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Electrical effects of non-uniform temperature distribution in current carrying conductors Anwar, Mohammad Zahural 1963

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E L E C T R I C A L EFFECTS OF NON-UNIFORM TEMPERATURE DISTRIBUTION I N CURRENT CARRYING CONDUCTORS by MOHAMMAD ZAHURUL ANWAR B . S c . ( H o n s . ) , U n i v e r s i t y o f D a c c a , 1955 M . S c . ,- U n i v e r s i t y o f D a c c a , 1956 A THESIS SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f PHYSICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA S e p t e m b e r , 1 9 6 3 . I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives., I t i s understood that copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Physics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date September 13, 1963. The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY MOHAMMAD ZAHURUL ANWAR B.Sc , U n i v e r s i t y of Dacca, Pakistan M.Sc, U n i v e r s i t y of Dacca, Pakistan TUESDAY, JANUARY 7, 1964, AT 4:00 P.M. IN ROOM 304, HENNINGS BUILDING (PHYSICS) of COMMITTEE IN CHARGE Chairman: F.H. Soward R.E. Burgess R. Barrie J . Grindlay J.C. Savage C.F. Schwerdtfeger C.A, Swanson External Examiner: A. van der Uni v e r s i t y of Minnesota Z i e l ELECTRICAL EFFECTS OF NON-UNIFORM TEMPERATURE DISTRIBUTION IN CURRENT-CARRYING CONDUCTORS ABSTRACT Anomalous e l e c t r i c a l behaviour may appear i n the d.c. and a.c. c h a r a c t e r i s t i c s of metals or semi-conductors i n which the current i s determined by both voltage and temperature. T h e o r e t i c a l investigations have been c a r r i e d out by assuming d i f f e r e n t models of heat flow and the conditions for the appearance of thermal breakdown and Negative Resistance (NR) have been obtained for both metals and semi-conductors. For purely l o n g i t u d i n a l heat flow, NR i s predicted for metals while f o r semi-conductors, the d.c. charac-t e r i s t i c i s of the'"breakdown" type without NR. On the other hand, the r a d i a l heat flow model predicts NR for semi-conductors and the conductivity modulation due to the r a d i a l temperature d i s t r i b u t i o n causes a concentration cpf current-density along the axis, giving r i s e to the "thermal pinch" e f f e c t . For metals, t h i s model does not predict NR and the r e s i s t i v i t y mo-dulation confines the current-density within a small depth from the surface giving r i s e to the "thermal skin e f f e c t " . NR i s also predicted for the model considering l o n g i t u d i n a l heat flow with surface heat loss, the d.c, /\. —K thermal theory, i n semi-conductors whereas for metals, the theory does not predict NR. For the a p p l i c a b i l i t y of the 'X-K thermal theory, the specimen must be thin enough to ensure an isothermal cross-section. The a.c. impedance of the specimen with a small a.c. voltage superimposed on the d.c. bias has been obtained for the yV-K thermal theory. Theoreti-c a l analysis shows that a non-zero surface loss parameter % i s e s s e n t i a l for a t t a i n i n g NR i n semi-conductors. . Experiments were performed with metals and semi-conductors i n an attempt to check the d.c. and a.c. 'A-K thermal theories. Comparison of the experiments with the theory shows that for semi-conductors, the ^X-K thermal theory i s v a l i d for current-density J 2 0 amps.cm"2 while for metals, i t i s v a l i d for J «5 x lO^ amps.cm"2. The measured a.c. characteris-t i c s at both low and high frequencies are interpreted on the basis of the a.c. 'X-K thermal theory but over the intermediate frequency region, the theory o f f e r s no explanation for the " c i r c u l a r arc" locus of impe-dances observed experimentally for both metals and semi-conductors. The present i n v e s t i g a t i o n enables one to determine the character of heat flow from measurements of the e l e c t r i c a l c h a r a c t e r i s t i c s of the specimen and also to d i s t i n g u i s h the thermal e f f e c t s which may be present i n other experiments (e.g.