UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

High field current fluctuations in n-type germanium Hart, Laurence Gilbert 1966

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1967_A1 H37.pdf [ 4.37MB ]
Metadata
JSON: 831-1.0085358.json
JSON-LD: 831-1.0085358-ld.json
RDF/XML (Pretty): 831-1.0085358-rdf.xml
RDF/JSON: 831-1.0085358-rdf.json
Turtle: 831-1.0085358-turtle.txt
N-Triples: 831-1.0085358-rdf-ntriples.txt
Original Record: 831-1.0085358-source.json
Full Text
831-1.0085358-fulltext.txt
Citation
831-1.0085358.ris

Full Text

THE UNIVERSITY-.OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES. PROGEAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of LAURENCE GILBERT HART B.Sc, Uni v e r s i t y of Alberta M.Sc, Uni v e r s i t y of Alberta FRIDAY, MARCH 31, 1967 AT 2:30 P.M. IN ROOM 304, HENNINGS BUILDING Chairman: I. McT. Cowan R. Barrie J. W. Bichard E. V. Bohn R. E. Burges F. W. Dalby D. L. Livese External Examiner: M. Lax B e l l Telephone Laboratories Murray H i l l N.J., U.S.A. Research Supervisor: R. E. Burges HIGH FIELD CURRENT FLUCTUATIONS IN N-TYPE GERMANIUM • ABSTRACT The work reported here i s an experimental and t h e o r e t i c a l i n v e s t i g a t i o n of high-frequency e l e c t r i -c a l noise generated in e x t r i n s i c s i n g l e - c r y s t a l n-type germanium at high e l e c t r i c f i e l d s . The e l e c t r i c f i e l d was pulsed so that the l a t t i c e temperature remained near 77°K„ During the pulse, the electrons quickly reach a nonequilibrium steady-state due to t h e i r gaining energy from the e l e c t r i c f i e l d and brought to a steady-state by means of c o l l i s i o n s with the l a t t i c e v i b r a t i o n s . Previous work has been concerned with noise measurements made at r i g h t angles to the e l e c t r i c f i e l d d i r e c t i o n , where aniso-tropic behaviour was observed„ The present measure-ments, made i n the d i r e c t i o n of the e l e c t r i c f i e l d , also show a high degree of anisotropy„" T h e . e l e c t r i c a l noise generated i s described by the noise temperature, T n, obtained by adapting the Nyquist formula to the non-equilibrium case., Measure-ment of T n J performed at frequencies of 70 Mc/s and 30 Mc/s, indicated,a .uniform,^oise spectrum i n this frequency range for a l l the samples used. The aniso-tropy of T n suggested that T n was explainable on the basis of the many-valley model of the conduction band of germanium, established by previous experi-mental investigations of the h i g h - f i e l d mobility anisotropy. A feature of the many-valley model i s that electrons i n d i f f e r e n t v a l l e y s of the conduction band, w i l l In general", e x h i b i t " d i f f e r e n t transport behaviour and as a r e s u l t , t r a n s i t i o n s between these valleys w i l l r e s u l t i n a noise phenomenon described as " i n t e r v a l l e y noise". However, for measurements of'T^ in the <^ 10c£> d i r e c t i o n , the " i n t e r v a l l e y noise" will, vanish, allowing a d i r e c t measure of the electron "heating" due to the e l e c t r i c f i e l d , the "hot electron noise". In the <Cm)> and <^ 1.1CT> " d i r e c t i o n s , both i n t e r v a l l e y and hot electron noise are.expected„ Both contributions to T n are evaluated by means of Barrie's extension to the case of many-va l l e y germanium of Stratton's h i g h - f i e l d transport theory„ GRADUATE STUDIES Physics Nuclear Physics Electromagnetic Theory S o l i d State Physics Noise i n Physics Systems Theory, of Measurements J. B. Warren R. Barrie J. B. Gunn R. E. Burgess A. Crooker Related Studies Analogue Computers Communication Theory Network Theory _,E. V. Bohn A. D. Moore A. D. Moore H I G H F I E L D C U R R E N T F L U C T U A T I O N S I N n - T Y P E G E R M A N I U M , b y L A U R E N C E G I L B E R T H A R T B . S c , U N I V E R S I T Y O F A L B E R T A , 1 9 5 4 M . S c , U N I V E R S I T Y O F A L B E R T A , 1 9 5 5 . A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F . T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n t h e D e p a r t m e n t o f > P h y s i c s . W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d - s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A • A u g u s t 1 9 6 6 • In present ing th is thes i s in p a r t i a l f u l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree tiiat tlu - j L ib rary sha l l make i t f ree l y ava i l ab le for reference and s t u d y I f u r ther agree that permission for extensive copying of th i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i c a t i on of th i s thes i s for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of 9\JrfSf The Un iver s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date Apg-fL- (j> )etC'] i i A B S T R A C T T h e w o r k r e p o r t e d h e r e i s a n e x p e r i m e n t a l a n d t h e o r e t i c a l i n v e s t i g a t i o n o f h i g h - f r e q u e n c y e l e c t r i c a l n o i s e g e n e r a t e d i n e x t r i n s i c s i n g l e - c r y s t a l n - t y p e g e r m a n i u m a t h i g h e l e c t r i c f i e l d s . T h e e l e c t r i c f i e l d w a s p u l s e d s o t h a t t h e l a t t i c e t e m p e r a t u r e r e m a i n e d n e a r 7 7 ° K . D u r i n g t h e p u l s e , t h e e l e c t r o n s q u i c k l y r e a c h a n o n -e q u i l i b r i u m s t e a d y - s t a t e d u e t o t h e i r g a i n i n g e n e r g y f r o m t h e e l e c t r i c f i e l d a n d b r o u g h t t o a s t e a d y - s t a t e b y m e a n s o f c o l l i s i o n s w i t h t h e l a t t i c e v i b r a t i o n s . P r e v i o u s w o r k : h a s b e e n c o n c e r n e d w i t h n o i s e m e a s u r e m e n t s m a d e a t r i g h t a n g l e s t o t h e e l e c t r i c f i e l d d i r e c t i o n , w h e r e a n i s o t r o p i c b e h a v i o u r w a s o b s e r v e d . T h e p r e s e n t m e a s u r e m e n t s , m a d e i n t h e d i r e c t i o n o f t h e e l e c t r i c f i e l d , a l s o s h o w a h i g h d e g r e e o f * a n i s o t r o p y . T h e e l e c t r i c a l n o i s e g e n e r a t e d i s d e s c r i b e d b y t h e n o i s e t e m p e r a t u r e , T „ , o b t a i n e d b y a d a p t i n g t h e N y q u i s t f o r m u l a t o t h e n o n - e q u i l i b r i u m c a s e . M e a s u r e m e n t s o f Tn , p e r f o r m e d a ' t f r e q u e n c i e s o f 7 0 M c / s a n d 3 0 M c / s , i n d i c a t e d a u n i f o r m n o i s e s p e c t r u m i n t h i s f r e q u e n c y r a n g e f o r a l l T M w a s e x p l a i n a b l e o n t h e b a s i s o f t h e m a n y - v a l l e y m o d e l o f t h e c o n d u c t i o n b a n d o f g e r m a n i u m / e s t a b l i s h e d b y p r e v i o u s e x p e r i m e n t a l i n v e s t i g a t i o n s o f t h e h i g h - f i e l d m o b i l i t y a n i s o t r o p y . A f e a t u r e o f t h e m a n y - v a l l e y m o d e l i s t h a t e l e c t r o n s i n d i f f e r e n t v a l l e y s o f t h e c o n d u c t i o n b a n d , w i l l i n t h e s a m p l e s u s e d . T h e a n i s o t r o p y o f T n s u g g e s t e d t h a t i i i , g e n e r a l , e x h i b i t d i f f e r e n t t r a n s p o r t b e h a v i o u r a n d a s a r e s u l t , t r a n s i t i o n s b e t w e e n t h e s e v a l l e y s w i l l r e s u l t i n a n o i s e p h e n o m e n o n d e s c r i b e d a s " i n t e r v a l l e y n o i s e " . H o w e v e r , f o r m e a s u r e m e n t s o f T n i n t h e (^00^ d i r e c t i o n , t h e " i n t e r v a l l e y n o i s e " w i l l v a n i s h , a l l o w i n g a d i r e c t m e a s u r e o f t h e e l e c t r o n " h e a t i n g " d u e t o t h e e l e c t r i c f i e l d , b o t h i n t e r v a l l e y a n d h o t e l e c t r o n n o i s e a r e e x p e c t e d . B o t h c o n t r i b u t i o n s t o T n a r e e v a l u a t e d b y m e a n s o f B a r r i e ' s t h e " h o t e l e c t r o n n o i s e rt I n t h e e x t e n s i o n t o t h e c a s e o f m a n y - v a l l e y g e r m a n i u m o f S t r a t t o n ' s h i g h - f i e l d t r a n s p o r t t h e o r y . v \ \ i v T A B L E O F C O N T E N T S C h a p t e r P a g e 1 I N T R O D U C T I O N 1 1 . 1 R e v i e w o f P r e v i o u s W o r k 1 1 . 2 : T r a n s p o r t P r o p e r t i e s o f a M a n y - 7 V a l l e y S e m i c o n d u c t o r 1 . 3 H i g h F i e l d T r a n s p o r t T h e o r y ' 1 1 ' 1 . 4 B a r r i e ' s E x t e n s i o n o f S t r a t t o n ' s 1 4 H i g h F i e l d T r a n s p o r t T h e o r y 1 . 5 P r e v i o u s D i s c u s s i o n s o f H i g h F i e l d 1 6 N o i s e • . 2 •' T H E O R Y O F C U R R E N T F L U C T U A T I O N S I N A 1 8 - M A N Y V A L L E Y S E M I C O N D U C T O R 3 • E X P E R I M E N T A L A P P A R A T U S A N D T E C H N I Q U E 2 5 3 . 1 . A p p a r a t u s D e s i g n C o n s i d e r a t i o n s 2 6 3 . 2 . S a m p l e H o l d e r D e s i g n 2 8 3 . 3 M e a s u r e m e n t s o f F i l t e r C o n s t a n t s 3 0 3 . 4 N o i s e T e m p e r a t u r e M e a s u r e m e n t 3 2 . P r o c e d u r e \ 3 . 5 M e a s u r e m e n t o f P u l s e V o l t a g e s 3 4 3 . 6 D e t e r m i n a t i o n o f S a m p l e E l e c t r i c 3 5 '., F i e l d , 3 . 7 S a m p l e P r e p a r a t i o n ^ 3 6 3 . 8 S a m p l e T e m p e r a t u r e R i s e D u e t o 3 7 • J o u l e H e a t i n g i 3 . 9 I n f l u e n c e o f S k i n E f f e c t o n S a m p l e 3 9 D i f f e r e n t i a l C o n d u c t a n c e TABLE OF CONTENTS Page EXPERIMENTAL APPARATUS AND TECHNIQUE(cont 'd ) 3.10 I n v e s t i g a t i o n s of S p u r i o u s Sou r ce s 40 o f N o i s e 3.11 Sample N o i s e Due t o Ho le G e n e r a t i o n 41 a t the C u r r e n t E l e c t r o d e s EXPERIMENTAL RESULTS 44 COMPARISON OP EXPERIMENTAL RESULTS . 45 WITH THEORY 5.1 D e t e r m i n a t i o n o f Group D r i f t 47 V e l o c i t i e s f r om B a r r i e Theo r y 5 .2 C a l c u l a t i o n o f Sample D i f f e r e n t i a l 51 Conductance 5.3 D e t e r m i n a t i o n o f C a r r i e r D e n s i t y 51 and E l e c t r o n P o p u l a t i o n 5.4 The Hot E l e c t r o n C o n t r i b u t i o n t o the 52: N o i s e Tempera tu re CONCLUSIONS- AND- SUGGESTIONS FOR FUTURE 54 , WORK BIBLIOGRAPHY 57 • \ APPENDICES F l u c t u a t i o n s i n E l e c t r o n P o p u l a t i o n of 58 Sample C a l c u l a t i o n o f Group P o p u l a t i o n S p e c t r a l 65 D e n s i t i e s TABLE OF CONTENTS APPENDICES C a l c u l a t i o n o f V a l l e y P o p u l a t i o n s a n d t h e I n t e r v a l l e y T r a n s i t i o n P r o b a b i l i t i e s C a l c u l a t i o n o f V e l o c i t y V a r i a n c e i n A r b i t r a r y D i r e c t i o n o f D i s p l a c e d M a x w e l l i a n D i s t r i b u t i o n . ' I n t e r v a l l e y T r a n s i t i o n R a t e a n d t h e A c o u s t o e l e c t r i c E f f e c t J L I S T O F F I G U R E S F i g u r e T i t l e F o i l o w i n g P a g e 1 . 1 E l l i p s o i d a l C o n s t a n t E n e r g y S u r f a c e s 1 0 o f C o n d u c t i o n B a n d E l e c t r o n s i n T T - s p a c e 5 . 1 T y p i c a l F o u r - P r o b e S a m p l e 3 1 3 . 2 S a m p l e H o l d e r I n s i d e S a m p l e B a t h , 3 1 • 3 . 3 • A p p a r a t u s f o r D e t e r m i n a t i o n o f N o i s e 3 1 T e m p e r a t u r e a n d D i f f e r e n t i a l C o n d u c t a n c e 3 . 3 a A p p a r a t u s U n i t T i t l e s 3 1 4 . 1 , N o i s e T e m p e r a t u r e v s . F i e l d i n < ( l Q 0 ^ 4 5 4 . 2 N o i s e T e m p e r a t u r e v s . F i e l d i n < ( l l 0 , > 4 5 4 . 3 N o i s e T e m p e r a t u r e v s . F i e l d i n ( i l l ) 4 5 4 . 4 - . D i f f e r e n t i a l C o n d u c t a n c e v s . F i e l d i n ( l 0 0 ) 4 5 4 . 5 ' ' D i f f e r e n t i a l C o n d u c t a n c e v s . F i e l d i n ^ l l d ) . 4 5 4 . 6 D i f f e r e n t i a l C o n d u c t a n c e v s . F i e l d i n . 4 5 \ • ' Acknowledgments T h e a u t h o r w i s h e s t o t h a n k P r o f e s s o r J . B . G u n n f o r s u g g e s t i n g t h e o r i g i n a l p r o b l e m a n d P r o f e s s o r R . E . B u r g e s s f o r a s s i s t a n c e w i t h t h e t h e o r y . F i n a n c i a l a s s i s t a n c e f r o m t h e D e f e n c e R e s e a r c h B o a r d o f C a n a d a i s g r a t e f u l l y a c k n o w l e d g e d . C H A P T E R 1 - I N T R O D U C T I O N 1 . 1 R E V I E W O F P R E V I O U S W O R K U n t i l r e c e n t l y , i n v e s t i g a t i o n s o f n o i s e i n c u r r e n t -c a r r y i n g s i n g l e c r y s t a l s e m i c o n d u c t o r s h a v e b e e n c o n c e r n e d w i t h b e h a v i o u r a t r e l a t i v e l y l o w f r e q u e n c i e s a n d f i e l d s , w h e r e e f f e c t s d u e t o c a r r i e r p o p u l a t i o n f l u c t u a t i o n s a r e d o m i n a n t . R e c e n t e x p e r i m e n t s ( E r l b a c h a n d G u n n , 1 9 6 2 ) • h o w e v e r , h a v e r e v e a l e d a s o u r c e o f n o i s e i n n - t y p e g e r m a n i u m p r e s e n t w h e n t h e e l e c t r o n s a r e s u b j e c t e d t o h i g h e l e c t r i c f i e l d s . T h i s n o i s e h a s b e e n i n t e r p r e t e d a s a n e n h a n c e m e n t o f t h e t h e r m a l o r N y q u i s t n o i s e , d u e t o t h e e l e c t r o n s b e i n g " h e a t e d " b y t h e f i e l d , a c o n c e p t u s e d e a r l i e r t o e x p l a i n t h e f i e l d d e p e n d e n c e o f t h e e l e c t r o n m o b i l i t y . A n a d d i t i o n a l s o u r c e o f n o i s e , p r e d i c t e d t h e o r e t i c a l l y t o b e m e a s u r a b l e u n d e r t h e h i g h - c u r r e n t e x p e r i m e n t a l c o n d i t i o n s o f t h i s e x p e r i m e n t , i s t h e s o - c a l l e d " i n t e r v a l l e y n o i s e " , w h i c h i s d u e t o c o n d u c t i o n e l e c t r o n t r a n s i t i o n s b e t w e e n s t a t e s o f d i f f e r e n t e f f e c t i v e m a s s . We s h a l l f i r s t r e v i e w t h e n o i s e w o r k d o n e o n c u r r e n t -c a r r y i n g s e m i c o n d u c t o r r e s i s t o r s , w h e r e t h e m a i n c o n c e r n i s w i t h c a r r i e r p o p u l a t i o n f l u c t u a t i o n s . T h e n , t h e h i g h - f i e l d t r a n s p o r t b e h a v i o u r o f n - t y p e g e r m a n i u m w i l l b e d i s c u s s e d a n d a p p l i e d t o t h e c a l c u l a t i o n o f . b o t h t h e " h o t . e l e c t r o n " a n d t h e " i n t e r v a l l e y n o i s e ' * . E l e c t r i c a l n o i s e i n a n y t w o - t e r m i n a l n e t w o r k i s t h e a p p e a r a n c e a c r o s s t h e t e r m i n a l s o f a r a n d o m f l u c t u a t i n g v o l t a g e a n d i s d e s c r i b e d q u a n t i t a t i v e l y b y m e a n s o f t h e T h e v e n i n a n d N o r t o n e q u i v a l e n t c i r c u i t s . T h e T h e v e n i n d e s c r i p t i o n i s a s f o l l o w s : I n a b a n d w i d t h a t f r e q u e n c y T t h e e q u i v a l e n t c i r c u i t i s a n i m p e d a n c e 2 (^0 i n s e r i e s w i t h a v o l t a g e g e n e r a t o r e C t ) , w h e r e 2 ( 0 i s t h e c i r c u i t i m p e d a n c e a t f r e q u e n c y - f a n d t h e v o l t a g e g e n e r a t o r e t t ) i s , b y c o n v e n t i o n , r e p r e s e n t e d b y i t s r o o t - m e a n - s q u a r e v a l u e , , t h e b a r i n d i c a t i n g a n a v e r a g e o v e r a t i m e i n t e r v a l m u c h g r e a t e r t h a n ( A ? ) ' . T h e f l u c t u a t i n g v o l t a g e e t t ) c o r r e s p o n d s t o w h a t w o u l d b e o b s e r v e d o n a c a t h o d e r a y o s c i l l o s c o p e p l a c e d a c r o s s t h e t e r m i n a l s , i f t h e o s c i l l o s c o p e b a n d w i d t h w e r e A ? . T h e s p e c t r a l d e n s i t y o f e C t ) i s d e f i n e d a s T h e m o s t b a s i c t w o - t e r m i n a l n o i s e g e n e r a t i n g n e t w o r k t o c o n s i d e r i s t h a t o f a p u r e r e s i s t a n c e K° i n t h e r m a l e q u i l i b r i u m w i t h i t s s u r r o u n d i n g s a t a n a b s o l u t e t e m p e r a t u r e T o . T h i s i d e a l c a s e w a s f i r s t c o n s i d e r e d b y N y q u i s t f r o m a t h e o r e t i c a l p o i n t o f v i e w . H e s h o w e d t h a t , t h e m e a n - s q u a r e v a l u e o f t h e v o l t a g e g e n e r a t o r i n a f r e q u e n c y r a n g e A{ , i s r e l a t e d t o K o , a n d T b y N y q u i s t ' s t h e o r e m : c ? « 4 - K T . R . \ > C 0 A f w h e r e K i s B o l t z m a n n ' s c o n s t a n t , a n d J>(4) i s t h e P l a n c k f a c t o r h b e i n g P l a n c k ' s c o n s t a n t . W e n o r m a l l y c o n s i d e r t h e 3 m u c h l e s s t h a n u n i t y . T h e n N y q u i s t ' s t h e o r e m T h e p r o o f o f t h i s c l a s s i c a l l i m i t o f N y q u i s t ' s t h e o r e m i s a c c o m p l i s h e d b y p u r e l y t h e r m o d y n a m i c a r g u m e n t s , u s i n g . t h e s e c o n d l a w o f t h e r m o d y n a m i c s a n d t h e e q u i p a r t i t i o n l a w , w h i c h g i v e s i t i t s g e n e r a l v a l i d i t y . H o w e v e r , f o r t h e c a s e o f a c o n d u c t o r i n w h i c h t h e f r e e e l e c t r o n s m o t i o n a r e d e s c r i b e d b y a v e r y s i m p l e m o d e l , B a k k e r a n d H e l l e r ( 1 9 3 9 ) w e r e a b l e t o d