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Microwave cyclotron resonance of hot electrons in p-type gallium antimonide Hill, Douglas Arthur 1972

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MICROWAVE CYCLOTRON RESONANCE OF HOT ELECTRONS IN P-TYPE GALLIUM ANTIMONIDE by DOUGLAS ARTHUR HILL B . S c , U n i v e r s i t y o f T o r o n t o , I 9 6 6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f 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 t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 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 s tudy . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copy ing o f 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 copy ing 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 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 . Depa rtment of 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 A microwave p h o t o c r e a t e d c y c l o t r o n r esonance s i g n a l i s o b s e r v e d i n p-type GaSb i n t h e t e m p e r a t u r e range 1-30 K . C i r c u l a r p o l a r i z a t i o n and o t h e r measurements i d e n t i f y t h e c a r r i e r s as e l e c t r o n s i n t h e (000) c o n d u c t i o n band. The e f f e c t s o f background a b s o r p t i o n , d i s p e r s i o n , t h e p r e s e n c e o f h o l e s at h i g h power and plasma s h i f t s a r e t a k e n i n t o a c c o u n t . The problem o f a s u r f a c e e f f e c t on t h e measured peak p o s i t i o n i s r e p o r t e d f o r t h e f i r s t t i m e , and i s a v o i d e d by b u l k c a r r i e r c r e a t i o n . The e l e c t r o n s a re h e a t e d by t h e s t r o n g microwave e l e c t r i c f i e l d t o e n e r g i e s up t o 30 + k meV above t h e c o n d u c t i o n band minimum. The s c a t t e r i n g mechanism i s s t u d i e d t h r o u g h t h e h a l f w i d t h o f t h e c y c l o t r o n r esonance l i n e . The measured oj-r o f 1.5 - ^ y i e l d s an e l e c t r o n c o l l i s i o n t i m e o f f ~ 10""^ s. The s c a t t e r i n g mechanism a t l i q u i d h e l i u m t e m p e r a t u r e s i s i d e n t i f i e d as b e i n g p a r t l y due t o n e u t r a l d e f e c t a c c e p t o r s c a t t e r i n g o f hot e l e c t r o n s . There i s a l s o an u n i d e n t i f i e d r e s i d u a l s c a t t e r i n g p r o c e s s . The e l e c t r o n p o l a r o n e f f e c t i v e mass i s measured t o be m»(p.olaron) _ 0 < 0 ^ 1 2 + , 0008 f o r e l e c t r o n s w i t h an average energy o f 15 meV. The a n g u l a r dependence o f t h e e f f e c t i v e mass i s l e s s t h a n 1%. When c o r r e c t i o n s f o r c o n d u c t i o n - b a n d n o n - p a r a b o l i c i t y and p o l a r o n e f f e c t s a re a p p l i e d , t h e band-edge f r e e e l e c t r o n mass i s c a l c u l a t e d t o be m p ( f r e e ) = 0.0396 m o + 0.0018 ( o r + The main e r r o r i n t h e f i n a l v a l u e d e r i v e s from t h e two c o r r e c t i o n s c a l c u l a t e d u s i n g t h e hot e l e c t r o n d i s t r i b u t i o n . » • • 1 1 1 TABLE OF CONTENTS Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v i i L i s t o f F i g u r e s v i i i Acknowledgements , x CHAPTER 1 - INTRODUCTION 1 .1 G e n e r a l I n t r o d u c t i o n 1 1 . 2 The M o t i v a t i o n f o r t h i s E x p e r i m e n t 3 1 . 3 T h e s i s O u t l i n e 4 CHAPTER 2 - THE MICROWAVE SPECTROMETER 2 . 1 The S p e c t r o m e t e r D e s i g n . , 5 2 . 2 S p e c t r o m e t e r O p e r a t i n g C o n d i t i o n s 10 2 . 3 The Problem o f D i s p e r s i o n 13 2A The C a v i t i e s 17 2 . 5 N o i s e S o u r c e s and S e n s i t i v i t y 23 CHAPTER 3 - THE CYCLOTRON RESONANCE SIGNAL 3 . 1 Theory 25 (a) The C l a s s i c a l T heory 25 (b) The S e m i c l a s s i c a l Theory 28 ( c) The Quantum Theory 31 3 . 2 The Plasma E f f e c t 32 i v Page CHAPTER 3 - C o n t i n u e d 3.3 G a l l i u m A n t i m o n i d e (GaSb) 35 3.4 The Measured A b s o r p t i o n S i g n a l (a) The Background.. 40 (b) The Treatment o f D i s p e r s i o n . . . , 41 ( c ) C i r c u l a r P o l a r i z a t i o n R e s u l t s 45 (d) E v i d e n c e o f " C y c l o t r o n Resonance"... 47 (e) S i g n a l C a l c u l a t i o n s . . . 48 ( f ) The S u r f a c e E f f e c t 52 (g) The Power Dependent S i g n a l I n t e n s i t y 57 (h) The Plasma S h i f t . , . . . 63 CHAPTER 4 - THE SCATTERING MECHANISM 4.1 The Measured U/-T 65 4.2 The A c c e p t o r Dependence o f t h e S c a t t e r i n g 72 4.3 S c a t t e r i n g Mechanism C a l c u l a t i o n s (a) The S c a t t e r i n g Time 77 (b) Phonon S c a t t e r i n g 79 (c-) 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 84 (d) N e u t r a l I m p u r i t y S c a t t e r i n g 87 (e) Impact I o n i z a t i o n o f t h e A c c e p t o r s . . 90 4.4 The Temperature Dependence o f t h e S c a t t e r i n g , , , 91 4.5 The Temperature Dependence 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 . . . , , 95 Page CHAPTER 5 - THE HOT ELECTRON MODEL 5.1 The Model 99 5.2 E v i d e n c e S u p p o r t i n g t h e Model (a) E x p e r i m e n t a l 100 (b) C a l c u l a t i o n s , 101 5.3 The E l e c t r o n Energy D i s t r i b u t i o n 102 5 A The Kane Band S t r u c t u r e Model 106 5.5 The E f f e c t i v e Mass Average 110 CHAPTER 6 - THE EFFECTIVE MASS 6.1 The Measured P o l a r o n E f f e c t i v e Mass 113 (a) The A n g u l a r Dependence o f t h e E f f e c t i v e Mass 113 (b) The Power Dependence o f t h e E f f e c t i v e Mass 116 "(c) The Temperature Dependence o f t h e E f f e c t i v e Mass 119 6.2 The Band-Edge P o l a r o n Mass... 120 6.3 The P o l a r o n C o r r e c t i o n and t h e Fr e e E l e c t r o n E f f e c t i v e Mass 120 6A The E f f e c t i v e Mass i n D i f f e r e n t Samples 121 6.5 Comparison W i t h Other Measurements 122 CHPATER 7 - CONCLUSIONS 126 BIBLIOGRAPHY ' 129 v i Page Appendix 2 . 1 D e r i v a t i o n o f t h e S i g n a l F o r m u l a 136 A p p e n d i x 2 . 2 The C r o s s - M o d u l a t i o n Technique 139 A p p e n d i x 3 . 1 Peak P o s i t i o n C a l c u l a t i o n s 1 4 1 A p p e n d i x 3 . 2 C y c l o t r o n A b s o r p t i o n C a l c u l a t i o n s 1 4 5 A p p e n d i x 3 . 3 The Background A b s o r p t i o n . . . . . 1 4 ? Appendix 3 . 4 Sample Data T a b l e s 1 4 8 A p p e n d i x 4 . 1 The GaSb C o n s t a n t s 1 5 3 Appendix 6 . 1 The Hol e E f f e c t i v e Mass V a l u e s 1 5 6 A p p e n d i x 6 . 2 The E f f e c t o f L i g h t H o l e s on t h e Measured Peak a t H i g h Power 1 5 8 A p p e n d i x 6 . 3 The Phonon C o r r e c t i o n s t o t h e E f f e c t i v e Mass 1 6 0 v i i L IST OF TABLES T a b l e Page 2.1 The microwave c a v i t i e s 18 3.1 The e f f e c t o f d i s p e r s i o n on the peak p o s i t i o n 45 4.1 The 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 t i m e 86 6.1 The e f f e c t i v e mass d a t a f o r t h r e e c r y s t a l l o g r a p h i c d i r e c t i o n s 116 6.2 The measured peak parameter as a f u n c t i o n o f microwave power and sample 118 6.3 S o u r c e s o f e r r o r i n t h e measured peak parameter 119 6.4 The te m p e r a t u r e dependence o f the e f f e c t i v e mass 121 6.5 Other d e t e r m i n a t i o n s o f t h e e l e c t r o n e f f e c t i v e mass i n GaSb.. 124 A3.1 L i n e a r p o l a r i z a t i o n peak p o s i t i o n c o r r e c t i o n s 143 A3.4-1 The sample c y c l o t r o n r e sonance d a t a 149 A3.4-2 The sample H a l l and r e s i s t i v i t y d a t a 150 A3.4-3 The mass s p e c t r o g r a p h i c a n a l y s i s o f f o u r samples , 151-2 A4.1 The GaSb c o n s t a n t s 154-5 A6.1 The measured h o l e e f f e c t i v e mass v a l u e s . . . . . . 157 v i i i L IST OF FIGURES F i g u r e Page 2.1 B l o c k d i a g r a m o f t h e microwave s p e c t r o m e t e r 6 2.2 The o p t i c a l system 9 2.3 P h a s o r diagram f o r t h e t o t a l microwave e l e c t r i c f i e l d a t t h e d e t e c t o r . I d e a l l y ft = = 90° Ik Z.k The p r o d u c t i o n o f c i r c u l a r p o l a r i z a t i o n i n t h e microwave system 20 2.5 C r o s s s e c t i o n o f t h e c i r c u l a r p o l a r i z a t i o n c a v i t y a s sembly-a c y l i n d r i c a l TE111 mode 21 3.1 T h e o r e t i c a l c u r v e s f o r a b s o r p t i o n and d i s p e r s i o n . 30 The l i n e a r p o l a r i z a t i o n c u r v e i s e q u i v a l e n t t o t h e sum o f t h e two c i r c u l a r p o l a r i z a t i o n components 30 3.2 The energy band s t r u c t u r e o f GaSb 36 3.3 The s e p a r a t i o n o f s i g n a l and background kZ 3.k The e f f e c t o f d i s p e r s i o n on t h e s i g n a l . . . . kk 3.5 The c i r c u l a r p o l a r i z a t i o n s i g n a l . . ., k6 3.6 The w a v e l e n g t h dependence of (a) t h e s i g n a l and background h e i g h t s , and (b) t h e p u b l i s h e d a b s o r p t i o n c o e f f i c i e n t , a 53 3.7 The w a v e l e n g t h dependence o f t h e peak parameter B Q f o r two samples, (a) 350/t t h i c k , (b) 100/u- t h i c k $k 3.8 The w a v e l e n g t h dependence o f (a) u / T , and (b) t h e s i g n a l h e i g h t , f o r t h e B a t t e l l e , sample.. 55 3.9 The power dependence o f t h e o b s e r v e d s i g n a l h e i g h t , S 58 i x F i g u r e Page 3 . 1 0 The power dependence o f t h e s i g n a l . i n t e n s i t y 60 3 . 1 1 The power dependence o f t h e r a t i o o f h o l e - t o -e l e c t r o n c o n c e n t r a t i o n s 61 3 . 1 2 The c a r r i e r c o n c e n t r a t i o n as a f u n c t i o n o f l i g h t i n t e n s i t y 63 3 . 1 3 The plasma s h i f t f o r a s i g n a l c o n t a i n i n g "both b u l k and s u r f a c e c a r r i e r s 64 4 . 1 (a) The o b s e r v e d s i g n a l a t h i g h e r powers....... 66 (b) The o b s e r v e d s i g n a l a t l o w e r powers 67 4 . 2 The power dependence o f U / T 69 4 . 3 The l i g h t i n t e n s i t y dependence o f u / r ......... 71 4 . 4 The r e l a t i v e s c a t t e r i n g r a t e v s . a c c e p t o r c o n c e n t r a t i o n 75 4 . 5 The t e m p e r a t u r e dependence o f usf f o r two samples 94 4 . 6 The t e m p e r a t u r e dependence 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 . 96 6 . 1 The a n g u l a r dependence o f t h e e f f e c t i v e mass... 114 X ACKNOWLEDGEMENTS I w i s h t o thank Dr. C. F. S c h w e r d t f e g e r , my r e s e a r c h s u p e r v i s o r , f o r h i s e n t h u s i a s t i c s u p p o r t o f t h e work, f o r i n s t r u c t i o n and t r a i n i n g i n e x p e r i m e n t a l p h y s i c s and f o r h i s p a t i e n c e t h r o u g h o u t t h e d u r a t i o n o f t h e r e s e a r c h . U s e f u l d i s c u s s i o n s w i t h Dr. R. B a r r i e , Dr. J . W. B i c h a r d , and Dr. R, R. Parsons a r e g r e a t l y a p p r e c i a t e d . I n p a r t i c u l a r , I w i s h t o acknowledge Dr. B a r r i e f o r p r o v i d i n g an example o f th e h i g h e s t i n t e l l e c t u a l s t a n d a r d s , I am p a r t i c u l a r l y g r a t e f u l t o my w i f e , Joan, f o r t y p i n g t h e t h e s i s , d o i n g t h e d r a w i n g s , p r o v i d i n g o t h e r sub-s t a n t i a l h e l p i n t h e t h e s i s , and f o r h e r p a t i e n c e , as a f e l l o w s c i e n t i s t , i n u n d e r s t a n d i n g t h e f r u s t r a t i o n s o f r e s e a r c h . F o r t h e c o n s t r u c t i o n o f much o f t h e a p p a r a t u s I am g r a t e f u l t o t h e men o f t h e P h y s i c s Shop under t h e a b l e s u p e r v i s i o n o f Mr. A l e c F r a s e r . I n p a r t i c u l a r , t h e c o n s t r u c t i o n o f numerous microwave c a v i t i e s by Mr. P e t e r Haas i s a p p r e c i a t e d . I a l s o w i s h t o thank Mr. Ro b i n H a l l i w e l l f o r h i s e x p e r i m e n t a l h e l p and i d e a s on s e v e r a l o c c a s i o n s . Two o f t h e samples used i n t h e e x p e r i m e n t s were g e n e r o u s l y donated by f e l l o w r e s e a r c h w o r k e r s , I would l i k e t o thank Dr, A. C. Beer o f t h e B a t t e l l e M e m o r i a l I n s t i t u t e , Columbus, Ohio, f o r t h e " B a t t e l l e " sample. I am a l s o g r a t e f u l t o Dr. L. Gouskov o f t h e F a c u l t e des S c i e n c e s de M o n t p e l l i e r , ( C e n t r e d'Etudes d ' E l e c t r o n i q u e des S o l i d e s ) F r a n c e , f o r s u p p l y i n g s e v e r a l " M o n t p e l l i e r " samples t o me t h r o u g h x i Mr. B r a d R i c k a r d s o f t h i s l a b o r a t o r y . The g r e a t l y a p p r e c i a t e d f i n a n c i a l s u p p o r t f o r t h i s work came from a 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 B u r s a r y , t h e H. R. M a c m i l l a n F a m i l y F e l l o w s h i p a t U. B. C , t h e Canadian Kodak F e l l o w s h i p , and Dr. S c h w e r d t f e g e r 1 s r e s e a r c h g r a n t . The t h e s i s r e s e a r c h was s u p p o r t e d by a 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 t o Dr. S c h w e r d t f e g e r (NRC A-2228) and by a Defence R e s e a r c h Board g r a n t t o Dr. B a r r i e (DRB 9510-35). 1 CHAPTER 1 INTRODUCTION Some books are t o be t a s t e d , o t h e r s t o be swallowed, and some few t o be chewed and d i g e s t e d . F r a n c i s Bacon. 1.1 G e n e r a l I n t r o d u c t i o n The c i r c u l a r p a t h o f a charged p a r t i c l e moving i n a p l a n e p e r p e n d i c u l a r t o a s t e a d y u n i f o r m m a g n e t i c f i e l d , B, i s c a l l e d c y c l o t r o n m o t i o n . The a n g u l a r f r e q u e n c y o f t h e eB m o t i o n i s afc = — where e i s t h e p a r t i c l e e l e c t r i c c h a rge, B = |B| and m i s t h e mass o f t h e p a r t i c l e ' . I f an a.c. e l e c t r i c f i e l d o f a n g u l a r f r e q u e n c y us i s a p p l i e d t o t h e p a r t i c l e i n t h e p l a n e o f t h e mo t i o n , a r e s o n a n t power a b s o r p t i o n may o c c u r when USC » u r . The c o n d i t i o n f o r t h e r e s o n a n t e f f e c t i s t h a t t h e p a r t i c l e t r a v e l an a p p r e c i a b l e d i s t a n c e around t h e c i r c l e b e f o r e b e i n g s c a t t e r e d by a c o l l i s i o n . T h i s c o n d i t i o n i s ufCjl where T i s t h e mean t i m e between c o l l i s i o n s . When t h i s e x p e r i m e n t i s performed i n a s e m i c o n d u c t o r t h e r e s o n a n c e c o n d i t i o n cVc = us p e r m i t s t h e d e t e r m i n a t i o n o f t h e c a r r i e r e f f e c t i v e mass i n t h e m a t e r i a l , m = m*. S i n c e t h i s r e q u i r e s o n l y a measurement o f a f r e q u e n c y us and a m a g n e t i c f i e l d B, t h i s i s a v e r y d i r e c t method o f mea s u r i n g t h e e f f e c t i v e mass. I t i s u s u a l l y t h e most p r e c i s e as w e l l . The c a r r i e r e f f e c t i v e mass f o r a n i s o t r o p i c m a t e r i a l s may a l s o be d e t e r m i n e d f o r any d i r e c t i o n i n t h e c r y s t a l . The v a l u e 2 o f t h e e f f e c t i v e mass i s v e r y u s e f u l i n t h e c a l c u l a t i o n s i n v o l v e d i n many s o l i d s t a t e p h y s i c s e x p e r i m e n t s , as w e l l as t o t e s t band s t r u c t u r e t h e o r i e s . The c y c l o t r o n resonance e x p e r i m e n t was f i r s t p e r f o rmed a t microwave f r e q u e n c i e s i n 1955. """* I n o r d e r t o measure t h e e f f e c t i v e mass i n d i f f e r e n t m a t e r i a l s , i . e . t o s a t i s f y t h e c o n d i t i o n U/T>1, t h e o p e r a t i n g f r e q u e n c y , us , has been e x t e n d e d t o t h e i n f r a - r e d r e g i o n o f t h e spectrum. S i n c e t h i s measurement d e t e r m i n e s t h e e f f e c t i v e mass a t a d i f f e r e n t e n e r g y i n t h e c o n d u c t i o n band, i t can be used t o complement t h e microwave measurement when t h e e f f e c t i v e mass depends on energy. The main d i f f i c u l t y i n p e r f o r m i n g t h e exp e r i m e n t a t i n f r a - r e d f r e q u e n c i e s i s t h e l a r g e m a g n e t i c f i e l d r e q u i r e d t o s a t i s f y t h e res o n a n c e c o n d i t i o n . The microwave measurement o f LUX as a f u n c t i o n o f t e m p e r a t u r e , microwave f r e q u e n c y and i m p u r i t y c o n c e n t r a t i o n has been used e x t e n s i v e l y t o s t u d y s c a t t e r i n g mechanisms. G a l l i u m a n t i m o n i d e has been s t u d i e d by many 2 e x p e r i m e n t a l t e c h n i q u e s . * The band s t r u c t u r e i s w e l l known and t h e e f f e c t i v e masses o f t h e l i g h t and heavy h o l e s have been measured by t h e c y c l o t r o n r esonance t e c h n i q u e . The " p u r i t y " o f GaSb i s l i m i t e d by t h e c o n c e n t r a t i o n o f 17 -3 ~ l - 2 x 10 ' cm o f r e s i d u a l s t o i c h i o m e t r i c d e f e c t a c c e p t o r s . F o r r e l a t i v e l y pure m a t e r i a l s w i t h i m p u r i t y c o n t e n t l e s s t h a n 1. The b e s t g e n e r a l r e f e r e n c e on c y c l o t r o n r e s o n a n c e , i n c l u d i n g t h e h i s t o r y o f t h e e x p e r i m e n t i s Lax and M a v r o i d e s (i960). 2. A summary o f some o f t h e GaSb l i t e r a t u r e i s p r e s e n t e d i n s e c t i o n 3.3. 3 10 ' cm"-^  t h e samples a r e always p - t y p e . The e l e c t r o n e f f e c t i v e mass has not p r e v i o u s l y been d e t e r -mined by t h e c y c l o t r o n r e s o n a n c e t e c h n i q u e . 1.2 The M o t i v a t i o n f o r t h i s E x p e r i m e n t The p r i m a r y purpose o f t h i s t h e s i s i s t o measure t h e e f f e c t i v e mass 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 h i s has been done by s t u d y i n g t h e e l e c t r o n c y c l o t r o n resonance s i g n a l i n GaSb, t h e o b s e r v a t i o n o f w h i c h i s r e p o r t e d f o r t h e f i r s t t i m e . The a n g u l a r dependence o f t h e e f f e c t i v e mass has a l s o been measured. A n o t h e r i m p o r t a n t purpose i s t o measure th e s c a t t e r i n g o f e l e c t r o n s from the n e u t r a l d e f e c t r e s i d u a l a c c e p t o r s . T h i s measurement i s o n l y p o s s i b l e f o r p h o t o c r e a t e d e l e c t r o n s (see s e c t i o n *+.l) and i s b e s t done by t h e c y c l o t r o n resonance t e c h n i q u e . O t h e r a s p e c t s o f t h i s s t u d y i n c l u d e : ( i ) t h e o b s e r v a t i o n o f a new s u r f a c e e f f e c t w h i c h may be i m p o r t a n t f o r o t h e r c y c l o t r o n resonance e x p e r i m e n t s . ( i i ) t h e c o n f i r m a t i o n o f t h e t h e o r y o f background plasma e f f e c t s . ( i i i ) t h e s t u d y o f hot e l e c t r o n e f f e c t s . 4 1.3 T h e s i s O u t l i n e A d e s c r i p t i o n o f t h e microwave s p e c t r o m e t e r i s g i v e n i n C h a p t e r 2 . I n C h a p t e r 3 t h e t h e o r y and o b s e r v a t i o n o f t h e c y c l o t r o n r esonance s i g n a l a r e p r e s e n t e d . The t h e o r y r e l a t e s t h e measured peak p o s i t i o n and h a l f w i d t h t o t h e e f f e c t i v e mass and s c a t t e r i n g t i m e , r e s p e c t i v e l y . The p o s s i b i l i t y o f plasma e f f e c t s i s a l s o d i s c u s s e d t h e o r e t i c a l l y . The obse r v e d s i g n a l i s t h e n p r e s e n t e d and many a s p e c t s o f i t are e x p l a i n e d and i n t e r p r e t e d . I n t h e n e x t c h a p t e r , t h e s c a t t e r i n g mechanism i s d i s c u s s e d . Measurements o f t h e s c a t t e r i n g r a t e as a f u n c t i o n o f power, t e m p e r a t u r e , and a c c e p t o r c o n c e n t r a t i o n a r e p r e s e n t e d . These, i n c o n j u n c t i o n w i t h t h e t h e o r e t i c a l c a l c u l a t i o n s a l s o p r e s e n t e d , i n d i c a t e t h a t a t low t e m p e r a t u r e s t h e s c a t t e r i n g p r o c e s s i s b e s t u n d e r s t o o d as b e i n g p r i n c i p a l l y due t o t h e s c a t t e r i n g o f hot e l e c t r o n s from n e u t r a l a c c e p t o r s . The e v i d e n c e f o r a hot e l e c t r o n e n e rgy d i s t r i b u t i o n , as w e l l as c a l c u l a t i o n s based on i t , a r e p r e s e n t e d i n C h a p t e r 5 . C h a p t e r 6 c o n t a i n s t h e measurements o f t h e e f f e c t i v e mass. C o r r e c t i o n s f o r p o l a r o n e f f e c t s and t h e c o n d u c t i o n band n o n - p a r a b o l i c i t y a r e a p p l i e d t o deduce t h e band-edge b a r e e l e c t r o n e f f e c t i v e mass. I n t h e f i n a l c h a p t e r , c o n c l u s i o n s and s u g g e s t i o n s f o r f u r t h e r e x p e r i m e n t s a r e p r e s e n t e d . The a p p e n d i c e s c o n t a i n f u r t h e r d a t a , some i m p o r t a n t c a l c u l a t i o n s and o t h e r i n f o r m a t i o n used i n t h e t h e s i s . 5 CHAPTER 2 THE MICROWAVE SPECTROMETER O f t e n one's f a i t h i n cause and e f f e c t i s put t o a s e v e r e t e s t by a p p a r a t u s t r o u b l e s , and t h e weak are l i k e l y t o f a l l back on t h e i r a n c e s t r a l i n h e r i t a n c e o f f a i t h i n g r e m l i n s and p s y c h i c phenomena, from "An I n t r o d u c t i o n t o S c i e n t i f i c R e s e a r c h " by E . B r i g h t W i l s o n , J r . 2.1 The S p e c t r o m e t e r D e s i g n The microwave s p e c t r o m e t e r has been c a r e f u l l y d e s i g n e d t o meet t h e s p e c i a l r e q u i r e m e n t s o f t h i s e x p e r i m e n t b u t i s , a t t h e same t i m e , v e r y v e r s a t i l e . W ith t h i s i n s t r u m e n t i t i s p o s s i b l e t o ; ( i ) change samples a t h e l i u m t e m p e r a t u r e s , ( i i ) r o t a t e t h e sample 360° and t h e magnet 180° w i t h r e s p e c t t o t h e c a v i t y , ( f o r t h e a n g u l a r dependence o f t h e e f f e c t i v e mass) ( i i i ) measure t h e s i g n a l i n t h e t e m p e r a t u r e range 1-^0K, ( i v ) d e t e r m i n e t h e c a r r i e r s i g n i n c y c l o t r o n resonance o r t h e s i g n o f t h e g - f a c t o r i n e l e c t r o n s p i n r e s o n a n c e (ESR). The b l o c k diagram o f t h e s p e c t r o m e t e r i s shown i n f i g u r e 2.1. I t i s a d o u b l e arm homodyne d e t e c t i o n system. The "Ka-band" (26-*K) GHz) f r e q u e n c y band i s chosen t o s a t i s f y t h e c o n d i t i o n U/T>1. A 6 dB d i r e c t i o n a l c o u p l e r i s used t o p r o v i d e d e t e c t o r b i a s power i n s t e a d o f t h e u s u a l magic t e e i n o r d e r t o i n c r e a s e t h e amount of k l y s t r o n power a v a i l a b l e a t t h e c a v i t y . Some o f t h e measurements r e q u i r e t h e maximum p o s s i b l e power. A c i r c u l a t o r i s a l s o used i n s t e a d o f a magic t e e t o t r a n s m i t a l l t h e " c a v i t y arm" power t o t h e c a v i t y and a l s o t o a l l o w a l l o f t h e s i g n a l from t h e c a v i t y t o r e a c h t h e d e t e c t o r f o r maximum s e n s i t i v i t y . W i t h t h e 20 dB cross-arm, c o u p l e r O s c i l l o -scope Mode D i s p l a y D e t e c t o r C u r r e n t Meter A u t o m a t i c Frequency C o n t r o l (A.F.C.) ON F i g u r e 2.1 B l o c k diagram o f t h e microwave s p e c t r o m e t e r . 7 reference arm one can vary the power to the ca v i t y while keeping the detector current constant. Operated i n t h i s manner one can deduce the power dependence of the cyclotron resonance s i g n a l as seen i n section 2.2. The dewar top i s constructed with two p a r a l l e l waveguides running through i t down to a flange about 10" from the magnet center. In the method chosen to produce the c i r c u l a r p o l a r i z a t i o n i t i s necessary to transmit power past the cavi t y to the detector and t h i s i s accom-plished with the two waveguides. When the spectrometer i s operated with one of the other three c a v i t i e s (see section 2.4) only one waveguide i s used. The waveguides are constructed of t h i n walled s t a i n l e s s s t e e l material to minimize "heat leak" to the ca v i t y from the room temperature dewar top. The sample i s positioned near the t i p of a fused quartz " l i g h t -pipe" which enters the dewar through an 0-ring seal between the two waveguides. In the case of one cavit y the sample s i t s i n a v e r t i c a l t e f l o n holder on the li g h t - p i p e and can be removed and changed at helium temperatures. The spectrometer i s almost always operated with l i q u i d helium i n the ca v i t y but pumped to a temperature below the X point to avoid the frequency changes and noise owing to helium bubbles i n the cavity. These problems are e s p e c i a l l y large at t h i s high frequency, and therefore, small cavi t y volume. To change samples one simply brings the vapour pressure above the helium bath to atmospheric pressure by opening the return l i n e , changes sample and pumps out the helium vapour again. With t h i s method the sample can be 8 changed b e f o r e t h e l i q u i d h e l i u m warms up t o t h e X p o i n t . Hence i t can be pumped down a g a i n q u i c k l y ( r e a c h i n g e q u i l i b r i u m i n about 15 m i n u t e s ) and v e r y l i t t l e l i q u i d h e l i u m i s l o s t i n t h e p r o c e s s . Up t o f o u r samples can be b r i e f l y measured w i t h one l i t r e o f l i q u i d h e l i u m . The microwave d e t e c t o r i s a 1N53D S y l v a n i a p o i n t c o n t a c t d i o d e . The microwave s o u r c e i s an OKI model 35V10 r e f l e x k l y s t r o n g e n e r a t i n g about 50 mW o f power. I t i s f r e q u e n c y s t a b i l i z e d by an 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 c i r c u i t (AFC) d e s c r i b e d by S l a g s v o l d (1966). A Magnion model HS - I365B magnet w i t h a 2 1/2" p o l e gap s u p p l i e s t h e magn e t i c f i e l d , t h e maximum f i e l d b e i n g 15.6 KG. A p a i r o f pancake c o i l s mounted on t h e p o l e f a c e s g i v e s up to 30 G m o d u l a t i o n a t 400 Hz. f o r t h e purpose o f ESR t e s t s o f t h e equipment. A r o t a t i n g c o i l gaussmeter i s used t o measure t h e f i e l d w i t h measurement e r r o r s owing p r i m a r i l y t o t h e l o n g term c a l i b r a t i o n d r i f t o f ± 0 . 1 5 $ . F r e q u e n c y measurements a r e made w i t h a M i c r o l i n e model 35 c a v i t y f r e q u e n c y meter. I t i s c a l i b r a t e d by t h e f r e q u e n c y t r i p l i n g t e c h n i q u e o f S l a g s v o l d (1966) and has an e r r o r o f 0.05%. The c y c l o t r o n r esonance c a r r i e r s a r e g e n e r a t e d by band-gap i r r a d i a t i o n o f t h e sample. The o p t i c a l system used i s shown i n f i g u r e 2.2. A s e p a r a t e t a b l e i s used f o r t h e o p t i c a l system t o p r e v e n t v i b r a t i o n s f rom t h e chopper and f a n r e a c h i n g t h e microwave a p p a r a t u s . The l i g h t i s chopped a t a f r e q u e n c y o f 270 Hz t o g a i n s e n s i t i v i t y by t h e use o f l o c k - i n a m p l i f i e r s i g n a l d e t e c t i o n . Lower f r e q u e n c i e s would t a k e t h e e x p e r i m e n t below t h e low f r e q u e n c y c u t -o f f l i m i t f o r t h e p r e a m p l i f i e r . A l l measurements O p t i c a l Bench on T a b l e #1 ( w i t h microwave s p e c t r o m e t e r ) Low F r e q . ( I . R . ) Pass F i l t e r S h u t t e r > <r A d j u s t a b l e M i r r o r t o p o s i t i o n t h e f o c u s e d l i g h t on t i p o f q u a r t z l i g h t - p i p e Lens A p e r t u r e N e u t r a l D e n s i t y F i l t e r F i g u r e 2.2 The o p t i c a l s y s t e m . Removable M i r r o r Monochromator 11 Mercury Lamp & Housing R e f e r e n c e Gem r a t o r T ungsten L i g h t Source t o "Ref." on L o c k - i n A m p l i f i e r T a b l e #2 (Not t o u c h i n g T a b l e #1) 10 d e s i g n e d t o s e p a r a t e b u l k and s u r f a c e e f f e c t s were done w i t h t h e t u n g s t e n l i g h t s o u r c e t o make use o f i t s approx-i m a t e l y w a v e l e n g t h independent o u t p u t . The mercury s o u r c e has b r o a d e m i s s i o n l i n e s superimposed on a f l a t back-ground. ( G a g l a r d i , 1972) 2.2 S p e c t r o m e t e r O p e r a t i n g C o n d i t i o n s The t h e o r y f o r t h e o b s e r v e d s i g n a l f o r a s p e c t r o m e t e r d e t e c t i n g pure a b s o r p t i o n i s d e r i v e d more f u l l y i n Appendix 2.1. I t can e a s i l y be g e n e r a l i z e d t o i n c l u d e t h e case o f an a d m i x t u r e o f d i s p e r s i o n (see s e c t i o n 2,3). The r e s u l t g i v e s , d r a Q 0 ( l - f 2 ) o(P 0)Ti where, p = c a v i t y r e f l e c t i o n c o e f f i c i e n t d T = p h o t o i n d u c e d change i n P P o microwave power i n t h e c a v i t y c a v i t y "unloaded" Q a microwave p h o t o c o n d u c t i v i t y e f f e c t i v e f i l l i n g f a c t o r . The o b s e r v e d s i g n a l v o l t a g e i s , 6 v . s i g a where, s l o p e o f c r y s t a l d e t e c t o r law a t t h e o p e r a t i n g p o i n t ( o ) . r.m.s. e l e c t r i c f i e l d a t t h e sample p o s i t i o n i n t h e c a v i t y , b u t i n t h e absence o f t h e sample. 11 cS E . = t h e change i n e l e c t r i c f i e l d r e f l e c t e d from s i g t h e c a v i t y due t o sample p h o t o c o n d u c t i v i t y . I f r i s k e p t c o n s t a n t , P Q a E Q . T h e r e f o r e , * v . i « a n % <-Hv<p„> p o / 2 d-rt To o p e r a t e a t maximum s p e c t r o m e t e r s e n s i t i v i t y we o p t i m i z e t h e r a t i o o f s i g n a l v o l t a g e t o n o i s e v o l t a g e . I n p r a c t i c e t h i s means a d j u s t i n g most o f t h e v a r i a b l e s t o o p t i m i z e t h e s i g n a l v o l t a g e . T h e r e f o r e we o p e r a t e w i t h ? ( i ) P = 0 , t h e c a v i t y i s c r i t i c a l l y c o u p l e d , ( i i ) Q Q = maximum, i . e . h i g h c o n d u c t i v i t y copper c a v i t i e s a r e used where p o s s i b l e , ( i i i ) maximum power u n l e s s a ( P ) d e c r e a s e s more r a p i d l y -1/2 ° t h a n Prt as PN i s i n c r e a s e d . F o r ESR where a ( P ) o o o i s r e p l a c e d by "X"(P ) and t h e l a t t e r shows s a t u r a t i o n , t h i s may be i m p o r t a n t , ( i v ) maximum (-^-T?) . The g a i n i n s e n s i t i v i t y from a & o i n c r e a s i n g t h i s f a c t o r i s l i m i t e d by t h e a p p r o x i m a t e l y f l a t c r y s t a l c u r r e n t dependence o f t h e r a t i o o f s i g n a l and n o i s e v o l t a g e s above ~ 30 jj.k found b o t h i n t h i s l a b o r a t o r y ( S l a g s v o l d , 1966) and by I s h i i and B r a u l t ( 1 9 6 2 ) . An o p e r a t i n g c u r r e n t o f 100JJLK was u s u a l l y chosen because t h e e x t r a c u r r e n t improves t h e performance o f t h e AFC. (v) maximum . T h i s c o n d i t i o n i s s u b j e c t t o t h e r e s t r i c t i o n t h a t Q Q n o t be re d u c e d by t h e sample l o s s e s t o l e s s t h a n h a l f i t s v a l u e i n t h e absence o f t h e sam p l e . 1 . The v a l u e o f n\ i s a f f e c t e d by b o t h sample p o s i t i o n and shape as d i s c u s s e d i n t h e n e x t two p a r a g r a p h s . 1. T h i s i s shown i n Po o l e ( 1 9 6 7 ) . 