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UBC Theses and Dissertations

A theoretical and experimental investigation of metal-semiconductor contacts Horita, Robert Eiji 1962

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A T H E O R E T I C A L AND 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 OF M E T A L - S E M I C O N D U C T O R CONTACTS b y ROBERT. E I J I HORITA 3 . A . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , i 9 6 0 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 OF THE R E Q U I R E M E N T S FOR THE DEGREE OF MASTER OF A P P L I E D S C I E N C E i n t h e D e p a r t m e n t o f P H Y S I C S We a c o 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 OF B R I T I S H C O L U M B I A S e p t e m b e r , 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PHYSICS The University of British Columbia, Vancouver 8, Canada. Date September 25. 1962 ABSTRACT Metal-semiconductor contacts have been known empir i -c a l l y to obey a modified diode equation 1 = 1 (exp qV/akT - 1) a where the parameter ' a ' often took values greater than two. However, previous theor ies , which are b r i e f l y summarized i n the thes i s , could not simply account for the anomaly where ' a ' i s greater than two. Previous theories are extended by considering one-dimensional b ipo lar flow of c a r r i e r s and neglect ing recombi-nat ion i n a homogeneous.semiconductor fi lament with a r e c t i f y i n g and an ohmic contact at opposite ends. The zero -e l ec tron-current theory by Borneman et . a l . (1955) I s extended to high i n j e c t i o n l eve l s by using the junct ion r e l a t i o n s of Misawa (1955)* Then the non-zero-e lectron-current theory i s developed. This theory shows that 'a ' i s unity for low In jec t ion Into e x t r i n s i c semiconductors and that a = (3b-M)/(b-M) for a r b i t r a r y i n j e c t i o n Into i n t r i n s i c semiconductors and, for high i n j e c t i o n Into e x t r i n s i c semiconductors, where 'M' Is the e lec tron to hole current r a t i o and 'b' i s the e lectron to hole mobi l i ty r a t i o . -Thus, ' a ' can take any value depending on the magnitude of M/b. To check the non-zero-e lectron-current theory, exper i -ments were performed only on n-type germanium. Rect i fy ing metal-semiconductor contacts were made by e l e c t r o p l a t i n g copper and rhodium on germanium and ohmic contacts were made by a l l o y i n g antimony-doped gold wire In a ni trogen atmosphere. A side-arm probe adjacent to the plated-contact measured the voltage across the contact . i l l F o r t h e r h o d i u m - c o n t a c t s p e c i m e n , t w o s i d e - a r m s a d j a c e n t t o t h e r h o d i u m c o n t a c t o n o p p o s i t e s i d e s o f t h e g e r m a n i u m f i l a m e n t w e r e u s e d t o m e a s u r e t h e t r a n s v e r s e a - c r e s i s t a n c e a s a f u n c t i o n o f l o n g i t u d i n a l d - c c u r r e n t . T h i s m e a s u r e m e n t d e m o n s t r a t e d t h e o c c u r r e n c e o f i n j e c t i o n a n d • 1 e x t r a c t i o n a t t h e c o n t a c t a n d t h a t t h e l e v e l o f i n j e c t i o n was f r o m m o d e r a t e t o h i g h i n t h e v i c i n i t y o f t h e c o n t a c t f o r l o n g i t u d i n a l c u r r e n t d e n s i t y b e t w e e n a b o u t 10 mA/cm a n d 1000 mA/cra . C o m p a r i s o n o f t h e I - V m e a s u r e m e n t s o f t h e c o n t a c t s w i t h t h e n o n - z e r o - e l e c t r o n - c u r r e n t t h e o r y i n d i c a t e d t h a t a n i n t e r m e d i a t e l e v e l o f I n j e c t i o n o c c u r r e d a t t h e r h o d i u m c o n t a c t 2 f o r l o n g i t u d i n a l c u r r e n t d e n s i t y b e t w e e n a b o u t 0.1 mA/cm a n d 2 100 mA/cm a n d t h a t h i g h i n j e c t i o n o c c u r r e d , w i t h M = 4.94, a t 2 t h e c o p p e r c o n t a c t f o r c u r r e n t d e n s i t y b e t w e e n a b o u t 1 mA/cm 2 a n d 50 mA/cm . T h u s , t h e l e v e l o f I n j e c t i o n c a n b e c a l c u l a t e d b y c o m p a r i s o n o f t h e e x p e r i m e n t a l v a l u e o f ' a ' w i t h t h e n o n -z e r o - e l e c t r o n - c u r r e n t t h e o r y . Iv CONTENTS p a g e C H A P T E R 1. I N T R O D U C T I O N 1 1.1 H i s t o r i c a l R e v i e w 1 1.2 P u r p o s e o f t h i s T h e s i s 2 C H A P T E R 2. T H E O R I E S FOR M E T A L - S E M I C O N D U C T O R CONTACTS 4 2.1 U n i p o l a r T h e o r i e s 4 2.2 B i p o l a r T h e o r i e s 10 2.3 Z e r o - E l e c t r o n - C u r r e n t T h e o r y 14 2.4 E x t e n s i o n o f t h e Z e r o - E l e c t r o n - C u r r e n t T h e o r y t o A r b i t r a r y I n j e c t i o n L e v e l s 18 2.5 N o n - Z e r o - E l e c t r o n - C u r r e n t T h e o r y 22 2.51 E x t r i n s i c S e m i c o n d u c t o r s 22 2.52 I n t r i n s l o S e m i c o n d u c t o r s 29 C H A P T E R H' E X P E R I M E N T A L T E C H N I Q U E S 32 3.1 G e o m e t r y o f S p e c i m e n s 32 3.2 P r e p a r a t i o n o f S p e c i m e n s 33 3.3 C i r c u i t s f o r I - V M e a s u r e m e n t s 36 3.31 D-C M e a s u r e m e n t s j6 3.32 P u l s e M e a s u r e m e n t s 37 3.4 T r a n s v e r s e A - C R e s i s t a n c e M e a s u r e m e n t 38 C H A P T E R 4. E X P E R I M E N T A L R E S U L T S AND I N T E R P R E T A T I O N 39 4.1 T r a n s v e r s e A - C R e s i s t a n c e M e a s u r e m e n t . 39 4.2 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 o f R h o d i u m -C o n t a c t S p e c i m e n 4 4 4.3 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 o f C o p p e r -C o n t a c t S p e c i m e n 4 6 V 4.4 Comparison o f E x p e r i m e n t a l R e s u l t s w i t h Theory 47 CHAPTER 5,. CONCLUSIONS 49 BIBLIOGRAPHY 51 ) v i I L L U S T R A T I O N S F i g u r e F a c i n g P a g e i 2.1 T h e M o t t B a r r i e r 5 2.2 F o r m a t i o n o f S c h o t t k y B a r r i e r 6 2.3 E q u i l i b r i u m e l e c t r o n - e n e r g y d i a g r a m s o f m e t a l t o s e m i c o n d u c t o r c o n t a c t s 8 2.4 S u r f a c e - S t a t e B a r r i e r 10 2.5 E q u i l i b r i u m e l e c t r o n - e n e r g y d i a g r a m o f a P-N j u n c t i o n > 11 2.6 E l e c t r o n i c - e n e r g y d i a g r a m o f a m e t a l - n + - n c o n t a c t 14 2.7 M o d e l f o r m e t a l - s e m i c o n d u c t o r c o n t a c t s 15 2.8 T h e d e p e n d e n c e o f ' a ' o n t h e e l e c t r o n t o h o l e c u r r e n t r a t i o d i v i d e d b y t h e e l e c t r o n t o h o l e m o b i l i t y r a t i o f o r t h e i n t r i n s i c c a s e 29 2.9 T h e d e p e n d e n c e o f t h e s a t u r a t i o n c u r r e n t d e n s i t y , J s i ' o n M f o r t h e i n t r i n s i c c a s e 30 2.10 T h e d e p e n d e n c e o n M / b o f t h e s i g n s o f c u r r e n t s a n d v o l t a g e s w h e n t o t a l c u r r e n t J i s p o s i t i v e i n a n i n t r i n s i c s e m i c o n d u c t o r 31 3.1 E q u i p o t e n t l a l c o n f i g u r a t i o n d i s t o r t e d by s i d e -a r m p r o b e 33 3.2 G e o m e t r y o f S p e c i m e n s 3^ 3.3 C i r c u i t s f o r I - V m e a s u r e m e n t s 36 3.4 T y p i c a l P u l s e F o r m s 37 3.5 T r a n s v e r s e a - c r e s i s t a n c e m e a s u r e m e n t o f c o n d u c -t i v i t y m o d u l a t i o n 38 T r a n s v e r s e a - c r e s i s t a n c e o f r h o d i u m - c o n t a c t s p e c i m e n f o r t h r e e d e c a d e s o f l o n g i t u d i n a l c u r r e n t T r a n s v e r s e a - c r e s i s t a n c e o f r h o d i u m - c o n t a c t s p e c i m e n a s a f u n c t i o n o f l o n g i t u d i n a l c u r r e n t D-C r e v e r s e c h a r a c t e r i s t i c o f r h o d i u m - c o n t a c t s p e c i m e n D-C f o r w a r d c h a r a c t e r i s t i c o f r h o d i u m - c o n t a c t s p e c i m e n G r a p h i c a l a n a l y s i s o f c h a r a c t e r i s t i c f o r r h o d i u m - c o n t a c t s p e c i m e n P u l s e r e v e r s e c h a r a c t e r i s t i c o f c o p p e r - c o n t a c t s p e c i m e n P u l s e f o r w a r d c h a r a c t e r i s t i c o f c o p p e r - c o n t a c t s p e c i m e n G r a p h i c a l a n a l y s i s o f c h a r a c t e r i s t i c f o r c o p p e r c o n t a c t s p e c i m e n ACKNOWLEDGMENT I w i s h t o t h a n k P r o f e s s o r R . E . B u r g e s s , w s u p e r v i s e d t h e r e s e a r c h r e p o r t e d i n t h i s t h e s i s , f o r h e l p f u l c r i t i c a l d i s c u s s i o n s a n d s u g g e s t i o n s . T h e r e s e a r c h w a s f i n a n c e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l I n t h e f o r m o f a B u r s a r y a n d b y t h e U n i t e d S t a t e s A i r F o r c e G r a n t AFOSR 65 - 0 2 4 0 . 1 C H A P T E R 1 I N T R O D U C T I O N 1.1 H i s t o r i c a l R e v i e w M e t a l - s e m i c o n d u c t o r c o n t a c t s h a v e b e e n t h e o b j e c t o f m u c h t h e o r e t i c a l a n d e x p e r i m e n t a l s t u d y e v e r s i n c e t h e o b s e r -v a t i o n o f t h e i r n o n - o h m i c b e h a v i o u r . B e f o r e S h o c k l e y ' s (1949) c l a s s i c t r e a t m e n t o f p - n j u n c t i o n s , t h e e x i s t i n g t h e o r i e s , a l t h o u g h c o n t a i n i n g many f e a t u r e s i n common w i t h m o d e r n t h e o r i e s , c o n s i d e r e d o n l y c o n d u c t i o n e l e c t r o n s g i v i n g r i s e t o c u r r e n t f l o w . T h e s e t h e o r i e s a s s u m e d a p o t e n t i a l b a r r i e r t o e x i s t a t t h e 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 . E l e c t r o n t u n n e l i n g t h r o u g h t h i s p o t e n t i a l b a r r i e r a t t h e J u n c t i o n was i n d e p e n d e n t l y p r o p o s e d b y W i l s o n (1932) a n d N o r d h e i m (1932) a s t h e m e c h a n i s m f o r r e c t i f i c a t i o n . B u t D a v y d o v (1938) s h o w e d t h a t t u n n e l i n g g a v e t h e d i r e c t i o n o f r e c t i -f i c a t i o n o p p o s i t e t o t h a t u s u a l l y o b s e r v e d a n d s u b s e q u e n t l y I n t e r e s t i n t u n n e l i n g a s t h e p r i m a r y means o f c u r r e n t t r a n s p o r t a c r o s s t h e c o n t a c t a l m o s t d i s a p p e a r e d ( a l t h o u g h i t was r e c o g n i z e d t h a t t u n n e l i n g was r e l e v a n t t o t h i n l a y e r s — e . g . H o l m 1951) u n t i l E s a k i ' s (1958) r e c e n t d e m o n s t r a t i o n o f I t s i m p o r t a n c e i n t h i n , h e a v i l y - d o p e d j u n c t i o n s . N e x t , M o t t (1938) p r o p o s e d a n d S c h o t t k y ( 1 9 3 9 , 19^2) f u r t h e r d e v e l o p e d a t h e o r e t i c a l m o d e l w h i c h c o n -s i d e r e d e l e c t r o n s s u r m o u n t i n g t h e p o t e n t i a l b a r r i e r by t h e r m a l e x c i t a t i o n . B u t t h e f o r e g o i n g m o d e l s w e r e u n s a t i s f a c t o r y b e c a u s e t h e y d e p e n d e d c r i t i c a l l y o n t h e d i f f e r e n c e s b e t w e e n t h e t h e r m i o n i c w o r k f u n c t i o n s o f t h e m e t a l a n d t h e s e m i c o n d u c t o r w h e r e a s e x p e r i -m e n t s o n s i l i c o n a n d g e r m a n i u m s h o w e d no o b v i o u s c o r r e l a t i o n b e t w e e n t h e d i f f e r e n c e s i n w o r k f u n c t i o n s a n d t h e r e c t i f y i n g p r o p e r t i e s o f v a r i o u s m e t a l - s e m i c o n d u c t o r c o n t a c t s . A s t u d y o f t h i s n o n - c o r r e l a t i o n l e d B a r d e e n t o p r o p o s e t h e s u r f a c e s t a t e t h e o r y w h i c h l e d h i m a n d B r a t t a i n ( 1 9 4 8 ) t o t h e d i s c o v e r y o f c a r r i e r I n j e c t i o n . S h o c k l e y ' s ( 1 9 4 9 ) d e f i n i t i v e a r t i c l e f o l l o w e d s h o r t l y . S u b s e q u e n t t h e o r e t i c a l w o r k s h a v e g i v e n e x t e n s i o n s a n d m o d i f i c a t i o n s t o S h o c k l e y ' s a r t i c l e . T h o s e r e l e v a n t t o t h i s t h e s i s h a v e b e e n g i v e n b y J M l s a w a ( 1 9 5 5 ) a n d B o r n e m a n , S c h w a r z , a n d S t i c k l e r ( 1 9 5 5 ) . R e f e r e n c e s t o o t h e r w o r k a n d a d e t a i l e d r e v i e w o f t h e g r o w t h o f r e c t i f i c a t i o n t h e o r i e s c a n b e f o u n d i n a t e x t b y H e n l s c h ( 1 9 5 7 ) . 1 . 2 P u r p o s e o f t h i s T h e s i s U n d e r s t a n d i n g o f c o n t a c t s i s h a m p e r e d b y t h e i r I n h e r e n t u n c e r t a i n t y o f p r e p a r a t i o n a n d n o r i - r e p r o d u c i b i l l t y d u e t o t h e i m p o r t a n c e o f t h e n a t u r e o f t h e s e m i c o n d u c t o r s u r f a c e o n t h e p r o p e r t i e s o f t h e c o n t a c t ( H a r r l c k 1 9 5 9 ) . F u r t h e r m o r e , t h e c o n -t a c t s a r e n o t a l w a y s s t a b l e , c h a n g i n g t h e i r p r o p e r t i e s w h e n a l a r g e c u r r e n t i s p a s s e d t h r o u g h t h e m . T h e l a c k o f u n d e r s t a n d i n g o f m e t a l - s e m i c o n d u o t o r c o n t a c t s i s r e f l e c t e d I n t h e i r u n s a t i s -f a c t o r y t h e o r i e s . B e c a u s e o f t h e c o m p l e x i t y o f t h e p r o b l e m I n u n d e r s t a n d i n g c o n t a c t s , t h e w o r k p r e s e n t e d I n t h i s t h e s i s a t t e m p t e d t o make a s many v a l i d s i m p l i f y i n g a s s u m p t i o n s a s p o s s i b l e a n d t o p r o v i d e ' t h e s i m p l e s t e x p e r i m e n t a l c o n d i t i o n s a n d t e c h n i q u e s . H e n c e , t h e t h e o r y p r e s e n t e d i n t h i s t h e s i s w i l l a s s u m e o n e - d l m e n s l o n a l f l o w o f c a r r i e r s t h r o u g h a h o m o g e n e o u s s e m i c o n d u c t o r f i l a m e n t w i t h t h e c o n t a c t u n d e r s t u d y a n d a n o h m i c c o n t a c t a t o p p o s i t e e n d s . I t w i l l n e g l e c t t h e e f f e c t s o f t r a p p i n g , t u n n e l i n g , r e c o m b i n a t i o n , i m a g e e f f e c t s , f i e l d - d e p e n d e n t m o b i l i t y , 3 a v a l a n c h e b r e a k d o w n , f i e l d e m i s s i o n , d e g e n e r a c y , n o n - I s o t h e r m a l c o n d i t i o n s , a n d p h o t o - e l e c t r i c e f f e c t s . To c o n f o r m w i t h t h e t h e o r e t i c a l a s s u m p t i o n s , o n l y a r e a c o n t a c t s o n g e r m a n i u m w e r e s t u d i e d a n d t h e s p e c i m e n s w e r e p l a c e d i n a l i g h t - t i g h t c o n s t a n t t e m p e r a t u r e o i l b a t h . T h e r e f o r e , t h e p u r p o s e o f t h i s t h e s i s I s t o p r e s e n t a b r i e f summary o f m e t a l - s e m i c o n d u c t o r c o n t a c t t h e o r i e s , t o d e v e l o p a n e x t e n s i o n t o o n e o f t h e p r e s e n t t h e o r i e s , a n d , i n a n a t t e m p t t o v e r i f y t h i s e x t e n s i o n , t o o b t a i n e x p e r i m e n -t a l d a t a o n m e t a l - s e m i c o n d u c t o r c o n t a c t s . * CHAPTER 2 THEORIES FOR METAL-SEMICONDUCTOR CONTACTS 2 . 