i o n s . They a n a l y s e d t h e i r r e s u l t s a s s u m ing low compensation and h o p p i n g e x c l u s i v e l y between p a i r s o f m a j o r i t y s i t e s . Under the f u r t h e r a s s u m p t i o n t h a t t h e r e s u l t s c o u l d be e x p r e s s e d a s a s u p e r p o s i t i o n o f r e s p o n s e s c o r r e s p o n d i n g t o v a r i o u s h o p p i n g t i m e s 1 t h e y wrote oo r i . ^ ffieKc.hj GftJJTzk^* (H.29) o w h i c h t h e y a p p l i e d t o t h e i r e x p e r i m e n t a l d a t a ( e q u a t i o n II.2.8) t o f i n d a v a l u e o f the w e i g h t i n g f u n c t i o n G ( t ) y i e l d i n g G(T) oC X " 1 ' 8 ( I I . 3 0 ) T h i s was t h e n compared w i t h t h e o r e t i c a l c a l c u l a t i o n s . F o r the p r e s e n t i n v e s t i g a t i o n t h e r e l e v a n t f a c t i s t h a t a t low f r e q u e n c i e s t h e donor e l e c t r o n s may behave a s though t h e y were l o c a l i z e d , i . e . t h e y c o n t r i b u t e n e g l i g i b l y t o the c o n d u c t i v i t y s i n c e the system can r e a d j u s t i t s e l f i n a t i m e s h o r t compared t o t h e p e r i o d o f the a p p l i e d f r e q u e n c y , whereas a t h i g h f r e q u e n c i e s t h e same e l e c t r o n s may respond a s i f t h e y were n o n - l o c a l i z e d . 2 - C o n d u c t i v i t y Temperature Dependence There a r e two s o u r c e s o f the t e m p e r a t u r e dependence o f t h e c o n d u c t i v i t y . The f i r s t i s t h e v a r i a t i o n o f t h e c a r r i e r c o n c e n t r a t i o n N and t h e second e n t e r s t h r o u g h t h e m o b i l i t y t e m p e r a t u r e dependence w h i c h i n t u r n r e f l e c t s changes i n t h e s c a t t e r i n g t i m e . C a r r i e r C o n c e n t r a t i o n A t h i g h t e m p e r a t u r e s , v i r t u a l l y a l l donor e l e c t r o n s , n o t i n a c c e p t o r s t a t e s , a r e i n the c o n d u c t i o n band. T h i s i s c a l l e d the e x h a u s t i o n t e m p e r a t u r e range. D e c r e a s i n g the t e m p e r a t u r e c a u s e s the number o f e l e c t r o n s i n t h e c o n d u c t i o n band t o d e c r e a s e a s e, D p r o v i d e d N A«N « N D where E b i s the donor b i n d i n g energy measured from the-c o n d u c t i o n band. N. i s t h e c o n c e n t r a t i o n o f a o o e p t o r s , 4. 42. " r O p t i c a l a b s o r p t i o n by h o p p i n g has been t r e a t e d and p o s s i b l y i n d i c a t e s photon a c t i v a t e d h o p p i n g . 19 N I s t h e c o n c e n t r a t i o n o f c o n d u c t i o n e l e c t r o n s and i s t h e c o n c e n t r a t i o n of donor c e n t e r s . As t h e t e m p e r a t u r e i s d e c r e a s e d f u r t h e r , N d e c r e a s e s t o a v a l u e much l e s s t h a n N A . The number o f e l e c t r o n s i n the c o n d u c t i o n band, -E /kT N 3 t h e n v a r i e s a s e . A t an i n t e r m e d i a t e t e m p e r a t u r e the number o f c o n d u c t i o n band c a r r i e r s i s e q u a l t o t h e a c c e p t o r c o n c e n t r a t i o n and t h i s t e m p e r a t u r e can be e s t i m a t e d f r o m the change i n t h e t e m p e r a t u r e dependence of the H a l l e f f e c t . An e s t i m a t e o f t h e a c c e p t o r c o n c e n t r a t i o n can t h u s be o b t a i n e d and hence the com p e n s a t i o n r a t i o . I f t he donor c o n c e n t r a t i o n i s s u f f i c i e n t , t h e donor e l e c t r o n s t a t e s o v e r l a p and merge w i t h the c o n d u c t i o n band i m p l y i n g t h a t E D goes t o z e r o . A t t h e s e c o n c e n t r a t i o n s t h e s e m i c o n d u c t o r r e s e m b l e s a v e r y d i l u t e m e t a l and t h e number o f e l e c t r o n s c o n t r i b u t i n g t o t h e e l e c t r i c a l c o n d u c t i v i t y i s a l m o s t i n d e p e n d e n t o f t e m p e r a t u r e . The c o n d u c t i v i t y i s t h e n l i m i t e d by i m p u r i t y s c a t t e r i n g . a s s o c i a t e d w i t h the n o n - p e r i o d i c i t y o f t h e l a t t i c e . b- M o b i l i t y The t e m p e r a t u r e dependence o f the s c a t t e r i n g t ime i s more complex. I t depends on the dominant s c a t t e r i n g mechanism and t h e degeneracy. F o r non-degenerate CdS the m o b i l i t y f o r T>150 K i s governed by p o l a r o p t i c a l phonon s c a t t e r i n g w h i l e a t l o w e r t e m p e r a t u r e s p i e z o e l e c t r i c s c a t t e r i n g 25 d o m i n a t e s . L i t t l e i s known about t h e t e m p e r a t u r e dependent 33 m o b i l i t y of Cdlr t ^ S ^ but the work of Sudo e t a l . (1970) 20 i n d i c a t e s t h a t a c o u s t i c p h o n o n s c a t t e r i n g d o m i n a t e s a t h i g h t e m p e r a t u r e s . F r o h l i c h ( 1 9 5 4 ) e s t i m a t e d t h e c o n t r i b u t i o n t o t h e m o b i l i t y o w i n g t o o p t i c a l p h o n o n s c a t t e r i n g . H e f o u n d j A a ? -x c o n s t a n t x | e x p - l j ( 1 1 . 3 1 ) w h e r e +v ^ot i s t h e a p p r o x i m a t e o p t i c a l p h o n o n e n e r g y ( • K i o L ~ 1 . 5 k T a t r o o m t e m p e r a t u r e f o r C d S ) . The c o n s t a n t a l s o c o n t a i n s a s l o w l y v a r y i n g t e m p e r a t u r e d e p e n d e n c e f o r "k^/ iOT > ^ - . The p i e z o e l e c t r i c s c a t t e r i n g c o n t r i b u t i o n t o t h e m o b i l i t y e x h i b i t s a t e m p e r a t u r e d e p e n d e n c e g i v e n a p p r o x i m a t e l y b y .A ^/Xp ~ c o n s t a n t x T 2 ^>2) The t e m p e r a t u r e d e p e n d e n c e o f t h e m o b i l i t y w h i c h i s l i m i t e d b y a c o u s t i c p h o n o n s c a t t e r i n g i s g i v e n f o r n o n - d e g e n e r a t e s t a t i s t i c s b y 2 , 7 -3/2 ~ c o n s t a n t x T ( 1 1 . 3 3) 3- H a l l E f f e c t I f a c u r r e n t d e n s i t y J * i s p a s s e d t h r o u g h a s e m i c o n d u c t o r i n t h e p r e s e n c e o f a m a g n e t i c f i e l d H z a p o t e n t i a l g r a d i e n t E Y i s g e n e r a t e d . T h i s i s c a l l e d t h e H a l l e f f e c t . F o r l o w f i e l d s t h e H a l l e l e c t r i c f i e l d i s p r o p o r t i o n a l t o t h e c u r r e n t d e n s i t y a n d t h e m a g n e t i c f i e l d a s E y = R J * H z ( 1 1 . 3 4) T h e c o n s t a n t R i s c a l l e d t h e H a l l c o n s t a n t . A s i m p l e 21 c l a s s i c a l a n a l y s i s o f t h e c u r r e n t i n a magn e t i c f i e l d l e a d s t o R - l / N e (11.35) where K i s the c a r r i e r c o n c e n t r a t i o n i n t h e specimen and e i s t h e charge o f t h e c a r r i e r s . E q u a t i o n (11.35) i s d e r i v e d on t h e a s s u m p t i o n t h a t a l l c a r r i e r v e l o c i t i e s a r e th e same. I t can be c o r r e c t e d t o a l l o w f o r a d i s t r i b u t i o n of c a r r i e r v e l o c i t i e s by R = r/Ne (II.36) where r i s t h e H a l l s c a t t e r i n g f a c t o r and i s g i v e n by r = eg >•'* (H • 37) where ? i s t h e e l e c t r o n r e l a x a t i o n t i m e . The s c a t t e r i n g f a c t o r r i s a number o f the o r d e r o f u n i t y v/hich depends on the m a g n e t i c f i e l d , t h e s c a t t e r i n g mechanism and. the degeneracy o f the c a r r i e r s , r becomes e q u a l t o u n i t y a t h i g h m a g n e t i c f i e l d (H»mVe'c ) and/or f o r h i g h l y d e g e n e r a t e c a r r i e r s . The low f i e l d v a l u e o f r f o r a c o u s t i c phonon 2.7 s c a t t e r i n g i s r = 3^8 and f o r i o n i z e d i m p u r i t y s c a t t e r i n g Z7 i s r=1.93 i n t h e non-degenerate l i m i t . D e v l i n (1967) t r e a t e d t h e case o f o p t i c a l phonon s c a t t e r i n g by s o l v i n g t he Boltzmann t r a n s p o r t e q u a t i o n u s i n g a v a r i a t i o n a l t e c h n i q u e . He p o i n t e d out t h a t the energy o f the o p t i c a l phonon i s n ot n e g l i g i b l e . H i s c a l c u l a t e d s c a t t e r i n g f a c t o r v a r i e s from 1.0 t o 1.23 as a 22 f u n c t i o n o f the temp e r a t u r e and i s . i n q u a l i t a t i v e agreement w i t h t h e measurements of S t i l l m a n , W o l f e , and Dimmock Ztl970) on p o l a r n -type G-aAs. Q u a n t i t a t i v e agreement i s reached a t room t e m p e r a t u r e . D e v l i n ' s c a l c u l a t i o n s p r e d i c t a low f i e l d s c a t t e r i