12 The sample p o s i t i o n i s t h e r e s u l t o f a compromise "between two f a c t o r s . On t h e one hand t h e sample s h o u l d be as n e a r as p o s s i b l e t o t h e maximum e l e c t r i c f i e l d , t h a t i s , n e a r t h e c a v i t y c e n t e r . I t s h o u l d a l s o be n e a r t h e t i p o f t h e l i g h t - p i p e s i n c e t h e l i g h t e m i t t e d f rom t h e l i g h t - p i p e d i v e r g e s q u i c k l y . But i f t h e l i g h t - p i p e i s i n s e r t e d t o o f a r i n t o t h e c a v i t y i t s d i e l e c t r i c l o s s e s l o w e r Q Q. T h e r e f o r e , compromise p o s i t i o n s a r e r e a c h e d f o r b o t h t h e sample and l i g h t - p i p e . The optimum sample shape i s a l a r g e - a r e a t h i n sample. The l a r g e a r e a i s r e q u i r e d because, o* a ( c a r r i e r c o n c e n t r a t i o n ) a ( l i g h t absorbed) a (sample a r e a i l l u m i n a t e d ) . T h i s assumes t h e t i p o f t h e l i g h t - p i p e cannot be made s m a l l e r t h a n , say, 1 mm d i a m e t e r . I t cannot be t o o l a r g e , e i t h e r , o r t h e l i g h t - p i p e h o l e i n t h e c a v i t y w a l l would be s u f f i c i e n t l y l a r g e t o t r a n s m i t much o f t h e power out o f t h e c a v i t y l i k e a waveguide. The sample s h o u l d be t h i n , on t h e o t h e r hand, t o reduce t h e sample volume and hence dark c o n d u c t i v i t y l o s s e s . The sample t h i c k n e s s was k e p t g r e a t e r t h a n 0 . 2 5 mm t o ensure a b u l k measurement o f t h e c y c l o t r o n r e s onance s i g n a l , but l e s s t h a n 0.4 mm where d a r k c o n d u c t i v i t y l o s s e s became a p p r e c i a b l e f o r some o f t h e samples. A t y p i c a l sample was o f d i m e n s i o n s 1 . 0 x 1 . 4 x 0 . 3 utan? . The t h e o r e t i c a l P^ dependence o f t h e f o r m u l a V2 f V . a a ( P ) P was t e s t e d f o r t h i s a p p a r a t u s i n t h e ° s i g v 0' 0 ^ f o l l o w i n g manner. The f o r m u l a i s e q u i v a l e n t t o t h e E S R 13 formula &YSip- A *X ' ( P 0 ) P o ' R o o m " f c e m P e r a " t u r e 2 , measurements of the ESR s i g n a l i n a t e s t sample of DPPH were performed. The sample showed no s a t u r a t i o n . 0.48+.05 (X ( p 0 ) = constant) and the r e s u l t was °vsj_g cc P -2 « 3 The Problem of D i s p e r s i o n As shown below, the measured s i g n a l i s p r o p o r t i o n a l to a R + pcjj , where a R and are the r e a l and imaginary p a r t s of the complex c o n d u c t i v i t y and 3 i s a constant.. 0 R and Oj are c a l l e d the ab s o r p t i o n and d i s p e r s i o n , and 0 w i l l be c a l l e d the d i s p e r s i o n parameter. The peak p o s i t i o n » f o r the measured curve, B„ , d i f f e r s from the d e s i r e d o q u a n t i t y , B Q, the peak p o s i t i o n of the pure a b s o r p t i o n o"R. B - B The r e l a t i v e peak s h i f t - — — i s approximately B o i n v e r s e l y p r o p o r t i o n a l to the Q of the ab s o r p t i o n l i n e (Appendix 3.1). S l a g s v o l d ( I 9 6 6 ) observed a p a r t l y d i s p e r s i v e ESR s i g n a l ||3| 4> °« 2 a n d showed t h a t f o r h i s high Q ESR l i n e s the r e l a t i v e peak s h i f t had n e g l i g i b l e e f f e c t on_the measured g-value. I n t h i s experiment we u s u a l l y have |g| ^ 0 . 2 but the Q of the l i n e i s only ~ 2 or 3 . The r e s u l t i n g r e l a t i v e peak s h i f t can be as high as 5% (see s e c t i o n 3 . 4 ( b ) ) and a c o r r e c t i o n must be made. The phasor diagram f o r the measured s i g n a l i s shown i n f i g u r e 2 . 3 . Both the e l e c t r i c f i e l d r e f l e c t e d from the c a v i t y and th a t from the b i a s arm have phase angles 1. "X i s the imaginary part of the complex magnetic s u s c e p t i b i l i t y . 2. DPPH i s d i - p h e n y l - p i c r y l - h y d r a z i l , r e c r y s t a l l i z e d from benzene. 1 4 F i g u r e 2 . 3 P h a s o r diagram f o r t h e t o t a l microwave e l e c t r i c f i e l d a t t h e d e t e c t o r . I d e a l l y 0=0'= 9 0 0 d i f f e r e n t from the i d e a l case = 0 = 90°. The observed s i g n a l i s proportional to SE . . s i g <Kig - <*Eabs c o s ^ + <*Edisp s i n < * = coscT ( J E a b s + P c ^ E d i s p ) a cos 6 (<rR + pa ) where & i s shown on the diagram and 8 = tan<£ . An upper l i m i t on the value of 8 i s |8| 4 0.3 which r e s u l t s i n j& | 4 1?° and cos S > 0.95. The e f f e c t of the cosine term on absolute s i g n a l heights was ignored since |8| £ 0.2 usually. There are several sources of dispersion i n the system. A l l the Ka-band microwave components cause appreciable r e f l e c t i o n s . The c i r c u l a t o r has a voltage standing wave r a t i o (VSWR) of 1.3 with matched loads (of VSWR < 1.05) on the ports. Also, a standing wave pattern i s set up between the c i r c u l a t o r and the cavity. The e f f e c t of the r e f l e c t i o n s involved i s therefore frequency dependent. The apparent cav i t y frequency as seen on the mode disp l a y i s not the true frequency and the apparent condition " c r i t i c a l l y coupled" may also not be accurate. The p r e c i s i o n c a l i b r a t e d attenuator produces a large phase s h i f t i n the range of the f i r s t 6 dB of attenuation. An adjustment of the phase shifter compensates f o r t h i s . The frequency dependence of the r e f l e c t i o n s enters i n another way. When a rectangular sample i s rotated i n the cavity the cavity frequency s h i f t s . This i s e s s e n t i a l l y due to the changes i n the average d i e l e c t r i c f i l l i n g factor when the e l e c t r i c f i e l d i s changed from a p o s i t i o n p a r a l l e l 16 t o t h e sample l e n g t h t o one p e r p e n d i c u l a r . T h i s frequency-s h i f t changes t h e r e f l e c t i o n s w h i c h change 8 and t h e r e f o r e t h e measured peak p o s i t i o n . An attempt t o measure t h e a n g u l a r dependence o f t h e e f f e c t i v e mass by r o t a t i n g t h e sample w i t h o u t c o r r e c t i o n s f o r d i s p e r s i o n , r e s u l t s i n a smooth v a r i a t i o n o f due t o d i s p e r s i o n . Because t h i s i s a l a r g e e f f e c t t h e a n g u l a r dependence o f t h e e f f e c t i v e mass was measured by r o t a t i n g t h e mag n e t i c f i e l d , n o t t h e sample, i n a d d i t i o n t o a p p l y i n g d i s p e r s i o n c o r r e c t i o n s . O t h e r s o u r c e s o f d i s p e r s i o n i n c l u d e AFC o f f s e t e r r o r (due t o k l y s t r o n f r e q u e n c y d r i f t ) , and improper c a v i t y c e n t e r i n g on t h e k l y s t r o n mode o r t h e use o f an asymmetric mode. A l t h o u g h t h e r e i s no easy way t o a c c u r a t e l y measure t h e phase a n g l e , £ , i t can be a d j u s t e d t o be a p p r o x i m a t e l y zero by w a t c h i n g t h e mode d i s p l a y . T h i s r e s u l t s i n |B| < 0 . 2 and a c o r r e c t i o n f o r t h e r e m a i n i n g d i s p e r s i o n i s made i n t h e a n a l y s i s o f t h e s i g n a l shape. One o t h e r problem a f f e c t s t h e s i g n a l shape and h e i g h t . W i t h l i q u i d h e l i u m i n t h e waveguide l a r g e o s c i l l a t i o n s ( ^ JOfo) and a s t e a d y d r i f t i n t h e d e t e c t o r c u r r e n t a r e measured as t h e h e l i u m l e v e l d e c r e a s e s . The o s c i l l a t i o n p e r i o d has been shown t o be e q u a l t o t h e ti m e f o r t h e h e l i u m l e v e l t o f a l l e x a c t l y one h a l f g u i d e -w a v e l e n g t h , A s i m p l e impedance mismatch c a l c u l a t i o n u s i n g a h e l i u m d i e l e c t r i c c o n s t a n t o f 1 . 0 2 5 ( T a b l e s , 1953) cannot p r e d i c t s u c h a l a r g e o s c i l l a t i o n . I t was c o n c l u d e d t h a t t h e r e must be a c o m p l i c a t e d s t a n d i n g wave p a t t e r n i n v o l v e d . 17 S i n c e t h e o s c i l l a t i o n p e r i o d o f ~ 3 minutes i s comparable t o t h e t i m e t o measure one s i g n a l t h i s r e s u l t e d i n s i g n a l d i s t o r t i o n . The waveguide was f i l l e d w i t h p o l y u r e t h a n e t o e x c l u d e t h e l i q u i d h e l i u m . T h i s i n t r o d u c e s an a d d i t i o n a l l o s s o f 1.2 dB. The l o s s e s i n t h e s t a i n l e s s s t e e l waveguide g o i n g t o t h e c a v i t y a r e 1.9 dB and i n t h e c i r c u l a t o r , 0.4 dB, I n a l l , t h e power r e a c h i n g t h e c a v i t y i s reduced 3.5 dB o r a f a c t o r o f 2.2. S i n c e o n l y t h r e e - q u a r t e r s o f t h e 50 mW o u t p u t from t h e k l y s t r o n e n t e r s t h e c a v i t y arm, t h e a t t e n u a t i o n f a c t o r o f 2.2 r e s u l t s i n . a maximum power o f 17 mW a t the c a v i t y . 2.4 The C a v i t i e s The f e a t u r e s o f t h e f o u r microwave c a v i t i e s used a r e t a b u l a t e d i n T a b l e 2.1. Whether t h e e l e c t r i c f i e l d n e a r t h e sample f a c e i s p a r a l l e l o r p e r p e n d i c u l a r t o t h e s u r f a c e a f f e c t s t h e r a t i o o f the, i n t e r n a l t o e x t e r n a l e l e c t r i c f i e l d s , ( s e e s e c t i o n 3.4(e)) The maximum i n t e r n a l e l e c t r i c f i e l d i s a c h i e v e d i n c a v i t y #1. A l l c a v i t i e s o p e r a t e n e a r a f r e q u e n c y o f 3^ .6" GHz a t t h e t e m p e r a t u r e o f t h e pumped l i q u i d h e l i u m , 1.6 K. The v a r i a b l e c o u p l e r i s d e s i g n e d a f t e r Gordon (1961). T h i s c o u p l e r c r e a t e s an u n d e s i r a b l e second r e s o n a n t c a v i t y between t h e t e f l o n p l u n g e r and t h e c a v i t y i r i s w i t h a " c o u p l i n g dependent" r e s o n a n t f r e q u e n c y . S i n c e t h i s i s n o t c l o s e t o t h e o p e r a t i n g f r e q u e n c y i t i s not a s e r i o u s p r o b l e m . The c o u p l e r does, however, cause a s m a l l a p parent C a v i t y C a v i t y Number Purpose M a t e r i a l Q, Mode V a r i a b l e C o u p l i n g Sample Change and R o t a t i o n E II o r ± t o Sample G e n e r a l Purpose Copper 4000 C y l i n d r i c a l T E111 yes yes II V a r i a b l e C o u p l i n g Copper 7000 R e c t a n g u l a r T E103 yes yes Temperature C o n t r o l Copper 7000 R e c t a n g u l a r T E103 yes no 4 C i r c u l a r B r a s s P o l a r i z a t i o n 1000 C y l i n d r i c a l T E111 no, o v e r -c o u p l e d no T a b l e 2.1 The Microwave C a v i t i e s 19 c a v i t y f r e q u e n c y change and asymmetry when moved from a c r i t i c a l l y c o u p l e d t o an o v e r c o u p l e d p o s i t i o n . O p e r a t i n g i n a s l i g h t l y u n d e r c o u p l e d p o s i t i o n a v o i d e d t h i s problem but caused e x t r a r e f l e c t i o n s from t h e c a v i t y and a phase change i n E r e f l # j C a v ^ . The p r i n c i p l e b e h i n d t h e c o n s t r u c t i o n o f t h e c i r c u l a r p o l a r i z a t i o n c a v i t y i s i l l u s t r a t e d i n f i g u r e 2.4. I t shows t h e t i m e dependence o f t h e microwave m a g n e t i c f i e l d (HI) seen t h r o u g h a c o u p l i n g i r i s mounted o f f c e n t e r on t h e b r o a d f a c e o f t h e wave g u i d e . The m a g n e t i c f i e l d i s s een t o be c i r c u l a r l y p o l a r i z e d . S i n c e i t c o u p l e s t h e power i n t o t h e c a v i t y , b o t h t h e m a g n e t i c and e l e c t r i c f i e l d s i n s i d e t h e c a v i t y a r e a l s o c i r c u l a r l y p o l a r i z e d . The i r i s i s c e n t e r e d on t h e end w a l l o f t h e c a v i t y t o p r e s e r v e symmetry about t h e c y l i n d r i c a l a x i s o f t h e c a v i t y . F i g u r e 2.5 i l l u s t r a t e s t h e c hosen c a v i t y d e s i g n . The c a v i t y i s mounted on t h e i n s i d e o f a U-shaped waveguide f o l l o w i n g G a i t e t a l . (1959) as t h i s was t h e e a s i e s t method o f p r o d u c i n g c i r c u l a r p o l a r i z a t i o n f o r a h o r i z o n t a l c a v i t y a x i s . ( T h i s i s r e q u i r e d so t h a t t h e a v a i l a b l e h o r i z o n t a l m a g n e t i c f i e l d can be u sed b o t h p a r a l l e l and a n t i - p a r a l l e l t o t h e c a v i t y a x i s . ) The p o s i t i o n o f t h e c o u p l i n g i r i s can be c a l c u l a t e d by t h e g e n e r a l f o r m u l a o f Cohn and C o a l e (1956). The o p e r a t i n g f r e q u e n c y chosen a l l o w s t h e use o f t h e s p e c i a l case c a l c u l a t e d by K a s t l e r (1954). The c o u p l i n g i r i s i s p o s i t i o n e d h a l f way from t h e c e n t e r o f t h e waveguide w a l l t o t h e i n n e r edge o f t h e waveguide. E x p e r i m e n t a l l y , t h e e x a c t p o s i t i o n was f o u n d n o t t o be c r i t i c a l . I t i s c r i t i c a l , 20 Time Waveguide w i t h i r i s I r i s , e n l a r g e d t = 0 t = T A t = T/2 r—^ ( V ' f—^ \ J t = 3 A ( T ) •e-t = T same as t = 0 F i g u r e 2,4 The p r o d u c t i o n o f c i r c u l a r p o l a r i z a t i o n i n t h e microwave system. ( A f t e r P o o l e , 196?) 21 C o u p l i n , A p e r t u r e C a v i t y Symmetry A x i s GaSb sample ( d i s c on c a v i t y a x i s ) B r a s s c a v i t y and waveguide w a l l f u s e d q u a r t z l i g h t p i p e B r a s s m i r r o r s u r f a c e f u s e d q u a r t z sample r o d T e f l o n sample h o l d e r 0 h + .1 S c a l e : 1 cm = 50 t h o u F i g u r e 2.5 C r o s s s e c t i o n o f t h e c i r c u l a r p o l a r i z a t i o n c a v i t y a s s embly - a c y l i n d r i c a l TE111 mode. 22 however t o p r e s e r v e c i r c u l a r symmetry i n t h e c a v i t y . D i f f i c u l t i e s i n t h i s r e g a r d were caused by samples o f square shape ( u n l e s s o f s i z e l e s s t h a n 0 . 5 x 0 . 5 x 0.2 mnr), samples p o s i t i o n e d o f f c e n t e r , and problems w i t h t h e t h i n c a v i t y end w a l l . I n each case t h e s e caused t h e c a v i t y mode t o s p l i t o r t e n d t o s p l i t i n t o two c a v i t y modes o f v e r y c l o s e f r e q u e n c y . One mode would be l i n e a r l y p o l a r i z e d v e r t i c a l l y and t h e o t h e r h o r i z o n t a l l y , as was shown by t h e a n g u l a r dependence o f t h e i n t e n s i t y o f an ESR t e s t s i g n a l . The e f f e c t can be u n d e r s t o o d by r e c a l l i n g t h a t a c i r c u l a r l y p o l a r i z e d f i e l d c an be decomposed i n t o two l i n e a r l y p o l a r i z e d components o s c i l l a t i n g 90° out o f phase. The asymmetry i n t h e c a v i t y c a uses t h e f r e q u e n c i e s o f t h e two dege n e r a t e l i n e a r p o l a r i -z a t i o n modes t o d i f f e r . The problems due t o t h e sample were s o l v e d by u s i n g a c a r e f u l l y p o s i t i o n e d c i r c u l a r d i s c sample o f d i a m e t e r .£1.0 mm.-The problem o f t h e end w a l l a r i s e s from t h e r e q u i r e m e n t t h a t t h e c o u p l i n g h o l e be s u f f i c i e n t l y wide and t h i n t o a l l o w enough power i n t o t h e c a v i t y . I t cannot be made t o o wide o r i t w i l l degrade the c i r c u l a r p o l a r i z a t i o n . V/ith a d i a m e t e r o f 0.080" t h i s r e q u i r e s a t h i c k n e s s o f 0.001-0.002". I f t h e whole end w a l l i s t o be t h i s t h i c k n e s s i t must be made o f b r a s s shim s t o c k s o l d e r e d on. T h i s t e n d s t o b u c k l e i n an asymmetric f a s h i o n . The s o l u t i o n was t o make t h e end w a l l 0.010" t h i c k a t t h e edges and t a p e r i t t o t h e r e q u i r e d s i z e w i t h a s p e c i a l t o o l and t h e s p a r k e r o s i o n c u t t e r . The maximum power and l i g h t i n t e n s i t y on t h e sample a c h i e v e d i n t h i s c a v i t y a r e b o t h e s t i m a t e d t o be one o r d e r o f magnitude l o w e r t h a n t h o s e i n t h e l i n e a r p o l a r i z a t i o n 23 c a v i t y #1. Other a u t h o r s u s i n g t h e G a i t method ( H u t c h i s o n and Weinstock i960, Chang 1964, and T s u k i o k a and Tamura 1972) have had t h e same problems and u s u a l l y s o l v e d them by p u t t i n g two o r t h o g o n a l t u n i n g screws i n t o t h e c a v i t y s i d e n e a r t h e c e n t e r . Because o f space problems ( t h e e n t i r e a s s embly f i t s i n t o a 1.00" dewar) i t was o n l y p o s s i b l e t o t r y t h e t u n i n g screws n e a r t h e end w a l l . There, t h e o n l y e f f e c t o b t a i n e d was a s p l i t t i n g o f t h e c a v i t y mode. Other good c i r c u l a r p o l a r i z a t i o n schemes which have been used a r e ; ( D r e s s e l h a u s e t a l . 1955» Artman and Tannenwald, 1955. and Tinkham and S t r a n d b e r g 1955). 2.5 N o i s e S o u r c e s and S e n s i t i v i t y The main " n o i s e " s o u r c e i s k l y s t r o n f r e q u e n c y d r i f t . The APC does n o t c o m p l e t e l y compensate f o r t h e s e d r i f t s , w h i c h a r e due t o t e m p e r a t u r e changes. The d i s p e r s i o n (o"j) was measured once t o c o n s i d e r u s i n g t h e c r o s s - o v e r p o i n t as a measure o f t h e e f f e c t i v e mass. The s i g n a l p r o v e d much more n o i s y t h a n t h e a b s o r p t i o n s i g n a l , w h i c h would be e x p e c t e d i f t h e f r e q u e n c y d r i f t were t h e main n o i s e s o u r c e . A n o t h e r i m p o r t a n t n o i s e s o u r c e i s m i c r o p h o n i c s o r s m a l l a m p l i t u d e l o w f r e q u e n c y v i b r a t i o n s . They a f f e c t t h e k l y s t r o n , t h e c r y s t a l d e t e c t o r and t h e c a v i t y c o u p l i n g . I n o r d e r t o reduce v i b r a t i o n s t h e k l y s t r o n i s c o o l e d by a st r e a m o f p r e s s u r i z e d a i r i n s t e a d o f a f a n . The chopper, chopper motor, and l i g h t s o u r c e f a n were put on a s e p a r a t e o p t i c a l system t a b l e n o t t o u c h i n g t h e microwave t a b l e , t h u s e l i m i n a t i n g t h e s e s o u r c e s o f v i b r a t i o n . A l s o , s l i d e screw t u n e r s were a v o i d e d i n t h e microwave c i r c u i t because o f t h e i r i n h e r e n t n o i s e c h a r a c t e r i s t i c s . The s p e c t r o m e t e r s e n s i t i v i t y , o r t h e minimum number o f d e t e c t a b l e s p i n s , f o r an ESR s i g n a l o f l i n e w i d t h A B 12 Gauss i s —(10 s p i n s / G a u s s ) AB. T h i s i s measured w i t h t h e f o l l o w i n g c o n d i t i o n s ; ( i ) Q 0 = 5000 ( i i ) S i g n a l / N o i s e e x t r a p o l a t e d t o 1 ( i i i ) l o c k - i n a m p l i f i e r t i m e c o n s t a n t o f 1 second ( i v ) optimum m o d u l a t i o n (v) a t e m p e r a t u r e o f 300 K ( v i ) a " p o i n t " sample p o s i t i o n e d on t e f l o n ( o f d i e l e c t r i c c o n s t a n t 2) ( v i i ) t h e sample used was DPPH o f w e l l known s p i n c o n c e n t r a t i o n . 25 CHAPTER 3 THE CYCLOTRON RESONANCE SIGNAL 3.1 Theory I n A u s t r a l i a , f o r i n s t a n c e , each s t a t e has b u i l t i t s own r a i l w a y , h i r i n g an e x p e r t t o t e l l i t what was t h e i d e a l gauge f o r t h e t r a c k s . I n New South Wales t h e answer was f o u r f e e t , e i g h t and a h a l f i n c h e s . I n South A u s t r a l i a i t was f i v e f e e t , t h r e e i n c h e s . I n Western A u s t r a l i a t h r e e f e e t , s i x i n c h e s . The o n l y t h i n g t h e e x p e r t s agreed on was t h a t t h e r a i l s s h o u l d be p a r a l l e l . f rom "A Herd o f Yaks" by E r i c N i c o l Three t h e o r e t i c a l approaches are used t o a n a l y s e the c y c l o t r o n resonance s i g n a l . Each t r e a t m e n t p r e d i c t s a r e s o n a n t power a b s o r p t i o n v e r y n e a r t h e c o n d i t i o n uj'= u / c , where UJ i s t h e microwave a n g u l a r f r e q u e n c y and LU~C i s t h e c y c l o t r o n f r e q u e n c y d e f i n e d l a t e r . I n a d d i t i o n , t h e c l a s s i c a l t h e o r y p r e d i c t s t h e l i n e s h a p e and the s m a l l c o r r e c t i o n t o t h e r e s o n a n c e c o n d i t i o n , t h e s e m i c l a s s i c a l t r e a t m e n t r e l a t e s t h e e f f e c t i v e mass t o t h e band s t r u c t u r e and t h e quantum c a l c u l a t i o n p r e d i c t s t h e e x i s t e n c e o f weaker t r a n s i t i o n s a t f i e l d s d i f f e r e n t from t h e f u n d a m e n t a l t r a n s i t i o n f i e l d . 3.1 (a) The C l a s s i c a l Theory C l a s s i c a l l y , a f r e e charged p a r t i c l e o f a r b i t r a r y i n i t i a l v e l o c i t y moving i n a s t e a d y u n i f o r m m a g n e t i c f i e l d , B, w i l l t r a c e out a h e l i c a l p a t h around t h e f i e l d d i r e c t i o n . The a n g u l a r f r e q u e n c y o f t h e c i r c u l a r m o t i o n i n t h e p l a n e p e r p e n d i c u l a r t o B i s t h e c y c l o t r o n f r e q u e n c y , 26 UJ- ='-^—, where q i s t h e p a r t i c l e charge, B = IBI and 0 m m i s t h e p a r t i c l e ' s mass. The same e f f e c t can be seen i n a s o l i d w i t h m r e p l a c e d by m*, t h e c a r r i e r e f f e c t i v e mass, and q = -e o r +e ( e l e c t r o n s o r h o l e s ) . E l e c t r o n s and h o l e s move i n h e l i c e s o f o p p o s i t e handedness due t o t h e i r o p p o s i t e c h a r g e s . I f a l i n e a r l y p o l a r i z e d microwave e l e c t r i c f i e l d o f a n g u l a r f r e q u e n c y u r « ^ u / ' c i s a p p l i e d t o t h e p a r t i c l e p e r p e n d i c u l a r t o 3 i t s e f f e c t can be a n a l y z e d as t h e r e s u l t o f two c o u n t e r r o t a t i n g c i r c u l a r l y p o l a r i z e d f i e l d s . One component i s i n phase w i t h t h e o r b i t o f t h e charge and hence a p p l i e s a c o n s t a n t outward f o r c e . S i n c e ur us t h e c u m u l a t i v e e f f e c t o f t h i s f o r c e can be l a r g e even though t h e f o r c e may be s m a l l . The c o n d i t i o n f o r t h e e f f e c t t o be l a r g e i s t h a t t h e p a r t i c l e complete about h a l f a r e v o l u t i o n b e f o r e any c o l l i s i o n , i . e . UJX> l . 1 " T h i s case o f r e s o n a n t power a b s o r p t i o n a t t h e c y c l o t r o n f r e q u e n c y i s c a l l e d c y c l o t r o n r e s o n a n c e . The e f f e c t o f the o t h e r c i r c u l a r l y p o l a r i z e d e l e c t r i c f i e l d component i s a s m a l l n o n - r e s o n a n t power a b s o r p t i o n which w i l l be d i s c u s s e d i n more d e t a i l l a t e r . 1. The c o n d i t i o n o / t > l i s o f t e n d i f f i c u l t t o s a t i s f y e x c e p t f o r pure samples a t low t e m p e r a t u r e s . O b v i o u s l y t h e l a r g e r UJ t h e b e t t e r , commensurate w i t h the a v a i l a b l e m agnetic f i e l d and e f f e c t i v e mass o f t h e c a r r i e r . . . The p r e s e n t experiment was performed a t Ka-band ( f = ^ = 35 GHz). At "this f r e q u e n c y u / f > 1 i s e q u i v a l e n t t o M — y 10 m 0 where t h e m o b i l i t y JU. = ^j^T i s e x p r e s s e d i n cm^v-ls""-'- . For t h i s e x p e r i m e n t w i t h m*/m = 0.04, t h e c o n d i t i o n r e q u i r e s JJ. > 250,000 c m 2 V " 1 s " 1 . 27 The t h e o r y f o r a s i n g l e e l e c t r o n can be e a s i l y d e r i v e d ( D r e s s e l h a u s e t a l . , 1955). C o n s i d e r t h e m o t i o n i n a s e m i c o n d u c t o r o f an e l e c t r o n o f charge -e (e> 0 ) , e f f e c t i v e mass m*, and c o l l i s i o n t i m e T" . Assuming t h e e f f e c t i v e mass and c o l l i s i o n t i m e t o be i s o t r o p i c and energy i n d e p e n d e n t and t a k i n g t h e t i m e dependent s t e a d y s t a t e r e s p o n s e o f t h e e l e c t r o n ' s v e l o c i t y , v ( t ) , t o t h e l i n e a r l y p o l a r i z e d microwave e l e c t r i c f i e l d , E ( t ) = R e ( E 0 e i a r t ) , as v ( t ) = R e ( v 0 e ^ " ^ ) , t h e e q u a t i o n o f m o t i o n i s g i v e n b y , 1 * m*( _±_)v(t) = - e ( E ( t ) + v ( t ) x B l (3-D Take B = ( 0 , 0 ,B) and E = ( E v e , 0 , 0 ) i n r e c t a n g u l a r C a r t e s i a n c o - o r d i n a t e s . P o r t h i s e x p e r i m e n t t h e two f o r c e s on t h e r i g h t hand s i d e o f e q u a t i o n 3.1 a r e comparable, w h i l e t h e e f f e c t o f t h e microwave m a g n e t i c f i e l d B-j_ 4* 0.15 G i s n e g l i g i b l e s i n c e B = 500 G a t r e s o n a n c e . By s o l v i n g t h e component e q u a t i o n s o f e q u a t i o n 3.1 f o r v x ( t ) and v y ( " t ) a ^ d e l i m i n a t i n g v y ( t ) t h e complex c o n d u c t i v i t y , g, i s o b t a i n e d : a = Ne v x E„ 1 + i J / 1 + UR - + 2 where V - UJT , V = Ufc T and ov i s t h e d.c. c o n d u c t i v i t y . 1. T h i s e q u a t i o n o f m o t i o n can be d e r i v e d r i g o r o u s l y from t h e B oltzmann e q u a t i o n , i f f i s energy i n d e p e n d e n t , by i n t e g r a t i n g t h e d i s t r i b u t i o n f u n c t i o n o v e r momentum space (Ziman, i 9 6 0 ) . 28 = N e 2 T o m* S i n c e a i s complex, g = a R + iCj , and a R 1 + V 2 + y c 2 a o kVz + ( l + i V c 2 - vz) 2 o-j v ( ^ 2 - y 2 -: l) (3.2) ao 4 y 2 + (l + j^. 2 - if 2) 2 The components a R and a-j- ace c a l l e d t h e a b s o r p t i o n and t h e d i s p e r s i o n . They a r e shown i n f i g u r e 3.1 a l o n g w i t h t h e c i r c u l a r p o l a r i z a t i o n components ( L a x and M a v r o i d e s , i960) d e f i n e d by, a * . = \{ <JR(+) + c * R ( - ) ) where, j ( L . P . ) j 2 ( ± ) = crR(+) + i ax(±) o o o 1 + + J ^ ) * 1 + iV + V C ) + i (3.3) 3.1(b) The S e m i c l a s s i c a l T h e o r y I n t h e s e m i c l a s s i c a l t h r e a t m e n t t h e e l e c t r o n m o t i o n i s c a l c u l a t e d by o ! t ~ ^ — ^ ~ + Xx5) where v = ^ ' 7 1P(k) and £(k) i s t h e c o n d u c t i o n band energy-crystal-momentum r e l a t i o n s h i p . The c y c l o t r o n e f f e c t i v e mass i s c a l c u l a t e d by D r e s s e l h a u s e t a l . (1955 a) t o be 29 .% ^2TT m* = 4 - f ? ¥ 6 ^ where k i s e x p r e s s e d i n a c y l i n d r i c a l c o - o r d i n a t e system as , 0) w i t h B || k . T h i s f o r m u l a r e l a t e s t h e measured e f f e c t i v e mass t o t h e band s t r u c t u r e . F o r an i s o t r o p i c band th e r e s u l t can be s i m p l i f i e d t o _1_ _ 1 , ^£ m' * = ' i f - ' s i n c e t h e n ( - r j — ) does n o t depend on 0, U s i n g t h e r e l a t i o n s h i p 2 2 2 k = <o + k z and u s i n g a power s e r i e s e x p a n s i o n f o r £ (k) i t can e a s i l y be shown t h a t t h e r e s u l t , f u r t h e r r e d u c e s t o F o r a p a r a b o l i c c o n d u c t i o n band t h e e f f e c t i v e mass i s a c o n s t a n t , i n d e p e n d e n t o f k. F o r n o n - i s o t r o p i c bands m* i s r e a l l y a t e n s o r depending on t h e d i r e c t i o n o f B i n t h e c r y s t a l . I n c y c l o t r o n r e s o n a n c e e x p e r i m e n t s t h e s e components can be deduced s e p a r a t e l y w h i l e i n o t h e r measurements o n l y w e i g h t e d a v e r a g e s o f t h e d i f f e r e n t mass components are o b t a i n e d . F u r t h e r m o r e , i n some cases t h e s e component masses may be r e l a t e d t o o r ^ f r a t h e r t h a n k^ ^ k ^ K bk T h e r e f o r e , c a u t i o n i n comparing e f f e c t i v e mass v a l u e s from d i f f e r e n t e x p e r i m e n t s i s r e q u i r e d . A good d i s c u s s i o n o f t h e d i f f e r e n t methods used t o d e t e r m i n e e f f e c t i v e mass v a l u e s i s g i v e n by Lax (1962). 30 Figure 3.1 Theoretical curves for absorption and dispersion. The l i n e a r p o l a r i z a t i o n c u r v e i s e q u i v a l e n t t o 31 3.1 ( c ) The Quantum Theory-Landau (1930) has s o l v e d S c h r o e d i n g e r ' s e q u a t i o n f o r the m o t i o n o f a f r e e p a r t i c l e i n a magnetic f i e l d and i n a box. The problem o f a n e a r l y f r e e e l e c t r o n i n a s o l i d has been s o l v e d by D i n g l e (1952). The energy l e v e l s i n c l u d i n g t h e Zeeman s p i n term a re g i v e n by, £ ( n , m g , k z ) = ( n + ) ) n u / . + g B m^ + ft2*.2 5) T> u r + g / * B B m s + —JJ,: where "t\ = P l a n c k ' s constant/2 - r T g = Lande g - f a c t o r B = m a g n e t i c i n d u c t i o n n = 0, 1, 2, . . . m s = ± I and k i s q u a s i c o n t i n u o u s (Wannier i960). T h i s assumes B i s p a r a l l e l t o t h e z a x i s i n a c u b i c c r y s t a l . The p a r t i c l e m o t i o n i n t h e ( t w o - d i m e n s i o n a l ) (k ,k ) - p l a n e , p e r p e n d i c u l a r t o B i s q u a n t i z e d w h i l e t h e m o t i o n a l o n g B i s u n a f f e c t e d by B. The " t w o - d i m e n s i o n a l " e n e r g y l e v e l s o f s p a c i n g *f\ VJ a r e c a l l e d Landau l e v e l s and have t h e p r o p e r t i e s o f one d i m e n s i o n a l s i m p l e harmonic o s c i l l a t o r l e v e l s . The e l e c t r i c d i p o l e s e l e c t i o n r u l e s f o r a p a r a b o l i c c o n d u c t i o n band a r e : A n = + 1, Am_ = 0, Ak„ = 0 (McCombe e t a l . I967), The c o n n e c t i o n w i t h t h e c l a s s i c a l model has been c a l c u l a t e d by Feldman and Kahn (1970). They c o n s t r u c t e d t h e c y c l o t r o n - m o t i o n wave p a c k e t from t h e " c o h e r e n t s t a t e s " o f t h e p r oblem and showed i t f o l l o w s t h e c l a s s i c a l o r b i t . Because t h e r e are many Landau l e v e l s i n an energy band f o r r e a s o n a b l e f i e l d s , t h e r e i s e f f e c t i v e l y no upper bound on t h e e n ergy l e v e l s . Hence t h e r e i s no s a t u r a t i o n o f c y c l o t r o n r e s o n a n c e a t h i g h powers. Because o f t h e n a t u r e o f t h e t r a n s i t i o n m a t r i x e l e m e n t s , t h e i n t e g r a t e d i n t e n s i t y i s a l s o independent o f t e m p e r a t u r e (see Appendix 3.2). When s p i n - o r b i t i n t e r a c t i o n i s i n c l u d e d i n t h e H a m i l t o n i a n t h e s p i n and Landau l e v e l s are mixed. Then, e l e c t r i c d i p o l e i n d u c e d t r a n s i t i o n s o f f r e q u e n c i e s Cwn + 2 US + uj- and us may be w e a k l y a l l o w e d ( "h u r = gyUgB) . I n t h e case o f t h i s e x periment on GaSb t h e s e "combined r e s o n a n c e " t r a n s i t i o n s were n e v e r o b s e r v e d . T h i s i s i n agreement w i t h t h e c a l c u l a t e d i n t e n s i t y e s t i m a t e s f o l l o w i n g McCombe e t a l . (1967). 3.2 The Plasma E f f e c t s To o b s e r v e t h e c y c l o t r o n r e s o n a n c e s i g n a l , as w e l l as t h e c o n d i t i o n UJT>1, i t i s a l s o n e c e s s a r y t o have us » u / p 0 where uj"^0, t h e plasma f r e q u e n c y i s g i v e n by u r 2 = N e 2 po K m* w i t h N = c a r r i e r d e n s i t y K = d i e l e c t r i c c o n s t a n t m* = c a r r i e r e f f e c t i v e mass. 3 3 When t h e c a r r i e r d e n s i t y i s s u f f i c i e n t l y l a r g e d e p o l a r i z a t i o n f i e l d s a r e s e t up by the s u r f a c e c h a r g e s . They depend on t h e shape o f t h e sample and t h e f r e q u e n c y . The i n t e r n a l e l e c t r i c f i e l d E i i s r e l a t e d t o t h e e x t e r n a l e l e c t r i c f i e l d Eo t h r o u g h t h e e l e c t r i c p o l a r i z a t i o n P and t h e g e o m e t r i c a l d e p o l a r i z a t i o n f a c t o r L by E i = Eo - LP . F o r an e l l i p s o i d a l sample P i s p a r a l l e l t o Eo and we have where X i s t h e e l e c t r i c s u s c e p t i b i l i t y . 