UBC Theses and Dissertations

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UBC Theses and Dissertations

ESR study of antimony doped cadmium sulphide Halliwell, Robin Ernest 1969

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ESR STUDY OF ANTIMDNY DOPED CADMIUM SULPHIDE by ROBIN ERNEST HALLIWELL A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS' FOR THE DEGREE OF .MASTER OF APPLIED SCIENCE i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA . October, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date Asf. /2 ,/<?6<j ABSTRACT The e l e c t r o n i c g - t e n s o r i n a n t i m o n y d o p e d cadmium s u l p h i d e has been m e a s u r e d a t 1.1°K. A sample d o p e d t o a room t e m p e r a t u r e r e s i s t i v i t y o f 3.3 ohm-cm e x h i b i t e d an a n i s o t r o p i c g - t e n s o r w i t h g/( = 1.788 and q± = 1.770. A s i n g l e a s y m m e t r i c l i n e was o b s e r v e d . The asymmetry o f t h i s l i n e was f o u n d t o d e c r e a s e w i t h d e c r e a s i n g i n c i d e n t power. F u r t h e r e x p e r i m e n t s t o s t u d y t h i s l i n e s hape a r e i n d i c a t e d . i i i TABLE OF CONTENTS Page Abstract i i Table of Contents i i i L i s t of Figures v L i s t of Tables v i Acknowledgements v i i CHAPTER I Introduction 1 CHAPTER 2 Principles of Operation 3 Apparatus 6 . CHAPTER 3 Samples 10 CHAPTER 4 Theory- 11 Experimental Results 18 Discussion 22 Linewidth 28 CHAPTER 5 Conclusions . 30 iv Page APPENDIX A Calculations of the Valence Band States in Wurtzite . 32 APPENDIX B Calculation of the Valence Band Contribution to the g-shift 37 Bibliography 40 V L I S T OF FIGURES Page F i g u r e 2 . 1 S e r i e s E q u i v a l e n t C i r c u i t o f a R e s o n a n t C a v i t y 4 F i g u r e 2 . 2 B l o c k D i a g r a m o f S p e c t r o m e t e r 7 F i g u r e 4 . 1 The B a n d s t r u c t u r e a t k = 0 i n CdS a n d Ge 1 2 F i g u r e 4 . 2 S p i n R e s o n a n c e S i g n a l o f CdS.Sb a t 1 . 1 ° K . . 1 9 F i g u r e 4 . 3 The g - v a l u e a s a F u n c t i o n o f O r i e n t a t i o n . 2 0 v i L I S T OF TABLES Page T a b l e 4.1 The V a l e n c e Band S t a t e s i n W u r t z i t e . . . . 17 T a b l e 4.2 R e s u l t s f o r t h e C o n d u c t i o n Band g - T e n s o r i n CdS 21 T a b l e 4.3 Band P a r a m e t e r s f o r CdS 25 T a b l e 4.4 C o m p a r i s o n o f A ' s O b t a i n e d f r o m Q u a s i - c u b i c M odel a n d f r o m F i t o f t h e g - T e n s o r 27 T a b l e 4.5 S p i n s a n d M a g n e t i c Moments o f Donor N u c l e i 29 ACKNOWL EDGMENTS I w i s h t o t h a n k D r . C. F. S c h w e r d t f e g e r f o r h i s g u i d a n c e t h r o u g h o u t t h i s work. I w o u l d a l s o l i k e t o t h a n k D r . J . Marko f o r t h e d i s c u s s i o n s h e l d w i t h h i m on t h e r e s u l t s o f t h e e x p e r i m e n t , and D r . J . B i c h a r d f o r h i s a d v i c e and a s s i s t a n c e d u r i n g t h e i n i t i a l s t a g e s o f t h e p r e p a r a t i o n o f t h i s t h e s i s . The r e s e a r c h f o r 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 , g r a n t number A-2228. CHAPTER I I n t r o d u c t i o n T h i s t h e s i s i s c o n c e r n e d w i t h e l e c t r o n s p i n r e s o n a n c e (ESR) i n a n t i m o n y d o p e d cadmium s u l p h i d e . The s u l p h i d e s a n d s e l e n i d e s o f cadmium and z i n c a r e s e m i c o n d u c t o r s w i t h a w u r t z i t e s t r u c t u r e , a n d a r e e s p e c i a l l y i m p o r t a n t as p h o t o c o n d u c t o r s a n d p h o s p h o r s . E l e c t r o n s p i n r e s o n a n c e e x p e r i m e n t s have p r o v i d e d v a l u a b l e i n f o r m a t i o n on t h e s t r u c t u r e o f t h e e n e r g y bands a n d on t h e e f f e c t o f i m p u r i t i e s i n s e m i c o n d u c t o r s . A r e v i e w o f t h e r e s u l t s f r o m e x t e n s i v e s t u d i e s o f s e m i c o n d u c t o r s by ESR t e c h n i q u e s h a s been g i v e n by L u d w i g and W o o d b u r y . ^ ESR s i g n a l s due t o s h a l l o w d o n o r s i n s i l i c o n and germanium e x h i b i t a h y p e r f i n e s t r u c t u r e due t o t h e i n t e r a c t i o n ( 2 ) o f t h e e l e c t r o n s p i n w i t h t h e d o n o r n u c l e a r m a g n e t i c moment. On t h e o t h e r hand, s h a l l o w d o n o r s i n compound s e m i c o n d u c t o r s u s u a l l y do n o t show a h y p e r f i n e s t r u c t u r e . T h i s i s t o be e x p e c t e d s i n c e t h e a v a i l a b l e compound s e m i c o n d u c t o r s n o r m a l l y h a v e i m p u r i t y c o n c e n t r a t i o n s i n a r e g i o n where i m p u r i t y b a n d i n g t a k e s p l a c e . The e l e c t r o n s a r e m o b i l e e v e n a t l o w t e m p e r a t u r e s a n d t e n d t o a v e r a g e o u t any h y p e r f i n e s t r u c t u r e . A s i n g l e l i n e ESR s p e c t r u m was o b s e r v e d a t a p p r o x i m a t e l y 1.1°K i n cadmium s u l p h i d e d o p e d w i t h a n t i m o n y . T h i s i s b e l i e v e d t o f o r m a s h a l l o w d o n o r i n CdS a n d t h e p r e s e n t measurements 1 fwere made t o d e t e r m i n e t h e g - t e n s