n g f a c t o r o f 1.16 f o r CdS a t room t e m p e r a t u r e . L e w i s and Sondhiemer (1954) c a r r i e d a s i m i l a r a n a l y s i s t o t h e f i r s t a p p r o x i m a t i o n o n l y and o b t a i n e d q u a l i t a t i v e agreement w i t h the te m p e r a t u r e dependence of t h e s c a t t e r i n g f a c t o r found by S t i l l m a n e t a l . L e w i s and Sondhiemer's c a l c u l a t i o n y i e l d s a s c a t t e r i n g f a c t o r of 1.06 f o r CdS. 9 B a l k a n s k i and H o p f i e l d i n t e r p r e t e d t h e i r CdS H a l l d a t a u s i n g a s c a t t e r i n g f a c t o r o f 1.0 and K r o g e r , V i n k and V o l g e r (1955) used 3Tr/8=1.18 . The p r e s e n t measurements (300°K) were i n t e r p r e t e d u s i n g r-1.10±.10 (11.38) The u n c e r t a i n t y i n the s c a t t e r i n g f a c t o r i s the l a r g e s t s o u r c e o f e r r o r i n the p r e s e n t e x p e r i m e n t . F o r C d l ^ S ^ a f u r t h e r c o r r e c t i o n t o ac c o u n t f o r t h e n o n - s p h e r i c a l c h a r a c t e r o f t h e e q u a l energy s u r f a c e s o f the c o n d u c t i o n band may be n e c e s s a r y , but i t i s i m p o s s i b l e t o p r e d i c t t h e form o f t h i s c o r r e c t i o n s i n c e t h e e x a c t n a t u r e o f t h e band i s not e s t a b l i s h e d . 39 The measurements o f Sudo e t a l . (1970) on CdIn 2S i i f i n d i c a t e t h a t a c o u s t i c phonon s c a t t e r i n g l i m i t s t he e l e c t r o n m o b i l i t y and t h e r e f o r e t h e s c a t t e r i n g f a c t o r 23 would be 1.18 f o r non-degenerate samples and f o r t h e p r e s e n t measurements w i t h \ — E F / k T ^ 1 t h e a p p r o p r i a t e v a l u e would be 1.10 ±.10 - t h e same a s used f o r the CdS d a t a . There a r e t h r e e a d d i t i o n a l s o u r c e s o f e r r o r i n making H a l l measurements i f t e m p e r a t u r e g r a d i e n t s a r e c r e a t e d . These a r e t h e E t t i n g s h a u s e n , N e r n s t , and R i g h i - L e d u c e f f e o t s . Jones (1961) gave a b r i e f d i s c u s s i o n o f t h e s e t h r e e w h i c h i n v o l v e t h e c r e a t i o n o f a t e m p e r a t u r e g r a d i e n t ( E t t i n g s h a u s e n e f f e c t ) o r a v o l t a g e r e s u l t i n g from one. The E t t i n g s h a u s e n t e m p e r a t u r e g r a d i e n t causes an e l e c t r i c f i e l d p a r a l l e l t o t h e H a l l e l e c t r i c f i e l d , t h e s t r e n g t h o f w h i c h i s d e t e r m i n e d by t h e t h e r m o - e l e c t r i c power c o e f f i c i e n t (Seebeck e f f e c t ) . T h i s i n c r e m e n t i n t h e t r u e H a l l c o n s t a n t i s n e g l i g i b l e compared t o t h e H a l l c o n s t a n t s measured i n t h e p r e s e n t work. The e f f e c t s o f t h e s e c o n t r i b u t i o n s can be m i n i m i z e d f u r t h e r by m a i n t a i n i n g t h e samples i n good t h e r m a l c o n t a c t w i t h a b a t h o r by making t h e measurements a t f r e q u e n c i e s such t h a t u)? T>i where ? T i s the t h e r m a l r e l a x a t i o n t ime and i s o f t h e o r d e r o f seconds f o r t h e p r e s e n t m a t e r i a l s . M a g n e t o - r e s i s t i v e e f f e c t s can be n e g l e c t e d i f t h e H a l l p r o b e s a r e d i r e c t l y o p p o s i t e one a n o t h e r and i f t h e measurement draws n e g l i g i b l e c u r r e n t . 24 CHAPTER I I I EXPERIMENTAL ARRANGEMENT The equipment n e c e s s a r y t o p r e f o r m t h e e x p e r i m e n t s was assembled d u r i n g t h e c o u r s e o f the p r e s e n t work. S i n c e t h e a p p a r a t u s f o r t h e F a r a d a y e f f e c t measurements was d i f f e r e n t f r o m t h a t used f o r the e l e c t r i c a l measurements i t w i l l be d i s c u s s e d s e p a r a t e l y . The a p p a r a t u s f o r d e t e r m i n i n g t h e i n d e x o f r e f r a c t i o n a t i n f r a r e d f r e q u e n c i e s i s d i s c u s s e d i n s e c t i o n C. A-A p p a r a t u s f o r F a r a d a y E f f e c t Measurements The e x p e r i m e n t c o n s i s t e d o f m e a s u r i n g t h e r o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n o f p l a n e p o l a r i z e d l i g h t a s a f u n c t i o n o f t h e m a g n e t i c f i e l d . The l i g h t was o b t a i n e d f r o m t h e i n f r a r e d s p e c t r a l l i n e s o f a m e r c u r y a r c and was d i s p e r s e d by a t r i p l e s p r i s m monochromator. I n d i v i d u a l s p e c t r a l l i n e s were p l a n e p o l a r i z e d and t r a n s m i t t e d t h r o u g h t h e sample i n t h e d i r e c t i o n o f the a p p l i e d m a g n e t i c f i e l d . A b l o c k d i a g r a m o f t h e e x p e r i m e n t a l a p p a r a t u s i s shown i n f i g u r e I I I - l . 1-Source The s o u r c e was a P E K - 5 0 0 m e r c u r y a r c . I t s ne a r i n f r a r e d e m i s s i o n spectrum was measured on an E b e r t 25 / I Source L i g h t P a t h Monochromator Mercury-Manometer S c a l e : 1 cm= 5 i n c h e s S u p p o r t s f o r D e t e c t i o n Assembly. . I n i t i a l P o l a r i z e r ^ C r y o s t a t V \ ( —' Magnet S 3 , R o t a t i n g P o l a r i z e r and. D e t e c t o r F i g u r e I I I - l B l o c k diagram of F a r a d a y e x p e r i m e n t a l arrangement. 26 s p e o t r o m e t e r ( r e s o l u t i o n ~*10A) and c o n s i s t s i n p a r t of s i x l i n e s o f h a l f w i d t h a p p r o x i m a t e l y 50A superimposed on a o o n s t a n t background. The peak i n t e n s i t y v a r i e d from 2 t o 3 t i m e s t h e background i n t e n s i t y . The s o u r c e h o u s i n g c o n t a i n e d a f o c u s i n g arrangement e n s u r i n g t h a t most of t h e l i g h t was d i r e c t e d i n t o t h e monochromator. 2- Monochromator The o u t p u t o f the PEK-500 a r c was f o c u s e d on t h e e n t r a n c e s l i t (1.5 mm) of an i n f r a r e d t r i p l e p r i s m 3 2 , . monochromator b u i l t by G. B r e a l e y (1951 )• I n o r d e r t o o b t a i n a p a r a l l e l beam o u t p u t , t h e monochromator was m o d i f i e d so t h a t l i g h t from t h e p r i s m was i n c i d e n t on a p l a n e m i r r o r i n s t e a d o f t h e o f f a x i s p a r a b o l o i d f o c u s i n g m i r r o r a s t h e i n i t i a l c o n s t r u c t i o n r e q u i r e d . A l t h o u g h t h i s m o d i f i c a t i o n reduced t h e r e s o l v i n g power from"^x^l500 t o ~ 40, t h i s was s u f f i c i e n t t o r e s o l v e the l i n e s e m i t t e d by t h e s o u r c e . F i g u r e I I I - 2 shows t h e m o d i f i e d monochromator and t h e l i g h t p a t h . The motor s c a n was o n l y used t o o b t a i n scans of t h e spectrum a s shown i n F i g u r e I I I - 3 . 3- I n i t i a l P o l a r i z e r A f t e r b e i n g r e f l e c t e d f rom the p l a n e m i r r o r t h e quasi-monochromatic l i g h t l e a v e s t h e monochromator and i s i n c i d e n t upon the i n i t i a l Glan-Thompson p o l a r i z e r o r i e n t e d such t h a t t h e e l e c t r i c v e c t o r of t h e t r a n s m i t t e d p l a n e p o l a r i z e d beam i s v e r t i c a l . T h i s o r i e n t a t i o n was 2 7 To s o u r c e E n t r a n c e " s l i t J E x t e r i o r w a l l o f monochromator o f f a x i s m i r r o r M o t o r s c a n uKr-r/J ! I ..Plane m i r r o r tip I 1 T f ^ i i p f l T I I - 2 L a y o u t o f t h e monochromator F i g u r e I I I - 3 I n f r a r e d spectrum o f s o u r c e : a s measured by t h e s p e c t r o m e t e r w i t h no sample i n the p a t h . 29 ohosen t o ensure maximum r e f l e c t e d , i n t e n s i t y from the s m a l l p l a n e m i r r o r s i n t h e o r y o s t a t . The p o l a r i z e r was mounted i n the h o u s i n g shown i n f i g u r e I I I - 4 w h i c h p e r m i t t e d i t t o be r o t a t e d f o r a l i g n m e n t p u r p o s e s . 