-on the "magnetic pinch"). -GRADUATE STUDIES F i e l d of Study: Physics Electromagnetic Theory Theory of Measurement Low Temperature Physics Physics of the S o l i d State Waves G.M. Volkoff J.R...Prescott J.B, Brown R.E. Burgess J.C. Savage Related Studies: C r y s t a l Structures J . T r o t t e r K.B-. Harvey D i f f e r e n t i a l Equations . C./ Fr.oese Functions of a Comples Variable H.A...Thurston - i i -ABSTRACT Anomalous e l e c t r i c a l b e h a v i o u r may a p p e a r i n t h e d . c a n d a . c . c h a r a c t e r i s t i c s o f m e t a l s o r s e m i - c o n d u c t o r s i n w h i c h t h e c u r r e n t i s d e t e r m i n e d b y b o t h v o l t a g e and tempe a t u r e . T h e o r e t i c a l i n v e s t i g a t i o n s h a v e been c a r r i e d o u t b y a s s u m i n g d i f f e r e n t m o d e l s o f h e a t f l o w and t h e c o n d i -t i o n s f o r t h e a p p e a r a n c e o f t h e r m a l b r e a k d o w n and N e g a -t i v e R e s i s t a n c e (NR) h a v e been o b t a i n e d f o r b o t h m e t a l s a n d s e m i - c o n d u c t o r s . F o r p u r e l y l o n g i t u d i n a l h e a t f l o w , NR i s p r e d i c t e d f o r m e t a l s w h i l e f o r s e m i - c o n d u c t o r s , t h e d . c . c h a r a c -t e r i s t i c i s o f t h e " b r e a k d o w n " t y p e w i t h o u t NR. On t h e o t h e r h a n d , t h e r a d i a l h e a t f l o w m o d e l p r e d i c t s NR f o r s e m i - c o n d u c t o r s and t h e c o n d u c t i v i t y m o d u l a t i o n due t o t h e r a d i a l t e m p e r a t u r e d i s t r i b u t i o n c a u s e s a c o n c e n -t r a t i o n o f c u r r e n t - d e n s i t y a l o n g t h e a x i s , g i v i n g r i s e t o t h e " t h e r m a l p i n c h " e f f e c t . F o r m e t a l s , t h i s m o d e l d o e s n o t p r e d i c t NR a n d t h e r e s i s t i v i t y m o d u l a t i o n c o n -f i n e s t h e c u r r e n t - d e n s i t y w i t h i n a s m a l l d e p t h f r o m t h e s u r f a c e g i v i n g r i s e t o t h e " t h e r m a l s k i n e f f e c t . ? . NR i s a l s o p r e d i c t e d f o r t h e m o d e l c o n s i d e r i n g l o n g i t u d i n a l h e a t f l o w w i t h s u r f a c e h e a t l o s s , t h e d . c . X~K t h e r m a l t h e o r y , i n s e m i - c o n d u c t o r s w h e r e a s f o r m e t a l s t h e t h e o r y does n o t p r e d i c t NR. F o r t h e a p p l i c a b i l i t y o f t h e X-K t h e r m a l t h e o r y , t h e s p e c i m e n must be t h i n enough - i i i -t o e n s u r e an i s o t h e r m a l c r o s s - s e c t i o n . The a . c . impedance o f t h e s p e c i m e n w i t h a s m a l l a . c . v o l t a g e s u p e r i m p o s e d o n t h e d . c . b i a s h a s been o b t a i n e d f o r t h e A~K t h e r m a l t h e o r y . T h e o r e t i c a l a n a l y s i s shows t h a t a n o n - z e r o s u r f a c e l o s s p a r a m e t e r X i s e s s e n t i a l f o r a t t a i n i n g NR i n s e m i -c o n d u c t o r s . E x p e r i m e n t s were p e r f o r m e d w i t h m e t a l s a n d s e m i -c o n d u c t o r s i n an a t t e m p t t o c h e c k t h e d . c , and a . c . A.-K t h e r m a l t h e o r i e s . 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 s w i t h t h e t h e o r y shows t h a t f o r s e m i - c o n d u c t o r s , t h e A.