1 . The e q u a t i o n s o f m o t i o n have been s o l v e d by D r e s s e l h a u s e t a l , ( 1 9 5 5 ( b ) ) f o r an e l l i p s o i d w i t h t h e two axes p e r p e n d i c u l a r t o B o f e q u a l l e n g t h . The r e s u l t i s a r e s o n a n t c o n d u c t i v i t y i d e n t i c a l t o t h e c y c l o t r o n r esonance r e s u l t o f e q u a t i o n 3 . 2 w i t h V r e p l a c e d by , P = V ( 3 . 5 ) where, V - UJ T P P L i Ne 2\ 2. m" L i - x + L % ( 3 . 6 ) The use o f y' f o r i> r e s u l t s i n b o t h a peak p o s i t i o n r e d u c t i o n and an "ixr-x" r e d u c t i o n by t h e same f r a c t i o n , y ' / v » f o r y p < y . F o r l/^y V a magnetoplasma resonance 1. R e c a l l i n g t h a t K = 1 + 4ITX t h i s e q u a t i o n r e d u c e s t o more f a m i l i a r ones i n two c a s e s . F o r a t h i n d i s k we have ( i ) E i = Eo f o r Eo p a r a l l e l t o t h e d i s c s u r f a c e ( L = 0 ) , ( i i ) E i = Eo(i) f o r Eo p e r p e n d i c u l a r t o t h e d i s c s u r f ace (L=4IT) 2. I n t h e case o f a t h i n d i s c t h i s f o r m u l a r e d u c e s t o U/p = u/" p o, t h e more f a m i l i a r r e s u l t , s i n c e L=4IT and K=1+4TTX. 34 d e v e l o p s i n w h i c h t h e peak p o s i t i o n b e a r s l i t t l e r e l a t i o n t o t h e c y c l o t r o n r esonance peak p o s i t i o n . The case o f i n t e r e s t i n t h i s e x p e r i m e n t i s t h a t o f a t h i n d i s c w i t h b o t h t h e mag n e t i c f i e l d and microwave e l e c t r i c f i e l d i n t h e p l a n e o f t h e d i s c . T a k i n g B = (0,0,B) and E = (E,0,0) t h e reso n a n c e c o n d i t i o n i s W H E R E UJ-'X = o / ( i — L i - ) . y UJ-L x The q u a n t i t i e s U/ are d e f i n e d by L i x = - v ^ . p y y 1 + LXA-I n t h i s c a s e f o r a v e r y t h i n d i s c t h e v a l u e s t o use would be L = 4TT and L = 0. The samples used i n t h e s e e x p e r i m e n t s y x a r e t r e a t e d as b e i n g a p p r o x i m a t i o n s t o a t h i n d i s c . They a r e t h i n r e c t a n g u l a r samples o f d i m e n s i o n s 1 x 1.4 x 0.30 mm^. To use t h e c a l c u l a t i o n o f D r e s s e l h a u s e t a l . (1955 ( b ) ) we e s t i m a t e | = = 0.25. T h i s g i v e s L x = 1.9 and L y = 4TT SO t h a t L i x = 0.70 and L^y = 0.84. F o r t h e s m a l l plasma s h i f t s o b s e r v e d i n t h e p r e s e n t e x p e r i m e n t t h e e q u a t i o n s 3.5 and 3.6 may be used w i t h Lj, = 0.77. The above a u t h o r s have a l s o c a l c u l a t e d t h e e f f e c t o f t h e p r e s e n c e o f a h i g h c o n c e n t r a t i o n o f m a j o r i t y c a r r i e r s on t h e c y c l o t r o n r esonance o f a low c o n c e n t r a t i o n o f m i n o r i t y c a r r i e r s . T h i s w i l l be d i s c u s s e d and a p p l i e d i n s e c i o n 4.5. 3.3 G a l l i u m A n t i m o n i d e (GaSb) A l t h o u g h GaSb i s one o f t h e I I I - V compound semi-c o n d u c t o r s w h i c h has been s t u d i e d q u i t e e x t e n s i v e l y , t o d a t e no one has measured t h e e l e c t r o n e f f e c t i v e mass by t h e c y c l o t r o n r e s o n a n c e t e c h n i q u e e i t h e r a t microwave o r i n f r a - r e d f r e q u e n c i e s . A d i s c u s s i o n o f t h e e f f e c t i v e mass v a l u e s d e t e r m i n e d by o t h e r e x p e r i m e n t s i s g i v e n i n s e c t i o n 6.4. The c y c l o t r o n r e s o n a n c e s i g n a l o f t h e r m a l l y g e n e r a t e d h o l e s has been measured by S t r a d l i n g (1966) and by Cronburg e t a l . ( 1 9 7 0 ) . They have found a l i g h t h o l e e f f e c t i v e mass m l h . 1 * o f — — — = 0 . 0 5 . S i n c e t h i s i s comparable t o t h e roc-e l e c t r o n e f f e c t i v e mass o f m*/mQ = 0.04 i t was n e c e s s a r y t o c o n f i r m t h e c a r r i e r s i g n f o r t h e o b s e r v e d s i g n a l by m i c r o -wave c i r c u l a r p o l a r i z a t i o n measurements. The a p p r o x i m a t e energy band s t r u c t u r e f o r GaSb i s shown i n f i g u r e 3.2. T h i s f i g u r e and t h e d a t a on i t a r e d e r i v e d from s e v e r a l s o u r c e s b u t m a i n l y from Higginbothara e t a l . (1968). S i n g l e group n o t a t i o n i s used w i t h d o u b l e group n o t a t i o n i n b r a c k e t s . The energy gap i s a d i r e c t t r a n s i t i o n a t k = 0 ( t h e P p o i n t ) . I n t h i s e x p e r i m e n t n e a r - i n f r a - r e d r a d i a t i o n o f energy hV E i s used t o g e n e r a t e e l e c t r o n - h o l e p a i r s a c r o s s t h i s gap. The e l e c t r o n s a r e s u r e l y c r e a t e d i n t h e T p o i n t c o n d u c t i o n band minimum. They cannot r e a c h t h e n e x t l o w e s t c o n d u c t i o n band minimum 1, A p p e n d i x 6.1 c o n t a i n s a d i s c u s s i o n o f t h e measured h o l e e f f e c t i v e masses. 36 k=-f(U1) k=(O.QO) k^ d.0.0) F i g u r e 3 . 2 The energy band s t r u c t u r e o f GaSb. 37 e x c e p t by s t r o n g microwave h e a t i n g w h i c h i s p r e v e n t e d by t h e c r e a t i o n o f l o n g i t u d i n a l o p t i c a l (L.O.) phonons (see l a t e r ) . B o t h l i g h t and heavy h o l e s a r e c r e a t e d , as shown by t h e two d i f f e r e n t c u r v a t u r e s o f t h e v a l e n c e bands, de g e n e r a t e a t t h e r p o i n t . The l o w e s t o r " s p l i t - o f f " v a l e n c e band i s l o w e r t h a n t h e o t h e r v a l e n c e bands by an amount A 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 Kane (1957) band s t r u c t u r e c a l c u l a t i o n i s one o f t h e b e s t f o r t h e energy bands n e a r t h e V p o i n t . I n i t t h e i n t e r a c t i o n between t h e l o w e s t c o n d u c t i o n band and t h e t h r e e h i g h e s t v a l e n c e bands i s c a l c u l a t e d d i r e c t l y . I t was shown t h a t a t t h e F p o i n t t h e c o n d u c t i o n band s t a t e s a r e " s - l i k e " w h i l e away from t h a t p o i n t some " p - l i k e " c h a r a c t e r i s mixed i n . I t i s t h i s f a c t o r w h i c h produces a n o n - p a r a b o l i c c o n d u c t i o n band. The v a l e n c e bands are " p - l i k e " . The model assumes t h a t t h e o t h e r l o w e r v a l e n c e bands are " v e r y f a r away" i n e n e r g y and a l s o t h a t t h e second c o n d u c t i o n band i s " f a r away". That i s , t h e e f f e c t o f t h e second c o n d u c t i o n band e n t e r s v i a second o r d e r p e r t u r b a t i o n t h e o r y and t h e e f f e c t o f t h e l o w e r v a l e n c e bands i s n e g l e c t e d . As i s e v i d e n t i n f i g u r e 3 . 2 , t h i s i s a p p r o x i m a t e l y t r u e . The p u r e s t a v a i l a b l e GaSb i s always p - t y p e w i t h an a c c e p t o r c o n c e n t r a t i o n , N A = 1-2 x 10^ /cm?. These " r e s i d u a l a c c e p t o r s " a r e a l m o s t c e r t a i n l y due t o a non-s t o i c h i o m e t r i c c r y s t a l w i t h a d e f i c i e n c y o f Sb. S i n c e t h i s e x p e r i m e n t measures t h e s c a t t e r i n g r a t e o f e l e c t r o n s from t h e s e a c c e p t o r s i n t h e n e u t r a l s t a t e , a b r i e f summary o f 38 t h e i r p r o p e r t i e s w i l l be made. The e v i d e n c e t h a t t h e a c c e p t o r s are s t o i c h i o m e t r i c d e f e c t s i s : ( E f f e r and E t t e r 1964, Van d e r Meulen 1967t A l l e g r e e t a l . 1970) ( i ) The r e s i d u a l a c c e p t o r c o n c e n t r a t i o n i s l a r g e r t h a n t h e i m p u r i t y c o n t e n t o f t h e e l e m e n t a l m a t e r i a l used t o grow t h e c r y s t a l s . Pour o f t h e samples used i n t h i s e x p e r i m e n t were a n a l y s e d f o r i m p u r i t i e s by t h e s p a r k s o u r c e mass s p e c t r o g r a p h i c t e c h n i q u e by Dr. D. S. R u s s e l l o f t h e N a t i o n a l R e s e a r c h C o u n c i l ( P u r e C h e m i s t r y D i v i s i o n ) . The r e s u l t s , i n A ppendix 3 A show t h a t t h e o n l y i m p u r i t i e s f o u n d t o have a c o n s i s t e n t c o n c e n t r a t i o n above 1 ppm (4 x 10 cm~J) i n t h e two c y c l o t r o n r e s o n a n c e samples measured were s i l i c o n and oxygen. S i l i c o n has been s p e c i f i c a l l y e x c l u d e d as a p o s s i b i l i t y by measurements o f t h e above a u t h o r s . Oxygen c o u l d be an a c c e p t o r i m p u r i t y were i t not f o r t h e o t h e r e v i d e n c e . ( i i ) The l i m i t i n g a c c e p t o r c o n c e n t r a t i o n has been found t o be t h e same by many r e s e a r c h e r s u s i n g d i f f e r e n t c r y s t a l g r o w t h t e c h n i q u e s and sample v e s s e l s . ( i i i ) The r e s i d u a l a c c e p t o r c o n c e n t r a t i o n i s l o w e r i n GaSb grown from an S b - r i c h m e l t . T h i s would be e x p e c t e d f o r an Sb v a c a n c y o r a d e f e c t r e l a t e d t o i t . The a u t h o r s c i t e d above have a d d i t i o n a l e v i d e n c e t h a t t h e a c c e p t o r s are l i k e l y t h e compound d e f e c t o f a g a l l i u m atom on an antimony s i t e and a v a c a n c y on t h e g a l l i u m s i t e . T h i s i s r e a s o n a b l e s i n c e Ga i s s m a l l e r t h a n Sb, but t h e t o t a l e v i d e n c e f o r t h i s i s n o t c o m p l e t e l y c o n v i n c i n g . The e n e r g y l e v e l s o f t h e a c c e p t o r s have been i n v e s t i g a t e d by a v a r i e t y o f measurements. H a l l e f f e c t measurements on "as grown" m a t e r i a l ( L e i f e r and Dunlap 1954. E f f e r and E t t e r 1964, Amirkanov and Amirkanova 1968) show l i t t l e c o mpensation and y i e l d a main a c c e p t o r l e v e l o f 32-37 meV w i t h 1-2 x I O 1 ? / c i P a c c e p t o r s and a second l e v e l a t 11 o r 24 meV w i t h about 16 3 1 x 10 / c n r a c c e p t o r s . Amirkanov e t a l . have o b s e r v e d e v i d e n c e o f i m p u r i t y band c o n d u c t i o n a t h e l i u m t e m p e r a t u r e s . H a l l e f f e c t measurements on s t r o n g l y compensated m a t e r i a l (D'Olne Campos e t a l . I969, B a x t e r e t a l . I967) and lu m i n e s c e n c e measurements on "as grown" and doped m a t e r i a l ( P o k r o v s k i i e t a l . I967, Johnson and Fan I965, D'Olne Campos e t a l . 1970, G o l o v i n a and Y u n o v i c h 1971, B e n o i t a l a G u i l l a u m e and L a v a l l a r d 1970) a l l show t h a t t h e a c c e p t o r s have two a c t i v e e n e rgy l e v e l s , o r t h a t t h e y a re d o u b l y i o n i z a b l e . The deduced e n e r g y l e v e l s a r e 21-34 meV and 56-62 meV. P h o t o c o n d u c t i v i t y measurements (Habegger, 1964) y i e l d t h e f o l l o w i n g e n e rgy l e v e l s : 1 l e v e l o f energy ~30 meV 1 o r 2 l e v e l s o f energy ~60 -80 meV and 1 l e v e l ~100 meV above t h e v a l e n c e band. These a r e a l l a t t r i b u t e d t o t h e r e s i d u a l a c c e p t o r s . O s c i l l a t i o n s i n t h e p h o t o c o n d u c t i v i t y as a f u n c t i o n o f e l e c t r o n c r e a t i o n e n e rgy have a l s o been r e p o r t e d (Habegger and Fan 1964, Fan I968) w i t h an energy s p a c i n g o f one l o n g i t u d i n a l o p t i c a l (L.O.) phonon energy ( d e n o t e d t\ cifQ). An e x p l a n a t i o n was proposed whereby t h e e l e c t r o n s e i t h e r s t a y a t t h e energy i n t h e band a t w h i c h t h e y were c r e a t e d o r , i f p o s s i b l e , d r o p i n energy by an i n t e g r a l number o f L,0. phonon q u a n t a . Habegger and Fan r e p o r t t h a t t h e o s c i l l a t i o n s d i s a p p e a r when t h e e l e c t r o n d i s t r i -b u t i o n i s a f f e c t e d by a d.c. e l e c t r i c f i e l d o f 0.1 v/cm. S i n c e t h e l o w e s t microwave e l e c t r i c f i e l d s employed here were g r e a t e r t h a n t h i s t h e non-observance o f such o s c i l l a t i o n s i n t h e p r e s e n t e x p e r i m e n t s i s r e a s o n a b l e . P a r s o n s (1971) has r e p o r t e d o s c i l l a t i o n s i n t h e degree o f p o l a r i z a t i o n o f t h e r e c o m b i n a t i o n r a d i a t i o n from o p t i c a l l y pumped e l e c t r o n s . He measured an e q u i l i b r i u m e l e c t r o n -9 l i f e t i m e a t h e l i u m t e m p e r a t u r e s o f t e = 7 x 10 ys. T h i s i s compared w i t h our measurement below. B e n o i t a* l a G u i l l a u m e and L a v a l l a r d (1970) deduced from l u m i n e s c e n c e work t h a t t h e p h o t o c r e a t e d e l e c t r o n c o n c e n t r a t i o n i s p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y and t h a t t h e r e c o m b i n a t i o n r a t e i s d e t e r m i n e d by n o n - r a d i a t i v e t r a n s i t i o n s . The same l i g h t i n t e n s i t y dependence i s o b s e r v e d h e r e . 3.4 The Measured A b s o r p t i o n S i g n a l F a c t s and Figures*. Put 'em down*. from "The Chimes, F i r s t Q u a r t e r " , by C h a r l e s D i c k e n s 3.4 (a) The Background The c y c l o t r o n r e s o n a n c e s i g n a l i s superimposed on a f i e l d i n d e p e n d e n t background. The r a t i o o f t h e s i g n a l a m p l i t u d e t o t h e background a m p l i t u d e ( H i / B a c k g r o u n d i n f i g u r e 3.3) i s ^ 1 f o r an u n e t c h e d sample and = 20 f o r an e t c h e d sample a t h i g h microwave power and l i g h t i n t e n s i t y ( d e n o t e d P and I ) , 41 The e t c h used c o n s i s t e d o f n i t r i c a c i d , h y d r o -f l u o r i c a c i d and d i s t i l l e d w a ter i n t h e p r o p o r t i o n s HNOyHFtHgO = 15:1«10 by volume. The e t c h g i v e s a s h i n y m e t a l l i c , b u t rough s u r f a c e t o samples c l e a n e d i n d i s t i l l e d w a t e r . The use o f a n o t h e r e t c h o r c l e a n i n g p r o c e d u r e i n s t e a d o f t h i s one makes no d i f f e r e n c e t o t h e r e s u l t s . The background has a d i f f e r e n t power and l i g h t i n t e n s i t y dependence from t h e s i g n a l . T h i s and o t h e r d e t a i l s a r e c o n t a i n e d i n Appendix 3 . 3 . F i g u r e 3 . 3 p r o v e s t h a t t h e background a m p l i t u d e c o n t i n u e s t o be f i e l d i ndependent under t h e c y c l o t r o n r e s o n a n c e s i g n a l , s i n c e t h e h e i g h t s HO and HI, d e f i n e d on t h e diagram, have t h e same power dependence, w h i c h i s d i f f e r e n t from t h a t o f t h e background. The h e i g h t HO i s c o r r e c t e d f o r t h e power dependence o f o/T and f o r t h e measured d i s p e r s i o n - i n d u c e d asymmetry. The curve f i t t i n g p r o c e d u r e adopted t o measure u/T and 6 i s d e s c r i b e d i n s e c t i o n 4 . 1 . 3 . 4 (b) The Treatment o f D i s p e r s i o n As e x p l a i n e d i n s e e t i o n 2 . 3 t h e i m p e r f e c t i o n s i n t h e Ka-band microwave s p e c t r o m e t e r r e s u l t i n t h e a d m i x t u r e o f d i s p e r s i o n i n t o t h e s i g n a l . T h i s e f f e c t i s e a s i e s t t o e l i m i n a t e t h r o u g h a n a l y s i s o f t h e d a t a . The o b s e r v e d s i g n a l , S, as 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 , B, i s S(B) a a R ( B ? u / t - ) + Bo-jCBj LUX) The d i s p e r s i o n parameter 8 d e t e r m i n e s t h e asymmetry log I F i g u r e 3.3 The s e p a r a t i o n o f s i g n a l and background. o f t h e o b s e r v e d s i g n a l as shown on f i g u r e 3.^ .* I t a l s o d e t e r m i n e s t h e c o r r e c t i o n t o t h e measured peak, B Q , t o deduce t h e "peak parameter", B Q, f o r pure a b s o r p t i o n . The peak 2 p a r a m e t e r B. i s t h e v a l u e o f B f o r Uf = u/„ * I n t h e case o f c i r c u l a r p o l a r i z a t i o n t h e peak o f t h e symmetric a b s o r p t i o n l i n e s a t i s f i e s t h i s c o n d i t i o n . I n t h e l i n e a r p o l a r i z a t i o n a b s o r p t i o n s i g n a l t h e r e i s a c o r r e c t i o n due t o t h e e f f e c t o f th e n o n - r e s o n a n t c i r c u l a r p o l a r i z a t i o n component w h i c h makes t h e l i n e a s ymmetric. T h i s c o r r e c t i o n i s 0,7% f o r U/T = 2 and l e s s f o r l a r g e r UJX, When d i s p e r s i o n i s mixed i n , t h e a d d i t i o n a l c o r r e c t i o n s can be as h i g h as 5$. These c o r r e c t i o n s a r e computed n u m e r i c a l l y t o v e r y h i g h a c c u r a c y , u s i n g e q u a t i o n s 3.2,and a p p l i e d t o a l l t h e d a t a , A t a b l e o f t h e s e c o r r e c t i o n s and t h e d e r i v a t i o n o f app r o x i m a t e c o r r e c t i o n f o r m u l a e a re g i v e n i n Appendix 3.1. When t h e d i s p e r s i o n c o r r e c t i o n i s much l a r g e r t h a n t h e o t h e r c o r r e c t i o n we have B Q 2 cu" - r B Q and B Q a r e a l s o c o r r e c t e d f o r microwave f r e q u e n c y v a r i a t i o n s s u c h t h a t t h e y b o t h r e f e r t o a s t a n d a r d e f f e c t i v e microwave f r e q u e n c y o f 35.43 GHz. That i s , t h e e f f e c t i v e mass i s deduced from o r = UJ where ur = e B " and u/= 2TT( 35.43) 10 9 s " 1 . c c m* The r e s u l t s o f t h e a n a l y s i s o f t h e measured c u r v e s o f f i g u r e 3 A are p r e s e n t e d i n t a b l e 3.1. 1. F o r t h e d i s p e r s i o n t e s t o f f i g u r e 3.4, 0 i s v a r i e d e x p e r i -m e n t a l l y by a d j u s t i n g t h e microwave phase i n t h e b i a s arm. 2. The peak parameter B 0 , w h i c h i s t h e f i g u r e used t h r o u g h o u t t h i s t h e s i s , w i l l sometimes be r e f e r r e d t o as t h e peak p o s i t i o n . I 1 I I I ' I I ' I I I L Magnetic Field (arb. units) F i g u r e 3A The e f f e c t o f d i s p e r s i o n on t h e s i g n a l . 4 5 T a b l e 3 . 1 The e f f e c t o f d i s p e r s i o n on t h e peak p o s i t i o n . PEAK POSITION F i g u r e u n c o r r e c t e d c o r r e c t e d T r a c e UJ T 3 B o (a) 3 . 1 1 + . 0 3 + . 0 2 4 9 2 . 5 + 2 . 0 5 2 3 . 0 + 3 . 5 (b) 3 . 0 2 + .04 - . 0 1 . 0 1 5 1 8 . 0 ± 1 . 0 5 2 4 . 0 + 1 . 5 ( c ) 3 . 1 0 + .04 + . 1 9 + . 0 2 5 3 6 . 5 + 2 . 5 5 2 4 . 5 + 4 . 0 Avg. 3.08 + . 0 6 5 2 3 . 7 + 2 . 5 The a t t r i b u t i o n o f t h e s i g n a l asymmetry t o d i s p e r s i o n i s s u p p o r t e d by t h e f o l l o w i n g e v i d e n c e : ( i ) By a d j u s t i n g one parameter, 3 , b o t h t h e f i t t e d s i g n a l shape and t h e c o r r e c t e d peak p o s i t i o n a r e g r e a t l y improved o v e r t h e u n c o r r e c t e d shape and peak p o s i t i o n (see f i g u r e 3 . 4 and t a b l e 3 . 1 ) . ( i i ) T h i s c o r r e c t i o n r e s u l t e d i n c o m p l e t e l y r e p r o d u c i b l e c o r r e c t e d peak p o s i t i o n s i n a l l measurements. ( i i i ) I t i s e x p e c t e d t h a t t h e a p p r e c i a b l e r e f l e c t i o n s and o t h e r i m p e r f e c t i o n s i n t h e microwave system would cause an a d m i x t u r e o f d i s p e r s i o n . Such an e f f e c t has been r e p o r t e d by S l a g s v o l d ( 1 9 6 6 ) f o r t h i s equipment. 3 . 4 ( c ) C i r c u l a r P o l a r i z a t i o n R e s u l t s The c i r c u l a r p o l a r i z a t i o n s i g n a l i s shown on f i g u r e 3 . 5 . The s i g n a l f o r B > 0 f o r t h i s "handedness" o f c i r c u l a r p o l a r i z a t i o n i s due t o e l e c t r o n s . T h i s f a c t was c o n f i r m e d by d o i n g c y c l o t r o n r e s o n a n c e on an o r i e n t e d s i n g l e Ul C 0 Magnetic Field , B (arb. units) > F i g u r e 3.5 The c i r c u l a r p o l a r i z a t i o n s i g n a l . 4 7 c r y s t a l o f germanium w i t h b o t h e l e c t r o n and h o l e s i g n a l s ( o f known peak p o s i t i o n s ) p r e s e n t . The s m a l l s i g n a l f o r B < 0 i s m a i n l y due t o l i g h t h o l e s but has a s m a l l c o n t r i -b u t i o n from e l e c t r o n s due t o t h e i m p e r f e c t i o n o f t h e c i r c u l a r p o l a r i z a t i o n . That t h e s m a l l s i g n a l i s m a i n l y due t o h o l e s has been d e t e r m i n e d b o t h from t h e power dependence o f t h e two s i g n a l h e i g h t s and from an ESR d e t e r m i n a t i o n t h a t = 60, where P + i s e q u a l t o t h e power i n t h e c a v i t y i n t h e p o s i t i v e and n e g a t i v e c i r c u l a r p o l a r i z a t i o n components. Because o f t h e c a v i t y Q and c o u p l i n g c o n d i t i o n t h e h i g h e s t power and b e s t s e n s i t i v i t y a v a i l a b l e from t h i s c a v i t y are b o t h l e s s t h a n f o r t h e copper l i n e a r p o l a r i z a t i o n c a v i t i e s . Compared t o t h o s e c a v i t i e s t h e s e measurements are a t medium powers ( .4 -9 d B ) . The power dependence o f t h e h o l e c o n c e n t r a t i o n was c o n f i r m e d a t h i g h e r powers i n a l i n e a r l y p o l a r i z e d c a v i t y by a d i f f e r e n t t e c h n i q u e ( s e e s e c t i o n 3 . 4 ( g ) ) 3 . 4 (d) E v i d e n c e o f " C y c l o t r o n Resonance" The o b s e r v e d s i g n 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 c y c l o t r o n r e s o n a n c e because: ( i ) The s i g n a l shape can be f i t v e r y w e l l by t h e s i m p l e c y c l o t r o n r e s o n a n c e t h e o r y f o r o*R + Bo^. ( i i ) The t r a n s i t i o n i s e l e c t r i c d i p o l e induced. , T h i s was c o n f i r m e d by m e a s u r i n g t h e s i g n a l s t r e n g t h as a f u n c t i o n o f t h e a n g l e between t h e microwave e l e c t r i c f i e l d and t h e d.c. m a g n e t i c f i e l d . I n t h e c a v i t y used t h e microwave e l e c t r i c and m a g n e t i c f i e l d s a r e p e r p e n d i c u l a r t o each o t h e r and b o t h a r e h o r i z o n t a l . I n t h i s way e l e c t r i c and 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 can be d i s t i n g u i s h e d . ( i i i ) The c i r c u l a r p o l a r i z a t i o n measurements show t h e c o r r e c t s e l e c t i o n r u l e f o r a b s o r p t i o n by m a i n l y one ty p e o f c a r r i e r , namely e l e c t r o n s . ( i v ) The d e r i v e d e f f e c t i v e mass v a l u e i s v e r y r e a s o n a b l e compared t o o t h e r e l e c t r o n e f f e c t i v e mass d a t a ( see s e c t i o n 6 . 5 ) . 3.4 (e) S i g n a l C a l c u l a t i o n s C e r t a i n c a l c u l a t i o n s a r e u s e f u l t o b e t t e r u n d e r s t a n d t h e measured s i g n a l . F i r s t t h e power absorbed by t h e e l e c t r o n s w i l l be c a l c u l a t e d . L e t t h i s a b s o r p t i o n a t t h e 2 r e s o n a n c e peak be P s . Then P g =  a e f f f s ^ i i where, V g = t h e e f f e c t i v e sample volume o"e£f = 1/2 aQ a t t h e resonance peak c = Ne 2 , C f ( P ) = t h e d.c. p h o t o c o n d u c t i v i t y w i t h m * f ( p ) f= 1 (see l a t e r ) E^ = t h e r.m.s. e l e c t r i c f i e l d i n s i d e t h e sample. T h i s i s j u s t t h e J o u l e h e a t i n g l o s s f o r m u l a . We have E^ = a E Q where E Q i s e q u a l t o t h e r.m.s. v a l u e o f t h e e l e c t r i c f i e l d a t t h e average p o s i t i o n i n t h e c a v i t y , i n t h e absence o f t h e sample and a i s t h e e m p i r i c a l c o n s t a n t f o r t h e sample p o s i t i o n and geometry used. F o r c a v i t y #1 t h e e l e c t r i c f i e l d i s p a r a l l e l t o t h e sample s u r f a c e and f o r a v e r y t h i n wide sample a = 1. F o r our sample a = 0 . 7 ? " T h i s n e g l e c t s t h e f a c t t h a t t h e p r e s e n c e o f t h e sample, o f d i e l e c t r i c c o n s t a n t = 15i d i s t o r t s t h e c a v i t y e l e c t r i c f i e l d d i s t r i b u t i o n . The r e s u l t i n g e l e c t r i c f i e l d n o n - u n i f o r m i t y w i t h i n t h e sample i s a l s o n e g l e c t e d . 1. T h i s e s t i m a t e comes from t h e sample d i m e n s i o n s and from measurements o f u / T ( P ) i n two c a v i t i e s w i t h E l l and _L t o t h e p l a n e o f t h e sample. 49 The e l e c t r i c f i e l d E Q can he c a l c u l a t e d from t h e d e f i n i t i o n o f the Q o f t h e c a v i t y . T h i s g i v e s f o r a c r i t i c a l l y c o u p l e d c a v i t y o f f r e q u e n c y = 3 5 GHz, Q = 4 5 0 0 , c a v i t y •a volume = 0 , 5 cm"? / — r E Q = 22 y/p V/cm where, P = power i n t o t h e c a v i t y i n mW. The maximum power o f 17 mW g i v e s E Q = 90 v/cm and E^ = 63 V/cm. The a b s o l u t e power absorbed i s b e s t measured a t f u l l l i g h t i n t e n s i t y . The l i g h t i n t e n s i t y f a l l i n g on t h e sample was 1 . 5 mW ( w i t h i n a f a c t o r o f 2) as measured by a r o u g h l y c a l i b r a t e d l e a d s u l p h i d e d e t e c t o r . The a b s o l u t e power a b s o r b e d was measured u s i n g t h e p r e c i s i o n c a l i b r a t e d a t t e n u a t o r and t h e o b s e r v a b l e p h o t o i n d u c e d change o f t h e c a v i t y c o u p l i n g on t h e o s c i l l o s c o p e mode d i s p l a y . The measured power a b s o r b e d was 0 . 2 3 mW ( + 1 0 $ ) . The e f f e c t i v e sample volume t o be used i n t h e c a l c u l a t i o n i s an e s t i m a t e o f t h e average volume o c c u p i e d by t h e e l e c t r o n s . I n t h i s measurement t h e e l e c t r o n s were c r e a t e d p a r t l y a t t h e s u r f a c e and p a r t l y i n t h e b u l k . I n v i e w o f t h e d i f f u s i o n l e n g t h o f ~30/A ( c a l c u l a t e d l a t e r ) and t h e average b u l k c r e a t i o n d e p t h o f ~100yu. t h e e f f e c t i v e t h i c k n e s s o f t h e volume o c c u p i e d by t h e e l e c t r o n s i s e s t i m a t e d t o be 60yx . 2 U s i n g t h i s t h i c k n e s s and an a r e a o f 1 mm t h e e f f e c t i v e sample volume i s V g = 0 . 0 6 mm^. T h i s g i v e s a d.c. p h o t o c o n d u c t i v i t y o f 0* = 0 . 0 2 0 (_a c m)" 1. T h i s i s comparable t o r e p o r t e d d a r k c o n d u c t i v i t i e s o f about 0 , 1 - 0 , 0 0 1 (-A-cm)"1. ( E f f e r and E t t e r 1964-, Amirkanov and Amirkanova 1 9 6 8 ) T h e r e f o r e t h e p h o t o c o n d u c t i v i t y does n o t a p p r e c i a b l y a f f e c t t h e microwave s k i n d e p t h , w h i c h i s about 1 cm. ' From t h e measured aQ and T = 1.5 x 10" s (ur-T = 3) t h e c a r r i e r c o n c e n t r a t i o n i s c a l c u l a t e d t o be 11 -3 N = 1.5 x 10 cm J w i t h i n a f a c t o r o f f o u r . The number o f e l e c t r o n s i n v o l v e d i s n = NV = 9 x 10 e l e c t r o n s , s U s i n g t h i s v a l u e o f n and t h e measured power a b s o r p t i o n , t h e r a t e o f energy g a i n p e r c a r r i e r was c a l c u l a t e d . F o r e v e r y i n t e r v a l o f t i m e i n w h i c h t h e e l e c t r o n has one c o l l i s i o n (on t h e average) t h e r e i s a n e t energy g a i n o f -~ 40 f\ u/ Q, o r f o r t y microwave q u a n t a . T h i s means t h a t t h e u s u a l quantum m e c h a n i c a l c a l c u l a t i o n o f t h e power absorbed, w h i c h uses f i r s t o r d e r t i m e dependent p e r t u r b a t i o n t h e o r y f o r t h e e l e c t r i c d i p o l e t r a n s i t i o n r a t e , does n o t a p p l y . The c o n d i t i o n t h a t t h e t ime o f t h e t r a n s i t i o n be much l o n g e r t h a n the t ime c o r r e s p o n d i n g t o the energy l e v e l d i f f e r e n c e , ( S l a t e r 1968) i s v i o l a t e d . The quantum c a l c u l a t i o n r e q u i r e d here, w h i c h must i n c l u d e t h e e f f e c t o f c o l l i s i o n s , 2 2 has not been done. * Because o f t h i s , t h e f a c t o r f ( E ^ ), o r f ( P ) , has been i n t r o d u c e d i n t h e c o n d u c t i v i t y f o r m u l a . f ( P ) can be e s t i m a t e d a t -6 dB from th e plasma s h i f t d i s c u s s e d i n s e c t i o n 3,4 ( g ) . The c a l c u l a t e d c a r r i e r d e n s i t y (assuming f ( P ) = 1) g i v e s a plasma f r e q u e n c y o f ur = 2.5 x 10 1 0 r a d / s . T h i s s h o u l d r e s u l t i n a plasma s h i f t o f 1.5%. The o b s e r v e d s h i f t i s 3,0+1.0%, 1. The s e m i c o n d u c t o r s k i n d e pth i s l a r g e r t h a n t h e m e t a l l i c s k i n d e pth ( F r e i and S t r u t t i960) because d i s p l a c e m e n t c u r r e n t s are l a r g e r t h a n c o n d u c t i v i t y c u r r e n t s ((K€Q)O/ > a) . 2. The e x a c t quantum problem i n v o l v i n g a r b i t r a r i l y l a r g e o s c i l l a t i n g e l e c t r i c f i e l d s , b u t i g n o r i n g c o l l i s i o n s , has been s o l v e d by Budd (1967). 50a w h i c h i s c o n s i s t e n t w i t h f ( P ) = 1, c o n s i d e r i n g t h a t N i s measured o n l y w i t h i n a f a c t o r o f f o u r . F u r t h e r m o r e , t h e plasma s h i f t shows t h a t n o t a l l o f the dependence Y(P) i s c o n t a i n e d i n t h e f a c t o r f ( P ) ( r e c a l l Y(P) a f ( P ) N ( P ) ) . That i s , t h e c a r r i e r c o n c e n t r a t i o n i s power dependent. Two more d e d u c t i o n s can he made from th e v a l u e n = 9 x 10 6. ( i ) S i n c e t h e measured s i g n a l - t o - n o i s e r a t i o i s -~5000, t h e minimum d e t e c t a b l e number o f e l e c t r o n s i s -~2000. ( i i ) From t h e r a t e e q u a t i o n s f o r t h i s system ( P a r s o n s 1971) one can show t h a t t h e s t e a d y s t a t e number o f e l e c t r o n s i s r e l a t e d t o t h e e l e c t r o n g e n e r a t i o n r a t e , G, and l i f e t i m e , t e , by n = G t g . The g e n e r a t i o n r a t e i s c a l c u l a t e d t o be G = 5.2 x IO 1-' e l e c t r o n - h o l e p a i r s / s e c o n d . Assuming f ( P ) = 1 t h i s g i v e s t = 1.7 x 10 7 s. T h i s i s s m a l l e r t h a n t h e e q u i l i b r i u m v a l u e r e p o r t e d by Parsons o f t = 7 x 10 s but l a r g e r t h a n Habegger's (1964) e s t i m a t e o f 10~9 - 10" 1 0 s. 51 I n t h e hot e l e c t r o n model (see C h a p t e r 5) t h e e l e c t r o n s are assumed t o be h e a t e d by t h e microwave f i e l d t o an average e n e r g y o f 15 meV. At t h a t energy t h e average c y c l o t r o n - m o t i o n r a d i u s i s f " c = 1 . 3 J X . The d i f f u s i o n l e n g t h i s e s t i m a t e d f o r hot e l e c t r o n s o f average energy — 1 ~~2 € = m*v by t h e f o l l o w i n g method. The d i f f u s i o n c o n s t a n t i s r e l a t e d t o t h e p a r t i c l e speed d i s t r i b u t i o n and s c a t t e r i n g t i m e by D Q = i v 2 f ( R e i f 1 9 6 5 ) . Prom t h e d i f f u s i o n e q u a t i o n t h e mean squared d i s p l a c e m e n t i n t i m e t a l o n g one a x i s f o r a t h r e e d i m e n s i o n a l problem i s Z 2 ( t ) = 2 D Q t i f Z ( 0 ) = 0 . S i n c e t h e q u a n t i t y o f i n t e r e s t h ere i s an e s t i m a t e o f t h e d i s t a n c e t r a v e l l e d by d i f f u s i o n a l o n g one d i r e c t i o n on one a x i s , we t a k e L ( t ) = \/D Qt f o r t h e l e n g t h t r a v e l l e d , on t h e average, by c a r r i e r s towards o r away from t h e sample s u r f a c e . W i t h "e" = 15 meV, X = 1 . 0 x 1 0 " 1 1 s ( u / T = 2) and t = t j = ? x 1 0 " 9 s one o b t a i n s L = 6 0 / t . A p o s s i b l y i m p o r t a n t c o r r e c t i o n t o t h i s i s t h e dependence o f t h e d i f f u s i o n c o n s t a n t on t h e m a g n e t i c f i e l d . F o r a m a g n e t i c f i e l d p a r a l l e l t o t h e sample s u r f a c e t h e d i f f u s i o n a l m o t i o n i n t o t h e sample ( p e r p e n d i c u l a r t o B) i s r e d u c e d by t h e c u r v a t u r e i n t h e p a t h caused by B. The d i f f u s i o n c o n s t a n t becomes (Gershenzon e t a l . 