o r . (3) R o t h has g i v e n a d e r i v a t i o n o f t h e g - v a l u e f r o m (4) t h e band s t r u c t u r e , and u s i n g t h e s e methods C a r d o n a has c a l c u l a t e d t h e g - v a l u e f o r t h e common g e r m a n i u m - l i k e s e m i -c o n d u c t o r s . By t r e a t i n g w u r t z i t e t y p e m a t e r i a l s a s c u b i c , a n d n e g l e c t i n g t h e a n i s o t r o p y , he was a b l e t o a c c o u n t f o r a b o u t o n e - h a l f o f t h e o b s e r v e d s h i f t f r o m t h e f r e e e l e c t r o n g - v a l u e . C h a p t e r s 2 and 3 g i v e a d e s c r i p t i o n o f t h e e x p e r i m e n t a l a r r a n g e m e n t . C h a p t e r 4 c o n t a i n s an a t t e m p t , (5) b a s e d on t h e q u a s i - c u b i c model o f H o p f i e l d v , t o c a l c u l a t e t h e v a l e n c e band c o n t r i b u t i o n t o t h e g - s h i f t t a k i n g a n i s o t r o p y i n t o a c c o u n t . 3 CHAPTER 2 P r i n c i p l e s o f O p e r a t i o n M a g n e t i c d i p o l e t r a n s i t i o n s c a n be i n d u c e d i n a p a r a m a g n e t i c m a t e r i a l by p l a c i n g i t i n a m i c r o w a v e c a v i t y , i n t h e m i c r o w a v e m a g n e t i c f i e l d , a n d a p p l y i n g a s u i t a b l e D.C. m a g n e t i c f i e l d p e r p e n d i c u l a r t o t h e m i c r o w a v e f i e l d . T h e impedance o f a m i c r o w a v e c a v i t y a t a d e t u n e d s h o r t p o s i t i o n i s g i v e n b y : ^ Z = Z Q e x 2.1 ° " z s R s where Z q i s t h e c h a r a c t e r i s t i c impedance o f t h e t r a n s m i s s i o n l i n e c o n n e c t i n g t h e c a v i t y t o t h e s i g n a l s o u r c e and Q q i s t h e Q - v a l u e o f t h e u n l o a d e d c a v i t y . The e x t e r n a l Q - v a l u e , Q , m e a s u r e s t h e power r e r a d i a t e d t h r o u g h t h e c o u p l i n g h o l e . R and Z a r e t h e r e s i s t a n c e and t o t a l i mpedance s s ^ r e s p e c t i v e l y o f t h e s e r i e s e q u i v a l e n t c i r c u i t ( s e e F i g . 2.1), A- s m a l l p a r a m a g n e t i c sample p l a c e d i n t h e c a v i t y c h a n g e s t h e s e l f i n d u c t a n c e f r o m L t o L where o L = L Q ( 1 + FX ) . 2.2 The f i l l i n g f a c t o r , F, i s a f u n c t i o n o f t h e r a t i o o f t h e s a m p l e t o c a v i t y v o l u m e s and t h e c a v i t y s h a p e . The c o m p l e x r f s u s c e p t i b i l i t y c • F i g u r e 2.1 S e r i e s E q u i v a l e n t C i r c u i t o f a R e s o n a n t C a v i t y 5 X = X« - i X " 2.3 i s a f u n c t i o n o f t h e D.C. m a g n e t i c f i e l d f o r any g i v e n f r e q u e n c y . T u n i n g t h e s i g n a l s o u r c e t o t h e r e s o n a n c e f r e q u e n c y o f t h e empty c a v i t y , W = / 1 , t r a n s o r m s . t h e L C i m p e d a n c e t o 0 o ^ e x Z = Z — ° 1 + i F Q Q ( X' - i X " ) 2.4 The r e a l a n d i m a g i n a r y p a r t s o f t h e s u s c e p t i b i l t y c a n be m e a s u r e d s e p a r a t e l y by u s i n g a m i c r o w a v e b r i d g e . S i n c e t h e r e a l p a r t r e p r e s e n t s a d e t u n i n g o f t h e c a v i t y f r o m r e s o n a n c e , i t i s p o s s i b l e t o c o m p e n s a t e f o r X 1 by a l l o w i n g t h e s i g n a l s o u r c e f r e q u e n c y t o f o l l o w t h a t o f t h e c a v i t y . I n p r a c t i c e t h i s i s a c c o m p l i s h e d by l o c k i n g t h e k l y s t r o n f r e q u e n c y t o t h e r e s o n a n t f r e q u e n c y o f t h e c a v i t y . T he s u p e r p o s i t i o n o f a s m a l l m o d u l a t i o n f i e l d on t h e D.C. m a g n e t i c f i e l d w i l l r e s u l t i n a c o r r e s p o n d i n g m o d u l a t i o n o f t h e impendance Z a t t h e same f r e q u e n c y . The a m p l i t u d e o f t h e impedance m o d u l a t i o n i s p r o p o r t i o n a l t o dX" ^ , t h e d e r i v a t i v e o f t h e i m a g i n a r y p a r t o f t h e s u s c e p t i b i l i t y w i t h r e s p e c t t o t h e m a g n e t i c f i e l d . dX" A d i s p l a y o f -55— c a n be o b t a i n e d by t u n i n g t h e Ctrl d e t e c t o r c i r c u i t t o t h e m o d u l a t i o n f r e q u e n c y a n d s l o w l y s w e e p i n g t h e D.C. m a g n e t i c f i e l d . The o u t p u t o f t h e d X " ( 7 ) d e t e c t i o n s y s t e m w i l l t h e n g i v e dH 6 Apparatus The main features of the spectrometer used are shown in Figure 2.2. The microwave power source is a Varian Associates Reflex Klystron, type V-153/6315. It generates approximately 70mw over the frequency range 8.5 - 10 GHz. The operating frequency of the klystron was measured with a Hewlett Packard 5245L electronic counter equipped with a 5255 A frequency converter plug-in unit . This unit was capable of measuring frequencies from 3 GHz up to 12.4 GHz. A T E^Q2 r e c t a n S u - ' - a r cavity was used. It was made of standard brass waveguide and gold plated with an immersion • type gold solution produced by Sigmund Cohn Co. Inc.