e r i v e t h e N y q u i s t f o r m u l a f r o m a h e x a m i n a t i o n o f t h e m i c r o s c o p i c e l e c t r o n m o t i o n s . I n a d d i t i o n t o t h e t h e r m o d y n a m i c d e r i v a t i o n o f N y q u i s t a n d t h e m i c r o s c o p i c d e r i v a t i o n o f B a k k e r a n d H e l l e r , t h e N y q u i s t f o r m u l a h a s b e e n v e r i f i e d e x p e r i m e n t a l l y b y J o h n s o n ( 1 9 2 8 ) . A l s o , M o u l l i n d e t e r m i n e d B o l t z m a n n . ' s c o n s t a n t b y m e a n s o f t h e r m a l n o i s e m e a s u r e m e n t s a n d o b t a i n e d a g r e e m e n t t o w i t h i n o n e p e r c e n t o f v a l u e s d e t e r m i n e d b y o t h e r m e t h o d s . A n o t h e r v e r y b a s i c n o i s e s o u r c e i s t h e s h o t n o i s e i n t h e t e m p e r a t u r e - l i m i t e d v a c u u m d i o d e , w h i c h c o n s i s t s o f a h o t e l e c t r o n e m i t t i n g c a t h o d e s u r r o u n d e d b y a n a n o d e . T h e d e v i c e i s o p e r a t e d a t h i g h p o s i t i v e a n o d e v o l t a g e s i n t h e c u r r e n t - s a t u r a t e d r e g i o n o f i t s c u r r e n t - v o l t a g e c h a r a c t e r -i s t i c . I n t h i s c o n d i t i o n t h e c u r r e n t i s t e m p e r a t u r e -l i m i t e d , b e c a u s e a l l e l e c t r o n s e m i t t e d b y t h e c a t h o d e t r a v e l d i r e c t l y t o t h e a n o d e . U n d e r t h e s e c o n d i t i o n s , t h e d i o d e c u r r e n t f l u c t u a t i o n s a t f r e q u e n c i e s m u c h l e s s t h a n t h e r e c i p r o c a l o f t h e t r a n s i t - t i m e t a k e s o n a s i m p l e f o r m g i v e n b y t h e S c h o t t k y t h e o r e m , i f w e a s s u m e t h a t t h e e m i s s i o n p r o c e s s a t t h e c a t h o d e i s P o i s s o n . T h e S c h o t t k y t h e o r e m s t a t e s , w h e r e Sx i s t h e d i o d e f l u c t u a t i n g c u r r e n t s p e c t r u m a n d X i i s t h e m e a n d i o d e c u r r e n t . T h e e q u i v a l e n t c i r c u i t i s t h e n b e s t d e s c r i b e d a s a c u r r e n t g e n e r a t o r g i v e n b y t h e S c h o t t k y t h e o r e m i n p a r a l l e l w i t h a f r e q u e n c y -d e p e n d e n t a d m i t t a n c e w h i c h i s n o r m a l l y v e r y s m a l l i n t h e r a d i o f r e q u e n c y r a n g e . F o r p r a c t i c a l n o i s e d i o d e s , t h e S c h o t t k y t h e o r e m d o e s n o t e x p l a i n t h e o b s e r v e d n o i s e a t a l l f r e q u e n c i e s . A t t h e l o w e r a u d i o f r e q u e n c i e s , f l i c k e r e f f e c t e n h a n c e s t h e o b s e r v e d n o i s e a n d a t f r e q u e n c i e s i n t h e m i c r o w a v e r e g i o n t r a n s i t - t i m e e f f e c t l o w e r t h e n o i s e b e l o w t h a t g i v e n b y t h e S c h o t t k y t h e o r e m . A n o t h e r n o i s e s o u r c e , w h i c h i s t h e m a i n t o p i c o f t h i s t h e s i s , i s t h e n o n - e q u i l i b r i u m c a s e o f a c u r r e n t - c a r r y i n g s i n g l e c r y s t a l s e m i c o n d u c t o r r e s i s t a n c e . I t s f l u c t u a t i o n e m f . w h e n i n t h e r m a l e q u i l i b r i u m w i t h i t s e n v i r o n m e n t i s g i v e n b y t h e N y q u i s t t h e o r e m . H o w e v e r , t h e p a s s a g e o f d i r e c t c u r r e n t r e s u l t s i n a d d i t i o n a l n o i s e , w h i c h i n g e n e r a l i s a f u n c t i o n o f t h e b i a s i n g c u r r e n t , f r e q u e n c y , c r y s t a l o r i e n t a t i o n a n d t h e s u r f a c e e n v i r o n m e n t o f t h e r e s i s t o r . M o s t e x p e r i m e n t a l a n d . t h e o r e t i c a l w o r k h a s b e e n c o n c e r n e d w i t h t h e s t r o n g l y f r e q u e n c y d e p e n d e n t l o w f r e q u e n c y n o i s e , w h i c h m a y b e o r d e r s o f m a g n i t u d e g r e a t e r t h a n t h e t h e r m a l n o i s e e v e n f o r r e l a t i v e l y s m a l l v a l u e s o f c u r r e n t d e n s i t y . W e c h a r a c t e r i z e t h e n o n - e q u i l i b r i u m n o i s e o f t h e s e m i c o n d u c t o r b y a p a r a m e t e r , T n , t h e n o i s e t e m p e r a t u r e . , o b t a i n e d b y u s i n g N y q u i s t ' s f o r m u l a f o r t h e n o n - e q u i l i b r i u m 5 c a s e a n d l e t t i n g T n s u b s t i t u t e f o r t h e t h e r m o d y n a m i c t e m p e r a t u r e . B y d e f i n i t i o n o f T n , t h e m e a n - s q u a r e v o l t a g e g e n e r a t o r . i n t h e s e r i e s r e p r e s e n t a t i o n i s : w h e r e R' i s t h e d i f f e r e n t i a l r e s i s t a n c e ; s i m i l a r l y i n t h e c u r r e n t g e n e r a t o r r e p r e s e n t a t i o n : V- - 4 - K T n G ^ f w h e r e G i s t h e d i f f e r e n t i a l c o n d u c t a n c e . W h e n n o c u r r e n t f l o w s , t h e n o i s e t e m p e r a t u r e e q u a l s t h e t h e r m o d y n a m i c t e m p e r a t u r e , f r o m N y q u i s t ' s t h e o r e m . M e a s u r e m e n t s p e r f o r m e d o v e r t h e f r e q u e n c y r a n g e o f o n e K c / s t o s i x M c / s b y v a n < d e r Z i e l a n d a s s o c i a t e s ( v a n d e r Z i e l , 1 9 5 4 ) o n a c r y s t a l o f n e a r - i n t r i n s i c n - t y p e g e r m a n i u m r e v e a l e d t h e e x i s t e n c e o f t w o f r e q u e n c y - d e p e n d e n t c o m p o n e n t s i n t h e n o i s e t e m p e r a t u r e , s o t h a t H e r e Ai a n d a r e c o n s t a n t s f o r t h e s a m p l e u s e d , I i s t h e d i r e c t c u r r e n t a n d i s a c o n s t a n t i n d e p e n d e n t o f t h e d i r e c t c u r r e n t . T h e f i r s t t e r m o n t h e r i g h t - h a n d s i d e i s d u e t o t h e r m a l n o i s e . T h e l a s t t e r m , b e c a u s e o f i t s f r e q u e n c y d e p e n d e n c e , i s i n t e r p r e t e d a s " g e n e r a t i o n -r e c o m b i n a t i o n n o i s e 1 * . T h e r e m a i n i n g t e r m i s c a l l e d " e x c e s s n o i s e " . \ • T h e g e n e r a t i o n - r e c o m b i n a t i o n n o i s e i s d u e t o f l u c t u a t i o n s i n t h e e l e c t r o n a n d h o l e g e n e r a t i o n r a t e s . A s s u m i n g t h a t e l e c t r o n a n d h o l e p o p u l a t i o n f l u c t u a t i o n s o c c u r d u e t o h o l e -e l e c t r o n p a i r c r e a t i o n a n d h o l e - e l e c t r o n r e c o m b i n a t i o n , n e g l e c t i n g t r a p p i n g a n d t a k i n g t h e t r a n s i t - t i m e m u c h g r e a t e r t h a n t h e l i f e t i m e , the n o i s e t e m p e r a t u r e due t o . t h i s p r o c e s s i s g i v e n by T« = lF a , Qu«. -»>v jV , , ? c Here F i s t he e l e c t r i c f i e l d s t r e n g t h , M-*. iP-* "the e l e c t r o n and h o l e m o b i l i t y , r e s p e c t i v e l y and fc, i s t he e l e c t r o n o r h o l e l i f e t i m e . The c o n d u c t i v i t i e s 0«. and o- ,^ a r e g i v e n by C \ - n ^ . A r v 0"V = IP ^ M r -where |? and Y\ a r e t h e h o l e and e l e c t r o n d e n s i t i e s , r e s p e c t i v e l y . Ta lc ing T 6 f r om the measured v a l u e o f -f, , c o u l d — g-r t h e n be d e t e r m i n e d f r o m t h e e x p e r i m e n t a l v a l u e of IM . Ob From * • and t he measured v a l u e o f G~p H - 0 T V , t he r e s i s t i v i t y o f i n t r i n s i c germanium was c a l c u l a t e d and found t o be i n good agreement w i t h t h e a c c e p t e d v a l u e . The exce s s n o i s e has been ob se r ved t o have the f o l l o w i n g b e h a v i o u r : 1) The f r e q u e n c y dependence i s r o u g h l y -f o ve r a range f r om the o r d e r o f one Mc/s t o iO c y c l e s pe r s e c o n d . 2) Over a t e m p e r a t u r e range f r om 4°K t o 240°K, i t i s r e l a t i v e l y c o n s t a n t . 3) I t i s e x t r e m e l y s e n s i t i v e t o the s u r f a c e c o n d i t i o n and ambient e n v i r o n m e n t . 4) I t s dependence on t h e d i r e c t c u r r e n t i s q u a d r a t i c . The "j? s p e c t r um may be o b t a i n e d i n a p u r e l y m a t h e m a t i c a l way i f a d i s t r i b u t i o n 9C^c) o f l i f e t i m e s i s u sed such t h a t f )Ctc) i s p r o p o r t i o n a l t o t c ' . The e x i s t e n c e o f b o t h ."-'slow t r a p p i n g " ana " f a s t t r a p p i n g " 1 r e s u l t s i n a w ide range o f l i f e t i m e s n e c e s s a r y t o o b t a i n 7 t h e s p e c t r u m o v e r t h e w i d e b a n d o f f r e q u e n c i e s i n d i c a t e d " s l o w s u r f a c e s t a t e s " ' e x i s t i n g i n t h e o x i d e l a y e r o n t h e s e m i c o n d u c t o r s u r f a c e . a s s u m i n g a d i s t r i b u t i o n o f l i f e t i m e s i f w e a s s u m e t h e c o n d u c t i o n e l e c t r o n s r e a c h t h e s u r f a c e t r a p s b y t u n n e l l i n g t h r o u g h a p o t e n t i a l b a r r i e r o f v a r i a b l e w i d t h . T h i s a l s o r e s u l t s i n a w i d e t e m p e r a t u r e - i n d e p e n d e n t l i f e t i m e d i s t r i b u t i o n , a s r e q u i r e d b y e x p e r i m e n t . S o f a r , o n l y l o w f r e q u e n c y , l o w f i e l d n o i s e i n c u r r e n t -c a r r y i n g s e m i c o n d u c t o r s h a s b e e n d i s c u s s e d . T h e h i g h - f i e l d , h i g h f r e q u e n c y n o i s e w i l l b e d i s c u s s e d l a t e r , a f t e r t h e r a t h e r c o m p l e x t r a n s p o r t p r o p e r t i e s o f g e r m a n i u m h a v e b e e n e x a m i n e d . 1 .2: T R A N S P O R T P R O P E R T I E S O F A M A N Y - V A L L E Y S E M I C O N D U C T O R T r a n s p o r t b e h a v i o u r i n v o l v e s i n t e r a c t i o n s b e t w e e n t h e e l e c t r i c f i e l d , t h e e l e c t r o n s , a n d t h e l a t t i c e . T h e e l e c t r i c f i e l d a c t s o n t h e e l e c t r o n s t o i n c r e a s e t h e i r e n e r g y a n d m o m e n t u m a n d t h e e l e c t r o n s i n t e r a c t w i t h t h e l a t t i c e v i b r a t i o n s t o l o s e e n e r g y a n d m o m e n t u m . T h e t r a n s p o r t p r o p e r t i e s t h e n d e s c r i b e a s t e a d y - s t a t e i n w h i c h t h e e l e c t r o n s g a i n e n e r g y a n d m o m e n t u m f r o m t h e f i e l d a t t h e s a m e r a t e t h e v l o s e i t t o t - h e l a t t i c e v i b r a t i o n s . T h e i n t e r a c t i o n b e t w e e n t h e f i e l d a n d a s i n g l e e l e c t r o n d e p e n d s o n t h e e f f e c t i v e m a s s o f t h e e l e c t r o n f o r m o t i o n i n t h e f i e l d d i r e c t i o n . T h e e f f e c t i v e m a s s i s f o u n d f r o m t h e s o l u t i o n o f t h e ' S c h r o e d i n g e r e q u a t i o n f o r t h e e l e c t r o n a b o v e . T h e " s l o w t r a p p i n g " h a s b e e n a s s o c i a t e d w i t h t h e I t i s p o s s i b l e t o e x p l a i n t h e s p e c t r u m w i t h o u t i n the p e r i o d i c f i e l d of the l a t t i c e . The energy eigen-values EC^ O f°r "the conduction band s t a t e s are f u n c t i o n s of the wave v e c t o r -k which i n t u r n are r e s t r i c t e d to v e c t o r s having the values where jp, ? j * 3 a r e i n t e g e r s and k , , ^ , * ^ L 3 are the p r i m i t i v e t r a n s l a t i o n v e c t o r s of the r e c i p r o c a l l a t t i c e . The c r y s t a l i s assumed to be a r e c t a n g u l a r p a r a l l e l e p i p e d having edges S| a*, , S 2 a t ; S^o^ • Let one minimum of the conduction band occur at -8?„ which i s d i f f e r e n t from zer o . I f the ~x} y, 2 c o - o r d i n a t e system i s chosen so that the o f f - d i a g o n a l components of the e f f e c t i v e mass t e n s o r are zero, then the T a y l o r s e r i e s expansion of about the minimum energy s t a t e !?0 i s f o r s m a l l v a l u e s of ij£-JL) = ^ 4 X > hl^ , k i ^ ; where WU , the e f f e c t i v e e l e c t r o n mass f o r motion i n the i - d i r e c t i o n , [l-t^,^ ) i s gi v e n by For the lowest energy minimum of germanium, the valu e s of are obtained from c y c l o t r o n resonance data at l i q u i d helium temperatures. The most re c e n t data are those of L e v i n g e r and F r a n k l ( 1 9 6 1 ) , quoted as: ^ = w i t = (o.08ISZ * .oooog) rYlo W } s = (U588 * .oo«r) mo where YY)0 i s the f r e e e l e c t r o n mass. A c o m p l e t e p i c t u r e o f t h e b a n d s t r u c t u r e o f g e r m a n i u m h a s b e e n p u b l i s h e d b y H e r m a n ( 1 9 5 5 ) a s a r e s u l t o f h i s o w n t h e o r e t i c a l w o r k a n d d a t a o b t a i n e d f r o m c y c l o t r o n r e s o n a n c e m e a s u r e m e n t s . C o n s i d e r i n g o n l y t h e c o n d u c t i o n b a n d , i t i s f o u n d t h a t t h e r e a r e t w o c l a s s e s o f v a l l e y : 1 ) F o u r i n t h e ( i l l ) d i r e c t i o n o f X s p a c e a t t h e z o n e e d g e . T h e c o n s t a n t e n e r g y s u r f a c e s a r e e l l i p s o i d s o f r e v o l u t i o n g i v e n b y e q u a t i o n 1 . 2 . 1 a b o v e , w i t h , t h e z - d i r e c t i o n b e i n g i n t h e ( i l l ) . 2 ) O n e m i n i m u m a t k-Q a b o u t w h i c h t h e c o n s t a n t e n e r g y s u r f a c e s a r e s p h e r i c a l w i t h e f f e c t i v e m a s s • T h e e n e r g y o f t h i s m i n i m u m i s 0 . 8 ^ t e . v . a b o v e t h e v a l e n c e b a n d a t 4 ° K s o t h a t o n l y a t h i g h t e m p e r a t u r e s i s t h e r e a n y d e g r e e o f o c c u p a n c y o f t h i s m i n i m u m . T h e s c a t t e r i n g o f e l e c t r o n s b e t w e e n s t a t e s i n t h e s a m e v a l l e y ( i n t r a v a l l e y s c a t t e r i n g ) a n d b e t w e e n s t a t e s o f d i f f e r e n t v a l l e y s ( i n t e r v a l l e y s c a t t e r i n g ) o c c u r s i n a c c o r d a n c e w i t h t h e s e l e c t i o n r u l e s : S' = t±t w h e r e M i s t h e f i n a l a n d Jk t h e i n i t i a l e l e c t r o n w a v e v e c t o r a n d i s t h e w a v e v e c t o r o f t h e p h o n o n i n v o l v e d i n t h e t r a n s i t i o n ( + f o r a b s o r p t i o n a n d — f o r e m i s s i o n ) . C o n s e r v a t i o n o f e n e r g y o f t h e s y s t e m r e q u i r e s t h a t E ( i ' ) - E ( £ ) ± t . a l ^ w i t h t h e + . a n d — m e a n i n g t h e s a m e a s a b o v e a n d U . i s t h e v e l o c i t y o f s o u n d i n t h e c r y s t a l , w h i c h m a y b e a f u n c t i o n o f \ . T h e p r o b a b i l i t y P i j o f a t r a n s i t i o n i n u n i t t i m e f r o m a s t a t e i n v a l l e y i t o a s t a t e i n v a l l e y j i s ' R i - H i r K E ' ) w h e r e i s t h e d e n s i t y o f s t a t e s a t t h e s t a t e t o w h i c h t h e e l e c t r o n i s s c a t t e r e d . H e r e i t i s a s s u m e d t h a t v a r i e s l i t t l e a l o n g t h e E s u r f a c e . T h e m a t r i x e l e m e n t H t j = [ n ( ^ > j ] ' i ' D i j f o r a b s o r p t i o n f o r e m i s s i o n w h e r e i s t h e n u m b e r o f p h o n o n s p r e s e n t i n i t i a l l y i n t h e m o d e i n v o l v e d , a n d D L J i s i n d e p e n d e n t o f t h e o c c u p a t i o n o f t h e p h o n o n s t a t e s . F o r i n t e r v a l l e y s c a t t e r i n g Di} i s n e a r l y i n d e p e n d e n t o f t h e l o c a t i o n s o f t h e i n i t i a l a n d f i n a l s t a t e s i n t h e i r r e s p e c t i v e v a l l e y s . H e n c e , f o r a b s o r p t i o n , t h e t r a n s i t i o n p r o b a b i l i t y f o r a t r a n s i t i o n t o a s t a t e o f e n e r g y t*i i n v a l l e y j i s wo. w —^rn^ e W b e i n g c o n s t a n t * w , K T t -1 F o r e m i s s i o n , U ^ U U E - W ) f o r a E > U =*0 \ f o r AE^I^U) T h e d e n s i t y o f s t a t e s i n b o t h c a s e s i s p r o p o r t i o n a l t o t h e s q u a r e - r o o t o f t h e f i n a l s t a t e e n e r g y . F i g . 1 . 1 A ooi E l l i p s o i d a l c o n s t a n t - e n e r g y s u r f a c e s of c o n d u c t i o n band e l e c t r o n s of germanium i n K - s p a c e . Only two of t h e f o u r minima are shown. BC and AC r e p r e s e n t , r e s p e c t i v e l y , i n t r a v a l l e y and i n t e r v a l l e y t r a n s i t i o n s . 1 . 3 H I G H F I E L D T R A N S P O R T T H E O R Y M a n y p a p e r s h a v e b e e n p u b l i s h e d o n t h e t h e o r y o f h i g h f i e l d t r a n s p o r t i n n - t y p e g e r m a n i u m , t h e m o s t r e c e n t ' o f w h i c h h a v e d e a l t w i t h t h e a n i s o t r o p y o f t h e m o b i l i t y a n d t h e c h a n g e s i n v a l l e y p o p u l a t i o n w i t h f i e l d . O t h e r i m p o r t a n t c o n s i d e r a t i o n s i n c l u d e t h e f o r m o f t h e v e l o c i t y d i s t r i b u t i o n a t n o n - z e r o f i e l d s a n d t h e e f f e c t o f 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 o n t h e ' f o r m o f t h e d i s t r i b u t i o n . A n y t h e o r y u s e d t o e x p l a i n t h e o b s e r v e d c u r r e n t f l u c t u 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 m u s t d e t e r m i n e t h e v a r i a n c e o f t h e f i e l d c o m p o n e n t o f e l e c t r o n v e l o c i t y a n d t h e d r i f t v e l o c i t i e s o f t h e v a r i o u s v a l l e y s a s a f u n c t i o n o f t h e e l e c t r i c f i e l d . T h e i d e a l a p p r o a c h t o t h e p r o b l e m i s t h e d e t e r m i n a t i o n o f t h e v e l o c i t y d i s t r i b u t i o n f u n c t i o n b y s o l v i n g t h e B o l t z m a n n t r a n s p o r t e q u a t i o n . f o r h i g h f i e l d s , a s s u m 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 a n d i m p u r i t y , a c o u s t i c a l , o p t i c a l • a n d i n t e r v a l l e y s c a t t e r i n g . T h e n e a r e s t a p p r o a c h t o t h i s i d e a l i s t h e w o r k ; o f R e i k a n d R i s k e n ( 1 9 6 2 ) , w h o s o l v e d t h e B o l t z m a n n e q u a t i o n f o r a m a n y - v a l l e y s e m i -c o n d u c t o r u n d e r t h e r e s t r i c t i o n o f h i g h e l e c t r o n e n e r g y o n l y , w h i c h w o u l d a p p l y o n l y f o r v e r y h i g h e l e c t r i c f i e l d s . T h e y f o u n d t h e e n e r g y d i s t r i b u t i o n t o b e M a x w e l l i a n , e v e n i g n o r 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 . 