1 9 7 0 ) D DT> = . At r e s o n a n c e t h i s g i v e s 1 + ( ar cr) 3 . 4 ( f ) The S u r f a c e E f f e c t A d i s t i n c t t r a n s i t i o n from b u l k t o s u r f a c e c y c l o t r o n r e s o n a n c e has been o b s e r v e d i n s i g n a l h e i g h t ( f i g u r e 3.6), peak p o s i t i o n ( f i g u r e 3 . 7 ) . and o > T ( f i g u r e 3 . 8 ) . The p u b l i s h e d a b s o r p t i o n c o e f f i c i e n t d a t a ( J o h n s o n and Fan, 1 9 6 5 ) i n f i g u r e 3 . 6 ( b ) show t h a t a " 1 > 2 0 / t f o r t h e w a v e l e n g t h range 1.54yx < \ < 1 . 5 7 y x . The v a l u e o f a - 1 i s a p p r o x i m a t e l y t h e mean d e p t h o f c a r r i e r c r e a t i o n , a - 1 > 2Cy*. i s a r e g i o n o f b u l k c a r r i e r c r e a t i o n as compared t o s u r f a c e c a r r i e r c r e a t i o n a t a " 1 = 2jx f o r X < 1 .525yU ( h y > E g ) . F i g u r e 3 . 7 shows t h a t t h e peak parameter i s w a v e l e n g t h independent f o r X ^ 1.545y<x , i n d i c a t i n g t h a t t h e t r a n s i t i o n t o t h e e n t i r e l y b u l k s i g n a l i s complete. F i g u r e 3 . 7 a l s o shows t h a t t h e peak p a r a m e t e r i n t h e b u l k i s t h e same f o r samples o f t h i c k n e s s 100yU and 3 5 0 / * . T h i s a l s o i n d i c a t e s a t r u e b u l k measurement. The peak parameter c o u l d n o t be measured a t X = I . 5 6 / * f o r t h e 1 0 0/c t h i c k sample as t h e s i g n a l h e i g h t a t t h a t w a v e l e n g t h was s u b s t a n t i a l l y r e d u c e d . At t h i s t h i c k n e s s most o f t h e r a d i a t i o n p a s s e s t h r o u g h t h e sample, as e x p e c t e d from t h e a b s o r p t i o n c o e f f i c i e n t . The measured b u l k peak p a r a m e t e r i s a l s o i n d e p e n d e n t o f s u r f a c e t r e a t m e n t , b e i n g t h e same f o r b o t h ground and e t c h e d samples. A l l samples r o u t i n e l y u sed i n t h e measurements a r e t h i c k e r t h a n 250yU- . U n l e s s o t h e r w i s e n o t e d , a l l r e p o r t e d measurements a r e i n t h e b u l k . The s i g n a l h e i g h t was measured w i t h a v e r y narrow s l i t w i d t h f o r one sample ( f i g u r e 3 . 8 ) . T h i s was done t o 53 F i g u r e 3 . 6 The w a v e l e n g t h dependence o f (a) t h e s i g n a l and background h e i g h t s , and (b) t h e p u b l i s h e d a b s o r p t i o n c o e f f i c i e n t , a ( a f t e r J ohnson and Fan (I965)). 5 4 0 r 5 3 0 5 2 0 Ul Ul a 51'0f CD CO 5 0 0 4 9 0 h 5 2 8 • 4 - -* I / (b) 1 0 0 JX 1.50 1-51 1-52 1-53 . . -1 .54 1-55 1-56 1-57 X(/JL) 1-51 1-52 1-53 1-54 1-55 F i g u r e 3*7 The w a v e l e n g t h dependence o f t h e peak parameter B Q f o r two samples, (a) 350 u t h i c k , (b) 100JA t h i c k . 55 F i g u r e 3.8 The w a v e l e n g t h dependence o f (a ) "UTT ' , and (b) t h e s i g n a l h e i g h t , f o r t h e B a t t e l l e sample. 56 compare i t t o t h e p u b l i s h e d a b s o r p t i o n c o e f f i c i e n t ( f i g u r e 3.6(b)) wh i c h e x h i b i t s e x c i t o n a b s o r p t i o n . At w a v e l e n g t h s where e x c i t o n s a r e c r e a t e d a t an a p p r e c i a b l e r a t e , one might e x p e c t a r e d u c t i o n i n t h e s t e a d y s t a t e e l e c t r o n d e n s i t y . The f a c t t h a t t h i s i s n o t ob s e r v e d i n d i c a t e s t h a t t h e c r e a t e d e x c i t o n s a r e q u i c k l y i o n i z e d i n t o f r e e e l e c t r o n s and h o l e s . A n o t h e r p r o o f t h a t e x c i t o n s have no e f f e c t on t h e s i g n a l i s t h a t t h e measured b u l k peak parameter i s t h e same i n two samples w i t h d i f f e r e n t e x c i t o n p r o p e r t i e s . F o r example, t h e B e l l and H o w e l l sample showed s h a r p e x c i t o n a b s o r p t i o n peaks and t h e Monsanto sample d i d n o t , i n measurements done i n t h i s l a b o r a t o r y a t 1.7K by R i c k a r d s (1972). I t i s n o t t h e purpose o f t h i s t h e s i s t o s t u d y t h e s u r f a c e a f f e c t e d c y c l o t r o n r esonance s i g n a l . I t has been shown t h a t t h i s e f f e c t can be a v o i d e d t o measure a t r u e b u l k e f f e c t i v e mass. N e v e r t h e l e s s , some f u r t h e r remarks on t h i s e f f e c t can be made: ( i ) S i n c e t h e t r a n s i t i o n t o t h e b u l k s i g n a l i s complete i n a sample o f 100/t t h i c k n e s s , t h e s u r f a c e a f f e c t e d ? d e p t h i s much l e s s t h a n IOOJUL. ( i i ) One measurement o f t h e s u r f a c e s i g n a l was; made on a sample o f t h i c k n e s s ^.lOyu.. The sample had a h i g h l y p o l i s h e d s u r f a c e p r e p a r e d by g r i n d i n g and was mounted on g l a s s . T h i s s i g n a l peaked a t 300 G and was v e r y asymmetric as though 8 = - 0.5. T h i s c o n f i r m s t h a t t h e s u r f a c e e f f e c t i n c r e a s e s as sample t h i c k n e s s d e c r e a s e s . ( i i i ) Any t h e o r y o f t h i s e f f e c t must t a k e i n t o a c c o u n t t h e l e n g t h s i n v o l v e d i n t h e problem. The d e p t h o f c a r r i e r c r e a t i o n i s a " 1 - 2jl. F o r hot e l e c t r o n s , t h e average r a d i u s o f a c y c l o -t r o n o r b i t a t ctr^ = uf i s rQ = 1.3yA, and t h e average d i f f u s i o n l e n g t h i s L = 30yU. I t i s a l s o i m p o r t a n t t o note t h a t t h e e f f e c t measured i s an average which can be c a l c u l a t e d u s i n g - r (B) - m * v and L( B) - V3 T T T , c x ' eB l + {uso<rr) 2 ( i v ) To t h e a u t h o r ' s knowledge, t h i s i s t h e f i r s t r e p o r t o f a s u r f a c e e f f e c t on t h e peak p o s i t i o n i n c y c l o t r o n r e s o n a n c e . T h i s t e c h n i q u e has been used t o s t u d y b u l k and s u r f a c e r e c o m b i n a t i o n r a t e s ( Gershenzon e t a l . 1 9 7 0 ). I t may be t h a t t h e s u r f a c e e f f e c t depends on h a v i n g a hot e l e c t r o n d i s t r i b u t i o n , s i n c e t h e n r ^ a " 1 . A l s o , r e l a t i v e l y few s t u d i e s o f hot c a r r i e r c y c l o t r o n resonance have been r e p o r t e d . 3.4(g) The Power Dependent S i g n a l I n t e n s i t i e s The power dependence o f t h e s i g n a l h e i g h t , S ( P ) ( = H1(P) on f i g u r e 3 . i ) i s shown on f i g u r e 3.9. The s i g n a l w i l l be i n t e r -p r e t e d as c o n t a i n i n g c o n t r i b u t i o n s from b o t h e l e c t r o n and h o l e c o n c e n t r a t i o n s o f Ne(P) and Nh(P) r e s p e c t i v e l y . The t o t a l c a r r i e r c o n c e n t r a t i o n i s N(P) = Ne(P) + Nh(P). As shown i n s e c t i o n 2 . 2 , 1/2 S(P) a c R ( P ) P ' f o r a s p e c t r o m e t e r o p e r a t e d a t c o n s t a n t c r y s t a l c u r r e n t . D e f i n e t h e s i g n a l i n t e n s i t y as -oo oo , — Y(P) = J 0 R ( B s P ) dB = f ( P ) J oR°(B) dB 0 0 2 o N P "Ty where o R i s t h e c l a s s i c a l o R g i v e n by e q u a t i o n 3 . 2 and oQ = — — Oj°(B) dB a N f o r one t y p e o f c a r r i e r . 0 K Assuming f ( P ) i s t h e same f o r b o t h e l e c t r o n s and h o l e s Y(P) a f ( P ) N ( P ) . Now f a R(B»P) dB a ( a R h e i g h t ) ( a R w i d t h ) a a R ( B 0 , j P ) ( a / T ( P ) ) " 1 . The use o f t h i s f o r m u l a assumes t h a t t h e a r e a under t h e o b s e r v e d F i g u r e 3.9 The power dependence o f t h e o b s e r v e d s i g n a l h e i g h t , S. c y c l o t r o n resonance s i g n a l i s t h e same as t h e a r e a under t h e f i t t e d c u r v e . The maximum d i f f e r e n c e i n v o l v e d here i s 7% a t h i g h powers and i s n e g l e c t e d . The o t h e r a p p r o x i m a t i o n , as i n d i c a t e d , i n v o l v e s a maximum e r r o r o f 5% f o r U/^C =1 . 8 and i s a l s o n e g l e c -t e d . The c a l c u l a t e d q u a n t i t y Y(P) a S(B^;P) P " 1 / / 2 ( u / T ( P ) ) ~ 1 i s p l o t t e d on f i g u r e 3-10. Y(P) undergoes a change o f s l o p e a t a p p r o x i m a t e l y - 6 dB. T h i s i s where uJ'T. ( p ) , p l o t t e d on f i g u r e 4.2, s t a r t s t o d e c r e a s e a p p r e c i a b l y as P i n c r e a s e s . The low power s t r a i g h t l i n e o f f i g u r e 3.10 (and i t s e x t e n s i o n ) i s i n t e r -p r e t e d as t h e power dependent e l e c t r o n s i g n a l i n t e n s i t y Y e ( P ) . The e x c e s s c a r r i e r s c o n t a i n e d i n t h e h i g h power s i g n a l a r e i n t e r -p r e t e d as l i g h t h o l e s . The d i f f e r e n t h o l e peak p o s i t i o n and LA/'V r e s u l t i n a d e c r e a s e o f t h e measured UJ-T:, as c a l c u l a t e d i n s e c t i o n 6.1(b). T h i s i n t e r p r e t a t i o n i s s u p p o r t e d by t h e d a t a o f f i g u r e 3 . H . The h i g h power d a t a p o i n t s (x) are deduced from Y(P) and t h e low power d a t a p o i n t s (o) are t a k e n from t h e c i r -c u l a r p o l a r i z a t i o n measurements, . The a b s o l u t e power f o r t h e second s e t o f d a t a , t a k e n i n a d i f f e r e n t c a v i t y o f l o w e r Q and more u n d e r c o u p l e d , was a d j u s t e d t o make t h e two c u r v e s c o i n c i d e . The a d j u s t e d v a l u e o f t h e power i s i n good agreement w i t h t h e e s t i m a t e d c o r r e c t i o n . The measured power dependence i s Ye(P) a p 0 , 29±.05 a pO.831.16^ 1. T h e r e f o r e y ( p ) a p1.12+.21e and ^ I f f ( P ) c o u l d be c a l c u l a t e d t h e n t h e e l e c t r o n c o n c e n t r a t i o n Ne(P) c o u l d be c a l c u l a t e d from Y e ( P ) . I f t h e e l e c t r o n g e n e r a t i o n r a t e were f u r t h e r assumed t o be independent o f power t h e n t h e e l e c t r o n l i f e t i m e , t e ( P ) c o u l d be c a l c u l a t e d . The same c a l -1* Y e ( p j = Ne(P'} s ^ n c e ^ s assumed the same f o r b o t h c a r r i e r t y p e s . 3 0 0 n f i o o £ O n o >-"(/> c o +-> - 10 o c Y <* P n n= . 2 9 Point X o - 2 4 L i g h t Intensity 1.00 . 2 8 8 .101 .017 n = .571.05, n= - 5 0 n= . 4 6 n = . 4 7 -18 -12 - 6 P o w e r , P ( d B ) - 3 0 F i g u r e 3.10 The power dependence o f t h e s i g n a l i n t e n s i t y . 61 Power, P ( d B) F i g u r e 3.11 The power dependence o f t h e r a t i o o f h o l e - t o -e l e c t r o n c o n c e n t r a t i o n s . The d a t a p o i n t s (x) a r e d e t e r m i n e d from t h e a r e a u nder t h e cu r v e a t h i g h power i n t h e l i n e a r p o l a r i z a t i o n c a v i t y . The d a t a p o i n t s (o) a r e d e t e r m i n e d d i r e c t l y from t h e c i r c u l a r p o l a r i z a t i o n measurements. c u l a t i o n c o u l d n o t be a p p l i e d t o t h e h o l e s . The power dependence o f t h e h o l e g e n e r a t i o n r a t e has n o t been e s t a b l i s h e d . I f h o l e s were c r e a t e d by impact i o n i z a t i o n o f n e u t r a l a c c e p t o r s , as c a l -c u l a t e d i n s e c t i o n 4.3(e), t h e g e n e r a t i o n r a t e would be power dependent. No p r e s e n c e o f heavy h o l e s was e v e r d e t e c t e d . Because o f t h e i r l a r g e r mass t h e y have a s m a l l e r m o b i l i t y . I f t h e heavy h o l e s are n o t h e a t e d by t h e microwave f i e l d and have t h e aft: v a l u e measured by S t r a d l i n g (1966), t h e n t h e y would o n l y become d e t e c t a b l e a t a c o n c e n t r a t i o n e q u a l t o Ne. I t i s n o t u n r e a s o n a b l e t h a t t h e y were not o b s e r v e d . F i g u r e 3.12 shows t h e c a r r i e r c o n c e n t r a t i o n as a f u n c t i o n o f l i g h t i n t e n s i t y . F o r f i x e d power f ( P ) = c o n s t a n t and Y ( I ) a N ( l ) . 3.4(h) The Plasma S h i f t F o r measurements i n t h e b u l k , a narrow s p e c t r o m e t e r s l i t w i d t h and hence l o w e r l i g h t i n t e n s i t y i s u s e d . These measurements show a b a r e l y d e t e c t a b l e plasma s h i f t at -6 dB and none a t l o w e r powers. To s e a r c h f o r a plasma s h i f t more c a r r i e r s must be g e n e r a t e d by a h i g h e r l i g h t i n t e n s i t y . T h i s u n f o r t u n a t e l y r e q u i r e s a s l i t w i d t h ^500 A5 and hence th e o b s e r v a t i o n o f a m i x t u r e o f b u l k and s u r f a c e s i g n a l s . Under t h i s c o n d i t i o n t h e plasma s h i f t o f f i g u r e 3.13 was o b s e r v e d . The e x p e c t e d peak p a r a m e t e r s h i f t i s , from e q u a t i o n 3.5* — - g ^ - a . ur a N(P,I) a I . The o b s e r v e d s h i f t a t -6 dB i s d i r e c t l y p r o p o r t i o n a l t o t h e r e l a t i v e l i g h t i n t e n s i t y . S i n c e t h e plasma s h i f t i s s m a l l e r a t l o w e r powers, i t i s c o n c l u d e d t h a t t h e r e i s a r e a l dependence N ( P ) , c o n t a i n e d i n Y(P) a f ( P ) N ( P ) . ; -2 log I -1 0 F i g u r e 3.12 The c a r r i e r c o n c e n t r a t i o n as a f u n c t i o n o f l i g h t i n t e n s i t y . Relative Light Intensity (%>) £ F i g u r e 3.I3 The plasma s h i f t f o r a s i g n a l c o n t a i n i n g b o t h b u l k and s u r f a c e c a r r i 65 CHAPTER 4 THE SCATTERING MECHANISM There was no "One, two, t h r e e and. away," but t h e y began r u n n i n g when t h e y l i k e d and l e f t o f f when t h e y l i k e d , so t h a t i t was n o t e a s y t o know when t h e r a c e was o v e r . L e w i s C a r r o l l . 4.1 The Measured a / T F i g u r e s 4.1(a) and (b) show f o u r e x p e r i m e n t a l a b s o r p t i o n c u r v e s t a k e n a t d i f f e r e n t microwave power l e v e l s , and t h e t h e o r e t i c a l f i t t o each. The b a s e l i n e i s drawn t o e x c l u d e t h e f i e l d i n dependent background and t h e s i g n a l a t h i g h e r m a g n e t i c f i e l d s i s n o t shown because i t c o n t a i n s no s t r u c t u r e . The adopted c u r v e f i t t i n g p r o c e d u r e a d j u s t s UJX and B t o s a t i s f y two c o n d i t i o n s : ( i ) The t h e o r e t i c a l c u r v e f i t t h e e x p e r i m e n t a l c u r v e e x a c t l y a t t h e m a g n e t i c f i e l d s B = B^, 1/2 B^ and 3 / 2 B^ where B^ i s t h e measured peak p o s i t i o n . ( i i ) I t i s f u r t h e r a d j u s t e d t o make t h e d i f f e r e n c e between t h e o r y and e x p e r i m e n t t h e same a t B = 0 and 2 B Q . The f i t t i n g cannot be done a n a l y t i c a l l y ( e s s e n t i a l l y because B^ ^ B Q, t h e eB f i e l d f o r t h e r e s o n a n c e c o n d i t i o n , -~9 • = o r ) b u t i s done n u m e r i c a l l y t o v e r y h i g h a c c u r a c y . T h i s method o f c u r v e f i t t i n g was chosen f o r s e v e r a l r e a s o n s . F i r s t o f a l l , t h e r e are two problems i n s i m p l y m e a s u r i n g t h e w i d t h a t h a l f maximum. The 1. Q o f t h e l i n e i s o n l y t h e same as t h e u/X v a l u e f o r c i r c u l a r p o l a r i z a t i o n measurements, where t h e l i n e s h a p e i s 1. The Q i s the.peak p o s i t i o n d i v i d e d by t h e f u l l w i d t h a t h a l f h e i g h t . J 1 I I 1 Magnetic Field (arb. units) F i g u r e 4.1 (a) The o b s e r v e d s i g n a l a t h i g h e r powers. 67 T 1 1 ! 1 1 experiment theory J 1 i i " i Magnetic Field (arb. units) F i g u r e 4.1 (b) The o b s e r v e d s i g n a l a t l o w e r powers. 68 L o r e n t z i a n . A l s o , a h a l f - w i d t h cannot he d e f i n e d f o r 6 t / f < 1.4 s i n c e i n t h a t case a R ( 0 ) > ^ ^ R ^ Q ) F O R l i n e a r p o l a r i z a t i o n s i g n a l . F u r t h e r m o r e , s i n c e a d e t e r m i n a t i o n o f 8 i s a l s o r e q u i r e d , i t i s n e c e s s a r y t o use more t h a n two p o i n t s from t h e e x p e r i m e n t a l c u r v e . The e r r o r quoted f o r t h e measured UTT i s t a k e n t o be t h e l a r g e r o f two q u a n t i t i e s ; ( i ) The minimum e r r o r r e s u l t i n g from t h e n o i s e on one s i g n a l r e c o r d i n g , and ( i i ) t h e e r r o r t a k e n as h a l f t h e d i f f e r e n c e between t h e v a l u e s d e t e r m i n e d from two s i g n a l measurements. U s u a l l y t h e l a t t e r d o m i n a t e s . The measurement o f OJT as a f u n c t i o n o f s e v e r a l p a r a m e t e r s 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 s c a t t e r i n g mechanism. A p a r t from i t s i m p o r t a n c e i n e s t i m a t i n g t h e e f f e c t i v e mass non-p a r a b o l i c i t y c o r r e c t i o n , t h e s c a t t e r i n g mechanism i s o f i n t e r e s t i n i t s own r i g h t . The r e s u l t shows t h a t t h e s c a t t e r i n g o f t h e e l e c t r o n s i s a t l e a s t p a r t l y from n e u t r a l d e f e c t a c c e p t o r s . The measurement o f t h i s s c a t t e r i n g p r o c e s s can o n l y be done w i t h p h o t o c r e a t e d e l e c t r o n s . I f e l e c t r o n s were c r e a t e d by c o m p e n s a t i n g t h e a c c e p t o r s w i t h a donor c o n c e n t r a t i o n NQ> N^, t h e n t h e s c a t t e r i n g would no l o n g e r be due t o n e u t r a l a c c e p t o r s b u t would be from i o n i z e d i m p u r i t i e s . F i g u r e 4.2 d e p i c t s t h e power dependence o f u / r f o r one sample. I t i s s i m i l a r f o r t h e f o u r samples measured. ur - f i s a p p r o x i m a t e l y independent o f power f o r low powers w h i l e i t d e c r e a s e s a t h i g h powers. The h i g h power r e s u l t i s due t o t h e p r e s e n c e o f l i g h t h o l e s . The h o l e UST. i s deduced t o be -^-2.5 f o r t h e sample i n v o l v e d . T h i s v a l u e r e s u l t s from t h e measured r a t i o ^) N e ( P ) F i g u r e 4.2 T h e power d e p e n d e n c e o f a r t . OS N O 70 d e t e r m i n e d i n s e c t i o n 3.Mg) and from t h e a n a l y s i s o f t h e measured peak s h i f t i n s e c t i o n 6.1(b). The v a l u e d e t e r m i n e d i s f o r hot h o l e s and i s t o be compared t o t h e e q u i l i b r i u m v a l u e o f UJIZ = 0.25 deduced from S t r a d l i n g ' s (1966) c y c l o t r o n r e s o n a n c e r e s u l t s . The d i f f e r e n c e i s a t t r i b u t e d t o t h e s t r o n g microwave h e a t i n g o f t h e l i g h t h o l e s . The l i g h t i n t e n s i t y dependence o f a r t i s p l o t t e d on f i g u r e 4.3. The s o u r c e o f t h e v e r y weak dependence found has no t been i d e n t i f i e d , b u t c o u l d p o s s i b l y be due t o a weak e l e c t r o n - e l e c t r o n s c a t t e r i n g . The r e s u l t s r e p o r t e d i n t h i s t h e s i s a r e from f o u r d i f f e r e n t GaSb samples o f good s i g n a l s t r e n g t h and "-rT>1.2 ("group A " ) . A n o n - r e s o n a n t p h o t o c o n d u c t i v i t y s i g n a l w i t h UJ"C < 0.8 was seen i n t h r e e o t h e r samples ("group B " ) . I n each o f t h e l a t t e r t h e s i g n a l s t r e n g t h was o n l y about 1% o f t h a t f o r t h e group A samples i n d i c a t i n g a s h o r t e r r e c o m b i n a t i o n t i m e . F u r t h e r m o r e t h e group B samples had s i g n i f i c a n t l y more da r k c o n d u c t i v i t y l o s s e s s i n c e t h e c a v i t y Q was a f f e c t e d by them t o a g r e a t e r e x t e n t . The d i f f e r e n c e s between t h e group A and B samples i n d i c a t e d t h e p r e s e n c e o f a g r e a t e r c o n c e n t r a t i o n o f a c t i v e i m p u r i t i e s i n t h e l a t t e r . T h e r e f o r e t h e s e samples w i l l l i k e l y have a d i f f e r e n t s c a t t e r i n g mechanism. S i n c e we are i n t e r e s t e d i n d e t e r m i n i n g t h e s c a t t e r i n g mechanism m a i n l y i n t h e samples used t o measure t h e e f f e c t i v e mass (gro u p A ) , t h e group B samples w i l l n o t be s t u d i e d f u r t h e r . The p r o p e r t i e s o f a l l samples examined a re l i s t e d i n Appendix J.k. The measured c u f i s independent o f t h e d i r e c t i o n o f t h e ma g n e t i c f i e l d i n t h e c r y s t a l . T h i s i s e x p e c t e d even f o r an log I t—1 F i g u r e 4 . 3 The l i g h t i n t e n s i t y dependence of-tiTTT. 72 a n i s o t r o p i c s c a t t e r i n g mechanism because t h e c y c l o t r o n s c a t t e r i n g t i m e c o n t a i n s c o n t r i b u t i o n s from a l l d i r e c t i o n s i n t h e c r y s t a l . 4.2 The Dependence o f t h e S c a t t e r i n g on A c c e p t o r C o n c e n t r a t i o n A l l samples s t u d i e d had an a/"f (P) dependence s i m i l a r t o t h a t o f f i g u r e 4.2. The low power v a l u e o f a r c a t t h e b r o a d peak o f t h e c u r v e i s denoted US-Vr and i t s i n v e r s e ( u/-£r ) ~ 1 L L r e p r e s e n t s t h e low power r e l a t i v e s c a t t e r i n g r a t e . The l a t t e r w i l l be compared 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 d e t e r m i n e d by d.c. H a l l measurements a t t h e two t e m p e r a t u r e s o f 300 K and 77 K. The d.c. H a l l measurements were t a k e n as th e average o f f o u r measurements i n c l u d i n g b o t h m a g n e t i c f i e l d d i r e c t i o n s and b o t h c u r r e n t d i r e c t i o n s . T h i s average e l i m i n a t e s two o f t h e t h r e e t h e r m o - g a l v a n o - m a g n e t i c e f f e c t s w h i c h a f f e c t d.c. measurements ( P u t l e y , 1968). B o t h o f t h e s e e f f e c t s were f o u n d t o be s m a l l and t h e t h i r d , w h i c h i s u s u a l l y s m a l l e r , was n e g l e c t e d . The f i v e - l e a d d u m b e l l shaped samples were c u t w i t h t h e s p a r k e r o s i o n c u t t e r and t h e l e a d s were s o l d e r e d onto f r e s h l y ground s u r f a c e s i n a i r . The c o n t a c t s were' ohmic up t o lOV/cm a t b o t h t e m p e r a t u r e s . The c a r r i e r c o n c e n t r a t i o n i s r e l a t e d t o t h e measured H a l l c o e f f i c i e n t t h r o u g h t h e H a l l s c a t t e r i n g f a c t o r , r . At room t e m p e r a t u r e t h e main s c a t t e r i n g mechanism i s p o l a r mode phonon s c a t t e r i n g . F o r a Debye t e m p e r a t u r e o f 9 Q = 346 K we have Q^/l = 1.15}' T h i s g i v e s r = 1.04 + .04 ( S t i l l m a n e t a l . 1970). We w i l l t a k e r ^ n n = 1.0. The s c a t t e r i n g a t 1. See Appendix 4.1 f o r a t a b l e o f d a t a used i n t h e c a l c u l a t i o n s . 73 77 K f o r "as grown" p-GaSb i s m a i n l y p o l a r mode s c a t t e r i n g w i t h a p o s s i b l e s m a l l c o n t r i b u t i o n from 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 , r ^ r , i s 1.05 f o r o p t i c a l mode s c a t t e r i n g ( S t i l l m a n e t a l . 1970) so a s a f e e s t i m a t e i s 1 <rr,y < 1.4. The maximum change i n N A r e s u l t i n g from a l t e r i n g r ^ r , from 1 t o 1.4 i s o n l y 3%» w h i c h i s not i m p o r t a n t h e r e . r3oo w a s i n d e P e n ( * e n t o f m a g n e t i c f i e l d t o 15 DG but d e c r e a s e d l i n e a r l y w i t h f i e l d and rea c h e d a v a l u e a t 15 KG 22% s m a l l e r t h a n t h a t a t 0 KG. The measurements were t a k e n a t 15 KG. E f f e r and E t t e r (1964) assume, i n t h e i r model, t h a t a c c e p t o r s o f c o n c e n t r a t i o n have a s i n g l e a c t i v a t i o n energy of about 30 meV, and t h a t t h e y are p a r t l y compensated by a s m a l l e r donor c o n c e n t r a t i o n N^. The r e s u l t i n g m a s s - a c t i o n law i s N A - ND - P _ e ^ " c T " 3 / 2 P ( N D + p) C J ? where p = h o l e c o n c e n t r a t i o n i n t h e v a l e n c e band c = r a t i o o f t h e a c c e p t o r s t a t e s p i n d e generacy t o t h e te m p e r a t u r e independent p a r t o f t h e v a l e n c e band e f f e c t i v e d e n s i t y o f s t a t e s . c " 1 was d e t e r m i n e d by E f f e r and E t t e r t o be 4.39 x 10 /cm^. Us i n g t h e i r model and assuming N Q << p r , 7 , r3Q0 = 1 a n d r ^ y = 1-1.4, v a l u e s were deduced f o r N A and E A f o r t h e samples used. i s about 10% l a r g e r t h a n P^o* T h e r e f o r e , even i f t h e model i s o n l y a p p r o x i m a t e l y c o r r e c t t h e deduced r e l a t i v e v a l u e s o f N A s h o u l d be q u i t e a c c u r a t e . The d a t a f o r E A ( N A ) show t h e e x p e c t e d r e d u c t i o n o f a c t i v a t i o n energy 1/3 g i v e n by t h e f o r m u l a E f l = E^ - q ( N A - N p) where E j L i s 1. The d a t a E A ( N A ) are t a b u l a t e d i n Appendix 3.4 but are n o t f u r t h e r a n a l y s e d because o f t h e problem d i s c u s s e d below. 74 t h e d i l u t e a c t i v a t i o n e n e rgy and a i s a c o n s t a n t . (Van Mau e t a l . 1970 f o r GaSb, P e a r s o n and Bardeen 1949 f o r S i , Debye and C o n w e l l 195^ f o r Ge) The measurements a t two t e m p e r a t u r e s cannot be used t o c a l c u l a t e t h e t h r e e q u a n t i t i e s NA' ND a n d EA^ NA' ND^* Fu r t h e r m o r e t h e model may be t o o s i m p l i f i e d ( s e e s e c t i o n 3.3). T h e r e f o r e one cannot d e r i v e t h e donor c o n c e n t r a t i o n from t h e s e measurements. F i g u r e 4.4 shows the r e l a t i v e s c a t t e r i n g r a t e ( c c - f ^ ) " 1 p l o t t e d v s . t h e a c c e p t o r c o n c e n t r a t i o n . The v a l u e s o f o r t - ^ are each t h e average o f s e v e r a l measurements. The l i n e a r dependence o f t h e s c a t t e r i n g r a t e on a c c e p t o r c o n c e n t r a t i o n i s e v i d e n c e t h a t t h e s c a t t e r i n g i s a t l e a s t p a r t l y due t o n e u t r a l a c c e p t o r s . An i m p o r t a n t c o r r e c t i o n t o t h e measured curve i s t h e c o r r e c t i o n f o r t h e l i g h t i n t e n s i t y dependence o f u / T w h i c h r e p r e s e n t s an a d d i t i o n a l s c a t t e r i n g mechanism. I n o r d e r t o d e t e r m i n e t h e s c a t t e r i n g r a t e due t o a c c e p t o r s i t i s r e a s o n a b l e t o s u b t r a c t o f f t h i s c o n t r i b u t i o n . F o r t h e B a t t e l l e sample, an e s t i m a t e o f t h e a d d i t i o n a l s c a t t e r i n g r a t e i s o b t a i n e d from t h e d a t a o f f i g u r e s 4.2 and 4.3. The a d d i t i o n a l s c a t t e r i n g r a t e i s s een t o be a p p r o x i m a t e l y independent o f power and i s e s t i m a t e d as A ( o i / T ^ ) ~ ^ = 0.060 f o r a d e c r e a s e o f l i g h t i n t e n s i t y by a f a c t o r o f 60. I f t h e a d d i t i o n a l s c a t t e r i n g depends on t h e c a r r i e r con-c e n t r a t i o n t h e same c o r r e c t i o n s h o u l d a p p l y t o each o f t h e f o u r samples s i n c e each has t h e same e l e c t r o n c o n c e n t r a t i o n ( w i t h i n a f a c t o r o f two) a t a g i v e n l i g h t i n t e n s i t y . The c o r r e c t e d c u r v e o b t a i n e d by s u b t r a c t i n g t h e same s c a t t e r i n g r a t e from each p o i n t i s shown as t h e dashed c u r v e o f f i g u r e 4.4. 75 3 .2 .1 0 0 4 C O R R E C T E D J i i i ' i i i i ' i 8 12 16 20 24 16 - 3 N ( 10 c m ) Figure 4 . 4 The r e l a t i v e s c a t t e r i n g r a t e vs. the acceptor c o n c e n t r a t i o n . The s o l i d l i n e represents the four measured p o i n t s and the dashed l i n e i n c l u d e s the c o r r e c t i o n s from U / f ( I ) , 76 I t appears t h e r e i s s t i l l a r e s i d u a l s c a t t e r i n g p r o c e s s w h i c h i s independent o f a c c e p t o r c o n c e n t r a t i o n . One way t o a c c o u n t f o r i t i s t o e x t r a p o l a t e t h e uJT ( I ) c o r r e c t i o n . I t i s a l s o p o s s i b l e t h e r e i s a n o t h e r s c a t t e r i n g mechanism n o t t a k e n i n t o a c c o u n t . 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 i s u n l i k e l y because t h e e s t i m a t e o f s e c t i o n 4 . 3 p r e d i c t s t h a t i t s h o u l d be s m a l l and t h a t i t s h o u l d d i f f e r f o r each sample. There a r e two i n h o m o g e n e i t y e f f e c t s which might be e x p e c t e d t o broaden t h e s i g n a l . Sample i n h o m o g e n e i t y has been r e p o r t e d by S t r a d l i n g ( I966). F o r v e r y s m a l l samples c u t from t h e same m a t e r i a l t h e UJT v a l u e s d i f f e r e d by up t o 5 0 % . I t has been shown by computer s t u d i e s t h a t t h e average b r o a d e n i n g from t h i s e f f e c t would be s m a l l and v e r y n e a r l y t h e same f o r each v a l u e o f N.. A The second i n h o m o g e n e i t y e f f e c t i s t h e inhomogeneous b r o a d e n i n g o f t h e l i n e due t o t h e s p r e a d i n peak p o s i t i o n s . T h i s e f f e c t , w h i c h i s due t o c o n d u c t i o n band n o n - p a r a b o l i c i t y and p o l a r o n mass c o r r e c t i o n s , has a l s o been shown t o be s m a l l , and i s n e g l e c t e d . A n o t h e r approach t o e x p l a i n t h e ( UJT'j)-1(NA) c u r v e i s t o assume t h a t a l l t h e s c a t t e r i n g i s due t o a c c e p t o r s and t h a t f o r some r e a s o n t h e s c a t t e r i n g r a t e i s n o t p r o p o r t i o n a l t o N A. One p o s s i b i l i t y i s s u g g e s t e d i n s e c t i o n 5 . 3 » i n v o l v i n g t h e f e a t u r e s o f t h e hot e l e c t r o n model. A n o t h e r p o s s i b i l i t y a r i s e s f r om t h e h i g h a c c e p t o r c o n c e n t r a t i o n . F o r N A = I O 1 ? /cxr? t h e mean d i s t a n c e between a c c e p t o r s i s 1 2 0 i\. The a c c e p t o r r a d i u s i s e s t i m a t e d t o be - ^ 5 0 (see s e c t i o n 4 . 3 ( d ) ) The 77 a c c e p t o r s a re c l o s e enough t o g e t h e r t o i n t e r a c t a p p r e c i a b l y on t h e a v e r a g e , 1 . T h i s c o u l d i n f l u e n c e t h e s c a t t e r i n g t h r o u g h e i t h e r m u l t i p l e s c a t t e r i n g p r o c e s s e s o r t h r o u g h t h e i n c r e a s e d i m p o r t a n c e o f a c c e p t o r p a i r and t r i p l e t s c a t t e r i n g a t h i g h e r a c c e p t o r c o n c e n t r a t i o n s . M u l t i p l e s c a t t e r i n g has been shown t o i n c r e a s e t h e s c a t t e r i n g f o r t h e case o f 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 (Moore I967). On t h e o t h e r hand., an attempt was 16 3 made t o f i t t h e c o r r e c t e d c u r v e ( f r o m N A = 6-20 x 10 / c n r ) by a c o m b i n a t i o n o f s c a t t e r i n g from s i n g l e and p a i r e d a c c e p t o r s . T h i s c o u l d be done by assuming t h e p a i r s c a t t e r i n g c r o s s - s e c t i o n t o be about o n e - f i f t h t h a t o f a s i n g l e a c c e p t o r . T h i s i s u n r e a s o n a b l e . I t i s c o n c l u d e d t h a t t h e e l e c t r o n s c a t t e r i n g mechanism i s i n p a r t n e u t r a l a c c e p t o r s c a t t e r i n g and i n p a r t an undeter m i n e d mechanism. 4.3 S c a t t e r i n g Mechanism C a l c u l a t i o n s 4.3 (a) The S c a t t e r i n g Time The c l a s s i c a l i n t e r p r e t a t i o n o f t h e s c a t t e r i n g p t i m e ', T~ , i s e s s e n t i a l l y c o n t a i n e d i n t h e Boltzmann e q u a t i o n ( i . e . e q u a t i o n 3.1). The quantum m e c h a n i c a l i n t e r p r e t a t i o n o f T" has been e l u c i d a t e d by Kawabata (I967) u s i n g t h e Kubo l i n e a r r e s p o n s e f o r m u l a . I t was shown t h a t t h e s c a t t e r i n g t i m e i s i n g e n e r a l r e l a t e d t o t h e t r a n s p o r t r e l a x a t i o n t i m e and n o t t o 1. E v i d e n c e o f i m p u r i t y band c o n d u c t i o n has been r e p o r t e d . (Amirkhanov and Amirkhanova 1968) 2. The s c a t t e r i n g t i m e , w h i c h w i l l a l s o be c a l l e d t h e r e l a x a t i o n t i m e o r t h e c o l l i s i o n t i m e , r e f e r s t o t h e momentum r e l a x a t i o n t i m e w h i c h i s , i n g e n e r a l , d i f f e r e n t from t h e energy r e l a x a t i o n t i m e . 