* This cavity was undercoupled at room temperature, but was very nearly c r i t i c a l l y coupled at the operating temperature of 1.1 °K. The samples were placed against the end wall of the cavity and held in place with vacuum grease. The crystal detector was a 1N23B low noise diode from Microwave Associates. The detector was an integral part of the preamplifier. The preamplifier had an A . C . gain of 40 db + 3 db over the range 100 Hz to 1 MHz, and was constructed around the crystal mount to reduce the noise introduced in transferring the signal to the Lock-in Amplifier. *Mount Vernon, N. Y . , U.S.A. ERROR 2>\&NAV_ •£ • > PovVER 5 U P P L X KLYSTRON OSC\LUaOR POvWER OSCILLO-SCOPE. AlTCWliWctR •DIRECTIONAL. T E E . \50VAtER C O U P U E R MRKHED L-OftO •SUCE. 3CRON TUNER F R E Q U E N C E C O U N T E R MODULATION OSCILLATOR CR^'aTftu DEFECTOR 4O0 H i AUDIO OSCU-USTOiR ftMPUfASR SECTOR OSCILLO-SCOPE L O C K - I N A M P L I F I E R " R E C O R D E D 40O H Z • R E F E - R E H C E . M/VaMET "BLOCK T>iAsv?ftM o p SPECTROMETER 8 The r e f l e c t o r v o l t a g e o f t h e k l y s t r o n was m o d u l a t e d a t 10 KHz. T h i s f r e q u e n c y was d e t e c t e d by t h e c r y s t a l a n d p a s s e d on t o t h e A.F.C. ( A u t o m a t i c F r e q u e n c y C o n t r o l ) where i t was a m p l i f i e d and r e c t i f i e d by a p h a s e s e n s i t i v e d e t e c t o r . A n y s i g n a l a t 10 KHz w o u l d t h e n r e s u l t i n a c o r r e c t i o n v o l t a g e b e i n g a p p l i e d t o t h e r e f l e c t o r v o l t a g e . I n t h i s manner t h e k l y s t r o n f r e q u e n c y was made t o f o l l o w t h a t o f t h e c a v i t y . T he o u t p u t o f t h e p r e a m p l i f i e r a l s o c o n t a i n e d t h e 400 Hz f i e l d m o d u l a t i o n f r e q u e n c y w h i c h c a r r i e d t h e r e s o n a n c e s i g n a l . T h i s was f e d i n t o an E l e c t r o n i c s , M i s s i l e s and C o m m u n i c a t i o n s I n c . Mo d e l RJB L o c k - i n A m p l i f i e r , w i t h t h e o u t -p u t g o i n g t o a M o s e l e y 680 c h a r t r e c o r d e r . The magnet was a Newport I n s t r u m e n t s T y p e D 8" e l e c t r o m a g n e t . T he m a g n e t i c f i e l d was m o d u l a t e d a t 400 Hz by a p a i r o f c o i l s mounted on t h e p o l e f a c e s and d r i v e n by an a u d i o o s c i l l a t o r and a m p l i f i e r . The maximum a m p l i t u d e o f t h i s m o d u l a t i o n f i e l d was a b o u t 12 g a u s s . The m a g n e t i c f i e l d was m e a s u r e d w i t h a m a r g i n a l o s c i l l a t o r s i m i l a r t o one p r o d u c e d by M a g n i o n A s s o c i a t e s . A g l y c e r i n e p r o t o n s a m p l e was mounted on one o f t h e p o l e f a c e s . C a l i b r a t i o n , u s i n g DPPH a s a s t a n d a r d , showed t h a t t h e r e was a d i f f e r e n c e o f « 4 g a u s s between t h e sample p o s i t i o n i n t h e c a v i t y and t h e p r o t o n s a m p l e . The f r e q u e n c y o f t h e m a r g i n a l o s c i l l a t o r was c o u n t e d w i t h t h e H e w l e t t P a c k a r d c o u n t e r , w h i l e t h e r e s o n a n c e i t s e l f was m o n i t o r e d 9 on an o s c i l l o s c o p e . A pen marker on the E.M.C. L o c k - i n A m p l i f i e r made i t p o s s i b l e t o mark the r e c o r d e r output as the proton resonance c r o s s e d the c e n t r e of the o s c i l l o s c o p e t r a c e . The c a v i t y and sample were immersed i n l i q u i d helium c o n t a i n e d i n a g l a s s dewar. I t was p o s s i b l e t o a t t a i n temperatures of ~ 1.1°K by pumping on the helium t o reduce i t s vapour p r e s s u r e . CHAPTER 3 S a m p l e The c r y s t a l u s e d f o r t h i s e x p e r i m e n t was cadmium s u l p h i d e d o p e d w i t h a n t i m o n y , s u p p l i e d by E a g l e P i c h e r , M i a m i , O k l a h o m a . The c r y s t a l was X - r a y e d t o l o c a t e t h e c - a x i s and a 1 mm t h i c k p l a t l e t was c u t f r o m i t s u c h t h a t t h e c - a x i s l a y i n t h e p l a n e o f t h e p l a t l e t . Room t e m p e r a t u r e r e s i s t i v i t y measurements i n d i c a t e d t h a t t h e s a mple had a r e s i s t i v i t y o f ~ 3.3 ohm cm. I f a 2 m o b i l i t y o f 100 cm / v o l t s e c . i s assumed, t h e c a r r i e r c o n c e n t r a t i o n s h o u l d be ~ 2 x 1 0 ^ p e r c c . T h e s a mple u s e d f o r t h e s e e x p e r i m e n t s had a volume _2 o f 2.5 x 10 c c a n d h a d a s i g n a l t o n o i s e r a t i o o f 10:1. A b s o l u t e s e n s i t i v i t y m easurements f o r t h e s p e c t r o p h o t o m e t e r 12 i n d i c a t e d t h a t i t was c a p a b l e o f d e t e c t i n g ~ 10 s p i n s / g a u s s . T h i s w o u l d t h e n g i v e a c a r r i e r c o n c e n t r a t i o n o f ~- 1 0 ^ / c c f o r t h e s a mple u s e d . 11 CHAPTER 4 Theory The e f f e c t i v e mass approximation can be used t o de s c r i b e an e l e c t r o n which i s near a point of minimum energy i n the conduction band. For the case of a non-degenerate band the e l e c t r o n i c g-value f o r a donor e l e c t r o n whose i o n i z a t i o n energy i s small w i t h respect t o the band gap, i s given b y ^ : 4 Y_ <°f/Px( n>< n!p y[°?