4 - C r y o s t a t The l i n e a r l y p o l a r i z e d i n f r a r e d beam e n t e r s t h e c r y o s t a t , shown i n f i g u r e I I I - 5 , t h r o u g h a f u s e d q u a r t z window ( t r a n s m i s s i o n range 0.2-4.5/<-)> and i s r e f l e c t e d a t 45° a n g l e o f i n c i d e n c e f rom a g o l d p l a t e d m i r r o r making 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 p a r a l l e l t o t h e m a g n e t i c f i e l d . The r a d i a t i o n p a s s e s t h r o u g h a 3/4 i n c h d i a m e t e r h o l e i n t h e copper n i t r o g e n t e m p e r a t u r e s h i e l d i n t o t h e g l a s s h e l i u m dewar, and f o l l o w s a s i m i l a r p a t h l e a v i n g the c r y o s t a t . ^ - D e t e c t i o n System A f t e r l e a v i n g t h e c r y o s t a t , t h e l i g h t was i n c i d e n t upon a n o t h e r Glan-Thompson p o l a r i z e r r o t a t i n g a t 14 r e v / s e c about a n a x i s p a r a l l e l t o t h e d i r e c t i o n o f l i g h t p r o p a g a t i o n . The r o t a t i n g p o l a r i z e r m o d u l a t e d t h e t r a n s m i t t e d l i g h t i n t e n s i t y a t 28 H e r t z . The m o d u l a t e d beam was i n c i d e n t upon a B3-SA19M l e a d s u l p h i d e c e l l f r o m . I n f r a r e d I n d u s t r i e s g e n e r a t i n g a 28 H e r t z s i g n a l . F i g u r e I I I - 6 c o n t a i n s a s c h e m a t i c o f t h e d e t e c t i o n e l e c t r o n i c s . Two 45 v o l t b a t t e r i e s i n p a r a l l e l were used t o b i a s t h e l e a d s u l p h i d e c e l l . A 28 H e r t z r e f e r e n c e s i g n a l was g e n e r a t e d by c h o p p i n g 30 gears Polarizer To dewar To monochromator Calibrated manual drive F i g u r e I I I - 4 I n i t i a l p o l a r i z e r h o u s i n g . The g e a r d r i v e i s a l s o shown. 31 F i g u r e I I I - 5 L i q u i d h e l i u m o r y o s t a t L i q u i d h e l i u m i n p u t t o vaciuum pump U l i g h t p a t h Copper / S e c t i o n A-A n i t r o g e n t e m p e r a t u r e s h i e l d • . ' ' 32 + 6 V o l t s 33fcja. Keferenoe Signal Output L. I Schematic of reference s i g n a l 45 V o l t s 2 e 5 M j T 5 _ Scale Expansion 25 P ? PAR #121 PbS o e l l 2.5M.rt-B r i s t o l 590 Reference input Schematic of detection e l e c t r o n i c s Figure III-6 Schematic of Faraday e l e c t r o n i c s . 33 l i g h t f r o m an Armaco PLB6 l i g h t i n c i d e n t on a P h i l i p s 0CP71 p h o t o t r a n s i s t o r . The l i g h t was chopped by two b l a d e s a t t a c h e d t o t h e r o t a t i n g p o l a r i z e r a s i n f i g u r e I I I - 7 0 F i g u r e I I I - 6 a l s o c o n t a i n s a s c h e m a t i c o f the r e f e r e n c e e l e c t r o n i c s . A 6 v o l t c a r b a t t e r y p r o v i d e d t h e power f o r p h o t o t r a n s i s t o r and the PLB6 b u l b . The s i g n a l and r e f e r e n c e were f e d i n t o a P r i n c e t o n A p p l i e d R e a s e a r c h L o c k i n #121 a m p l i f i e r s e t t o measure t h e phase d i f f e r e n c e between t h e i n p u t and r e f e r e n c e s i g n a l s . The o u t p u t v o l t a g e , p r o p o r t i o n a l t o the phase d i f f e r e n c e , was r e c o r d e d on a B r i s t o l 590 r e c o r d e r . 6- Temperature D e t e r m i n a t i o n The t e m p e r a t u r e o f the l i q u i d h e l i u m b a t h was d e c r e a s e d f r o m i t s b o i l i n g p o i n t (4.17~K) t o 2°K by pumping on t h e b a t h w i t h a l a r g e S t o k e s pump. The p r e s s u r e i n the h e l i u m dewar was measured w i t h a m e r c u r y manometer and t h e t e m p e r a t u r e i n f e r r e d u s i n g the 1958 t e m p e r a t u r e d a t a of t h e N a t i o n a l Bureau of S t a n d a r d s . The t e m p e r a t u r e was m a i n t a i n e d t o w i t h i n 0.1 K o f 2 K by a d j u s t i n g a s m a l l v a l v e i n t h e pumping l i n e t o keep t h e p r e s s u r e between 1.7 cm and 3*1 cm o f m e r c u r y . A c a r e f u l a d j u s tment a t t h e b e g i n n i n g o f an e x p e r i m e n t was found t o be s u f f i c i e n t . A C a r t e s i a n d i v e r p r e s s u r e r e g u l a t o r i n s t a l l e d i n t h e pumping l i n e was found t o be u n n e c e s s a r y . 7- Magnet A P a c i f i c E l e c t r i c M o t o r Company w a t e r c o o l e d magnet 34 E x p l a n a t i o n o f F i g u r e I I I - 7 The r e f e r e n c e , e l e c t r o n i c s (1) and t h e r e f e r e n c e g e a r (3) r o t a t e i n the h o u s i n g (2) by means o f t h e worm ge a r d r i v e (4). The m o d u l a t i n g Glan-Thompson p o l a r i z e r ( 7 ) was mounted i n s i d e t h e chopper ( 5 ) w h i c h was r o t a t e d a t 14 r e v / s e c i n h o u s i n g (6 ) by means o f a 3/8 t h h o r s e sychronous motor suspended below. The chopper b l a d e s (5) s p i n i n s i d e the groove shown i n (3) c h o p p i n g t h e l i g h t f rom t h e PLB6 b u l b i n c i d e n t upon t h e p h o t o t r a n s i s t o r The l i g h t and p h o t o t r a n s i s t o r were mounted on e i t h e r s i d e o f the chopper g r o o v e . F i g u r e I I I - 7 . R o t a t i n g p o l a r i z e r a s s e m b l y , (see o p p o s i t e page f o r d e t a i l s ) 36 model 12V-AT-L1 was used w i t h a 2.97 i n c h p o l e gap. The p o l e f a c e d i a m e t e r s were 4 i n c h e s . The magnet was powered by a i o o horsepower motor d r i v e n 300 Amp, 250 v o l t g e n e r a t o r w h i c h was l o c a t e d ^ 100 f e e t from the magnet t o e l i m i n a t e v i b r a t i o n s . The maximum m a g n e t i c f i e l d o b t a i n e d v/as 20.2 kG. 8- M a g n e t i o F i e l d D e t e r m i n a t i o n s A Rawson-Lush r o t a t i n g c o i l t y p e 820/B gaussmeter was p l a c e d n e x t t o t h e p o l e t i p about l j " from the sample. U s i n t h e n u l l t e c h n i q u e , t h e f i e l d was d e t e r m i n e d t o ± 2%. I n i t i a l c a l i b r a t i o n showed t h a t f i e l d s a t the r o t a t i n g c o i l and the sample were t h e same t o w i t h i n t h i s u n c e r t a i n t y . The g r a d i e n t i n the f i e l d n e a r t h e sample was -jL. | ^ ~ l $ / c m p a r a l l e l t o I f and T r | ^ - y | ^  0.5%/cm n o r m a l t o H. T y p i c a l sample t h i c k n e s s e s were 0 . 4 cm* 9 - Sample P r e p a r a t i o n The n-type CdS was p u r c h a s e d from E a g l e P i c h e r (samples #2 and #3) and from C l e v i t e C o r p o r a t i o n (samples CI, C2-A, C2-B, and C 3 ) . The C d I n 2 S ^ samples were g e n e r o u s l y donated by Dr. W. C z a j a o f RCA L a b o r a t o r i e s i n Z u r i c h . The c - a x i s o f t h e CdS s i n g l e c r y s t a l s and a p r i n c i p a l a x i s o f t h e C d l n 2 S^ s i n g l e c r y s t a l s were l o c a t e d by t h e back r e f l e c t i o n Laue x - r a y p a t t e r n , and s l i c e s were c u t no r m a l t o t h e s e a x e s w i t h a w i r e saw. The o r i e n t a t i o n o f t h e . c r y s t a l s - w a s checked a f t e r each cut t o ensure t h a t the f a c e s o f t h e samples were w i t h i n 2 degrees o f b e i n g n o r m a l 37 t o t h e chosen a x e s . The s l i c e s v a r i e d i n t h i c k n e s s from 0 . 6 0 cm f o r samples # 2 and # 3 t o 0 . 2 2 om f o r sample C 2 - A . S u r f a c e damage due t o sample c u t t i n g was removed by g r i n d i n g and p o l i s h i n g . The samples were ground t o a f l a t f i n i s h w i t h # 6 0 0 carborundum on g l a s s and t h e n p o l i s h e d w i t h # 6 0 0 carborundum on A s t r o m a t c l o t h t o a m i r r o r f i n i s h . B oth f a c e s o f the c r y s t a l were t r e a t e d i n t h i s manner. A f t e r p o l i s h i n g , t h e samples were mounted i n a sample h o l d e r and the mounted sample was x-rayed t o det e r m i n e t h e r e l e v a n t a x i s o r i e n t a t i o n r e l a t i v e t o t h e c r y s t a l f a c e t o w i t h i n -| d e g r e e . The sample i n i t s h o l d e r was t h e n mounted on t h e end of t h e t h i n w a l l nonmagnetic s t a i n l e s s s t e e l p r o b e e x t e n d i n g i n t o t h e p o l e gap o f t h e magnet. 1 0 - A l i g n m e n t W i t h t h e d e t e o t o r and r o t a t i n g p o l a r i z e r removed, a l a s e r beam was used t o t r a c e out t h e o p t i c a l p a t h . The l a s e r was l o c a t e d 1 1 3 i n c h e s from t h e sample. The beam, back r e f l e c t e d from the sample f a c e was used t o a l i g n t h e sample such t h a t the o r i e n t a t i o n o f the p o l i s h e d f a c e p o s i t i o n e d t h e d e s i r e d a x i s a l o n g t h e o p t i c a x i s . The o r i e n t a t i o n c o u l d be checked a c c u r a t e l y s i n c e one degree o f r o t a t i o n moved the r e f l e c t e d image 2 i n c h e s . The f i n a l a d j u s t m e n t t o t h e sample a l i g n m e n t was made w i t h t h e d e t e c t o r and r o t a t i n g p o l a r i z e r i n p l a c e . The sample o r i e n t a t i o n was changed s l i g h t l y so t h a t t h e r e l a t i v e 3 8 phase of the reference and s i g n a l was the same w i t h and without the sample i n p l a c e . T h i s adjustment was done to the CdS samples only since CdIn 2S / ! f i s not b i r e f r i n g e n t . 