3 t r a t t o n ( 1 9 5 7 ) h a s c a r r i e d o u t a n a n a l y s i s o f t h e r o l e o f 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 i n d e t e r m i n i n g t h e v e l o c i t y a n d e n e r g y d i s t r i b u t i o n s . H i s o r d e r o f m a g n i t u d e e s t i m a t e s o f t h e a v e r a g e e n e r g y e x c h a n g e d i n e l e c t r o n - e l e c t r o n a n d 12 a c o u s t i c a l mode s c a t t e r i n g l e a d him t o a s s e r t t h a t a c r i t i c a l e l e c t r o n d e n s i t y , Ho, , g i v e n by e x i s t s , where T u i s t he l a t t i c e t empe ra t u r e and E t he e l e c t r o n ene r gy . F o r d e n s i t i e s g r e a t e r t h a n YV , t he energy d i s t r i b u t i o n w i l l t e n d toward s a M a x w e l l i a n f o r e n e r g i e s n e i g h b o r i n g on E . I n o r d e r t h a t t he momentum d i s t r i b u t i o n be M a x w e l l i a n , d i s p l a c e d i n the ca se o f h i g h f i e l d s , t he v a l u e o f in must be g r e a t e r t h a n a t a l a t t i c e t e m p e r a t u r e o f 77°K. I n t he p r e s e n t e x p e r i m e n t , t h e c a r r i e r d e n s i t y was about oxio cw . A c c o r d i n g t o S t r a t t o n ' s a n a l y s i s t h e n , tne a s s u m p t i o n o f a d i s p l a c e d M a x w e l l i a n momentum d i s t r i b u t i o n f o r t h i s ca se i s no t v a l i d , f o r e l e c t r o n e n e r g i e s much g r e a t e r t h a n . What f o rm t h e d i s t r i b u t i o n t a k e s t h e n i s n o t known. However, t he r o l e o f 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 i n d e t e r m i n i n g t he d i s t r i b u t i o n may be f ound by compar ing t h e r e s u l t s o f S t r a t t o n ' s a n a l y s i s w i t h t h o s e o f Yamash i t a and Watanabe(1954) , who found the f o rm of the v e l o c i t y d i s t r i b u t i o n a t h i g h f i e l d s by i g n o r i n g t h e ' e f f e c t s of 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 . T h e i r s o l u t i o n , t he P i s a r e n k o d i s t r i b u t i o n , d i f f e r s f r om t h e M a x w e l l i a n except a t z e r o f i e l d . A l t h o u g h 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 may i n f l u e n c e t h e shape of t h e v e l o c i t y d i s t r i b u t i o n , as f a r as t he i n t e r p r e t a t i o n o f t he p r e s e n t e xpe r imen t i s c o n c e r n e d , i t i s n e c e s s a r y t o know o n l y t he e x t e n t t o wh ich such c o l l i s i o n s a f f e c t t he mean v a l u e and the mean-square v a l u e o f the v e l o c i t y d i s t r i b u t i o n . T h i s was done by c a l c u l a t i n g t he mean and mean-square v e l o c i t y a t an a r b i t r a r y f i e l d o f 87 v o l t s p e r cm. f o r b o t h the d i s p l a c e d M a x w e l l i a n and the P i s a r e n k o d i s t r i b u t i o n s . I n t h e s e c a l c u l a t i o n s , a c o u s t i c a l mode s c a t t e r i n g and a s c a l a r e f f e c t i v e mass were u s e d . The r e s u l t s of t he c a l c u l a t i o n s were t h a t t h e mean v a l u e s d i f f e r by 5% and the mean-square by 11$. Hence i t was c o n c l u d e d t h a t the d i s p l a c e d M a x w e l l -i a n d i s t r i b u t i o n p r o v i d e s a r e a s o n a b l e v a l u e o f t h e f i r s t and second, moments, even f o r the case of a t e n s o r e f f e c t i v e mass as i s u sed i n B a r r i e ' s e x t e n s i o n o f S t r a t t o n ' s t h e o r y . The t h e o r y of B a r r i e was used t o p r o v i d e a l l f i r s t . a n d . second moments o f the v a r i o u s v a l l e y v e l o c i t y d i s t r i b u t i o n s needed i n the t h e o r e t i c a l e x p r e s s i o n s f o r t he c u r r e n t -f l u c t u a t i o n s . As w e l l , i t p r o v i d e d t h e v a l l e y e l e c t r o n t e m p e r a t u r e s , wh i ch were used t o c a l c u l a t e the h i g h f i e l d v a l l e y p o p u l a t i o n s . The B a r r i e t h e o r y has been t e s t e d e x p e r -i m e n t a l l y as f a r as t he dependence o f d r i f t v e l o c i t y on f i e l d ( B a r r i e and 3 u r g e s s , 1 9 6 2 ) . The t h e o r y p r e d i c t s c o r r e c t l y t h e o c c u r r e n c e of m o b i l i t y a n i s o t r o p y a t 77*K, but t he a c t u a l va lues^ o f d r i f t v e l o c i t y a r e i n o n l y f a i r agreement w i t h t h e t h e o r y . A n o t h e r app roach i n i n t e r p r e t i n g h i g h f i e l d t r a n s p o r t b e h a v i o u r i s t h a t o f P a i g e ( 1 9 6 0 ) . He assumed a M a x w e l l i a n energy d i s t r i b u t i o n i n each v a l l e y and by emp loy i n g s c a l a r e f f e c t i v e mass t h e o r y i n h i s e x p r e s s i o n s f o r m o b i l i t y . was a b l e t o deduce t h e v a l l e y e l e c t r o n t e m p e r a t u r e s and 14 v a l l e y m o b i l i t i e s f o r t h e c u r r e n t d i r e c t i o n c l o s e t o the ^111^ d i r e c t i o n . P u b l i s h e d d a t a on the a n g l e between c u r r e n t and f i e l d , was used i n h i s a n a l y s i s . F o r the p r i n c i p a l d i r e c t i o n s t h i s method of a n a l y s i s i s of no v a l u e s i n c e the c u r r e n t and f i e l d d i r e c t i o n s ^ ' a r e the same. 1.4 BARBIE'S'EXTENSION OF STRATTQN'S HIGH FIELD THEORY I n t h i s t h e o r y t h e band scheme of germanium d i s c u s s e d i n S e c t i o n 1.2 i s assumed. A l s o , the mean e l e c t r o n energy i s assumed low so t h a t o n l y t h e f o u r ( i l l ) minima are o c c u p i e d and c o n t r i b u t e t o the c u r r e n t . Other assumptions and a p p r o x i m a t i o n s used a r e t h e f o l l o w i n g : 1) The d i s t r i b u t i o n . •fUO o f e l e c t r o n s i n one v a l l e y i n A-space i s M a x w e l l i a n at a l l f i e l d s and i s d i s p l a c e d about a f i e l d - d e p e n d e n t p a r t J^o . , t h a t i s ; _ E $ - £ O . ) / K T + U ) * e Here t h e o r i g i n i n -K-space i s t a k e n at the zone edge i n t h e ( i l l ) d i r e c t i o n . 2) The s t e a d y - s t a t e e q u a t i o n s from which the parameters JKo and T are f o u n d c o n t a i n t h e f a c t o r s \ <t L c and /BJF(JC)\ V "3t~"/o>» > " t i i e r a _ t e °f change of the d i s t r i b u t i o n f u n c t i o n due t o a c o u s t i c a l and o p t i c a l mode s c a t t e r i n g , r e s p e c t i v e l y . The above f a c t o r s may be e x p r e s s e d as sums i n v o l v i n g t h e t r a n s i t i o n p r o b a b i l i t i e s and ^ > w h i c h r e p r e s e n t r e s p e c t i v e l y , t r a n s i t i o n s by a b s o r p t i o n o f a phonon ^, t o a s t a t e and t r a n s i t i o n s by e m i s s i o n of a phonon ^. t< a s t a t e k O r d i n a r y p e r t u r b a t i o n t h e o r y i s used t o c a l c u l a t e (k_ (X , £ + and ^ £ (j£ 4-^ .-, JCj . T h i s i m p l i e s use of the Born a p p r o x i m a t i o n and the r e s u l t a n t e x p r e s s i o n f o r the t r a n s i t i o n p r o b a b i l i t y , where E ' and E are the f i n a l and i n i t i a l e n e r g i e s , r e s p e c t i v e l y . BG^) i s the square of the a b s o l u t e v a l u e of the m a t r i x component o f the p e r t u r b i n g p o t e n t i a l , V , c o n n e c t i n g t h e two s t a t e s , and i s d e f i n e d by 4>£. and b e i n g t h e wave f u n c t i o n s of the i n i t i a l and f i n a l s t a t e s . ft(^) i s t he average number o f phonons i n the <£-mode, and i s g i v e n by: i s assumed t o be i s o t r o p i c i n ^,-space. S i n c e i n t e r v a l l e y s c a t t e r i n g i s i g n o r e d i n the d e t e r -m i n a t i o n of the v a l l e y m o b i l i t i e s , we need c o n s i d e r o n l y t h e i n t r a v a l l e y o p t i c a l and a c o u s t i c a l s c a t t e r i n g . A c c o r d i n g t o . H e r r i n g ( 1 9 5 5 ) , i n the case o n l y o f i n t r a -v a l l e y a c o u s t i c a l ^ s c a t t e r i n g may B(^3 be a n i s o t r o p i c - d u e t o the p o s s i b i l i t y of shear s t r a i n s , p r o d u c i n g d e f o r m a t i o n p o t e n t i a l s . The B a r r i e e q u a t i o n s of S e c t i o n 5.1 reduce t o t h e o r i g i n a l e q u a t i o n s of S t r a t t o n ( 1 9 5 8 ) i f vtt-t and W\i a r e .equal. S i n c e t h e o r i g i n a l S t r a t t o n t h e o r y p r e d i c t s an i s o t r o p i c m o b i l i t y a t a l l f i e l d s , i t i s q u i t e i n a p p r o p r i a t e f o r the i n t e r p r e t a t i o n of t h e p r e s e n t e x p e r i m e n t . 1.5 PREVIOUS DISCUSSIONS OF HIGH FIELD NOISE P . J . P r i c e ( 1 9 5 9 ) f i r s t d i s c u s s e d the e x t e n s i o n of the N y q u i s t theorem t o the case of a c o n d u c t o r b i a s e d a t s t r o n g e l e c t r i c f i e l d s such t h a t the r e l a t i o n between t h e c u r r e n t d e n s i t y and e l e c t r i c f i e l d i s no l o n g e r l i n e a r . He c o n s i d e r e d the f l u c t u a t i o n of v e l o c i t y of a s i n g l e e l e c t r o n , n e g l e c t i n g any 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 and was a b l e t o r e l a t e the s p e c t r a l d e n s i t y of the f i e l d component of e l e c t r o n v e l o c i t y f l u c t u a t i o n t o a. component of the d i f f e r e n t i a l d i f f u s i o n d y a d i c . He a p p l i e s h i s r e s u l t t o two h i g h f i e l d c a s e s : one i n w h i c h the c o n s t a n t " i f * energy s u r f a c e s i n Jk-space are s p h e r i c a l and s c a t t e r i n g o n l y by a c o u s t i c a l mode i n t e r a c t i o n s ; the o t h e r , a p p l i c a b l e o n l y i n the case of v e r y h i g h f i e l d s , where o n l y s c a t t e r -i n g by e m i s s i o n of o p t i c a l phonons i s i m p o r t a n t . I n b o t h cases t h e n o i s e t e m p e r a t u r e i s of the same o r d e r as , where i s t h e mean e l e c t r o n energy. The f r e q u e n c y dependence o f T n i s u n i f o r m up to f r e q u e n c i e s the o r d e r o f the r e c i p r o c a l of the mean-free-time, c o r r e s p o n d i n g t o about 10 c y c l e s p e r second. Gurevich(196£} d i s c u s s e d t h e c u r r e n t f l u c t u a t i o n s i n the non.-equilip.rium. s t a t e of a sem i c o n d u c t o r a t h i g h f i e l d . He assumed s p h e r i c a l energy s u r f a c e s i n A -space, a c o u s t i c a l mode s c a t t e r i n g and no 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 . H i s e x p r e s s i o n f o r Si^) i n "the d i r e c t i o n of t h e a p p l i e d f i e l d , where S x(io) i s the s p e c t r a l d e n s i t y of c u r r e n t f l u c t u a t i o n s due t o the e l e c t r o n t h e r m a l m o t i o n s , c o n t a i n s a parameter "Ts , wh i c h r e s u l t s i n d i s p e r s i o n i n S x(w) f o r f r e q u e n c i e s o f the o r d e r tV* , where -to i s of t h e o r d e r [0 seconds. F o r f r e q u e n c i e s much l e s s t h a n ^ s ' , h i s t h e o r y p r e d i c t s the c u r r e n t spectrum t o be f r e q u e n c y independent and t o be p r o p o r t i o n a l t o the s q u a r e - r o o t of t h e a p p l i e d f i e l d . A n other s o u r c e of n o i s e a l s o p r e d i c t e d t o extend out t o h i g h f r e q u e n c i e s i s the i n t e r v a l l e y n o i s e , f i r s t d i s c u s s e d by P r i c e ( 1 9 6 0 ) . T h i s s o u r c e of c u r r e n t f l u c t -u a t i o n i s due t o t h e random r a t e of e l e c t r o n t r a n s i t i o n s between t h e v a r i o u s v a l l e y s of the c o n d u c t i o n band and i s p r e s e n t o n l y when the v a l l e y s a r e " q u a s i - i s o l a t e d " , i n the sense t h a t an e l e c t r o n l o c a l i z e d i n one v a l l e y tends t o execute many c o l l i s i o n s w i t h i n one v a l l e y b e f o r e making a t r a n s i t i o n t o another v a l l e y . A n e c e s s a r y c o n d i t i o n f o r f o r t he e x i s t e n c e of t h i s e f f e c t i s the v a r i a t i o n of e l e c t r o n m o b i l i t y between the v a l l e y s . - U s i n g t h e e x p e r i m e n t a l v a l u e s of t h e i n t e r v a l l e y t r a n s -i t i o n r a t e o b t a i n e d by W e i n r e i c h , Sanders, and White(1959) at 77°K, P r i c e u r e d i c t e d t h a t t h e n o i s e measured i n the tyooy d i r e c t i o n w i t h t h e d i r e c t c u r r e n t i n t h e ^ L i o ) d i r e c t -i o n s h o u l d be d e t e c t a b l e w i t h a j o u l e h e a t i n g due t o t h e a p p l i e d f i e l d of t h e o r d e r o f 100 w a t t s per c u b i c c e n t i m e t e r and t h a t the s p e c t r a l d e n s i t y s h o u l d drop o f f at microwave f r e q u e n c i e s , s i n c e the mean l i f e t i m e of an e l e c t r o n i n a v a l l e y i s of the o r d e r of 10 seconds. CHAPTER 2 - THEORY OF CURRENT FLUCTUATIONS I N A  MANY-VALLEY SEMICONDUCTOR L e t us suppose the se m i c o n d u c t o r sample t o be i n th e form o f a homogeneous s i n g l e - c r y s t a l b a r of l e n g t h L and of u n i f o r m c r o s s - s e c t i o n w i t h c u r r e n t l e a d s a t t a c h e d t o i t s two ends. E l e c t r o n s are. assumed t o be the m a j o r i t y c a r r i e r and the h o l e p o p u l a t i o n t a k e n t o be . n e g l i g i b l e . We assume a u n i f o r m e l e c t r i c f i e l d -F e x i s t s i n the sample r e s u l t i n g i n a s t a t i o n a r y f l u c t u a t i n g c u r r e n t , 1 ^ 0 , whose s p e c t r a l d e n s i t y , S^fa) 3 we w i s h t o c a l c u l a t e . The c u r r e n t I ( t ) i s due t o t h e motion of t h e f r e e e l e c t r o n s of t h e sample and i s g i v e n by ' wnere N(t) i s the f r e e e l e c t r o n p o p u l a t i o n a t time t -and Uj(t) i s t h e v e l o c i t y component of the j t h e l e c t r o n i n .the d i r e c t i o n of t h e l o n g i t u d i n a l sample a x i s . L e t us now assume t h a t the t o t a l p o p u l a t i o n N(t) i s composed of two s e m i - i s o l a t e d groups of p o p u l a t i o n N*(t) and N^Ct) w i t h each group h a v i n g d r i f t v e l o c i t i e s i n the f i e l d w h i c h are,, i n g e n e r a l , d i f f e r e n t - . By s e m i - i s o l a t e d we mean t h a t e l e c t r o n t r a n s i t i o n s . o c c u r between t h e groups but not at a r a t e s u f f i c i e n t t o a f f e c t t h e i n d i v i d u a l group energy or v e l o c i t y d i s t r i b u t i o n s . N a t ) and w i l l f l u c t u a t e due t o random t r a n s i t i o n s o c c u r r i n g between t h e groups. w i l l a l s o f l u c t u a t e , but i t s v a r i a n c e i s s u f f i c i e n t l y s m a l l (Appendix l j ' t h a t MCt) has been t a k e n c o n s t a n t , The c u r r e n t i n t h e sample at time t i s t h e n : rt.it) jwt) j 3 i A=t where vrjL-t) and w^tt) a r e , r e s p e c t i v e l y , the i n s t a n t -aneous a x i a l components of v e l o c i t y of t h e j t h e l e c t r o n i n group 1 and of the k t h e l e c t r o n i n group 2. The c u r r e n t may be ex p r e s s e d i n terms of t h e v a r i a b l e s Nl.lt), Wa.fcfc) , AU^.Ct) , and AU3^(t), the l a t t e r two b e i n g t h e a r i t h m e t i c - m e a n v e l o c i t y f l u c t u a t i o n s , d e f i n e d by i-» Nz(.t)4_ where and A VTj U ) - U j I t ) - T r A i S ^ C t ) = u T ^ C t ) - U ? Then l ( t ) = .•^-[N , C t ) ( v r + A v t t t ) ) + N z ( t ) ( w + AW - *C - t | We assume t h a t c o v L ^ ^ ^ u r ^ C t + * ) ] e o u a l s z e r o . T h i s means t h a t t h e r e i s assumed no a p p r e c i a b l e v e l o c i t y " c a r r y -o v e r " from one v a l l e y t o a n o t h e r and seems r e a s o n a b l e s i n c e the m a t r i x element f o r i n t e r v a l l e y s c a t t e r i n g i s n e a r l y independent of the' l o c a t i o n s of t h e i n i t i a l and f i n a l states'* i n t h e i r r e s p e c t i v e v a l l e y s ( H e r r i n g , 1955 ). T h i s l e a d s t o the f o l l o w i n g e x p r e s s i o n : 4- i? ur |cov[N,ft) ) M x Ct+r)| + c o v [ M x ( - t ) , M , ( t t t j ]J + N i l t ) N i ( t + * ) c o M ^ t t ) , u ^ C t + t ) ] * Wa.Ct)N x(t+r) c o v [ o r a a ) , u r j t + r j The c u r r e n t s p e c t r a l d e n s i t y , $x(u5) , i s g i v e n by the W e i n e r - K h i n t c h i n e theorem, + 1 wo . 