1 U n i p o l a r T h e o r i e s Before Che d i s c o v e r y of c a r r i e r I n j e c t i o n ( d e f i n e d I n s e c t i o n 2.2), t h e o r i e s developed to e x p l a i n the phenomenon of r e c t i f i c a t i o n at metal-semiconductor c o n t a c t s c o n s i d e r e d mobile charges of one s i g n o n l y . Although much of the work now c o n s i d e r s c a r r i e r s of both s i g n s , u n i p o l a r t h e o r i e s are by no means o b s o l e t e f o r they are v a l i d f o r n o n - i n j e c t i o n semiconductors and are v a l i d approximations f o r other semiconductors i n s p e c i a l cases such as i n h eavily-doped e x t r i n s i c semiconductors or at low r e l a t i v e temperatures of o p e r a t i o n . Semiconductors l i k e cuprous oxide, t i t a n i u m d i o x i d e , and selenium have g i v e n no evidence of c a r r i e r i n j e c t i o n whereas other semiconductors l i k e germanium, s i l i c o n , and compounds of the group I I I and V elements of the p e r i o d i c t a b l e e x h i b i t pronounced c a r r i e r i n j e c t i o n . T h e r e f o r e , the Important f e a t u r e s of the u n i p o l a r t h e o r i e s proposed by Wilson ( 1 9 3 2 ) , Mott ( 1 9 3 9 ) , and Schottky ( 1 9 3 9 , 1 9 ^ 2 ) w i l l be b r i e f l y d e s c r i b e d . Wilson c o n s i d e r e d a p o t e n t i a l energy hump f o r e l e c t r o n s e x i s t i n g between the metal and the semiconductor. He then o b t a i n e d e x p r e s s i o n s f o r the number of e l e c t r o n s In the semi-conductor h i t t i n g u n i t area of the p o t e n t i a l b a r r i e r per second with t h e i r v e l o c i t i e s i n the d i r e c t i o n p e r p e n d i c u l a r to the b a r r i e r and l y i n g w i t h i n an elemental v e l o c i t y range. M u l t i p l i -c a t i o n of these e x p r e s s i o n s by the t r a n s m i s s i o n c o e f f i c i e n t of the p o t e n t i a l hump and I n t e g r a t i o n over the v e l o c i t y ranges gave e l e c t r o n e n e r g y c o n d u c t i o n b a n d d o n o r s SEMICONDUCTOR / / / / / / a l e n c e b a n d d i s t a n c e F i g u r e 2 . 1 The M o t t B a r r i e r — A n a p p l i e d f o r w a r d v o l t a g e V r a i s e s t h e c o n d u c t i o n a n d v a l e n c e b a n d by q V , b u t t h e b a r r i e r t h i c k n e s s r e m a i n s t h e s a m e . T h e e l e c t r i c f i e l d a c r o s s t h e b a r r i e r ( g i v e n by s l o p e o f l i n e ) i s u n i f o r m i n t h e b a r r i e r . 5 t h e n u m b e r o f e l e c t r o n s p e r u n i t a r e a p a s s i n g f r o m t h e s e m i -c o n d u c t o r i n t o t h e m e t a l p e r s e c o n d . T h e n u m b e r o f e l e c t r o n s p a s s i n g i n t h e r e v e r s e d i r e c t i o n was s i m i l a r l y o b t a i n e d a n d t h e d i f f e r e n c e o f t h e t w o o p p o s i t e l y m o v i n g e l e c t r o n f l o w s g a v e t h e n e t e l e c t r o n f l o w a c r o s s t h e m e t a l - s e m i c o n d u c t o r c o n t a c t . How-e v e r , t u n n e l i n g was s h o w n t o b e o f s e c o n d a r y I m p o r t a n c e i n o r d i n a r y m e t a l - s e m i c o n d u c t o r c o n t a c t s w h e n D a v y d o v (1938) p r o v e d t h a t i t g a v e t h e d i r e c t i o n o f r e c t i f i c a t i o n o p p o s i t e t o t h a t u s u a l l y o b s e r v e d i n m e t a l - s e m i c o n d u c t o r d i o d e s . S h o r t l y a f t e r W i l s o n ' s t h e o r y was s h o w n t o g i v e t h e i n v e r s e d i r e c t i o n o f r e c t i f i c a t i o n , M o t t (1939) a n d S c h o t t k y (1939, 19^2) d e v e l o p e d s i m p l e t h e o r i e s f o r t h e c o n t a c t b e t w e e n a m e t a l a n d a s e m i c o n d u c t o r . M o t t p r o p o s e d t h a t t h e r e e x i s t s a t t h e c o n t a c t a b a r r i e r d e p l e t e d o f d o n o r l e v e l s w h i c h i s s u f f i c i e n t l y h i g h s o t h a t t h e e l e c t r i c f i e l d i s u n i f o r m i n t h e b a r r i e r a n d t h e b a r r i e r t h i c k n e s s 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 a p p l i e d v o l t a g e ( F i g u r e 2.1). T h e S c h o t t k y b a r r i e r , h o w e v e r , h a s a v o l t a g e - d e p e n d e n t t h i c k n e s s . To e x p l a i n t h e n a t u r e o f t h e S c h o t t k y b a r r i e r , c o n s i d e r a n i d e a l c o n t a c t b e t w e e n a m e t a l o f w o r k f u n c t i o n 0m a n d a n n - t y p e s e m i c o n d u c t o r w i t h a n e l e c t r o n a f f i n i t y % a n d w o r k f u n c t i o n 0 s < 0 m « N o t e f r o m F i g u r e 2.2 t h a t t h e e l e c t r o n a f f i n i t y a n d ' t h e s e m i c o n d u c t o r w o r k f u n c t i o n a r e r e l a t e d by 0 g - X = E c - E p w h e r e E Q i s t h e e l e c t r o n e n e r g y a t t h e b o t t o m o f t h e c o n d u c t i o n b a n d a n d E p i s t h e F e r m i l e v e l . Now c o n s i d e r a n I d e a l c a s e w h e r e a m e t a l a n d s e m i c o n d u c t o r w i t h I n f i n i t e p l a n e s u r f a c e s a p p r o a c h e a c h o t h e r a n d come i n t o c l o s e c o n t a c t . F i g u r e 2.2a s h o w s t h e e q u i l i b r i u m e n e r g y p r o f i l e o f t h e s y s t e m b e f o r e c o n t a c t h a s o c c u r r e d w h e r e e l e c t r o n e n e r g y I s 0ffl G a p 0 m V////// G a p -e- -e- -e-( a ) M e t a l a n d s e m i c o n d u c t o r I n e q u i l i b r i u m s e p a r a t e d b e f o r e c o n t a c t ( b ) E q u i l i b r i u m a f t e r c l o s e a p p r o a c h ( c ) E q u i l i b r i u m i n c l o s e c o n t a c t F i g u r e 2 . 2 F o r m a t i o n o f S c h o t t k y B a r r i e r p l o t t e d v e r t i c a l l y a n d a p o s i t i o n c o - o r d i n a t e h o r i z o n t a l l y . . T h e F e r m i l e v e l s a r e a l i g n e d s i n c e t h e t w o m a t e r i a l s a r e i n t h e r m a l e q u i l i b r i u m ( C . H e r r i n g a n d M . H . N i c h o l s 1 9 4 9 ) . I t s h o u l d b e m e n t i o n e d t h a t some a u t h o r s a r b i t r a r i l y c o n s i d e r a n o n - e q u l l l b r l u m s i t u a t i o n i n w h i c h t h e v a c u u m l e v e l s o f t h e m e t a l a n d s e m i c o n d u c t o r a r e c o i n c i d e n t . B e c a u s e t h i s s i t u a t i o n i s v e r y a r b i t r a r y a n d h a s n o i m p o r t a n t f e a t u r e d i f f e r e n t f r o m o t h e r n o n - e q u i l i b r i u m s i t u a t i o n s , l t i s i r r e l e v a n t i n e n e r g y b a n d d i a g r a m s d e m o n s t r a t i n g t h e f o r m a t i o n o f a S c h o t t k y b a r r i e r . Some a u t h o r s a l s o s t a t e t h a t b e f o r e e q u i l i b r i u m c a n b e e s t a b l i s h e d b e t w e e n t h e m e t a l a n d t h e s e m i c o n d u c t o r s o t h a t t h e F e r m i l e v e l s i n t h e t w o m a t e r i a l s a r e a l i g n e d a n d a c o n t a c t p o t e n t i a l ( d i f f e r e n c e o f t h e t h e r m i o n i c w o r k f u n c t i o n s ) e x i s t s I n ' t h e g a p b e t w e e n t h e i n f i n i t e p l a n e f a c e s , c o n t a c t m u s t b e made b e t w e e n t h e m e t a l a n d t h e s e m i c o n d u c t o r a t some o t h e r s u r f a c e . T h i s i s i n c o r r e c t . T h e r m a l e q u i l i b r i u m , a n d n o t p h y s i c a l c o n t a c t , i s t h e n e c e s s a r y •> a n d s u f f i c i e n t c o n d i t i o n f o r c o i n c i d e n t F e r m i l e v e l s . A s t h e t w o i n f i n i t e p l a n e s u r f a c e s o f t h e m e t a l a n d s e m i c o n d u c t o r a p p r o a c h e a c h o t h e r , t h e m e t a l a c q u i r e s a n e g a t i v e s u r f a c e c h a r g e w h i l e t h e s e m i c o n d u c t o r c h a r g e s u p a n e q u a l a m o u n t p o s i t i v e l y . B e c a u s e o f t h e l o w d e n s i t y o f d o n o r s I n t h e s e m i c o n d u c t o r , t h e d o n o r s w i l l b e c o m e I o n i z e d o v e r a r e g i o n w h i c h e x t e n d s w e l l i n t o t h e s e m i c o n d u c t o r . T h u s , a s p a c e - c h a r g e l a y e r i s f o r m e d . As t h e g a p d i m i n i s h e s t o l n t e r - a t o m l c d i s t a n c e s , l t b e c o m e s t r a n s -p a r e n t t o e l e c t r o n s . M e a n w h i l e , t h e s p a c e c h a r g e i n t h e s e m i -c o n d u c t o r i n c r e a s e s a n d 0 ( s e e F i g u r e 2 . 2 b ) a p p r o a c h e s t h e l i m i t i n g v a l u e o f 0ffl - X , w h i c h I s t h e h e i g h t o f t h e c o n t a c t b a r r i e r ( F i g u r e 2 . 2 c ) . C o n t a c t s t o p - t y p e s e m i c o n d u c t o r s c a n a l s o b e a n a l y z e d I n a s i m i l a r m a n n e r . F r o m P o i s s o n ' s e q u a t i o n a n d t h e a s s u m p t i o n t h a t a l l t h e d o n o r l e v e l s a r e I o n i z e d I n t h e s p a c e - c h a r g e r e g i o n , t h e t h i c k n e s s o f t h e S c h o t t k y b a r r i e r , x Q , I s f o u n d t o v a r y a s t h e s q u a r e r o o t o f t h e d i f f e r e n c e I n w o r k f u n c t i o n s o f t h e m e t a l a n d s e m i c o n d u c t o r : x o \]K ~ *s N o t e t h a t t h e t h i c k n e s s o f t h e s p a c e - c h a r g e l a y e r , x 0 , d e p e n d s o n 0 m - 0 S r a t h e r t h a n o n 0 m - X b e c a u s e l t i s t h e p o t e n t i a l e n e r g y c h a n g e o f a n e l e c t r o n t r a v e r s i n g t h e s p a c e - c h a r g e l a y e r w h i c h I s i n v o l v e d i n t h e d e r i v a t i o n o f x Q . I f a n e x t e r n a l l y a p p l i e d v o l t a g e , V , c h a n g e s t h e p o t e n t i a l d r o p a c r o s s t h e b a r r i e r b y q V , w h e r e q i s t h e e l e c t r o n i c c h a r g e , t h e t h i c k n e s s o f t h e b a r r i e r c h a n g e s . I t i n c r e a s e s i f a n n - t y p e s e m i c o n d u c t o r i s b i a s e d p o s i t i v e l y a n d d e c r e a s e s i f l t i s b i a s e d n e g a t i v e l y w i t h r e s p e c t t o t h e m e t a l . T h u s ( T h e r e v e r s e I s t r u e f o r a c o n t a c t o n a p - t y p e s e m i c o n d u c t o r . ) W i t h b o t h t h e M o t t a n d t h e S c h o t t k y b a r r i e r s , c u r r e n t -v o l t a g e r e l a t i o n s may be o b t a i n e d b y e i t h e r t h e d i f f u s i o n o r t h e d i o d e t h e o r y . T h e d i f f u s i o n t h e o r y a s s u m e s t h e t h i c k n e s s o f t h e b a r r i e r l a y e r t o b e l a r g e c o m p a r e d w i t h t h e mean f r e e p a t h f o r s c a t t e r i n g o f e l e c t r o n s by l a t t i c e v i b r a t i o n s . T h e d i o d e t h e o r y ( B e t h e 19^2) n e g l e c t s c o l l i s i o n s w i t h i n t h e b a r r i e r r e g i o n a n d c o n s i d e r s two t h e r m i o n i c e m i t t e r s f a c i n g e a c h o t h e r . T h e s e t h e o r i e s g i v e t h e I - V r e l a t i o n M e t a l — • — • — • E p N - t y p e S e m i c o n d u c t o r 777777 P - t y p e S e m i c o n d u c t o r . . E p M e t a l ( a ) 0 m > 0 S ( b ) 0 m < 0 s 7 M e t a l E t S e m i c o n d u c t o r ( c ) 0 = 0 v ' *m v s V,. / M e t a l — E r N - t y p e S e m i c o n d u c t o r P - t y p e S e m i c o n d u c t o r ( d ) 0 < 0 s ' v m ( e ) 0 > 0 o m s F i g u r e 2.3 E q u i l i b r i u m e l e c t r o n - e n e r g y d i a g r a m s o f m e t a l t o s e m i c o n d u c t o r c o n t a c t s 8 I = I 8 ( e x p q V / k T - 1) (2.1.1) w h e r e I i s t h e c u r r e n t , V t h e v o l t a g e ( I a n d V a r e p o s i t i v e w h e n t h e ' m e t a l i s b i a s e d p o s i t i v e l y w i t h r e s p e c t t o a n n - t y p e s e m i -c o n d u c t o r o r w h e n t h e m e t a l i s b i a s e d n e g a t i v e l y w i t h r e s p e c t t o a p - t y p e s e m i c o n d u c t o r ) , q t h e e l e c t r o n i c c h a r g e , k t h e B o l t z m a n n c o n s t a n t , T t h e a b s o l u t e t e m p e r a t u r e , a n d I a c o m p l i c a t e d f u n c t i o n s I n v o l v i n g t h e p a r a m e t e r s o f t h e m e t a l a n d s e m i c o n d u c t o r . I t s h o u l d b e s t r e s s e d t h a t t h e S c h o t t k y b a r r i e r a r i s e s e i t h e r f r o m t h e c o n t a c t o f a n n - t y p e s e m i c o n d u c t o r w i t h a m e t a l o f h i g h e r w o r k f u n c t i o n o r f r o m t h e c o n t a c t o f a p - t y p e s e m i -c o n d u c t o r w i t h a m e t a l o f l o w e r w o r k f u n c t i o n . D i f f e r e n t e q u i l i b r i u m e n e r g y p r o f i l e s r e s u l t w h e n t h e w o r k f u n c t i o n o f a n n - t y p e s e m i c o n d u c t o r i s e q u a l t o o r g r e a t e r t h a n t h a t o f t h e m e t a l w i t h w h i c h l t m a k e s c o n t a c t . A p - t y p e s e m i c o n d u c t o r i n c o n t a c t w i t h a m e t a l o f h i g h e r w o r k f u n c t i o n c a n b e t r e a t e d s i m i l a r l y ( F i g u r e 2.3e). F o r t h e c a s e w h e n t h e w o r k f u n c t i o n s o f t h e m e t a l a n d s e m i c o n d u c t o r a r e e q u a l a n d t h e F e r m i l e v e l s a r e a l r e a d y m a t c h e d i n t h e n e u t r a l c o n d i t i o n , n o s p a c e - r c h a r g e l a y e r i s f o r m e d ( F i g u r e 2.3c). F u r t h e r m o r e , i f i n f i n i t e l y h i g h t h e r m a l g e n e r -a t i o n a n d r e c o m b i n a t i o n r a t e s o f c a r r i e r s a r e a s s u m e d i n t h e m e t a l w h i c h a r e a b l e t o r e - e s t a b l i s h a n y d i s t u r b e d e q u i l i b r i u m o f c a r r i e r c o n c e n t r a t i o n i n t h e s e m i c o n d u c t o r , t h e c o n c e n t r a t i o n o f e l e c t r o n s a n d h o l e s a t t h e m e t a l - s e m i c o n d u c t o r i n t e r f a c e w i l l b e f i x e d a n d i n d e p e n d e n t o f t h e c u r r e n t f l o w i n g . T h u s , t h e c a r r i e r c o n c e n t r a t i o n r e m a i n s u