-K t h e r m a l t h e o r y i s v a l i d f o r c u r r e n t - d e n s i t y J £ 20 amps.cm 2 w h i l e f o r m e t a l s , i t i s v a l i d f o r J - 5 x 10^ amps, cm . The m e a s u r e d a . c . c h a r a c t e r i s t i c s a t b o t h l o w a n d h i g h f r e q u e n c i e s a r e i n t e r p r e t e d on t h e b a s i s o f t h e a . c . X - K t h e r m a l t h e o r y b u t o v e r t h e i n t e r m e d i a t e f r e q u e n c y r e g i o n , t h e t h e o r y o f f e r s no e x p l a n a t i o n f o r t h e " c i r c u l a r arc** l o c u s o f impedances o b s e r v e d e x p e r i -m e n t a l l y f o r b o t h m e t a l s a n d s e m i - c o n d u c t o r s . The p r e s e n t i n v e s t i g a t i o n e n a b l e s one t o d e t e r m i n e t h e c h a r a c t e r o f h e a t f l o w f r o m measurements o f t h e e l e c t r i c a l c h a r a c t e r i s t i c s o f t h e s p e c i m e n and a l s o t o d i s t i n g u i s h t h e t h e r m a l e f f e c t s w h i c h may be p r e s e n t i n o t h e r e x p e r i m e n t s ( e . g . on t h e " m a g n e t i c p i n c h " ) . x i i -ACKNOWLEDGEMENTS I s h o u l d l i k e t o e x p r e s s my g r a t i t u d e t o P r o f e s s o r R . E . B u r g e s s , my t h e s i s s u p e r v i s o r , f o r h i s i n v a l u a b l e g u i d a n c e a n d a s s i s t a n c e d u r i n g t h e p r e p a r a t i o n o f t h i s w o r k . I am a l s o i n d e b t e d t o t h e E x t e r n a l A i d O f f i c e o f Canada f o r p r o v i d i n g t h e f i n a n c i a l s u p p o r t i n t h e f o r m o f a s c h o l a r s h i p u n d e r t h e a u s p i c e s o f t h e t e c h n i c a l c o - o p e r a t i o n scheme o f t h e Colombo P l a n a n d a l s o t o t h e D e f e n s e R e s e a r c h B o a r d o f Canada f o r summer s u p p l e m e n t . ~ i v -CONTENTS CHAPTER 1. 1.1 1 .2 CHAPTER 2 . 2 , 1 2 . 2 2 . 3 2 . 4 2.5 CHAPTER 3_. 3 . 1 3 . 2 INTRODUCTION O b j e c t and Scope o f t h e T h e s i s N e g a t i v e R e s i s t a n c e (NR) and T h e r m a l I n s t a b i l i t y D . C . PROPERTIES FOR DIFFERENT MODELS OF HEAT FLOW L o n g i t u d i n a l H e a t F l o w : E x p o n e n t i a l M o d e l ( S e m i - C o n d u c t o r s a n d M e t a l s ) L o n g i t u d i n a l H e a t F l o w : L i n e a r M o d e l ( S e m i - C o n d u c t o r s a n d M e t a l s ) R a d i a l H e a t F l o w : E x p o n e n t i a l M o d e l ( S e m i - C o n d u c t o r a n d M e t a l s ) R a d i a l H e a t F l o w : L i n e a r M o d e l ( S e m i - C o n d u c t o r a n d M e t a l s ) X - K M o d e l : L o n g i t u d i n a l H e a t F l o w w i t h S u r f a c e H e a t L o s s ( S e m i -c o n d u c t o r s a n d M e t a l s A . C . PROPERTIES OF THE X - K MODEL A . C . T h e r m a l T h e o r y f o r S e m i -c o n d u c t o r s A . C . T h e r m a l T h e o r y f o r M e t a l s Page 1 1 6 12 16 19 27 27 35 =v~ Page CHAPTER 4 . STEP-FUNCTION BEHAVIOUR OF THE 4 . 1 4 . 2 A - K MODEL 3 8 T h e r m a l T ime C o n s t a n t a s a F u n c t i o n o f C u r r e n t a n d E x t e r n a l C i r c u i t (Lumped M o d e l ; S e m i - C o n d u c t o r s and M e t a l s ) 3 8 T h e r m a l Time C o n s t a n t a s a F u n c t i o n o f t h e B i a s i n g C u r r e n t a n d E x t e r n a l C i r c u i t ( D i s t r i b u t e d M o d e l ; S e m i - C o n d u c t o r s a n d M e t a l s ) 42 CHAPTER 5 . 5 . 1 5 . 2 5 . 3 SEMI-CONDUCTORS 49 P r e p a r a t i o n o f t h e S p e c i m e n a n d D . C . M e a s u r e m e n t s 49 A . C . M e a s u r e m e n t s 56 Measurement o f t h e E f f e c t i v e T h e r m a l Time C o n s t a n t o f t h e F i l a m e n t 58 CHAPTER 6 . 6 . 1 6 . 2 6 . 3 EXPERIMENTAL TECHNIQUES; METALS 60 S e l e c t i o n o f S p e c i m e n a n d D . C . M e a s u r e m e n t s 60 A . C . E x p e r i m e n t a l T e c h n i q u e 64 Measurement o f t h e E f f e c t i v e T h e r m a l Time C o n s t a n t o f t h e S p e c i m e n 65 ~ v i -Page CHAPTER 7 . EXPERIMENTAL RESULTS AND INTERPRETATIONS 67 7 . 1 S e m i - C o n d u c t o r s 67 7 . 2 M e t a l s 74 CHAPTER 8 . CONCLUSIONS 80 BIBLIOGRAPHY 8 3 - v i i -ILLUSTRATIONS F i g u r e F a c i n g Page 1 . 