> and % =2 + lmm^ E—rt 4-1 n o o n g = 2 + im i Z F T T {<°T|p y |n><n|P z|0i> + 1 m n ^ o ° n Y 4.2 <0i | p y [ n > < n [ p z ( 0 t > ] In these expressions fo} r e f e r s t o the band f o r which g i s being c a l c u l a t e d , j n r e f e r s t o a l l other bands, and m i s the mass of a f r e e e l e c t r o n . To apply these formulae t o the conduction band of cadmium s u l p h i d e some knowledge of the p o s i t i o n and s t r u c t u r e of nearby bands i s r e q u i r e d . The extrema of the conduction — ^ / O N and valence bands of CdS are at or very c l o s e t o k = 0 . 12 E E r r E n = 0 . 0 l 6 ev 1 ?| E 2=0.057 ev 3.05 e v f 1 2« fr k = 0 1c = 0 (a) (b) F i g u r e 4.1 The band s t r u c t u r e o f k = o i n CdS. ( R e y n o l d s , L i t t o n a n d W h e e l e r 1964) (a) a n d Ge (Cardona 1963a) ( b ) . fl end /g a r e i r r e d u c i b l e r e p r e s e n t a t i o n s o f t h e W u r t z i t e d o u b l e g r o u p . , and Z^ - a r e i r r e d u c i b l e Is r e p r e s e n t a t i o n s o f t h e f u l l c u b i c g r o u p ( s i n g l e g r o u p ) , and ^15c r, a n d l c 15v a r e t h e c o r r e s p o n d i n g r e p r e s e n t a t i o n s o f t h e z i n c b l e n d e s t r u c t u r e . 13 F i g u r e 4.1 shows t h e b a n d s t r u c t u r e a t t h i s p o i n t . The band s t r u c t u r e o f germanium, t h e c o r r e s p o n d i n g i s o e l e c t r o n i c h o m o p o l a r s e m i c o n d u c t o r , i s a l s o shown. The c o n d u c t i o n band minimum i n CdS b e l o n g s t o t h e i r r e d u c i b l e r e p r e s e n t a t i o n f l , o f t h e w u r t z i t e d o u b l e g r o u p . I t i s s e p a r a t e d by a g a p o f 3.58 e v f r o m t h e u p p e r m o s t o f t h e t h r e e v a l e n c e b a n d s . T h e s e t h r e e s p i n d e g e n e r a t e bands a r e c l o s e t o g e t h e r and b e l o n g t o t h e f^. , and fl^ r e p r e s e n t a -t i o n s r e s p e c t i v e l y . A n o t h e r g r o u p o f s t a t e s w i t h t h e same t r a n s f o r m a t i o n p r o p e r t i e s i s l o c a t e d 6.1 ev a b o v e t h e v a l e n c e (9) bands a r e : T h e b a s e - f u n c t i o n s f o r t h e and r e p r e s e n t a t i o n s : c s f + d z f + ^ f ( x + i y )l c s l + d z l + ^ f ( -x + i y )T 4.3 ^ : y | ( x + i y ) f ( x - i y )>(• where x, y, z a r e p - o r b i t a l s ; s an s - f u n c t i o n ; c, d, a n d f a r e c o n s t a n t s ; and T a n d 4 a r e t h e s p i n f u n c t i o n s . Some s i m p l i f i c a t i o n o f t h e s e b a s e - f u n c t i o n s c a n be e f f e c t e d s i n c e t h e c o n d u c t i o n band i s a l m o s t p u r e l y s - l i k e a n d t h e Rj v a l e n c e bands a r e d o m i n a n t l y p - l i k e ^ * ^ . T h i s means t h a t f o r t h e c o n d u c t i o n band, c = 1 and f = d = 0, and f o r t h e v a l e n c e bands, c = 0 a s a f i r s t a p p r o x i m a t i o n . The c a s e 14 o f c fc 0 f o r t h e v a l e n c e b a n d s w i l l be c o n s i d e r e d i n t h e ( 5 ) d i s c u s s i o n . The q u a s i - c u b i c m o d e l o f H o p f i e l d r e l a t e s t h e r e m a i n i n g d's a n d f " s t o t h e o b s e r v e d v a l e n c e b a n d s p l i t t i n g s . T h i s m o d e l c o n s i d e r s t h e w u r t z i t e s t r u c t u r e a s a c u b i c c r y s t a l t h a t h a s be e n u n i a x i a l l y s t r a i n e d a l o n g a ( 111 ) - d i r e c t i o n . The v a l e n c e b a n d o f a c u b i c s e m i c o n d u c t o r , w i t h v a n i s h i n g s p i n - o r b i t e n e r g y , c o n s i s t s o f s i x s t a t e s . T h e i r r o t a t i o n p r o p e r t i e s c a n be r e p r e s e n t e d by t h e J = 3/2, a n d J = 1/2 s t a t e s ( I D / 3 / 2 , 3 / 2 ) = f 3/2,1/2 > = / 3/2, -1/2 > = / 3 / 2 , - 3 / 2 ) = / 1 / 2 , 1 / 2 ) = ( 1 / 2 , - 1 / 2 ) = ( x + i y ) f y i . [ 2 z t + ( x + i y H ] ^ { 2z j - ( x - i y ) f ) _^ ( x - i y ) \ ^ | { z T - ( x + i y H } ^ { z ^  + ( x - i y ) f } 4.4 I n t h i s r e p r e s e n t a t i o n t h e s p i n o r b i t e n e r g y i s d i a g o n a l , a n d t h e J = 1/2 a n d J = 3/2 s t a t e s a r e s p l i t by an e n e r g y % . The s p i n - o r b i t H a m i l t o n i a n m a t r i x i s g i v e n b y : / o o o o o o \ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 0 ,0 0 0 0 0 § H = s-o 4.5 1 5 The s t a t e s t o w h i c h t h e rows c o r r e s p o n d have t h e same o r d e r a s t h e y have i n e q . 4.4. I f t h e c r y s t a l were c u b i c , w i t h no h e x a g o n a l c r y s t a l f i e l d , $ w o u l d be t h e v a l e n c e band s p l i t t i n g . T h e a p p l i c a t i o n o f a u n i a x i a l s t r a i n i n t h e z - d i r e c t i o n w i l l l e a v e t h e x T , x 4 , y f , a n d y i s t a t e s u n c h a n g e d b u t w i l l s h i f t t h e e n e r g y o f t h e z T and z i s t a t e s by an amount A w i t h r e s p e c t t o them. I f t h e r e were no s p i n - o r b i t c o u p l i n g A w o u l d be t h e v a l e n c e band s p l i t t i n g a n d i n t h i s r e p r e s e n t a t i o n t h e s t r a i n H a m i l t o n i a n w o u l d be: 0 0 0 o o o \ 0 0 0 0 0 0 0 0 ^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \ 0 0 0 0 0 A T h e o r d e r o f t h e rows o f t h i s m a t r i x i s x f , y t , z i , x j , y J , z f . H a n d H , . may now be d i a g o n a l