n c h a n g e d t h r o u g h o u t t h e s e m i -c o n d u c t o r u n d e r c u r r e n t f l o w . S u c h a c o n t a c t i s c a l l e d a n o h m i c 9 c o n t a c t ( s e e s e c t i o n 2.2). When c h e n - t y p e s e m i c o n d u c t o r ' s w o r k . f u n c t i o n e x c e e d s t h a t o f t h e m e t a l i n c o n t a c t w i t h i t , i t c a n be s e e n t h a t t h e e l e c t r o n s f l o w f r o m t h e m e t a l t o t h e s e m i c o n d u c t o r , i n t r o d u c i n g a n e g a t i v e s p a c e c h a r g e , t o e s t a b l i s h c o i n c i d e n c e o f t h e F e r m i l e v e l s o f t h e m e t a l a n d t h e s e m i c o n d u c t o r ( F i g u r e 2.3d). W i t h t h i s t y p e o f c o n t a c t a n a n a l o g y i s made w i t h a t h e r m i o n i c c a t h o d e e m i t t i n g e l e c t r o n s i n t o a v a c u u m , s e t t i n g u p a s p a c e - c h a r g e -l i m i t e d c u r r e n t ( H e l j n e i960). I n t h e v a c u u m c a s e , t h e c u r r e n t v a r i e s a s t h e 3/2 p o w e r o f t h e v o l t a g e ( C h i l d ' s L a w ) , w h i l e t h e d e p e n d e n c e i n a s e m i c o n d u c t o r i s q u a d r a t i c I n p l a n e - p a r a l l e l g e o m e t r y . I n o r d i n a r y s i t u a t i o n s t h e s p a c e - c h a r g e - l i m i t e d c u r r e n t I s n e g l i g i b l e t o t h e n o r m a l d r i f t c u r r e n t i n t h e s e m i -c o n d u c t o r , b u t i t may b e d o m i n a n t w h e n t h e c o n d u c t i v i t y i s l o w , t h e a p p l i e d f i e l d h i g h , a n d t h e s e p a r a t i o n o f t h e t w o o p p o s i t e c o n t a c t s s m a l l . H e n c e , i n p r a c t i c a l s i t u a t i o n s w h e r e t h e l e n g t h a n d c o n d u c t i v i t y o f t h e s a m p l e a r e s u f f i c i e n t l y h i g h a n d t h e f i e l d s t r e n g t h I s l o w , t h e s e c o n t a c t s a c t a s g o o d o h m i c c o n t a c t s . B e c a u s e t h e M o t t a n d S c h o t t k y b a r r i e r m o d e l s p r e d i c t e d a s t r o n g d e p e n d e n c e o f t h e c o n t a c t p r o p e r t i e s o n t h e w o r k f u n c t i o n d i f f e r e n c e b e t w e e n t h e m e t a l a n d t h e s e m i c o n d u c t o r a n d b e c a u s e 1 V c o n t a c t s o n g e r m a n i u m w e r e f o u n d t o h a v e p r o p e r t i e s i n d e p e n d e n t o f t h e w o r k f u n c t i o n o f t h e m e t a l u s e d , B a r d e e n (194?) p o s t u l a t e d t h e p r e s e n c e o f s u r f a c e s t a t e s w h i c h i m m o b i l i z e s t h e e l e c t r o n s a t t h e s e m i c o n d u c t o r s u r f a c e a n d p r o d u c e s a l a y e r o f d e p l e t e d c o n d u c t i v i t y , t h e i n v e r s i o n l a y e r , J u s t b e l o w t h e s u r f a c e o f t h e s e m i c o n d u c t o r . S u c h s u r f a c e s t a t e s w h i c h may l i e w i t h i n t h e ( b ) E q u i l i b r i u m ; t h e s u r f a c e c h a r g e i s e q u a l a n d o p p o s i t e t o t h e s p a c e c h a r g e F i g u r e 2.4 S u r f a c e - s t a t e B a r r i e r n o r m a l l y f o r b i d d e n e n e r g y g a p may b e d u e t o t h e d i s c o n t i n u i t y o f t h e c r y s t a l l a t t i c e p e r i o d i c i t y a t t h e s u r f a c e o r , p a r t l y , d u e t o a d s o r b e d a t o m s . F i g u r e 2.4a s h o w s t h e s u r f a c e s t a t e s f i l l e d u p t o t h e e n e r g y l e v e l E Q i n t h e a b s e n c e o f a n e t s u r f a c e c h a r g e . Now s i n c e t h e F e r m i l e v e l a s s o c i a t e d w i t h t h e s u r f a c e s t a t e s m u s t c o i n c i d e w i t h t h a t o f t h e b u l k m a t e r i a l u n d e r e q u i l i b r i u m c o n d i t i o n s , e l e c t r o n s f r o m t h e c o n d u c t i o n b a n d w i l l t e n d t o f i l l u p t h e s u r f a c e s t a t e s u n t i l t h e h i g h e s t f i l l e d s u r f a c e s t a t e c o i n c i d e s w i t h t h e b u l k F e r m i l e v e l ( F i g u r e 2.4b). T h u s , s p a c e - c h a r g e l a y e r s may b e p r e s e n t a t t h e f r e e s u r f a c e s o f s e m i c o n d u c t o r s s u c h a s g e r m a n i u m a n d s i l i c o n I n d e p e n d e n t o f a m e t a l c o n t a c t . E x p e r i m e n t s w h i c h w e r e p e r f o r m e d t o c o n f i r m t h e s u r f a c e s t a t e t h e o r y o f B a r d e e n l e d t o t h e d i s c o v e r y o f h o l e i n j e c t i o n . 2 . 2 B i p o l a r T h e o r i e s W i t h t h e d i s c o v e r y o f c a r r i e r i n j e c t i o n ( B a r d e e n a n d B r a t t a i n 1948) i t b e c a m e p o s s i b l e f o r b u l k c o n t a c t t h e o r i e s t o b e d e v e l o p e d . T h e p r e v i o u s o n e - c a r r i e r t h e o r i e s a s s o c i a t e d r e c t i f i c a t i o n w i t h a s p a c e - c h a r g e l a y e r w h i c h i s p a r t l y d e p l e t e d o r c o m p l e t e l y e x h a u s t e d o f m o b i l e c h a r g e c a r r i e r s i n t h e s e m i -c o n d u c t o r n e x t t o t h e m e t a l a n d t h e b u l k o f t h e s e m i c o n d u c t o r w a s p r e s u m e d t o b e o h m i c . I n b u l k c o n t a c t t h e o r i e s , p a r t o f t h e b u l k o f t h e s e m i c o n d u c t o r c o n t r i b u t e s , a l o n g w i t h t h e s p a c e -c h a r g e l a y e r , t o t h e r e c t i f i c a t i o n e f f e c t . A c l a s s i c e x a m p l e i s S h o c k l e y ' s t h e o r y f o r t h e p - n J u n c t i o n (1949, 1950). He c o n -s i d e r e d a s i m p l e o n e - d i m e n s i o n a l m o d e l w h e r e t h e p a n d n r e g i o n s a r e s e p a r a t e d b y a t r a n s i t i o n r e g i o n a s s u m e d t o b e s o n a r r o w t h a t F i g u r e 2 . 5 E q u i l i b r i u m e l e c t r o n - e n e r g y d i a g r a m o f a P - N J u n c t i o n 11 r e c o m b i n a t i o n i n i t i s n e g l i g i b l e ( F i g u r e 2.5). A s s u m i n g t h a t t h e i n j e c t e d c a r r i e r d e n s i t y i s s m a l l he o b t a i n e d t h e e x p r e s s i o n P < ° ) = P o n e x P q V / k T (2.2.1) w h e r e p ( 0 ) i s t h e d e n s i t y o f h o l e s a t t h e b o u n d a r y b e t w e e n t h e t r a n s i t i o n r e g i o n a n d t h e n - r e g l o n , p Q n i s t h e t h e r m a l e q u i l i b r i u m d e n s i t y o f h o l e s I n t h e n - r e g i o n , a n d V i s t h e a p p l i e d v o l t a g e , p o s i t i v e w h e n t h e p - r e g l o n i s b i a s e d p o s i t i v e l y w i t h r e s p e c t t o t h e n - r e g l o n . T h e c u r r e n t d e n s i t y a c r o s s t h e J u n c t i o n , J , i s g i v e n b y J = J s ( e x p q V / k T - 1) (2.2.2) w h e r e J g i s t h e s a t u r a t i o n c u r r e n t d e n s i t y g i v e n b y J s - <1 ( P 0 n V L p + n o p D n / L n > w h e r e n 0 p i s t h e t h e r m a l e q u i l i b r i u m e l e c t r o n d e n s i t y i n t h e p -r e * g l o n , a n d D n a r e t h e h o l e d l f f u s l v i t y I n t h e n - r e g i o n a n d t h e e l e c t r o n d l f f u s l v i t y i n t h e p - r e g i o n r e s p e c t i v e l y , a n d L p a n d L n a r e r e s p e c t i v e l y t h e h o l e d i f f u s i o n l e n g t h i n t h e n - r e g i o n a n d t h e e l e c t r o n d i f f u s i o n l e n g t h i n t h e p - r e g l o n . T h e d i f f u s i o n l e n g t h s a r e r e l a t e d t o t h e l i f e t i m e f o r h o l e s i n t h e n - r e g i o n , f , P a n d t h e l i f e t i m e f o r e l e c t r o n s l n t h e p - r e g i o n , * ^ n i b y L 2 = D ~f P p P 2 L = D "C n n Si E q u a t i o n (2.2.2) i s o f g r e a t I m p o r t a n c e b e c a u s e l t p r e d i c t s t h e d - c 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 l n t e r m s o f b a s i c 12 m e a s u r a b l e s e m i c o n d u c t o r p a r a m e t e r s . T h e e q u a t i o n i s r e l e v a n t t o m e t a l - s e m i c o n d u c t o r c o n t a c t s s i n c e l t i s b e l i e v e d t h a t i n many c a s e s t h e c o n t a c t s a r e e f f e c t i v e l y p - n J u n c t i o n s . A c t u a l l y , m e t a l - s e m i c o n d u c t o r c o n t a c t s h a v e b e e n f o u n d t o f o l l o w t h e I - V r e l a t i o n j I » I ( e x p q V / a k T - 1) (2.2 .3 ) s w h e r e ' a * h a s b e e n o b s e r v e d t o h a v e v a l u e s f r o m o n e t o a s h i g h a s t e n . P r e v i o u s t h e o r i e s h a v e a t t e m p t e d t o e x p l a i n t h i s a n o m a l y b y c o n s i d e r i n g t h e c o n t a c t t o b e p a t c h y , t h a t i s , t h e n a t u r e o f t h e c o n t a c t i s d i f f e r e n t f r o m p o i n t t o p o i n t . B u t i t w i l l b e s h o w n i n S e c t i o n 2.5 t h a t l t I s p o s s i b l e t o o b t a i n a n y v a l u e o f * a ' , w i t h o u t a s s u m i n g a p a t c h y c o n t a c t , i f p l a u s i b l e h y p o t h e s e s a r e m a d e . B e f o r e p r o c e e d i n g a n y f u r t h e r w i t h b i p o l a r t h e o r i e s , l t i s n e c e s s a r y t o d e f i n e v a r i o u s t e r m s r e l a t e d t o t w o - c a r r i e r t h e o r i e s . T h e f o r w a r d d i r e c t i o n o f c u r r e n t f l o w r e f e r s t o t h a t d i r e c t i o n o f e a s y c u r r e n t f l o w t h r o u g h t h e c o n t a c t w h e r e a s t h e r e v e r s e d i r e c t i o n r e f e r s t o t h a t d i r e c t i o n i n w h i c h l i t t l e c u r r e n t c a n f l o w a n d t h e c o n t a c t r e s i s t a n c e i s h i g h . F o r c o n t a c t s o n n - t y p e m a t e r i a l , t h e f o r w a r d d i r e c t i o n o f c u r r e n t f l o w o c c u r s w h e n t h e m e t a l i s b i a s e d p o s i t i v e l y w i t h r e s p e c t t o t h e s e m i -c o n d u c t o r a n d f o r p - t y p e m a t e r i a l i t o c c u r s w h e n t h e m e t a l i s b i a s e d n e g a t i v e l y w i t h r e s p e c t t o t h e s e m i c o n d u c t o r . A l a r g e i n c r e a s e i n m i n o r i t y c a r r i e r d e n s i t y w i t h f o r w a r d c u r r e n t i s k n o w n a s i n j e c t i o n a n d t h e c o n t a c t s u p p l y i n g t h e e x c e s s c a r r i e r s i s k n o w n a s a n i n j e c t i n g c o n t a c t . A d e c r e a s e i n m i n o r i t y c a r r i e r c o n c e n t r a t i o n w i t h r e v e r s e c u r r e n t I s c a l l e d e x t r a c t l o n . S i m i l a r 13 i n c r e a s e s a n d d e c r e a s e s i n m i n o r i t y c a r r i e r c o n c e n t r a t i o n , b u t w i t h t h e c u r r e n t d i r e c t i o n s r e v e r s e d f r o m t h o s e a b o v e , a r e k n o w n a s a c c u m u l a t i o n a n d e x c l u s i o n r e s p e c t i v e l y (Low 1955)• An o h m i c  c o n t a c t i s d e f i n e d a s a c o n t a c t w h o s e I-V r e l a t i o n i s n o t o n l y l i n e a r b u t h a s a c o n d u c t i v i t y g i v e n by ^ n n o +>V>o> ( 2 * 2 * 4 ) w h e r e j u n , ,Up a n d n Q , p Q a r e t h e e l e c t r o n a n d h o l e m o b i l i t i e s a n d e q u i l i b r i u m d e n s i t i e s . T h i s e q u a t i o n i m p l i e s t h a t t h e r e i s no p o t e n t i a l b a r r i e r a t t h e c o n t a c t a n d t h a t t h e s u r f a c e r e c o m b i n a -t i o n - g e n e r a t i o n r a t e o f c a r r i e r s a t t h e c o n t a c t I s i n f i n i t e s o t h a t An = Ap = 0 ( e q u a t i o n (2.3.1)) i n t h e s e m i c o n d u c t o r ( F i g u r e 2.3c). T h e r e f o r e , a n I-V r e l a t i o n w h i c h i s l i n e a r d o e s n o t i n d i -c a t e a n o h m i c c o n t a c t n e c e s s a r i l y . A r e c t i f y i n g c o n t a c t i s m e r e l y a s p e c i a l c a s e o f t h e n o n - o h m i c o n e w h e r e t h e r e s i s t a n c e i n o n e d i r e c t i o n o f c u r r e n t f l o w i s a l w a y s g r e a t e r t h a n t h a t i n t h e o t h e r d i r e c t i o n o f c u r r e n t f l o w t h r o u g h t h e c o n t a c t . O h m i c c o n t a c t s h a v e b e e n f o u n d i n p r a c t i c e t o be m o r e e a s i l y made t o s t r o n g l y e x t r i n s i c m a t e r i a l s t h a n t o h i g h r e s i s -t i v i t y o n e s . The r e a s o n f o r t h i s may b e t h a t t h e t h i c k n e s s o f t h e p o t e n t i a l b a r r i e r i s s o s m a l l t h a t i t b e c o m e s t r a n s p a r e n t t o c a r r i e r f l o w . A t t h e same t i m e , l t i s f o u n d t h a t o h m i c c o n t a c t s t o n - t y p e s e m i c o n d u c t o r s c a n o f t e n b e made w i t h a m e t a l t h a t w o u l d , i f u s e d t o d o p e a n i n t r i n s i c s e m i c o n d u c t o r , g i v e r i s e t o d o n o r l e v e l s i n t h e s e m i c o n d u c t o r , t h a t i s , make i t n - t y p e . A s i m i l a r s i t u a t i o n a r i s e s a t a c o n t a c t o f a p - t y p e s e m i c o n d u c t o r t o a n " a c c e p t o r " m e t a l . T h e r e f o r e K r o g e r , D l e m e r , a n d K l a s e n s (1956) I n f e r r e d t h a t t h e c o n t a c t m a t e r i a l a l w a y s d i f f u s e s i n t o F i g u r e 2.6 E l e c t r o n i c - e n e r g y diagram of a m e tal - n + - n c o n t a c t 14 t h e s e m i c o n d u c t o r m a k i n g i t s t r o n g l y e x t r i n s i c . T h u s , a n n+ -l a y e r w o u l d be f o r m e d ( F i g u r e 2 . 6 ) . T h e c a r r i e r s w o u l d t u n n e l t h r o u g h t h e t h i n s p a c e - c h a r g e l a y e r s o t h a t t h e c o n t a c t i s e s s e n t i a l l y l i k e t h a t o f F i g u r e 2.3d. H e n c e , a l l o y e d o r d i f f u s e d n - n + - m e t a l a n d p - p + - m e t a l c o n t a c t s a r e o f t e n f a b r i c a t e d i n d e v i c e s w h e r e o h m i c c o n t a c t s a r e r e q u i r e d . B e s i d e s t h e c o n d i t i o n t h a t n o p o t e n t i a l b a r r i e r e x i s t a t t h e c o n t a c t , g o o d o h m i c c o n t a c t s r e q u i r e I n f i n i t e g e n e r a t i o n a n d r e c o m b i n a t i o n r a t e s o f c a r r i e r s a t t h e j u n c t i o n . I n p r a c t i s e i t a p p e a r s t h e g e n e r a t i o n a n d r e c o m b i n a t i o n r a t e s may be i n c r e a s e d by t h e a d d i t i o n o f r e c o m b i n a t i o n c e n t r e s i n t h e c o n t a c t a r e a . T h i s i s a c c o m p l i s h e d by d i f f u s i o n o f t h e c o n t a c t m a t e r i a l o r by m e c h a n i c a l t r e a t m e n t o f t h e s e m i c o n d u c t o r s u r f a c e b e f o r e a p p l i c a t i o n o f t h e m e t a l . 