0 ) -v- 4 Ni(t)N»(t+t) COY [vr^oUt-rcjJ cos cordis + tixLt)$xlUr) cov[w&.Ci\uk(t+r)jcosior The group p o p u l a t i o n s p e c t r a are found i n Appendix 2. The f i e l d and c r y s t a l - o r i e n t a t i o n dependent d r i f t v e l o c i t i e s , vr and M~ T are d e t e r m i n e d from some a p p r o p r i a t e t r a n s p o r t t h e o r y . There remains o n l y the c a l c u l a t i o n of cov(ykCt) k U"o. C-t ^ and covjwktt), Wa.(-t-t^)j . These may be o b t a i n e d d i r e c t l y from t h e i r d e f i n i t i o n s , -as f o l l o w s , f o r group 1: M u l t i p l y i n g t h e s e ' two ^equations t o g e t h e r and a v e r a g i n g ; Nj(t) Nil (*+'*) C o V where i s the f r a c t i o n of e l e c t r o n s p r e s e n t i n groun 1 a t time t t h a t r e m a i n i n group 1 u n t i l t ime "fc + t . \ r ( t ) i s t h e a x i a l v e l o c i t y component of any group 1 e l e c t r o n . , The d e p a r t u r e o f e l e c t r o n s from group 1 t o group 2. i s governed by the e q u a t i o n (Appendix 2),. where P12.N1 i s the p r o b a b i l i t y per u n i t time of a t r a n s i t i o n from group 1 to group 2. Hence has the form £(.t)= e",p,iY . cov^vrCt),vr(t may be found i f we make the assumption that i n t r a v a l l e y s c a t t e r i n g i n v o l v e s only e l a s t i c c o l l -i s i o n s . Then, i f *^ \, i s the angle of d e f l e c t i o n as a r e s u l t of a s i n g l e s c a t t e r i n g event, where the r e l a x a t i o n time f i i s giv e n by (Wannier): •Si i s the time between c o l l i s i o n s f o r e l e c t r o n s i n group 1. The average i n d i c a t e d by the angular b r a c k e t s i s performed over a l l p o s s i b l e i n t r a v a l l e y t r a n s i t i o n s . S i m i l a r e x p r e s s i o n s may a l s o be w r i t t e n f o r group 2 e l e c t r o n s , i n which case ^a. and Sa, r e p l a c e t\ and S i r e s p e c t i v e l y . Assuming t h a t ^ i s much g r e a t e r than >^ia, , the time dependence of the r i g h t - h a n d - s i d e of equation 2.2 i s determined by c o v ^ t f ( t ) , v ( t +-c0] so that equation 2.2 takes the form S i m i l a r l y i f " r ^ " i s much g r e a t e r than Pa,i , N*tt) N»Ct -**) c o v ^ t ^ ^ c C t - K c ) ) = Na.vav.ure" The c u r r e n t s p e c t r a l d e n s i t y , as gi v e n by .equation 2.1, with the above expressions along w i t h the p o p u l a t i o n s p e c t r a t a k e n from Appendix 2, now t a k e s the f o r m : — r o C a r r y i n g oat t h e i n t e g r a t i o n s g i v e s . The d i f f e r e n t i a l , c o n d u c t a n c e , G i t a ) , due t o the e l e c t r o n s o f t h e i t h group i s , (Wannier) r \ _ Zi  where W \ i i s the e f f e c t i v e mass of the i t h e l e c t r o n i n the f i e l d d i r e c t i o n and Xl i s t h e i n t r a v a l l e y v e l o c i t y -r e l a x a t i o n t i m e . We may check t h e s e f o r m u l a s by c o n s i d e r i n g t h e e q u i l i b r i u m z e r o f i e l d c ase. Then V , and ur are z e r o and t h e i n t r a v a l l e y v e l o c i t y r e l a x a t i o n t i m e s a r e e q u a l t o ' t o , t h e same f o r a l l v a l l e y s . I f we assume the t o t a l conductance i s e q u a l t o the sum of t h e - s e p a r a t e group con d u c t a n c e s , i g n o r i n g t h e c o n t r i b u t i o n of i n t e r - g r o u p t r a n s i t i o n s t o Gt<*0 , the'n, i n e q u i l i b r i u m The c u r r e n t s p e c t r a l d e n s i t y i s , a t e q u i l i b r i u m , I n the above e x p r e s s i o n , a n& \'^ \ are t a k e n as z e r o , w h i c h i s c o n s i s t e n t w i t h the e a r l i e r a s sumption t h a t G°(w) = G ° ^ ) 4 G°xCw) I n t h e e q u i l i b r i u m c a s e , y&.r\r and yavur obey the. s i m p l e r e l a t i o n s g i v e n i n Appendix 3, Wi varvr = K T U w x v u v u j = . K T U The c u r r e n t s p e c t r a l d e n s i t y t h e n c o r r e c t l y assumes the form -given by the- N y q u i s t f o r m u l a , KTu = 4 GT(co) KTu R e t u r n i n g t o the n o n - e q u i l i b r i u m c a s e , we make the f o l l o w i n g i n e q u a l i t i e s i n accordance w i t h . t h e e x p e r i m e n t a l c o n d i t i o n s : The c u r r e n t spectrum t h e n assumes i t s l o w - f r e q u e n c y form, 1 L* (fc* + fcu? The n o i s e t e m p e r a t u r e , T n , of the sample under non-e q u i l i b r i u m c o n d i t i o n s i s a c o n v e n i e n t way of d e s c r i b i n g i t s b e h a v i o u r . I t i s found u s i n g the. N y q u i s t f o r m u l a S x = 4 K T n G where • G i s the sample d i f f e r e n t i a l conductance, g i v e n by t h e sum of the c o n t r i b u t i o n s from each group of the e l e c t r o n p o p u l a t i o n . where Xi. i s t h e mean c u r r e n t c a r r i e d by group i e l e c t r o n s and V i s the a p p l i e d v o l t a g e . From t h e f i e l d dependence of and "uj , Giy and G^-x. may be found. The t o t a l d i f f e r e n t i a l conductance i s t h e n and the n o i s e t e m p e r a t u r e k " l?G> (^+K,?K r KG, Thus we f i n d t h a t the n o i s e t e m p e r a t u r e c o n t a i n s a c o n t r i b u t i o n due t o t h e d i f f e r e n c e i n d r i f t v e l o c i t y of the d i f f e r e n t v a l l e y s . T h i s c o n t r i b u t i o n , termed i n t e r v a l l e y n o i s e by P r i c e (1965) i s , due t o t h e (vr—ur)* ' f a c t o r , s t r o n g l y dependent on the sample o r i e n t a t i o n . The second c o n t r i b u t i o n , t h e "hot e l e c t r o n " c o n t r i b u t -i o n , r e p r e s e n t s t h e i n c r e a s e i n sample n o i s e due t o t h e i n c r e a s e i n the v a r i a n c e of the v e l o c i t y d i s t r i b u t i o n i n each v a l l e y . CHAPTER 3 - EXPERIMENTAL APPARATUS AND TECHNIQUE A s e r i e s of determinations of noise temperature,T n > and d i f f e r e n t i a l conductance, G , were made as f u n c t -ions of a p p l i e d e l e c t r i c f i e l d on r e c t a n g u l a r , s i n g l e c r y s t a l n-type germanium samples whose le n g t h s were i n the ( i l l ) , ^lio) > a n ( i ^1Qo) c r y s t a l d i r e c t i o n s . The samples i n each case were cooled to l i q u i d n i t r o g e n temperatures and measurements were performed at f r e q u e n c i e s of 30 Mc/s and 70 Mc/s . The same measure-ment technique was; used f o r a l l samples. B a s i c a l l y , the apparatus c o n s i s t e d of: 1) A low output impedance v o l t a g e p u l s e generator f o r s u p p l y i n g l O ^ s e c . p u l s e s to samples under study. The s a m p l e . r e s i s t a n c e s were of the order of one hundred ohms. 2.) A v a r i a b l e frequency m u l t i v i b r a t o r f o r t r i g g e r i n g the v o l t a g e generator and the d e l a y m u l t i v i b r a t o r ( s e e 4 below). 3) A low-noise gated 30 Mc/s a m p l i f i e r , a l o n g with'30 Mc/s and 70 Mc/s p r e a m p l i f i e r s and output .meter f o r measur-i n g the sample nois e only d u r i n g a 5>isec i n t e r v a l centered on the sample v o l t a g e p u l s e . 4) One delay m u l t i v i b r a t o r and square-wave v o l t a g e gener-a t o r f o r s u p p l y i n g the g a t i n g p u l s e t o the a m p l i f i e r . 5) A t e m p e r a t u r e - l i m i t e d n o i s e diode which served as a n o i s e r e f e r e n c e source. 6) A set of d e p o s i t e d carbon r e s i s t o r s whose conductance at the two measuring f r e q u e n c i e s were known. 7) A sample mount c o n s i s t i n g of a p p r o p r i a t e f i l t e r s and designed f o r f o u r - t e r m i n a l sample n o i s e measure-ments. The mount had'to have p r o v i s i o n f o r h o l d i n g l i q u i d n i t r o g e n and, yet be compact so as to minimize the l e a d inductance between the sample and the input t e r m i n a l s of the p r e a m p l i f i e r . 3.1 APPARATUS DESIGN CONSIDERATIONS The apparatus served to determine the n o i s e equiv-a l e n t c i r c u i t of the sample under h i g h f i e l d c o n d i t i o n s but with minimal sample h e a t i n g . For the d e s i r e d i m p u r i t y is .3 range, of the order of 10 cm used i n the experiment, p r a c t i c a l sample s i z e s r e s u l t s i n r e s i s t a n c e s i n l i q u i d n i t r o g e n of the order of 100 ohms. Since e l e c t r i c f i e l d s of at l e a s t s e v e r a l hundred v o l t s per centimeter • are. r e q u i r e d to produce a measurable amount of n o i s e , j o u l e h e a t i n g i s extreme u n l e s s very short p u l s e s are used. Hence, the average power d i s s i p a t e d by the sample must be very s m a l l . With these requirements i n mind a p u l s e d u r a t i o n of l O ^ s e c . was chosen, with a r e p e t i t i o n r a t e of from nine p u l s e s per second to 210 per second. The next c o n s i d e r a t i o n i s the measurement of the sample n o i s e . Since the n o i s e must be measured on l y under-p u l s e d f i e l d c o n d i t i o n s , the measurement frequency should be much g r e a t e r than the r e c i p r o c a l of the p u l s e d u r a t i o n so t h a t a number of c y c l e s of the n o i s e should be v i s i b l e . T h i s i n d i c a t e s a value of 10 Mc/s t o 100 Mc/s. A choice of frequency i n t h i s range i s a l s o d e s i r a b l e s i n c e the l / f and the excess n o i s e terms would be expected to be smal l , as d i s c u s s e d i n Chapter 1. As a check that the frequency-dependent noi s e sources are n e g l i g i b l e , measure-ment of n o i s e temperature at two widely d i f f e r e n t f r e q -uencies i n t h i s range are d e s i r a b l e . The lowest of the two f r e q u e n c i e s was chosen at 30 Mc/s, because of the a v a i l -a b i l i t y of commercial a m p l i f i e r s at t h i s frequency. A higher frequency of 70Mc/s was chosen a r b i t r a r i l y . The 30 Mc/s a m p l i f i e r c o n s i s t e d of a low n o i s e cascode p r e a m p l i f i e r f o l l o w e d by f i v e synchronously tuned 30 Mc/s pentode stages. Vacuum diode d e t e c t i o n was used. The d e t e c t -or was f o l l o w e d by a two stage v i d e o . a m p l i f i e r w i t h i t s l a s t stage a cathode f o l l o w e r . The 70 Mc/s a m p l i f e r c o n s i s t e d of the same apparatus as used at 30 Mc/s except the 50 Mc/s p r e a m p l i f i e r was r e p l a c e d by a 70 Mc/s cascode and a mixer stage. In order to achieve maximum s e n s i t i v i t y of the output meter to the pulse' sample n o i s e , the a m p l i f i e r gain was gated so t h a t o n l y d u r i n g the "pulse on" time was the ga i n non-zero. T h i s was achieved b y - r e p l a c i n g the l a s t . r a d i o -frequency stage of the a m p l i f i e r , normally a 6AK5 pentode, by a 6AS6, which i s s i m i l a r to the 6AK5 except that i t s suppressor g r i d i s net i n t e r n a l l y connected to the cathode. A g a t i n g p u l s e s u f f i c i e n t to "'cut-off'" the 6AS6 was then a p p l i e d to i t s suppressor g r i d . The d u r a t i o n of the g a t i n g p u l s e was set at about one-half that of the sample pulse so as to avoid any t r a n s i e n t v o l t a g e s from e n t e r i n g the a m p l i f i e r due to the switch-on and s w i t c h - o f f of the sample c u r r e n t . . 28 Such t r a n s i e n t s , i f p r e s e n t , would be v i s i b l e on the m o n i t o r i n g o s c i l l o s c o p e . T h e i r p o s s i b l e p r e s e n c e was a l s o checked by p l a c i n g a m e t a l r e s i s t o r a c r o s s t h e a m p l i f i e r i n p u t t e r m i n a l s i n p l a c e o f the germanium sample and p u l s i n g i t w i t h the same v o l t a g e and c u r r e n t magnitudes. S i n c e no i n c r e a s e i n the a m p l i f i e r output meter r e a d i n g o c c u r r e d , i t was conclu d e d t h a t the system was f r e e from t r a n s i e n t e f f e c t s . Due t o t h e low r e p e t i t i o n r a t e and t h e s h o r t d u r a t i o n of the n o i s e p u l s e from the a m p l i f i e r , the measurement o f th e noise.power of the p u l s e d sample was put of t h e q u e s t i o n . I n s t e a d , a " p e a k - r e a d i n g " v o l t m e t e r was used f o r the output meter. S i n c e the measuring proc e d u r e c a l l e d f o r a comparison of two n o i s e s o u r c e s , the p e a k - r e a d i n g v o l t m e t e r was used t o i n d i c a t e the e q u i v a l e n c e of two n o i s e s o u r c e s . 3. 2. SAMPLE HOLDER DESIGN The f i r s t e x p e r i m e n t a l sample h o l d e r t r i e d was a s t y r o f o a m b o t t l e w i t h s t a i n l e s s s t e e l c o n n e c t i o n s t o the two c u r r e n t l e a d s of the sample. The c u r r e n t l e a d s s e r v e d a l s o as the n o i s e s e n s i n g e l e c t r o d e s . The sample was cut i n the form of a r e c t a n g u l a r c r o s s - s e c t i o n bar w i t h g o l d w i r e l e a d s a l l o y e d t o the two ends, so t h a t t h e p u l s e c u r r e n t and the sample n o i s e were b o t h conducted by the same e l e c t r o d e s . T h i s arrangement ' i s p e r f e c t l y s a t i s f a c t o r y i f no a d d i t i o n a l n o i s e i s i n t r o d u c e d at the j u n c t i o n s between the s e m i c o n d u c t o r and the w i r e l e a d . However, s i n c e p r e -c a u t i o n s a g a i n s t j u n c t i o n n o i s e were t a k e n by van der Z i e l and a s s o c i a t e s , ( v a n der Z i e l , 1 9 5 4 } i n t h e i r low f r e q u e n c y measurements r e f e r r e d , t o i n Chapter 1, j u n c t i o n n o i s e may a l s o be p r e s e n t at the h i g h e r f r e q u e n c i e s as w e l l . 'Prelim-i n a r y t w o - t e r m i n a l n o i s e t e m p e r a t u r e measurements i n ( i l l ) samples showed c o n s i s t e n c y between samples, independ-ent a l s o of the d i r e c t i o n of p u l s e c u r r e n t . I n the (iOO) samples, however, n o i s e much i n excess of the ( i l l ) s a m p l e n o i s e and dependent on the d i r e c t i o n of c u r r e n t f l o w . th r o u g h the sample was observed. T h i s b e h a v i o u r was i n t e r -p r e t e d as e v i d e n c e of j u n c t i o n n o i s e and the t w o - t e r m i n a l measurements were d i s c o n t i n u e d i n f a v o u r of f o u r - t e r m i n a l d e t e r m i n a t i o n s . The f o u r - t e r m i n a l measurements were made on samples cut as shown i n F i g . 3 . 1 . The p u l s e c u r r e n t l e a d s a re a t t a c h e d t o the two ends and the n o i s e measurements made between the two s i d e c o n t a c t s . I n t h i s manner, any n o i s e g e n e r a t e d by the passage of c u r r e n t t h r o u g h a m e t a l - s e m i -c o n d u c t o r j u n c t i o n does not appear d i r e c t l y a c r o s s the' i n p u t t e r m i n a l s ' of the a m p l i f i e r . However, p a r t of the n o i s e g e n e r a t e d at t h e c u r r e n t e l e c t r o d e s may s t i l l e n t e r t h e a m p l i f i e r u n l e s s e f f o r t s a r e made t o i s o l a t e the c u r r e n t e l e c t r o d e s from ground p o t e n t i a l . T h i s was done by means of f i l t e r s , a s d e s c r i b e d i n S e c t i o n 3.3, tuned t o the c e n t e r of the a m p l i f i e r pass band. I t i s the n n e c e s s a r y t o check t h a t any j u n c t i o n n o i s e e n t e r i n g the a m p l i f i e r i s s m a l l compared t o the n o i s e a p p e a r i n g a c r o s s t h e s i d e c o n t a c t s of the sample. T h i s was done by u s i n g two v a l u e s of impedance f o r ' each f i l t e r . A l l f i l t e r s had band-widths much l e s s t h a n t h e a m p l i f i e r pass band and hence t h e a b s o l u t e v a l u e of impedance, \Z\, of each f i l t e r over the pass band was determined hy i t s c a p a c i t a n c e , C , and i n d u c t a n c e , L , v a l u e s . The r e l a t i o n f o r Z i s 1^ 1 « (si where _uL i s the f r e q u e n c y and uJo. i s the r e s -^ T T " air onant f r e q u e n c y of the f i l t e r . The f i l t e r c a p a c i t a n c e , C o ,is composed o f an e x t e r n a l v a r i a b l e p a r t and the d i s t -r i b u t e d c o i l c a p a c i t a n c e . F i l t e r s h a v i n g d i f f e r e n t v a l u e s of Co were used t o v a r y t h e amount of a t t e n u a t i o n of any j u n c t i o n n o i s e p r e s e n t at the c u r r e n t e l e c t r o d e s . S i n c e no dependence of n o i s e temperature on f i l t e r c a p a c i t -a nc e was found , i t was c o n c l u d e d t h a t a l l f i l t e r s were s u c c e s s f u l i n a t t e n u a t i n g the j u n c t i o n n o i s e . w e l l below t h e s i d e e l e c t r o d e n o i s e . 