i z e d t o g i v e J s-o s t r a i n J ^ ^ t h e s t a t e s and t h e i r e n e r g i e s r e l a t i v e t o t h e /"L l e v e l . y T h e r e s u l t s a r e g i v e n i n T a b l e 4.1. S i n c e E^ a n d E^ a r e s y m m e t r i c u n d e r i n t e r c h a n g e o f A and - $, s o l v i n g f o r S and A i n t e r m s o f E^ and E^ w i l l g i v e two s e t s o f s o l u t i o n s : H s t r a i n 4.6 16 5 = ( E L + JsE 2 ) i h Y ( E 2 - E 1 ) 2 - 3 E 1 2 A= - ( E1 + hE2 ) ± h VTE2 -E^)2 - 3 E 1 2 4*7 T h e s e two s o l u t i o n s a r e r e f e r r e d t o as t h e H o p f i e l d model ( % > / A ( ) a n d t h e B i r m a n model ( $ < /A [ ) . U s i n g t h e f o r m u l a e 4.1 and 4.2 and t h e wave f u n c t i o n s g i v e n i n T a b l e 4.1 i t i s p o s s i b l e t o compute t h e v a l e n c e band c o n t r i b u t i o n t o t h e g - s h i f t a t t h e b o t t o m o f t h e c o n d u c t i o n band. The r e s u l t s o f t h i s c a l c u l a t i o n a r e : ' = 2 - 2 lf f ^  ~ 2 A ' " 2 A ' r 4.8 L g E „ + E, E ^ + E, + E„ g = 2 + 2 & ( 2 ( 2 ^2 / 3 E 1 - 2 ) + 2 j \ 2 E g + E l + E2 3 ( E1 + E 2 ) _ 2 . . 4 > 9 where /if 2 = /< ?I V C^)} f 4.10 T a b l e 4 . 1 -The V a l e n c e Band S t a t e s i n w u r t z i t e E n e r g y B a s i s F u n c t i o n W = 0 zjk- ( x + i y ) f ( x - i y W ^2 ff^h - 2J z r - ( x + i y ) ^ j )?j x - i y + 2 j 3 } \ ( ( 2 f i , - . ) . l - , « . . 7 « } i ! ^ i ! r } _ 2 ] z ^ + ( x _ i y ) t ^ { 2 + - 2 y } -* r-1 18 E x p e r i m e n t a l R e s u l t s T he e l e c t r o n s p i n r e s o n a n c e s p e c t r u m o f CdS:Sb a t 1.1°K c o n s i s t s o f a s i n g l e l i n e w i t h no a p p a r e n t h y p e r f i n e s t r u c t u r e . The asymmetry o f t h e l i n e was f o u n d t o d e c r e a s e a s t h e m i c r o w a v e power i n c i d e n t on t h e sample was r e d u c e d . T y p i c a l t r a c e s f o r v a r i o u s power l e v e l s a r e shown i n F i g u r e 4.2. The g - v a l u e was m e a s u r e d a s a f u n c t i o n o f t h e a n g l e 0 between t h e m a g n e t i c f i e l d a n d t h e c - a x i s . The r e s u l t s a n d t h e i r f i t t o 2 2 2 2 2 g = g s i n Q + g c o s 0 4.11 x It a r e shown i n F i g u r e 4.3. The r e s u l t s a r e t a b u l a t e d i n T a b l e 4.2 and a r e compared w i t h t h e r e s u l t s o f o t h e r w o r k e r s . 19 I n c i d e n t power = 15 mw F i g u r e 4.2 S p i n Resonance o f CdS.Sb a t 1.1°K F i g u r e 4.3 The g - v a l u e as a F u n c t i o n o f t h e A n g l e between t h e M a g n e t i c F i e l d and t h e C - a x i s ro o Table 4.2 Results for the Conduction Band g-tensor in CdS Crystal g Width Between Approximate Conditions Inflection True Half Number of of Points Width Spins Observation (gauss) (gauss) CdS :5b a CdS:Br a CdS:I 1.788+.0005 1.770+.0005 1.789+.001 1.771+.001 1.785+.001 1.767+.001 a CdS(10%zn):I 1.796+.001 1.780+.001 b CdS:Cl 1.79 c CdS(Ga > Cu) 1.792 d CdS 1.78+.05 1.78 1.775 1.72+.10 15 34 12 15 5 10 * 20 * 50 - 17 * 20 8 10 16 3x10 17 ^ 2x10 17 2x10 a 10 16 17 1.1°K 1.7°K 1.7°K 1.7°K • 4.2°K 77 °K exciton spectrum a) Slagsvold and Schwerdtfeger (1965) v / b) Lambe and Kikuchi (1958) ( 1 8 ) c) Dieleman (1962) ( 1 9 ) d) Hopfield and Thomas (1961) ^ 2 0 ^ 22 Discussion Calculations in the hydrogen-like approximation (12) show that the donor binding energy is given by : E = 13.6 ~ ±2. (eV) 4.12 where X i s the static d ie lec tr ic constant of the Berni-ni* w conductor. For the case of CdS — ~ 0.2 and f\ ^ 10 m giving a value of ~ 0.028 eV for the binding energy. This (13) agrees with photoconductivity measurements , which indicate that the halogens give r ise to localized levels 0.03 eV below the conduction band. Since the g-values measured for CdS:Sb are nearly the same as for CdS doped with halogens i t is reasonable to assume that Sb also enters the CdS substitutionally as a shallow donor and w i l l have nearly the same binding energy. The wave functions of the donor states are composed of Bloch functions from the bottom of the conduction band with a hydrogen-like envelope function centred about the donor s i te . On this basis one would expect that the g-value of the donor level should be quite close to that of the conduction band. The difference would appear only in the replacing of E^ with E^ minus 0.028 eV, a typical shallow donor binding energy. The spin-orbit and crystal f i e l d parameters can be calculated from equations 4.7, using E, = 0.016 eV and E_ = 0.057 eV "(see Figure 4.1) to be: 23 0.0596 eV A = -0.0294 eV (Hopfield model) S= 0.0294 eV -0.0596 eV (Birman model) 4.13 The Birman model ( $ < / A ( ) would account for less than a quarter of the observed g-shifts when only the valence band (14) contributions are considered. Observations by Sobolev indicate that the shift in energies, upon deformation, for optical transitions between the two upper valence bands and the conduction band, in CdS and CdSe, are different. This (15) led Sandomirskii to conclude that the Hopfield model ( %yi&l ) is the preferable model. For these reasons our discussion of the g-shifts w i l l be based upon the Hopfield model. (4) Cardona has considered the effect of an ant i -symmetric perturbation on the