78 t h e l i f e t i m e o f t h e Landau s t a t e s . The b a s i c c o n c e p t i s t h a t any p r o c e s s which d i s t u r b s t h e c i r c u l a r m o t i o n o f t h e e l e c t r o n c o n t r i b u t e s t o t h e o b s e r v e d t . U s u a l l y o n l y s c a t t e r i n g p r o c e s s e s c o n t r i b u t e t o t , bu t i f t h e e l e c t r o n l i f e t i m e i n the c o n d u c t i o n band, t e , i s comparable t o T t h e n t h e e l e c t r o n r e c o m b i n a t i o n p r o c e s s may a l s o d i s t u r b t h e m o t i o n . T h i s has been t h o r o u g h l y i n v e s t i g a t e d i n a c y c l o t r o n r e s o n a n c e s t u d y o f s i l i c o n ( O t s u k a e t a l . 1968). The r e c o m b i n a t i o n t i m e b r o a d e n i n g i s u n i m p o r t a n t i n t h e p r e s e n t 8 11 e x p e r i m e n t s i n c e t 1 0 " s >) t - ^ 1 0 " s. One purpose o f t h i s d i s c u s s i o n i s t o e s t i m a t e t h e o r e t i c a l l y t h e s c a t t e r i n g t ime f o r each p r o c e s s i n o r d e r t o i d e n t i f y t h e dominant p r o c e s s . The c a l c u l a t e d t i m e f o r t h a t p r o c e s s w i l l be compared t o t h e measured v a l u e . The c a l c u l a t i o n s a l s o p e r t a i n t o t h e hot e l e c t r o n m odel, F o r t h i s r e a s o n t h e s c a t t e r i n g t i m e w i l l be c a l c u l a t e d as a f u n c t i o n o f e l e c t r o n energy, t{€-), and t h e n a p p l i e d t o t h e case o f b o t h h o t and c o l d e l e c t r o n s . The c a l c u l a t i o n s p r o v i d e a b a s i s f o r assuming t h e hot e l e c t r o n model i n i n t e r -p r e t i n g t h e r e s u l t s . I n c o n n e c t i o n w i t h t h e model, b o t h t h e momentum r e l a x a t i o n t i m e , w h i c h i s measured by o r f , and t h e energy r e l a x a t i o n t i m e , w h i c h g o v e r n s the e l e c t r o n energy d i s t r i b u t i o n , w i l l be c a l c u l a t e d . . The s c a t t e r i n g due t o phonons, i o n i z e d i m p u r i t i e s and n e u t r a l a c c e p t o r s , i n c l u d i n g t h e impact i o n i z a t i o n p r o c e s s , w i l l be d i s c u s s e d . The GaSb c o n s t a n t s used i n t h e c a l c u l a t i o n s a r e t a b u l a t e d and d i s c u s s e d i n Appendix 4 . 1 . The c a l c u l a t i o n s and d i s c u s s i o n f o l l o w C o n w e l l ( 1 9 6 7 ) , u n l e s s o t h e r w i s e s t a t e d . 79 The r e s u l t s o f the r a t h e r l e n g t h y c a l c u l a t i o n s w i l l be summarized h e r e . I t i s shown t h a t the e l e c t r o n s n e v e r a t t a i n e n e r g i e s a p p r e c i a b l y g r e a t e r t h a n t h e v a l u e £^-30+4 meV. At l i q u i d h e l i u m t e m p e r a t u r e s when £ < the main s c a t t e r i n g p r o c e s s i s c a l c u l a t e d t o be n e u t r a l a c c e p t o r s c a t t e r i n g . The c a l c u l a t e d c o l l i s i o n t i m e i s i n good agreement w i t h t h e measured one and t h e r e s u l t depends on t h e e l e c t r o n d i s t r i b u t i o n b e i n g " h o t " . The c a l c u l a t e d t f o r n e u t r a l a c c e p t o r s c a t t e r i n g o f c o l d e l e c t r o n s r e s u l t s i n uS-V(cold) < 1. When £ > £ M t h e main s c a t t e r i n g p r o c e s s i s e i t h e r l o n g i t u d i n a l o p t i c a l (LO) phonon c r e a t i o n o r impact i o n i z a t i o n o f t h e a c c e p t o r s . E i t h e r o f t h e s e p r o c e s s e s r e s u l t s i n a l o n g s c a t t e r i n g t i m e and a l a r g e r a t e o f e n e r g y l o s s . A t a t e m p e r a t u r e o f 25 K t h e 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 t i m e i s e s t i m a t e d t o be comparable t o t h e n e u t r a l a c c e p t o r s c a t t e r i n g t i m e and t o t h e measured t i m e , f o r G < I t i s r e j e c t e d as t h e main s c a t t e r i n g mechanism a t t h a t t e m p e r a t u r e because i t has t h e wrong t e m p e r a t u r e dependence and s h o u l d n o t be t h e same f o r each sample. 4 . 3 (b) Phonori S c a t t e r i n g There a r e e i g h t p o s s i b l e i n t r a - v a l l e y phonon s c a t t e r i n g mechanisms. These r e s u l t from a l l c o m b i n a t i o n s o f t h e t h r e e f a c t o r s i n v o l v e d j ( i ) t h e p o l a r o r n o n - p o l a r n a t u r e o f t h e s c a t t e r i n g p r o c e s s , ( i i ) a c o u s t i c o r o p t i c a l phonons ( a d j a c e n t atoms moving i n phase o r out o f phase, r e s p e c t i v e l y ) , ( i i i ) t r a n s v e r s e o r l o n g i t u d i n a l phonon p o l a r i z a t i o n . These w i l l be d i s c u s s e d , by t r e a t i n g b o t h p o l a r i z a t i o n s t o g e t h e r , i n t h e o r d e r ? n o n - p o l a r a c o u s t i c , p o l a r a c o u s t i c , n o n - p o l a r Sc-eptical and polar optical mode scattering. It w i l l be shown that in each case the calculated is at least one order of magnitude larger than the measured value, and that the energy-loss rates are negligible except for the optical phonon creation process when £ >tvco"0. 1* The non-polar acoustic mode scattering, or deformation potential scattering, has been calculated by Shockley (1950) and discussed by Conwell (1967). The electron is scattered only by the longitudinal modes. The momentum co l l i s i o n time is ±.4 -2 ~Vz given by ? a c ( £ ) = " * / V £ <V> where 3 = the material density u = an average longitudinal speed of sound E^ n = the conduction band deformation potential constant k = the Boltzmann constant, o The temperature enters through the equilibrium phonon population. For GaSb one obtains, T" ( 6 ) = 5.6 x 10" 9 ( e') "^T" 1 where G' is the electron energy, £, expressed in units of 10 meV. This allows quick estimates by taking £' ^  1 for hot electrons. For hot electrons at the highest temperature pertaining to .the experiment ( £' = 1 . 5 and T = 30 K) the scattering time is " l " „ = 1.6 x 1 0 ~ 1 0 s. As w i l l be seen later, SIC of a l l the phonon scattering mechanisms this one comes closest to the measured value of t = 1.0 x 1 0 " 1 1 s ( artr = 2 ) . For hot electrons at lower temperatures or for cold electrons at any temperature attained in the experiment, the scattering is completely negligible. 1. i s "the LO phonon energy at k = 0. 81 The energy r e l a x a t i o n t i m e , which d i f f e r s from t h e momentum r e l a x a t i o n t i m e i s t e m p e r a t u r e independent and i s 8 _ 1 / 2 g i v e n by T o r v o r,( & ) = 6 x 10" ( £ ' ) s. The c o r r e s p o n d i n g energy l o s s r a t e i s n e g l i g i b l e compared t o t h e o b s e r v e d n e t r a t e o f energy g a i n . T h i s i d e a was used by Habegger and Fan (1964) t o e x p l a i n t h e o b s e r v e d o s c i l l a t o r y p h o t o c o n d u c t i v i t y as a f u n c t i o n o f e l e c t r o n e n e rgy when t h e e l e c t r i c f i e l d was v e r y s m a l l . The p o l a r a c o u s t i c mode s c a t t e r i n g , o r p i e z o e l e c t r i c s c a t t e r i n g a r i s e s from t h e e l e c t r i c f i e l d accompanying an a c o u s t i c phonon i n a p i e z o e l e c t r i c c r y s t a l , i . e . i n any non-m e t a l l i c s i n g l e c r y s t a l w h i c h l a c k s i n v e r s i o n symmetry. F o r example, when an a c o u s t i c phonon d i s t o r t s t h e p o s i t i o n s o f t h e two d i f f e r e n t atoms i n t h e GaSb u n i t c e l l , t h e r e s u l t i n g charge p o l a r i z a t i o n c r e a t e s a l o c a l e l e c t r i c f i e l d . A c a l c u l a t i o n o f t h e problem by Hutson uses a s p h e r i c a l average o f t h e v e r y a n i s o t r o p i c p i e z o e l e c t r i c and e l a s t i c c o n s t a n t s t o o b t a i n an a p p r o x i m a t e v a l u e f o r t h e s c a t t e r i n g t i m e . Most o f t h e s c a t t e r i n g i s due t o t h e l o n g i t u d i n a l e l e c t r i c f i e l d s accompanying b o t h 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 a c o u s t i c phonons. The P3/2 - f c 2 £ r e s u l t i s T r p z (£ ) =  2 « * ^- k Q T Q m* ^av A where ^ = t h e d i e l e c t r i c p e r m i t t i v i t y , (MKS u n i t s are employed i n t h i s c a l c u l a t i o n ) e = t h e e l e m e n t a r y charge p / . 2\ and K = v e i j ' a v where e-j \ i s t h e p i e z o e l e c t r i c "e" av A n ) " £ Q °av 82 t e n s o r and C a v i s an average e l a s t i c c o n s t a n t . From Hutson's 2 1 2 c a l c u l a t i o n we e s t i m a t e ( e . [ j ) a v ^ ^ e i V T h i s y i e l d s 8 1/2 1 TT («£.) = 2 x 10" (e') T s. T h i s s c a t t e r i n g p r o c e s s pz i s n e g l i g i b l e f o r h o t o r c o l d e l e c t r o n s a t any t e m p e r a t u r e used i n t h i s e x p e r i m e n t . S i n c e t h e s c a t t e r i n g i s e l a s t i c , t h e r a t e o f energy l o s s i s a l s o c o m p l e t e l y n e g l i g i b l e . The o p t i c a l phonon mode s c a t t e r i n g i s a l s o o f b o t h p o l a r and n o n - p o l a r c h a r a c t e r . Because o f t h e l a r g e energy l o s s when an e l e c t r o n c r e a t e s one o p t i c a l phonon, a r e l a x a t i o n t i m e cannot be p r o p e r l y d e f i n e d f o r use i n t h e Boltzmann t r a n s p o r t e q u a t i o n . I t i s p o s s i b l e , however, t o c a l c u l a t e t h e d £ r a t e o f energy change, (g^ -)p 0» a n c* t h e n t o e s t i m a t e a c h a r a c t e r i s t i c t i m e f o r t h i s s c a t t e r i n g p r o c e s s by t h e f o r m u l a 1 _ p_o T p o *urQ ' The n o n - p o l a r mode s c a t t e r i n g i s e s s e n t i a l l y a deforma-t i o n p o t e n t i a l s c a t t e r i n g and e x i s t s i n b o t h h e t e r o p o l a r and homopolar s e m i c o n d u c t o r s . F o r 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 o f GaSb, because t h e w a v e - f u n c t i o n i s s - l i k e , a z e r o s c a t t e r i n g p r o b a b i l i t y r e s u l t s . Even the a p p r e c i a b l e a d m i x t u r e o f p - l i k e c h a r a c t e r away from k = 0 i s not enough t o make the s c a t t e r i n g r a t e comparable t o t h a t f o r p o l a r o p t i c a l mode s c a t t e r i n g . The l a t t e r mechanism, w h i c h o n l y e x i s t s i n h e t e r o p o l a r m a t e r i a l s , i s s t r o n g e s t f o r t h e l o n g i t u d i n a l modes. U s i n g t h e above f o r m u l a , t h e s c a t t e r i n g r a t e i s c a l c u l a t e d t o be 83 ^po ( 2m* G )" 2 e E 0 r N sinh -1 C ^ +(N„+l)sinh" 1' USrl 1 q where eE_ s m * e ( ^ ^ o ) / 1_ _ 1 \ K a n d K a r e t h e s t a t i c and o p t i c a l d i e l e c t r i c constants, and N the thermal equilibrium phonon population. That i s , i t i s assumed that the phonon system i s i n thermodynamic equilibrium with the l i q u i d helium at temperature, T. The second term, which represents the rate of creation of l o n g i t u d i n a l o p t i c a l (LO) phonons, i s only to be included of £ ) ^ u / , The evaluation of the expression f o r T = 30 K and £ = 1.10 gives, f o r the term i n the square bracket, N (.88) + (N + 1) {Tio = 10"5(.88) + .33 = 0.33. In t h i s case the creation of phonons dominates. In fact the sharp r i s e of the second term causes a sudden onset of appreciable s c a t t e r i n g . This basic idea was reported by Callen (19^9) where a s l i g h t l y d i f f e r e n t c a l c u l a t i o n of the problem gave f o r the square bracket -~ 4 not 0.33. Using the f u l l formula the time i s calculated to be -^- p 0 = 10" 1 2 s (Conwell) or l O " 1 ^ s (Callen) f o r e > t \ u r When € < t\ UJQ the scattering rate i s reduced to the f i r s t term, and i s approximately energy independent 1. 1 at = 2 x 10 1 2 N = 2 x 10 7 s " 1 for 30 K. This rate i s Tpo q n e g l i g i b l e i n the present experiment. The phonon sc a t t e r i n g c a l c u l a t i o n s may be summarized 1. The r e s u l t i s energy independent since x" 1 s i n h - 1 x 1 f o r 0 < x « l . 84 as f o l l o w s : ( i ) F o r 6 < kusQ t h e s c a t t e r i n g r a t e i s n e g l i g i b l e f o r h ot o r c o l d e l e c t r o n s a t a l l t e m p e r a t u r e s c o n s i d e r e d . ( i i ) F o r £ r > t s u / t h e p o l a r o p t i c a l s c a t t e r i n g r a t e would 12 -13 r e s u l t i n t h e c r e a t i o n o f one LO phonon i n a time -~-10 -10 ^ s . The p r o c e s s i n v o l v e s b o t h momentum and energy l o s s and i s independent o f t e m p e r a t u r e . 4 , 3 (c) 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 The c o l l i s i o n t i m e f o r the s c a t t e r i n g o f an e l e c t r o n from an i o n i z e d i m p u r i t y has been c a l c u l a t e d by Brooks (1955) and H e r r i n g (1955). Long and Myers (1959) and o t h e r w o rkers have v e r i f i e d t h e d e r i v e d f o r m u l a e x p e r i m e n t a l l y . The case o f a random d i s t r i b u t i o n o f i m p u r i t i e s w i t h a s c r e e n e d Coulomb p o t e n t i a l was c o n s i d e r e d . The c a l c u l a t i o n a l s o assumed s p h e r i c a l e l e c t r o n e n e rgy s u r f a c e s and n e g l i g i b l e e l e c t r o n - e l e c t r o n i n t e r a c t i o n . The most i m p o r t a n t a s s u m p t i o n i n v o l v e d was t h e use o f t h e Born a p p r o x i m a t i o n i n t h e c a l c u l a t i o n o f t h e s c a t t e r i n g c r o s s - s e c t i o n . The a p p r o x i m a t i o n i s b » 1 where b = (2ka) ; k i s t h e c a r r i e r wave number and a i s t h e e f f e c t i v e d i s t a n c e a t w h i c h t h e s c a t t e r i n g p o t e n t i a l i s c u t o f f . The f a i l u r e o f t h i s a p p r o x i m a t i o n has been s t u d i e d by B l a t t (1957 a and b) and t h e c o r r e c t s c a t t e r i n g c r o s s - s e c t i o n c a l c u l a t e d u s i n g t h e p a r t i a l wave s c a t t e r i n g a n a l y s i s . The s c a t t e r i n g r a t e as a f u n c t i o n o f e l e c t r o n energy i s g i v e n by ^ - y = N i ^ * V 2 : ( 2 e ) " 3 / 2 K - 2 ( _ L ) - f ( b ) 85 where b 8TTm«Kk QT € f ( b ) = l n ( l + b ) - ( b 1 + b p' = p + (p + N n) 1 -N A _ I n t h e f o r m u l a K i s t h e d i e l e c t r i c c o n s t a n t , t h e i m p u r i t y c o n c e n t r a t i o n , o and o*g t h e r e a l and Born s c a t t e r i n g c r o s s -s e c t i o n s , p' t h e e f f e c t i v e s c r e e n i n g c a r r i e r c o n c e n t r a t i o n , p t h e v a l e n c e band h o l e c o n c e n t r a t i o n , N Q and t h e donor and a c c e p t o r c o n c e n t r a t i o n s r e s p e c t i v e l y . F o r t h e low t e m p e r a t u r e s and low donor c o n c e n t r a t i o n s o f t h e p r e s e n t e x p e r i m e n t we have p « N Q « N A > T h i s r e s u l t s i n p* = N Q and N-j. = 2 N D s i n c e t h e r e a r e NQ i o n i z e d donors and an e q u a l c o n c e n t r a t i o n o f i o n i z e d a c c e p t o r s . B l a t t ' s p a p e r s s i n c e t h e c o n d i t i o n s i n t h i s sample g i v e v a l u e s f o r h i s R' and Q o f R 1 ^ 0 and Q <3rlcP. T h i s p u t s t h e exp e r i m e n t j u s t o u t s i d e t h e range o f v a r i a b l e s p r e s e n t e d i n h i s c u r v e s . b o t h f o r t h e case o f hot and c o l d e l e c t r o n s i n t a b l e 4.1. The 15 3 a s s u m p t i o n N Q = 4 x 10 cm i s made and €- i s t a k e n t o be 10 meV o r 3/2 kT f o r hot o r c o l d e l e c t r o n s r e s p e c t i v e l y . From t h e d a t a i n t h e t a b l e i t i s seen t h a t a t 1.6 K t h e 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 p r o c e s s s h o u l d make o n l y a s m a l l c o n t r i b u t i o n t o t h e t o t a l s c a t t e r i n g r a t e . A t 25 K t h e c a l c u l a t i o n shows a p p r e c i a b l e s c a t t e r i n g f o r e i t h e r hot o r c o l d e l e c t r o n s . 12 The c a l c u l a t e d t r = 4 x 10 s i s t o be compared t o t h e The r a t i o (ov/a) can o n l y be e s t i m a t e d r o u g h l y from The r e s u l t s o f t h e c a l c u l a t i o n a re a g a i n p r e s e n t e d 86 T a b l e 4.1 The 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 Time E l e c t r o n D i s t r i b u t i o n "Hot" " C o l d " • T(K) 1.6 25 1.6 25 b 0.9 13 0.009 1.9 a B / a 10 1 10 4 f (s) 4xl0- 1 0 4 x l 0 ~ 1 2 1x10"9 4 x l 0 " 1 2 measured 7 x 10 s. S i n c e t h e o b s e r v e d t e m p e r a t u r e dependence 2 0 1 a t 25 K i s T and n o t ( l o g T) , as e x p e c t e d f o r i o n i z e d 15 -3 i m p u r i t y s c a t t e r i n g , one can c o n c l u d e t h a t <. 4 x 10 cm" , a p p r o x i m a t e l y , and t h a t t h e main s c a t t e r i n g mechanism at t h a t t e m p e r a t u r e i s n o t 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 . The v a l u e o f N D used i n t h e c a l c u l a t i o n i s r e a s o n a b l e f o r t h e s e samples. A v a l u e o f N D = 1.3 x l O 1 ^ cm"-^  i s o b t a i n e d f o r t h e B a t t e l l e sample i n s e c t i o n 4.5, from t h e t e m p e r a t u r e dependence 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 . A n o t h e r argument s u p p o r t i n g t h e i d e a t h a t 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 i s s m a l l a t 1.6 K i s t h a t each o f t h e f o u r samples l i k e l y c o n t a i n s a d i f f e r e n t donor c o n c e n t r a t i o n , and t h e r e f o r e f i g u r e 4.4 would n o t be a s t r a i g h t l i n e i f t h i s s c a t t e r i n g p r o c e s s were i m p o r t a n t . . S i n c e 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 i s e l a s t i c t h e r a t e o f e n e r g y l o s s i s z e r o . 87 4.3 (d) N e u t r a l I m p u r i t y S c a t t e r i n g E r g i n s o y (1950) has c a l c u l a t e d a f o r m u l a f o r t h e s c a t t e r i n g r a t e o f e l e c t r o n s from n e u t r a l h y d r o g e n i c d o n o r s . I t assumes t h e e l e c t r o n e n e r g y t o be l e s s t h a n h a l f t h e i m p u r i t y i o n i z a t i o n energy, and uses a t o m i c s c a t t e r i n g d a t a t o d e s c r i b e t h e s i t u a t i o n . The r e s u l t i s 1 _ C(e )a«-K N TTZJ mj where C ( £ ) = 20, ( E r g i n s o y ) a* = (m/m|)Ka o, t h e e f f e c t i v e Bohr r a d i u s K = t h e d i e l e c t r i c c o n s t a n t a Q = 0.52 A* = t h e Bohr r a d i u s m* = t h e e l e c t r o n e f f e c t i v e mass e N = n e u t r a l i m p u r i t y c o n c e n t r a t i o n . S i n c e C i s ind e p e n d e n t o f £. i n t h i s case t h e measured t h e r m a l -e q u i l i b r i u m s c a t t e r i n g t i m e i s independent o f t e m p e r a t u r e . A l l t h e n e u t r a l i m p u r i t y s c a t t e r i n g f o r m u l a s t o be d i s c u s s e d w i l l r e s u l t from changes i n C ( S ). Sometimes t h e d e r i v e d t e m p e r a t u r e dependent s c a t t e r i n g i s d e s c r i b e d by a C ( T ) . Two m o d i f i c a t i o n s o f t h e above f o r m u l a f o r n e u t r a l donor s c a t t e r i n g have been c a l c u l a t e d 1 ( i ) c o r r e c t i o n s t o t h e E r g i n s o y c a l c u l a t i o n w h i c h g i v e r i s e t o a t e m p e r a t u r e dependent s c a t t e r i n g t i m e ( S i n h a 1971, and B l a g o s k l o n s k a y a e t a l . 1970). ( i i ) c o r r e c t i o n s due t o an a p p r e c i a b l e f r e e e l e c t r o n p o l a r i z a t i o n i n t e r a c t i n g w i t h t h e i m p u r i t y - e l e c t r o n p o l a r i z a t i o n . (Ohyama e t a l . 1968). T h i s r e s u l t s i n C = 20 - 9.6 P - ^ where ? 1 and P 2 a r e t h e f r e e and bound s p i n p o l a r i z a t i o n s r e s p e c t i v e l y . These e f f e c t s a r e o n l y i m p o r t a n t f o r e q u i l i b r i u m systems when "ft ur yy kT. They were o b s e r v e d by Ohyama e t a l . below 2 K s i n a c y c l o t r o n r esonance e x p e r i m e n t . I n t h i s e x p e r i m e n t , i f t h e same i d e a a p p l i e s t o n e u t r a l a c c e p t o r s c a t t e r i n g , t h e e f f e c t c o u l d p r o b a b l y n o t be o b s e r v e d because o f t h e v e r y low f) US v a l u e o f — — a t t h e c y c l o t r o n resonance f i e l d o f ^- 500 G. T h i s would make P g ~ 0 even though a P-^  o f -~- 0.5 c o u l d be o b t a i n e d by o p t i c a l pumping ( P a r s o n s 1971). The s c a t t e r i n g o f e l e c t r o n s from n e u t r a l h y d r o g e n i c a c c e p t o r s i s much weaker. The b a s i c i d e a i s e x p l a i n e d by Otsuka e t a l . (1966). The s c a t t e r i n g o f an e l e c t r o n from a n e u t r a l h y d r o g e n i c a c c e p t o r has a d i f f e r e n t charge c o n f i g u r a t i o n t h a n f o r t h e case o f a donor. Here a n e g a t i v e l y c h a r g e d e l e c t r o n i s s c a t t e r e d from a n e g a t i v e c e n t e r s c r e e n e d by a moving p o s i t i v e c h a r g e . By r e v e r s i n g t h e s i g n s o f a l l t h e ch a r g e s i n v o l v e d , , t h i s i s e q u i v a l e n t t o t h e s c a t t e r i n g o f a p o s i t r o n from a n e u t r a l hydrogen atom. By u s i n g t h e a t o m i c d a t a f o r t h a t s i t u a t i o n a p p r o x i m a t e s c a t t e r i n g f o r m u l a s have been d e r i v e d by Otsuk a e t a l . (1966) and by B l a g o s k l o n s k a y a e t a l . (1969). The f i r s t c a l c u l a t i o n p r e d i c t s a r e d u c e d t e m p e r a t u r e dependent s c a t t e r i n g w i t h C(T) = 3.4 a t a t e m p e r a t u r e o f ^ ~ OK and a s c a t t e r i n g d e c r e a s e w i t h i n c r e a s i n g t e m p e r a t u r e . The f o r m u l a was compared w i t h c y c l o t r o n r esonance measurements o f a r t r ( c o r r e c t e d f o r phonons) i n germanium doped w i t h t h e a c c e p t o r s g a l l i u m and i n d i u m , and w i t h t h e donor antimony. The a c c e p t o r a* v a l u e s o f 35 A* and 55 A* r e s p e c t i v e l y , were u n i q u e l y d e t e r m i n e d from f i t t i n g t h e t e m p e r a t u r e dependence o f UJf. U s i n g t h e s e v a l u e s t h e c a l c u l a t e d a c c e p t o r s c a t t e r i n g 89 a g r e e d w e l l w i t h t h e measurements, w h i l e t h e measured donor s c a t t e r i n g agreed w e l l w i t h t h e E r g i n s o y f o r m u l a . Otsuka e t a l . must be c r e d i t e d w i t h t h e p r e d i c t i o n o f t h e r e s u l t s o f t h i s e x p e r i m e n t . T h e i r a b s t r a c t t a k e s note o f "a p r o m i s i n g p o s s i b i l i t y o f o b s e r v i n g e l e c t r o n c y c l o t r o n resonance i n h i g h l y doped p-type m a t e r i a l s " . 1 . The second c a l c u l a t i o n by B l a g o s k l o n s k a y a e t a l . g i v e a g r a p h o f C ( £ ) and t h e d e r i v e d G ( T ) . C ( 6 ) peaks at a v a l u e o f C = 2.7 f o r £ = .01 where i s t h e a c c e p t o r i o n i z a t i o n energy. I t d e c r e a s e s almost l i n e a r l y t o £ = ,2£^, The d e s c r i p t i o n t a £ m w i l l be used i n c h a p t e r 5 f o r c e r t a i n c a l c u l a t i o n s . F o r 0.01 < ^ 0 . 2 t h e graph shows m C i s c o n s t a n t a t 0.2 f o r 0.2 < € /£ i <.0.5 i . e . here m<=^0. No r e s u l t s f o r 0 . 5 < £ / £ ^ < l are g i v e n . T h i s f o r m u l a was v e r i f i e d by c y c l o t r o n resonance measurements on g a l l i u m doped germanium by t h e a u t h o r s . A c a l c u l a t i o n o f t h e s c a t t e r i n g t i m e f o r e l e c t r o n s s c a t t e r e d f r om n e u t r a l h y d r o g e n i c a c c e p t o r s w i l be made and used as an e s t i m a t e f o r n e u t r a l d e f e e t a c c e p t o r s c a t t e r i n g i n GaSb. The main purpose o f t h e c a l c u l a t i o n i s t o e s t i m a t e a ro u g h v a l u e f o r t h e s c a t t e r i n g t i m e t o be compared w i t h o t h e r c a l c u l a t i o n s and t h e e x p e r i m e n t s . T h i s a p p r o x i m a t i o n may n o t be u n r e a s o n a b l e because t h e main d i f f e r e n c e s i n t h e two s c a t t e r i n g p r o c e s s e s w i l l o n l y t a k e p l a c e i n t h e u n i t c e l l . 17 -3 1. A sample c o n t a i n i n g 10 cm a c c e p t o r s would be c o n s i d e r e d h i g h l y doped as f a r as c y c l o t r o n resonance measurements are c o n c e r n e d . 7 V S i n c e t h e i m p u r i t y s i z e 2a* ^ 100 X i s much l a r g e r t h a n t h e s i z e o f t h e u n i t c e l l ( — 10 #) most o f t h e s c a t t e r i n g does n o t t a k e p l a c e i n t h e l a t t e r . The e s t i m a t e d e f f e c t i v e Bohr r a d i u s i s an average o f t h e v a l u e s 25 X and 150 A* c a l c u l a t e d from t h e f o r m u l a a* = m QKa 0/m h u s i n g t h e two d i f f e r e n t h o l e masses m h = 0.30 m Q 17 -3 and 0.05 m 0 r e s p e c t i v e l y . F o r N = 10 ' cm J t h e E r g i n s o y f o r m u l a , e q u a t i o n 4.1, u s i n g C = 20 y i e l d s T" = 0.4 x 10-1^ s. The B l a g o s k l o n s k a y a f o r m u l a u s i n g C = 0.2 g i v e s = 0.4 x 10"11 s. 17 -3 ~ _11 The measured v a l u e f o r = 10 ' cm J i s ^ = 1.2 x 10 s ( u r - c = 2.6) from f i g u r e 4.4. The c a l c u l a t e d n e u t r a l a c c e p t o r s c a t t e r i n g t i m e i s i n much b e t t e r agreement w i t h t h e ex p e r i m e n t t h a n any o t h e r c a l c u l a t e d mechanism. I f t h e c a l c u l a t i o n i s done f o r t h e r m a l e q u i l i b r i u m c a r r i e r s at h e l i u m t e m p e r a t u r e 12 C = 2 and V - 0.4 x 10 s. I n t h i s model t h e h e a t i n g o f t h e c a r r i e r s i s n e c e s s a r y t o have uff } 1 and hence t o see c y c l o t r o n r e s o n a n c e . 4.3'(e) Impact I o n i z a t i o n o f t h e A c c e p t o r s S i n c e t h e a c c e p t o r i o n i z a t i o n energy i s a p p r o x i m a t e l y e q u a l t o t h e L.0 phonon energy o f 30 meV t h e impact i o n i z a t i o n o f a c c e p t o r s may compete w i t h t h e c r e a t i o n o f phonons as t h e ene r g y l i m i t i n g p r o c e s s . The r a t e o f t h i s p r o c e s s can be e s t i m a t e d from - ~ = N v a (4.2) where v i s t h e e l e c t r o n speed ( a t 30 meV) and o i s t h e s c a t t e r i n g c r o s s s e c t i o n . S i n c e t h e a c c e p t o r s a re c l o s e t o g e t h e r t h e e x c i t e d s t a t e s form an i m p u r i t y - b a n d e d t a i l t o t h e v a l e n c e band. That i s , we w i l l assume t h e r e a re no l o c a l i z e d e x c i t e d s t a t e s t o c o n s i d e r . 1 . The e x c i t a t i o n c r o s s s e c t i o n can be e s t i m a t e d t o be t h e g e o m e t r i c a l c r o s s s e c t i o n a - t r a * 2 . T h i s i s shown t o be a p p r o x i m a t e l y t r u e f o r t h i s p r o c e s s by K a c h l i s h v i l i (1971) U s i n g a* = 50 ft, a = 7.8 x 1 0 ~ 1 3 cm 2, N = 1 0 1 7 cm" 3, and v = 5.2 x 1 0 7 cm s " 1 g i v e s T = 2.5 x 1 0 ~ 1 3 s. T h i s p r o c e s s does compete w i t h t h e phonon p r o c e s s . The two cannot be d i s t i n g u i s h e d by t h e measurements done. The o n l y p r a c t i c a l e f f e c t o f t h i s u n c e r t a i n t y i s t h a t t h e power dependence o f t h e h o l e c o n c e n t r a t i o n cannot be i d e n t i f i e d as th e power dependence o f t h e h o l e r e c o m b i n a t i o n t i m e , "th(P)« I f t h e h o l e g e n e r a t i o n r a t e i s g h ( P ) we have, i n s t e a d y s t a t e , N h ( P ) = g h ( P ) ' t h ( P ) . The measurements g i v e N h ( P ) a p1.12±.21 s e c t i o n 3.4(g)). I f t h e h o l e g e n e r a t i o n i s s o l e l y due t o impact i o n i z a t i o n i t can be shown t h a t g h ( P ) a p 1 " 0 ( f r o m t h e c y c l i n g t ime i n t h e h ot e l e c t r o n model) and t h i s g i v e s t h ( P ) a p.12±.2i whic h i s s i m i l a r t o t h e t e ( P ) a p.29+.05# i f t h e h o l e g e n e r a t i o n i s s o l e l y due t o t h e o r i g i n a l c a r r i e r g e n e r a t i o n t h e n g h ( P ) i s a c o n s t a n t and t h ( P ) a P ' - . 4.4 The Temperature Dependence o f t h e C o l l i s i o n Time The t e m p e r a t u r e dependence o f u>T was measured i n an a p p a r a t u s d e s i g n e d t o warm up s l o w l y from 4.2 K. A copper microwave c a v i t y (#3) was s i t u a t e d on a v e r t i c a l copper r o d w i t h t h e l o w e r end i n l i q u i d h e l i u m and t h e upper end a t t a c h e d 1. No o p t i c a l a b s o r p t i o n measurements o f t h e a c c e p t o r s t a t e s have been r e p o r t e d . 92 t o a copper c o n t a i n e r f i l l e d w i t h 300 g. o f " m o l e c u l a r s i e v e " m a t e r i a l . A h e a t e r warmed t h e a p p a r a t u s t o t h e d e s i r e d temp-e r a t u r e where i t would r e m a i n s i n c e t h e heat l e a k was j u s t b a l a n c e d by t h e c o o l i n g e f f e c t o f t h e l i q u i d h e l i u m . Temperature f l u c t u a t i o n s were s m a l l because o f t h e l a r g e heat c a p a c i t y o f th e system. With 4" o f l i q u i d h e l i u m i n t h e dewar t a i l measurements c o u l d be performed s l o w l y from 4.2 K t o 35 K ov e r a t i m e span o f t h r e e h o u r s . The t e m p e r a t u r e was measured w i t h a c a l i b r a t e d germanium r e s i s t a n c e thermometer s i t u a t e d 1 cm below t h e c a v i t y . The measurement was made a t zer o m a g n e t i c f i e l d t o a v o i d t h e m a g n e t o r e s i s t a n c e e f f e c t o f t h e thermometer. A t e m p e r a t u r e g r a d i e n t i s e x p e c t e d because o f t h e a p p a r a t u s d e s i g n . The maximum e r r o r i n c a v i t y t e m p e r a t u r e due t o t h i s e f f e c t i s e s t i m a t e d t o be + 0.2 K a t 25 K. 1* A t t h a t t e m p e r a t u r e t h e a b s o l u t e c a l i b r a t i o n e r r o r i s + 0.5 K, whi c h i s t h e l a r g e s t c a l i b r a t i o n e r r o r i n t h i s e x p e r i m e n t . The problem o f t h e r m a l c o n t a c t between t h e sample and t h e copper s u r f a c e , o r t h e l i q u i d h e l i u m , i s acu t e a t low t e m p e r a t u r e s (Berman 1956, L i t t l e 1959, P o l l a c k 1969). The s o l u t i o n i s t o use a t h i n l a y e r o f s i l i c o n e vacuum g r e a s e between t h e sample and t h e copper. T h i s i n t r o d u c e s t h e p o s s i b i l i t y o f sample s t r a i n . The s i g n a l h e i g h t was measured v s . w a v e l e n g t h under t h i s c o n d i t i o n t o de t e r m i n e t h e s t r a i n v i a t h e s h i f t i n t h e a b s o r p t i o n band-edge. The measured s h i f t i s l e s s t h a n 3 meV. When t h i s i s combined w i t h t h e ( u n p u b l i s h e d ) 1. The t e m p e r a t u r e g r a d i e n t was measured on a s i m i l a r a p p a r a t u s from 4-30 K. 93 dE measured s h i f t o f — = -14 e V / u n i t s t r a i n d ( s t r a i n ) ( R i c k a r d s 1972) f o r b i a x i a l s t r a i n , the d e r i v e d s t r a i n i s l e s s -4 t h a n 2 x 10 . The v a l u e o f t h e s t r a i n p r e d i c t e d by t h e d i f f e r e n t i a l t h e r m a l c o n t r a c t i o n o f t h e copper and GaSb i s 1.7 x 10 . One can c o n c l u d e t h a t t h e s i l i c o n e g r e a s e does n o t bond t h e two m a t e r i a l s t o g e t h e r f i r m l y u n t i l most o f t h e t h e r m a l c o n t r a c t i o n has t a k e n p l a c e . F i g u r e 4.5 d e p i c t s t h e t e m p e r a t u r e dependence o f cuv f o r two samples. The e r r o r b a r s on t h e t e m p e r a t u r e i n d i c a t e t h e t e m p e r a t u r e change d u r i n g t h e measurement and do n o t i n c l u d e t h e two e r r o r s c i t e d e a r l i e r w h i c h a r e a t most +0.7 K. A t t h e low microwave power l e v e l employed i n t h e measurement o n l y e l e c t r o n s c o n t r i b u t e t o t h e s i g n a l . The low t e m p e r a t u r e d a t a f o r t h e two samples can be f i t by two p a r a l l e l s t r a i g h t l i n e s a t t r i b u t e d i n p a r t t o n e u t r a l a c c e p t o r s c a t t e r i n g and denoted c f T ^ . An attempt was made t o f i t t h e h i g h t e m p e r a t u r e measurements f o r b o t h samples by a s i n g l e s t r a i g h t l i n e , i . e . by an i n t r i n s i c s c a t t e r i n g mechanism (e.g. phonon s c a t t e r i n g ) . T h i s i s s u p p o r t e d by t h e o b s e r v a t i o n ( n o t shown on f i g u r e 4.5) t h a t a t low t e m p e r a t u r e s cuz depends on microwave power w h i l e a t t e m p e r a t u r e s g r e a t e r t h a n 10 K i t does n o t . The d e r i v e d c u r v e , L U T 2 does n o t produce a good f i t t o any o f t h e c a l c u l a t e d phonon s c a t t e r i n g t i m e s . The measured T Z = 5 x 10~ 9 r p - 2 * 0 ! 