2.3 MEASUREMENT OF FILTER CONSTANTS I t was d e c i d e d t o mount t h e f i l t e r s a l o n g w i t h the sample i n the l i q u i d n i t r o g e n , s i n c e c o o l i n g the c o i l s i n c r e a s e d t h e v a l u e of the p a r a l l e l r e s i s t a n c e , R, , and hence d e c r e a s e d the f i l t e r bandwidth. We assume each f i l t e r t o c o n s i s t of a p a r a l l e l c o m b i n a t i o n of the i n d u c t -ance L 0 , d i s t r i b u t e d c a p a c i t a n c e C c , and e x t e r n a l c a p a c i t a n c e C\ , w i t h shunt r e s i s t a n c e R4 L-o and Cj f o r each f i l t e r i n l i q u i d n i t r o g e n were found by a d d i n g known v a l u e s of f i x e d mica condensers f o r Ci and d e t e r m i n i n g the r e s o n a n t f r e q u e n c y f o r each v a l u e of Ci . The re s o n a n t f r e q u e n c y was found u s i n g a g r i d - d i p meter whose f r e q u e n c y was compared a g a i n s t a Marconi Meter as s t a n d a r d . Lo was t h e n found; from the s l o p e of a p l o t of (S0 v s . Cx , and was o b t a i n e d hy s u b t r a c t i n g C\ from the t o t a l c a p a c i t a n c e « 0 L0 , f o r each v a l u e of C\ . The f i n a l v a l u e of C<j was found by a v e r a g i n g each of th e s e v a l u e s . The shunt r e s i s t a n c e , R, , was found i n the f o l l o w i n g way. The v a r i a b l e mica condensers were now p l a c e d i n p a r a l l e l w i t h each c o i l , t he c o m b i n a t i o n now c o n s t i t u t i n g the form of the f i l t e r s used i n the l a t e r n o i s e t e m p e r a t u r e measurements. B o t h c o i l l o s s e s and d i e l e c t r i c l o s s e s i n the v a r i a b l e condenser w i l l c o n t r i b u t e t o Rj The f i l t e r was mounted i n a s t y r o f o a m c o n t a i n e r h o l d -i n g l i q u i d n i t r o g e n . E l e c t r i c a l c o n n e c t i o n s c o n s i s t i n g of Vs i n c h by 3/s i n c h b r a s s s t r i p s were s e a l e d i n t o the b a t h bottom so t h a t t h e f i l t e r , c o o l e d t o l i q u i d n i t r o g e n t e m p e r a t u r e s .could be connected d i r e c t l y a c r o s s the Q-M e t e r - t e r m i n a l s . The c o n n e c t i o n s b e i n g , s h o r t and of low i n d u c t a n c e c o n t r i b u t e d l i t t l e t o the c o i l impedance. The measurement procedure c o n s i s t e d of r e s o n a t i n g the Q-Meter w i t h a h i g h Q c o i l at the f i l t e r r e s o n a n t f r e q u e n c y . The f i l t e r was t h e n connected a c r o s s the condenser t e r m i n a l s of the Q-Meter and the Q-Meter c a p a c i t o r changed s l i g h t l y , i f n e c e s s a r y , t o b r i n g i t i n t o r e s o n a n c e . Then the dec-re a s e of the Q-Meter r e a d i n g , due t o the shunt r e s i s t a n c e of t h e f i l t e r , a l l o w s the v a l u e of t h e shunt r e s i s t a n c e t o be f o u n d . F i g . 3 . 1 - T y p i c a l f o u r probe sample - S c a l e 17:1 t o a m p l i f i e r yrofoam c o n t a i n e r l u c i t e base l i q u i d n i t r o g e n v o l t a g e p u l s e i n f i l t e r c o i l F i g . 3 . 2 . Sample h o l d e r and n i t r o g e n b a t h - S c a l e 1.66 ® A p p a r a t u s used i n the d e t e r m i n a t i o n o f sample n o i s e temperature and d i f f e r e n t i a l conductance. Numbers and c o r r e s p o n d i n g u n i t t i t l e s f o l l o w i n F i g . 5 . 5 a . i • TP ~ <•/ m. o \ £ 1 g , <J . ^  d. Apparatus U n i t J I i t l e s ( r e f e r t o F i g . 3 . 3 ) Number D e s c r i p t i o n 1 V a r i a b l e f r e q u e n c y m u l t i -v i b r a t o r 2 l O j j L s e c . r e c t a n g u l a r p u l s e g e n e r a t o r 3 P u l s e a m p l i f i e r 4 F i x e d r e s i s t o r 5 P u l s e a t t e n u a t o r 6 O s c i l l o s c o p e 7 2c 8 F i l t e r s 9 Germanium sample 10 L i q u i d n i t r o g e n b a t h • 11 5722 N o i s e d i o d e 12 30 Mc/s or 70 Mc/s p r e -a m p l i f i e r 13 100 Mc/s o s c i l l a t o r 14 • Peak, r e a d i n g v o l t m e t e r . 15 30 Mc/s gated a m p l i f i e r 16 ^ 4 ^ 3 6 0 . r e c t a n g u l a r p u l s e g e n e r a t o r 17 , • 3^csec. d e l a y i n g m u l t i -v i b r a t o r 32 3.4 NOISE TEMPERATURE MEASUREMENT PROCEDURE The f i r s t s t e p i n the' d e t e r m i n a t i o n o f the sample n o i s e t e m p e r a t u r e c o n s i s t e d i n d e t e r m i n i n g t h e d i f f e r -e n t i a l sample conductance under th e p u l s e d e l e c t r i c f i e l d . The f o l l o w i n g procedure a c c o m p l i s h e d t h i s . .Set the n o i s e d i o d e c u r r e n t near i t s maximum r a t e d v a l u e and a c r o s s the i n p u t t e r m i n a l s of the a m p l i f i e r p l a c e a known conductance, G. A d j u s t t h e a m p l i f i e r g a i n t o g i v e a r e a d i n g X on the a m p l i f i e r output meter,' w h i c h we d e s i g n a t e hy OM. Then, r e p l a c e G by the sample, p u l s e d by the e l e c t r i c f i e l d . N o t i n g the r e a d i n g of OM, VJ© i n c r © 8 . s© "til© p u l s e d f i e l d a p p l i e d t o the sample u n t i l . OM reads X. I f t h e n o i s e diode c u r r e n t has been chosen h i g h enough t o "swamp o u t " a l l o t h e r n o i s e s o u r c e s i n the sample or the known conductanc G , then the sample conductance w i l l now e q u a l the known conductance. To check t h a t a l l o t h e r n o i s e s o u r c e s have been overwhelmed by t h e n o i s e d i o d e , i t i s o n l y n e c e s s a r y t o r e p e a t the above pr o c e d u r e f o r a s m a l l e r v a l u e of t h e n o i s e d i o d e c u r r e n t . The same f i e l d s h o u l d be o b t a i n e d i n b o t h c a s e s . Once the sample d i f f e r e n t i a l conductance has been foun d , t h e measurement of the sample n o i s e temperature p r oceeds i n e i t h e r of two ways, denoted by cases A and 3. Case A (Used f o r n o i s e t e m p e r a t u r e g r e a t e r t h a n .  room te m p e r a t u r e ) C o n s i d e r i n g the sample's N o r t o n n o i s e e q u i v a l e n t • i c i r c u i t , the mean-square n o i s e c u r r e n t g e n e r a t o r i n the f r e q u e n c y range i s i | " and i s g i v e n by I t i s i n p a r a l l e l w i t h t h e conductance G. With the sample a c r o s s t h e a m p l i f i e r i n p u t t h e . o u t p u t meter r e a d i n g i s some v a l u e , say X .The known conductance a l o n g w i t h i t s n o i s e c u r r e n t g e n e r a t o r , denoted by i£ , i s t h e n s u b s t i t u t e d f o r the sample and the n o i s e d i o d e c u r r e n t v a r i e d u n t i l OM reads X once a g a i n . Then, the two c u r r e n t g e n e r a t o r s a r e the same, s i n c e the measured n o i s e i s the same i n b o t h cases and bo t h g e n e r a t o r s a re i n p a r a l l e l w i t h the same v a l u e of conductance. S i n c e the n o i s e d i o d e c u r r e n t g e n e r a t o r and t h e sample n o i s e g e n e r a t o r are independent, we w r i t e I f = I F + I F where in i s the n o i s e diode< c u r r e n t g e n e r a t o r . S u b s t i t u t i n g t h e f o r m u l a s f o r t h e above c u r r e n t g e n e r a t o r s , l e a d s t o where i s the mean n o i s e d i o d e c u r r e n t . The n o i s e t e m p e r a t u r e i s t h e n Case B (Used f o r n o i s e t e m p e r a t u r e s l e s s t h a n room te m p e r a t u r e ) I n t h i s c a s e , t h e known conductance i s f i r s t connected a c r o s s t h e a m p l i f i e r i n p u t and' the g a i n a d j u s t e d so t h a t OM reads some c o n v e n i e n t v a l u e X . The sample under f i e l d , i s then connected to a m p l i f i e r input and the no i s e diode c u r r e n t ad/justed u n t i l OM reads X . Equating c u r r e n t g e n e r a t o r s . g i v e s , and the n o i s e temperature i s T - T - l i . U l o 2.KG T r i a l and e r r o r alone determine whether case A or 3 i s a p p r o p r i a t e . For example, i f Tn i s g r e a t e r than To and case B i s t r i e d , then OM w i l l read g r e a t e r than X when the sample i s a p p l i e d to a m p l i f i e r input and i t i s obvious that the noise temperature of the sample i s g r e a t e r than . room temperature, and case A procedure must be used to determine the noise temperature. 3.5 MEASUREMENT OF PULSE VOLTAGES For each de t e r m i n a t i o n of noise temperature, the sample c u r r e n t I and the v o l t a g e V a p p l i e d across i t s c u r r e n t t e r m i n a l s were a l s o found. I was found from the v o l t a g e drop across a f i x e d 118 ohm wire r e s i s t o r i n s e r i e s with the sample but at room temperature. A l l v o l t a g e measure-ments were obtained by a p p l y i n g the unknown v o l t a g e p u l s e s to a 6.15 to 1 pul s e a t t e n u a t o r to which a v a r i a b l e L.C. v o l t a g e , Vj>.c. , was a p p l i e d at an i n t e r n a l point> through a switch. U s i n g the o s c i l l o s c o p e as a d e t e c t o r , the switch' was opened and c l o s e d , while VB.0. was v a r i e d u n t i l i t e q u a l l e d the attenuated p u l s e v o l t a g e . The r e s i s t o r s used i n the compensated p u l s e a t t e n -uator were of deposited carbon c o n s t r u c t i o n and of low v o l t a g e c o e f f i c i e n t . The a t t e n u a t i o n was checked a t a 280 v o l t p u l s e a m p l i t u d e and found t o be w i t h i n one per cent of t h e low v o l t a g e v a l u e . 5.6 DETERMINATION 0? SAMPLE ELECTRIC FIELD • E x p e r i m e n t a l l y i t was found t h a t f o r the sample geometry used, a s i d e e l e c t r o d e and the c e n t e r l i n e of the s i d e - a r m extended t h r o u g h the sample form an e c u i -p o t e n t i a l s u r f a c e . The e l e c t r i c f i e l d F i n t h e " a c t i v e " p a r t o f t h e sample, t h a t i s , t h e p a r t between the s i d e probes was th e n t a k e n as V A - V 3 F « L where Yx and V 3 are the s i d e e l e c t r o d e v o l t a g e s and. L i s the c e n t e r l i n e s e p a r a t i o n of the s i d e - a r m s . E m p i r i c a l l y , i t was found t h a t where V, i s the v o l t a g e a c r o s s the c u r r e n t e l e c t r o d e s , and C^ i s c o n s t a n t f o r each sample. T h i s means t h a t the e l e c t r i c , f i e l d may be det e r m i n e d from a measurement of V, a l o n e . That i s , tr = V*-V3 ^ VL. The l e n g t h C^L i s found t o be l e s s t h a n t h e -sample c u r r e n t e l e c t r o d e s e p a r a t i o n ; the d i f f e r e n c e i s l i k e l y due t o p e n e t r a t i o n of the c u r r e n t e l e c t r o d e s i n t o the sample i n t e r i o r and t h e consequent l o w e r i n g c f the c u r r e n t e l e c t r o d e s e p a r a t i o n . 3.7 SAMPLE PREPARATION' ' The s i n g l e c r y s t a l from w h i c h t h e samples were p r e -pared was grown from the -melt by the C z o c h r a l s k i t e c h -n i q u e . The a x i s of t h e c r y s t a l was i n the ( l l O ^ d i r e c t i o n . A v a l u e of i m p u r i t y d e n s i t y was chosen so as t o produce a r e s i s t i v i t y of about f i v e ohm-cm a t the c e n t e r of t h e c r y s t a l . T h i s was t h e v a l u e of r e s i s t i v i t y f o r which the most r e c e n t e x p e r i m e n t a l work on a n i s o t r o p y of hot e l e c t r o n c o n d u c t i v i t y had been done and so f o r the sake of c o n t i n -u i t y i n t h e experiments t h i s seemed the most r e a s o n a b l e c h o i c e . The grown c r y s t a l e x h i b i t e d s i x f a c e s a l o n g the d i r e c t i o n of growth. Laue x - r a y a n a l y s i s e s t a b l i s h e d f o u r of t h e s e f a c e s as ^111^ and two as ^LOO^. The ( l l O ^ d i r e c t -ion-was found as w e l l . Next, s l i c e s were cut a t r i g h t a n g l e s t o t h e growth a x i s of t h e c r y s t a l by means of a 0.003 i n c h d i a m e t e r t u n g s t e n w i r e , d r i v e n i n a r e c i p r o c a t i n g w i r e saw. Carborundum was used as t h e c u t t i n g a b r a s i v e . The s l i c e s were;then l a p p e d t o a t h i c k n e s s of about 0.4 m i l l i m e t e r w i t h a v a r i a t i o n i n t h i c k n e s s over each s l i c e of about 5$. Four t e r m i n a l samples' ( F i g . 5.1) were cut from the s l i c e s u sing 1 1 an u l t r a s o n i c c u t t i n g t o o l and carborundum as the c u t t i n g a b r a s i v e . The samples were t h e n e t c h e d i n medium C?4 s o l u t i o n f o r about 45 seconds t o c l e a n and p r e p a r e the s u r f a c e f o r the attachment o f t h e f o u r g o l d w i r e l e a d s . These l e a d s c o n s i s t e d of 0.005 i n c h d i a m e t e r g o l d , doped w i t h 0.6% a n t i m o n y , • a c c o r d i n g t o the Sigmund Cohn Co., t h e m a n u f a c t u r e r . When b r o u g h t . i n t o . c o n t a c t w i t h t h s etched germanium i n a n i t r o g e n gas atmosphere a t a t e m p e r a t u r e above the gold-germanium e u t e c t i c temp-e r a t u r e , a l i q u i d phase c o n s i s t i n g of.germanium, a n t -imony, and g o l d i s formed. The temperature i s t h e n l o w e r -ed and the l i q u i d phase s o l i d i f i e s t o form an a l l o y region, between the g o l d . w i r e and t h e sample. Upon s o l i d i f i c a t i o n , a germanium r e g r o w t h l a y e r , h i g h l y doped w i t h antimony forms on t h e sample s u r f a c e . T h i s j u n c t i o n has been termed a n^-n j u n c t i o n , t h e P r e f e r r i n g t o the h i g h l y doped r e g r o w t h l a y e r , and the n t o the sample c o n d u c t i v i t y . E l e c t r i c a l l y , t h e s e j u n c t i o n s p e r f o r m i n such a manner t h a t t h e y are h i g h l y i m p e r v i o u s t o h o l e c u r r e n t w h i l e a l l o w i n g e l e c t r o n c u r r e n t passage. E x p r e s s i o n s have been d e r i v e d f o r the h o l e c u r r e n t but depend I n p a r t on the d i f f i c u l t t o d e t e r m i n e p r o p e r t i e s of the h i g h l y doped r e g r o w t h l a y e r of the j u n c t i o n . I n t h i s e x p e r i m e n t , the p o s s i b i l i t y i s always assumed t h a t h o l e i n j e c t i o n t a k e s p l a c e and i n d i r e c t , means adopted t o a s s e s s i t s i m p o r t a n c e . As a f i n a l s t e p , the samples were etc h e d f o r f i v e seconds i n a medium CP4 s o l u t i o n . 3.8 SAMPLE TEMPERATURE RISE DUE TO JOULE SEATING W i t n no v o l t a g e p u l s e a p p l i e d t o the sample, i t w i l l r emain i n t h e r m a l e q u i l i b r i u m w i t h the l i q u i d n i t r o g e n b a t h s u r r o u n d i n g i t a t the ambient t e m p e r a t u r e , Go. . We now c o n s i d e r the sample t e m p e r a t u r e when a s u c c e s s i o n of v o l t a g e p u l s e s of d u r a t i o n . <* and f r e q u e n c y , where p i s t h e o f f - t i m e between p u l s e s . A f t e r a s u f f i c i e n t l y l o n g time t h e sample w i l l be i n ' a s t e a d y - s t a t e when the heat g e n e r a t e d by a s i n g l e p u l s e i s removed from th e sample d u r i n g t h e o f f t i m e . T h i s heat removal can be hy c o n d u c t i o n and c o n v e c t i o n t o the l i q u i d n i t r o -gen and by c o n d u c t i o n t h r o u g h t t h e sample l e a d w i r e s . Vve assume t h a t the t h e r m a l c o n d u c t i v i t y of the sample i s h i g h enough so t h a t t h e r a d i a l t e m p e r a t u r e g r a d i e n t i s s m a l l and t h a t t h e heat conducted by the l e a d s i s s m a l l compared t o the l o s s e s from th e s u r f a c e s t o t h e l i q u i d n i t r o g e n . The sample then may be c h a r a c t e r i z e d by a u n i f o r m t e m o e r a t u r e , act) w h i c h , i n the s t e a d y - s t a t e i s c y c l i c i n t ime t . I n the s t e a d y - s t a t e , 8— Gi. at the s t a r t of the v o l t a g e p u l s e , r i s e s t o 0-f at the end of the p u l s e , t h e n decays back t o Qi at the s t a r t of the next p u l s e . Assuming Newton's law of c o o l i n g , we g e t , f o r t i n the range from z e r o t o , at n 0 c b ' t o where P i s the p u l s e power d i s s i p a t e d d u r i n g the p u l s e , V\o i s the sample mass. C\> i s i t s s p e c i f i c h e a t , and "t0 i s i t s t h e r m a l time c o n s t a n t . I n t h e i n t e r v a l <x ^ t , oLt to By e x p e r i m e n t , to was found t o be about 0.05 seconds. _5 F o r a l l measurements, i s c o n s t a n t and e q u a l t o 10 sec. Then, f o r much l e s s t h a n u n i t y , P u t t i n g i n extreme values of P = -Xoo watt —-5 M 0 = 4-4-x / 0 Cfr = 0-153 Jo"'e Then 9 t - 0 * ^ 0-4° - a . 3 0 So, the maximum " background" temperature r i s e i s l e s s than 0.5 degrees, and the r i s e d u r i n g the pulse of the order of 3 degrees. Taking the case of low e l e c t r i c f i e l d and high pulse r e p e t i t i o n r a t e r e s u l t s i n the f o l l o w i n g values:: Then Q i - Q ^ l-4-° and d $ - Q ± — \ - S ° 3.. 9 INFLUENCE OF SKIN EFFECT ON SAMPLE DIFFERENTIAL  CONDUCTANCE The s k i n depth, & , i s g i v e n by-where i s the frequency, c r the c o n d u c t i v i t y , and jx0 i s the p e r m e a b i l i t y of the medium. For a c y l i n d r i c a l sample of r a d i u s T0 , the r a t i o of the r e s i s t a n c e at frequency f to the zero-frequency r e s i s t a n c e , £ 0 , i s approximated as The above a p p r o x i m a t i o n h o l d s o n l y l o r s k i n depth much g r e a t e r t h a n the r a d i u s . At 77° K, sample r e s i s t i v i t y was about 0.3 ohm-cm. and the mean c r o s s - s e c t i o n a l a r e a of t h e o r d e r 0.1 ramx. Then, o _7 and s k i n e f f e c t i s c o m p l e t e l y n e g l i g i b l e . I n the case of the f i n e g o l d w i r e l e a d s , we f i n d s k i n e f f e c t i s a p p r e c i a b l e , but can be n e g l e c t e d anyway. The r e s i s t i v i t y of the g o l d w i r e was about 10 ohm-cm. F o r a d i a m e t e r of 0.005 i n c h e s , t h e r a t i o i s about f i v e . F o r a t y p i c a l l e n g t h of 3 mm,, the e f f e c t i v e r e s i s t a n c e i s about 0.01 ohms, which i s s m a l l compared t o the sample r e s i s t a n c e and can t h e r e f o r e be n e g l e c t e d . 