bandstructure of a homopolar semiconductor l ike germanium. The perturbed wave functions for such a system are: V r i s c ' - • r < r i 5 > - b * < r25. > f P< r i 5 V ) - b Y< r 1 5 ) + a t ( r 2 5 , > 4 . i 5 where a and b are constants which may be expressed in terms of the "unperturbed" energies. This perturbation has negligible effect on the Y ( /"!,, ) state. Hence: P l c The wave function for the polar material can be substituted into equation 4.10, using the information that the 4.16 24 transit ion f~^l * * is forbidden, to give: (ll2 = a 2 ll. n / 2 4.17 1 ' ' homopolar ' A good overall agreement between predicted and observed values for effective masses and g-values of wurtzite-type 2 2 materials is obtained using p = 21 eV, where p is defined by: 2 2 / I I2 P = ' homopolar! 4 l g The expression for 'a' (equation 4, ref . 4) can be used, with the known band spl i t t ings for Ge and CdS, to obtain 2 a = 0.75. Equations 4.17 and 4.18 then give for the transit ion to the valence bands: 2 [ll 2 = 15.7 eV 4.19 m The g-shifts computed for equations 4.8 and 4.9 are ^ gff= -0.10 and ^ g = -0.06. These values are less than half the observed g-shifts, and predict the wrong anisotropy. Cadmium selenide, having a band gap of only 1-^ -84 eV, should give a much better agreement i f the discrepancies are due to improper consideration of the structure of the upper conduction bands. A comparison of the parameters of CdS and CdSe are shown in Table 4.3. 25 Table 4.3 Band Parameters for CdS c/a g g $ (eV) & (eV) E (eV) E (eV) 1 g g c CdS 1.623 1.785 1.767 0.0596 -0.0294 2.58 a 3.52 a CdSe 1.63 O ^ O . l 1 3 0.51^0.05 0 . 4 4 ° - 0 . 0 4 ° 1.84a 4.26 3 a) Cardona (1963a) b) Sobolev (1964) ( 1 4 ) c) Dimmock & Wheeler (1961) g c E^ is the energy sepa-ration between conduction bands. The contribution to the g-shift from a group of states separated by an energy E, w i l l be approximately 2 111 2 proportional to k—• , where / I / is the matrix element mE between the states and the band for which the g-shift is being calculated. The relat ive importance of the valence and (7) conduction bands is given by a 2 $ -^f c g „ 4.20 Although —— is not known, i t should be of the order of unity. The expression (equation 4.20) has a value of 6 for CdS and 26 16 for CdSe. This gives a g-value of ~0.84 for CdSe. From equations 4.8 and 4.9 i t is found that ^ g = -1.15 and Zi = -1.06. This also has the incorrect anisotropy. These results ' indicate that the quasi-cubic model is inadequate for calculations of the g-tensor. If a l l the constants in expressions 4.3 are allowed to be non-zero there w i l l be additional terms in equations 4.8 and 4.9. Since these terms are d i f f i c u l t to estimate only a modification of the existing expressions w i l l be considered here. (15) Hopfield has shown .that the value of 'c' (equation 4.3) for the valence band should be at most of A U the order of ( —— ) . This then gives for the normalizing Eg constants ^ 2 a n t ^ ^ 3 (see Table 4.1), that: A 2 = * \ g A The values of - — for CdS and CdSe are ^0.01 and 0.02 E g respectively, indicating 1% and 2% lower values for A 's. Using the g-tensors of Table 4.3 and ignoring any possible contribution from the upper conduction bands, i t is possible to estimate the amount of s-part admixture to the valence bands by treating 7\^ and as parameters. The results , showing the expected order of magnitude admixtures, are given in Table 4.4. 27 Table 4.4 Comparison of A> • s Obtained from Quasi-cubic Model and from F i t of the g-tensor TV2 "A2 % A2 A2 % quasip- f i t reduction qua sir-. f i t reduction ^ cubic ^ cubic CdS 0.292 0.289 1 0.208 0.200 4 CdSe 0.1875 0.1875 0 0.321 0.289 10 It should be noted at this point that Hopfield and Thomas allowed a 15% s-part admixture to the valence states to account for the intensities of the exciton lines in CdS. These considerations show that a generalization of the expression for cubic germanium-like semiconductors within the framework of the quasi-cubic model does not adequately explain the observed g-shift . Moreover, i t gives the wrong sign for the anisotropy of the g-value for both CdS and CdSe. Allowing small s-part admixtures to the valence band states though, does make i t possible to f i t the observed g-tensors to the formulae. This suggests that more complete knowledge of the band structure is required to give a good correspondence between theory and experiment for the g-tensor in CdS and CdSe. 28 The L i n e w i d t h S i n c e t h e h y p e r f i n e s t r u c t u r e due t o t h e m a g n e t i c moments o f t h e d o n o r n u c l e i d i s a p p e a r s i n t h e i m p u r i t y band (2) r e g i o n , i t i s n o t u n e x p e c t e d t h a t t h e o b s e r v e d l i n e s show no s t r u c t u r e . I f t h e r e s o n a n c e l i n e f o r CdS.Sb i s compared t o t h o s e o f CdS d o p e d w i t h h a l o g e n s , t h e n T a b l e 4.5 w o u l d s u g g e s t t h a t t h e l i n e w i d t h f o r t h e a n t i m o n y doped sample s h o u l d be o f t h e same o r d e r , o r s l i g h t l y g r e a t e r t h a n t h a t o f t h e i o d i n e o r b r o m i n e d o p e d s a m p l e s . T h i s -is i n f a c t t h e c a s e , a l t h o u g h , b e c a u s e o f t h e a s y m m e t r i c l i n e s t h e l i n e s h a p e s a r e n o t r e l i a b l e enough t o make a good c o m p a r i s o n . 