11- Measurement of the Faraday Angle The Faraday angle was detected as the change of the r e l a t i v e phase of the output of the PbS c e l l and the reference s i g n a l . The wavelength was sel e c t e d by manually a d j u s t i n g the prism o r i e n t a t i o n w h i l e m o n i t o r i n g the output of the PbS o e l l . By s t a r t i n g at e i t h e r end of the spectrum and counting the output maxima the wavelenth was i d e n t i f i e d . The vo l t a g e output of the L o c k i n (phase mode) was i n t e r n a l l y biased to near zero m i l l i v o l t s w i t h o n l y the r e s i d u a l magnetic f i e l d H 4 0 Gauss) a c t i n g on the sample. The a p p l i e d f i e l d was then r a i s e d i n s i x steps o f ~ 1 kG to ~ 6„kG w h i l e n o t i n g the f i e l d s e t t i n g s and the output v o l t a g e . The a p p l i e d f i e l d was then reduced to zero and the procedure repeated. The same sequence was fol l o w e d f o r each of the other f i v e wavelengths. Each set of„six data p o i n t s was l e a s t squares f i t t e d to o b t a i n the change i n output v o l t a g e per u n i t magnetic f i e l d change. The s p e c i f i c r o t a t i o n was f i n a l l y obtained by comparing the l e a s t square r e s u l t s . 1 2 - M i r r o r C o r r e c t i o n The e x i t m i r r o r of the c r y o s t a t a f f e c t s the plane of 39 p o l a r i z a t i o n of t h e r e f l e c t e d beam. T h i s was a c c o u n t e d f o r by c a l i b r a t i n g t h e phase change a s measured a g a i n s t a c t u a l r o t a t i o n s i n the p l a n e o f p o l a r i z a t i o n . A d i s c o f p o l a r o i d was mounted i n the probe shown i n f i g u r e I I I - 8 and lowered i n t o t h e dewar t o i n t e r s e c t t h e o p t i c a x i s . The p o l a r o i d c o u l d be r o t a t e d 20 d e g r e e s about t h e o p t i c a x i s by means o f the h a n d l e ( i t e m A i n f i g u r e I I I - 8 ) . One complete r e v o l u t i o n o f t h e h a n d l e produced a 2.54 degree r o t a t i o n o f t h e p o l a r o i d . The c a l i b r a t i o n was made by comparing the PAR #121 o u t p u t v o l t a g e ( L o c k i n s e t t o measure phase d i f f e r e n c e s ) a s the p o l a r o i d was r o t a t e d and as t h e r e f e r e n c e was a d j u s t e d . The c o m p a r i s o n p r o c e d u r e was a n a l a g o u s t o t h e p r o c e d u r e f o r o b t a i n i n g t h e F a r a d a y a n g l e . F i g u r e III-8 C a l i b r a t i n g a p p a r a t u s 41 B -Apparatus f o r E l e c t r i c a l Measurements The e l e c t r i c a l measurements were made u s i n g the same dewar, L o c k i n a m p l i f i e r , and magnet as used f o r t h e o p t i c a l s t u d i e s . They a r e d i s c u s s e d i n p a r t A o f t h i s c h a p t e r . The PAR #121 L o c k i n a m p l i f i e r was used as a t u n a b l e f r e q u e n c y v o l t m e t e r . 1- E l e c t r i c a l Probe The samples were mounted on a b a k e l i t e probe a s shown i n f i g u r e 111=9 w h i c h was d e s i g n e d t o g i v e minimum c a p a c i t a n c e between t h e sample l e a d s . T h i s c a p a o i t a n c e was k e p t t o ~ 9 pF by u s i n g #37 copper w i r e (0.005 i n c h d i a m e t e r ) and m a i n t a i n i n g a l e a d s e p a r a t i o n o f 1 i n c h . An e i g h t i n c h p i e c e o f coax j o i n e d t h e l e a d s t o t h e L o c k i n p r o v i d i n g a t o t a l o f 25 pF c a p a c i t a n c e between the r e s i s t i v i t y l e a d s . A t 10^ Hz t h i s i s e q u i v a l e n t t o a 70 k i t . r e s i s t a n c e i n p a r a l l e l w i t h t h e sample and s e r v e s t o l i m i t t h e sample r e s i s t a n c e t h a t can be measured w i t h o u t a p p r e c i a b l e (10%) c o r r e c t i o n s t o <16 k,a. 2- Temperature Measurements Two thermometers were used t o c o v e r t h e t e m p e r a t u r e range f r o m 300° K t o 2°K. A copper r e s i s t a n c e thermometer, wound w i t h #37 oopper w i r e from Canadian Wire and C a b l e Co., was c a l i b r a t e d a t l i q u i d n i t r o g e n t e m p e r a t u r e u s i n g the d a t a o f Dauphinee and Preston-Thomas 3 5(1954) • The r e s u l t s were r e p r o d u c i b l e and p e r m i t t e d the d e t e r m i n a t i o n o f the t e m p e r a t u r e t o w i t h i n 42 O o ^ S h i e l d e x t e n t T e r m i n a l s . 'Sapphire sample base .Sample ^Copper h e a t c o n d u c t i o n s t r i p * Carbon r e s i s t o r , * Copper c o i l (thermometer)—~_ ( j I Germanium thermometer 1 O D F i g u r e I I I - 9 Sample H o l d e r f o r e l e c t r i c a l s t u d i e s : The d i m e n s i o n s o f t h e b a k e l i t e were 31 i n c h e s l o n g , 1 i n c h wide and 1/8 i n c h t h i c k . 43 ±2°K o v e r the range 300 °K t o 65°K. The r e s i s t a n c e was measured w i t h a Gray I n s t r u m e n t s Co. Wheat stone B r i d g e model #33107. The 77°K r e s i s t a n c e was 17.88 ohms. A S o l i t r o n #1753 germanium r e s i s t a n c e thermometer was used below 45°K. The recommended measurement c i r c u i t shown i n f i g u r e 111-10 was assembled u s i n g 1% m e t a l f i l m r e s i s t o r s and a H e w l e t t P a c k a r d model 419A D.C. n u l l v o l t m e t e r . Temperatures a c c u r a t e t o the l a r g e r o f ±1% o r 0.1°K were o b t a i n e d w i t h t h i s f o u r p o i n t measurement. A p p r o x i m a t e l y one h a l f pound of m o l e c u l a r s i e v e was packed around the sample chamber t o slow t h e warm up ti m e from l i q u i d h e l i u m t e m p e r a t u r e s . W i t h t h i s amount i t was n e c e s s a r y t o heat t h e chamber t o a v o i d e x c e s s i v e l y l o n g s r u n s . 3-Sample P r e p a r a t i o n F o r t h e H a l l e f f e c t measurements, dumbbell shaped samples were s p a r k c u t fr o m s l i c e s o f the F a r a d a y e f f e c t samples a s shown i n f i g u r e I I I - l l . I t was n e c e s s a r y t o s p a r k c u t them s i n c e b o t h CdS and Cdln^S, a r e too b r i t t l e ^ 4 f o r u l t r a s o n i c impact c u t t i n g . The low t e m p e r a t u r e measurements on a l l CdS samples e x c e p t #2 and C2-B were made w i t h such samples. I n o r d e r t o keep t h e r e s i s t a n c e a t low t e m p e r a t u r e s t o <16 kXL. f o r samples #2 and C2-B i t was n e c e s s a r y t o use a d i f f e r e n t geometry and match t h e low t e m p e r a t u r e r e s u l t s t o t h e h i g h e r t e m p e r a t u r e r e s u l t s on the dumbbell shaped samples. , 130 k.o -\— —vyvv -30 k X l trimmer C u r r e n t l i m i t i n g r e s i s t a n c e 1.5 v o l t s N-x-10 k-A 1 k n - \ — l oo r v A — Germanium r e s i s t a n c e <t5 51 w i d t h 6 l e n g t h l e n g t h = 4.59 mm w i d t h ~ 2.53 mm t y p i c a l t h i c k n e s s = 2.50 mm o v e r a l l l e n g t h - 10.70 mm F i g u r e I I I - l l H a l l Sample: The shape and d i m e n s i o n s a r e shown. 46 4- E l e c t r i c a l C o n t a c t s The e n t i r e sample a r e a on w h i c h a c o n t a c t was not t o be made was p a i n t e d w i t h p o l y s t y r e n e d i s s o l v e d i n t o l u e n e . A f t e r t h i s hardened (~15 m i n u t e s under h e a t ) t h e samples were etched i n a nea r b o i l i n g s o l u t i o n o f chromic a c i d (see Appendix-A) f o r 10 m i n u t e s and q u i c k l y t r a n s f e r r e d t o a warmed i n d i u m p l a t i n g s o l u t i o n ( A p p e n d i x - B ) . Two p r e s s u r e c o n t a c t s were made t o the etched samples and a c u r r e n t o f ~3 mA was passed f o r 2-3 h o u r s . P u r e i n d i u m (99.999%) was used a s t h e anode. The c o n t a c t s were r o t a t e d p e r i o d i c a l l y so t h a t a l l t h e e t c h e d s u r f a c e was i n d i u m p l a t e d . The p o l y s t y r e n e s e r v e d t o l i m i t t h e c o n t a c t a r e a . P r e t i n n e d ( w i t h i n d i u m ) g o l d l e a d s were s o l d e r e d t o the samples u s i n g p u r e i n d i u m a s s o l d e r , a s soon a s t h e y were removed f r o m the p l a t i n g s o l u t i o n . The s o l d e r i n g i r o n t e m p e r a t u r e was k e p t j u s t above the i n d i u m m e l t i n g p o i n t o f 156°C. T h i s was found t o be v e r y i m p o r t a n t f o r good c o n t a c t s . Ohmic c o n t a c t s were made i n t h i s manner. 