2.3 Z e r o - E l e c t r o n - C u r r e n t T h e o r y B o r n e m a n , S c h w a r z , a n d S t i c k l e r ( 1 9 5 5 ) o f t h e P h i l c o C o r p o r a t i o n d e r i v e d c u r r e n t - v o l t a g e r e l a t i o n s f o r m e t a l - s e m i -c o n d u c t o r c o n t a c t s b a s e d o n a m o d e l h a v i n g a s u r f a c e b a r r i e r -t o e l e c t r o n s ( F i g u r e 2.7a). To d e r i v e t h e s e I - V r e l a t i o n s , t h e y made t h e f o l l o w i n g a s s u m p t i o n s . ( a ) O n l y h o l e s c a n c r o s s t h e m e t a l c o n t a c t s o t h a t t h e e l e c t r o n c u r r e n t d e n s i t y , J n , i s z e r o . ( b ) T h e l i f e t i m e o f t h e h o l e s , ¥ , I s i n f i n i t e . ( c ) T h e r e i s n o b u l k s p a c e c h a r g e so t h a t A n = n - n 0 = p - p 0 s A p (2.3 . 1 ) S p a c e -c h a r g e ' e g i o n ( a ) R e c t i f y i n g c o n t a c t O h m i c c o n t a c t ( b ) F i g u r e 2.7 M o d e l f o r m e t a l - s e m i c o n d u c t o r c o n t a c t s . ( a ) D i a g r a m o f e l e c t r o n b a r r i e r a t r e c t i f y i n g c o n t a c t o n a n n - t y p e s e m i c o n d u c t o r w i t h a n o h m i c c o n t a c t ( b ) S e m i c o n d u c t o r f i l a m e n t — o n e - d i m e n s i o n a l f l o w o f c a r r i e r s ( m a g n i t u d e o f x 0 i s e x a g g e r a t e d ) 15 where n and p are the electron and hole densities respectively and n Q and p 0 are their thermal equilibrium values. (d) The hole density at the opposite contact i s p Q. (e) In the bulk, the electron and hole current d e n s i t i e s , J n and Jp, are given by J n = ^ n n E + k T ^ n g r a d n J p = ^HpPE ~ k T u p S r a d P (2.3.2) (2.3.3) where j j p and p n are the constant m o b i l i t i e s of the holes and electrons respectively, q i s the electronic charge, k i s Boltzmann's constant, T i s the absolute temperature, and E i s the e l e c t r i c f i e l d . (f) The voltage across the b a r r i e r , V c, i s given by v c - kT/q l n P<0> Po (2.3.4) where p(0) i s the hole density at the boundary of the space-charge region i n the semiconductor. From the above assumptions, Borneman, Schwarz, and S t i c k l e r have derived I-V c h a r a c t e r i s t i c s i n generalized ortho-gonal co-ordinates. The hole current crossing the metal contact i s Ip = CkT^ p where i p = p(0) - p 0 2<$p - (n Q-p 0) In ( 1 + £p/n 0) (2.3-5) (2.3.6) - "o+Po , 2 -1 + 4ppn 0 1 + ( n 0 + P o ) ^ < e xP ^/M - 1) where V = V Q + Vg i s the t o t a l voltage across the specimen, Vg 16 i s the v o l t a g e a c r o s s the bulk r e g i o n , and C i s a constant depending only on the geometry of the diode. For a c i r c u l a r c o n t a c t of r a d i u s 1 r 1 on a s e m i - i n f i n i t e s l a b , C = kr. I f the hole flow i s one-dimensional along p o s i t i o n c o - o r d i n a t e 'x', then C = A / ( L - x Q ) where A i s the area of the co n t a c t and L - x Q i s the l e n g t h of the bulk r e g i o n ( F i g u r e 2.7b). From equations (2.3-5) and (2.3.6) < ( n G + Po) 1 + j l + 4 n 0 p 0 ( n 0 + p o r 2 ( e x p qV/kT - 1) J l + 4 n o p o ( n o + p 0 ) " 2 ( e x p qV/kT -1) J - 1 no+Po - 1 (2.3.7)-I f V i s not too l a r g e , the f o l l o w i n g l i m i t i n g cases r e s u l t . , (a) n Q / p 0 » 1 (n-type) I p = CkTju pp 0 (exp qV/kT - 1) (2.3.8) (b) n 0 / p 0 = 1 ( i n t r i n s i c ) I p = 2CkTu pp 0 (exp qV/2kT - 1) (2.3-9) (c) n 0 / p Q « 1 (p-type) I p = C q u p P o V (2.3.10) Equation (2.3.8) has the form of the us u a l diode e q u a t i o n . I t c o u l d a l s o have been d e r i v e d by assuming t h a t the hole d r i f t c u r r e n t i s n e g l i g i b l e compared to the hole d i f f u s i o n c u r r e n t i n equation (2.3.3) when we note t h a t kTu = qD P P ( E i n s t e i n r e l a t i o n ) . E quation (2.3.9) p r e d i c t s that f o r I n t r i n s i c m a t e r i a l s the s a t u r a t i o n c u r r e n t would be double t h a t f o r n-type semiconductors and that the r e c t i f i c a t i o n r a t i o 17 w o u l d b e d e c r e a s e d . By a s s u m i n g t h e h o l e d i f f u s i o n c u r r e n t i s n e g l i g i b l e c o m p a r e d t o t h e h o l e d r i f t c u r r e n t i n e q u a t i o n ( 2 . 3 . 3 ) 1 e q u a t i o n ( 2 . 3 . 1 0 ) c a n b e o b t a i n e d . I t i s a n o h m i c r e l a t i o n i n t h e s t r i c t s e n s e b e c a u s e o f t h e l i n e a r I - V r e l a t i o n a n d b e c a u s e t h e c o n d u c t i v i t y h a s t h e c o r r e c t v a l u e f o r h i g h l y e x t r i n s i c p -t y p e m a t e r i a l s . T h e r e f o r e , e q u a t i o n ( 2 . 3 . 7 ) p r e d i c t s t h a t t h e I - V c u r v e s s h o u l d v a r y i n f o r m f r o m t h a t o f a n i d e a l d i o d e t o o h m i c l i n e s a s t h e r e s i s t i v i t y v a r i e s f r o m h i g h l y e x t r i n s i c n -t y p e m a t e r i a l s t h r o u g h i n t r i n s i c t o h i g h l y e x t r i n s i c p - t y p e m a t e r i a l s . B o r n e m a n , S c h w a r z , a n d S t i c k l e r (1955) h a v e f o u n d t h a t q u a l i t a t i v e l y a n d q u a n t i t a t i v e l y , t h e i r e x p e r i m e n t a l c u r v e s b e h a v e d v e r y much a s p r e d i c t e d b y e q u a t i o n ( 2 . 3 . 7 ) e x c e p t t h a t t r i e r e c t i f i c a t i o n r a t i o was u s u a l l y s o m e w h a t l e s s t h a n p r e d i c t e d a n d t h a t , r e l a t i v e t o t h e c u r r e n t s i n t h e p - t y p e s a m p l e s , t h e c u r r e n t s i n t h e n - t y p e s a m p l e s w e r e s l i g h t l y l a r g e r t h a n e x p e c t e d t h e o r e t i c a l l y . T h e y n o t e d t h a t i f r e c o m b i n a t i o n w e r e i n c l u d e d i n t h e t h e o r y t h e t h e o r e t i c a l c u r v e s w o u l d c h a n g e l n s u c h a m a n n e r a s t o r e d u c e t h e d i s c r e p a n c i e s w i t h t h e e x p e r i -m e n t a l d a t a . H o w e v e r , t h e d i s c r e p a n c i e s s t r o n g l y s u g g e s t e d t h a t some e l e c t r o n c u r r e n t e x i s t e d a n d t h e e n s u i n g t h e o r e t i c a l I n v e s t i g a t i o n l e d t o t h e d e v e l o p m e n t o f t h e n o n - z e r o - e l e c t r o n -c u r r e n t t h e o r y . A l s o , d u r i n g t h e c o u r s e o f t h i s t h e o r e t i c a l i n v e s t i g a t i o n , t h e z e r o - e l e c t r o n - c u r r e n t t h e o r y was e x t e n d e d t o a r b i t r a r y i n j e c t i o n l e v e l s . T h e s e f o r m t h e s u b s t a n c e o f t h e r e m a i n i n g s e c t i o n s o f t h i s c h a p t e r . 18 2.4 E x t e n s i o n o f t h e Z e r o - E l e c t r o n - C u r r e n t T h e o r y £p_ A r b i t r a r y I n j e c t i o n L e v e l s T h e p r e v i o u s z e r o - e l e c t r o n - c u r r e n t t h e o r y i s v a l i d o n l y f o r l o w i n j e c t i o n s i n c e S h o c k l e y ' s r e l a t i o n , e q u a t i o n ( 2 . 3 . 4 ) , w h i c h i s v a l i d o n l y f o r l o w i n j e c t i o n i s u s e d i n t h e d e r i v a t i o n o f t h e f o r m u l a e p r e s e n t e d i n t h a t t h e o r y . T h i s t h e o r y c a n b e e x t e n d e d t o a r b i t r a r y i n j e c t i o n l e v e l s b y u s i n g t h e r e l a t i o n d e r i v e d b y M l s a w a ( 1 9 5 5 ) . P r o m S h o c k l e y , n(0) = n i e x p q C ^ - ^ J / k T p ( 0 ) = n i e x p q ( f p - ^ ) / k T w h e r e n(0) a n d p(0) a r e t h e e l e c t r o n a n d h o l e d e n s i t i e s a t x = x 0 , n ^ i s t h e i n t r i n s i c c a r r i e r d e n s i t y , ^ i s t h e e l e c t r o -s t a t i c p o t e n t i a l , a n d n a n d ^ a r e t h e e l e c t r o n a n d h o l e q u a s l - F e r m i l e v e l s ( i m r e f s ) . By m u l t i p l y i n g t h e a b o v e e q u a t i o n s , M l s a w a o b t a i n e d n(0) p ( 0 ) » n x 2 e x p q V Q / k T ( 2 . 4 . 1 ) a n d by a s s u m i n g s p a c e c h a r g e n e u t r a l i t y h e o b t a i n e d t h e s y m -m e t r i c a l f o r m s p ( 0 ) n(0) n 0 + p(0) - p 0 p 0 + n(0) - n 0 = n i 2 e x p q V c / k T = x ^ 2 e x p qV / k T ( 2 . 4 . 2 ) ( 2 . 4 . 3 ) I t i s r e m a r k a b l e - t h a t t h e s e M l s a w a r e l a t i o n s w h i c h a r e o f t r e m e n d o u s t h e o r e t i c a l s i g n i f i c a n c e a r e s o e a s i l y d e r i v e d f r o m t h e d e f i n i t i o n s o f t h e q u a s l - F e r m l l e v e l s . P r e v i o u s t o t h e d e r i v a t i o n o f t h e s e e q u a t i o n s , l o w i n j e c t i o n was I n v a r i a b l y 19 a s s u m e d a t t h e J u n c t i o n s . B u t w i t h t h e d e v e l o p m e n t o f e q u a t i o n s (2.4.1), (2.4.2), a n d (2.4.3), i n j e c t i o n a t a r b i t r a r y l e v e l s c o u l d be t r e a t e d t h e o r e t i c a l l y . F o r l o w i n j e c t i o n w h e n o p « n 0 > e q u a t i o n (2.4.2) r e d u c e s t o S h o c k l e y ' s r e l a t i o n ( s e e e q u a t i o n (2.2.1) a n d (2.3.4)) p(0) = p 0 e x p q V c / k T (2.4.4) A n e q u i v a l e n t e q u a t i o n t o S h o c k l e y ' s r e l a t i o n f o r l o w i n j e c t i o n w h e n i n <<p0 i s g i v e n by ( n o t e W = n(0) - nQ = op) n(0) - n Q e x p q V c / k T (2.4.5) A t h i g h i n j e c t i o n l e v e l s , t h a t I s , w h e n t h e c a r r i e r d e n s i t i e s p(0) a n d n(0) » s e m i c o n d u c t o r m a t e r i a l n 0 - P 0 we h a v e f o r b o t h t y p e s o f n(0) p(0) » n1 e x p qV c/2kT (2.4.6) a d d e d t o E q u a t i o n (2.4.2) c a n be s o l v e d f o r V c w h i c h i s t h e n V B - JE d x « ( k T / q l n (1 + i p / n Q ) t o o b t a i n V = V f i + V c w h i c h s a t i s f i e s t h e e q u a t i o n e x p q V / k T - (1 + < * p / n 0 ) 2 (1 + h/p0) (2.4.7) Now we h a v e f r o m t h e p r e v i o u s s e c t i o n t h a t I p = CkT.Mp 2^p - ( n 0 - p 0 ) l n (1 + ( $ p / n 0 ) (2.3.5) w h e r e £p = p(0) - p 0 . E q u a t i o n s (2.4.7) a n d (2.3.5) c a n now b e u s e d t o s o l v e f o r t h e d - c I - V r e l a t i o n . T h e i n t r i n s i c c a s e c a n 20 be solved exactly to y i e l d an I-V r e l a t i o n holding at a l l i n j e c t i o n l e v e l s . However, the e x t r i n s i c cases cannot be solved exactly and approximations must be made. The cases are; (a) H Q / P O > > l (n-type material) i) Low i n j e c t i o n ( o p « n 0 ) I p = CkTu pp 0 (exp qV/kT - 1) 11) High i n j e c t i o n ( i p » n Q ) ( 2 . 4 . 8 ) Ip = CkTu pn Q Ip = 2 C k T j u p p 0 (exp qV/3kT - 1) 2 ( p b / n 0 ) l / 3 exp qV/3kT - qV/3kT - 1/3 In p 0 / n 0 (2.4.9) (b) n 0/p Q = 1 ( I n t r i n s i c material) ( 2 . 4 . 1 0 ) (c) no/Po ^ 1 (p-type material) 1) Low Injection (<Sn«p 0) Ip = |Cqu pp 0V i i ) High i n j e c t i o n (<£n»p 0) ( 2 . 4 . 1 1 ) Ip = CkfJ)ipPo 2 ( n Q / p 0 ) 2 / 3 exp qV/3kT + qV/3kT + 1/3 In p Q / n 0 ( 2 . 4 . 1 2 ) These results are s i g n i f i c a n t because the value of 'a' i n the exponential term (see equation (2.2.3)) i s 3 for high i n j e c t i o n l n a l l types of semiconductors. This value of 'a' has been observed frequently in p-n Junctions under high Injection conditions. The high i n j e c t i o n formulae for the current 21 i n e x t r i n s i c m a t e r i a l s a l s o c o n t a i n terms l i n e a r l n V and Independent of V while the formula f o r the c u r r e n t a t any l e v e l of I n j e c t i o n i n i n t r i n s i c m a t e r i a l depends only exponen-t i a l l y on the v o l t a g e . In the low i n j e c t i o n case, the u s u a l diode equation a p p l i e s to n-type m a t e r i a l s and a l i n e a r I-V r e l a t i o n a p p l i e s f o r p-type m a t e r i a l s . When the low i n j e c t i o n equations (2.4.8) and (2.4.11) and the exact equation (2.4.10) are compared with equations (2.3.8), (2.3.9) and (2.3.10) i t i s found that the diode e q u a t i o n f o r n-type m a t e r i a l i s i d e n t i c a l but that the c u r r e n t f o r a given v o l t a g e across p-type and i n t r i n s i c m a t e r i a l i s l e s s than p r e d i c t e d i n the z e r o - e l e c t r o n -c u r r e n t theory. The reason f o r t h i s r e d u c t i o n i n c u r r e n t can be found by r e f e r r i n g to the Mlsawa equation (2.4.1). Notice t h a t V Q i s g i v e n i n the z e r o - e l e c t r o n - c u r r e n t theory by equation (2.3.4) . However, l n the e x t e n s i o n , although the Mlsawa equa t i o n reduces to equation (2.3.4) f o r n-type m a t e r i a l , f o r p-type m a t e r i a l at low I n j e c t i o n l e v e l s when o n « P 0 , the Mlsawa equation reduces to the symmetrical form of Shockley's low l e v e l approximation. T h i s symmetrical form i s e q u a t i o n (2.4.5) . For i n t r i n s i c m a t e r i a l no terms need be dropped from the Mlsawa equation. The e f f e c t of the use of the symmetri-c a l form of Shockley's approximation f o r p-type m a t e r i a l and the exact Mlsawa equation f o r i n t r i n s i c m a t e r i a l l n the e x t e n s i o n Is to Increase, f o r a g i v e n p(0), the value of V c o b t a i n e d by Shockley's low l e v e l approximation. Thus, the e f f e c t on the e x t e n s i o n i s to y i e l d , f o r a given V, a s m a l l e r value of c u r r e n t than l n the z e r o - e l e c t r o n - c u r r e n t theory. T h i s Is seen l n equations (2.4.10) and (2.4.11). 22 The above low Injec t ion equations are v a l i d i n the reverse d i r e c t i o n . However, the high i n j e c t i o n equations are not v a l i d i n the reverse d i r e c t i o n since there i s a l i m i t to the number of minority c a r r i e r s that can be extracted. This l i m i t occurs i n n-type materia l when p(0) = 0, and in p-type mater ia l when n(0) = 0. Equation (2.3.5) then gives the current i n n-type material at sa turat ion: Hence, the magnitude of the saturat ion current i s greater i n n-type materia l i f the amount of doping i s less and, furthermore, the saturat ion current for i n t r i n s i c mater ia l i s double that for the highly doped n-type mater ia l . No saturat ion current ex i s t s for p-type mater ia l since i t s a t i s f i e s a l i n e a r I - V r e l a t i o n . 2.5 Non-Zero-Electron-Current Theory 2.51 E x t r i n s i c Semiconductors s idered the case of zero e lec tron current only . But since some e lectron current always e x i s t s , i t s influence on the I - V c h a r a c t e r i s t i c w i l l now be derived for the one-dimensional case with no recombination. For convenience the relevant equations I p = - C k T p p 2p 0 + ( n 0 - p Q ) In (1 - p 0 /no) (2.4.13) The previous two sections of th i s chapter have con-are l i s t e d . J. n qu n nE + k T u n grad n (2.51.1) J P qu p pE - kT;j p grad p (2.51.2) J = P (2.51.3) 23 'dn/ot = -(n-n 0)/^ n + q" 1 d i v J n ( 2 . 