29 T a b l e 4.5 S p i n s and M a g n e t i c Moments o f Donor N u c l e i N u c l e u s N a t u r a l A b u n d a n c e S p i n M a g n e t i c Moment i n N u c l e a r B o h r Magnetons c i 3 5 75.4% 3 2 0. 82 c i 3 7 24.6% 0.68 B r 7 9 50.52% 3 2 2.10 " B r 8 1 49.48% 3 2 2.26 -,.127 100% 5 2 2.79 s b 1 2 1 57.25% 5 2 3.34 s b 1 2 3 42.75% 7 2 2.53 ( a ) T . . V a r i a n A s s o c i a t e s NMR T a b l e , F i f t h E d i t i o n , 1965. CHAPTER 5 C o n c l u s i o n s A 3 cm m i c r o w a v e s p e c t r o m e t e r e m p l o y i n g s t r a i g h t c r y s t a l d e t e c t i o n a n d 400 Hz f i e l d m o d u l a t i o n has been a s s e m b l e d and shown t o have a s u f f i c i e n t s e n s i t i v i t y t o a l l o w d e t e c t i o n o f v a r i o u s ESR s i g n a l s i n s e m i c o n d u c t o r s s u c h as s i l i c o n and cadmium s u l p h i d e . I n p a r t i c u l a r , g o o d s i g n a l t o n o i s e r a t i o s were o b t a i n e d a t l i q u i d He t e m p e r a t u r e s f r o m an a n t i m o n y - d o p e d s i n g l e c r y s t a l o f CdS w i t h a room t e m p e r a t u r e 16 —1 c a r r i e r c o n c e n t r a t i o n a r o u n d 2 x 10 cm" . The s i n g l e ESR l i n e o b s e r v e d i n t h i s c r y s t a l i s b e l i e v e d t o be due t o e l e c t r o n s i n an i m p u r i t y band a b o u t 0.03 eV b e l o w t h e b o t t o m o f t h e c o n d u c t i o n band. The w i d t h o f t h i s l i n e was a b o u t 3 t i m e s l a r g e r t h a n t h a t o f a l i n e r e p o r t e d s e e n i n C d S : C l . T h i s i s l e s s t h a n t h e r a t i o o f t h e n u c l e a r m a g n e t i c moments o f t h e d o n o r n u c l e i w h i c h i s ^T*( ~ 4. The g - v a l u e i n CdS:Sb d i f f e r s o n l y s l i g h t l y f r o m t h o s e o f C d S : B r a n d C d S : I , and l i k e them e x h i b i t s a s m a l l a n i s o t r o p y i n t h e u n i a x i a l l y s y m m e t r i c g - t e n s o r , w i t h q/( g^ = 0.018. I t was f o u n d t h a t k n o w l e d g e o f t h e b a n d s t r u c t u r e o f CdS i s i n s u f f i c i e n t l y a c c u r a t e t o a l l o w an unambiguous e x p l a n a t i o n o f t h e g - t e n s o r . Good a g r e e m e n t w i t h e x p e r i m e n t f o r CdSe and CdS c a n be o b t a i n e d by c o n s i d e r i n g o n l y t h e 31 valence bands i f small corrections are made to the quasi-cubic model. The reason for the highly asymmetric l i n e i s not known at t h i s stage but further work i s planned on t h i s problem. 32 APPENDIX A C a l c u l a t i o n o f t h e V a l e n c e B and S t a t e s i n W u r t z i t e The t o t a l wave- f u n c t i o n o f t h e v a l e n c e band edge f o r a c u b i c s e m i c o n d u c t o r w i t h o u t i n v e r s i o n symmetry t r a n s f o r m s a s [Zj + HQ. The b a s i s f u n c t i o n s may be c h o s e n a s : r • ' 8* (3/2,3/2 > ( x + i y ) f = | l > (3/2,1/2 } = ^ ( 2 z t + ( x + i y ) j ) ( 3 / 2 , - l / 2 > = y | ( 2zi _ ( x - i y ) f ) = 12) | 3 > (3/2,-3/2 7 = ^ ( x - i y )l r ? : (1/2,1/2 > = Y 3 { z t - ( x + i y ) ^ ) = ( l / 2 , - 1 / 2 ) = ^ ( zi + ( x - i y ) T ) = |6> The and Pg s t a t e s a r e s p l i t by an amount % due t o t h e s p i n - o r b i t i n t e r a c t i o n . The s p i n - o r b i t e n e r g y i s d i a g o n a l i n t h i s r e p r e s e n t a t i o n a n d has t h e m a t r i x : /o 0 0 0 0 o \ 0 0 0 0 0 0 H s-o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S o \ o 0 0 0 0 S 1 0 |2> |3> U > / 5 > 1 6 ) A t t h i s p o i n t i t i s c o n v e n i e n t t o t r a n s f o r m t o a r e p r e s e n t a t i o n w i t h t h e b a s i s xT, yt, zi, xi, yi', z f s i n c e s t r a i n i s d i a g o n a l i n t h i s r e p r e s e n t a t i o n . I f t h e o r i g i n a l b a s i s i s u , a n d t h e new b a s i s i s v , t h e n we c a n w r i t e : n n u = B . v. n i n 1 where B. a r e e l e m e n t s o f t h e u n i t a r y m a t r i x B. T h i s t h e n i n J g i v e s t h e r e s u l t : u 1 H = B H V B o r H V = B H U B f C o n s i d e r a t i o n o f t h e b a s i s f u n c t i o n s g i v e s : / 1/-T2 0 -1/V6 0 0 1/1/3 / ±/i2 0 i/V6 0 0 - i / / 3 0 0 2/V6 0 0 1/V3 B = 0 1/V6 0 1/V2 - 1//3 0 0 i/T/6 0 -i/lf2 - i / V 3 0 2/Y6 0 0 1/V3 0 / a s t h e t r a n s f o r m a t i o n m a t r i x . T h i s c a n be a p p l i e d t o H s-o t o g i v e : H t r a n = B H B f so where, 34 H t r a n 0 0 0 0 0 0 - i -1 1 - i i 1 A p p l i c a t i o n o f s t r a i n a l o n g t h e z - d i r e c t i o n w i l l l e a v e t h e s t a t e s xT , x ^ , yT , y <l unchanged, b u t w i l l c h a n g e t h e e n e r g y o f zT and z i by £ w i t h r e s p e c t t o t h e o t h e r s t a t e s . The t o t a l H a m i l t o n i a n , n e g l e c t i n g a d m i x t u r e s , t h e n becomes: H = 0 0 i Y Y i Y 0 0 0 Y - i Y o o o o o o i Y - Y o 0 0 Y i Y where V = - \ T h e e i g e n v a l u e s o f H may be f o u n d by s o l v i n g t h e s e c u l a r d e t e r m i n a n t /H.. - W I = 0 . The s o l u t i o n s a r e : W l , 2 = 0 W 3 , 4 + H1) + S A A l l o f w h i c h a r e