5- R e s i s t i v i t y Measurement P r o c e d u r e The p r o c e d u r e f o r d e t e r m i n i n g t h e A.C. and D.C. r e s i s t i v i t i e s was the same. The v o l t a g e V between l e a d s 3 and 4, ( f i g u r e 11-12) was measured w i t h a H e w l e t t P a c k a r d D.C. n u l l v o l t m e t e r f o r D.C. measurements and w i t h a PAR #121 L o c k i n a m p l i f i e r f o r A.C. measurements. The v o l t a g e V a c r o s s t h e sample was compared t o the 47 u O U -P <a a O rH d «H CO •P o © o rH - H © d W cd F i g u r e 111-12 S c h e m a t i c o f H a l l e l e c t r o n i c s : F o r D.C. m e a s u r e m e n t s t h e a u d i o o s c i l l a t o r was r e p l a c e d w i t h a b a t t e r y a n d a c u r r e n t l i m i t i n g r e s i s t o r . 48 v o l t a g e Vs a c r o s s the 1% m e t a l f i l m r e s i s t o r R s and the r e s i s t a n c e R between the sample l e a d s d e t e r m i n e d u s i n g R- R s x V/V s ( I I I . 1 ) T h i s was r e p e a t e d 2-3 t i m e s a t d i f f e r e n t v o l t a g e l e v e l s , up t o ~5>0 mV and the average r e s i s t a n c e c a l c u l a t e d . A t y p i c a l v a l u e was 0.5 ohms a t room t e m p e r a t u r e . The sample d i m e n s i o n s were measured on a t r a v e l l i n g m i c r o s c o p e w i t h r e s o l u t i o n ^1 m i c r o n and t h e r e s i s t i v i t y c a l c u l a t e d u s i n g /•:R"'A/1 ( I I I . 2 ) where A i s the c r o s s s e c t i o n a l a r e a and 1 i s t h e l e n g t h between t h e l e a d s . The l a t t e r was chosen as i n d i c a t e d i n f i g u r e I I I - l l and was the major source o f e r r o r i n t h e r e s i s t i v i t y d e t e r m i n a t i o n s . 6 - H a l l E f f e c t The H a l l e f f e c t a p p a r a t u s i s shown s c h e m a t i c a l l y i n f i g u r e 111-12. The r e s i s t a n c e p o t was a d j u s t e d t o g i v e a z e r o t r a n s v e r s e v o l t a g e w i t h minimum magnetic f i e l d . The t r a n s v e r s e v o l t a g e was r e c o r d e d a s a f u n c t i o n o f t h e m a g n e t i c f i e l d s t r e n g t h a s t h e l a t t e r was i n c r e a s e d t o 12-13 kG. Fo u r o r f i v e d e t e r m i n a t i o n s were made and the av e r a g e H a l l v o l t a g e p e r k i l o g a u s s was o b t a i n e d . The c u r r e n t I was d e t e r m i n e d by m e a s u r i n g t h e v o l t a g e a c r o s s 49 a l/o p r e c i s i o n r e s i s t o r i n s e r i e s w i t h the H a l l samples. The number o f c a r r i e r s N p e r c u b i c c e n t i m e t e r was o b t a i n e d f rom t h e H a l l v o l t a g e p e r k i l o g a u s s Vg and t h e c u r r e n t I u s i n g the r e l a t i o n N - 2.75 x l O 1 ^ X(amps) (TTT ^ V B ( v o l t s / k G ) ( I I I , 3 J where t h e t h i c k n e s s o f the samples was 2.50 mm and t h e s c a t t e r i n g f a c t o r was t a k e n t o be 1.10 . 50 C-Apparatus f o r I n d e x o f R e f r a c t i o n Measurements The a p p a r a t u s c o n s i s t e d o f a t u n g s t e n source (3500°K) a chopper, Bausch and Lomb monochromator ( c a t a l o g u e #33-86-25) and a C d l n S p r i s m mounted a t the c e n t e r o f a c a l i b r a t e d 2 4 t u r n t a b l e . A f i x e d t e l e s c o p e f o c u s e d on the e x i t s l i t s o f th e monochromator p r o v i d e a p a r a l l e l beam o f q u a s i -monochromatic l i g h t i n c i d e n t on a p r i s m and a PbS c e l l d e t e c t o r (same d e t e c t o r used f o r the F a r a d a y s t u d y ) . The c e l l was mounted on a r o t a t a b l e arm and d e t e c t e d t h e r e f r a c t e d beam. The o u t p u t o f the PbS c e l l was sent t o the i n p u t o f t h e PAR #121 L o c k i n a m p l i f i e r o p e r a t i n g as a tuned f r e q u e n c y v o l t m e t e r . The monochromator s l i t s were s e t t o g i v e a r e s o l v i n g power o f -^ x * 100 (AX-75A) and t h e a n g u l a r w i d t h o f the r e f r a c t e d beam a t t h i s s e t t i n g was 44 m i n u t e s of a r c . The a n g l e s o f minimum d e v i a t i o n v a r i e d f rom 49° 5l' a t X-1.60yW t o 54 46 a t A = 0.60/*. The apex a n g l e o f the p r i s m was 28 31 • • I n o r d e r t o p r e v e n t i n t e r f e r e n c e from o v e r l a p p i n g o r d e r s from the monochromator a f i l t e r was used t o e l i m i n a t e t h e r a d i a t i o n w i t h A < 0.9 w h i l e t h e a n g l e s o f minimum d e v i a t i o n were d e t e r m i n e d f o r 0.90^ " A < 1.60^ /U , The f i l t e r was removed f o r t h e 0.70$ A $ 1.00/* measurements. The r e s u l t s f o r t h e r e g i o n of o v e r l a p were t h e same t o w i t h i n 5 m i n u t e s o f a r c . A. v i s u a l check was a l s o made i n the o p t i c a l r e g i o n and the r e s u l t s extended t o 0.60/* by o b s e r v i n g t h e v i s i b l e r a d i a t i o n i n t h e second o r d e r . 51 The p r i s m apex a n g l e was det e r m i n e d by r e f l e c t i n g a l a s e r beam from the p r i s m f a c e s and m e a s u r i n g t h e r o t a t i o n o f the p r i s m n e c e s s a r y t o g i v e t h e same r e f l e c t i o n from b o t h f a c e s o f t h e p r i s m w h i c h d e f i n e the p r i s m apex a n g l e . 52 CHAPTER IV  RESULTS AND ANALYSIS The r e s u l t s f o r CdS and C d I n 2 S ^ a r e p r e s e n t e d s e p a r a t e l y i n s e c t i o n s A and B r e s p e c t i v e l y . A- CdS 1- F a r a d a y E f f e c t R e s u l t s S i n c e i t has been shown i n c h a p t e r I I t h a t the m a j o r c o n t r i b u t i n g e f f e c t s t o the F a r a d a y r o t a t i o n i n t h e p r e s e n t i n v e s t i g a t i o n e x h i b i t e i t h e r a A x or aX w a v e l e n g t h dependence, t h e d a t a were l e a s t s q u a r e s f i t t o -JL- - v \ Z + S A _ 2 , ( i v . i ) I u The c o e f f i c i e n t s \r and § a r e t a b u l a t e d i n T a b l e IV-1 and IV-2. The r o t a t i o n o bserved f o r t h e sample w i t h l o w e s t (#2) and h i g h e s t (03) f r e e c a r r i e r c o n c e n t r a t i o n i s shown i n f i g u r e I V - 1 . 2- D i s c u s s i o n of 300°K R e s u l t s The terms i m p o r t a n t t o t h e F a r a d a y r o t a t i o n a t room t e m p e r a t u r e ar6 i ) f r e e c a r r i e r c o n t r i b u t i o n and i i ) i n t e r b a n d c o n t r i b u t i o n s . S p i n e f f e c t s a r e n e g l i g i b l e a t room t e m p e r a t u r e s i n c e k T » g ^ H . A t room t e m p e r a t u r e t h e donor e l e c t r o n s n o t i n a c c e p t o r s t a t e s w i l l be t h e r m a l l y e x c i t e d i n t o the c o n d u c t i o n band where t h e y w i l l c o n t r i b u t e t o t h e f r e e 53 T a b l e IV- 1 F a r a d a y R o t a t i o n a t 300°K Sample •u-(xl0 8) deg rt t ( x l O )deg-cm kG-cm3 kG #2 0.045± .013 2 .00 ± .04 C2-B 0.093 * . 0 1 1 1.87 ± .03 #3 0.267 ±.010 2 . 0 8^ . 0 3 C2-A 0.400 +.013 1.92 ±-.03 C l 0.890 ±.014 2.02 ±.03 C3 1.35 ±.04 2.09 ±.06 T a b l e I V - 2 F a r a d a y R o t a t i o n a t 2°K Sample ^ - ( x l O 8 ) deg t ( x l O )deg-cm kG-cm3 kG #2 0.090 * .017 2 . 0 5 ^ . 0 5 C2-B 0.131 ±.010 1.98 ± .02 #3 0.323 ±-012 2.00 i .03 C2-A 0.445 ± . 0 1 4 1.94 ± .04 C l 0 . 9 5 7 ± .019 2 . 0 3 * .06 C3 1.62 ± .04 1 . 9 1 ± .07 5k I.I 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 1 . 7 W A V E L E N G T H (/A) Q AYMf carl TP a v»o A o tr -r»o. +- o 4- A ^ 55 c a r r i e r r o t a t i o n . Those i n a c c e p t o r l e v e l s w i l l c o n t r i b u t e t o t h e i n t e r b a n d t e r m . F o r 1.01^< \ < 1.70^ t h e F a r a d a y r o t a t i o n i s composed o f a f r e e c a r r i e r term (<: Xz) and an i n t e r b a n d term X " 2 ) . a- X Component The c o e f f i c i e n t o f the X * term o f (I V . 1 ) r e p r e s e n t s t h e c o n t r i b u t i o n of f r e e c a r r i e r s . The r e s u l t s o f s i x n-ty p e CdS samples were o b t a i n e d and a n a l y s e d i n t h i s manner and the c o e f f i c i e n t i n t e r p r e t e d w i t h -\J ~ — 7-TT IV.2 2f(nc4m*2 u s i n g the f r e e c a r r i e r c o n c e n t r a t i o n N d e t e r m i n e d from H a l l e f f e c t measurements. The c o n d u c t i o n band e f f e c t i v e mass v a l u e s a r e summarized i n Table IV-3. T a b l e IV-3: C o n d u c t i o n Band E f f e c t i v e Mass Sample F r e e C a r r i e r mVm0 C o n c e n t r a t i o n #2 3.0x10^ /cc 0.196±.040 C2-B 5 . 