5 1 . 4 ) •ap/at « - (p-p 0 )/ f p - q " 1 d i v J p ( 2 . 5 1 . 5 ) p ( 0 ) n ( 0 ) = nr2 exp (qV c/kT) ( 2 . 5 1 . 6 ) n 0 p Q = n ^ 2 ( 2 . 5 1 . 7 ) A n s n - n 0 = p - p Q s Ap ( 2 . 5 1 . 8 ) p = p ( 0 ) a t x = x Q ( 2 . 5 1 . 9 ) p = p Q a t x = L ( 2 . 5 1 . 1 0 ) The f i r s t t h r e e e q u a t i o n s a r e the e x p r e s s i o n s f o r the e l e c t r o n c u r r e n t d e n s i t y , the h o l e c u r r e n t d e n s i t y , and the t o t a l c u r r e n t . E q u a t i o n s ( 2 . 5 1 . 4 ) and ( 2 . 5 1 . 5 ) a r e r e s p e c t i v e l y the c o n t i n u i t y e q u a t i o n s f o r e l e c t r o n s and h o l e s . E q u a t i o n ( 2 . 5 1 . 6 ) , the Mlsawa r e l a t i o n , was d e r i v e d a t the b e g i n n i n g o f S e c t i o n 2 . 4 . E q u a t i o n ( 2 . 5 1 . 7 ) s t a t e s t h a t the p r o d u c t o f the t h e r m a l e q u i l i -b r ium c a r r i e r d e n s i t i e s I s a c o n s t a n t . Space charge n e u t r a l i t y i s assumed when e q u a t i o n ( 2 . 5 1 . 8 ) i s a p p l i e d . F i n a l l y , t he l a s t two r e l a t i o n s a r e the boundary c o n d i t i o n s i n the semiconductor, Now i f the l i f e t i m e s o f h o l e s and e l e c t r o n s , Xp and X n , a r e assumed i n f i n i t e , then f o r the steady s t a t e J n - c o n s t a n t J = c o n s t a n t P From e q u a t i o n s ( 2 . 5 I . I ) t o ( 2 . 5 1 . 3 ) g r a d p = PJ(1VP - 1 - N/p) g a P kTu p ( 2 p+N)(l+M) ( 2 . 5 1 . I D 24 where M = J n / J p , b = J%/)Jp, and N = n 0 - p Q . Integrat ing and applying the boundary condit ions we obtain j = kTu n(M+l) (L-x 0 ) (M-b) c M+b / po + bN(b-M)" 1 • 2 ^P + l n (p(0) + bN(b-M)- A ( 2 . 5 L 1 2 ) Note that when p(0) = 0, J = - J s so that for n-type materia l j = ^ n ( l + M )  s (L-x Q ) (b-M) 2p c b+M 1 + (b-M)pp bN ( 2 . 5 1 . 1 3 ) and If (b-M)p Q/bN « 1 J s = k T u p p 0 ( l + M ) ( L - x 0 ) -1 An equivalent expression for the saturat ion current can be derived for p-type material by se t t ing n(0) = 0 i n the current equation ( 2 . 5 1 . 1 2 ) , and not ing that the current at saturat ion equals - J g . An a l t e r n a t i v e procedure for der iv ing J s for p-type materia l is to apply the symmetry transformations, to be discussed l a t e r in th i s sec t ion , to equation (2.5I.I3). From E = - dV/dx, equations ( 2 . 5 1 . 2 ) , ( 2 . 5 I . H ) , and the boundary condi t ions , the voltage across the bulk, Vg, i s found to be ,-1 \ w _ kT M+b , V B " q M-b l n Pp + bN(b-M)' p(0) + bN(b-M) -1 (2 .51 .14) Equation ( 2 . 5 1 . 6 ) gives the voltage across the b a r r i e r v c - kT/q In (1 + < * p / p 0)(l + 5n/nQ) ( 2 . 5 1 . 1 5 ) i n which, of course, Sp = p(0) - p Q = n(0) - n0 = Sn from equation ( 2 . 5 1 . 8 ) . Equations ( 2 . 5 1 . 1 2 ) , ( 2 . 5 1 . 1 4 ) , and ( 2 . 5 1 . 1 5 ) cannot be appl ied exactly to give the I-V c h a r a c t e r i s t i c 25 e x p l i c i t l y . N o t e t h a t e q u a t i o n s ( 2 . 5 1 . 1 2 ) . a n d ( 2 . 5 1 . 1 4 ) g i v e t h e e x a c t r e l a t i o n N o w , a s i n t h e p r e v i o u s s e c t i o n , , low a n d h i g h i n j e c t i o n c a s e s m u s t be c o n s i d e r e d , ( a ) Low I n j e c t i o n 1) n 0 / p 0 » l ( n - t y p e m a t e r i a l ) I f j o p / n 0 | « l a n d ( b - M ) p ( 0 ) / b n , *o e q u a t i o n s ( 2 . 5 1 . 1 4 ) a n d ( 2 . 5 1 . 1 5 ) a p p r o x i m a t e t o « 1 , t h e n V B * f i p ( 2 . 5 1 . 1 7 ) a n d V c s i i i n ( j + iE) ( 2 . 5 1 . 1 8 ) Q PQ Now f r o m e q u a t i o n s ( 2 . 5 1 . 1 6 ) t o ( 2 . 5 1 . 1 8 ) we o b t a i n q u n n ( M + l ) J ° ( L - X o ) ( i l t b ) V B ( 2 . 5 1 . 1 9 ) k T p _ p - ( M + l ) J = ? L - x 0 ) — { e x p ^ V c / k T - U ( 2 . 5 L 2 0 ) w h i c h s h o w s a l i n e a r a n d a l s o a n o h m i c I - V r e l a t i o n f o r t h e b u l k a n d t h e u s u a l d i o d e e q u a t i o n f o r t h e s p a c e - c h a r g e r e g i o n . 11) n 0 / p 0 « 1 ( p - t y p e m a t e r i a l ) I f 4n/p 0 « 1 a n d | ( M - b ) n ( 0 ) / M p o « 1 , t h e n e q u a t i o n s ( 2 . 5 1 . 1 4 ) a n d ( 2 . 5 1 . 1 5 ) a p p r o x i m a t e t o V B * - f | ^ i » ( 3 . 5 1 . 2 1 , 26 and V In (1 + *&) (2.51.22) c q n 0 Applying equation (2.51.16) to the foregoing equations gives j _ W Q C H - D V ' ( 2 5 1 # 2 3 ) J (L-x 0)(M+b) V B U , 5 i , O J and kTu nn r t(M+l) J = tr \u "" ( e x P 1 v c A T " 1) (2.51.24) \ ti— XQ y n which shows a l i n e a r and also an ohmic I-V r e l a t i o n for the bulk and usual diode equation for the space-charge region. Note should be taken of the symmetry of the above equations for low i n j e c t i o n into n- and p-type mater ia l . In f a c t , a l l the equations for the low i n j e c t i o n of c a r r i e r s into p-type material can be obtained from those for the n-type by simply making the symmetry transformations n p n C o r o l l a r i e s of these transformations are of course b >-l/b M 1/M Thus, for example, equation (2.51.17) transforms into equation (2.51.21), (2.51.18) into (2.51.22), and (2.51.19) and (2.51.20) Into (2.51.23) and (2.51.24). Other examples are the expressions for the d r i f t and d i f f u s i o n currents i n Sect ion 2.52. The reason why th i s symmetry d id not occur in the zero-e lec tron-current 27 t h e o r y was b e c a u s e s e t t i n g M = 0 I m m e d i a t e l y I n t r o d u c e d a n a s y m m e t r y i n t o t h e a n a l y s i s . B u t w h e n M ^ 0, t h e n t h e s y m m e t r y t r a n s f o r m a t i o n r u l e s s h o u l d h o l d . (b) H i g h I n j e c t i o n F o r s u f f i c i e n t l y h i g h i n j e c t i o n s u c h t h a t a n d <Wn0 >>1, s o t h a t ( b - M ) p ( 0 ) / b N <£P/P0 » 1 » 1 , e q u a t i o n s (2.5L14) a n d (2.51.15) a p p r o x i m a t e t o (2.51.25) a n d k T ( M + b ) / ( M - b ) q I n - In 1 + ( b - M ) p 0 / b N V c «• (2kT/qj l n ( p ( 0 ) / n 1 ) ( b - M ) p ( 0 ) / b N u (2.51.26) s o t h a t V = V g + V p i s g i v e n by V = 3b-M "b+FT V B j . 2 k T 1 + ——- l n 1 + ( b - M ) p 0 / b N ( b - M ) n i / b N (2.51.27) F i n a l l y , f r o m e q u a t i o n s (2.51.16), (2.51.25), a n d (2.51.27), t h e h i g h i n j e c t i o n I - V r e l a t i o n f o r e x t r i n s i c s e m i c o n d u c t o r s i s J ( L - X p ) ( b - M ) „ 2 H ( b + M ) / ( 3 b - M ) p T n ( a V _ t o = M » k T ^ n ( M + l ) e x p ( k T 3b-M } q N b + M v + 2 N b+M k T 3b-M 3b-M w h e r e u _ b N + ( b - M ) p 0  H ~ ( b - M ) n i (2.5L28) T h i s h i g h i n j e c t i o n e q u a t i o n p r e d i c t s a n ' a ' v a l u e ( s e e e q u a t i o n (2.2.3)) g i v e n by a _ 3b - M  a b - M 28 which reduces to a = 3 when M = 0. This high i n j e c t i o n equation reduces to equations (2.4.9) and (2.4.12) when M = 0 and the type of semiconductor i s taken into account. Note that as'the l e v e l of i n j e c t i o n increases i n e x t r i n s i c materials the value of 'a ' increases from unity to a value determined by the value of 1H' . Thus, i t i s seen that the value of 1M 1 plays a major ro le i n the form of the I-V c h a r a c t e r i s t i c . It can be seen that the e lec tron to hole current r a t i o , M = J n / J p , i s s i g n i f i c a n t i n the above equations, and i t s s i g n i -f icance can be seen more c l e a r l y in the i n t r i n s i c case i n the next sect ion . 'M' Is re la ted to the i n j e c t i o n r a t i o , Y, the r a t i o of hole current to the t o t a l current by V J P _ i * = j ~ F T T (2.51.29) Its s ign i f i cance can also be seen l n the fo l lowing ana lys i s . Examining J n = (<Vdrlft + ( J n>dlf f = JM/(M+1) (2.51.30) J P = (Vdrlft + <Jp>dlff = J / ( W + D (2.51.3D where < Jn>drift = 1 F n n E ( J p) d r i f t = ^ p P E  ( J n } d i f f = k T M n 6 r a d n ( J p ^ d i f f = ~ k T ^ p § r a d P I f -1 < M <0, l t i s found that J n < 0 and that the e lec tron d i f f u s i o n current dominates the e lec tron d r i f t current while F i g u r e 2.8 T h e d e p e n d e n c e o f ' a ' o n t h e e l e c t r o n t o h o l e c u r r e n t r a t i o d i v i d e d b y t h e e l e c t r o n t o h o l e m o b i l i t y r a t i o f o r t h e i n t r i n s i c c a s e ( d a s h e d l i n e s a r e a s y m p t o t e s ) 29 J p > 0 a n d t h e h o l e d i f f u s i o n c u r r e n t d o m i n a t e s t h e h o l e d r i f t c u r r e n t i f t h e s e m i c o n d u c t o r i s n - t y p e o r w e a k l y p - t y p e a n d t h e r e v e r s e i s t r u e i f i t I s s t r o n g l y p - t y p e . I f M = 0 , t h e n J n = 0 a n d t h e e l e c t r o n d r i f t a n d d i f f u s i o n c u r r e n t s a r e e q u a l a n d o p p o s i t e . The h o l e d i f f u s i o n c u r r e n t d o m i n a t e s t h e h o l e d r i f t c u r r e n t i f t h e s e m i c o n d u c t o r i s n - t y p e a n d t h e r e v e r s e I s t r u e i f i t i s p - t y p e . F o r M > 0 , t h e e l e c t r o n d r i f t c u r r e n t a l w a y s d o m i n a t e s t h e e l e c t r o n d i f f u s i o n c u r r e n t . T h e h o l e d r i f t c u r r e n t d o m i n a t e s t h e h o l e d i f f u s i o n c u r r e n t i f t h e s e m i c o n d u c t o r I s p - t y p e o r w e a k l y n - t y p e w h i l e t h e r e v e r s e i s t r u e i f i t i s . s t r o n g l y n - t y p e . A s p e c i a l f e a t u r e t o b e n o t i c e d i s M/b = n 0 / p Q w h i c h i s t h e o h m i c c a s e w h e r e J = q ( ^ n n 0 + ; j p P 0 ) . 2.52 I n t r i n s i c S e m i c o n d u c t o r s The p r e v i o u s e x t r i n s i c c a s e c o u l d n o t b e s o l v e d e x a c t l y d u e t o t h e t r a n s c e n t a l n a t u r e o f t h e e q u a t i o n s . H o w e v e r , t h e i n t r i n s i c c a s e c a n b e s o l v e d e x p l i c i t l y and r e v e a l s many i n t e r e s t i n g f e a t u r e s o f t h i s t h e o r y . F i r s t , f r o m t h e t w o c u r r e n t e q u a t i o n s (2.51.30) a n d (2.51.31) i t c a n b e s h o w n t h a t ( J n ) d r i f t ( , V d l f f ( " V d r i f t ( J p ) d l f f = = t J M+T (2.52.1) M - b ' * J M+7 (2.52.2) t J b(M+l) (2.52.3) £ J b ( i ! l ) (2.52.4) a n d t h e r e f o r e we o b t a i n t h e r a t i o s F i g u r e 2.9 T h e d e p e n d e n c e o f t h e s a t u r a t i o n c u r r e n t d e n s i t y , J s i , o n M ( t a k i n g b=2) f o r t h e i n t r i n s i c c a s e . H e n c e , t h e d e p e n d e n c e o f t h e I - V c h a r a c t e r i s t i c o n M = J n / J p ( d a s h e d l i n e s a r e a s y m p t o t e s ) M 30 n ' d r l f t _ * Jn>diff M+b M-b ( A d r i f t p ' d l f f (2.52.5) Now when the equations (2.51.1) to (2.51.15) are modified for t h i s i n t r i n s i c case, we obtain g r a d p " 2 k T ^ U + M ) J = ^M-bHL-x Q) [ P ° ' p ( 0 ) (2.52.6) (2.52.7) (2.52.8) 2kT l n ( p ( 0 ) / n i ) „ kT Ib-M , , ,n\/ \ J = J s l (exp qV/akT - 1) where a = (3b-M)/(b-M) and ' s i = 2 k T u nn 1(l+M) (b-M)(L-x Q) (2.52.9) (2.52.10) (2.52.11) Thus, we have obtained equation (2.52.11) which predicts values of 'a' of any magnitude, depending on the value of M. (Figure 2.8) Analysis of these foregoing equations indicates that the factor M/b i s important l n determing the nature of the I-V c h a r a c t e r i s t i c (Figure 2.9). It can be shown by examining equation (2.52.11) that the regions of interest where physically r e a l i z a b l e I-V c h a r a c t e r i s t i c s are obtained are F i g u r e 2 . 1 0 T h e d e p e n d e n c e o n M/b o f t h e s i g n s o f c u r r e n t s a n d v o l t a g e s w h e n t o t a l c u r r e n t J I s p o s i t i v e i n a n i n t r i n s i c s e m i c o n d u c t o r 31 ( a ) - l / b < M/b < 1 ( r e g i o n 1) ( b ) M / b = 1 ( r e g i o n 1-2) ( c ) 1 < M / b < 3 ( r e g i o n 2) T h e v a l u e s o f M / b ( w h e r e M = J n / J n ) o u t s i d e t h e s e r e g i o n s l e a d t o p h y s i c a l l y i m p o s s i b l e I - V c h a r a c t e r i s t i c s , t h a t i s , p o s i t i v e c u r r e n t w i t h n e g a t i v e v o l t a g e . I n r e g i o n 1, t h e I - V c h a r a c t e r i s t i c i s o f t h e f o r m J = J 0 ( e x p q V / p k T - 1) (2.52.12) w h e r e JQ > 0 a n d ^3 >0. I n r e g i o n 1-2, t h e I - V r e l a t i o n i s t h e o h m i c o n e J = q ( > i n n i + ^ p ^ j V (2.52.13). a n d f i n a l l y i n r e g i o n 2 l t h a s t h e f o r m J = J Q (1 - e x p ( - q V / / 3 k T ) ) (2.52 . 14) w h e r e , a s a b o v e , J 0 a n d a r e b o t h p o s i t i v e . N o t e f r o m t h e a b o v e t h a t t h e d i r e c t i o n o f r e c t i f i c a t i o n h a s r e v e r s e d i n g o i n g f r o m r e g i o n 1 i n t o r e g i o n 2. ( S e e F i g u r e 2.10) Now e x a m i n i n g t h e d r i f t a n d d i f f u s i o n c o m p o n e n t s o f t h e h o l e a n d e l e c t r o n c u r r e n t s g i v e n a t t h e b e g i n n i n g o f t h i s s e c t i o n , l t c a n b e s e e n t h a t i n r e g i o n o n e d i f f u s i o n i s d o m i n a n t f o r - l / b < M / b < 0, d i f f u s i o n a n d d r i f t c u r r e n t s a r e e q u a l a t M / b = 0, a n d d r i f t I s d o m i n a n t f o r 0 < M/b < 1. I n r e g i o n 1-2 t h e d i f f u s i o n c u r r e n t i s z e r o . I n r e g i o n 2 t h e d r i f t c u r r e n t i s d o m i n a n t . A c u r i o u s f e a t u r e o f r e g i o n 2 i s t h a t V Q < 0 when V a n d V g a r e b o t h p o s i t i v e . A n o t h e r c u r i o u s f e a t u r e i s t h a t I n r e g i o n 1, J n < 0 f o r J > 0 a n d - l / b < M/b < 0. ( F i g u r e 2.10) 32 C H A P T E R 2i E X P E R I M E N T A L T E C H N I Q U E S 3•1 G e o m e t r y o f S p e c i m e n s I t I s e x p e r i m e n t a l l y d i f f i c u l t t o m e a s u r e t h e v o l t a g e a c r o s s t h e c o n t a c t u n d e r s t u d y w i t h o u t a l w a y s I n c l u d i n g some o h m i c b u l k o f t h e s e m i c o n d u c t o r . One may c h o o s e t o make a s p e c i m e n ^so s h o r t t h a t t h e o h m i c b u l k i s n e g l i g i b l e b u t s u c h a s p e c i m e n w i l l r e q u i r e a g o o d ' o h m i c ' c o n t a c t o p p o s i t e t h e c o n t a c t s t u d i e d b e c a u s e o t h e r w i s e t h e two c o n t a c t s w i l l m u t u a l l y i n t e r a c t . S u c h g o o d ' o h m i c ' c o n t a c t s a r e d i f f i c u l t e x p e r i m e n -t a l l y t o o b t a i n . H e n c e , a l o n g s p e c i m e n w i t h t h e two c o n t a c t s s e p a r a t e d by t h e o h m i c b u l k o f t h e s e m i c o n d u c t o r w i t h a p r o b e n e a r t h e c o n t a c t t o b e m e a s u r e d was d e c i d e d u p o n t o b e t h e b e s t c o n f i g u r a t i o n . I t i s r e c o g n i z e d , a l t h o u g h i t i s i m p o s s i b l e t o e l i m i n a t e t h e n e c e s s i t y f o r a n ' o h m i c ' c o n t a c t s i n c e a c o m p l e t e e l e c t r i c a l c i r c u i t I s r e q u i r e d t o p a s s c u r r e n t t h r o u g h t h e c o n t a c t s t u d i e d , t h a t t h e ' o h m i c ' c o n t a c t may y e t i n f l u e n c e t h e e l e c t r i c a l p r o p e r t i e s o f t h e c o n t a c t m e a s u r e d . W i t h a p r o b e t e c h n i q u e d e c i d e d u p o n , i t was t h e n n e c e s s a r y t o a v o i d f l o a t i n g p o t e n t i a l s w h i c h e x i s t a t r e c t i f y i n g p o t e n t i a l p r o b e s ( S h o c k l e y 1949). T h e r e f o r e , m e t a l p o i n t -c o n t a c t s w e r e a v o i d e d a s p o t e n t i a l p r o b e s . A l s o a v o i d e d w e r e g o l d - a n t i m o n y a l l o y e d ' o h m i c ' c o n t a c t s o n t h e s i d e o f t h e s p e c i m e n b e c a u s e o f t h e i n h o m o g e n e l t l e s w h i c h may r e s u l t d u e t o t h e d i f f u s i o n o f I m p u r i t i e s i n t o t h e s p e c i m e n a t t h e h i g h a l l o y i n g t e m p e r a t u r e s . T h e r e f o r e , s i d e - a r m s w e r e d e c i d e d u p o n a s p o t e n t i a l p r o b e s w h e r e t h e ' o h m i c ' c o n t a c t s c o u l d b e a l l o y e d O < I 00 1 I 1 1 i. ON o O o o O o m O o o O X X X * X < < < < I 8 o X < o < Figure 3.1 Equlpotent ia l conf igurat ion d i s t o r t e d by side-arm probe • * • 7 • • • 75 u V -50 u V -30 uV -20 UV -10 u v -o o X" < to o o T V < v>> o c5 X < \ \ . 