d o u b l y d e g e n e r a t e as r e q u i r e d by K r a m e r ' s t h e o r e m . I n t h e m a t e r i a l s o f i n t e r e s t $ i s a p o s i t i v e q u a n t i t y , so i t i s c o n v e n i e n t t o c h a n g e i t s s i g n , t o g i v e : W l , 2 = 0 w - 4 - S 1 / s' + a 2 J A W3,6 2 V 2 " 3 To f i n d t h e new b a s i s f u n c t i o n s i t i s n e c e s s a r y t o s o l v e t h e e q u a t i o n : S ( H - W I ) = S W T h i s c a n be b r o k e n down i n t o a number o f s e t s o f s i m u l t a n e o u s e q u a t i o n s : S i l H l l + S i 2 H 2 1 + S i 3 H 3 1 = W i S i l S i l H 1 2 + S i 2 H 2 2 + S i 3 H 3 2 = V i 2 1 = 1 , 3 , 5 . i i 13 i 2 23 i 3 33 l i 3 i 4 44 i 5 54 16 64 l i 4 i 4 45 i 5 55 16 65 l i 5 l = 2,4,6 14 46 i 5 56 16 66 i 16 w h i c h can be s o l v e d t o g i v e : s = / 1/V2 -i/V2 0 0 0 0 0 0 0 1//2 i / l / 2 0 - i A 2 - ( W 2 - 2 ) A Y 0 0 0 0 0 0 - 1 A 2 . - ( W 2 t A 3 i ? \ 3 ( W3-2)X-if 3 0 0 0 \ o 0 0 _ * 3 (W 3 i " < 2 + ( f . 2 i 2 >-•» b a s i s f u n c t i o n s c a n now be o b t a i n e d t h r o u g h v ~ n S. -0. i n ' I 0± = S " 1 v = np n ( S r ) v np n = S* v pn n where 0^ a r e t h e new b a s i s f u n c t i o n s . The f i n a l r e s u l t s o f t h i s c a l c u l a t i o n a r e g i v e n , i n T a b l e 4 . 1 . 3 7 APPENDIX B C a l c u l a t i o n o f t h e V a l e n c e Band C o n t r i b u t i o n t o t h e q - s h i f t The e l e c t r o n i c g - v a l u e a t a band edge i s g i v e n b y : ^ g,= 2 + i M where < 0 ( p v | n> < n | p i 0 > M -n^O E o - E n The momentum o p e r a t o r s a r e d e f i n e d by: T h e s t a t e s t o be c o n s i d e r e d a r e : C o n d u c t i o n b a n d : s t + a z T + ^b ( x + i y H = I 0^  Pg v a l e n c e b a n d : - ~fh ( x + i y ) t = ( l ^ ih ( x -. i y U = (2 > U p p e r /"^  v a l e n c e s^T + a ^ z f + ^b^ ( x + i y )^ = I 3 ? ba nd: ^ + a Jr + igb^ ( -x + i y ) T = J 4 > Lower P v a l e n c e s 2 ? + a 2 z 7 + ^ x + i y ) >£• = b a n d : s 2^+ a 2 z I + X2&2 ( -x + i y ) f = / 6 ^ 38 A p p l y i n g t h e momentum o p e r a t o r s t o t h e s e s t a t e s g i v e s : P X U ? - l/f2f P |1) y ' ' = -1/-/2 T P X ! 2 > - -±/f2i P ( 2> * y f = - 1 / / 2 I P X I 3 > = - K * P I 3> + P X 14? - - i s , + § y P (O y + P X I 5 ? . - i b * 2 2 p [ 5 ? * y • ' + P X I 6 ) = + | b 2 t . P I 6 ) ^ y » - -*2 (J)* + I f t h e s e a r e now u s e d i n t h e e x p r e s s i o n f o r M, one g e t s M = i m £ § - [ § ( s * d t ] [ - i s j s dT ] + . l - 3 - T - E 7 [ ^ ( - i ) / s 1 dT + f b J s M t ] ^ b 2 ( s * d t ] [ - ^ / s ^ ^ J d t + ^ 2 / s d t ] ] D e f i n i n g I Q = fsd? , I x = J s ^ f - J d t = / ^ ( j f - j d t , a n d s u b s t i t u t i n g them i n t o t h e e q u a t i o n f o r M, one o b t a i n s : 3 9 M = f - < f» | Io/ 2 + v i T , ' [ l b ' 2 ' I i l 2 + / b i / 2 / U 2Re b ^ I ^ } ' + h [ W 2 | l 2 / 2 + / b 2 | 2 / l 0 | 2 2 Re b * b 2 I o X 2 } ,M = l b i l E E + E, g g 1 ' b 2 ' 2 ) - |bf 2 ( E g + E 1 + E 2 E + E, g i + E g + E l + E 2 b * b n l I b * b _ i i _ _ 2 Re ( 1 ° 1 + 2 ° 2 ) E + E n g i E g + E 1 + E 2 J I f t h e c o n d u c t i o n band i s assumed t o be a p u r e s - s t a t e , a=b=0 ** ~ 2 1 E E +E n E + E,+E„ ; g g 1 g 1 2 C o m p a r i s o n o f t h e s t a t e s ln^ w i t h t h o s e i n T a b l e 4.1 g i v e s : h | b j 2 = 2 A2 a n d H \b.J2 = 2 r\\ A n d h e n c e : _ ' 2 2/1 | 2 . 2 *l 2 ^ y " m v E E +E-, E +E,+E 0 ' g g 1 g l 2 A s i m i l a r c a l c u l a t i o n c a n be done t o g i v e : 2 / I o / 2 r 2 ? i 2 3 E i v q 1 9 2 ? \ 2 3 ( E 1 + E 2 ) } 40 BIBLIOGRAPHY 1. Ludwig, G. W. and Woodbury, H. H. 1962. Sol id State Physics, ed. by F. Seitz and D. Turnbull, Vo l . 13 ( A c a d e m i c Press, N. Y . ) , p. 223. 2. Feher, G. 1959. Phys. Rev. 114, 1219. 3. Roth, L . M. 1960. Phys. Rev. 118, 1534. 4. Cardona, M. 1963a. J . Phys. Chem. Solids. 24, 1543. 5. Hopfield, J . J . I960. J . Phys. Chem. Solids. 15, 97. 6. Ginzton, E. L . 1957. Microwave Measurements (McGraw-H i l l Book Co. , Inc . ) . 7. Slagsvold, B. J . 1966. Doctoral Thesis, U. B. C. (Unpublished.) 8. Reynolds, D. C , Li t ton , C. W. and Wheeler, R. G. 1964. Proc. 7th Internat. Conf. on Semiconductor Physics. Paris 1964. (Academic Press, N. Y . ) , p. 148. 9. Cardona, M. 1963b. Phys. Rev. 129, 1068. 10. Hopfield, J . J . 1961. J . Appl. Phys. 32' Suppl. 2277. 11. K i t t e l , C. 1963. Quantum Theory of Solids (John Wiley & Sons, N. Y . ) . 12. Kohn, W. 1957. Sol id State Physics. Vol . 5, ed. by F. Seitz and D. Turnbull . (Academic Press, N. Y . ) . 13. Bube, R. H. and Thomsen, S. M. 1955. J . Chem. Phys. 23, 15. 14. Sobolev, V. V. 1964. Soviet Phys. Sol id State. 6, 697. 15. Sandomirskii, V. B. 1964. F i z . Tverdogo Tela. 6, 324. 16. Birman, J . L.. 1959. Phys. Rev. 1L4, 1490. 17. Slagsvold, B. J . and Schwerdtfeger, C. F. 1965. Can. J . Phys. 43, 2092. 18. Lambe, J . and Kikuchi, C. 1958. J . Phys. Chem. Solids. 9, 492. 41 19. Dieleman, J . 1962. Proc. 11th Colloque Ampere, Eindhoven (North-Holl. Publ. Comp. Amsterdam, 1963), p. 412. 20. Hopfield, J . J . & Thomas, D. G. 1961. Phys. Rev. 122 35. 21. Dimmock, J . 0. and Wheeler, R. G. 1961. J . Appl. Phys. 32, s u p p l . 2271. 

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