0 x l 0 n /cc 0.181*.016 #3 1 . 4 x l 0 L J / c c 0.190 ±.011 C2-A 2 . 1 x 1 0 1 6 / c c 0.181± .010 CI 6 . 0 x l 0 1 8 /cc 0.210±-.013 C3 7 . 6 x 1 0 l b /oc 0.202 ^ .015 The a v e r a g e v a l u e o f rnVm^- 0 . 1 9 3 i . 0 2 3 i s i n good agreement w i t h t h a t c a l c u l a t e d from t h e e f f e c t i v e mass a p p r o x i m a t i o n o f m/m^ - 0.19. F o l l o w i n g t h e method o f Cardona (1961), a k.p 56 p e r t u r b a t i o n c a l c u l a t i o n f o r t h e c o n d u c t i o n b a n d y i e l d s a n e f f e c t i v e m a s s a s a f u n c t i o n o f t h e c o n c e n t r a t i o n g i v e n b y • r , -r f x ^ H V I _ _ J m \ O U T j x ^ k x - n M * I - (IV.3) w h e r e f i s t h e F e r m i f u n c t i o n a n d ' ^ - E ? / k T i s t h e d e g e n e r a c y f a c t o r . T h i s p r e d i c t s a 2% i n c r e a s e i n m*" f o r t h e c o n c e n t r a t i o n r a n g e 3.0xl017cm"'J t o 7,6xl0 1 8cm~3. T h e r e s u l t s i n T a b l e I V - 3 h a v e a 7% s c a t t e r t h o u g h i n c r e a s i n g f o r t h e h i g h e r d o p e d s a m p l e s . However, s i n c e t h e a s s i g n e d e r r o r i s 7%, no f i r m c o n c l u s i o n c a n be d r a w n a b o u t t h i s p r e d i c t i o n . The h i g e s t d o p e d s a m p l e h a s a d e g e n e r a c y f a c t o r v ^ - i 3 i m p l y i n g t h a t t h e m m e a s u r e d i s a n a v e r a g e o f t h e m a s s e s up t o E<L25 meV a b o v e t h e b o t t o m o f t h e c o n d u c t i o n b a n d b u t s t r o n g l y w e i g h t e d b y ^ t o g i v e t h e m a s s c h a r a c t e r i s t i c o f e l e c t r o n s w i t h e n e r g y E-^Ej, . S i n c e t h e e r r o r s o f m* o v e r l a p f o r t h e v a r i o u s c o n c e n t r a t i o n s and t h u s y i e l d a c o n s t a n t m*, t h e d a t a w i l l be a n a l y s e d b y 1 1 |T) *• - ~TT ^ - c o n s t a n t (IV.4) T h u s t o w i t h i n e x p e r i m e n t a l e r r o r t h e e f f e c t i v e mass i s c o n s t a n t up t o a F e r m i l e v e l of-73 meV a b o v e t h e b o t t o m o f t h e c o n d u c t i o n b a n d . O t h e r m e a s u r e m e n t s o f m7m chave b e e n made f o r f r e e c a r r i e r s i n CdS b u t t h e o n l y o t h e r m e a s u r e m e n t s a s a f u n c t i o n 57 o f t h e c o n c e n t r a t i o n i s t h e r e f l e c t i v i t y s t u d y by P i p e r and M a r p l e (1961). T h e i r r e s u l t s show a c o n s i d e r a b l e s c a t t e r (m%i0-= 0.14* .05 t o 0.235* .02) o r a 35$ v a r i a t i o n and t h e i r a v e r a g e v a l u e o f m*/m0 = 0.22 ± .01 i s l a r g e r t h a n l a t e r i n v e s t i g a t i o n s . T h e i r s t u d y was made a t n e a r i n f r a r e d f r e q u e n c i e s and y i e l d s a bare mass as does the F a r a d a y e f f e c t used i n the p r e s e n t e x p e r i m e n t s . 10 S i n c e t h e n a number o f d i f f e r e n t t e c h n i q u e s have been a p p l i e d t o measure m on s e l e c t e d samples but t h e r e s u l t s a l s o v a r y c o n s i d e r a b l y . Most o f t h e s e measurements a r e s e n s i t i v e t o the L0 p h o n o n - e l e c t r o n c o u p l i n g and hence y i e l d a m o d i f i e d mass c a l l e d a p o l a r o n mass. b- A ^  Component The c o e f f i c i e n t § o f the AT 2 component o f ( I V . 1) -8 is(2.00± . 0 8 ) x l 0 deg-cm/kG a t room t e m p e r a t u r e . T h i s 35 v a l u e may be compared w i t h t h e r e s u l t s o f S b i n a e t a l . (1965) a s shown i n f i g u r e I V - 2 . The a n a l y s i s of S b i n a 35" e t a l . i s based on t h e h i g h f r e q u e n c y f i t t o the i n t e r b a n d r o t a t i o n . They f i n d a d i s c r e p a n c y between the e x t e n s i o n o f t h e h i g h f r e q u e n c y f i t and t h e i r e x p e r i m e n t a l r e s u l t s a t l o w e r f r e q u e n c i e s where the i n t e r b a n d e f f e c t v a r i e s l e s s r a p i d l y . The p r e s e n t v a l u e o f J can be c o n s i d e r e d an e x t e n s i o n o f t h e i r work and i s a l s o shown i n f i g u r e I V - 2 . I t i s e v i d e n t t h a t t h e e x p e r i m e n t a l v a l u e s a r e i n agreement and b o t h ^ 3 0 ^ l a r g e r than the t h e o r e t i c a l f i t . The p r e s e n t work i n d i c a t e s t h i s d i s c r e p a n c y i s not due t o f r e e c a r r i e r s PHOTON S N 3 R G Y (eV) 35 F i g u r e IV-2 Comparison w i t h the r e s u l t s o f E b i n a e t a l . The r e s u l t s above 1.7eV i n d i c a t e d a s ( ) a r e a s t a k e n f r o m t h e i r f i g u r e 2. The r e s u l t s i n d i c a t e d by (•*-*-*-) a r e t h o s e o b t a i n e d i n t h e p r e s e n t s t u d y . The l i n e ( ) i s ) 6 the f i t of t h e p r e d i c t i o n o f Boswarva, Howard and L i d i a r d a s i n d i c a t e d by E b i n a et a l . i n t h e i r f i g u r e l+. 5 9 s i n c e t h a t c o n t r i b u t i o n was s u b t r a c t e d . CdS samples from the same s o u r c e a s t h a t used h e r e c were mass s p e c t r o g r a p h i c a l l y a n a l y s e d and the r e s u l t s showed l a r g e amounts o f a c c e p t o r and deep donor elements ( e . g . L i , Cr, F e ) . The e x t r a r o t a t i o n may be due t o e l e c t r o n s o c c u p i n g a c c e p t o r and deep donor s t a t e s . 3 - D i s c u s s i o n o f t h e 2°K R e s u l t s The r o t a t i o n o b s e r v e d a t 2°K was l a r g e r t h a n t h a t o b s e r v e d a t room t e m p e r a t u r e . The l e a s t s q u a r e s a n a l y s i s o f t h i s r o t a t i o n t o (IV„'l) c o n s i s t e n t l y gave a l a r g e r X ^ component, and a X" component w h i c h was a p p r o x i m a t e l y e q u a l t o the room t e m p e r a t u r e X component. The magnitude of t h e e x t r a r o t a t i o n observed a t 2°K i s g i v e n i n -table IV-4 Ta b l e IV-4 E x t r a R o t a t i o n a t 2 1 Sample Average of e(2°K)-9(300°K) B (2"PC] J 2 C2-B #3 C2-A CI C 3 •855 10% 4% 5 % 4% 8% a- X Component The c o e f f i c i e n t I of the X *" component of (IV„1} i s -8 (1,99- .08)xl0 deg-cm/kG w h i c h i s t h e same a s t h e room t e m p e r a t u r e v a l u e . A d e c r e a s e i n the i n t e r b a n d r o t a t i o n w i t h t e m p e r a t u r e from 300°K t o 100°K f o r CdS was obs e r v e d 60 3 5 by E b i n a e t a l . (1965) near t h e band gap f r e q u e n c i e s . S i m i l a r measurements by B a l k a n s k i , A m z a l l a g and Langer (1965) show a t e m p e r a t u r e i n s e n s i t i v e i n t e r b a n d r o t a t i o n down t o 77°K i n t h e f r e q u e n c y range used f o r the p r e s e n t s t u d y . T h i s term c o n t a i n s an e s t i m a t e d 3$ e x t r a r o t a t i o n a s s o c i a t e d w i t h the change i n t h e v e l o c i t y m a t r i x e l e m e n t s due t o s p i n - o r b i t i n t e r a c t i o n . .2 b- A Component -Samples C2-A, C l and 03 S i n c e the donor l e v e l s have merged- w i t h t h e c o n d u c t i o n band, t h e F a r a d a y d i s p e r s i o n due t o donor e l e c t r o n s was a s s o c i a t e d w i t h two e l e c t r i c d i p o l e a b s o r p t i o n p r o c e s s e s : i ) f r e e c a r r i e r a b s o r p t i o n and i i ) c o m b i n e d r e s o n a n c e a b s o r p t i o n . The f i r s t c o n t r i b u t i o n should i n c r e a s e by 1% from 300° K o t o 2 K s i n c e t h e t h r e e t e m p e r a t u r e dependent p a r a m e t e r s of e q u a t i o n ( I I . 