3.10 INVESTIGATIONS OF SPURIOUS SOURCES OF NOISE Hundred v o l t p u l s e s r e s u l t i n g i n c u r r e n t s of t h e o r d e r of one ampere were a p p l i e d t o the sample. The poss-i b l e e x i s t e n c e of t r a n s i e n t v o l t a g e s as w e l l as n o i s e g e n e r a t e d i n the s o l d e r e d j u n c t i o n s and p r e s s u r e m e t a l - t o -metal' c o n t a c t s e n t e r i n g the a m p l i f i e r must be i n v e s t i g a t e d . P o s i t i o n i n g t h e ,!on"' p u l s e t o the a m p l i f i e r at the c e n t e r of the v o l t a g e p u l s e s u p p l i e d t o the' sample minim-i z e s the t r a n s i e n t s . Any l a r g e t r a n s i e n t s p i c k e d up by the a m p l i f i e r s h o u l d be v i s i b l e on t h e o s c i l l o s c o p e monitor-i n g the output n o i s e of the a m p l i f i e r . None was observed i n the course of the e x p e r i m e n t s . S m a l l t r a n s i e n t s a s . w e l l as s p u r i o u s n o i s e g e n e r a t e d a t the. s o l d e r e d j o i n t s and p r e s s u r e c o n t a c t s were i n v e s t i g a t e d i n t h e f o l l o w i n g way. A m e t a l r e s i s t o r i n t h e f o r m o f a f i n e w i r e was s u b -s t i t u t e d f o r t h e s a m p l e i n t h e b a t h o f l i q u i d n i t r o g e n . The i m p e d a n c e o f t h e w i r e was r e s i s t i v e o f t h e same o r d e r as a t y p i c a l g e rmanium s a m p l e and. had as w e l l a s m a l l i n d u c t i v e component i n s e r i e s . No a d d i t i o n a l n o i s e was o b s e r v e d f r o m t h e a m p l i f i e r when t h e w i r e s a m p l e was p u l s e d . H e n c e , i t was c o n c l u d e d t h a t t h e s y s t e m was f r e e f r o m t r a n s i e n t e f f e c t s f r o m t h e s a m p l e p u l s e as w e l l as s p u r i o u s e l e c t r o d e n o i s e e x t e r n a l t o t h e s a m p l e . A n o t h e r p o s s i b l e s o u r c e o f n o i s e i s due t o o p t i c a l g e n e r a t i o n o f h o l e - e l e c t r o n p a i r s i n t h e s a m p l e cue t o t h e room i l l u m i n a t i o n . T h i s was c h e c k e d and f o u n d t o be n o n - o b s e r v a b l e s i n c e t h e p u l s e d s a m p l e n o i s e d i d n o t v a r y when t h e i l l u m i n a t i o n was c h a n g e d from- t h e n o r m a l amount t o z e r o , when a l l l i g h t s w e r e t u r n e d o f f and t h e s a m p l e b a t h c o v e r e d w i t h a "txlack c l o t h . . 3.11 SAMPLE NOISE DUE TO HOLE GENERATION AT THE CURRENT ELECTRODES The p o s s i b l e e x i s t e n c e o f a d d i t i o n a l n o i s e due t o h o l e i n j e c t i o n a t t h e p o s i t i v e s a m p l e c u r r e n t e l e c t r o d e c a n o n l y be i n v e s t i g a t e d i n d i r e c t l y i n t h e c a s e o f t h e f o u r t e r m i n a l s a m p l e s . F o r t h e • t w o t e r m i n a l n o i s e m e a s u r e -ments a more d i r e c t c h e c k was p o s s i b l e . The f i r s t n o i s e t e m p e r a t u r e m e a s u r e m e n t s , p e r f o r m e d . on r e c t a n g u l a r b a r s i n t h e ( i l l ) d i r e c t i o n were a b o u t t h e same as o b t a i n e d l a t e r w i t h the' f o u r t e r m i n a l s a m p l e s . To e l i m i n a t e p o s s i b l e h o l e i n j e c t i o n n o i s e i n ,the two t e r m -i n a l m e a s u r e m e n t s , a s p e c i a l s a m p l e c o n s i s t i n g o f a r e c t -a n g u l a r f i l a m e n t i i i t h e ( i l l ) d i r e c t i o n w i t h a l a r g e b l o c k end was c u t . The u s u a l g o l d w i r e l e a d s were a l l o y e d t o one c o r n e r o f t h e b l o c k and t o t h e end o f t h e f i l a m e n t . The b l o c k e l e c t r o d e d i m e n s i o n s were c h o s e n so t h a t any h o l e s i n i e c t e d . i n t o t h e b l o c k e l e c t r o d e w o u l d d r i f t on t h e a v e r a g e o n l y o n e - h a l f t h e b l o c k e l e c t r o d e l e n g t h d a r -i n g t h e p u l s e d u r a t i o n . H e n c e , i f t h e b l o c k e l e c t r o d e w e r e p o s i t i v e , d u r i n g t h e v o l t a g e p u l s e t h e h i g h r e s i s t a n c e p a r t o f t h e s a m p l e w o u l d be r e l a t i v e l y f r e e o f i n j e c t e d h o l e s . M e a s u r e m e n t s o f n o i s e t e m p e r a t u r e on t h i s s a m p l e showed no d e p e n d e n c e on p u l s e , p o l a r i t y and were t h e same as o b t a i n e d w i t h t h e p r e v i o u s two t e r m i n a l s a m p l e s . Hence i t was c o n c l u d e d , t h a t t h e n o i s e m e a s u r e d i n t h e ( i l l " ) d i r e c t i o n was n o t due t o h o l e i n j e c t i o n . I n t h e c a s e o f t h e f o u r t e r m i n a l s a m p l e s , f u r t h e r -i n f o r m a t i o n o n h o l e - i n j e c t i o n was o b t a i n e d w i t h n o i s e t e m p e r a t u r e m e a s u r e m e n t s on t h e same s a m p l e a t b o t h 70 Mc/s and 30 Mc/s. Any n o i s e due t o h o l e i n j e c t i o n s h o u l d e x h i b i t a s t r o n g f r e q u e n c y d e p e n d e n c e o v e r t h i s r a n g e , s i n c e t h e h o l e c u r r e n t s p e c t r a l d e n s i t y s h o u l d be o f t h e f o r m S r (00) = 4 - 1 ^ l l - c o s u a i ^ where Ij> i s t h e h o l e c u r r e n t and • tj> , t h e h o l e t r a n s i t -7 t i m e i s o f t h e o r d e r o f 10 s e c o n d s . Here i t i s assumed t h a t h o l e r e c o m b i n a t i o n may be n e g l e c t -ed due".to t h e i r b r i e f t r a n s i t t i m e . The l i f e t i m e o f h o l e s was d e t e r m i n e d f o r a t y p i c a l s a m p l e and f o u n d t o be a b o u t 10"'*' s e c o n d s , which i s s e v e r a l o r d e r s o f m a g n i t u d e g r e a t e r t h a n t h e t r a n s i t t i m e . The a b s e n c e o f f r e q u e n c y d e p e n d e n t n o i s e i n t h e s a m p l e u s e d t h e n i n d i c a t e s t h a t h o l e i n j e c t i o n n o i s e c o u l d be c o n s i d e r e d s m a l l . CHAPTER 4 - EXPERIMENTAL RESULTS The e x p e r i m e n t a l r e s u l t s c o n s i s t o f n o i s e t e m p e r a t u r e , T n , and d i f f e r e n t i a l c o n d u c t a n c e , G , m e a s u r e d a t b o t h 30 Mc/s and 70 Mc/s, a s a f u n c t i o n o f e l e c t r i c f i e l d , E, f o r t h e s a m p l e a x i s i n t h e ( i l l ) , <(llo) , and ^100) d i r e c t -i o n s . The s a m p l e m a t e r i a l - , o f n - t y p e 'germanium was o f one v a l u e o f i m p u r i t y d e n s i t y . The l a t t i c e t e m p e r a t u r e was w i t h i n t h e r a n g e o f 77°£ t o 80°K. The most p r o m i n e n t f e a t u r e s o f t h e . n o i s e t e m p e r a t u r e a r e i t s a p p a r e n t l a c k o f f r e q u e n c y d e p e n d e n c e o v e r t h e r a n g e 50 Mc/s t o 70 Mc/s a s w e l l as i t s h i g h d e g r e e o f a n i s o t r o p y . The l a c k o f f r e q u e n c y d e p e n d e n c e s u g g e s t s t h a t t h e e v e n t s c o n t r i b u t i n g t o t h e c u r r e n t f l u c t u a t i o n s h a v e a mean d u r a t i o n 'X , s u c h t h a t MX i s much l e s s t h a n u n i t y . A t 70-Mc/s t h e n , - X i s l e s s t h a n Sy-io''" s e c o n d s . C e r t a i n l y t h e c a r r i e r mean f r e e t i m e i s l e s s t h a i ! s o t h a t t h e r m a l n o i s e w o u l d be e x p e c t e d t o be f r e q u e n c y i n d e p e n d e n t i n t h e r a n g e o f m e a s u r e m e n t s . A l s o , s i n c e t h e a v e r a g e t i m e an- e l e c t r o n ' s p e n d s i n one v a l l e y o f t h e c o n d u c t i o n b and i s o f t h e o r d e r o f £xl0 s e c o n d s , ( A p p e n d i x 5)- a ny m o d u l a t i o n o'f t h e - c u r r e n t a r i s i n g f r o m s u c h v a l l e y -v a l l e y t r a n s i t i o n s \ w o u l d a l s o - b e i n d e p e n d e n t o f f r e q -u e n c y i n t h e m e a s u r e d r a n g e . S r l b a c h and G u n n ( 1 9 6 2 ) , by m e a s u r i n g t h e c u r r e n t f l u c t u a t i o n s a t r i g h t a n g l e s t o t h e e l e c t r i c f i e l d a l s o f o u n d a n i s o t r o p i c b e h a v i o u r a t a l a t t i c e t e m p e r a t u r e o f 77 K. However, the dependence oi' the n o i s e t e m p e r a t u r e , \ n r e p r e s e n t i n g the c u r r e n t f l u c t u a t i o n s p e r p e n d i c u l a r t o zhe f i e l d , on f i e l d was found t o be l i n e a r w i t h e l e c t r i c f i e l d over most of the range i n v e s t i g a t e d by them, which was o f the r e g i o n up t o one thousand v o l t s per c e n t i m e t e r . For our e x p e r i m e n t a l r e s u l t s , b o t h l o g T * v s . F and l o g i n v s . l o g ? p l o t s were made t o see i f any s i m p l e r e l a t i o n s h i p e x i s t e d between I n and F but none was found. Most of the e x p e r i m e n t a l e r r o r i n both ' Tn and G was due t o f l u c t u a t i o n s i n the meter m o n i t o r i n g the out-put n o i s e o f the a m p l i f i e r , the maximum e s t i m a t e d e r r o r i n a s i n g l e r e a d i n g b e i n g about 10$> f o r the n o i s e temperat-ure and about 7% f o r the d i f f e r e n t i a l conductance. An e r r o r of about 4yo was p r e s e n t i n the v a l u e s of f i e l d . 400-a) u •p cd a? © co o >7» 2 5 o 200 \ ISO) . : : E x £ . * , l N o i s e Temperature vs.' E l e c t r i c F i e l d f o r sample a x i s i n ^00} and l a t t i c e t e m p e r a t u r e o f 77°K. — -. e x p e r i m e n t a l t h e o r e t i c a l y y y y y y y y y y y y y y y \OQY y y y So too I So 2.00 z$o ooo 35b 4-co E l e c t r i c F i e l d , ( y / c m ) W 7 fn ZS -P co <D (o • P. £•! CO E-< 0 co O c 5 N o i s e Temperature v s . E l . e c t r i c F i e l d f o r sample a x i s i n (Lid) and l a t t i c e t e m p e r a t u r e of 77° K. e x p e r i m e n t a l t h e o r e t i c a l / / 100 ZOO 260 3QQ 3SO E l e c t r i c F i e l d , ( v / c m ) 400 \lo<- ? j r-.A . 7, 1560. 1300 SH in CD 0 EH CD I I O O CO • H O <?00 loo 5 0 0 300 N o i s e T e m p e r a t u r e v s . E l e c t r i c F i e l d . f o r s a m p l e a x i s i n ( i l l ) and l a t t i c e t e m p e r a t u r e o f 77° K. 7 e x p e r i m e n t a l t h e o r e t i c a l |O0 i n t e r v a l l e y & / / h o t e l e c t r o n / / 5o ico |$o 2oo 3oo 350 E l e c t r i c F i e l d , ( v / c m ) 4-oo \ \ \ \ \ \ s. D i f f e r e n t i a l Conductance v s . E l e c t r i c F i e l d f o r sample a x i s i n <^.od]>. and l a t t i c e t e m p e r a t u r e of 77°K. e x p e r i m e n t a l t h e o r e t i c a l l i 0 S o joo 1^ 0 2.00 z$o 300 3£o 4<?o E l e c t r i c F i e l d , ( v / c m ) SO )00 jSO XOo 250 3 0 0 35o 400 E l e c t r i c F i e l d , ( v / c m ) C H A P T E R 5 - C O M P A R I S O N OF EXPERIMENTAL R E S U L T S v / I T H THEORY I n t h i s c h a p t e r , t h e r e s u l t s o f C h a p t e r 4 a r e c ompared w i t h t h e t h e o r e t i c a l v a l u e s o f Tn and G- o b t a i n e d f r o m t h e f o r m u l a s o f C h a p t e r 2 a f t e r t h e a p p r o p r i a t e f i e l d -d e p e n d e n t and o r i e n t a t i o n - d e p e n d e n t p a r a m e t e r s a r e f o u n d . The f i n a l e x p r e s s i o n f o r t h e s a m p l e n o i s e t e m p e r a t u r e , as d e r i v e d i n C h a p t e r 2, shows two i n d e p e n d e n t t e r m s : t h e i n t e r v a l l e y and t h e h o t e l e c t r o n c o n t r i b u t i o n s . To e v a l u a t e t h e i n t e r v a l l e y c o n t r i b u t i o n , t h e f o l l -o w i n g i s r e q u i r e d : 1) The d r i f t , v e l o c i t y o f e l e c t r o n s i n any v a l l e y , f o r t h e e l e c t r i c f i e l d i n any one 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 o r i e n t a t i o n s . B a r r i e T s e x t e n s i o n o f S t r a t t o n ' s t h e o r y ( B a r r i e and B u r g e s s , 1962) a s d i s c u s s e d l a t e r , s u p p l i e s t h e r e q u i r e d d r i f t v e l o c i t i e s . 2) The a b s o l u t e v a l u e s o f t h e i n t e r g r o u p t r a n s i t i o n p r o b a b i l i t i e s , ptj , a t a n y f i e l d s t r e n g t h i n . t h e t h r e e p r i n c i p a l c r y s t a l d i r e c t i o n s . F o r t h e c a s e o f z e r o f i e l d t h e j p ^ may be t a k e n f r o m t h e d a t a o f Y c e i n r e i c h e t a l , ( A p p e n d i x 5 ) . F o r non z e r o f i e l d s , the' d i f f e r e n t v a l l e y e l e c t r o n t e m p e r a t u r e s , a s f o u n d f r o m t h e B a r r i e t h e o r y , must be u s e d t o d e d u c e t h e s t e a d y - s t a t e v a l l e y p o p u l a t i o n s and t h e i n t e r v a l l e y t r a n s i t i o n p r o b a b i l i t i e s . T h i s i s done i n A p p e n d i x 2. 3) The s a m p l e d i f f e r e n t i a l c o n d u c t a n c e , c a l c u l a t e d f r o m t h e v a l u e s o f d r i f t - v e l o c i t y g i v e n by t h e B a r r i e t h e o r y . F r o m t h e d r i f t v e l o c i t i e s t h e m o b i l i t i e s a r e f o u n d : 47 t h e Q u a n t i t i e s f i n a l l y used f o r t h e d i f f e r e n t i a l con-ductance are A-C^ifO and <L 0^?-^) The n o i s e temperature c o n t r i b u t i o n due t o "hot e l e c t r o n s " r e q u i r e s the d i f f e r e n t i a l conductance of each of the v a l l e y groups; t h i s i s o b t a i n e d from the m o b i l i t y r e s u l t s , ' as i n 3) above. As w e l l as the d i f f e r e n t i a l c onductance, the v a l l e y e l e c t r o n t e m p e r a t u r e s a r e needed; t h e s e are . g i v e n by the B a r r i e t h e o r y . 5.1 DETERMINATION OF GROUP DRIFT VELOCITY FROM THE BARRIE THEORY B a r r i e ' s e x t e n s i o n of S t r a t t o n ' s h i g h f i e l d t h e o r y ( B a r r i e and Burgess,1962) c o n s i s t s of two e q u a t i o n s r e l a t i n g the d r i f t v e l o c i t y , v T * v of e l e c t r o n s i n a s i n g l e v a l l e y t o the e l e c t r i c f i e l d , , where the s t a r denotes v a l u e s i n the t r a n s f o r m e d &-space. The e q u a t i o n s a r e : 4- V-A? K! W o 4 ^ K T L 7 e + X where H o •= e 0/ T u . V « ©a. T _ d 0 C and 3 Q are c o n s t a n t s and V ' i s the c r y s t a l volume.' ;Ience, i f vr** i s known, the d r i f t v e l o c i t y , IT 01 the v a l l e y e l e c t r o n s may he found from the r e l a t i o n s : W o . and . 1 Wo * ^ J r t t U V J T M I A I t s h o u l d be noted t h a t t h e B a r r i e e q u a t i o n s a r e r a t h e r easy t o a p p l y i f t h e d i r e c t i o n of the e l e c t r i c " f i e l d i s known, as w e l l as i t s magnitude. T h i s i s t h e s i t u a t i o n i n . the. p r e s e n t e x p e r i m e n t , i n ' w h i c h t h e samples were o r i e n t a t e d a l o n g the p r i n c i p a l c r y s t a l " d i r e c t i o n s - Then, because o f the s y m m e t r i c a l , arrangement of the v a l l e y s about t h e l o n g i t u d i n a l sample a x i s , t h e f i e l d v:as known t o be-i n the same d i r e c t i o n as the l o n g i t u d i n a l a x i s . The complete procedure i n s o l v i n g t h e f i n a l e q u a t i o n s of the B a r r i e - t h e o r y then i s as f o l l o w s . T h i s s o l u t i o n . w i l l g i v e the e l e c t r o n t e m p e r a t u r e , Tt , f o r each v a l l e j r as w e l l as t h e d r i f t . v e l o c i t i e s f o r any v a l u e of a p p l i e d f i e l d , p r o v i d e d the f i e l d and sample c u r r e n t a r e i n t h e p r i n c i p a l c r y s t a l d i r e c t i o n s . ' 1) S o l v e the e q u a t i o n s f o r vr** and by i n s e r t i n g v a l u e s o f e l e c t r o n t e m p e r a t u r e , T , i n t h e • e q u a t i o n s , s t a r t i n g w i t h T e q u a l t o t h e . . . l a t t i c e t e m p e r a t u r e and p r o c e e d i n g upwards. 2) , Knowing the a p p l i e d f i e l d r , a l l o w s T t o be found. From the dependence of AT** on F* nd knowing /V** and r * a r e p a r a l l e l , - g i v e s i r * * g i v e s T .,. and hence the v a l l e y pop-u l a t i o n s may be found. From AT*"* the d r i f t v e l o c i t y V of each v a l l e y i s o b t a i n e d . Then, t a k i n g comp-onent's of VT i n t h e f i e l d r d i r e c t i o n g i v e s t h e : group d r i f t v e l o c i t i e s , v f and W The r e l a t i o n s h i p between some of t h e s e q u a n t i t i e s i s shown i n a t a b l e on the f o l l o w i n g page. The f o l l o w i n g t a b l e summarizes the i n f o r m a t i o n cn the group t r a n s p o r t p r o p e r t i e s : Samples. A x i s \^'. D i r e c t i o n s . F.* F c * F _vr \)-«"* ur 100 54°. 8 P| --- P i ---' . 110 Gold V a l l e y s Hot V a l l e y s 35°. 2 '90° 111 Cold V a l l e y Hot V a l l e y s 70°. 5 0° ? 