1 \ . -. 1 I 75 uV 50 UV 30 UV 20 uv 10 uV o o < c$ o x* < ON p o x - o o X 00 o o X vO o q X < A l f o i l length = 100 cm Probe length = 20 cm Current = 500 mA t o t h e e n d s o f t h e s i d e - a r r a s . S i n c e t h e r e was some d o u b t a s t o w h i c h e q u l p o t e n t i a l was b e i n g s a m p l e d i n t h e s p e c i m e n b y t h e p r o b e , a p r o b e - a n a l o g e x p e r i m e n t was p e r f o r m e d o n t h i n a l u m i n u m f o i l . T h e f o i l was c u t i n t o a l o n g s t r i p w i t h a p e r p e n d i c u l a r p r o j e c t i o n s i m u l a t i n g a p o t e n t i a l p r o b e n e a r t h e c e n t r e . The e n d s o f t h e s t r i p w e r e c l a m p e d i n b r a s s p l a t e s a n d a c o n s t a n t c u r r e n t , 5 0 0 mA, was p a s s e d t h r o u g h t h e f o i l . A d - c VTVM p r o b e was u s e d co p l o t t h e e q u l p o t e n t i a l l i n e s ( F i g u r e 3.1). F r o m t h i s p r o b e - a n a l o g e x p e r i m e n t i t was f o u n d t h a t t h e s i d e - a r m p r o b e s m e a s u r e d t h e e q u l p o t e n t i a l o n t h e p l a n e b i s e c t i n g t h e s i d e - a r m . 3 . 2 P r e p a r a t i o n o f S p e c i m e n s B o t h s p e c i m e n s r e p o r t e d i n t h i s t h e s i s w e r e made f r o m a b o u l e o f n - t y p e g e r m a n i u m . F i r s t , a s l i c e was c u t f r o m t h e b o u l e o n a w i r e s a w . T h i s s l i c e was a b r a d e d i n r o u g h a n d f i n e c a r b o r u n d u m s l u r r y a n d t h e n r i n s e d i n d e - i o n i z e d w a t e r . N e x t , t h e w a f e r r e s i s t i v i t y was m e a s u r e d o n a B a i r d A t o m i c F o u r P o i n t P r o b e A s s e m b l y a n d s u i t a b l e c o r r e c t i o n s w e r e made t o t h e r e s i s -t i v i t y v a l u e o b t a i n e d ( V a l d e s 1 9 5 4 ) . T h e r e s i s t i v i t y o f t h e w a f e r f r o m w h i c h t h e c o p p e r - c o n t a c t s p e c i m e n was made i s 2 6 . 3 o h m - c m s a n d t h a t o f t h e w a f e r f r o m w h i c h t h e r h o d i u m - c o n t a c t s p e c i m e n was made i s 2 3 . 5 o h m - c m s . E a c h w a f e r was g l u e d t o p h o t o g r a p h i c p l a t e - g l a s s w i t h K o d a k E a s t m a n 9 1 0 A d h e s i v e a n d a l l o w e d t o d r y o v e r n i g h t . I t was I m p o r t a n t t o e n s u r e t h e e n t i r e s u r f a c e o f t h e w a f e r was a d h e r i n g t o t h e g l a s s p l a t e b e c a u s e d u r i n g t h e s u b s e q u e n t u l t r a s o n i c c u t t i n g o p e r a t i o n o n t h e R a y t h e o n U l t r a s o n i c I m p a c t G r i n d e r t h e w a f e r may e i t h e r b e c o m e T csi - o.soo o 00 <*• -M0-( a ) C o p p e r - c o n t a c t s p e c i m e n ( t h i c k n e s s = O . 8 6 5 ) 8.0 90 4.881 ( b ) R h o d i u m - c o n t a c t s p e c i m e n 4.890 — —5". 48 3 o in a* 1/ _ i 0.224 S i d e -v i e w F i g u r e 3 . 2 G e o m e t r y o f S p e c i m e n s S c a l e : 20 mm = 1 mm (20x a c t u a l s i z e ) ( t h e n u m b e r s a r e l n m i l l i m e t e r s ) 3 4 p i t t e d w h e r e t h e s u r f a c e was n o t a d h e r i n g t o t h e g l a s s o r l t may c r a c k . T h e s p e c i m e n s w e r e t h e n c u t i n t o t h e i r f i n a l f o r m o n t h e w i r e saw ( F i g u r e 3 . 2 a , b ) . T h e s p e c i m e n s w e r e t h e n s e p a r a t e d f r o m t h e g l a s s p l a t e b y d e c o m p o s i n g t h e a d h e s i v e b y h e a t i n g l t t o a p p r o x i m a t e l y 150° C . T h e y w e r e c l e a n e d w i t h h y d r o g e n p e r o x i d e , r i n s e d i n d e - i o n i z e d w a t e r , e t c h e d i n C P - 4 , r i n s e d , e t c h e d , a n d r i n s e d a g a i n . A f t e r t h e s p e c i m e n s h a d d r i e d o n c l e a n f i l t e r p a p e r , g o l d w i r e w i t h 0.6% a n t i m o n y d o p i n g was a l l o y e d i n a n i t r o g e n a t m o s p h e r e t o t h e e n d s o f t h e p r o b e s a n d t o t h e ' o h m i c ' e n d o f t h e s p e c i m e n s . T h e a l l o y i n g was p e r f o r m e d by p l a c i n g t h e p o r t i o n o f t h e s p e c i m e n t o be a l l o y e d i n t h e c e n t r e o f a h e a t i n g c o i l o f 5 m i l l i m e t e r d i a m e t e r a n d 3 m i l l i m e t e r l e n g t h a n d t h e n h e a t i n g t h e p o r t i o n t o a t e m p e r a t u r e j u s t a b o v e t h e e u t e c t i c t e m p e r a t u r e , a b o u t 3 5 6 ° C , o f t h e t h r e e p h a s e s y s t e m G e - A u - S b . M i c r o - m a n i p u l a t o r s w e r e u s e d t o b r i n g t h e g o l d ( a n t i m o n y - d o p e d ) w i r e i n c o n t a c t w i t h t h e g e r m a n i u m . P r o v i d e d t h a t t h e w i r e a n d t h e g e r m a n i u m w e r e k e p t s c r u p u l o u s l y c l e a n b y c l e a n i n g a l l a s s o c i a t e d t o o l s w i t h a c e t o n e , t h e w i r e a n d g e r m a n i u m m e l t e d a t t h e p o i n t o f c o n t a c t i m m e d i a t e l y . A s p r e s s u r e was k e p t o n t h e w i r e t o w a r d s t h e g e r m a n i u m , t h e h e a t i n g c o l l was s l o w l y c o o l e d t o r o o m t e m p e r a -t u r e . T h i s p r o c e d u r e was f o l l o w e d b y a q u i c k d i p i n C P - 4 t o r e m o v e a n y o x i d e w h i c h may h a v e f o r m e d o n t h e s p e c i m e n a n d a r i n s e i n d e - l o n l z e d w a t e r . T h e s p e c i m e n s w e r e t h e n r e a d y f o r e l e c t r o p l a t i n g . T u r n e r (1959) f o u n d a l l p l a t e d m e t a l s e x c e p t a n t i m o n y p r o d u c e d r e c t i f y i n g c o n t a c t s o n n - t y p e g e r m a n i u m a n d o h m i c c o n t a c t s o n p - t y p e g e r m a n i u m . A n t i m o n y b e h a v e d J u s t t h e o p p o s i t e . S i n c e r e c t i f y i n g c o n t a c t s w e r e d e s i r e d l n t h i s t h e s i s , c o p p e r a n d r h o d i u m w e r e c h o s e n a s t h e m e t a l s t o b e e l e c t r o p l a t e d t o t h e s p e c i m e n s . T h e c o p p e r a n d r h o d i u m c o n t a c t s w e r e f o r m e d by e l e c t r o p l a t i n g w i t h H a r n s h a w C h e m i c a l Company c o p p e r f l u o b o r a t e p l a t i n g s o l u t i o n a n d t h e S i g m u n d C o h n M f g . C o . r h o d i u m p l a t i n g s o l u t i o n r e s p e c c i v e l y . E a c h s p e c i m e n , h e l d i n a s m a l l m e t a l l i c c l a m p , was e l e c t r o p l a t e d by d i p p i n g I t s e n d i n t o t h e p l a t i n g s o l u t i o n a n d d r a w i n g i t u p u n t i l a m e n i s c u s made c o n t a c t t o o n l y t h e e n d . T h e c u r r e n t d e n s i t y p a s s i n g t h r o u g h t h e s p e c i m e n w a s g r a d u a l l y b u i l t u p s t a r t i n g f r o m a n i n i t i a l l y l o w v a l u e s o t h a t t h e c o n t a c t w o u l d d e p o s i t f i r m l y a n d e v e n l y o n t h e e n d o f t h e g e r m a n i u m f i l a m e n t . T h e c o p p e r c o n t a c t was p l a t e d a t 0.1 mA f o r 15 m i n u t e s , t h e n a t 0.5 mA f o r 45 m i n u t e s . T h e r h o d i u m c o n t a c t was p l a t e d a t 0.05 mA f o r 45 m i n u t e s a n d a t 0.10 mA f o r 5 m i n u t e s . B u t b e c a u s e e x a m i n a t i o n u n d e r a 32 - p o w e r m i c r o s c o p e r e v e a l e d p a t c h e s w h e r e t h e r h o d i u m p l a t i n g a p p e a r e d t h i n n e r t h e p l a t i n g was r e s u m e d a t 0.02 mA f o r 2 h o u r s . T h e t e n a c i t y o f t h e s e c o n t a c t s d i f f e r e d m a r k e d l y . W h e r e a s c o p p e r c o n t a c t s c o u l d be s c r a p e d o f f f a i r l y e a s i l y w i t h t h e t h u m b n a i l ( a l t h o u g h n o t a s e a s i l y a s l e a d c o n t a c t s c o u l d be w i p e d o f f ) , t h e r h o d i u m c o n t a c t c o u l d n o t b e s c r a p e d o f f e v e n by u s i n g a s h a r p m e t a l l i c p o i n t . F i n a l l y , t h e p h y s i c a l d i m e n s i o n s o f t h e s p e c i m e n s w e r e t h e n m e a s u r e d o n t h e C a r l Z e i s s M e a s u r i n g M i c r o s c o p e (0 - 50). F i g u r e 3.2a,b g i v e s t h e s e m e a s u r e m e n t s f o r t h e r h o d i u m - c o n t a c t a n d c o p p e r - c o n t a c t s p e c i m e n s . ( a ) D-C m e a s u r e m e n t o f c o n t a c t c h a r a c t e r i s t i c 1 X ^^yVW-P u l s e G e n e r a t o r c r u 10012-V V e r t i c a l —o D u a l beam ' s c o p e ( d i f f e r e n -t i a l I n p u t s ) V e r t i c a l 2 -o ( b ) P u l s e m e a s u r e m e n t o f c o n t a c t c h a r a c t e r i s t i c F i g u r e 3 . 3 C i r c u i t s f o r I - V m e a s u r e m e n t s 36 3.3 C i r c u i t s f o r I-V M e a s u r e m e n t s 3.31 D-C M e a s u r e m e n t s T h e s p e c i m e n s p r e p a r e d w e r e p l a c e d i n a l u c i t e s p e c i m e n h o l d e r w h i c h p r o v i d e d a p h o s p h o r b r o n z e p r e s s u r e c o n t a c t o n t h e a r e a c o n t a c t u n d e r s t u d y . S o f t l e a d f o i l was p l a c e d b e t w e e n t h e p r e s s u r e c o n t a c t a n d t h e a r e a c o n t a c t s o t h a t t h e m e t a l c o n t a c t , p a r t i c u l a r l y t h e c o p p e r o n e , w o u l d n o t b e i n a d v e r t e n t l y s c r a p e d o f f . T h e l u c i t e h o l d e r was d e s i g n e d so t h a t a l l f o u r s i d e s o f t h e s p e c i m e n w o u l d b e e x p o s e d t o t h e o i l b a t h o f p a r a f f i n o i l ( w h i t e , l i g h t , d o m e s t i c , v i s c o s i t y 125/135). T h i s o i l was c o n t a i n e d i n a l i g h t - t i g h t B a y l e y I n s t r u m e n t C o . c o n s t a n t t e m p e r a t u r e b a t h ( M o d e l 134). T h e c o n s t a n t t e m p e r a t u r e o f t h e o i l b a t h was a r b i t r a r i l y s e t a t 23.4° C . F o r m e c h a n i c a l s t a b i l i t y e a c h g o l d l e a d f r o m t h e s p e c i m e n was f a s t e n e d down by two s c r e w s , t h e s e c o n d s c r e w a l s o f a s t e n i n g a c o p p e r l e a d g o i n g t o t h e J u n c t i o n b o x . T h e c o p p e r l e a d s w e r e c o a t e d w i t h a n i n s u l a t i n g f i l m a n d e a c h l e a d was r u n t h r o u g h p l a s t i c t u b i n g b e f o r e b e i n g r u n t h r o u g h s h i e l d i n g t a k e n f r o m c o a x i a l c a b l e s . T h i s s h i e l d i n g e x t e n d e d f r o m t h e s p e c i m e n h o l d e r t o t h e J u n c t i o n b o x . W i t h i n t h e J u n c t i o n b o x t h e l e a d s w e r e s o l d e r e d t o A m p h e n o l UHF c o -a x i a l c o n n e c t o r s . The d - c c u r r e n t was s u p p l i e d f r o m a b a t t e r y i n s e r i e s w i t h r h e o s t a t s w h i c h p e r m i t t e d a r a n g e o f c u r r e n t f r o m 1 jxA t o 10 mA. The c u r r e n t was n o t a l l o w e d t o e x c e e d 1 mA l n t h e c o p p e r -c o n t a c t s p e c i m e n a n d 10 mA i n t h e r h o d i u m - c o n t a c t s p e c i m e n f o r i t was f o u n d i n p r a c t i c e t h a t c u r r e n t s much i n e x c e s s o f t h e s e v a l u e s c a u s e d a n i r r e v e r s i b l e c h a n g e t o t a k e p l a c e i n t h e c o n t a c t . ( a ) P u l s e f o r m s — f o r w a r d d i r e c t i o n ( I = 1.4 mA, V = 250 m V ) _ r h o d i u m - c o n t a c t ( b ) P u l s e f o r m s - - r e v e r s e d i r e c t i o n ( I =-8.7 JJA, V = -100 m V ) - r h o d i u m - e o n t a c t F i g u r e 3.4 T y p i c a l P u l s e F o r m s 37 T h e v o l t a g e a c r o s s t h e c o n t a c t was m e a s u r e d b y a d - c VTVM ( H e w l e t t P a c k a r d M o d e l 4 1 2 A ) . F i g u r e 3.3a s h o w s t h e s c h e m a t i c o f t h e c i r c u i t . T h e p r o c e d u r e f o l l o w e d l n p l o t t i n g t h e p o l n t -b y - p o i n t I - V c u r v e was t o i n c r e a s e t h e v o l t a g e , t a k e c u r r e n t a n d v o l t a g e r e a d i n g s , i n c r e a s e t h e v o l t a g e a g a i n , t a k e r e a d i n g s , e t c . A f t e r t h e maximum r e a d i n g s h a d b e e n t a k e n t h e p o i n t s o b t a i n e d w e r e c h e c k e d , a s t h e v o l t a g e was r e d u c e d , t o e n s u r e t h a t t h e I - V c h a r a c t e r i s t i c h a d n o t c h a n g e d . A s i m i l a r p r o c e d u r e was f o l l o w e d f o r t h e r e v e r s e d i r e c t i o n . 