0 * 1 s c o m e s c l o s e s t t o t h e a c o u s t i c phonon c a l c u l a t i o n (see s e c t i o n 4 . 3 ( b ) ) , T a c = 4 x 10" 9 T" 1 s but d i f f e r s from i t by one o r d e r o f magnitude a t 30 K. S c a t t e r i n g due t o t h e r m a l l y i o n i z e d 6-0 5.0 4.0 3.0 i 1 1 1 1 — i — r T r \ \ .1*1 \ 3 1.C» •7 .5 Theory Curves . _ cuT^ OC T - * — * ouT^ T .17 ± .02 -2.0 - .1 1 1 1 Experimental Points  Battelle Monsanto 1 J I I I I I L J ' i t 3 4 6 10 Temperature, T (K) 15 20 30 40 Figure 4.5 The temperature dependence of ur-r f o r two samples. \0 95 a c c e p t o r s i s n e g l i g i b l e because t h e c o n c e n t r a t i o n i s o n l y 12 -3 •—10 cm J a t 30 K. The t e m p e r a t u r e dependence o f such a c o n t r i b u t i o n would be one o r d e r o f magnitude more r a p i d t h a n A c o n t r i b u t i o n from t h e N D i o n i z e d donors and t h e same number o f i o n i z e d a c c e p t o r s i s more l i k e l y . The c a l c u l a t e d 1 5 - 3 v a l u e (see s e c t i o n 4 . 3 ( c ) ) u s i n g N p = 4 x 10 cm i s 12 -12 = 4 x 1 0 " s as compared t o t h e measured v a l u e , 7 x 10 s. 1 2 The e x p e c t e d t e m p e r a t u r e dependence i s ( l o g T ) ~ n o t T . I t i s r e a s o n a b l e t h a t t h i s mechanism s h o u l d c o n t r i b u t e t o t h e s c a t t e r i n g a t h i g h t e m p e r a t u r e b u t s i n c e t h e v a l u e o f N p i s u n c e r t a i n and t h e t e m p e r a t u r e dependence wrong, t h i s cannot be i d e n t i f i e d as t h e p r i n c i p a l mechanism. I t i s c o n c l u d e d t h a t t h e h i g h t e m p e r a t u r e s c a t t e r i n g mechanism remains u n i d e n t i f i e d , w h i l e t h e low t e m p e r a t u r e s c a t t e r i n g i s due i n p a r t t o t h e n e u t r a l a c c e p t o r s . 4 . 5 The Temperature Dependence 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 The c a r r i e r c o n c e n t r a t i o n was deduced from t h e a r e a under t h e a b s o r p t i o n c u r v e , a t c o n s t a n t power u s i n g a s i g n a l h e i g h t c o r -r e c t i o n f o r t h e s m a l l t e m p e r a t u r e dependence o f t h e c a v i t y Q. The r e s u l t i n g g r a p h i s p l o t t e d as l o g N vs T _ 1 on f i g u r e 4 . 6 From T = 26 K t o 10 K (and below) t h e f l a t c u r v e p r o v e s t h a t t h e e l e c t r o n l i f e t i m e i s independent o f t e m p e r a t u r e . Above 26 K t h e r e i s an e x p o n e n t i a l l o s s o f c a r r i e r c o n c e n t r a t i o n . The p o i n t s shown t h e r e may have a d d i t i o n a l e r r o r s due t o two f a c t o r s ? ( i ) t h e r e s o n a n c e c u r v e s appear s t r o n g l y d i s p e r s i v e , ( i i ) t h e l a s t two p o i n t s use o n r ( T ) v a l u e s o b t a i n e d by 9 7 e x t r a p o l a t i o n o f t h e measured c u r v e s o f f i g u r e 4 . 5 . Never-t h e l e s s , t h e r e i s an e x p o n e n t i a l l o s s o f c a r r i e r s w i t h an +9 a c t i v a t i o n energy o f 2 7 _ ^ meV w h i c h i s c l o s e t o t h e E A = 3 1 meV d e t e r m i n e d f o r t h i s sample from t h e H a l l measure-ments . p h o t o c o n d u c t i v i t y as t h e h o l e plasma f r e q u e n c y "becomes comparable t o t h e microwave f r e q u e n c y . The plasma resonance c a l c u l a t i o n s o f D r e s s e l h a u s e t a l . ( 1 9 5 5 b) can e x p l a i n t h i s q u a l i t a t i v e l y . They c a l c u l a t e d t h e e f f e c t o f t h e p r e s ence o f a h i g h m a j o r i t y - c a r r i e r c o n c e n t r a t i o n on t h e c y c l o t r o n r e s o n a n c e o f a low c o n c e n t r a t i o n o f m i n o r i t y c a r r i e r s . They c o n s i d e r t h e case o f ^ ^ ^ ^ p h w ^ e r e "-pe a n < * ^ p h a r e t h e ( p h o t o c r e a t e d ) e l e c t r o n and ( e q u i l i b r i u m ) h o l e p l a s m a f r e q u e n c i e s . By n e g l e c t i n g t h e e l e c t r o n c o n t r i b u t i o n t o t h e d e p o l a r i z i n g f i e l d and by n e g l e c t i n g m a g n e t i c e f f e c t s on t h e h o l e s , i . e . a / C h ' * " h"*^ 1* "they deduce a f o r m u l a f o r t h e measured e l e c t r o n p h o t o c o n d u c t i v i t y . When plasma e f f e c t s become dominant, i . e . UJ »UJ and |u/| T » 1 t h e s i g n a l s t r e n g t h becomes more d i s p e r s i v e w i t h 0 a p p r o a c h i n g - 1 . The s i g n a l r e d u c t i o n i s a p p r o x i m a t e l y p r o p o r t i o n a l t o T h i s t h e o r y f i t s q u a l i t a t i v e l y i n t h e p r e d i c t i o n o f t h e t e m p e r a t u r e dependence o f t h e s i g n a l h e i g h t and i n h a v i n g 0 approach - 1 . I t does not p r e d i c t q u a n t i t a t i v e l y t h e e x a c t way i n w h i c h t h e shape and s i g n a l h e i g h t change w i t h ul T h i s i s a t t r i b u t e d t o t h e e x p e c t e d l o s s o f e l e c t r o n a LW 2 The a p p r o x i m a t e f o r m u l a = 2 i s a p p l i e d f o r t h e 2 ULT 9« t e m p e r a t u r e T = 29 K where t h e s i g n a l s t r e n g t h i s r e d u c e d a f a c t o r o f 2. By a l s o u s i n g t h e H a l l model, p r e s e n t e d i n s e c t i o n 4.2, and t h e v a l u e o f E A = 27 meV, t h e v a l u e N Q = 1.3 x l O 1 ^ cm" 3 i s c a l c u l a t e d . C h anging t h e v a l u e o f E^ t o 31 meV has t h e e f f e c t o f d o u b l i n g N^, S i n c e t h e a s s u m p t i o n used i n t h e c a l c u l a t i o n ( C LT^T^ << 1) i s l i k e l y j i o t t r u e , t h e c a l c u l a t i o n c o u l d perhaps be b r o u g h t i n t o agreement w i t h e x p e r i m e n t by d e r i v i n g t h e c o r r e c t f o r m u l a . The same c a l c u l a t i o n i s a l s o o f i n t e r e s t i n u n d e r s t a n d -i n g why t h e o b s e r v e d p h o t o c o n d u c t i v i t y i s not a f f e c t e d by t h e h o l e s i n t h e i m p u r i t y band a t h e l i u m t e m p e r a t u r e s . T a k i n g = 10 1 7 cm" 3 t h e c a l c u l a t e d f r e q u e n c i e s are U / 2 = 6xl0 2^ s ~ 2 i • i 14 -1 and \oJ | = 3 x 10 s T a k i n g a h o l e i m p u r i t y band 2 -1 -1 m o b i l i t y l e s s t h a n 300 cm V s s a t i s f i e s t h e c o n d i t i o n s u^"C h « 1 and Icc'lT'k <£ 1. I n t h i s case t h e o b s e r v e d p h o t o c o n d u c t i v i t y i s j u s t t h a t due t o e l e c t r o n s ( D r e s s e l h a u s e t a l . 1955 b ) . T h e r e f o r e t h e measured peak parameter s h o u l d be independent o f a c c e p t o r c o n c e n t r a t i o n . 99 CHAPTER 5 THE HOT ELECTRON MODEL "World r u n n i n g out o f hot e l e c t r o n s , s a y s p h y s i c i s t . " The G e o r g i a S t r a i g h t 5.1 TJi§ Model The e v i d e n c e t h a t t h e e l e c t r o n s are not i n t h e r m a l e q u i l i b r i u m w i t h t h e l i q u i d h e l i u m , but r a t h e r a r e " h o t " , has been p r e s e n t e d . The r e l e v a n t d a t a i s summarized i n s e c t i o n 5.2.**" Q u a l i t a t i v e l y , t h e model o f t h e e l e c t r o n d i s t r i b u t i o n can be i n t r o d u c e d by c o n s i d e r i n g t h e t i m e dependence o f th e e n e r g y o f one e l e c t r o n . An e l e c t r o n s t a r t i n g n e a r t h e band edge i s a c c e l e r a t e d by t h e s t r o n g microwave e l e c t r i c f i e l d . I t g a i n s e n e r g y f a s t e r t h a n i t i s l o s t . The r a t e o f energy l o s s c a l c u l a t e d i n s e c t i o n 4.3 i s n o t l a r g e enough t o compensate f o r t h e s n e r g y g a i n e d t h r o u g h microwave h e a t i n g . The e l e c t r o n c o n t i n u e s t o g a i n e n e rgy u n t i l i t r e a c h e s an energy e q u a l t o one l o n g i t u d i n a l o p t i c a l phonon ( K c t r Q ) . Then -12 i t c r e a t e s one phonon q u i c k l y ( t L 0 — 10 s) and f a l l s t o t h e band edge t o r e p e a t t h e c y c l e . I t c y c l e s 10^ t i m e s d u r i n g i t s l i f e t i m e i n t h e c o n d u c t i o n band. T h i s means t h e d i s t r i -b u t i o n s h o u l d be i n s e n s i t i v e t o t h e microwave e l e c t r i c f i e l d w h i c h g o v e r n s t h e c y c l i n g t i m e . The phonon "energy b a r r i e r " s e t s an e f f e c t i v e upper l i m i t on t h e e l e c t r o n energy o f 1. A s i m i l a r c y c l o t r o n r e s o n a n c e s i g n a l has a l s o been a t t r i b u t e d t o hot c a r r i e r s i n InSb. ( G e r s h e n z o n e t a l . , 1971, " C y c l o t r o n Resonance o f Hot E l e c t r o n s i n p-type InSb.") 100 ~]r\ uf0 = 30 meV ( A p p e n d i x 4 . 1 ) . A p o s s i b l e c o mpeting p r o c e s s w h i c h a l s o l i m i t s t h e g a i n i n e n e r g y i s t h e impact i o n i z a t i o n o f n e u t r a l a c c e p t o r s . The e s t i m a t e d c o l l i s i o n t i m e f o r t h i s p r o c e s s i s comparable (see s e c t i o n 4.3) t o t L Q . The energy b a r r i e r a s s o c i a t e d w i t h impact i o n i z a t i o n i s e q u a l t o t h e a c c e p t o r a c t i v a t i o n energy, EA = 30 + 4 meV. S i n c e t h i s i s v e r y c l o s e t o f\ LWQ , t h e v a l u e o f t h e energy b a r r i e r , t h e c a l c u l a t e d d i s t r i b u t i o n o f e l e c t r o n s would n o t depend on w h i c h p r o c e s s p r e v a i l s . To i n c l u d e b o t h p o s s i b i l i t i e s t h e energy o f t h e b a r r i e r , w h i c h w i l l be denoted £. M, i s t a k e n t o be 30 + 4 meV. The c o n c e p t o f an e l e c t r o n t e m p e r a t u r e cannot be used i n t h e model because t h e e l e c t r o n s do not i n t e r a c t w i t h each o t h e r s u f f i c i e n t l y s t r o n g l y . That i s , t h e e l e c t r o n d e n s i t y i s below some c r i t i c a l d e n s i t y , N c. To d e f i n e an e l e c t r o n t e m p e r a t u r e a p p l i c a b l e t o t h e case o f e l e c t r o n energy l o s s by e m i s s i o n o f o p t i c a l phonons, S t r a t t o n (1958) has c a l c u l a t e d a f o r m u l a f o r N... T h i s has been used s u c c e s s f u l l y by Shah and L e i t e (1969) t o d e s c r i b e t h e r a d i a t i v e r e c o m b i n a t i o n • 14 / 3 o f p h o t o e x c i t e d h o t e l e c t r o n s i n GaAs. F o r GaSb, N c = 10 /cm . 11 "\ Comparing t h i s t o t h e measured v a l u e o f N < 10 / c n r we f i n d t h a t t h e c o n c e p t o f e l e c t r o n t e m p e r a t u r e does n o t a p p l y . 5.2 E v i d e n c e S u p p o r t i n g t h e Model 5.2(a) E x p e r i m e n t a l ( i ) The power dependence o f t h e e f f e c t i v e mass shows l e s s t h a n 1 % v a r i a t i o n ( s e e s e c t i o n 6.1), w h i l e m*(6 ) 1 0 1 changes 5 % i n t h e range 0 < £ < £ M . A c o m p a r i s o n o f t h e s e f i g u r e s shows t h a t t h e e l e c t r o n e n e rgy d i s t r i b u t i o n does n ot depend s t r o n g l y on power. Thus t h e e l e c t r o n s a r e e i t h e r " v e r y c o l d " o r " v e r y hot w i t h a power independent d i s t r i b u t i o n f u n c t i o n " . That i s , i f t h e y were "warm" t h e d i s t r i b u t i o n would change w i t h power, r e s u l t i n g i n a l a r g e r m*(P) v a r i a t i o n . ( i i ) The e l e c t r o n s i g n a l i n t e n s i t y i s power dependent and t e m p e r a t u r e i n d e p e n d e n t . The h o l e c o n c e n t r a t i o n i s a l s o power dependent. ( i i i ) No o s c i l l a t i o n s were seen i n t h e p h o t o c o n d u c t i v i t y as a f u n c t i o n o f l i g h t e x c i t a t i o n energy. T h i s i s i n t e r p r e t e d as p r o o f t h a t t h e microwave e l e c t r i c f i e l d a f f e c t s t h e e l e c t r o n e n e r gy d i s t r i b u t i o n . Habegger and Fan ( 1 9 6 4 ) o b s e r v e d o s c i l l a t i o n s i n t h e p h o t o c o n d u c t i v i t y as a f u n c t i o n o f l i g h t e x c i t a t i o n energy. The e x c i t a t i o n energy d e t e r m i n e s t h e e l e c t r o n ' s i n i t i a l e nergy i n t h e band, Habegger and Fan i n t e r p r e t e d t h e o s c i l l a t i o n i n terms o f a model whereby t h e e l e c t r o n n e i t h e r g a i n s n o r l o s e s energy a p p r e c i a b l y d u r i n g i t s l i f e t i m e i n t h e band. When t h e y a p p l i e d a d.c. e l e c t r i c f i e l d o f ~ 0 . 1 V/cm t h e o s c i l l a t i o n s d i s a p p e a r e d . T h i s was i n t e r p r e t e d as t h e e f f e c t o f t h e e l e c t r i c f i e l d on t h e energy d i s t r i b u t i o n . S i n c e t h e p r e s e n t e x p e r i m e n t u t i l i z e s h i g h e r e l e c t r i c f i e l d s t h e same i d e a s h o u l d a p p l y . 5 . 2 (b) C a l c u l a t i o n s Ci) I f t h e e l e c t r o n d i s t r i b u t i o n i s a p p r o x i m a t e l y t h e r m a l t h e n one can use t h e d e f i n i t i o n o f t h e m o b i l i t y , 102 t o w r i t e v = JJ- E i . ( 5 . 1 ) W i t h u = ^ - 2 - = 7 5 0 , 0 0 0 c m 2 A - s (to T = 3) and 7 m* E^ ^ = 60 V/cra one o b t a i n s v = 4 . 5 x 10? cm/s. The c o r r e s p o n d i n g e n e r g y f o r t h i s speed i s € = 1/2 m*v 2 = 23 meV: S i n c e t h i s c a l c u l a t i o n shows t h e e l e c t r o n s a r e h o t , one cannot r e l y on th e c a l c u l a t e d £ t o be a c c u r a t e because e q u a t i o n 5-1 i s no l o n g e r n e c e s s a r i l y t r u e f o r hot e l e c t r o n s . ( i i ) The average n e t power absorbed p e r e l e c t r o n d u r i n g one c o l l i s i o n t i m e i s ^ 40 ft LXJQ. (see s e c t i o n 3 . 4 ( e ) ) S i n c e t h e l a r g e s t l i k e l y l o s s o f energy d u r i n g a c o l l i s i o n i s 1 "h (JWC , ( f o r € M ) , t h e measured power a b s o r p t i o n r a t e s u p p o r t s t h e model. ( i i i ) S c a t t e r i n g t i m e c a l c u l a t i o n s ( s e c t i o n 4 . 3 ) show t h a t (a) t h e r e a r e no a p p r e c i a b l e e n ergy l o s e e s due t o s c a t t e r i n g f o r £ < € M , w h i l e f o r £ > ^ M t h e energy l o s s p r o c e s s i s v e r y r a p i d , (b) t h e o b s e r v e d c o l l i s i o n t i m e agrees w i t h t h e h o t e l e c t r o n s c a t t e r i n g t i m e f o r n e u t r a l h y d r o g e n i c a c c e p t o r s . 5 . 3 The E l e c t r o n E n e r g y D i s t r i b u t i o n U s i n g t h e Boltzmann e q u a t i o n , R a b i n o v i c h (1970 a) has c a l c u l a t e d t h e e l e c t r o n e n e rgy d i s t r i b u t i o n f u n c t i o n f o r e x p e r i m e n t a l c o n d i t i o n s v e r y s i m i l a r t o t h o s e o f t h e p r e s e n t e x p e r i m e n t . The c a l c u l a t i o n i s o f i n t e r e s t here t o e s t i m a t e t h e average energy t o be used i n t h e n o n - p a r a b o l i c i t y c o r r e c t i o n t o t h e e f f e c t i v e mass. The c a l c u l a t i o n i s extended 103 i n a second paper ( R a b i n o v i c h , 1970 b) where t h e e f f e c t o f th e o p t i c a l phonon s c a t t e r i n g on t h e c y c l o t r o n r e s o n a n c e h a l f w i d t h i s c a l c u l a t e d . I n t h e f i r s t paper t h e e l e c t r o n d i s t r i b u t i o n f u n c t i o n i s c a l c u l a t e d f o r t r a n s p o r t i n t h e presence o f c r o s s e d d.c. e l e c t r i c and mag n e t i c f i e l d s , assuming kT <^ f\ ULTQ and n e g l e c t i n g e l e c t r o n - e l e c t r o n c o l l i s i o n s . 1 " I t i s assumed t h a t e l e c t r o n s w i t h e n e rgy €> f\ OTQ i n t e r a c t w i t h o p t i c a l phonons w i t h a c o l l i s i o n t i m e T^ , whereas t h o s e w i t h £ < ft cxrQ i n t e r a c t w i t h i m p u r i t i e s and a c o u s t i c phonons w i t h c o l l i s i o n t i m e » T" 2 . I t i s shown t h a t t h e r e i s a range o f e l e c t r i c f i e l d s s u ch t h a t t h e e l e c t r o n energy i s d i s s i p a t e d m a i n l y by o p t i c a l phonon c r e a t i o n i n t h e r e g i o n & > "K u_rQ whereas t h e momentum i s d i s s i p a t e d m a i n l y by e l a s t i c c o l l i s i o n s i n t h e r e g i o n € < "ts LLTQ . The upper l i m i t on th e d.c. e l e c t r i c f i e l d i s governed by eE "Z^ « p Q where p Q i s t h e c r y s t a l momentum o f an e l e c t r o n o f energy E - T\ urQ . That c o n d i t i o n a p p l i e s t o t h e p r e s e n t e x p e r i m e n t f o r power l e v e l s P 4 - 6 dB. Because o f t h e l i m i t on E and t h e c o n d i t i o n » T 2 ^ e e l e c t r o n 3 n e v e r o b t a i n e n e r g i e s a p p r e c i a b l y g r e a t e r t h a n " f r u j Q O The e l e c t r o n d i s t r i b u t i o n d e r i v e d i s p r a c t i c a l l y i s o t r o p i c and i s independent o f e l e c t r i c f i e l d a l t h o u g h i t depends on t h e m a g n e t i c f i e l d . F o r f i x e d m agnetic f i e l d t h e average e n e r g y and r e l a x a t i o n t i m e a re a l s o i n d e p e n d e n t o f t h e e l e c t r i c f i e l d , and Ohm's law i s s a t i s f i e d 1. The use o f a d.c. e l e c t r i c f i e l d i n t h e c a l c u l a t i o n does n o t p r e v e n t t h e a p p l i c a t i o n o f the model t o t h e c y c l o t r o n r esonance e x p e r i m e n t i n v o l v i n g an a.c. e l e c t r i c f i e l d , because t h e c a l c u l a t e d c o n d u c t i v i t y can be e a s i l y t r a n s -formed t o t h a t c a s e . 104 f o r an a r b i t r a r y s c a t t e r i n g mechanism. The d e r i v e d d i s t r i b u t i o n f u n c t i o n i s V f ( e ) a £ 2 F( G ) where F ( z ) = D ^ l + A f - ^ z ) + A f 2 ( z ) ) z = D = ( l + A I X + A VI I 2 ) , V c = c P, o e E ^ ( -h u / Q ) » and where f - ^ z ) , f 2 ( z ) , and I 2 are i n t e g r a l s i n v o l v i n g T"^ ( £ ) . The i n t e g r a l s 1^ and I 2 , are a l s o i n v o l v e d i n t h e c y c l o t r o n r e s o n a n c e h a l f w i d t h c a l c u l a t i o n and are I± - f j f a I 2 - f 1 t ( z ) d z 0 ° > (5.2) T , ( e ) and, t ( z ) = — • / . r— X'1Cr\ur0) The dependence o f t h e d i s t r i b u t i o n on t h e m a g n e t i c f i e l d 2 e n t e r s t h r o u g h J/ . C The d i s t r i b u t i o n f u n c t i o n can be s i m p l i f i e d by assuming, f o r t h e purpose o f d i s c u s s i o n , t ( z ) a z m . (5.3) The c a s e s m = 0 and m = 1 w i l l be d i s c u s s e d as approximate l i m i t s f o r t h e n e u t r a l a c c e p t o r s c a t t e r i n g . F o r V 2 >) 1 1/2 t h e R a b i n o v i c h model y i e l d s ^ ( z ) a ( z ' - z) f o r m = l j 105 1/2 and f(z) a (1-z ' ) f o r m = 0 . I n t h e two ca s e s c o n s i d e r e d t h e average energy i s / f ( z ) z d z 7 _ 0  z _ j P ( z ) d z 0' ; = 0 . 3 0 and 0.40 r e s p e c t i v e l y . The a s s u m p t i o n s o f t h e model do n o t a p p l y e x a c t l y t o t h i s e x p e r i m e n t , s i n c e t h e e l e c t r o n - e l e c t r o n c o l l i s i o n s and c a r r i e r g e n e r a t i o n and r e c o m b i n a t i o n were n e g l e c t e d . The measured weak dependence o f OJT on l i g h t i n t e n s i t y may be due t o t h e f o r m e r , w h i l e t h e power dependent r e c o m b i n a t i o n t i m e i n d i c a t e s t h a t t h e l a t t e r must be c o n s i d e r e d . The weak power dependence o f a r t r , and t h e changes i n s i g n a l shape as a f u n c t i o n o f power ( w h i c h a re n o t f u l l y measured by w r ) a r e a l s o n o t e x p l a i n e d by t h e model. I n h i s second paper ( R a b i n o v i c h , 1970 b) t h e a u t h o r a p p l i e s t h e c a l c u l a t i o n t o t h e case o f c y c l o t r o n r e s o n a n c e . The c o n d u c t i v i t y t e n s o r a t t h e microwave f r e q u e n c y , u/ i s deduced from t h a t a t z e r o f r e q u e n c y by t h e t r a n s f o r m a t i o n , °\^Ur' 5) = aij(°' l ' ) where, B = B (1 - ~j~r). T h i s r e l a t i o n s h i p can be d e r i v e d 0 from t h e Boltzmann e q u a t i o n assuming o n l y i s o t r o p y i n t h e absence o f t h e ma g n e t i c f i e l d ( B ass e t a l . , I969). The 2 2 2 2 ass u m p t i o n s USQ )>) 1 and UJQ T"2 « 1 a r e a l s o made. I t i s shown t h a t t h e UJZ v a l u e deduced from t h e h a l f w i d t h i n t h i s case i s ^ 1 T h i s i s t r u e f o r any momentum r e l a x a t i o n t i m e 6 ), i n t h e a p p r o x i m a t i o n s o f t h e model. U s i n g e q u a t i o n 5.2 and 1 1 / I \ 2 2 5.3 one o b t a i n s ( " i f " ) = ( i + " ) f o r ° ^ r a < 1 ' The a p p l i c a t i o n o f t h i s f o r m u l a t o n e u t r a l a c c e p t o r s c a t t e r i n g r e s u l t s i n t h e m u l t i p l i c a t i o n o f each CUT v a l u e by t h e same f a c t o r f o r a l l v a l u e s o f . T h i s cannot improve t h e agreement between t h e measured a c c e p t o r dependence o f UJT and t h e e x p e c t e d r e s u l t , ( Wfj)-1 a N A. S i n c e t h e i n e q u a l i t y UfZl7Z >> 1 i s n o t w e l l s a t i s f i e d , 1/2 i t i s p o s s i b l e t h a t t h e f a c t o r ( ^ / ^ l ^ depends on T"^Ch L t r ) and t h e r e f o r e on N A. Such a m o d i f i c a t i o n c o u l d improve t h e agreement w i t h e x p e r i m e n t . As an example, i f 16 3 t h e f a c t o r were 1/2 f o r N A = 6 x 10 cm"-3 and 1 f o r 17 -3 N A = 2 x 10• cm ^ t h e c o r r e c t e d e x p e r i m e n t a l c u r v e o f f i g u r e 4.4 would i n t e r s e c t t h e o r i g i n . 5.4 The Kane Band S t r u c t u r e Model Kane (1957) c a l c u l a t e d t h e band s t r u c t u r e f o r InSb and h i s c a l c u l a t i o n has been a p p l i e d t o many o f t h e othe I I I - V s e m i c o n d u c t o r s . As s t a t e d i n s e c t i o n 3.3 t h e ass u m p t i o n s used f o r InSb, c o n c e r n i n g t h e energy band s e p a r a t i o n s a t k = 0 are a l s o f a i r l y good f o r GaSb. Kane c a l c u l a t e d d i r e c t l y t h e i n t e r a c t i o n o f t h e f i r s t c o n d u c t i o n 107 band and t h e t h r e e h i g h e s t v a l e n c e bands. T h i s i n c l u d e d t h e k.p_ i n t e r a c t i o n and t h e k-independent s p i n - o r b i t i n t e r a c t i o n , The c a l c u l a t i o n has been a p p l i e d t o GaSb by Yep and B e c k e r (1966) who make t h e a p p r o x i m a t i o n 6 « ( - j A+Eg) where € i s t h e e l e c t r o n energy measured from t h e c o n d u c t i o n band edge and A and E„ are t h e 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 bands (-—0.8 eV) and t h e energy gap (— 0.8 eV) r e s p e c t i v e l y . W ith t h a t a p p r o x i m a t i o n t h e c o n d u c t i o n band i s g i v e n by, where in - f r e e e l e c t r o n mass o "hk = e l e c t r o n c r y s t a l momentum I _ p2< I A + Eg> ( A + E g ) P = t h e momentum m a t r i x element between t h e s - l i k e c o n d u c t i o n band s t a t e s and t h e p - l i k e v a l e n c e band s t a t e s . F o r s m a l l k t h e e x p a n s i o n o f t h e square r o o t g i v e s 6 - " S T " - a * k + 2 — Z T (5.5) 0 Eg where m* i s t h e band edge e f f e c t i v e mass o 1 0 8 T h i s g i v e s a c o n d u c t i o n "band w h i c h i s s p h e r i c a l "but non-p a r a b o l i c . The n o n - p a r a b o l i c i t y r e s u l t s from a l a r g e a d m i x t u r e o f t h e v a l e n c e band p - l i k e s t a t e s i n t o t h e c o n d u c t i o n band s t a t e s when k / 0. Note t h a t , s i n c e E g and A a r e w e l l known, t h e two q u a n t i t i e s a and m£ o n l y i n v o l v e one u n d e t e r -2 mined parameter y , o r e q u i v a l e n t l y P. S i n c e a d e t e r m i n e s t h 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 e f f e c t i v e mass, t h i s means t h a t b o t h t h e band edge and i t s n o n - p a r a b o l i c i t y a r e d e t e r m i n e d by t h e same parameter V Kane (195?) and o t h e r a u t h o r s have t r e a t e d t h e e f f e c t o f t h e h i g h e r c o n d u c t i o n bands on t h e f i r s t c o n d u c t i o n band by p e r t u r b a t i o n t h e o r y . The r e s u l t i s a p r e d i c t e d w a r p i n g o f t h e c o n d u c t i o n band, i . e . i t becomes n o n - s p h e r i c a l as w e l l as n o n - p a r a b o l i c a t s u f f i c i e n t l y h i g h e n e r g i e s g i v i n g a s m a l l a n i s o t r o p y i n t h e e f f e c t i v e mass (see s e c t i o n 6.1), A n o t h e r r e s u l t o f t h e p e r t u r b a t i o n i s a c o r r e c t i o n t o e q u a t i o n 5.6. Reine e t a l . (1972), u s i n g t h e coupled-band c a l c u l a t i o n o f P i d g e o n and Brown (1966) o b t a i n e d m 2m. y 2 - I T - l + + 2 f ( 5 - 7 ) O k g where f "fr /m - F = a m a t r i x element r e p r e s e n t i n g t h e i n t e r a c t i o n o f t h e c o n d u c t i o n band w i t h h i g h e r bands. F u r t h e r m o r e , t h e same wo r k e r s have d e r i v e d a n o n - p a r a b o l i c i t y c o e f f i c i e n t s l i g h t l y d i f f e r e n t from Yep and B e c k e r . The /+2 2\2 second term i n e q u a t i o n 5.5 can be w r i t t e n as ~ P y £ m j where p = + a4m2 . The r a t i o o f t h e two para m e t e r s i s 109 p R e l n e _ fj, + kQ, + 2Q' / m o l ) eq. 5.7 Reine T e p where Q = 3 .+ 5Q + 2Q' v m* l ) eq. 5 . 6 7 Yep ( 5 . 8 ) A. E g The e f f e c t i v e mass as a f u n c t i o n o f energy i s c a l c u l a t e d from e q u a t i o n 5 * 5 i g n o r i n g t h e t h i r d term. F o r e n e r g i e s < 30 meV, even t h e second term i n e q u a t i o n 5 - 5 g i v e s a t most a 3fo c o r r e c t i o n t o t h e p a r a b o l i c e n ergy term. The s e m i c l a s s i c a l f o r m u l a f o r t h e e f f e c t i v e mass i n t h e case o f an i s o t r o p i c e n e rgy band i s , f r o m s e c t i o n 3 . 1 , 1_ m* 1> k he. (5.9) U s i n g e q u a t i o n 5 . 5 t h i s y i e l d s m m* o l _ 1 . samf ( - J ^ ) o ( 5 . 1 0 ) S i n c e t h e second term i n e q u a t i o n 5 . 1 0 i s a s m a l l c o r r e c t i o n we can use t h e a p p r o x i m a t i o n , e = 2, 2 2m* ( 5 . 1 D The c y c l o t r o n r e s o n a n c e e x p e r i m e n t measures an average o f m*(<~ ) o v e r t h e energy d i s t r i b u t i o n (see s e c t i o n 5 . 5 ) . That i s , f r o m e q u a t i o n s 5 . 1 0 and 5 . H . m*(€ ) = m* ( 1 + A S + ...) ? m* 2 where A = 8 a m * * = 2 p ( — ) ( 5 . 1 2 ) The b e s t e x p e r i m e n t a l e v i d e n c e s u p p o r t i n g t h e n o n - p a r a b o l i c i t y model i s t h e v e r y r e c e n t work o f Rein e , A ggarwal and Lax ( 1 9 7 2 ) . They measured t h e s t r e s s - m o d u l a t e d 110 i n t e r b a n d m a g n e t o - r e f l e c t i v i t y a t a te m p e r a t u r e o f 30K. They o b s e r v e d many t r a n s i t i o n s between Landau l e v e l s i n t h e v a l e n c e band and t h o s e i n t h e c o n d u c t i o n band. By u s i n g a g e n e r a l i z e d l e a s t s q u a r e s f i t t i n g p r o c e d u r e t h e y d e t e r m i n e d f i v e band p a r a m e t e r s . I n p a r t i c u l a r t h e y d e t e r m i n e d t h e e l e c t r o n e f f e c t i v e mass a t t h e band edge as m* = 0.0418 + .0012. They a l s o found v e r y good e v i d e n c e o f t h e c o n d u c t i o n band n o n - p a r a b o l i c i t y and e x p e r i m e n t a l l y d e t e r m i n e d p = 0.43 meV"""1". U s i n g t h e 3 1 measured p and m* v a l u e s t h e c o e f f i c i e n t A i s 1.49 x 10"^ meV" w i t h an e s t i m a t e d e r r o r o f + 10%. 5.5 The E f f e c t i v e Mass Average C o n s i d e r t h e i n c r e m e n t a l c o n t r i b u t i o n t o t h e measured e f f e c t i v e mass o f a c o n c e n t r a t i o n o f e l e c t r o n s , w i t h energy i n t h e i n t e r v a l ( £ ± , € ± + ), W t = UJ"tr ^ m* = m* and d i s p e r s i o n parameter 8 . The measured peak parameter f o r t h e s e e l e c t r o n s , e x p r e s s e d as an e f f e c t i v e mass i s ra i = m i ^+-r&E-x (5.13) model and e q u a t i o n 5.12 we have m. = m * (1 + A < £ . ) ( ! + — J ^ - + E.) (5.14) I n t h i s e q u a t i o n t h e maximum v a l u e o f A 6 . i s 0.045 f o r € . = £ and t h e maximum v a l u e s o f and l E . I a r e l M 2 cu T 1 11 ~ 0 . 0 5 and 0 . 0 2 r e s p e c t i v e l y , f o r u / f > 2 and t h e c o n d i t i o n |0| < 0 . 2 . The terms on t h e r i g h t hand s i d e o f e q u a t i o n 5 . 1 ^ can t h e r e f o r e be expanded d r o p p i n g p r o d u c t s o f s m a l l q u a n t i t i e s w h i c h a r e o f t h e o r d e r 0 . 3 $ o r s m a l l e r . These q u a n t i t i e s a r e e x p e c t e d t o average t o a n e g l i g i b l e s i z e o v e r many measurements. The e x p a n s i o n g i v e s m t = m* ( 1 + A € i + -~Tr~ * E i } ( 5 ' 1 5 ) F o r t h e o b s e r v e d m* t h e c o r r e c t w e i g h t e d average o f ra^ o v e r i e t h e e l e c t r o n d i s t r i b u t i o n s l c ( s e e Appendix 3 . 1 ) i M 3 ) p(e ) r ( 6 ) m(€ ) d<s i«4= =5— = ( 5 . 1 6 ) 1 ' f (€ )T(6 ) d£ I n t h i s average 0 i s t h e same f o r a l l e l e c t r o n s s i n c e i t i s d e t e r m i n e d by t h e same microwave r e f l e c t i o n s . E q u a t i o n s 5 . 1 5 and 5 . 1 6 g i v e m = m* ( 1 + A £ + L^l + | ) ( 5 . 1 7 ) o 2 UJ f€m 3 w i t h }. = 0 J ( 5 . 1 8 ) J P(£ ) -C<e ) 3 d€ 0 J where X s 6<>r f " 1 ^ ) .or E (6 ) . D e f i n e <T> = (T1) . 1. The e x p e r i m e n t a c t u a l l y measures in and n o t m~l s i n c e B o i = m i u r / e , , a* re s o n a n c e " and t h e peak measurement g i v e s B Q , t h e w e i g h t e d average o f B Q^. 112 S i n c e E ( € ) i s v e r y s m a l l t h e d i f f e r e n c e between E and E( <T*> ) used i n t h e peak c o r r e c t i o n s w i l l be n e g l e c t e d . The e r r o r i n u s i n g t h e peak c o r r e c t i o n f a c t o r 8/2 6cr<tr> must a l s o be s m a l l s i n c e t h e c o r r e c t i o n s c a l c u l a t e d from r e p l a c i n g uf<T} by t h e measured U/-T average have been shown t o be s m a l l i n s e c t i o n 3.4(d). We w i l l assume t h e s e e r r o r s a r e i n c l u d e d i n t h e e s t i m a t e d + 0.4% e r r o r due t o d i s p e r s i o n c o r r e c t i o n s ( t a b l e 6.3). U s i n g t h e R a b i n o v i c h d i s t r i b u t i o n f u n c t i o n s f o r m = 0 and m = 1 one o b t a i n s £ / 6 ^ = 0.30 and 0.68 r e s p e c t i v e l y . Comparing t h e s e t o t h e average e n e r g i e s c a l c u l a t e d t o be e. /&M -0.30 and 0.40 r e s p e c t i v e l y , one can see t h a t t h e main e r r o r i n £ comes from t h e u n c e r t a i n t y i n m, n o t i n t h e d i s t r i b u t i o n . F o r t h i s r e a s o f i i t i s u n i m p o r t a n t t o know t h e e x a c t energy d i s t r i b u t i o n i f t h e e n e r g y l i m i t i n g p r o c e s s i s impact i o n i z a t i o n o f t h e a c c e p t o r s . I t w i l l be assumed t h a t t h e range 0^m4l r e p r e s e n t s a r e a s o n a b l e e s t i m a t e f o r n e u t r a l a c c e p t o r s c a t t e r i n g -T a k i n g t h e v a l u e € / € M = 0.50 + .20 a l o n g w i t h = 3 0 + 4 meV y i e l d s f = 16 + 8 meV. U s i n g t h e v a l u e A = 1.49 x 10"3 meV" 1 + 10% t h e c o r r e c t i o n t o t h e e f f e c t i v e mass f o r e q u a t i o n 5-18 i s A. I = 0.032 + 0.021. T h i s w i l l be used i n s e c t i o n 6.2. 