5 K?i i s the angle between the major a x i s of a constant energy e l l i p s o i d and the f i e l d d i r e c t i o n . • •© = / i s s . ^ « . = (Bus. + B a " 51 5.2 CALCULATION OF DIFFERENTIAL CONDUCTANCE The low f r e q u e n c y v a l u e of the d i f f e r e n t i a l conduct-ance f o r one v a l l e y i s g i v e n by b-u — — — mi L2-T h i s i s a l s o e a u a l t o ^Ti _ a Mi oU^H where Mx i s t h e group i m o b i l i t y , It i s the mean c u r r e n t of group i e l e c t r o n s , and F - v / i _ ±s -the a p p l i e d f i e l d . V a l u e s of ^ i F were o b t a i n e d from B a r r i e ' s t h e o r y . At 50 v o l t p e r cm. i n t e r v a l s , the f u n c t i o n ^ i F was numer-i c a l l y d i f f e r e n t i a t e d , u s i n g a f i v e - p o i n t d i f f e r e n t i a t i o n f o r m u l a . The e r r o r i n v o l v e d i n t h e d i f f e r e n t i a t i o n was e s t i m a t e d i n the f o l l o w i n g way. The d i f f e r e n t i a t i o n was r e p e a t e d u s i n g a t h r e e - p o i n t f o r m u l a , and the r e s u l t s compared w i t h the f i v e - p o i n t f o r m u l a v a l u e s . No d i f f -erence g r e a t e r t h a n 2.5% was found between them. The t o t a l conductance G was found 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 groups., 5.3 DETERMINATION OF CARRIER DENSITY AND ELECTRON  POPULATION In the ( l i d ) d i r e c t i o n sample , the r e s i s t i v i t y was found t o be 2.46 ohm-cm at 23°C, w i t h an e s t i m a t e d e r r o r of 6% due t o t h i c k n e s s and w i d t h v a r i a t i o n s a l o n g the sample l e n g t h . Taking, the m o b i l i t y as b e i n g due t o the l a t t i c e c o n t r i b u t i o n of s c a t t e r i n g o n l y w i t h the imp-u r i t y c o m p o n e n t s m a l l , r e s u l t s i n a v a l u e o f 6.5x10 m f o r the c a r r i e r d e n s i t y . To c h e c k t h a t the 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 at room t e m p e r a t u r e , the m o b i l i t y as g i v e n by the Debye-Conwell t h e o r y , which i n c l u d e s the e f f e c t of b o t h i m p u r i t y .and l a t t i c e s c a t t e r i n g , was c a l c u l a t e d and found t o be i n agreement w i t h the l a t t i c e m o b i l i t y a l o n e t o w i t h i n l e s s than one per c e n t . I n the (lOO*) and ( i l l ) d i r e c t i o n s , the measured r e s i s t i v i t i e s were 2.03 and 2.00 ohm-cm. r e s p e c t i v e l y , w i t h maximum e r r o r s of about &fo. The c a r r i e r d e n s i t i e s 710 -3 are e s s e n t i a l l y the same at 8.0x10 ra . In c a l c u l a t i n g the t o t a l sample e l e c t r o n p o p u l a t i o n , t h e n o i s e g e n e r a t i n g p a r t of the sample of i n t e r e s t i n the p r e s e n t i n v e s t i g a t i o n was c o n s i d e r e d t o c o n s i s t of the main body of the s a m p l e ( F i g . 3 . 1 ) between t h e c e n t e r l i n e s of. t h e s i d e probes and e x c l u d i n g the s i d e probes t h e m s e l v e s , which would be l a r g e l y low f i e l d r e g i o n s . The v a l u e s of t h e t o t a l p o p u l a t i o n o b t a i n e d were: <110> N = i.fc&y.io" (l00> N = ( i l l ) f\ - S--I7 x i o " 5.4 THE HOT ELECTRON C O N T R I B U T I O N TO THE 'NOISE TEiviPERATURE As g i v e n i n Chapter 2, t h i s c o n t r i b u t i o n i s T W ' ° ' — Ni WW w u - + l a . ^ v « r ur « L U x F ) • n —— c \ , r d i -KM d_('ju.,F + M*F) I n Appendix 3, i t i s shown t h a t f o r the d i s t r i b u t i o n i n A -space assumed by B a r r i e , the v e l o c i t y v a r i a n c e s are g i v e n by where vY\ t • and Wrt^ , a r e , r e s p e c t i v e l y " , the group 1 and group 2 e f f e c t i v e masses i n the f i e l d d i r e c t i o n and T-t i s the e l e c t r o n t e m p e r a t u r e of the i t h group. The f i n a l e x p r e s s i o n f o r t h e hot e l e c t r o n c o n t r i b u t -i o n i s t h e n T = dF , ; The v a l u e s of ^ ' / N and ^ V H a r e found from the v a l l e y p o p u l a t i o n r a t i o , ^ V Y \ % . For t h e sample a x i s i n the ( l l O ) d i r e c t i o n , - n a. F o r the ( i l l ' ) d i r e c t i o n , _Hv_ __ n X N 3 + a. V a l u e s of -711 .are. found u s i n g r e l a t i o n ' A£.l o f Appendix 2 and r e q u i r e o n l y the v a l u e of t h e group e l e c t r o n , temperat 1 r e s , T, . and T^ . CHAPTER 6 - CONCLUSIONS AND SUGGESTIONS FOR FUTURE w'ORK The work d e s c r i b e d i n the. p r e c e d i n g chapters i s the f i r s t experimental i n v e s t i g a t i o n of hi g h c u r r e n t , •'high-frequency n o i s e i n the f i e l d d i r e c t i o n in- an e x t r i n s i c semiconductor. Two independent sources o f . n o i s e have,been used to i n t e r p r e t the r e s u l t s , i n t e r v a l l e y n o i s e and hot e l e c t r o n n o i s e . An a d d i t i o n a l source, due to f l u c t u a t i o n s i n the t o t a l e l e c t r o n p o p u l a t i o n , has been shown t h e o r e t i c a l l y to be n e g l i g i b l e . In ^two c r y s t a l o r i e n t a t i o n s , <^lll) and - s'lio) , the i n t e r v a l l e y n o i s e i s " c a l c u l a t e d to dominate the hot e l e c t r o n n o i s e , with the r e s u l t i n g ( i l l ) n o i s e temp-er a t u r e g r e a t e r than'that of the- ( l i d ) . . The ' c a l c u l a t e d , values of n o i s e temperature agree w e l l w i t h the exper-imental r e s u l t s f o r the ( i l l ) d i r e c t i o n and .rather p o o r l y f o r the (lio) . . . -In the (lOO^ d i r e c t i o n , , the i n t e r v a l l e y n o i s e i s p r e d i c t e d to v a n i s h and only the hot. e l e c t r o n c o n t r i b u t -i o n appearing i n the n o i s e temperature. The c a l c u l a t e d hot e l e c t r o n n o i s e i s much s m a l l e r .than the i n t e r v a l l e y n o i s e i n the other two c r y s t a l o r i e n t a t i o n s . . T h i s agrees -roughly with the experimental r e s u l t s . • One p o s s i b l e source of e r r o r i n the experimental values of n o i s e temperature.is due to s l i g h t misalignment of the sample, a x i s with r e s p e c t t o the p r i n c i p a l ' c r y s t a l d i r e c t i o n T h i s would be p a r t i c u l a r l y u n d e s i r a b l e i n the "(lOO) samples where the n o i s e t e m p e r a t u r e i s so much s m a l l e r t n a n f o r the o t h e r o r i e n t a t i o n s . Any such m i s a l i g n m e n t would r e s u l t i n d i f f e r e n t t r a n s p o r t p r o p e r t i e s of the d i f f -e r e n t v a l l e y s and c o n s e q u e n t l y , i n t e r v a l l e y n o i s e . F o r . t h i s r e a s o n , i t i s somewhat s p e c u l a t i v e t o draw any c o n c l u s i o n s from th e d i s c r e p a n c i e s between t h e o r y and experiment f o r measurements i n t h i s d i r e c t i o n . D i s c r e p a n c i e s between a c t u a l and p r e d i c t e d n o i s e t e m p e r a t u r e s may be due t o s e v e r a l s i m p l i f i c a t i o n s c o n t a i n ed i n the t h e o r y . These a r e : 1) I n t h e ( i l l ) and ( l i o ) c u r r e n t d i r e c t i o n s , the i n t e r -v a l l e y n o i s e c a l c u l a t i o n s i n c l u d e t r a n s i t i o n s o n l y betw-een the f o u r ( i l l ) minima of the c o n d u c t i o n band'. However, a t h i g h f i e l d , s t r e n g t h s • when the e l e c t r o n t e m p e r a t u r e s of t h e s e . v a l l e y s are h i g h , t r a n s i t i o n s t o the (pOO) minimum t a k e p l a c e . Because of the low e f f e c t i v e mass of 0.034 e l e c t r o n masses, e l e c t r o n s i n t h i s minimum s h o u l d p o s s e s s h i g h m o b i l i t y and would c o n t r i b u t e t o t h e i n t e r v a l l e y n o i s e . 2) The i n t e r v a l l e y t r a n s i t i o n r a t e under e q u i l i b r i u m c o n d i t i o n s was found from r a t h e r s c a n t y d a t a of W e i n r e i e h et a l . Only f i v e v a l u e s of sample i m p u r i t y were i n v e s t -i g a t e d and i n t e r p o l a t i n g t h e i r d a t a was r a t h e r u n c e r t a i n . I n c o n c l u s i o n , i t may be added that, the e x p e r i m e n t a l v a l u e s of t h e d i f f e r e n t i a l conductance show q u i t e good agreement w i t h t h e p r e d i c t i o n s of the B a r r i e t h e o r y at the h i g h e r v a l u e s of f i e l d . The poor agreement at low f i e l d s may be a t t r i b u t e d t o the n e g l e c t of i m p u r i t y s c a t t e i n g i n *the t h e o r y . From the r e s u l t s of t h i s i n v e s t i g a t i o n i t has not been proven t h a t the a n i s o t r o p y of n o i s e temperature i s due t o the i n t e r v a l l e y n o i s e component a l o n e . Measurements made over a much g r e a t e r f r e q u e n c y range t h a n r e p o r t e d here a n d ' e x t e n d i n g i n t o t h e microwave r e g i o n would be expected t o show a f r e q u e n c y dependence i f the major s o u r c e of n o i s e were due t o i n t e r v a l l e y t r a n s i t i o n s whereas the hot e l e c t r o n n o i s e s h o u l d be u n i f o r m beyond the microwave spectrum, d r o p p i n g o f f w i t h f r e q u e n c y a t 10/ the o r d e r of 10 c y c l e s per second. Some a d d i t i o n a l i n f o r m a t i o n might a l s o be o b t a i n e d from n o i s e t e m p e r a t u r e d e t e r m i n a t i o n s made at d i f f e r e n t l a t t i c e t e m p e r a t u r e s . I n c r e a s i n g the l a t t i c e t e m p e r a t u r e from 77° K d e c r e a s e s the e l e c t r o n m o b i l i t y and i n c r e a s e s t h e i n t e r v a l l e y t r a n s i t i o n r a t e , r e s u l t i n g i n a r e d u c t i o n i n t he i n t e r v a l l e y n o i s e . Other s e m i c o n d u c t o r s p o s s e s s i n g , l e s s complex band s t r u c t u r e might a l s o be i n v e s t i g a t e d . I n p a r t i c u l a r , i n d i u m a n t i m o n i d e , which i s b e l i e v e d t o have s p h e r i c a l c o n s t a n t energy s u r f a c e s i n A - s p a c e would show no i n t e r v a l l e y n o i s e . N o i s e analogous t o i n t e r v a l l e y n o i s e , and due t o h o l e t r a n s i t i o n s between the degenerate v a l e n c e bands of p-type germanium might be o b s e r v a b l e w i t h t h e p r e s e n t e x p e r i m e n t a l arrangement. Since- the c o n s t a n t energy s u r f a c e s of the h o l e s ar e s p h e r i c a l , t h e n o i s e temperature s h o u l d be i s o t r o p i c . BIBLIOGRAPHY Ba k k e r , C . J . and H e l l e r , G. (1939) P h y s i c a 6, 263. B a r r i e , R. and B u r g e s s , R.R. (1962.) Can.J.Phys.40, 1056. B u r g e s s , R.E. (1965) Radio S c i e n c e J . 69D, 381. E r l b a c h , E. and Gunn, J.B. (1962) Semiconductor C o n f e r e n c e at E x e t e r p.128. G u r e v i c h , V.L. (1963) S o v i e t P h y s i c s - J.E.T.P. 16, 1252. Herman, F. (1955) P r o c . I . R.E. 43,. 1703. H e r r i n g , C. (1955) B.S.T.J. 34, 237. Johnson, J.B. (1928) Phys. Rev. 32, 97. L e v i n g e r , B.W. and F r a n k l , D.R.(1961) J.Phys.Chem.Solids 20, 261. P a i g e , E.G.3. (1960) P r o c . P h y s . S o c . 75, 174. P r i c e , ' P . J . (1959) I.B.M. J o u r n a l 3, 191. P r i c e , P . J . (1960) J.Appl.Phys. 31, 949. P r i c e , P . J . and Hart man, R.L. (1964) J . Phys .'Chem. S o l i d s 25, 567. R e i k , H.G. and R i s k e n , H.(1961) Phys.Rev. 124, 777. R e i k , I-I.G. and R i s k e n , H. (1962) Phys.Rev. 126, 1757. S t r a t t o n , R. (1957) P r o c . R o y a l Soc.London A242, 355. S t r a t t o n , R. (1958) S o l i d State- P h y s i c s i n E l e c t r o n i c s and T e l e c o m m u n i c a t i o n s - B r u s s e l s , p343. van der Z i e l , A. (1954) N o i s e P r e n t i c e - H a l l Wannier, G.H. Elements o f S o l i d S t a t e Theory - Cambridge U n i v e r s i t y P r e s s W e i n r e i c h , G. , Sanders, T.M. , and White, H.G. (1959) ' Phys.Rev. 114, 33. W i l s o n , A.H. (1950) Theory of P e t a l s - Cambridge U n i v e r s i t y P r e s s Y a m a s h i t a , J . and Watanabe, M. (1954) P r o g r . Theor. Phys. 12t 443. APPENDIX 1 - FLUCTUATIONS IN ELECTRON' POPULATION • ' .:• _ • OF 'SAMPLE ' We w i s h . u l t i m a t e l y t o c a l c u l a t e the. f l u c t u a t i o n s i n ' t he t o t a l e l e c t r o n p o p u l a t i o n of a t y p i c a l sample .under h i g h e l e c t r i c f i e l d c o n d i t i o n s . To do t h i s we must f i r s t c o n s i d e r the e q u i l i b r i u m p o p u l a t i o n f l u c t u a t i o n s . The system t o be c o n s i d e r e d i s a -homogeneous i m p u r i t y s e m i c o n d u c t o r c r y s t a l • c o n t a i n i n g a f i x e d number No of s t a t i o n a r y p o s i t i v e i o n s and a f l u c t u a t i n g number H&) o f e l e c t r o n s whose mean number N e q u a l s No so as t o p r e s e r v e space-charge n e u t r a l i t y . Such' a system c o n s t i t u t e s .a plasma i n . w h i c h the p o s i t i v e i o n s i n s t e a d of b e i n g f r e e a r e , r i g i d l y bound t o f i x e d l o c a t i o n s . The c h a r a c t e r i s t i c s o f such a system are determined by two i n t e r a c t i o n s : the coulomb i n t e r a c t i o n w h i c h ' t e n d s . t o c r e a t e a u n i f o r m d i s t r i b u t i o n of e l e c t r o n s amongst t h e f i x e d i o n s and the e f f e c t s of t h e r m a l a g i t a t i o n of the e l e c t r o n s .-which t e n d s '. t o . d i s r u p t t h e u n i f o r m d i s t r i b u t i o n . These two competing p r o c e s s e s r e s u l t "in e l e c t r o s t a t i c s h i e l d i n g of both i o n s and . e l e c t r o n s . I n t h e case-where the mean e l e c t r o n - e l e c t r o n coulomb i n t e r a c t i o n i s s m a l l compared t o t h e mean e l e c t r o n k i n e t i c energy, the D e b y e - E l i c k e l t h e o r y i s a p p l i c a b l e , and the e f f e c t of s h i e l d i n g s h i e l d i n g r e s u l t s i n the f o r m u l a " 4 - r v £ r f o r the average e l e c t r o s t a t i c p o t e n t i a l V(?) a t a d i s t -ance f from the. charge \ . The. l e n g t h D , the Debye l e n g t h , i s g i v e n by - . . . where C • i s the p e r m i t t i v i t y and 'Ho the mean e l e c t r o n d e n s i t y . Thus, e n c l o s i n g every f i x e d or m o b i l e charge i n the medium i s a - sphere of charge n e u t r a l i t y • whose r a d i u s i s of t h e o r d e r of the Debye l e n g t h . S i n c e the e l e c t r o n s a r e i n t her ma 3. e q u i l i b r i u m w i t h the L a t t i c e at a temp e r a t u r e T } t h e y possess -random .thermal motions- a p p r o p r i a t e t o ~T but c o n s t r a i n e d by the coulomb i n t e r a c t i o n . Such random motions' and t h e i r a s s o c i a t e d c u r r e n t d e n s i t y ' f l u c t u a t i o n s a r e d e s c r i b e d by the f l u c t u a t i o n - d i s s i p a t i o n theorem, w h i c h i s a 'gen e r a l -i z a t i o n of t h e N y q u i s t theorem of c u r r e n t and v o l t a g e f l u c t u a t i o n s i n a d i s s i p a t i v e network. By a p p l y i n g the f l u c t u a t i o n - d i s s i p a t i o n . t h e o r e m t o the l o n g i t u d i n a l modes of propagation,., Burgess (1965) found f o r the spectrum of charge d e n s i t y ' f l u c t u a t i o n s : The F o u r i e r component of wave v e c t o r P has the form ^ v w h i c h , when i n t e g r a t e d over t h e volume of the sample y i e l d s an e x p r e s s i o n f o r , t h e o o n u l a t i o n f l u c t u a t i o n due t o t h e F o u r i e r component. •^(AN^p i s th e n f o u n d , a f t e r a v e r a g i n g over an ensemble, 50 ((<\NT) • g ^ T £ ^ ^ (\-COS^L) ( l -COS r A A ) ( j - C O S P^B) the phase f a c t o r vf h a v i n g d i s a p p e a r e d . L , A , and B a r e t h e .dicaensions of the p a r a l l e l e p i p e d sample i n r e s p -e c t i v e l y t h e x,y, and z d i r e c t i o n s . The e x p r e s s i o n f o r v a rN i s o b t a i n e d hy i n t e g r a t i n g over a l l modes of V• ,-the mode d e n s i t y b e i n g per u n i t volume o f P space, t o y i e l d : oo co r r r varN K T £ . (i-cosr xL) (\- co6r sA)(i-cPsPzBN)AR,,in,ip. n a-1 "3 An upper, l i m i t t o va r N i s found i m m e d i a t e l y ' i f we i g n o r e the' one term i n the denominator of t h e ' i n t e g r a n d . T h i s c o r r e s p o n d s t o t h e case o f an i n f i n i t e pebye l e n g t h , o r the. absence of coulomb i n t e r a c t i o n . . T h i s ' - l i m i t is ' : -R e t u r n i n g t o t h e g e n e r a l c a s e , the i n t e g r a t i o n over one of the components of P " say'. Vx i s performed u s i n g t h e f o l l o w i n g r e l a t i o n , , which'' i s e s t a b l i s h e d e a s i l y by co n t o u r i n t e g r a t i o n : CO — CO The e x p r e s s i o n f o r v a r N t h e n becomes V d r M = K T L a> co For i n t e g r a t i o n over w e - r e q u i r e -the s o l u t i o n of t h e f o l l o w i n g i n t e g r a l : 0-cosn,A)af; 3 -00 L e t t i n g HjA = X and »<A = D^T^ ' r e s u l t s i n I n . t h e l i m i t ' of much l e s s than u n i t y , X| • .aooroaches an upper l i m i t ' , - • 00 o R e t u r n i n g t o the e x p r e s s i o n f o r varN a n d - i n t e g r a t i n g o v er 17 l e a d s t o an e x p r e s s i o n f o r the upper l i m i t of v a r N . '  '' '' . - - - • 00 - O O A u 1 , L7I % + + J£MJ&LS\ 0 - cos n ^ U r * The i n t e g r a t i o n over Q i s s i m i l a r t o t h a t f o r P y , \ except t h e parameter °< i s now e q u a l to. u n i t y . . F i n d i n g an upper l i m i t t o t h e . i n t e g r a t i o n l e a d s , t o - t h e . f i n a l e x p r e s s i o n f o r the upper l i m i t ' o f v a rN , w i t h the assumpt i o n t h a t a l l sample dimensions are much g r e a t e r than the Debye l e n g t h : . \L- A P u t t i n g i n the f o l l o w i n g v a l u e s ," which' a r e . a p p r o p r i a t e f o r a t y p i c a l sample i n the p r e s e n t experiment under e q u i l i b r i u m c o n d i t i o n s : . . Y\0= 8 x 1 0 *\ • 1= \Uo . A ^ 3 * / o m and u s i n g t h e r e s u l t f o r v a r N . r e s u l t s i n . . . . We see from t h i s the importance of the Debye l e n g t h : i n d e t e r m i n i n g the p o p u l a t i o n f l u c t u a t i o n s i n a volume V of t h e plasma. Of p a r t i c u l a r , importance, i s the minimum di m e n s i o n . o f V i n cases of samples shaped l i k e t h i n s l a b s or l o n g r e c t a n g u l a r b a r s , as i n t h e p r e s e n t "exper-iment . I n a p p l y i n g t h e s e r e s u l t s t o an e q u i l i b r i u m german-ium sample, we are o v e r e s t i m a t i n g the f l u c t u a t i o n s , s i n c e we have assumed a l l s u r f a c e s of V- ; t o be permeable t o t h e passage of, l o n g i t u d i n a l d e n s i t y f l u c t u a t i o n s , w h e r e a s . o n l y the ohmic end c o n t a c t s a l l o w unimpeded . e l e c t r o n f l o w , t o and' from the sample. '••.