3.32 P u l s e M e a s u r e m e n t s The c o n d i t i o n s u n d e r w h i c h t h e p u l s e m e a s u r e m e n t s w e r e made w e r e i d e n t i c a l w i t h t h o s e i n t h e d - c m e a s u r e m e n t s e x c e p t f o r t h e R u t h e r f o r d P u l s e G e n e r a t o r ( M o d e l B 7 B ) w h i c h r e p l a c e d t h e b a t t e r y a n d t h e r h e o s t a t s a n d f o r t h e T e k t r o n i x t y p e 5 0 2 d u a l -beam o s c i l l o s c o p e w h i c h was u s e d t o d i s p l a y t h e c u r r e n t a n d t h e v o l t a g e p u l s e f o r m s ( F i g u r e 3.3b). T h e p u l s e g e n e r a t o r was o p e r a t e d a t 20 p u l s e s p e r s e c o n d w i t h 5 0 0 j j s e c p u l s e w i d t h . On t h e o s c i l l o s c o p e , t h e c u r r e n t a n d v o l t a g e r e a d i n g s w e r e t a k e n a t t h e f l a t p o r t i o n o f t h e p u l s e f o r m a f t e r t h e t r a n s i e n t s h a d d i e d o u t ( F i g u r e 3.4). To o b t a i n t h e I - V c u r v e a s i m i l a r t e c h n i q u e a s u s e d l n t h e d - c m e a s u r e m e n t s was f o l l o w e d . The p o i n t - b y - p o i n t v o l t a g e a n d c u r r e n t r e a d i n g s w e r e t a k e n a s t h e v o l t a g e was I n c r e a s e d a n d t h e p o i n t s o n t h e I - V c u r v e w e r e c h e c k e d , a s t h e v o l t a g e was d e c r e a s e d , t o e n s u r e t h a t t h e c o n t a c t h a d n o t c h a n g e d . T h i s p r o c e d u r e was f o l l o w e d I n t h e f o r w a r d a n d r e v e r s e d i r e c t i o n s . Impedance B r i d g e R a-c 1 pF F i g u r e 3 . 5 T r a n s v e r s e a-c r e s i s t a n c e measurement o f c o n d u c t i v i t y m o d u l a t i o n 38 3 . 4 Transverse A-C R e s i s t a n c e Measurement The t r a n s v e r s e a-c r e s i s t a n c e was measured i n an attempt to determine whether i n j e c t i o n or e x t r a c t i o n occurs at the a r e a contact analogously to H a r r l c k ' s (1956) measurements with h i s i n f r a - r e d f r e e c a r r i e r a b s o r p t i o n technique. However, to ensure that i n j e c t i o n or e x t r a c t i o n d i d not occur at the 'ohmic' c o n t a c t s on the side-arms, t r a n s v e r s e d-c I-V measure-ments were made. The I-V r e l a t i o n was a l i n e a r one where the r e s i s t a n c e never v a r i e d more than 3% from 1 1 . 6 kXL as the c u r r e n t v a r i e d from -80 u^A to 20 ^uA. Then, the c i r c u i t shown i n F i g u r e 3 .5 was used to measure the t r a n s v e r s e a-c r e s i s t a n c e of the side-arms and the r e g i o n under the rhodium c o n t a c t . A G e n e r a l Radio I65OA Impedance Bridge measured the a-c r e s i s t a n c e at one k i l o c y c l e as the l o n g i t u d i n a l d-c c u r r e n t was v a r i e d . The 100 ohm r e s i s t o r and the a-c Hewlett Packard T r a n s i s t o r Voltmeter (Model 403A) measured the a-c c u r r e n t which was kept at about 5 JuA. 39 C H A P T E R 4 E X P E R I M E N T A L R E S U L T S AND I N T E R P R E T A T I O N 4.1 T r a n s v e r s e A - C R e s i s t a n c e M e a s u r e m e n t The I n i t i a l p u r p o s e o f t h e t r a n s v e r s e a - c r e s i s t a n c e m e a s u r e m e n t o n t h e r h o d i u m - c o n t a c t s p e c i m e n was t o d e t e r m i n e t h e o c c u r r e n c e o f e i t h e r i n j e c t i o n o r e x t r a c t i o n o f m i n o r i t y c a r r i e r s a t t h e c o n t a c t i n t o t h e b u l k o f t h e s e m i c o n d u c t o r . A n o t h e r p u r p o s e was t o d e t e r m i n e t h e l e v e l o f i n j e c t i o n t a k i n g p l a c e a t t h e c o n t a c t , t h e l e v e l o f i n j e c t i o n b e i n g r e q u i r e d i n o r d e r t o d e t e r m i n e t h e v a l i d i t y o f a s s u m i n g h i g h i n j e c t i o n a t t h e r h o d i u m c o n t a c t . H i g h i n j e c t i o n was p r o v i s i o n a l l y a s s u m e d f o r t h e e x p e r i m e n t a l I - V c h a r a c t e r i s t i c s b e c a u s e l t i s t h e o n l y c a s e f o r w h i c h t h e t h e o r e t i c a l a n a l y s i s f o r e x t r i n s i c s e m i c o n d u c t o r s i n S e c t i o n 2.51 g i v e s a n e x p l i c i t I - V r e l a t i o n w h e r e b y ' M 1 , t h e e l e c t r o n t o h o l e c u r r e n t r a t i o , c a n b e c a l c u l a t 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 s o f ' o d ' ( d e f i n e d i n S e c t i o n 4.2), a n d t h e s a t u r a t i o n c u r r e n t . F i g u r e 4.1 s h o w s t h e r e l a t i o n b e t w e e n t r a n s v e r s e a - c r e s i s t a n c e , R a _ c » a n d t h e l o n g i t u d i n a l d - c c u r r e n t , I , f l o w i n g t h r o u g h t h e r h o d i u m c o n t a c t . I t d e m o n s t r a t e s t h a t t h e c o n d u c -t i v i t y o f t h e s p e c i m e n i n c r e a s e s a n d t h e r e f o r e t h a t i n j e c t i o n o c c u r s when t h e r h o d i u m c o n t a c t i s b i a s e d p o s i t i v e l y w i t h r e s p e c t t o t h e s e m i c o n d u c t o r ( f o r w a r d d i r e c t i o n ) , a n d t h a t e x t r a c t i o n o c c u r s w h e n t h e c o n t a c t i s b i a s e d i n t h e r e v e r s e d i r e c t i o n . The same v a r i a b l e s o f F i g u r e 4.1 p l o t t e d o n a l o g - l o g g r a p h s h o w s e m p i r i c a l l y t h a t ( F i g u r e 4.2) 20 - f -fi a - c ( k A ) F i g u r e 4.2 T r a n s v e r s e a - c r e s i s t a n c e o f r h o d i u m - c o n t a c t s p e c i m e n a s a f u n c t i o n o f l o n g i t u d i n a l c u r r e n t 10 8 o o ( a ) I (|iA) "0 -~ ( b ) I (juA) x 1 0 ' H =3? I - 0-324 k S L a - c 0„ 2-r 1 H-H-I ( | 1 A ) 6 8 10 20 4 0 60 8 0 100 N o t e : c u r r e n t , s c a l e f o r c u r v e ( a ) ( s q u a r e s ) r a n g e s f r o m 1 t o 100 pA a n d f r o m 100 t o 1 0 , 0 0 0 jiA f o r c u r v e ( b ) ( c i r c l e s ) 40 R A _ = 11.6 k_Q_ f o r 1 ^ 1 uA a " ° ^ (4.1.1) -o ^?4 v 37 I U ° " H ICA. f o r 100 JJA < I < 10 mA A s u b t l e p o i n t to be noted here Is that the t r a n s v e r s e a-c r e s i s t a n c e , H a_ c, equals the t r a n s v e r s e d-c r e s i s t a n c e , (£<3_c)t , H a - c = < Bd-o>t (4.1.2) because no d-c c u r r e n t flows i n the t r a n s v e r s e d i r e c t i o n . I f a t r a n s v e r s e d-c c u r r e n t e x i s t e d , then H a_ c= dV^/dlj. ^ (^d-c^t where V*t and 1^ . are the d-c v o l t a g e and c u r r e n t i n the t r a n s v e r s e d i r e c t i o n . An attempt was made to f i n d the t h e o r e t i c a l l y expected r e l a t i o n between R a_ c and the l o n g i t u d i n a l d-c c u r r e n t , I. S e v e r a l assumptions were made here. I t was assumed that d i f f u s i o n i s dominant, that the side-arms do not a f f e c t the l o n g i t u d i n a l one-dlmenslonal v a r i a t i o n of hole d e n s i t y with l o n g i t u d i n a l d i s t a n c e , x, |and that the holes d i f f u s e i n t o the side-arms i n an e x p o n e n t i a l manner. Thus, Ap(x) = Ap(0) exp ( x 0 - x ) / L p (4.1.3) A p ( 0 ) = V j L j / q D p (4.1.4) ApU) = Ap(0) exp ( x Q - L ) / L (4.1.5) A P y = Ap{L) exp - y / L p (4.1.6) where Ap(x) = p(x) - p Q and Ap y = p y - p Q . Note that the same l e t t e r 1L• which stood f o r the l e n g t h of the diode l n the 41 theory of Chapter 2 Is purposely set equal to the distance between the plane b i s e c t i n g the side-arms and the edge of the space-charge region In the semiconductor. This was done since the voltage V (which corresponds to that at x = L i n the theory of Chapter 2) was measured experimentally at that plane. Le t t er 'y ' i s the transverse co-ordinate whose o r i g i n l i e s at the edge of the specimen. The d i f f u s i o n of minority c a r r i e r s into the side-arm can modulate i t s conduct iv i ty so that the a-c res i s tance of the side-arm Is, Invoking equation (4.1.2), (Ra-c>s = / dy/^A' *o-s 7* i» 1 + C j J 1 + C j J exp ( - 1 S/L ) .(4.1.7) P' ' where R 0 _ s i s the equi l ibr ium value of the side-arm a-c res i s -tance, l g i s the length of the side-arm, A' i s the c r o s s -sec t iona l area of the s ide-arm, (f i s the conduct iv i ty of the semiconductor, J i s the l o n g i t u d i n a l d-c current dens i ty , and ftp-sA'qupU+^yLp By using the E i n s t e i n r e l a t i o n and noting that fi0_sA'/ls = ^o"1 Vq(l+b)L D C i = A rough computation was made for the value of C 1 taking b = 2, Y = 1, q/kT = 40 v o l t " 1 , (TQ = 0.04 ohm"1 cm" 1 , and L P = 0.07 cm since the hole l i f e t i m e of the mater ia l i s approximately lO'^sec and i t was taken that Dp = 50 cm 2 sec" 1 , i t was found that as 42 the current , I , ranged from 0.1 mA to 10 mA, the fac tor C^J ranged from 3 to 300 since J = I /A where A, the area of the rhodium contact , i s approximately 7 .5 x 10 J cm . Also ln th i s current range, the factor C^J exp ( - 1 B / L p ) ranged from 0.5 to 5 0 . Hence, i t Is only when the current , I , exceeds 1 mA that the two factors above are large compared with unity so that equation (4.1.7) can be approximated by " W s _ W 0 ( e x p l s / L p - 1) ^ 7 > V S ( i " b ) i t 4-V 8 >, A rough analys is of the a-c res i s tance of the centre port ion of the specimen would assume that the transverse e l e c t r i c f i e l d i s uniform from one side of the specimen to the other. Then the a-c current i s given by, using equation ( 4 . 1 . 2 ) , T Xazc a-c ~ 1 „ tva-o f r - — J W P [ x l where V a _ c i s the transverse a-c voltage across the centre port ion of the specimen, t i s the specimen thickness , l c i s the distance across the centre port ion from one side-arm to the other, and Xj^ and x 2 are the l o n g i t u d i n a l distances from the rhodium contact to the sides of the s ide-probes . The other symbols such as p, the hole dens i ty , N = n 0 - p 0 the di f ference of the thermal equi l ibr ium values of the e lec tron and hole d e n s i t i e s , have been defined before. Thus, the a-c res is tance of the centre por t ion i s given by 43 >J + 1/R0- (4.1.9) o - c w h e r e R 0 - c i s t h e e q u i l i b r i u m v a l u e o f t h e c e n t r e a - c r e s i s t a n c e a n d t q f t ( l + b ) L p ° 2 = — m „ e x p / x 0 - X-L / x o -- e x p 1 N o w , a p p r o x i m a t e l y , C 2 =0.55 v o l t - 1 c m 2 w h e n t h e n u m e r i c a l v a l u e s u s e d I n t h e a n a l y s i s o f t h e s i d e - a r m r e s i s t a n c e a r e -t a k e n w i t h x 1 = O.0369 c m , x 2 = 0.0593 c m , t = 0.057 c m , a n d l c = 0 . 1 4 cm ( F i g u r e 3.2b). H e n c e , i n t h e c u r r e n t r a n g e o f 0.1 mA t o 10 mA, t h e m i n i m u m v a l u e o f C 2 J 1/150 ohm"" 1 e x c e e d s l / B o - c s 0 t h a t I n t h i s c u r r e n t r a n g e ( R a - c ) c ~ I/C2J (4.1.10) T h e r e f o r e , t h e t o t a l t r a n s v e r s e a - c r e s i s t a n c e i s o b t a i n e d f r o m e q u a t i o n s (4.1.7) a n d (4.1.9) s i n c e s a - c = ( R a - c ) c + 2 ^ R a - c ^ s A r o u g h c a l c u l a t i o n s h o w s t h a t t h e c e n t r e r e s i s t a n c e i s s m a l l c o m p a r e d t o t h a t o f t h e s i d e - a r m s s o t h a t w h e n t h e c u r r e n t , I , e x c e e d s 1 mA, t h e t o t a l a - c r e s i s t a n c e , H = 2(fi „)„, c a n be o b t a i n e d f r o m e q u a t i o n (4.1.8). T h u s , I n t h i s c u r r e n t r a n g e w h e r e e q u a t i o n (4.1'. 8) i s v a l i d , t h e t r a n s v e r s e a - c r e s i s t a n c e i s i n v e r s e l y p r o p o r t i o n a l t o t h e l o n g i t u d i n a l c u r r e n t . H o w e v e r , l t was o b s e r v e d t h a t t h e r e s i s t a n c e i s i n v e r s e l y p r o p o r t i o n a l t o a p p r o x i m a t e l y t h e c u b e r o o t o f t h e l o n g i t u d i n a l c u r r e n t . A p o s s i b l e r e a s o n f o r t h i s d i s c r e p a n c y c a n b e s e e n b y e x a m i n i n g a (a) 35-3 0 - - 3000, 6000 25 2 0 15 I-( b ) ( c ) 3500, 7000 2500, 5000 2 0 0 0 , 4 0 0 0 1500, 3000 1 0 — 1000, 2000 500, 1000 F i g u r e 4 . 4 D-C f o r w a r d c h a r a c t e r i s t i c o f r h o d i u m - c o n t a c t s p e c i m e n ( s c a l e s ( a ) , ( b ) , a n d ( c ) / c o r r e s p o n d t o c u r v e s °/ ( a ) , ( b ) , a n d ( c ) ) / / /O 25 ( a ) 250 ( b ) 500 ( c ) e q u a t i o n (4.1.8). I f t h e l i f e t i m e o f t h e c a r r i e r s , c h a n g e s s o t h a t t h e d i f f u s i o n l e n g t h f o r h o l e s , L , c h a n g e s a s t h e i n j e c t i o n l e v e l v a r i e s , t h e n t h e t r a n s v e r s e a - c r e s i s t a n c e w i l l n o t r e m a i n i n v e r s e l y p r o p o r t i o n a l t o t h e l o n g i t u d i n a l c u r r e n t . T h e l e v e l o f i n j e c t i o n may b e e x a m i n e d b y a r e l a t i o n o b t a i n e d f r o m t h e e x p r e s s i o n f o r C , a n d e q u a t i o n (4.1.4), a n d by i n s e r t i n g t h e p r e v i o u s n u m e r i c a l v a l u e s we f i n d t h a t Ap(0) = 4.5 x 10 1 3 JCX c m " 3 . By n o t i n g t h a t f o r t h e l o n g i t u -d i n a l c u r r e n t r a n g e ,0.1 mA t o 10 mA, J C ^ v a r i e s f r o m 3 t o 300, i t c a n b e s e e n t h a t t h e i n j e c t i o n l e v e l c h a n g e s f r o m m o d e r a t e t o h i g h . 4.2 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 o f R h o d i u m - C o n t a c t S p e c i m e n F i g u r e s 4.3 a n d 4.4 show t h e d - c 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 o f t h e r h o d i u m - c o n t a c t s p e c i m e n . T h e c h a r a c -t e r i s t i c d o e s n o t o b e y t h e d i o d e e q u a t i o n b e c a u s e o f t h e p r e s e n c e o f a l o w s e r i e s r e s i s t a n c e , R s , a n d a h i g h p a r a l l e l r e s i s t a n c e , R . I n o r d e r t o o b t a i n t h e 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 o f t h e c o n t a c t p r o p e r , i t i s n e c e s s a r y t o s u b t r a c t t h e e f f e c t s o f t h e s e a s s o c i a t e d r e s i s t a n c e s b y t h e r e l a t i o n s Ap(0) = (4.1.11) q ( l + b ) ^ p I = I B ( e x p <*V -1) (4.2.1) V, = V T I :fi S (4.2.2) m F i g u r e 4.5 G r a p h i c a l a n a l y s i s o f c h a r a c t e r i s t i c f o r r h o d i u m - c o n t a c t s p e c i m e n 45 I = I - V/R (4.2.3) m p where I and V are the experimentally measured values of the current and the voltage whereas I and V are the corrected & m m values of the current and the voltage of the contact. This, procedure immediately gives the value of the saturation current, Ig_ r» In the reverse d i r e c t i o n . Then for the values of.the cur-rent, I f f l, l n the reverse d i r e c t i o n , log ( I m + I s _ r ) i s plotted against V. (Figure 4.5). The i n i t i a l slope of the l i n e obtained i s o ( r . In order to obtain for the forward d i r e c t i o n and the value of I of equation (4.2.1) f o r the forward d i r e c t i o n ( I s _ t 0 , log I i s plotted against Vffl where, for values of >> kT/q, the curve becomes l i n e a r (Figure 4.5). The l i n e a r portion i s extended to intersect the v e r t i c a l axis and the intercept corresponds to I s _ f . Subsequently, log (I + I s _ f ) i s plotted against V"m and the slope of the l i n e y i e l d s o<f. Therefore, the corrected current-voltage c h a r a c t e r i s t i c has been reduced to the four parameters c< _, I_ „, o< , and I . The values of these I ' s-i» r' s-r parameters obtained for the rhodium-contact specimen by the d-c measurement are: o< f = 34.5 v o l t " 1 <=^ r = 31.2 v o l t " 1 I g _ f = 9.0 UA I = 8 . 