113 CHAPTER 6 THE EFFECTIVE MASS F i n d out t h e cause o f t h i s e f f e c t , Or r a t h e r say, t h e cause o f t h i s d e f e c t , F o r t h i s e f f e c t d e f e c t i v e comes by cause, from, "Hamlet" by W. Shakespeare 6.1 The Measured P o l a r o n E f f e c t i v e Mass I n e x p e r i m e n t s conducted a t f r e q u e n c i e s UJ <X US t h e e l e c t r o n moves i n t h e c r y s t a l s u r r o u n d e d by a l a t t i c e d e p o l a r i z a t i o n f i e l d due t o o p t i c a l phonons. The moving q u a s i p a r t i c l e , i . e . t h e " e l e c t r o n s u r r o u n d e d by a c l o u d o f v i r t u a l o p t i c a l phonons" i s c a l l e d a p o l a r o n ( K i t t e l 1963). I t has an e f f e c t i v e mass wh i c h d i f f e r s f r om t h e f r e e e l e c t r o n mass by an amount d e p e n d i n g on t h e average number o f phonons i n t h e c l o u d . I n t h e p r e s e n t e x p e r i m e n t t h e c o n d i t i o n us « t c r Q i s s a t i s f i e d and t h e r e f o r e t h e p o l a r o n e f f e c t i v e mass i s measured. I t w i l l be seen i n s e c t i o n 6.3 t h a t i n t h i s case t h e p o l a r o n mass does n o t d i f f e r g r e a t l y from t h e f r e e e l e c t r o n mass. The c o r r e c t i o n w i l l be c a l c u l a t e d t h e r e . 6.1 (a) The A n g u l a r Dependence o f t h e E f f e c t i v e Mass The e f f e c t i v e mass i s shown i n f i g u r e 6.1 t o be ind e p e n d e n t o f m a g n e t i c f i e l d d i r e c t i o n i n t h e c r y s t a l . The e r r o r i n t h e a n g l e 0 d e r i v e s m a i n l y from t h e d i f f i c u l t y i n mea s u r i n g t h e sample o r i e n t a t i o n i n s i d e t h e c a v i t y . C a v i t y #2 was u s e d i n t h i s measurement t o a v o i d d i s p e r s i o n problems. I n t h i s c a v i t y t h e m a g n e t i c f i e l d may be r o t a t e d w h i l e t h e s i g n a l s t r e n g t h remains c o n s t a n t s i n c e t h e e l e c t r i c f i e l d i s v e r t i c a l . 535h in to u CD 530-o m c o 1 0 ? 5 2 5 h CL OJ Q_ 5 2 0 0 10 20 3 0 Magnetic Field 4 0 5 0 6 0 7 0 Direction , 8 (degrees) 8 0 9 0 100 F i g u r e 6 . 1 The a n g u l a r dependence o f t h e e f f e c t i v e mass. The m a g n e t i c f i e l d l i e s i n a ( 1 1 0 ) p l a n e . 115 T h i s p r o c e d u r e a v o i d s r o t a t i n g t h e sample, c a u s i n g a s y s t e m a t i c d i s p e r s i o n change and t h e r e f o r e a p o s s i b l e b i a s even a f t e r t h e d i s p e r s i o n c o r r e c t i o n s are made. The sample i s h e l d i n p o s i t i o n by a t h i n l a y e r o f s i l i c o n e vacuum g r e a s e , which produces a sample s t r a i n l e s s t h a n • ^  = 2 x 10~^ (see s e c t i o n 4.5). Any s t r a i n p r e s e n t d i d n o t a f f e c t t h e a/T v a l u e measureably. The peak p o s i t i o n = 528.4 + 2.9 may have been a f f e c t e d s i n c e i t d i f f e r s from t h e v a l u e 524.3 +3.6 measured f o r t h e same sample ( u n s t r a i n e d ) a t a s i m i l a r power i n a d i f f e r e n t c a v i t y . I t i s assumed t h a t any s uch s m a l l e f f e c t would n o t change t h e a n g u l a r dependence o f t h e e f f e c t i v e mass. A s t a t i s t i c a l t e s t was a p p l i e d t o th e d a t a t o s e t an upper l i m i t on th e measured a n g u l a r dependence o f m*. The measured v a l u e s o f B Q a r e g i v e n i n t a b l e 6.1 f o r a l l d a t a t a k e n w i t h i n 8° o f t h e t h r e e p r i n c i p a l c r y s t a l l o g r a p h i c d i r e c t i o n s . A s t a t i s t i c a l t e s t , based on S t u d e n t ' s t d i s t r i b u t i o n ( f r o m C r o x t o n e t a l . 1967) was a p p l i e d t o the l a r g e s t d i f f e r e n c e i n th e mean v a l u e s o f th e t a b l e . The t e s t shows t h a t one can be 97$ c o n f i d e n t t h a t t h e d i f f e r e n c e i n t h e p o p u l a t i o n means f o r t h e [lio] and [ i l l ] d i r e c t i o n s i s l e s s t h a n 5.3 G, o r 1$. T h i s r e s u l t i s c o n s i s t e n t w i t h b o t h band s t r u c t u r e c a l c u l a t i o n s and o t h e r measurements (see l a t e r ) . Because o f th e i s o t r o p i c e f f e c t i v e mass most measurements were performed i n u n o r i e n t e d samples. 116 T a b l e 6.1 The e f f e c t i v e mass d a t a f o r t h r e e c r y s t a l l o g r a p h i c d i r e c t i o n s C r y s t a l l o g r a p h i c D i r e c t i o n [100] [111] [ n o ] Number o f . Measurements 7 5 7 B Q ( G ) 529.6 527.9 529.6 +S 1(G) 3,60 2.35 - 3.09 i s t h e s t a n d a r d e r r o r i n one measurement. 6.1 (b) The Power Dependence o f t h e E f f e c t i v e Mass At t h e h i g h e s t power o b t a i n e d , t h e ob s e r v e d s i g n a l cannot be d e s c r i b e d w e l l by t h e t h e o r e t i c a l c u r v e ( f i g u r e 4.1 ( a ) , P = 0 d B ) . T h i s i s due t o t h e pr e s e n c e o f l i g h t h o l e s h a v i n g a d i f f e r e n t e f f e c t i v e mass and u / T . The peak p o s i t i o n cannot be p r e c i s e l y d e t e r m i n e d i n t h i s case because o f t h e peak b r o a d e n i n g and t h e g r e a t e r e r r o r s i n h e r e n t i n th e d i s p e r s i o n c o r r e c t i o n . A f u r t h e r c o m p l i c a t i o n i s t h e s m a l l plasma s h i f t o f b o t h t h e e l e c t r o n and h o l e peaks. A d i s c u s s i o n o f t h e c o r r e c t i o n s f o r t h e p r e s e n c e o f h o l e s i s c o n t a i n e d i n Appendix 6.2. The r e s u l t s w i l l be b r i e f l y summarized h e r e . The h i g h power c u r v e i s f i t w i t h t h e f o l l o w i n g p a r a m e t e r s : N h/N e = 0.8 (measured) J -X ( m*lh/m0 = 0 . 0 5 2 ( f r o m Appendix 6 , 1 ) UST = 3 . 3 (measured a t low power) a / T l h = 2 . 5 ( a d j u s t e d ) The f i n a l v a l u e s o f t h e c a l c u l a t e d a/T and B Q a r e 2 . 0 and 5 4 5 G, r e s p e c t i v e l y , w h i c h are i n s a t i s f a c t o r y agreement w i t h t h e measured uJt = 1.8 + 0 . 1 and B Q = 537 + 8 G. The peak c o r r e c t i o n s a t l o w e r powers are shown t o be 0 + 4 G a t P = - 6 dB, and 0 + 2 G a t P = - 1 2 dB. The power dependence o f B Q was measured f o r f o u r samples and t h e r e s u l t s t a b u l a t e d i n t a b l e 6 . 2 . The e r r o r s q u o t e d i n t h e t a b l e a r e t h e l a r g e r o f t h e two q u a n t i t i e s : ( i ) t h e s t a n d a r d d e v i a t i o n i n t h e s e t o f measurements, ( i i ) t h e mean o f t h e measured peak p o s i t i o n and d i s p e r s i o n c o r r e c t i o n e r r o r s . The w e i g h t e d a v e r a g e s show t h a t t h e e l e c t r o n e f f e c t i v e mass i s independent o f power w i t h i n about 1%, i n agreement w i t h t h e hot e l e c t r o n model. The s o u r c e s o f e r r o r i n t h e measurement ar e l i s t e d i n t a b l e 6 . 3 . . The e r r o r i n t h e c h o i c e o f f i t t i n g p r o c e d u r e , + 0 . 5 % i s e s t i m a t e d from e x p e r i e n c e w i t h o t h e r peak f i t t i n g p r o c e d u r e s w h i c h a l s o produce r e a s o n a b l e l i n e s h a p e agreement. To deduce t h e e f f e c t i v e mass v a l u e th e measurements a t - 6 dB are e x c l u d e d because o f t h e u n c e r t a i n t y i n t h e c o r r e c t i o n f o r h o l e s ( + 4 G ). The w e i g h t e d average o f t h e 19 measurements a t powers o f - 1 2 dB and -18 dB i s •B~ = 5 2 1.8 + 4 . 6 G. The w e i g h t e d s t a n d a r d e r r o r i n t h e mean i s + 2 . 0 G o r + 0 . 4 % . The v a l u e s o f t h e l a s t f o u r c o r r e c t i o n s i n t a b l e 6 . 3 are n o t r e d u c e d by i n c r e a s i n g t h e number o f T a b l e 6,2 The measured peak parameter as a f u n c t i o n o f microwave power and sample. B Q (P, sample) (Gauss) Power(dB) B a t t e l l e B e l l & Monsanto M o n t p e l l i e r Weighted H o w e l l 21 Average -6 5 2 5.8 •f - 3 . 0 521.2 ± 6.1 519.0 ± 6.4 520.0 t 8.0 524.3 t 4.0 (19) (3) (3) (1) (26) -12 524.3 ± 3.6 522.0 i- 2 . 5 520.2 ± 4.2 5H.0 ± 9 . 5 *522.5 ± 4.1 (9) ( l ) (2) (1) (13) 18 5 2 2.6 * 4.1 514.5 ± 7 . 5 517.5 * 9 . 5 *520.4 * 5.6 (4) (1) (1) (6) (N) - Number o f measurements * The w e i g h t e d average o f t h e 19 measurements a t Powers o f -12 and -18 dB i s 521.8 t 4.6 . 119 measurements. T h e r e f o r e t h e t o t a l e r r o r i n the measurement i s + 2.0%. From t h e v a l u e o f and t h e e q u a t i o n eB^/m* = Uf ( u s i n g UT/2TT = 35.^3 GHz) one o b t a i n s m*/m0 = 0.04123 + .00082. T h i s i s t h e p o l a r o n mass f o r e l e c t r o n s w i t h an average e n e r g y ~C = 15 meV,, C o r r e c t i o n s f o r band n o n - p a r a b o l i c i t y and f o r t h e p o l a r o n s h i f t a r e g i v e n i n s e c t i o n 6.2 and 6.3 r e s p e c t i v e l y . T a b l e 6.3 S o u r c e s o f e r r o r i n t h e measured peak parameter Source E r r o r Reduced E r r o r -measurement o f peak p o s i t i o n ±.5%^ measured d i s p e r s i o n c o r r e c t i o n s + .4% > + .9% * 0.4% e r r o r i n t h e c o r r e c t i o n f o r h o l e s ±0.3% c h o i c e o f f i t t i n g p r o c e d u r e ±0.5% a b s o l u t e f i e l d and f r e q u e n c y e r r o r s +0.2% p o s s i b l e d i f f e r e n c e s i n samples +0.6% (see s e c t i o n 6,4) TOTAL +2.0% 6.1 ( c ) The Temperature Dependence o f t h e E f f e c t i v e Mass The e f f e c t i v e mass was shown t o be independent o f te m p e r a t u r e from 1.3 t o 18 K by t h r e e measurements i n t h e B a t t e l l e sample summarized i n t a b l e 6.4. 120 T a b l e 6 . 4 The t e m p e r a t u r e dependence o f t h e e f f e c t i v e mass Temperature B o ± s l (K) (Gauss) 1 . 2 7 + . 0 7 527 + 5 4 . 2 3 + . 0 5 5 2 6 + 4 1 7 . 5 ± 1 . 5 525 + 4 6.2 The Band-Edge P o l a r o n Mass T a k i n g t h e measured p o l a r o n mass o f m*(€ )/m e q u a l t o 0.04123 + .00082 ( s e c t i o n 6.1 ( b ) ) and t h e c a l c u l a t e d n o n - p a r a b o l i c i t y c o r r e c t i o n m*( € ) = m * ( l 4- . 0 3 2 + .021) ( s e c t i o n 5 - 5 ) t h e band-edge p o l a r o n e f f e c t i v e mass i s c a l c u l a t e d t o be m*/mQ = 0.0400 + 0.0017 ( o r + 4 . 1 % ) . H a l f t h e e r r o r i n t h e r e s u l t i s due t o t h e n o n - p a r a b o l i c i t y c o r r e c t i o n . 6.3 The P o l a r o n C o r r e c t i o n and t h e F r e e E l e c t r o n E f f e c t i v e Mass There a r e c o r r e c t i o n s t o t h e e l e c t r o n e f f e c t i v e mass a r i s i n g from t h r e e e l e c t r o n phonon i n t e r a c t i o n s } t h e a c o u s t i c d e f o r m a t i o n p o t e n t i a l , t h e a c o u s t i c p i e z o e l e c t r i c and t h e o p t i c a l phonon i n t e r a c t i o n s . These e f f e c t s a r e c a l c u l a t e d i n d e t a i l i n Appendix 6.3. where t h e f i r s t two e f f e c t s a r e shown t o be v e r y s m a l l . The o p t i c a l phonon c o r r e c t i o n v / i l l be b r i e f l y summarized h e r e . The d i m e n s i o n l e s s p o l a r o n c o u p l i n g c o n s t a n t f o r GaSb i s a = 0.0249. S i n c e a « 1 t h e w e a k - c o u p l i n g a p p r o x i m a t i o n can be used t o c a l c u l a t e t h e change i n e l e c t r o n energy? 121 £ ( k ) = * k - < X ( W ) g ( U ) 2m* 2 t\ 2 k 2 1 where ^ = "V,., . x-U s i n g -4 = — \ x one o b t a i n s -i = i» ( 1 - 7^(77 )) • • t 'O where F(r> ) = * L—* v 1 _ s l n " 1 ( ^ ) F o r ?? <^1, F( -xj ) = 1 w h i l e F( . 5 ) = 1.82 and F( .9) = 6 . 1 5 . As 1, F(T^ ) — o o but i n such a way t h a t t h e average o f F(T£ ) i s not d i v e r g e n t . S i n c e -g-- F(T£ ) i s s m a l l f o r most v a l u e s o f ->£ c o n s i d e r e d , t h e r e s u l t i s i n v e r t e d t o g i v e m^= m* ( 1 + ^F( ) ) . When t h i s i s averaged o v e r t h e d i s t r i b u t i o n f u n c t i o n t h e r e s u l t f o r F, d e f i n e d a n a l o g o u s l y t o e q u a t i o n 5.18, i s F = 2 . 6 + 0.8. The average i s t h e r e s u l t o f c a l c u l a t i o n s u s i n g t h e R a b i n o v i c h d i s t r i b u t i o n w i t h m = 0 and m = 1 as w e l l as a c a l c u l a t i o n u s i n g an e n e r g y - i n d e p e n d e n t T- ( £• ) and ^ ( € : ) . The c o r r e c t i o n i s g i v e n by mg(polaron) = m * ( f r e e ) ( 1 . 0 l 0 + . 0 0 3 ) . T h i s y i e l d s , f.or t h e band-edge f r e e e l e c t r o n e f f e c t i v e mass m * ( f r e e ) 0.0396 + 0.0018 ( o r + 4 . 4 % ) . S i x t y p e r c e n t o f t h e e r r o r i n the r e s u l t d e r i v e s from t h e hot e l e c t r o n d i s t r i b u t i o n t h r o u g h t h e n o n - p a r a b o l i c i t y and p o l a r o n c o r r e c t i o n s , 6 . 4 The E f f e c t i v e Mass i n D i f f e r e n t Samples As seen i n t a b l e 6 . 2 t h e r e i s a s m a l l dependence o f the measured peak parameter on t h e sample. The t h r e e samples on t h e r i g h t - h a n d s i d e o f t h e t a b l e w i l l be c a l l e d group C. At each 122 power l e v e l , B Q f o r group C i s about 6 G l e s s t h a n f o r t h e B a t t e l l e sample. T r e a t i n g t h e 13 measurements o f group C samples a t a l l t h r e e powers as a s e t , and u s i n g t h e s t a n d a r d e r r o r i n th e mean, t h e r e i s a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e between group C and t h e B a t t e l l e sample a t t h e 90% c o n f i d e n c e l e v e l . To a l l o w f o r t h i s e f f e c t an e r r o r o f + 0.6% ( + 3G) has been e s t i m a t e d f o r t a b l e 6,3. An e x p l a n a t i o n f o r t h e sample d i f f e r e n c e can be o f f e r e d by t a k i n g t h e e n ergy l i m i t i n g p r o c e s s t o be impact i o n i z a t i o n o f a c c e p t o r s . The energy b a r r i e r , E^, i s t h e n somewhat d i f f e r e n t f o r each sample (see Appendix 3.*0. T a k i n g = 1,4 a t 15 KG th e mean v a l u e o f E^ f o r t h e group C s a m p l e s 1 , i s 24.7 meV + 1.5meV. The l o w e r E A f o r group C r e s u l t s i n s m a l l e r c o r r e c t i o n s b o t h f o r n o n - p a r a b o l i c i t y (.025+,017) and f o r p o l a r o n e f f e c t s ( F = 1.8 + 0.8) s i n c e most o f th e p o l a r o n s h i f t o c c u r s a t h i g h e n e r g i e s as G — > ^ u r Q . The r e s u l t o f t h e s e c o r r e c t i o n s r e d u c e s t h e raw d a t a e f f e c t i v e mass r a t i o s ( i . e . q u o t i n g o n l y t h e s t a n d a r d e r r o r i n t h e mean o f th e measurements) from m*(e)/m 0 = .0414 +.0002 ( B a t t e l l e ) and .0408 + .0002 (Group C) t o th e same band edge v a l u e m * ( f r e e ) / m Q = .0398 + .0015 ( w i t h a l l t h e o t h e r e r r o r s r e s t o r e d ) . The o b j e c t i o n t o t h i s e x p l a n a t i o n i s t h a t a v a l u e r ? 7 = 1.4 a t 15 KG r e q u i r e s r 7 ? = 1.6 a t 0 KG from the measured f i e l d dependence o f r 7 7 . S i n c e t h e s c a t t e r i n g i s m a i n l y polar-mode phonon s c a t t e r i n g and s i n c e r ? 7 ^ 1 . 0 5 ( S t i l l m a n e t a l . 1970) f o r t h a t case t h e l a r g e v a l u e o f r 7 7 r e q u i r e d f o r t h i s e x p l a n a t i o n cannot be ac c o u n t e d f o r . 6 . 5 Comparison With O t h e r Measurements Other d e t e r m i n a t i o n s o f the e l e c t r o n e f f e c t i v e mass 1. There are t o o few low power measurements on t h e group C samples t o t r e a t them i n d i v i d u a l l y . 123 are p r e s e n t e d i n t a b l e 6,5 ' A l l t h e t e c h n i q u e s r e p o r t e d i n t h e t a b l e measure t h e " c y c l o t r o n resonance e f f e c t i v e mass", 1 1 F —sr- = —o—(• °- ) . The F e r m i e n e r g i e s l i s t e d were c a l c u l a t e d m 2 k * k by Yep and B e c k e r (1966). The d a t a conform t o t h e non-p a r a b o l i c i t y model. The most p r e c i s e measurement, r e p o r t e d by Reine e t a l . ( 1 9 7 2 ) r e s u l t s from m e a s u r i n g t h e reduced mass o f t h e e l e c t r o n s and heavy h o l e s , and d e r i v i n g f i v e band parameters by a g e n e r a l i z e d l e a s t s q u a r e s f i t t i n g p r o c e d u r e . The r e p o r t e d v a l u e o f 0.0418 + 0.0012 f o r t h e band-edge p o l a r o n mass i s i n agreement w i t h t h i s d e t e r m i n a t i o n o f 0.0400 + 0.0017, 2 w i t h i n e x p e r i m e n t a l e r r o r . * S e i l e r and B e c k e r (I969) made a c a r e f u l s t u d y o f t h e Shubnikov-de Haas o s c i l l a t i o n s . From t h e o b s e r v a t i o n o f a b e a t i n g e f f e c t i n t h e o s c i l l a t i o n s t h e y i n f e r r e d a warped F e r m i s u r f a c e . That i s , a t s u f f i c i e n t l y h i g h e l e c t r o n e n e r g i e s the. e f f e c t i v e mass i s a n i s o t r o p i c . I n o r d e r t o o b t a i n e f f e c t i v e mass v a l u e s t h e a m p l i t u d e o f t h e o s c i l l a t i o n s i s f i t t e d t o the t h e o r y . Two a d j u s t a b l e parameters a r e u s e d ; t h e n o n t h e r m a l b r o a d e n i n g t e m p e r a t u r e and t h e i n h o m o g e n e i t y b r o a d e n i n g t e m p e r a t u r e , T^. These par a m e t e r s have a g r e a t e f f e c t on th e deduced mass parameter. F o r example, f o r t h e i r 1. Two r e f e r e n c e s have been o m i t t e d from t h e t a b l e because t h e y m a i n l y i n v o l v e a r e c a l c u l a t i o n o f o t h e r d a t a (m*/mQ = . 041, H a l p e r n 1965, and m*/m0 = .045, A d a c h i 1968). 2. The f i n a l v a l u e r e p o r t e d here i s w i t h i n t h e quoted e r r o r i n t h e p u b l i s h e d a b s t r a c t ( H i l l and S c h w e r d t f e g e r 1970) w h i c h gave m*/m0 = 0.042 + 0.002. T a b l e 6 . 5 Other D e t e r m i n a t i o n s o f t h e E l e c t r o n E f f e c t i v e Mass i n GaSb Temperature (K) F e r m i Energy o r average energy (meV) m*( E p o r e ) Technique Type m 0 R e f e r e n c 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 7 7 , 300 n E p = 46 049 + * 0 0 5 * U 4 y - .003 P i l l e r ( 1 9 6 3 ) 20 n .048 + .002 Bordure and G u a s t a v i n o (1968) I n t e r b a n d Magneto-R e f l e c t i v i t y 1 . 5 P e = 67 + 3 .047 + . 0 0 3 Z w e r d l i n g e t a l . (1959) w i t h S t r e s s - M o d u l a t i o n 30 P € = 0 .0418 + .0012 R e i n e e t a l . (1972) M a g n e t o r e s i s t a n c e Measurements 4.2, 77, 3 0 0 n Ep = 80±5 • . 0 5 2 + .002 B e c k e r e t a l . (I96I) and Sagar (i960) Shubnikov-de Haas • O s c i l l a t i o n s 1.3-20 n Ep = 3 0 - 5 0 f o u r samples .041 + . 0 0 5 Yep and Beck e r (I966) " and t h e H a l l E f f e c t 1.3-^.2 n E w not c a l c u -* l a t e d whole s e t , S e i l e r and Becker(1969) see d i s c u s s i o n T h i s work 1.2-20 P e = 15 € = 0 "e = 0 .0412 ± .0008"^  .0400 + .0017J .0396 + .0018 ( p o l a r o n m*) ( f r e e m*) 125 sample 2JB i n t h e <^ L 1 i"> d i r e c t i o n t h e r e s u l t i s — = 0.0506 + .0006 f o r T. = 0 and m 0 - 1 'jjp = 0.0441 + .0007 f o r T i = 0.84 + .20 K. o A l t h o u g h t h e a b s o l u t e v a l u e s depend on T^j t h e r e l a t i v e mass v a l u e s f o r d i f f e r e n t c r y s t a l l o g r a p h i c d i r e c t i o n s a r e a p p r o x i m a t e l y independent o f T^. The e f f e c t i v e mass i n t h e [ i l l ] d i r e c t i o n was found t o be ^ 2% l a r g e r t h a n i n t h e p. 0 0] and [l 1 0] d i r e c t i o n s a t an energy o f E p = 80 meV. T h i s agrees w i t h t h e c a l c u l a t i o n o f Zhang (1970) w h i c h p r e d i c t s a l i n e a r d i v e r g e n c e from k = 0 o f t h e e f f e c t i v e mass parameter f o r d i f f e r e n t d i r e c t i o n s . The p r e d i c t e d d i f f e r e n c e i s 0.3% a t 15 meV and was n o t d e t e c t e d i n t h i s e x p e r i m e n t . Comparing t h e two b e s t measurements ( t h i s work and Reine e t a l . 1972) one can c o n c l u d e t h a t t h e band-edge p o l a r o n mass i s m* — — = 0.0409 + 0.0015. 126 CHAPTER 7 CONCLUSIONS "Caveat emptor." The e l e c t r o n c y c l o t r o n resonance s i g n a l i n GaSb i s r e p o r t e d f o r t h e f i r s t t i m e . The p h o t o c r e a t e d microwave s i g n a l i s measured i n t h e t e m p e r a t u r e range 1-30 K and t h e c a r r i e r s a r e i d e n t i f i e d as e l e c t r o n s by c i r c u l a r p o l a r i z a t i o n measurements. The e f f e c t s o f background a b s o r p t i o n , d i s p e r s i o n , s u r f a c e s h i f t s , t h e pr e s e n c e o f h o l e s ( a t h i g h power) and plasma s h i f t s are a l l t a k e n i n t o a c c o u n t . E v i d e n c e i s p r e s e n t e d t h a t t h e s i g n a l i s due t o e l e c t r o n s w h i c h a re s t r o n g l y h e a t e d by t h e microwave e l e c t r i c f i e l d . The e l e c t r o n s s t e a d i l y g a i n energy u n t i l t h e y r e a c h an "energy b a r r i e r " o f 30 + 4 meV above t h e c o n d u c t i o n band edge. The e l e c t r o n e n e r g y i s t h e n v e r y q u i c k l y l o s t by e i t h e r LO phonon c r e a t i o n o r t h e p r o c e s s o f impact i o n i z a t i o n o f an a c c e p t o r . The measured power dependence o f t h e e l e c t r o n s i g n a l i n t e n s i t y i s Ye(P) a P 0 - 2 9 + - 0 5 . The s c a t t e r i n g mechanism i n t h e p-type samples i s s t u d i e d t h r o u g h t h e measured LVT v a l u e o f t h e c y c l o t r o n r e s o n a n c e s i g n a l ( LOT = 1.5-4, T =*= I O " 1 1 s ) . At l i q u i d h e l i u m t e m p e r a t u r e s t h e s c a t t e r i n g p r o c e s s i s a t t r i b u t e d p a r t l y t o n e u t r a l a c c e p t o r s c a t t e r i n g and p a r t l y t o an u n i d e n t i f i e d r e s i d u a l p r o c e s s . The c o l l i s i o n t i me f o r e l e c t r o n s s c a t t e r e d from t h e n e u t r a l d e f e c t a c c e p t o r s i s i n good agreement w i t h t h e B l a g o s k l o n s k a y a (1969) f o r m u l a f o r t h e s c a t t e r i n g o f hot e l e c t r o n s from n e u t r a l h y d r o g e n i c a c c e p t o r s . I t i s o n l y 127 p o s s i b l e t o s t u d y t h i s s c a t t e r i n g p r o c e s s w i t h p h o t o c r e a t e d e l e c t r o n s s i n c e an n-type sample would c o n t a i n ~ 2 x I O 1 7 cm~^ i o n i z e d i m p u r i t i e s w h i c h would c o n s t i t u t e the main s o u r c e o f s c a t t e r i n g . The o b s e r v a t i o n o f a s c a t t e r i n g t i m e 'C a T~2'^ — a t 30 K i s not u n d e r s t o o d . The measured tu"T f o r " h o t " h o l e s , 2.5, i s t e n t i m e s t h a t r e p o r t e d f o r " c o l d " h o l e s and can n o t be e x p l a i n e d . The measured e l e c t r o n p o l a r o n e f f e c t i v e mass i s — = 0.0412 + .0008 f o r e l e c t r o n s h a v i n g an average energy m o — o f 15 meV. The a n g u l a r v a r i a t i o n o f t h e e f f e c t i v e mass i s l e s s t h a n 1% w i t h 97% c o n f i d e n c e . A weak dependence o f t h e measured e f f e c t i v e mass on sample i s o b s e r v e d and t a k e n i n t o a c c o u n t . I n agreement w i t h t h e hot e l e c t r o n model, t h e measured m* i s independent o f t e m p e r a t u r e and microwave power. C o r r e c t i o n s f o r t h e c o n d u c t i o n - b a n d n o n - p a r a b o l i c i t y y i e l d a band-edge p o l a r o n e f f e c t i v e mass o f m o ( P o l a r o n ) - 0.0400 + m_ — o 0.0017. When t h e p o l a r o n c o r r e c t i o n s a re t a k e n i n t o a c c o u n t , t h e band-edge f r e e e l e c t r o n e f f e c t i v e mass i s deduced t o be m * ( f r e e ) — = 0.0396 + .0018. S i x t y p e r c e n t o f t h e e r r o r i n o th e r e s u l t d e r i v e s from t h e hot e l e c t r o n d i s t r i b u t i o n t h r o u g h t h e n o n - p a r a b o l i c i t y and p o l a r o n c o r r e c t i o n s . The o b s e r v a t i o n o f a s u r f a c e e f f e c t on t h e peak p o s i t i o n i s r e p o r t e d f o r t h e f i r s t t i m e . I t may r e s u l t from t h e hot e l e c t r o n d i s t r i b u t i o n . The t h e o r y o f t h e plasma e f f e c t s due t o a "background" c a r r i e r c o n c e n t r a t i o n i s a p p l i e d . The absence o f any e l e c t r o n c y c l o t r o n r esonance plasma s h i f t due t o t h e h o l e s i n t h e i m p u r i t y band i s t h u s e x p l a i n e d . The o b s e r v e d e x p o n e n t i a l 128 l o s s o f e l e c t r o n p h o t o c o n d u c t i v i t y a t 27 K i s a l s o p r e d i c t e d by t h e t h e o r y . P o s s i b l e s u g g e s t i o n s f o r f u r t h e r e x p e r i m e n t s i n c l u d e : ( i ) t h e e x t e n s i o n o f t h e e f f e c t i v e mass measurement t o t h e f a r i n f r a - r e d r e g i o n o f t h e spectrum, where a more a c c u r a t e measurement o f m* may be p o s s i b l e , ( i i ) f u r t h e r s t u d y o f t h e s m a l l i m p u r i t y o r sample e f f e c t on m*, ( i i i ) i n v e s t i g a t i o n o f t h e s u r f a c e - a f f e c t e d c y c l o t r o n resonance s i g n a l as a p o s s i b l e t e c h n i q u e f o r t h e s t u d y o f s u r f a c e p r o p e r t i e s , ( i v ) t h e i d e n t i f i c a t i o n o f t h e s c a t t e r i n g mechanism a t 30K, w h i c h might y i e l d f u r t h e r i n f o r m a t i o n on t h e s c a t t e r i n g o f hot e l e c t r o n s t h r o u g h t h e measured t e m p e r a t u r e dependence o f u/TL . 1 2 9 BIBLIOGRAPHY Names impress a c c o r d i n g t o t h e square o f t h e i r i n i t i a l s , f rom "Gamesmanship", by Stephen P o t t e r E. 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S o l i d s 9, 320. 136 Appendix: a d j u n c t , a d d i t i o n , a f f i x appendage, augmentation, r e i n f o r c e m e n t , a c c e s s o r y , g a r n i s h , accompaniment, complement, supplement, e x t e n s i o n , p o s t s c r i p t , i n t e r l i n e a t i o n , c o r o l l a r y , o f f - s h o o t , t r a p p i n g s , s i d e i s s u e , c o l o p h o n , from Roget's Thesaurus (Longmans) Appendix 2.1 D e r i v a t i o n o f t h e S i g n a l F o r m u l a The change i n c a v i t y Q due t o t h e sample a b s o r p t i o n w i l l f i r s t be c a l c u l a t e d and t h e n t h e r e l a t i o n s h i p t o t h e o b s e r v e d s i g n a l w i l l be d e r i v e d . The l o s s e s a s s o c i a t e d w i t h t h e r e s o n a n t c a v i t y c o n s i s t o f l o s s e s i n t h e c a v i t y and t h e power l o s t from t h e c a v i t y out t h e c o u p l i n g a p e r t u r e . The c a v i t y " unloaded Q" and " e x t e r n a l Q", Q and Q_v, are d e f i n e d i n r e l a t i o n t o t h e s e two l o s s e s r e s p e c t i v e l y . I n p a r t i c u l a r , from P o o l e ( 1 9 6 ? ) , p 1 _ c a v i t y w a l l l o s s e s + o ( P 0 ) V (otE 0) Q ~ - o 3  ° aJ 6^ vo E o c o where o"(P Q) i s t h e sample p h o t o c o n d u c t i v i t y , V g and V c are t h e sample and c a v i t y volumes, E Q i s the r.m.s, c a v i t y e l e c t r i c f i e l d and a i s t h e r a t i o o f t h e r.m.s. e l e c t r i c f i e l d i n s i d e the sample t o E Q . When t h e p h o t o c o n d u c t i v i t y i s s w i t c h e d on, t h e e x t e r n a l Q does n o t change s i n c e i t i s r e l a t e d t o t h e c o u p l i n g i r i s , b ut Q Q changes. I t i s e a s i l y seen t h a t 2 r t 1 \ _ cr(Pn) V q a / T, . • S { % - ] = c 3 - e 0 \ ~ a a(po>n where V = a 2 Vg_ t h e e f f e c t i v e f i l l i n g f a c t o r . 137 The measured s i g n a l i s r e l a t e d t o & ( l / Q Q ) i n the f o l l o w i n g way. The c a v i t y r e f l e c t i o n c o e f f i c i e n t P i s d e f i n e d by E r = T E ^ n where E r i s the e l e c t r i c f i e l d r e f l e c t e d from t h e c a v i t y and E ^ n i s t h e e l e c t r i c f i e l d i n c i d e n t on t h e c a v i t y ( G o l d s b o r o u g h and M a n d e l i960). F o r a c a v i t y o p e r a t e d a t i t s r e s o n a n t f r e q u e n c y P i s r e a l , and t h e p h o t o i n d u c e d change i n E r , o r t h e " s i g n a l e l e c t r i c f i e l d " , i s c f S g ^ g = E ^ n SP. r can be r e l a t e d t o S(l/Q ) t h r o u g h t h e c o u p l i n g c o e f f i c i e n t , Q o 1 - r 1-B - 7 — and t h e r e l a t i o n s h i p B = , n ( F e h e r 1957). %x 1 * r The e q u a t i o n s r e l a t i n g £C to S ( l / Q Q ) a r e : ^ B = g i - . i ' Q 0 and = r ^ ^ Q 0 . ^ex ^ o ^ o These g i v e , u s i n g t h e d e f i n i t i o n o f 3, 2 y o a TJ' Q Q (1 - f 2 ) a(P 0) w h i c h i s t h e r e s u l t used i n s e c t i o n 2.2. Some p u b l i s h e d a n a l y s e s o f m a g n e t i c resonance spectrome-t e r s ( S t r a n d b u r g e t a l . 1956, Feher i960) d e r i v e t h e i n c o r r e c t r e s u l t t h a t t h e most s e n s i t i v e c o u p l i n g c o n d i t i o n i s n o t always P =0, b u t r a t h e r depends on whether t h e c r y s t a l d e t e c t o r 1. The e x p e r i m e n t a l d e t e r m i n a t i o n o f P and p i s d i s c u s s e d by Y a r i v and Clapp (1959). 138 r e s p o n s e i s l i n e a r o r square law. They e s s e n t i a l l y make th e m i s t a k e o f s q u a r i n g v o l t a g e s and a d d i n g t o g e t t h e t o t a l power, r a t h e r t h a n a d d i n g and t h e n s q u a r i n g . The b e s t p u b l i s h e d a n a y s i s , by G o l d s b o r o u g h and Mandel (i960), p o i n t s out t h i s m i s t a k e and t h e n makes a n o t h e r one i n t h e second l a s t l i n e o f the paper. 139 Appendix 2.2 The C r o s s - M o d u l a t i o n Technique An a l t e r n a t e t e c h n i q u e f o r t h e microwave c y c l o t r o n resonance e x p e r i m e n t has been d e v e l o p e d ( Z e i g e r e t a l , 1959, K a p l a n 1965, Mears and S t r a d l i n g 1969). I n some cases i t y i e l d s a s e n s i t i v i t y improvement o f two o r d e r s o f magnitude over t h e microwave p h o t o c o n d u c t i v i t y t e c h n i q u e . P e r p e n d i c u l a r d.c. m a g n e t i c and e l e c t r i c f i e l d s are a p p l i e d t o a sample i n t h e microwave system."*"' The microwave power i s used t o heat t h e c a r r i e r s and a change i n t h e d.c, m a g n e t o r e s i s t a n c e i s r e c o r d e d . When t h e m a g n e t i c f i e l d i s swept t h e d.c. c o n d u c t i v i t y change t r a c e s out t h e c y c l o t r o n resonance c u r v e . The t e c h n i q u e s u f f e r s from s e v e r a l problems. To a v o i d i n j e c t e d c a r r i e r problems ohmic l e a d s t o t h e sample are r e q u i r e d . A " t h e r m o e l e c t r i c peak r e v e r s a l " must be a v o i d e d by o p e r a t i n g a t s u f f i c i e n t l y h i g h power ( F i s h e r and Wagner 1968). There a r e a l s o , i n some c a s e s , u n e x p l a i n e d problems p o s s i b l y r e s u l t i n g from s u r f a c e and h y s t e r e s i s e f f e c t s ( Z e i g e r e t a l . 