1 6 ) N, n and m*" change i n t h e f o l l o w i n g manner: i ) a l l N c a r r i e r s r e m a i n i n the c o n d u c t i o n band i i ) n d e c r e a s e s by 2% f r o m 300° K t o 2°K and i i i ) nT i s the c o n d u c t i o n band e f f e c t i v e mass wh i c h v a r i e s by an e s t i m a t e d +0.5% from 300°K t o 2°K. The r e m a i n d e r o f t h e X r o t a t i o n s h o u l d be due t o combined r e s o n a n c e e f f e c t s . From the d a t a i t i s seen t h a t i t i s 4-8% o f t h e t o t a l r o t a t i o n o r -10% of. t h e f r e e c a r r i e r F a r a d a y r o t a t i o n . The e s t i m a t e based on t h e r e l a t i v e a b s o r p t i o n s t r e n g t h s i s 5%« (51 o~ A Component -Samples jf2 and C2-B The F a r a d a y d i s p e r s i o n owing t o donor e l e c t r o n s was a s s o c i a t e d w i t h t h r e e e l e c t r i c - d i p o l e a b s o r p t i o n p r o c e s s e s i ) f r e e c a r r i e r a b s o r p t i o n by e l e c t r o n s i n donor s t a t e s , i i ) e l e c t r i c - d i p o l e s p i n f l i p a b s o r p t i o n and i i i ) donor s t a t e t o c o n d u c t i o n band a b s o r p t i o n . The f i r s t i s t h e c o n t r i b u t i o n o f i n t e r e s t w h i l e t h e second i s e s t i m a t e d t o be o n l y ~ 5% of i t . The t h i r d c o n t r i b u t i o n t o t h e F a r a d a y d i s p e r s i o n was a p p r o x i m a t e l y e q u a l t o the c o n d u c t i o n band f r e e c a r r i e r F a r a d a y d i s p e r s i o n s i n c e t h e a p p r o p r i a t e e x p r e s s i o n f o r the r o t a t i o n i s ( I I . 1 6 ) and a l l N c a r r i e r s ' f r e e z e ' i n t o s h a l l o w donor s t a t e s and t h e o t h e r p a r a m e t e r s change i n t h e manner o u t l i n e d above. T h i s i m p l i e s t h a t t h e F a r a d a y d i s p e r s i o n a s s o c i a t e d w i t h donor to c o n d u c t i o n band a b s o r p t i o n s h o u l d be a p p r o x i m a t e l y Ifo g r e a t e r t h a n t h e 300°K f r e e c a r r i e r F a r a d a y d i s p e r s i o n w h i c h i s l e s s t h a n t h e u n c e r t a i n t y i n t h e l e a s t s quares a n a l y s i s . The 2°K d a t a was a n a l y s e d by s u b t r a c t i n g t h e 300°K f r e e c a r r i e r F a r a d a y r o t a t i o n and i n t e r p r e t i n g the r e s t as donor f r e e c a r r i e r F a r a d a y r o t a t i o n . The e s t i m a t e d -5% r o t a t i o n due t o combined resonanoe e f f e c t s c o n t r i b u t e d t o t h e e r r o r . The r e s u l t was t h a t t h e r a t i o N/m*2 a s s o c i a t e d w i t h t h e donor f r e e c a r r i e r F a r a d a y e f f e c t was the same t o w i t h i n a f a c t o r o f 2 a s t h e v a l u e e x p e c t e d i f a l l N c a r r i e r s were 62 i n the conduction band. Thus, the high frequency behaviour of e l e c t r o n s i n donor s t a t e s i s i n d i s t i n g u i s h a b l e from th a t of e l e c t r o n s i n the conduction bando 4-Results of E l e c t r i c a l Experiments Room temperature r e s i s t i v i t y and f r e e c a r r i e r c o n c e n t r a t i o n s are given i n t a b l e I V - 5 . The r e s i s t i v i t y temperature dependence i s shown i n f i g u r e I V - 3 . H a l l e f f e c t measurements at temperatures down to 45°K shown i n f i g u r e IV-4 f o r sample §2 i n d i c a t e the t r a n s i t i o n from the number of f r e e c a r r i e r s v a r y i n g as exp(-E D/2kT) t o exp(-E D/kT) a t l i q u i d n i t r o g e n temperatures. A compensation r a t i o N^/Njf 1/3 i s i n d i c a t e d from t h i s change i n the H a l l e f f e c t temperature dependence. The H a l l a c t i v a t i o n energy f o r sample #2, E^-12.0 meV i s i n good agreement w i t h the r e s i s t i v i t y a c t i v a t i o n energy of 11.8 meV f o r 40 < T>s 15°K. Thus, the r e s i s t i v i t y a c t i v a t i o n energy E-^  of t a b l e IV-6 may be i n t e r p r e t e d as the bin d i n g energy of the donor e l e c t r o n s r e l a t i v e to the conduction 38 band. L i and Huang (1970) suggested that the small a c t i v a t i o n energy E 2 of t a b l e IV-6 f o r T< 4.2°K was a s s o c i a t e d w i t h hopping conduction. Their c o n c l u s i o n appears to be based on the absence of any other s a t i s f a c t o r y e x p l a n a t i o n r a t h e r than any f i r m i d e n t i f i c a t i o n of hopping. F i g u r e IV-3 Temperature dependent r e s i s t i v i t y d a t a . 64 -^-X | 0 2 ( ° K ) " ' F i g u r e IV - 4 Temperature dependent H a l l c o n s t a n t . Data i s t h a t t a k e n on CdS sample #2. • 65 The r a d i o f r e q u e n c y r e s i s t i v i t y . m e a s u r e m e n t s a r e shown i n f i g u r e IV-5, f o r sample #2. A l l suoh measurements e x h i b i t e d a f r e q u e n c y i n d e p e n d e n t r e s i s t i v i t y up t o 1()5 Hz i m p l y i n g t h a t the maximum h o p p i n g time was l e s s t h a n 10"5 seconds. The l a r g e r e s i s t i v i t y of samples §2 and C2-A a t h e l i u m t e m p e r a t u r e s i m p l i e s t h a t most o f t h e f r e e c a r r i e r s d e t e r m i n e d a t 300°K a r e ' f r o z e n ' i n t o l o c a l i z e d s t a t e s . A t low f r e q u e n c i e s t h e number o f f r e e c a r r i e r s a t 2°K i s i o a t most 10 _ i f t i m e s the number a t 300 K. 66 10' I 0 2 I 0 3 ! 0 4 I 0 5 CJ ( H z ) F i g u r e IV -5 F r e q u e n c y dependent r e s i s t a n c e " , f o r sample #2 a t 4.2°K. The D.C. r e s i s t a n c e was 2.6 lea.. 67 T a b l e IV-5 R e s i s t i v i t y and F r e e C a r r i e r C o n c e n t r a t i o n Sample R e s i s t i v i t y (.a-cm) F r e e C a r r i e r s (300"K) (300°K) #2 0.080 3.0xl0'7_ cm"3 C2-B 0.043 5.0x10* cm" 3 #3 0.020 1.4xl0Jfa cm" 3 C2-A 0.014 - 2.1x10'!: cm" 5 CI 0.007 6.0x10 !* cm - 3 C3 0.002 7.6xl0,b cm'3 T a b l e I V - 6 R e s i s t i v i t y A c t i v a t i o n E n e r g i e s Sample Temperature Range A c t i v a t i o n Energy #2 40>T>12 E, -11.8 meV 4-2 T Ej_r 0.7 meV C2-B 40>T>12 E i i 6 . 9 meV 4.2 >T E z z 1.0 meV #3 40>T>12 E, = 5.3 meV 4.2 >T E z - 0.4 meV 68 B - C d I n 2 S ^ 1-Faraday E f f e c t R e s u l t s The F a r a d a y e f f e c t was observed f o r t h r e e samples o f C d l i ^ S ^ a t room, t e m p e r a t u r e and t h e r e s u l t s were a n a l y s e d u s i n g e q u a t i o n TV.1 JL - -vr Xz +- f (iv.i) The c o e f f i c i e n t s AT and § a r e t a b u l a t e d i n t a b l e IV-7. Table IV-7 F a r a d a y R o t a t i o n a t 300°K Sample ~vr(xl0 8) deg i (xl0~ 8)deg-cm kG-cm.3 kG 1 2 3 0.22± i02 0.13 ±..01 0.07 ± .02 1.58 ±.07 1.56*.03 1.55*-.05 2-Index o f R e f r a c t i o n The i n d e x o f r e f r a c t i o n of Cdln^S. d 4 (sample 3) i s shown i n f i g u r e IV-6 . I t v a r i e s s l o w l y from 2.70 a t 1.60/A-t o 2.57 a t 0 .60yU . The l o n g w a v e l e n g t h i n d e x of r e f r a c t i o n i s a p p r o x i m a t e l y 2.55 w h i c h i s c l o s e t o t h e v a l u e of 2.43 p r e d i c t e d from t h e e m p i r i c a l Moss r e l a t i o n . n ^ E g - c o n s t a n t = 77 eV (IV.5) where n i s the i n d e x o f r e f r a c t i o n and E_. i s t h e band gap 4o s energy. The v a l u e o f E~= 2.2 eV was used. INDEX OF R E F R A C T I O N ro ro ro % • • F i g u r e IV-6 R e f r a c t i v e i n d e x of C d I n 2 S 70 3 - E l e c t r i c a l R e s u l t s The H a l l e f f e c t and r e s i s t i v i t y were measured f o r samples 1 and 2. The r e s u l t s a r e g i v e n i n t a b l e IV-8 and a r e i n agreement v / i t h p r e v i o u s l y r e p o r t e d e l e c t r i c a l measurements of Sudo e t a l . (1970) . Ta b l e IV-8 C d l n ^ E l e c t r i c a l R e s u l t s Sample Nfcm"- 5) / ( i i - c m ) M o b i l i t y 1 1.1x10' S 0.078 80 2 6 . 9 x l 0 n 0.090 110 .2 cm v o l t - s e c 4 - A n a l y s i s o f R e s u l t s component o f the F a r a d a y r o t a t i o n w i t h ( I V . 2) 9 3 N x/ = > A „i (IV.2) the c o n d u c t i o n band e l e c t r o n i c e f f e c t i v e mass o f C d l n ^ ^ a t room t e m p e r a t u r e was d e t e r m i n e d t o be (0.17 ^  .02)m o f o r b oth samples 1 and 2. U s i n g t h e e f f e c t i v e mass and •' the average m o b i l i t y o f 95 c m ^ / v o l t - s e c t h e e l e c t r o n r e l a x a t i o n t i m e o f 9.3x±&^5 seconds was d e t e r m i n e d . From t h e F a r a d a y d a t a on sample 3 and assuming a p a r a b o l i c c o n d u c t i o n band, a f r e e c a r r i e r c o n c e n t r a t i o n o f ~3.3x10^-7 cm"3 can be i n f e r r e d . The e f f e c t i v e mass o b t a i n e d here can be' compared w i t h t h e e f f e c t i v e mass deduced from Seebeck c o e f f i c i e n t d a t a . The 71 Seebeck e f f e c t i v e mass i s g i v e n by i S d ^ 2 ) 1 / ^ ' ' 3 (iv.6). where S i s t h e number o f minima i n t h e c o n d u c t i o n band. U s i n g (IV.6 ) and (11-10) w h i c h i s t h e a p p r o p r i a t e e x p r e s s i o n f o r the Far a d a y e f f e c t i v e mass y i e l d s a n e g a t i v e m^ . f o r e i t h e r 1-3 or 6 as suggested by Endo e t a l . 