•'" , I n order, t o extend t h i s a n a l y s i s t o t h e , n o n - e q u i l -i b r i u m h i g h f i e l d c a s e , we must f i r s t e v a l u a t e the e f f e c t of e l e c t r o n d r i f t , due t o the a p p l i e d f i e l d , on the \ \ • . . . p o p u l a t i o n f l u c t u a t i o n s . T h i s may be done by c o n s i d e r i n g the p o p u l a t i o n f l u c t u a t i o n s " i n a volume i d e n t i c a l to. the sample volume, which; i s moving at the d r i f t v e l o c i t y - U. • i n , f o r c o n v e n i e n c e , the x - d i r e c t i o n of a s e t of axes at r e s t w i t h r e s p e c t t o the e q u i l i b r i u m plasma. As i n the e q u i l i b r i u m c a s e , we need. ^ ) , the V -spectrum i n the moving c o o r d i n a t e system. A p l a n e wave of c i r c u l a r f r e q u e n c y 60 and wave v e c t o r P as observed i n the f i x e d c o o r d i n a t e s w i l l appear i n the moving c o o r d i n a t e s t o have the same wave v e c t o r r but c i r c u l a r f r e q u e n c y CO = CO — n i x I n t he s t a t i o n a r y c o o r d i n a t e system, Burgess(1965) showed t h a t the T -spectrum of the l o n g i t u d i n a l space-charge d e n s i t y modes i s g i v e n by S i n c e i n the moving system o n l y a f r e q u e n c y s h i f t o c c u r s , where CO1 - G O - u Q . . I n t e g r a t i n g (^rJ) over bi' g i v e s ( ^ f * ) f ° r "the moving system. co 00 «" U"^. ur, The f i r s t term i s i d e n t i c a l t o t h e e q u i l i b r i u m v a l u e , and the second,due t o d r i f t , w e now c o n s i d e r . S i n c e ^ m ^_~£(p (J)\ i s a n e v e n f u n c"ti°n of t o , the 64 d r i f t term i s odd i n T% ; hence, i n c l u d i n g the d r i f t term i n . t h e ^ - s p a c e / i n t e g r a l i n the e v a l u a t i o n o f v a r N r e s u l t ' s i n no change from the e q u i l i b r i u m v a l u e . . . A second e f f e c t of the h i g h ' f i e l d on f l u c t u a t i o n s ••in .' t h e ' t o t a l p o p u l a t i o n i s i n .the p o s s i b l e enhancement of the t h e r m a l v e l o c i t i e s of the e l e c t r o n s as d i s t i n c t from the d r i f t . j u s t c o n s i d e r e d . T h i s may be . t a k e n i n t o , account.' . i f we adopt the concept o f . a n e l e c t r o n t e m p e r a t u r e t o d e s c r i b e t h e n o n - e q u i l i b r i u m v e l o c i t y d i s t r i b u t i o n , as i n the . B a r r i e t h e o r y , and assume a n o n - e q u i l i b r i u m Debye .. l e n g t h e x i s t s which i s p r o p o r t i o n a l to' the .square-root of t h e e l e c t r o n temperature.' i APPENDIX 2 - CALCULATION OF GROUP POPULATION . SPECTRAL DENSITIES Le t us c o n s i d e r the j t h group, s t i l l r e s t r i c t i n g o u r s e l v e s t o the case of- two groups i n a l l . Then, f o r s m a l l f l u c t u a t i o n s i n the group p o p u l a t i o n we may w r i t e the f o l l o w i n g two L a n g e v i n e q u a t i o n s f o r th e p o p u l a t i o n f l u c t u a t i o n s , A N 3 :: 1 A M j It) « k ANjCt) _ bjJk A Nj (t) + v|». (t) (A 2.1) tit J where j-taKj^ i s " t i l s p r o b a b i l i t y p e r u n i t time of a t r a n s i t i o n from group j t o group k. N^ Gt) i s a s t o c h a s t i c term r e p r e s e n t i n g t he f l u c t u a t i o n s i n eLANljfr). L e t us denote t h e F o u r i e r t r a n s f o r m of any s u i t a b l e f u n c t i o n xCt) o f time by 2 ( u ) ) 2 ( x t t J e i M t a t Then, F o u r i e r t r a n s f o r m i n g e q u a t i o n A2.1 y i e l d s t h e two e q u a t i o n s : w i t h the s o l u t i o n where .2. The group p o p u l a t i o n s p e c t r a l d e n s i t i e s a re d e f i n e d by and T-^co T From t h e s o l u t i o n of e q u a t i o n A2.1 we a r r i v e at the f o l l o w i n g e x p r e s s i o n f o r SN^ W) : The spectrum of M^ i i s r e l a t e d t o t h e events c o n s t i t -u t i n g by: i f the ev e n t s a r e m u t u a l l y independent. Here l> i s the mean r a t e of o c c u r r e n c e of e v e n t s , a t y p i c a l event h a v i n g the form v(-fc) , and commencing at time ' t e q u a l s z e r o . Two p r o c e s s e s c o n t r i b u t e t o V> : e l e c t r o n t r a n s i t i o n s between groups, o c c u r r i n g at t h e r a t e ft^Nv -V- J? \\x and e n t r y and e x i t from the sample b o u n d a r i e s due t o the passage of c u r r e n t , o c c u r r i n g a t t h e r a t e L = Hi where t j i s the • • t r a n s i t - t i m e . S i n c e i s much s m a l l e r t h a n j 3 ^ , the major con t -r i b u t o r t o V i s t h e i n t e r g r o u p t r a n s i t i o n r a t e . rfct) i s a d e l t a f u n c t i o n , ± £ ( £ - t o ) where' "to i s the time of t r a n s i t i o n , w i t h the p o s i t i v e s i g n . f o r e n t r y and the n e g a t i v e s i g n f o r e x i t . The spectrum S^ ; i s t h e n : S«¥ i (^ i s found by c o n s i d e r i n g ^YiIt) + ^ xtt) . I f we i g n o r e f l u c t u a t i o n s i n t h e t o t a l e l e c t r o n p o p u l a t i o n , as discussed, i n Appendix. 1, d - t d t Then, a d d i n g e q u a t i o n s A2.1 g i v e s The spectrum o f ty| + 4^ . i s r e l a t e d t o , and 9 f , A S i n c e S^ *.^  i s z e r o , The c o e f f i c i e n t s i n e q u a t i o n A2.2. have the v a l u e s : E q u a t i o n A2.2 t h e n y i e l d s f o r the p o p u l a t i o n spectrum of t h e j t h group: F o r f r e q u e n c i e s 60 , SN^ W) approaches SMJ , where S i n c e and Hence 68 \ \ APPENDIX 3 - CALCULATION OF VALLEY POPULATIONS AND THE  INTERVALLEY TRANSITION PROBABILITIES As s t a t e d i n Chapter 1, the p r o b a b i l i t y of an e l e c t r o n t r a n s i t i o n between v a l l e y s due to a b s o r p t i o n of a l a t t i c e phonon of energy ft to i s : \ I _. \ / (AE +\QS?X where i s the i n i t i a l energy of the e l e c t r o n above the v a l l e y minimum.. For emission of a phonon of energy fto) , the t r a n s i t i o n p r o b a b i l i t y i s We "WtAE-Uf f o r AE^*w - O A E < ^ W Assuming that the energy d i s t r i b u t i o n of the e l e c t r o n s i n a, s i n g l e v a l l e y i s Iviaxwellian at an e l e c t r o n temp-e r a t u r e T, the energy d i s t r i b u t i o n i s given by -f(Ae) = a ( T ) ( ^ e ^ e / K T o.tT)is r e l a t e d to the s i n g l e v a l l e y p o p u l a t i o n d e n s i t y , n,CT)by C o n s i d e r i n g now the t r a n s i t i o n s from an energy shelleCCde) i n one v a l l e y , by a b s o r p t i o n , we f i n d that the t r a n s i t i o n p r o b a b i l i t y per u n i t time i s p r o p o r t i o n a l t o e —I T h e c o r r e s p o n d i n g e x p r e s s i o n ^ f o r e m i s s i o n i s H e n c e , t h e t o t a l t r a n s i t i o n p r o b a b i l i t y , BICT), f r o m a v a l l e y a t e l e c t r o n t e m p e r a t u r e T t o a n o t h e r v a l l e y o f - a r b i t r a r y e l e c t r o n t e m p e r a t u r e , i s p r o p o r t i o n a l t o ACT), + T h e a b o v e i n t e g r a l s a r e r e a d i l y e v a l u a t e d i n t e r m s o f t h e B e s s e l f u n c t i o n r(V) ' \ T h e n , 0,Cr)rf ^ T ) f If.ggf) + e -I |-e E x p r e s s i n g a C T ) i n t e r m s o f v i t C T ) , 71 6,CT) * n , (T ) T K, k j e + _e Then t h e t r a n s i t i o n p r o b a b i l i t y p e r e l e c t r o n , denoted by to1' Suppose two v a l l e y s h a v i n g e l e c t r o n t e m p e r a t u r e s T, and T x are i n a s t e a d y - s t a t e of p o p u l a t i o n and t h a t a l l f o u r v a l l e y s a r e c h a r a c t e r i z e d by e i t h e r T 4 or T z. Then the s t e a d y - s t a t e c o n d i t i o n 8 \ C T 0 ~ ®»^ T».') a l l o w s the v a l l e y p o p u l a t i o n r a t i o ~Y\X t o be found. The r e l a t i o n i s where e ' I — e and has the bounds: f o r T e q u a l T u ( 9C*T,.)=cscL and f o r T much g r e a t e r t h a n i i u ) •> 3 ~ co+k, ^ T - ^ S i n c e t h e v a l l e y p o p u l a t i o n at z e r o f i e l d , when T e q u a l s T L. , i s o n e - q u a r t e r the t o t a l e l e c t r o n p o p u l a t i o n , the a b s o l u t e v a l u e s of rtt and Y\%. may now be found. The i n t e r g r o u p t r a n s i t i o n p r o b a b i l i t i e s |>,x and j ^ i i n t r o d u c e d i n Chapter 2 are found from the v a l u e s of ^it"1") f o r the v a l l e y c oncerned, t o g e t h e r w i t h t h e v a l u e of ^ C I T K ) and the a b s o l u t e v a l u e o f and ^ a t 77°K, w h i c h a r e e v a l u a t e d i n Append ix 5. F o r example, f o r t he case o f one v a l l e y a t e l e c t r o n t empe ra t u r e Tj and t h e o t h e r t h r e e a t e l e c t r o n t e m p e r a t -u r e s T^, we have , d e n o t i n g the s i n g l e v a l l e y by- s u b -s c r i p t 1 : where V£L and Vai a r e t he t r a n s i t i o n p r o b a b i l i t i e s p e r e l e c t r o n a t z e r o f i e l d f o r t h e s p e c i a l case o f one v a l l e y a t e l e c t r o n t e m p e r a t u r e T, and t h e o t h e r t h r e e v a l l e y s a t e l e c t r o n t e m p e r a t u r e ; T ^ . As shown i n Append ix 5 , V°ia. e q u a l s 3 ^ . ! \ \ APPENDIX 4 - CALCULATION OF VELOCITY VARIANCE IN ARBITRARY DIRECTION OF DISPLACED MAXWELLIAN DISTRIBUTION L e t i x , i a , I t be the components of a v e c t o r i n ' " - s p a c e , t a k i n g the o r i g i n of the c o o r d i n a t e system at the c e n t e r of one of t h e c o n s t a n t energy e l l i p s o i d s . The 4ta a x i s i s d i r e c t e d a l o n g the major a x i s of the e l l i p s o i d . T a k i n g an a r b i t r a r y d i r e c t i o n o( w i t h d i r e c t i o n c o s i n e s ,°lx , ^3 we r e q u i r e t h e v a r i a n c e o f t h e d i s p -l a c e d M a x w e l l i a n d i s t r i b u t i o n i n t h i s d i r e c t i o n . /oi e l e c t r o n i n s t a t e A has v e l o c i t y components <**.}v<i, v-a-> r e l a t e d t o M by : Wit **A. D e n o t i n g i t s component of v e l o c i t y i n the d i r e c t i o n <?C hy we have A v e r a g i n g over a l l e l e c t r o n s i n t h e v a l l e y , d e n o t i n g the average by A T ^ : The mean-square v a l u e i s \ where. 3 ^ ^ i s t n e e l e c t r o n v e l o c i t y d i s t r i b u t i o n . S i n c e ••• 74 U s i n g t h e e x p r e s s i o n f o r o f t h e d i s p l a c e d M a x w e l l -i a n d i s t r i b u t i o n , * t T ) c w i t h a n d T h e n , b y d i r e c t c a l c u l a t i o n , w h e r e ©<* i s t h e a n g l e ' b e t w e e n t h e d i r e c t i o n «< a n d t h e VT^ a x i s , a n d s W\ i s t h e i n e r t i a l e f f e c t i v e m a s s f o r a c c e l e r a t i o n i n t h e c< d i r e c t i o n , g i v e n b y ( W i l s o n ) l A j j o i n . Qcl + W * C o S dot 1 APPENDIX 5 ' - INTERVALLEY TRANSITION RATE AND THE  AGOUSTOELECTRIC EFFECT The i n t r o d u c t i o n of an a c o u s t i c a l wave i n t o a l a t t i c e of n-type germanium atoms w i l l s h i f t the f o u r minima or v a l l e y s of the conduction hand, whereas i n the undeformed l a t t i c e the minima have the same energy. T h i s means that the p r o b a b i l i t y of o c c u p a t i o n of s t a t e s i n the v a l l e y s ' w i l l change and as a consequence the v a l l e y p o p u l a t i o n s w i l l r e d i s t r i b u t e due to i n t e r v a l l e y t r a n s i t i o n s . I f the a c o u s t i c a l wave frequency i s h i g h enough, the r e d i s t r i b u t i o n w i l l not be complete, and i f i t i s much g r e a t e r than the i n t e r v a l l e y r a t e , l i t t l e or no r e d i s t -r i b u t i o n w i l l occur. Hence, measurements of the r e d i s t -r i b u t i o n as a f u n c t i o n of frequency a f f o r d a d i r e c t means of g a i n i n g ciata about the i n t e r v a l l e y t r a n s t i o n r a t e . . The a c o u s t o e l e c t r i c e f f e c t ( W e i n r e i c h , G. et a l . 1958), allows i n f o r m a t i o n to be determined about the r e d i s t r i b u t i o n . The e f f e c t i s the appearance of a d.c. v o l t a g e i n the d i r e c t i o n of p r o p a g a t i o n of a t r a v e l l i n g a c o u s t i c a l wave. As f a r as i n t e r v a l l e y t r a n s i t i o n s are concerned, the ' r e l e v a n t q u a n t i t y determined from the a c o u s t o e l e c t r i c e f f e c t i s the i n t e r v a l l e y t r a n s i t i o n time,t,„ , which the authors d e f i n e from the equation, o _ a ( n , - n j where K i a . i s the net r a t e of e n t r y of e l e c t r o n s i n t o group 2 from group 1. Here, each of the two groups c o n s i s t s o f two v a l l e y s , and Y\i i s the number of e l e c t r o n s per u n i t volume i n group i . The above r e l a t i o n a l l o w s t h e d e t e r m i n a t i o n of , the p r o b a b i l i t y per u n i t t i m e of an e l e c t r o n making a t r a n s i t i o n from group i t o group j , where groups i and j each c o n s i s t of two v a l l e y s , f o r the e q u i l i b r i u m case when = I n terms of \>*j > S i n c e , i n e q u i l i b r i u m , Y\t=-*\x t h e n = J?a* Hence R.a = \L (.n,-"*") E q u a t i n g R>% t o W e i n r e i c h ' s e x p r e s s i o n g i v e s Eor the case o f one v a l l e y i n one group and t h r e e i n the .0 o t h e r , we may a l s o determine the f^'s , denoted now by \>°? f o r t h i s case t o a v o i d c o n f u s i o n w i t h t h e p r e v i o u s case o f two v a l l e y s i n each group. I n t h i s c a s e , t h e g r o s s r a t e o f e n t r y o f , e l e c t r o n s t o t h e second group of t h r e e v a l l e y s from t h e o n e - v a l l e y f i r s t group i s , a t e q u i l i b r i u m where n i s the t o t a l f r e e e l e c t r o n c o n c e n t r a t i o n . E q u a t i n g t h i s t o the g r o s s r a t e from group 2 t o group 1: ft - s Kl C o n s i d e r i n g a s i n g l e v a l l e y i n the case of two groups,each c o n t a i n i n g two v a l l e y s , a t e q u i l i b r i u m the g r o s s r a t e from t h e o n e v a l l e y t o a l l t h e o t h e r v a l l e y s i s s i n c e *I i s t h e r a t e t o t h e t w o v a l l e y s o f t h e A- ' s e c o n d g r o u p . H e n c e , s u b s t i t u t i n g f o r t h e e x p r e s s i o n s o b t a i n e d f o r t h e g r o s s r a t e i n t e r m s o f \ ^ , g i v e s , a n d , i n t e r m s o f % v , \>°' - i -M e a s u r e d . v a l u e s o f t h e i n t e r v a l l e y t r a n s i t i o n r a t e , R , e q u a l t o , f o r t h e c a s e o f a n t i m o n y a s t h e i m p u r i t y h a v e b e e n o b t a i n e d b y ^ e i n r e i c h a n d T e l l , a n d r e p o r t e d b y ' P r i c e a n d H a r t m a n ( 1 9 6 - 4 ) . T h e i r v a l u e o f t h e d o n o r c o n t r i b u t i o n , Rjun«- t o R i s : f o r t h e : l a t t i c e t e m p e r a t u r e , T u , i n t h e r a n g e f r o m 2 0 " K t o 1 0 0 * K . N t i s t h e d o n o r d e n s i t y , w h i c h i n t h e p r e s e n t e x p e r i m e n t w a s , t h e o r d e r o f 1 0 c m . H e n c e , f o r t h i s e x p e r i m e n t Rjto*»»r = A'0& $l<? F r o m W e i n r e i c h e t a l ( 1 9 5 9 ) , t h e p h o n o n c o n t r i b u t i o n a t 7 7 * K , r e a d f r o m h i s g r a p h s i s S i n c e t h e d o n o r c o n t r i b u t i o n i s s m a l l a n d i s e x p e c t e d t o d e c r e a s e a t h i g h f i e l d s w h e n t h e v a l l e y s a r e a t e l e c t r o n t e m p e r a t u r e s h i g h e r t h a n t h e l a t t i c e t e m p e r a t u r e , t h e d o n o r c o n t r i b u t i o n w a s n e g l e c t e d a l t o g e t h e r . T h e z e r o f i e l d v a l u e o f t h e i n t e r v a l l e y r a t e t h e n i s t a k e n a s t h e p h o n o n c o n t r i b u t i o n a l o n e . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085358/manifest

Comment

Related Items