1 9 5 9 ) . F u r t h e r m o r e i t i s n o t c l e a r t h a t t h e measured peak p o s i t i o n s a t i s f i e s t h e c y c l o t r o n resonance c o n d i t i o n , o r = UJ, e s p e c i a l l y f o r n o n - p a r a b o l i c bands (Akopyan 1971 and 1972). S e v e r a l a d d i t i o n a l problems i n t h e t e c h n i q u e a r e i m p o r t a n t f o r t h i s e x p e r i m e n t . F i r s t , i t i s b e l i e v e d t h a t t h e e f f e c t r e q u i r e s an e n e r g y dependent c o l l i s i o n t i m e , T(£ ), ( Z e i g e r e t a l . 1959) w h i c h may n o t be t r u e i n t h e p r e s e n t e x p e r i m e n t . I n a d d i t i o n , s i n c e t h e c a r r i e r s are p h o t o c r e a t e d t h e h e a t i n g o f t h e s m a l l p h o t o - c o n d u c t i v i t y would have t o be d e t e c t e d a g a i n s t a l a r g e r background c o n d u c t i v i t y . A s e p a r a t e microwave c i r c u l a r 1. Hence t h e name " c r o s s - m o d u l a t i o n " t e c h n i q u e . 140 p o l a r i z a t i o n e x p e r i m e n t would be r e q u i r e d i n any case t o i d e n t i f y t h e c a r r i e r s i g n . Some means o f m o n i t o r i n g t h e c a r r i e r t y p e as a f u n c t i o n o f power i n t h e c r o s s - m o d u l a t i o n scheme would have t o be d e v e l o p e d . The a p p l i c a t i o n o f t h e hot e l e c t r o n model, w h i c h c o n t r i b u t e s h a l f o f t h e f i n a l e r r o r i n t h e p r e s e n t measure-ment, would a l s o be n e c e s s a r y . F o r t h e above r e a s o n s , i t i s s e e n t h a t t h e c r o s s - m o d u l a t i o n t e c h n i q u e does n o t o f f e r any advantages i n t h e p r e s e n t e x p e r i m e n t , 141 Ap p e n d i x 3.1 Peak P o s i t i o n C a l c u l a t i o n s ( i ) The e f f e c t o f d i s p e r s i o n on t h e c i r c u l a r p o l a r i z a t i o n peak p o s i t i o n i s c a l c u l a t e d . The ob s e r v e d s i g n a l i s p r o p o r t i o n a l t o f ( ^ c ; 2 > ) = O - R ( + N + 3 a i ( + ) • T h e p e a k P ° s i t i o n . = \) i s t h e s o l u t i o n t o t h e e q u a t i o n — = 0. c cp b v c U s i n g e q u a t i o n 3.3 t h e e x a c t s o l u t i o n i s ^ c p " V . = 28 )) (4 V2 + 3) V (4y 2 +1J2 F o r klJ2 » 1 t h i s r e d u c e s t o V -jJ t $ o r V — = 2 7 " B* - B p -°~-5—— = P , as s t a t e d i n s e c t i o n 3 .4(b). ( i i ) F o r t h e o b s e r v e d l i n e a r p o l a r i z a t i o n s i g n a l t h e e x a c t peak s h i f t due t o d i s p e r s i o n cannot be c a l c u l a t e d a n a l y t i c a l l y . The r e s u l t i s a p p r o x i m a t e d by k p V ( 4 V 2 n - 3 ) - 1  V kV 2(1 + 2 i)2) - (3 + 8 y 2 ) whic h r e d u c e s f o r 8 = 0 t o y c " ^  • -1 — - = 5 TT~ , T h i s term r e p r e s e n t s the . V 4 V (1 + 2 J/^) peak s h i f t i n t h e l i n e a r p o l a r i z a t i o n s i g n a l due t o t h e non-r e s o n a n t p o r t i o n o f t h e L o r e n z i a n c u r v e , a R ^ y F o r |3| l a r g e , t h e f o r m u l a r e d u c e s t o ^ c p " ^  _ p s i n c e t h e main peak c o r r e c t i o n i n t h i s case i s from t h e d i s p e r s i o n and t h e main d i s p e r s i o n c o r r e c t i o n i s due t o Ow.x T h i s can be seen on I ( + ) . 142 f i g u r e 3 . 1 where t h e s l o p e o f 0'j(+) at ^ c = V i s much s t e e p e r t h a n t h e s l o p e o f e i t h e r a R ^ _ ^ o r o"j(_)« The peak c o r r e c t i o n s t o t h e l i n e a r p o l a r i z a t i o n s i g n a l due t o t h e s e two f a c t o r s a re c a l c u l a t e d t o h i g h a c c u r a c y on t h e computer by a n u m e r i c a l s o l u t i o n o f t h e e q u a t i o n s i n v o l v e d . A b r i e f summary of t h e r e s u l t s i s p r e s e n t e d i n t a b l e A3.1. ^cAV ^ S ^ e e s t i m a t e o f V1 d e r i v e d from a v e r a g i n g the two l> v a l u e s o b t a i n e d a t t h e i n t e r s e c t i o n s o f t h e l i n e y = aH w i t h t h e a b s o r p t i o n c u r v e , o f peak h e i g h t H. E x p e r i m e n t a l l y t h e v a l u e o f a used i n d e t e r m i n i n g ^ c^y» and hence B' , i s m a i n t a i n e d i n t h e range 0 . 9 2 < a < 0 . 9 6 . The computer program c a l c u l a t e s w i t h a = 0 , 9 ^ and t h e e r r o r i n the r e s u l t due t o t h e v a r i a t i o n s i n a i s v e r y s m a l l . The l a s t column o f t h e t a b l e shows t h e q u a n t i t y 0/21^ , shown above t o be an e s t i m a t e o f cp ^ . ( i i i ) F o r an inhomogeneously broadened pure a b s o r p t i o n l i n e , an a p p r o x i m a t e f o r m u l a f o r t h e c o n t r i b u t i o n s o f t h e e f f e c t i v e mass components w i l l be d e r i v e d . I t i s assumed t h a t t h e averaged peak i n a l i n e a r p o l a r i z a t i o n s i g n a l i s a p p r o x i m a t e l y t h e same as t h e averaged peak i n a c i r c u l a r p o l a r i z a t i o n s i g n a l . I t i s a l s o assumed t h a t d i f f e r e n c e s between V c p and V c A V f o r t h e inhomogeneously broadened c i r c u l a r p o l a r i z a t i o n peak are s m a l l . The sum o f two c i r c u l a r p o l a r i z a t i o n pure a b s o r p t i o n l i n e s o f peak p o s i t i o n s B ^, B r 2 , c o l l i s i o n t i m e s T^, "£" 2, e f f e c t i v e mass v a l u e s m^, m*2, and peak h e i g h t s H-^ , H 2, i s g i v e n by H l H 2 S = ±_ • c l + ( 2 - E i ) 2 ( B - B r l ) 2 1 * ( ? ^ ) 2 ( B - B r 2 ) 2 mj i J- m2 Table A 3.1 Linear polarization peak position corrections. V P -.2 1.5 2.0 3.0 4.0 -10.39 - 6.49 - 3.76 - 2.71 -9.48 -5.98 -3.54 -2.56 -6.66 -5.00 -3.33 -2.50 0.0 1.5 2.0 3.0 4.0 2.35 .79 .17 .05 • 2.06 • .70 • .15 - .05 o . o . o . o . + .2 1.5 2.0 3.0 4.0 5.^5 4.77 3.41 2.60 5.25 4.55 3.24 2.47 6.66 5.00 3.33 2.50 144 The m a g n e t i c f i e l d v a l u e a t t h e peak i s c a l c u l a t e d t o be i s good f o r t h e range o f e l e c t r o n mass v a l u e s i n t h e expe r i m e n t b u t p o o r e r when a p p l i e d t o c a l c u l a t i n g t h e sum o f t h e e l e c t r o n p l u s l i g h t h o l e c u r v e s . I n t h i s case l e s s a c c u r a c y i s r e q u i r e d i n t h e answer. The above f o r m u l a can be g e n e r a l i z e d t o t h e sum o f many c u r v e s . U s i n g t h e r e l a t i o n s h i p f o r t h e a r e a under u s i n g a l s o t h e ass u m p t i o n , B 7 « 1. T h i s a s s u m p t i o n t h e c u r v e ( A p p e n d i x 3.2) N-, a 1 o r Hn = N-, t h e g e n e r a l i z e d answer i s B o r e q u i v a l e n t l y 145 Appendix 3.2 C y c l o t r o n A b s o r p t i o n C a l c u l a t i o n s ( i ) At low power, u s i n g t i m e dependent p e r t u r b a t i o n t h e o r y , t h e n e t a b s o r p t i o n r a t e from t h e n^*1 Landau l e v e l can be c a l c u l a t e d from t h e p u b l i s h e d e l e c t r i c d i p o l e m a t r i x element. The ( i n d u c e d ) t r a n s i t i o n r a t e between l e v e l s n and n' i s p r o p o r t i o n a l t o ( D i n g l e 1952b) VL „, = CE 2D 2 , where C i s a c o n s t a n t , E t h e e l e c t r i c f i e l d and D , i s t h e e l e c t r i c d i i D o l e n,n m a t r i x element. S i n c e t h e Landau l e v e l s a re e s s e n t i a l l y one-d i m e n s i o n a l s i m p l e harmonic o s c i l l a t o r s t a t e s , Dn,n« * \/n + 1 S n , ( I 1 + 1 , g i v i n g W n > n, a E 2 ( n + 1) S n , f n + 1 • t h The n e t a b s o r p t i o n from t h e n l e v e l i s p r o p o r t i o n a l t o [ wn.n+l " W n - l , n ] a E * + D = E * T h e r e f o r e t h e continuum energy d i s t r i b u t i o n can be d e s c r i b e d ^ I f a ( i i ) The a r e a under t h e a b s o r p t i o n curve w i l l now be c a l c u l a t e d c l a s s i c a l l y . -°° -0 B) dB F i r s t A' = f°a(i) c ) d V c i s c a l c u l a t e d u s i n g c R ( L p ) = ^ a R ( + ) ' f c r R ( _ ) ) 0 t o s i m p l i f y t h e i n t e g r a l , -OO I 0 1 i+( » - v c ) 2 i+cy+ ^ c ) 2 d ^ c i s t r a n s f o r m e d u s i n g \J ' = - p i n t h e second term and t h e n x = p - Vn t o oo g i v e d- x g = IT ( a l l i n t e g r a l s come from S p i e g e l 1968) . - 0 0 1 + x Now A i s calculated from A' using V* = US Xi = ^~ T~ . c c m*c 2 Using the d.c. conductivity a = L , the r e s u l t i s , o m* i n Gaussian units A = Neerr/2, ( i i i ) The same r e s u l t i s obtained quantum mechanically by Dingle (1952b) using the Fermi-Dirac d i s t r i b u t i o n function. I t i s su r p r i s i n g that the r e s u l t does not depend on temperature, and therefore the r e l a t i v e populations of states. The reason i t does not has to do with the s p e c i a l nature of D„ „, i n t h i s ^ n, n case. As shown i n section ( i ) , Wn n , a (n + 1) Sn, n + 1 Take the population of the n Landau l e v e l to be N , the t r a n s i t i o n rate between the states n, n + 1 to be P . Then the t o t a l absorption i s the sum of the net absorptions from each l e v e l , i . e . Absorption a N QP Q + N 1(P 1-P 0) + N 2(P 2-P 1) + ... = N o P o + N l P o + N 2 P o + ' •' = NPQ where N = Y~ N. and P .,-P_ = P„ have been used. The r—rr l n+1 n o 1=1 assumption of no upper bound on the le v e l s i s also required, or equivalently that N n —> 0 for large n. 147 Appendix 3.3 The Background A b s o r p t i o n S i n c e e t c h i n g removes most o f t h e background a b s o r p t i o n , t h e o r i g i n a l a b s o r p t i o n i s m o s t l y a s u r f a c e e f f e c t . The r e m a i n i n g a b s o r p t i o n i s p a r t l y a s u r f a c e e f f e c t and p a r t l y b u l k , s i n c e f i g u r e 3-6 (a) shows i t i s s l i g h t l y l a r g e r when b u l k c a r r i e r s a r e c r e a t e d . A n o t h e r p r o o f t h a t i t i s l a r g e l y a s u r f a c e e f f e c t i s t h a t i t i s a l m o s t as s t r o n g f o r X > 1.57M ( i ) It i s independent o f m a g n e t i c f i e l d s t r e n g t h and o r i e n t a t i o n , ( i i ) I t i s e q u a l l y s t r o n g f o r b o t h senses o f c i r c u l a r p o l a r i z a t i o n . ( i i i ) I t d i s a p p e a r s a l o n g w i t h t h e s i g n a l a t 27 K. ( i v ) I t i s i n d e p e n d e n t o f a c c e p t o r c o n c e n t r a t i o n . (v) The dependence on power and l i g h t i n t e n s i t y f o r e t c h e d - 25 7 samples i s Bkgd a P ' J I . The n e g a t i v e power dependence s u g g e s t s t h e p o s s i b i l i t y , w h i c h cannot be r u l e d o u t , t h a t c a r r i e r s a r e t r a n s f e r r e d from t h e background a b s o r p t i o n t o t h e r e s o n a n t a b s o r p t i o n as t h e power i n c r e a s e s , w h i l e t h e t o t a l number o f c a r r i e r s c o u l d r e m a i n c o n s t a n t . T h e r e f o r e , t h e c o r r e c t i n t e r p r e t a t i o n o f t h e measured t (P) i s t h e " e l e c t r o n l i f e t i m e i n t h e r e s o n a n t system". The s o u r c e o f t h e background a b s o r p t i o n i s unknown. The p o s s i b i l i t y o f a n o n - r e s o n a n t heavy h o l e background i s r u l e d out by a c a l c u l a t i o n . Assuming uST^ - .25 ( t h e t h e r m a l e q u i l i b r i u m v a l u e o f S t r a d l i n g (1966), c o r r e c t e d t o t h i s f r e q u e n c y ) t h e a b s o r p t i o n would be e x p e c t e d t o d e c r e a s e by a f a c t o r o f two from 3-5 t o 15 KG. The measured background a b s o r p t i o n was c o m p l e t e l y i n d e p e n d e n t o f f i e l d i n t h a t r e g i o n . (and up t o 162ju ), an energy range l e s s t h a n band-gap. The background a b s o r p t i o n has t h e f o l l o w i n g p r o p e r t i e s : 148 Appendix 3-4 Sample Data T a b l e s The c y c l o t r o n resonance d a t a i s p r e s e n t e d i n t a b l e A3.4-1. I n t h a t t a b l e , t h e s i g n a l h e i g h t a c c u r a c y i s w i t h i n a f a c t o r o f two. The l a r g e e r r o r i n t h e low power LUX v a l u e s U/T^, f o r group B samples d e r i v e s from th e poor agreement between t h e t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s . A power l e v e l o f - 6 dB, a l o n g w i t h f u l l l i g h t i n t e n s i t y , was used t o de t e r m i n e t h e r a t i o s o f t h e b u l k t o s u r f a c e LUX and the s i g n a l t o back-ground h e i g h t . The v a l u e s o f N A come from t h e H a l l a n a l y s i s w hich i s p r e s e n t e d i n t a b l e A3.4-2. The H a l l and r e s i s t i v i t y measurements have been d e s c r i b e d i n s e c t i o n 4,2. The a b s o l u t e e r r o r i n eRj^QO - ^ w h i l e t h e r e l a t i v e e r r o r i n t h e d i f f e r e n t N A v a l u e s deduced i s s l i g h t l y l e s s . The H a l l measurements f o r t h e B a t t e l l e sample were made by Dr. A., C. Beer, The v a l u e o f ©R^ -p-p i s c o r r e c t e d t o 15 KG from t h e v a l u e measured a t 8 KG. The d a t a N/ and a r e c a l c u l a t e d assuming b o t h r y y = x a n d r^=1.4 f o r comparison. The r e s u l t s o f t h e mass s p e c t r o g r a p h i c a n a l y s i s o f f o u r samples a r e p r e s e n t e d i n t a b l e A3.4-3. The a n a l y s i s was done by Dr. D. S. R u s s e l l o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada, A n a l y t i c a l C h e m i s t r y s e c t i o n , Ottawa.' The sp a r k s o u r c e mass s p e c t r o g r a p h i c t e c h n i q u e o n l y measures a b s o l u t e i m p u r i t y c o n t e n t w i t h i n a f a c t o r o f two a t th e l i m i t s o f d e t e c t a b i l i t y . Some o f t h e samples showed i n h o m o g e n e i t y beyond a f a c t o r o f two. A c o n c e n t r a t i o n o f 1 ppm at o m i c i s 4 x 10 cm" . F o r c o m p a r i s o n w i t h t h e model t h a t t h e a c c e p t o r s are d e f e c t s , a l l i m p u r i t i e s found t o be c o n s i s t e n t l y p r e s e n t a t a l e v e l -> 0 . 8 ppm are marked w i t h a d o u b l e arrow «—•. A l l o t h e r i m p u r i t i e s p r e s e n t i n t h e two group A samples i n c o n c e n t r a t i o n ->.2ppm are marked T a b l e A 3.k - 1 The sample c y c l o t r o n r e s o n a n c e d a t a . Sample # Name S i g n a l " ^ L " rBulk , , S i g n a l Ht, H e i g h t +.15 ^ f S u r f a c e " Bkgd Ht. " X N A ( l O l 6 c m " 3 ) ft B a t t e l l e B e l l and H o w e l l 21 M o n t p e l l i e r O r d i n a r y Monsanto C2 -345 l6oo 3300 1200 1500 3.04 2.47 2.12 1.89 1.3 1.3 1.2 1.3 25 16 5 7 .329+.016 .405+.023 6.4o 9.50 .472+.031 17.6 .529+.039 19.9 PQ ft O u o rS V e n t r o n A1437 6 M o n t p e l l i e r Long 7 B e l l and H o w e l l 17 20 20 20 .65+.2 .65+.2 1.0 .7 .08+.01 1.2 .6 ,06 1.55+.6 1.55+.6 13 + 2 10.9 23.7 9 .4 8 V e n t r o n A1433 n o t d e t e c t e d 11.5 T a b l e A 3 . 4 - 2 The sample H a l l and r e s i s t i v i t y d a t a . Sample # Name I m p u r i t y A n a l y s i s Done? H a l l e f f e c t ^ W 1 ) i o 1 6 ( e R H 7 7 ) _ 1 j cm 3 ?30o[ 0 p n A300) .cm2 i o 1 6 NA - 3 cm v E A meV A? / V-S r ? ? = l r 7 7 = 1 . 4 r ? 7 = l r 7 7—1.4 1 B a t t e l l e yes 5 . 9 0.926 .12 .17 880 4400 6.50 6 .40 34.9 31.2 2 B e l l and H o w e l l 21 no 8.6 1.42 .094 .138 980 3500 9.71 9.50 31.9 28.2 3 M o n t p e l l i e r O r d i n a r y J 15.1 2.38 .063 .132 710 2000 18.2 17.6 29.3 24.2 4 Monsanto C2-345 no 17.0 2 .99 .060 .128 61? 1630 20.6 19.9 27.0 21.8 5 V e n t r o n A1437 no 9.91 1.94 .090 .141 700 2300 11.2 10.9 28.5 23.3 6 M o n t p e l l i e r Long y 19.8 3.52 .050 .122 625 1460 24.5 23.7 26.0 20.? 7 B e l l and H o w e l l 17 yes 8.63 2.06 .101 .110 725 2750 9.51 9.4 26.3 20.8 8 V e n t r o n A1433 no 10.7 2.90 .081 .097 715 2220 11.9 11.5 23.O 1? 151 T a b l e A 3 A - 3 The mass s p e c t r o g r a p h i c a n a l y s i s o f f o u r samples. I m p u r i t y Content (p.p.m. a t o m i c ) J Sample #1 B a t t e l l e Sample #3 Montpe O r d i n a r y Sample #6 l l i e r Long Sample #7 B e l l and H o w e l l 17 Sample L i 0. 06 0. 01 0.005 ND (CLDQIi Be WD ( 0.01 ) N D ND ND B KD(0.02 ) iv!'; ND C 0. 5 0.1 l . 0.1 N 2 ND(O.Ol) ND ND ND 0 ? "1" » 50. 3 . 50. Hp . X -A. X A P 0.04 ND (0.02 ) ND ND Na I n t . I n t . i I n t . I n t . Mg 0. 006 NDi0.0051 ND ND A l 0.2 0. 3 0.05 0.05 *• S i tf. 3. 0.8 1. P 0.02 ND(0„02) KD ND S 0. OA. 0.1 KD 0.4 CI 0.07 0. 02 0. 01 ND (0.0.1 ) K 0. 5 0. 3 0.2 0.3 -* Ca 0.1 0.1 0.02 0. 04 Sc KD(0.01) KD ND ND T i rWO. 01 ) ND ND . N D V I;DCO. OI ) ND ND ND Cr 0. 0.1 ND (0.01 )• KD ND Mn ND(0. 01 ) KD KD ND Pe 0.1 0. 04 0. 006 0.0.1 Co ND(O.OI) ND KD KD . N i 0.03 ND(0.01 ) ND ND Cu 0.01 0.1 0.0*5 0. 03 • Zn KD ( 0 „ 0 2 ) ND i ND 0, OS Ga Ha i o r M a j o r ' '/.'.Major Ma j o r Ge KD(O>O2) ND 1 N D ND As i D ND(0.01 ) : ND ND Se ND( 0.005 ) ND ! ND ND Br KD(O. 02) ND ND KD Rb I n t . I n t . ND ( I n t . I n t . Sr K'D(0. 001 ) ND ND ND Y ND(0.01) FD KD ND Zr ND(0„02) ND ND KD Nb ND(O.Ol) ND ND KD Mo ND(0.03 ) ND ND KD Ru ND ( 0 „ 0 3 ) KD ND Nil-ND( j - Not d e t e c t e d and l e s s t h a n amount i n b r a c k e t s . X - Not l o o k e d f o r . M - Memory - c o n t a m i n a t i o n from i n s t r u m e n t . I n t . - L i n e i n t e r f e r e n c e . Table A 3.4 - 3 Continued 152 I m p u r i t y Content (p.p.m. atomic") Sample. #1 B a t t e l l e Sample #3 Montp Ordinary Sample #6 s l l i e r Long Sample #7 B e l l and Howell 1? Sample Rh ND(0.01 ) ND ND M D Pd N D ( 0 . 0 4 ) N D ND ND /, . 1 R 0.8 2. — 1—• — " Cd lvl)f o, 0 3 ) N I) ND ND I n I;]} (0.02) 50„ 1. ND Sn N D ( 0 „ 0 3 ) N D ND ND Sb Ma ,ior M ajor Major Ma j o r Te N D ( 0 , 0 3 ) N D ND N D I Ml; ( 0 . 0 1 ) r.n ND ND Cs N D ( 0 „ 0 0 1 ) NO ND ND Ba N i i f n . i ) ND ND La i ; : i(0.0l ) M n ND ND Ce N I K O . O I ) Mii ND ND Pr w i n ' 0 . 0 1 1 Ml) ND ND Nd Nn(o tm) KV, ND ND Sm Niif o„ m ) M n ND ND Eu N D ( 0 . 0 ? ) MD ND ND Gd N D ( 0 . 0 3 ) ND ND ND Tb w r i ( n n i \ Ml") N D Ni") Dy ND(0.03) N D ND ND Ho ND(0.01 ) ND ND N D Er ND(0.03) ND • ND ND Tm ND(O.Ol) N D ND ND Yb ND(0.03 ) N D ND ND Lu ND(0.01) N D ND ND Hf ND(0.03) ND Ni) ND Ta M M M M Re ND(0.02) ND ND ND Os ND(0.03) ND ND ND I r ND ( 0„02) N D ND ND Pt NJ) (&. 03 ) ND ND N JJ Au N D(0.1) ND ND ND Hg Kb(0.1) ND ND ND TI ND(0.01) ND ND ND Pb ND(O.Ol) ND ND ND B i ND(0. 01 ) N D ND ND Th ND(0„01) ND ND ND U ND(0.01) ND ND ND W ND(0.02) N D N D ND Not d e t e c t e d ' and l e s s than amount i n b r a c k e t s . Not l o o k e d f o r . Memory - c o n t a m i n a t i o n from i n s t r u m e n t . L i n e i n t e r f e r e n c e . Code: N D ( ) -X M I n t . Appendix 4.1 The GaSb Constants 153 The values for the GaSb constants used i n the thesis are presented i n table A4.1. In each case an attempt i s made to use the most r e l i a b l e data. The S z i g e t i e f f e c t i v e charge i s defined by S z i g e t i (1949 and 1950). The acoustic deformation p o t e n t i a l constant i s estimated by Rode (1970). The magnitude of the g-factor was measured by Hermann and Lampel (1971) and the sign i s taken as negative since t h i s i s well known. The two d i e l e c t r i c constants K and K have been determined by a v a r i e t y of authors (Oswald and Shade 1954, Picus et a l . 1959, Appel 1964, Hass 1967). The two values are u s u a l l y calculated from the s t a t i c constant and the r a t i o i n f r a - r e d measurements of the "reststrahlen" r e f l e c t i o n due to the o p t i c a l phonons (Picus et a l . 1959). Various authors use d i f f e r e n t values of the s t a t i c d i e l e c t r i c constant and quote d i f f e r e n t errors i n deductions from the data of Picus et a l . The values from the best analysis (Hass 1967) are quoted along with conservative error estimates i n - K~\ I t i s t h i s difference which determines the strength of the electro.n-optical-phonon interaction, a. o That r a t i o was determined from f a r 154 T a b l e A.4,1 The GaSb C o n s t a n t s C o n s t a n t Symbol V a l u e R e f e r e n c e e l a s t i c c o n s t a n t (T = 4.2K) p e i z o e l e c t r i c "e" c o n s t a n t S z i g e t i e f f e c t i v e charge k = 0 e n e r g y gap th e e n e r g y s e p a r a -t i o n t o t h e (111) minimum t h e a c o u s t i c d e f o r -m a t i o n p o t e n t i a l c o n s t a n t o f t h e c o n d u c t i o n band Lande g - f a c t o r d i e l e c t r i c c o n s t a n t s t a t i c «/ = 0 o p t i c a l UJ = oo >14 ( c l l ' c 1 2 ' c 4 4 ^ ~ L i n and Wong (9.092, 4.148, 4.440) (1972) x l O i : L d y n e / c m 2 s z E, E I n g K c K, oo 1 _ 1 KQQ K 0 . 1 2 6 C/m2 0 . 3 3 e 0 . 8 1 3 + . 0 0 1 eV E ( l l l ) - 84 meV E(000) ' 6 . 7 eV - 9 . 3 + . 3 15.69 14.44 A r l t and Q u a d f l i e g (1968) Hass and H e n v i s (1962) Z w e r d l i n g (1959) Wooley (1966) Rode (1970) Hermann and Lampel (1971) Hass (1967) ~ - £ - . 0 0 5 5 + . 0 0 0 5 e f f e c t i v e mass v a l u e s e l e c t r o n l i g h t h o l e heavy h o l e m? m i h m hh 3.74 x 1 0 " 2 9 g = .041 m Q . 0 4 5 m Q ~ ( . 3 + . l ) rn t h i s work (App. 6 i . l ) (App. 6.1) e l e c t r o n l i f e t i m e 7 x 1 0 " 9 s P a r s o n s ( 1 9 7 1 ) speed o f sound ( l o n g i t u d i n a l a c o u s t i c ) u 4 . 3 6 km/s Rode (1970) T a b l e A4.1 C o n t i n u e d 155 C o n s t a n t Symbol V a l u e R e f e r e n c e e l e c t r o n o p t i c a l phonon c o u p l i n g c o n s t a n t a 0.0249 + . 0 0 3 C a l c u l a t e d i n Appendix 6 . 3 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 bands maximum p o s s i b l e e l e c t r o n e n e r g y Debye t e m p e r a t u r e (a) from e l a s t i c c o n s t a n t s (b) from LO phonon energy d e n s i t y o = k o e D d e n s i t y microwave a n g u l a r f r e q u e n c y k = 0 phonon a n g u l a r f r e q u e n c y LO TO M 'D UJ MW UJ, . 7 4 9 + . 0 0 2 eV (T = 30 K) 31 + 3 meV Reine e t a l . ( 1 9 7 0 ) see s e c t i o n 5 . 1 270 + 3K (T=4.2K) L i n and Wong (1972) 346 + 5K 5 . 6 6 g.cm"^ P i c u s e t a l . ( 1 9 5 9 ) Rode ( 1 9 7 0 ) T 2 . 2 x 1 0 1 1 r a d s " 1 t h i s work ^ • ^ G ) x l 0 1 3 r a d s " 1 P i c u s e t a l . 4 . 2 5 J ( 1 9 5 9 ) Phonon E n e r g i e s a t k = 0 a/, T 240 2 3 1 29 28 3 -- 1 4 meV 156 Appendix 6.1 The Hole E f f e c t i v e Mass V a l u e s The h o l e e f f e c t i v e mass v a l u e s have been d e t e r m i n e d by a v a r i e t y o f a u t h o r s , and are t a b u l a t e d i n t a b l e A6.1. One p r o b l e m i n t h e m i l l i m e t e r c y c l o t r o n resonance d a t a has been d i s c u s s e d by Button(1970). He shows t h a t f o r m a t e r i a l s w i t h t h i s k i n d o f v a l e n c e band s t r u c t u r e t h e c y c l o t r o n r e s o n a n c e peak i s t h e envelope o f s e v e r a l t r a n s i t i o n s , and hence t h e peak o f t h e envelope does n ot y i e l d t h e h o l e mass. I t i s p o s s i b l e , t h e r e f o r e t h a t t h e l i g h t h o l e mass i s c l o s e r t o M J ^ / M 0 = 0.0^5. The m i l l i m e t e r c y c l o t r o n resonance v a l u e ^^t/mo ~ 0 ,°52 was used i n Appendix 6.2 t o s u c c e s s f u l l y p r e d i c t t h e e f f e c t o f l i g h t h o l e s on t h e c y c l o t r o n r esonance s i g n a l . Table A 6 . 1 The measured h o l e e f f e c t i v e mass v a l u e s , T echnique m * l h / m o <ioo> m * h h / m o <H1> <L10> R e f e r e n c e m i l l i m e t e r c y c l o t r o n r e sonance . 0 5 2 + . 0 0 4 .26+104 .36+.03 . 3 7 + . 0 4 S t r a d l i n g (1966) s u b m i l l i r a e t e r c y c l o t r o n resonance . 0 4 7 + . 0 0 5 .22+^ 02 .28+.0 3 Cronburg e t a l . (1970) s t r e s s - m o d u l a t e d .042+.002 i n t e r - b a n d and - 3 % . 2 9 + . 0 9 .40+.16 . 3 6 + . 1 3 R e i n e e t a l . (1972) m a g n e t o r e f l e c t i v i t y a n i s o t r o p y f r e e c a r r i e r r e f l e c t i v i t y and . 0 4 4 + .007 .33+.013 Walton and M i s h r a (1968) F a r a d a y r o t a t i o n h-1 158 Appendix 6.2 The E f f e c t o f L i g h t H o l e s on t h e Measured  Peak a t H i g h Power As d i s c u s s e d i n s e c t i o n 6.1 (b) t h e h i g h power a b s o r p t i o n c u r v e can be a p p r o x i m a t e l y f i t t e d by a d d i n g the c o n t r i b u t i o n s from b o t h e l e c t r o n s and l i g h t h o l e s . The d e t a i l s o f t h e pr o c e d u r e f o r t h e case o f t h e B a t t e l l e sample are now p r e s e n t e d . At a power o f 0 dB t h e parameters t a k e n a r e : N h/N Q = 0.8 (measured) m ^ ^ / m 0 ~ 0.052 ( f r o m Appendix 6.1) W = 3.3 (measured at low power) " ^ l h = 2 , 5 ( a d j u s t e d ) The t h e o r e t i c a l s i g n a l r e s u l t i n g from t h e sum o f t h e s e two c u r v e s i s c a l c u l a t e d as a computer sum. The shape d e r i v e d i s a b e t t e r f i t t o t h e e x p e r i m e n t a l c u r v e t h a n t h e t h e o r e t i c a l " e l e c t r o n s - o n l y c u r v e " o f f i g u r e 4.1 ( a ) . The r e s u l t i n g u/C i s measured from t h e computer sum t o be 2.16, and t h e r e s u l t i n g B Q i s c a l c u l a t e d from e q u a t i o n 5.14 "to be 560 G. To compare t h e s e v a l u e s w i t h t h e measured v a l u e s two c o r r e c t i o n s must be made. The u/T v a l u e must be c o r r e c t e d f o r t h e inhomogeneous b r o a d e n i n g due t o t h e ~ 5 % plasma s h i f t o b s e r v e d . T h i s g i v e s 6o"C 2.0, which i s t o be compared t o t h e measured v a l u e o f 1.8 +.1. The measured peak must be c o r r e c t e d b o t h f o r t h e plasma s h i f t and f o r t h e c h a n g i n g N^/N e r a t i o . From f i g u r e 3.10 we e s t i m a t e t h a t N u/N„ i s re d u c e d a f a c t o r o f 2 when I i s reduced h e a f a c t o r o f 10. T h i s means t h a t t h e plasma c o r r e c t e d peak measurement has N h/N g = 0.4. The c a l c u l a t e d v a l u e o f B Q would t h e n be 5^5 G. T h i s ag r e e s w i t h t h e measured v a l u e o f 537 + 8 G. The same p r o c e d u r e p r e d i c t s an W T o f 2.4 a t P = - 3 dB which i s c l o s e t o t h e measured v a l u e o f 2 . 2 5 + •!• Note t h a t we are assuming t h a t u/T i s a p p r o x i m a t e l y independent o f power. T h i s would be e x p e c t e d because t h e l i g h t h o l e s s h o u l d a l s o have a power-independent " h o t - h o l e " d i s t r i b u t i o n f u n c t i o n , f o r t h e same r e a s o n s as g i v e n i n s e c t i o n 5 . 2 . The h i g h power measured peak s h i f t i s 14 + 8 G from t h e low power measured v a l u e o f 523 G f o r t h i s sample. An e s t i m a t e o f t h e B Q c o r r e c t i o n f o r l o w e r powers has been made, t a k i n g i n t o a c c o u n t t h e d i f f e r e n t dependence o f t h e h o l e and plasma c o r r e c t i o n s on P and I . T h i s g i v e s a c o r r e c t i o n o f 0 + 4 G a t - 6 dB and 0 + 2 G a t - 12 dB. These c o r r e c t i o n s a re c o n s i s t e n t w i t h t h e o b s e r v e d absence o f any rneasureable peak s h i f t w i t h l i g h t i n t e n s i t y . i6o Appendix 6.3 The Phonon Corrections to the E f f e c t i v e Mass The corrections to the electron e f f e c t i v e mass w i l l now be calculated f o r each of the three electron-phonon i n t e r a c t i o n s . The c a l c u l a t i o n i s done by deriving the correction to the electron energy from t r e a t i n g the electron-phonon i n t e r a c t i o n as a perturbation. Then the cyclotron resonance It w i l l be 1 l \ f mass i s calculated using = —=-*• m-* -t d shown that only the o p t i c a l phonon i n t e r a c t i o n causes a s i g n i f i c a n t change i n the e f f e c t i v e mass. ( i ) The O p t i c a l Phonon Correction The dimensionless electron-optical-phonon coupling 2m* USQ_( 1 constant i s ( K i t t e l 1963) a = _. , which i s 0.0249 + .003 f o r GaSb, the main error coming from (j£zr - ir^ 1) • T n e energy perturbation i n t e g r a l i s shown ( K i t t e l 0 0 K, 1963) to be proportional to 1^, .00 r - 1 0 2kqy^ + q; where q p = if 2m*USQ ( T\ = 1) i s the electron wavevector f o r an electron of energy equal to the LO phonon energy. The i n t e g r a l i s transformed by x = and = — to H = V l 0 0 dx x + 1 - Zr)jx - 1 0 L From / dx one obtains, /-I 1 2 " f T X - 1 2 2 fA 2 + t a n " ! 161 The second term v a n i s h e s , b e i n g t h e i n t e g r a l o f an odd f u n c t i o n o v e r a symmetric i n t e r v a l . , From t h e f i r s t t e r m t h e r e s u l t is I = TT s i n ~ 1 ( , y l ) . The energy c o r r e c t i o n i s -ex "fturQ s i n * * 1 ^ ) . The e f f e c t i v e mass c o r r e c t i o n r e s u l t i n g from t h i s i s c a l c u l a t e d and d i s c u s s e d i n s e c t i o n 6 . 3 . F o r c o l d e l e c t r o n s ( « . 1) t h e r e s u l t r e d u c e s t o t h e more f a m i l i a r f o r m u l a , m* = m*( 1 + |) = m * ( 1 . 0 0 4 l ) . ( i i ) The C o r r e c t i o n from t h e A c o u s t i c P i e z o e l e c t r i c I n t e r a c t i o n T h i s c o r r e c t i o n has been c a l c u l a t e d s e m i c l a s s i c a l l y by Mahan and H o p f i e l d ( 1 9 6 4 ) . The e l e c t r o n e n e rgy i s g i v e n by , 2 , ^ 2 , 2 ^ (K f c) Q„ k T p (ir > - " k tr ' av. o t ( k ) - ^ i * " - i o a* k p where a* i s t h e e l e c t r o n e f f e c t i v e Bohr r a d i u s and (K ) i s av t h e average o f t h e square o f t h e e l e c t r o m e c h a n i c a l c o u p l i n g c o n s t a n t , g i v e n i n s e c t i o n 4 . 3 ( b ) . The r e s u l t i n g e f f e c t i v e 1 1 m o K a v k o T mass i s -sz = =-*( 1 + x) where x = ^ — — . m * m o 2 -h2 a* k 3 F o r CdS (Baer and D e x t e r 1964) x«= . 2 . I n t h i s case, however, 2 t h e s m a l l e r m* , s m a l l e r (K ) „ . and t h e much l a r g e r k f o r t h e h ot e l e c t r o n s g i v e s x ^  2 x 1 0 " ^ , w h i c h i s c o m p l e t e l y n e g l i g i b l e . ( i i i ) The C o r r e c t i o n from t h e A c o u s t i c D e f o r m a t i o n P o t e n t i a l  I n t e r a c t i o n The energy c o r r e c t i o n i s g i v e n by ( "K = 1) k 2 m*o G l ^ 0 J u where C-^  i s t h e a c o u s t i c d e f o r m a t i o n p o t e n t i a l c o n s t a n t , J t h e d e n s i t y , and u an average l o n g i t u d i n a l speed o f sound. [ 3 s i n e dq d8 d$ ( 2 T T ) 3 J JJ q + q q - 2kqcos6 where t h e i n t e g r a t i o n o v er q ex t e n d s t o q , the v a l u e o f q r e p r e s e n t i n g t h e edge o f t h e f i r s t B r i l l o u i n zone, q i s t h e magnitude o f t h e wav e v e c t o r f o r an e l e c t r o n w i t h speed u, l I 1 f** 3 -1 -g | d ( c o s e ) J _ai_d£_ (Z-rr)'' jS J q 2 + q q - 2kqcos9 c The i n t e g r a l i s made d i m e n s i o n l e s s by t h e t r a n s f o r m a t i o n q' = q / q c , k' = k / q c and = 1 . 1 n 2 T ' Then I = — — - 5 - q c x ( 2 T T ) £ r 1 where i ' = / <yt. j q ' 2 dq*  - 1 / 0 q' + ( 1 - 2ky± ) 1 1 = qm 2 " 2 < + l l 2 / q ; " ( 2 k > - 1 ) A I = / d / L ( 2 k « u - 1 ) l n ( — ) 1^ i s e v a l u a t e d as two i n t e g r a l s u s i n g ln(-g) = I n A — InB, t o g i v e I± = I 4 - ^ j^(2k- - l ) 3 |jLn(2k' - 1 ) - |j + (2k' + l ) 3 jjLn(2k' + 1 ) -f1 2 4 = J ( 2 k ^ - - l ) c l n ( a - 2 k ' ^ L ) w i t h I where a = q m + 1 » 2k'^-. I t i s e a s i e s t t o d i f f e r e n t i a t e w i t h r e s p e c t t o k under t h e 163 i n t e g r a l s i g n b e f o r e e v a l u a t i n g t h e i n t e g r a l . The r e s u l t , d e r i v e d u s i n g a s e r i e s e x p a n s i o n f o r I n £(a(l - 2 k ' ^ )j and f o r a ( l - QUL.), i s h- = ^ I n a + 0 ( k ' / a ) . a K ~h k' J l * h I t i s e a s i l y s een t h a t t h i s term dominates — ^ ,-• . U s i n g k' = k/q we have ^ ^ I - = ^ ? l n a . From t h e o r i g i n a l e q u a t i o n , w i t h "n r e s t o r e d , t h e r e s u l t i s 1 1 f , m S 2 c l 16 . ( x - ^ - 3 f - i n a U s i n g C, ^~ 5eV, q „ ^ 5 x 10 3 cm" 1, q m — 5 x 10 7 c m - 1 and a = q m/q r. ^ 10 , the c o r r e c t i o n i s c a l c u l a t e d t o be 5 x 10 , lm' c w h i c h i s n e g l i g i b l e . 

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