8 uA s-r / These values of o ( are less than the usually expected value of kT/q i n the ordinary diode equation. However, values of oi less than or equal to kT/q have been observed F i g u r e 4.7 P u l s e f o r w a r d c h a r a c t e r i s t i c o f c o p p e r - c o n t a c t s p e c i m e n ( v o l t a g e s c a l e s ( a ) a n d ( b ) c o r r e s p o n d t o c u r v e s ( a ) a n d ( b ) ) 46 f r e q u e n t l y b e f o r e b y o t h e r s a n d I s n o t s u r p r i s i n g h e r e . T h e p s a t u r a t i o n c u r r e n t d e n s i t y o f a b o u t 1.1 mA/cm o b s e r v e d f o r t h e r h o d i u m - c o n t a c t s p e c i m e n i n t h i s e x p e r i m e n t a g r e e s c l o s e l y w i t h 0.95 m A / c m 2 r e p o r t e d by B o r n e m a n , S c h w a r z , a n d S t i c k l e r (1955)» S i n c e t h e s a t u r a t i o n c u r r e n t o b t a i n e d f r o m t h e f o r w a r d a n d » r e v e r s e I - V m e a s u r e m e n t s a r e a p p r o x i m a t e l y e q u a l a n d s i n c e o ( f X c < r > i t a p p e a r s t h a t t h e r h o d i u m c o n t a c t s a t i s f i e s e q u a t i o n (4.2.1) w i t h t h e same p a r a m e t e r s I a n d o( f o r b o t h p o s i t i v e a n d n e g a t i v e c u r r e n t s . 4.3 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 o f C o p p e r - C o n t a c t S p e c i m e n F i g u r e s (4.6) a n d (4.7) show t h e 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 f o r t h e c o p p e r - c o n t a c t s p e c i m e n m e a s u r e d b y t h e p u l s e m e t h o d . The c h a r a c t e r i s t i c i s a n a l y z e d i n e x a c t l y t h e same m a n n e r a s was d o n e f o r t h a t o f t h e r h o d i u m - c o n t a c t s p e c i m e n i n S e c t i o n 4.2. The o n l y d i f f e r e n c e i n t h i s c a s e i s t h a t t h e v a l u e s o f V ^ i n t h e f o r w a r d d i r e c t i o n do n o t e x t e n d m u c h p a s t k T / q s o that t h e c u r v e o f l o g I a g a i n s t V d o e s n o t b e c o m e m l i n e a r . T h e r e f o r e , v a l u e s o f I s _ f w e r e a d d e d t o I u n t i l a b e s t s t r a i g h t l i n e o f l o g ( I + I s _ f ) a g a i n s t V m was o b t a i n e d . The v a l u e s o f t h e p a r a m e t e r s o b t a i n e d a r e o< f = 24.7 v o l t " 1 CX r = 10.0 V O l t " 1 I s - f = 9 0 MA  I s - r = 1 6 ? Vk 3 10 20 30 40 V f f l ( m V ) - - f o r w a r d V ( m V ) — r e v e r s e -200 -150 -100 -50 0 F i g u r e 4 . 8 G r a p h i c a l a n a l y s i s o f c h a r a c t e r i s t i c f o r c o p p e r - c o n t a c t s p e c i m e n 4 ? T h e c o p p e r c o n t a c t h a s v a l u e s o f o< much m o r e a p p r e c i a b l y l e s s t h a n q / k T t h a n t h e r h o d i u m c o n t a c t a n d t h e v a l u e s o b t a i n e d f r o m t h e r e v e r s e a n d f o r w a r d I - V m e a s u r e m e n t s a r e n o t e q u a l . S i m i l a r l y , t h e v a l u e s o f t h e s a t u r a t i o n c u r r e n t o b t a i n e d f r o m t h e f o r w a r d a n d r e v e r s e c h a r a c t e r i s t i c s a r e d i f f e r -2 e n t . The r e v e r s e s a t u r a t i o n c u r r e n t d e n s i t y o f 1 5 . 8 mA/cm a n d 2 • t h e f o r w a r d s a t u r a t i o n c u r r e n t d e n s i t y o f 8 . 5 mA/cm a g r e e i n o r d e r 2 o f m a g n i t u d e w i t h 9 . 2 mA/cm r e p o r t e d by B o r n e m a n , S c h w a r z , a n d S t i c k l e r ( 1 9 5 5 ) . B u t , u n l i k e t h e b e h a v i o u r o f t h e r h o d i u m c o n t a c t , t h e f o r w a r d a n d r e v e r s e c h a r a c t e r i s t i c s o f t h e c o p p e r c o n t a c t d o n o t s a t i s f y t h e same d i o d e e q u a t i o n n u m e r i c a l l y . 4 . 4 C o m p a r i s o n o f E x p e r i m e n t a l R e s u l t s w i t h T h e o r y T h e e x p e r i m e n t a l v a l u e s o b t a i n e d f o r t h e r h o d i u m a n d c o p p e r c o n t a c t s w e r e u s e d t o o b t a i n v a l u e s o f M , t h e e l e c t r o n t o h o l e c u r r e n t r a t i o . T h e f o r w a r d p a r a m e t e r s o<^, w e r e I n s e r t e d I n t o t a k i n g t h e v a l u e s o f t h e m o b i l i t i e s a t t h e t e m p e r a t u r e o f t h e = 1900 c m V v o l t - s e c . T h i s r e l a t i o n f o r M = J r / J p f r o m e q u a t i o n (2 . 5 1.28) i s v a l i d o n l y f o r h i g h i n j e c t i o n . H o w e v e r , a s was i n d i c a t e d a l s o f r o m t h e t r a n s v e r s e a - c r e s i s t a n c e m e a s u r e m e n t ( 4 . 4 . 1 ) w h e r e - a = q / o ( f k T . T h e v a l u e o f b = J J n / M r ) w a s o b t a i n e d b y f o r t h e c u r r e n t l e s s t h a n 1 mA, l t a p p e a r s t h a t t h e h i g h i n j e c t i o n a s s u m p t i o n , <£p/n 0 » 1, i s I n v a l i d f o r t h e r h o d i u m c o n t a c t s i n c e 48 M = -28.1 Is In the r e g i o n where p h y s i c a l l y i m p o s s i b l e I-V r e l a t i o n s r e s u l t . Moreover, s i n c e oC^ . = 34.5 v o l t * i s a valu e l y i n g between q/2kT and q/kT, i t appears that the l e v e l of I n j e c t i o n i n the experiment where I Is between 1 JAA and 1 mA 2 2 (J between 0.13 mA/cm and 130 mA/cm ) Is between the high and low l e v e l s . Thus, although M cannot be obtained f o r t h i s < Intermediate case, the above procedure shows i t i s p o s s i b l e to determine the approximate l e v e l o f i n j e c t i o n . The copper c o n t a c t , however, appears to s a t i s f y the high i n j e c t i o n c r i t e r i o n s i n c e 0<f = 24.7 v o l t " 1 i s s i g n i f i -c a n t l y l e s s than q/kT. M = 4.94 f u r t h e r i n d i c a t e s that high I n j e c t i o n occurs f o r I between 10 pk and 500 ]xk ( J between 2 2 1 mA/cm and 50 mA/cm ) s i n c e t h i s value leads to p h y s i c a l l y p o s s i b l e I-V r e l a t i o n s . From S e c t i o n 2.51, t h i s value of M r e v e a l s that the e l e c t r o n d r i f t c u r r e n t dominates the e l e c t r o n d i f f u s i o n c u r r e n t . 49 C H A P T E R j i C O N C L U S I O N S T h e w o r k I n t h i s t h e s i s h a s e x t e n d e d t h e t h e o r y o f m e t a l - s e m i c o n d u c t o r c o n t a c t s . I n S e c t i o n 2.4 t h e e x i s t i n g z e r o - e l e o t r o n - c u r r e n t t h e o r y i s e x t e n d e d t o a r b i t r a r y i n j e c t i o n l e v e l s . S e c t i o n 2.5 d e v e l o p s t h e n o n - z e r o - e l e c t r o n - c u r r e n t t h e o r y . T h e o u t s t a n d i n g f e a t u r e o f t h i s s e c t i o n i s t h a t f o r i n j e c t i o n i n t o i n t r i n s i c s e m i c o n d u c t o r s a n d f o r h i g h i n j e c t i o n i n t o e x t r i n s i c s e m i c o n d u c t o r s t h e v a l u e o f a = ( 3 b - M ) / ( b - M ) i n t h e m o d i f i e d d i o d e e q u a t i o n ( 2 . 2 . 3 ) c a n t a k e a n y v a l u e d e p e n d i n g o n t h e v a l u e o f M , t h e e l e c t r o n t o h o l e c u r r e n t r a t i o . E x p e r i m e n t s w e r e p e r f o r m e d i n a n a t t e m p t t o c h e c k t h e n o n - z e r o - e l e c t r o n - c u r r e n t t h e o r y . T r a n s v e r s e a - c r e s i s t a n c e m e a s u r e m e n t s made o n t h e r h o d i u m - c o n t a c t s p e c i m e n g a v e a t r a n s -v e r s e a - c r e s i s t a n c e , R a - c » v e r s u s l o n g i t u d i n a l c u r r e n t , I , r e l a t i o n w h i c h d i d n o t a g r e e w i t h t h e e x p e c t e d t h e o r e t i c a l r e l a t i o n . H o w e v e r , i f t h e c a r r i e r l i f e t i m e i s a f u n c t i o n o f t h e c a r r i e r d e n s i t y , t h e n t h e t h e o r e t i c a l r e l a t i o n may c h a n g e t o r e d u c e t h e d i s c r e p a n c y . R o u g h c a l c u l a t i o n s made f r o m t h i s d a t a s n o w e d t h a t m o d e r a t e t o h i g h I n j e c t i o n o c c u r r e d a t t h e 2 r h o d i u m c o n t a c t f o r c u r r e n t d e n s i t y J b e t w e e n 10 a n d 1000 mA/cm . T h e e x p e r i m e n t a l p a r a m e t e r s f o r t h e r h o d i u m - c o n t a c t s p e c i m e n I n d i c a t e d t h a t h i g h i n j e c t i o n d i d n o t o c c u r a t t h a t c o n t a c t f o r J b e t w e e n 0.1 a n d 100 m A / c m 2 , w h e r e a s t h e c o n v e r s e was t r u e f o r t h e c o p p e r - c o n t a c t s p e c i m e n f o r J b e t w e e n 1 a n d 50 mA/cm . • T h e r e f o r e , l t was f o u n d M = 4.94 f o r t h e c o p p e r c o n t a c t s p e c i m e n b y u s i n g t h e h i g h i n j e c t i o n e q u a t i o n ( 2 . 5 1 . 2 8 ) . 50 T h u s , m e t a l - s e m i c o n d u c t o r c o n t a c t s c a n b e c h e c k e d f o r h i g h I n j e c t i o n b y c a l c u l a t i n g t h e v a l u e o f t h e e l e c t r o n t o h o l e c u r r e n t r a t i o , M, f r o m t h e m e a s u r e d v a l u e s o f F i n a l l y , i t s h o u l d b e n o t e d t h a t a d i s c r e p a n c y may e x i s t b e t w e e n t h e t h e o r y a n d e x p e r i m e n t a l d a t a f o r s e v e r a l r e a s o n s . T h e r e I s no g u a r a n t e e t h a t M r e m a i n s c o n s t a n t t h r o u g h o u t a l l v o l t a g e r a n g e s . I n f a c t , t h e i n j e c t i o n r a t i o , ( w h i c h i s r e l a t e d t o M b y e q u a t i o n ( 2 . 5 1 . 2 9 ) ) i s e x p e c t e d t o d i m i n i s h w i t h i n c r e a s i n g f o r w a r d c u r r e n t , s i n c e , d u e t o s p a c e c h a r g e n e u t r a l i t y , a s m o r e m i n o r i t y c a r r i e r s a r e i n j e c t e d , t h e l o c a l c o n c e n t r a t i o n o f m a j o r i t y c a r r i e r s a l s o i n c r e a s e s so t h a t t h e c o n t r i b u t i o n o f t h e m a j o r i t y c a r r i e r s t o t h e t o t a l c u r r e n t I s I n c r e a s e d . B a n b u r y a n d H o u g h t o n ( 1 9 5 4 ) h a v e o b s e r v e d t h i s e x p e c t e d d i m i n u t i o n i n X w i t h i n c r e a s e d f o r w a r d c u r r e n t t h r o u g h p o i n t c o n t a c t s o n n - t y p e g e r m a n i u m . R e c o m b i n a t i o n , w h i c h was a s s u m e d n e g l i g i b l e l n t h e t h e o r y , i s a n o t h e r s o u r c e o f d i s c r e p a n c y b e t w e e n t h e t h e o r y a n d t h e e x p e r i m e n t a l r e s u l t s . To s a t i s f y t h e t h e o r e t i c a l a s s u m p t i o n s , i t I s n e c e s s a r y t o h a v e t h e h o l e d e n s i t y e q u a l t o i t s t h e r m a l e q u i l i b r i u m v a l u e a t t h e p l a n e b i s e c t i n g t h e s i d e - a r m s , b u t l t I s d o u b t f u l w h e t h e r t h i s was t r u e I n t h e e x p e r i m e n t s . I n c o n c l u s i o n , f u t u r e u n d e r s t a n d i n g o f m e t a l - s e m i c o n -d u c t o r c o n t a c t s c a n b e a p p r o a c h e d by d e t e r m i n i n g t h e m e c h a n i s m w h i c h s e t s t h e v a l u e o f M a t t h e c o n t a c t . F u r t h e r m o r e , r e c o m b i -n a t i o n s h o u l d b e c o n s i d e r e d s i n c e t h i s i m p l i e s a p o s i t i o n -d e p e n d e n t M . F i n a l l y , i t m u s t b e s t r e s s e d t h a t t h e t h e o r y p r e s e n t e d i n t h i s t h e s i s a s s u m e d a n o h m i c c o n t a c t o p p o s i t e t h e r e c t i f y i n g o n e . 51 B I B L I O G R A P H Y B a n b u r y , P . C . a n d H o u g h t o n , J . P h y s l c a 20, 1050 U954) B a r d e e n , J . P h y s . R e v . 21, 717-727 (1947) B a r d e e n , J . a n d B r a t t a l n , W . H . P h y s . R e v . 2k, 230-231 a n d 231-232 (1948) * B e t h e , H . A . M a s s a c h u s e t t s I n s t , o f T e c h . , R a d i a t i o n L a b . R e p o r t 43/12 (1942) B o r n e m a n , E . H . , S c h w a r z , R . F . , a n d S t i c k l e r , J . J . J . A p p l . P h y s . 26, 1021-1028 (1955) " D a v y d o v , B . T e c h . P h y s . ( U S S R ) 2, 87-95 (1938) E s a k i , L . P h y s . R e v . 1 0 2 , 603-604 (1958) H a r r i c k , N . J . P h y s . R e v . 102, H 7 3 - H 8 1 (1956) H a r r i c k , N . J . P h y s . R e v . H i , 876-882 (1959) H e i j n e , L . T h e s i s , U n i v e r s i t y o f A m s t e r d a m (i960); ( P h i l i p s R e s e a r c h R e p t s . S u p p l . N o . 4, (I96D) H e n i s c h , H . K . R e c t i f y i n g S e m i c o n d u c t o r C o n t a c t s . O x f o r d U n i v . P r e s s , L o n d o n (1957) H e r r i n g , C . a n d N i c h o l s , M . H . R e v . M o d . P h y s . 21, 185-270 (1949) H o l m , R . J . A p p l . P h y s . 22, 569-574 (195D K r o g e r , F . A . , D l e m e r , G . , a n d K l a s e n s , H . A . P h y s . R e v . 102, 279 (1956) L o w , G . G . S . P r o c . P h y s . S o c . 68B, 310-314 (1955) M l s a w a , T . J . P h y s . S o c . J a p a n 10, 362-367 (1955) M o t t , N . F . P r o c . R o y . S o c . ( L o n d o n ) 1 7 1 A . 27-38 (1939) * N o r d h e i m , L . W . Z . P h y s i k 7_5_, 434-441 (1932) » S c h o t t k y , W. Z . P h y s i k U2, 367-414 (1939) * S c h o t t k y , W. Z . P h y s i k 1 1 8 , 539-592 (1942) * S c h o t t k y , W. a n d S p e n k e , E . W l s s . V e r o f f . a . d . S i e m e n s -W e r k e n s 1 8 , 225-291 (1939) S h o c k l e y , W. B e l l S y s . T e c h . J . 28, 435-489 (1949) 52 Shockley, W. Turner, D.B. Valdes, L.B. Wilson, A.H. E l e c t r o n s and Holes In Semiconductors, D. Van Nostrand Co. Inc., P r i n c e t o n , N.J. (1950) J . E l e c t r o c h e m i c a l Soc. 106, 786-790 (1959) Proc. I.R.E. 42, 420-427 (1954) Proc. Roy. Soc. (London) 136, 487-498 (1932) The t e x t by H.K. Henlsch Is the source of i n f o r m a t i o n on these r e f e r e n c e s . 

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