3 (1970). A n e g a t i v e e f f e c t i v e mass i s u n r e a l i s t i c f o r t h e p r e s e n t r e s u l t s . A l s o , s u b s t i t u t i n g the c o m b i n a t i o n s o f m^ and m^ quoted by Endo e t a l . i n t o (11.10) g i v e s 0.06srift0.09 where m* i s the Fa r a d a y e f f e c t i v e mass. These v a l u e s a r e f a r from t h e measured v a l u e of mymrfO . 1 7 * . 0 2 . The f a c t t h a t the F a r a d a y e f f e c t i v e mass i s m a r k e d l y d i f f e r e n t f r om t h a t p r e d i c t e d by t h e a n i s o t r o p i c masses quoted by Endo e t a l . (1970) i n d i c a t e s t h a t one o r both o f t h e i r a s s u m p t i o n s namely: a) t h e s c a t t e r i n g f a c t o r i s a p p r o x i m a t e l y i s o t r o p i c b) t h e c o n d u c t i o n band c o n s i s t s o f e i t h e r t h r e e o r s i x e q u i v a l e n t minima may be i n c o r r e c t . 72 CHAPTER V CONCLUSIONS A - G e n e r a l C o n c l u s i o n s When a donor i m p u r i t y i s added t o CdS i t s e l e c t r o n i c ground s t a t e l i e s ~30 meV below t h e c o n d u c t i o n band e n e r g y minimum. As more donors a r e added t h e s e donor s t a t e s broaden due t o e l e c t r o n i c i n t e r a c t i o n , f o r m i n g a r e l a t i v e l y narrow s e t o f energy l e v e l s . A t low f r e q u e n c i e s o f e x c i t a t i o n , e l e c t r o n s i n t h e s e l e v e l s r e s pond a s i f t h e y were l o c a l i z e d - - i . e . , t h e y c o n t r i b u t e n e g l i g i b l y t o t h e c o n d u c t i v i t y . A t microwave f r e q u e n c i e s (lo'° Hz) a f r a c t i o n o f t h e e l e c t r o n s behave a s i f t h e y were f r e e — i . e . , t h e y e x h i b i t a s i n g l e l i n e ESR spectrum c h a r a c t e r i s t i c o f c o n d u c t i o n e l e c t r o n s w i t h a g v a l u e o f 1.76. At n e a r i n f r a r e d f r e q u e n c i e s , the donor e l e c t r o n s behave the same a s f r e e c o n d u c t i o n band e l e c t r o n s i n CdS. The a p p a r e n t m o b i l i t y of the e l e c t r o n s a t h i g h f r e q u e n c i e s may be due t o e l e c t r o n s h o p p i n g from o c c u p i e d l o c a l i z e d s t a t e s t o v a c a n t s t a t e s . I n t h i s way t h e e l e c t r o n s c o u l d i n t e r a c t w i t h a l a r g e number o f n u c l e a r s p i n s r e s u l t i n g i n o n l y a s i n g l e ESR a b s o r p t i o n l i n e . A l t h o u g h t h e e f f e c t i v e mass co n c e p t as u s u a l l y a p p l i e d would have no meaning i n t h i s model, t h e f a c t t h a t t h e 73 r a t i o N/m*2 i s e q u a l t o the v a l u e observed, a t room t e m p e r a t u r e f o r N e l e c t r o n s i n the c o n d u c t i o n band s u g g e s t s t h a t a t s u f f i c i e n t l y h i g h f r e q u e n c i e s h o p p i n g c o n d u c t i o n i s i n d i s t i n g u i s h a b l e from band c o n d u c t i o n . B - S p e c i f i o C o n c l u s i o n s The F a r a d a y e f f e c t p r o v i d e s a u s e f u l t o o l f o r a band parameter s t u d y , but the e f f e c t must be measured o v e r an extended f r e q u e n c y range t o a c c u r a t e l y a s s e s s t h e v a r i o u s c o n t r i b u t i o n s . A l t h o u g h t h e f r e q u e n c y range was l i m i t e d f o r t h e p r e s e n t e x p e r i m e n t , some s p e c i f i c c o n c l u s i o n s a r e p o s s i b l e . a) The room t e m p e r a t u r e v a l u e o f the c o n d u c t i o n band e l e c t r o n i c e f f e c t i v e mass i s (0.193- .023)md f o r CdS. The main s o u r c e s of e r r o r a r e i ) the u n c e r t a i n t y i n the H a l l - s c a t t e r i n g f a c t o r t a k e n t o be 1.10±.10 and i i ) the e r r o r i n v o l v e d i n t h e l e a s t s q u a r e s a n a l y s i s o f the d a t a . The f r e e c a r r i e r F a r a d a y e f f e c t measurement of B a l k a n s k i and H o p f i e l d a t one c o n c e n t r a t i o n y i e l d e d (0.20±.01)mo but d i d n ' t i n c l u d e any u n c e r t a i n t y a s s o c i a t e d w i t h t h e s c a t t e r i n g f a c t o r . b) The c u r v a t u r e o f t h e 2 v s "ic r e l a t i o n s h i p of the c o n d u c t i o n band o f CdS i s c o n s t a n t t o w i t h i n 7$ t o a Fer m i l e v e l o f 73 meV above the c o n d u c t i o n band energy minimum. c) The o b s e r v e d i n t e r b a n d r o t a t i o n f o r CdS was l a r g e r t h a n e x p e c t e d i n t h e X r e g i o n but i n agreement w i t h p r e v i o u s work quoted i n t h e l i t e r a t u r e . The 74 p r e s e n t a n a l y s i s shows t h a t t h i s e x t r a r o t a t i o n i s n o t due t o f r e e c a r r i e r s . The t e m p e r a t u r e independence o f t h e component d i s p u t e s t h e s u g g e s t i o n of E b i n a e t a l . (1965) t h a t t h e l a r g e r i n t e r b a n d r o t a t i o n i s due t o phonon a s s i s t e d t r a n s i t i o n s . I t i s p r o b a b l y due t o t h e l a r g e number of a c c e p t o r s and deep donors shown t o be p r e s e n t i n CdS by mass s p e c t r o g r a p h s a n a l y s i s . d) The a d d i t i o n a l m a g n e t o - d i s p e r s i o n . o b s e r v e d a t o 2 K f o r CdS i s a consequence of the s o - c a l l e d combined resonance a b s o r p t i o n . T h i s e x p l a n a t i o n c o r r e c t l y p r e d i c t s t h e w a v e l e n g t h dependence and g i v e s a good e s t i m a t e o f i t s magnitude. e ) The room t e m p e r a t u r e v a l u e o f the c o n d u c t i o n band e l e c t r o n i c e f f e c t i v e mass i n C d I n 2 S i i f i s m7m0= 0.17 * .02 . The m ajor s o u r c e o f e r r o r i s the u n c e r t a i n t y i n t h e H a l l s c a t t e r i n g f a c t o r w h i c h was t a k e n t o be 1.10 * .10 -based on t h e t h e o r e t i c a l v a l u e of a c o u s t i c phonon s c a t t e r i n g * (1.18) and c o n s i d e r i n g t h a t t h e degeneracy f a c t o r was a p p r o x i m a t e l y u n i t y . f ) The i n d e x o f r e f r a c t i o n o f C d I n 2 S ^ from 0.6yu t o 1.6yu i s g i v e n i n f i g u r e ( I V - 6 ) . g) The proposed m o b i l i t y o f the e l e o t r o n s a t microwave f r e q u e n c i e s would e x p l a i n t h e l a c k o f h y p e r f i n e s t r u c t u r e i n t h e ESR a b s o r p t i o n spectrum o f s h a l l o w donors i n CdS. 75 APPENDIX SOLUTIONS A- Chromic A c i d E t c h The e t o h was made i n one l i t e r q u a n t i t i e s and remade when the c o l o u r changed f r o m r e d d i s h brown t o a l i g h t g r e e n . A s a t u r a t e d w a t e r s o l u t i o n o f p o t a s s i u m d i c h r o m a t e ( ^ a ^ r J . 0 1 ) was p r e p a r e d u s i n g a p p r o x i m a t e l y 7 grams K j C r ^ O ^ . One l i t e r o f c o n o e n t r a t e d s u l p h u r i o a c i d was t h e n s l o w l y added t o 35 cc s a t u r a t e d s o l u t i o n t h u s f o r m i n g t h e e t o h . A p i e c e o f cadmium s u l p h i d e examined under a m i c r o s c o p e a f t e r b e i n g i n t h e n e a r b o i l i n g e t c h f o r t e n m i n u t e s showed a f r e s h s u r f a c e w i t h t r i a n g u l a r e t c h p i t s . 76 •B-indium P l a t i n g S o l u t i o n ^ ' Five and one half grams of 99.999% pure indium wire were out Into small pieces to increase the surface area and dissolved in a boiling solution of 4 oc concentrated sulphuric acid diluted to 75 cc with d i s t i l l e d water. The volume was maintained at approximately 75 oo by adding d i s t i l l e d water periodically. The resulting solution was f i l t e r e d and diluted to 250 co with d i s t i l l e d water and f i n a l l y 2.5 grams of hydrated sodium sulphate (tOaSo. • vo UjD) was added. This indium plating solution : was reusable but some indium precipitated out as an indium salt in time and the solution had to be replaced. 77 REFERENCES 1 Lax B . , Z w e r d l i n g S . , ' P r o g r e s s i n S e m i c o n d u c t o r s ' 5,221 (I960) 5_, 221 (I960) Keywood and Company B a l k a n s k i M.,Amzallag E., Phys. Stat. S o l . 30, 234 (1967) S m i t h S.D.,'Hanbuch d e r P h y s i k ' XX7/2a. 234 (1967) S p r i n g e r - V e r l a g 2 S l a g s v o l d B . J . , U3C Ph.D. t h e s i s u n p u b l i s h e d (1966). 3 K e r r R., UBC Ph.D. t h e s i s u n p u b l i s h e d (1971). 4 H o p f i e l d J . J . , Thomas D.G., Phys. Rev. 122, 35 (1961). 5 M u l l e r K.A. , S c h n i e d e r J"., Phys. L e t t . 4, 428 (1963) 6 Lambe J . , K i k u c h i C., J. Phys. Chem. S o l i d s 9, 492 (1958). 7. 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