1a C u r r e n t - c o n t r o l l e d NR e x h i b i t e d by s e m i - c o n d u c t o r s 3 1 .1b V o l t a g e - c o n t r o l l e d NR e x h i b i t e d by m e t a l s 3 2 . 1 a 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 s e m i -c o n d u c t o r s h o w i n g " b r e a k d o w n " c h a r a c t e r -i s t i c w i t h o u t NR 8 2 . 1 b D . C . c h a r a c t e r i s t i c f o r m e t a l s h o w i n g v o l t a g e - c o n t r o l l e d t y p e o f NR 8 2 . 2 Dependence o f <5 T L on r 22 2 . 3 V a r i a t i o n o f R<r as a f u n c t i o n o f r 22 Ra 2 . 4 a D . C . c h a r a c t e r i s t i c s o f s e m i - c o n d u c t o r s w i t h c o n s t a n t X a n d v a r i a b l e K 23 2 . 4 b D . C . c h a r a c t e r i s t i c s f o r s e m i - c o n d u c t o r s w i t h f i x e d K and v a r i a b l e X 23 2 . 5 V a r i a t i o n o f T ( 0 ) a s a f u n c t i o n o f t h e T a c u r r e n t - d e n s i t y f o r m e t a l s 26 c % 2 . 6 Dependence o f o on c u r r e n t d e n s i t y 26 2 . 7 a 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 s w i t h c o n s t a n t K and v a r i a b l e X ( m e t a l s ) 26 2 . 7 b 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 s w i t h f i x e d X a n d v a r i a b l e K ( m e t a l s ) 26 - v i i i -F i g u r e F a c i n g Page 3 . 1 Dependence o f a . c . r e s i s t a n c e on f r e q u e n c y f o r s e m i - c o n d u c t o r b i a s e d i n t h e NR r e g i o n o f i t s d . c . c h a r a c -t e r i s t i c 31 3 . 2 Dependence o f r e a c t a n c e o n f r e q u e n c y on a l o g - l o g p l o t 31 3 . 3 Low f r e q u e n c y e q u i v a l e n t c i r c u i t f o r s e m i - c o n d u c t o r s 36 3 . 4 v a r i a t i o n o f a . c . r e s i s t a n c e w i t h f r e q u e n c y f o r m e t a l s 36 3 . 5 V a r i a t i o n o f r e a c t a n c e ( c a p a c i t i v e ) on a l o g - l o g p l o t f o r m e t a l s I 36 4 . 1 C i r c u i t f o r t h e r m a l s t e p - f u n c t i o n a n a l y s i s 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 s o f s e m i -c o n d u c t o r s a s a f u n c t i o n o f 47 5 . 1 P u l s e measurement o f 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 s p e c i m e n 49 5 . 2 B r i d g e f o r m e a s u r i n g t h e l i f e - t i m e o f t h e m i n o r i t y c a r r i e r s 52 5 . 3 B r i d g e - c i r c u i t f o r m e a s u r i n g t h e d . c . r e s i s t a n c e as a f u n c t i o n o f c u r r e n t o r t e m p e r a t u r e 53 5 . 4 D . C . 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 s e m i - c o n d u c t o r s p e c i m e n immersed i n a t e m p e r a t u r e b a t h 54 - i x -F a c i n g Page V a r i a t i o n o f t h e r e s i s t a n c e o f t h e s e m i -c o n d u c t o r s p e c i m e n a s a f u n c t i o n o f t e m p e r a t u r e 55 5 . 6 Dependence o f l o g R on T"" 1 55 5 . 7 C i r c u i t f o r c u r r e n t - v o l t a g e measurements 56 5 . 8 A . C . B r i d g e c i r c u i t f o r m e a s u r i n g t h e impedance o f t h e s p e c i m e n ( s e m i - c o n d u c t o r ) a s a f u n c t i o n o f f r e q u e n c y 57 5 . 9 B r i d g e f o r m e a s u r i n g t h e e f f e c t i v e t h e r m a l t i m e c o n s t a n t o f t h e f i l a m e n t 58 6 . 1 B r i d g e f o r m e a s u r i n g t h e d . c , r e s i s t a n c e o f t h e s p e c i m e n ( m e t a l ) a s a f u n c t i o n o f c u r r e n t o r t e m p e r a t u r e 61 6 . 2 C i r c u i t f o r c u r r e n t - v o l t a g e measurements 61 6 . 3 V a r i a t i o n o f r e s i s t a n c e o f t h e m e t a l s p e c i m e n a s a f u n c t i o n o f e x c e s s t e m p e r -a t u r e 63 6 . 4 A . C . B r i d g e f o r m e a s u r i n g t h e Impedance o f t h e s p e c i m e n (metal) as a f u n c t i o n o f f r e q u e n c y 64 6 . 5 B r i d g e f o r m e a s u r i n g t h e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n ( m e t a l ) 65 7 . 1 P l o t o f t h e d e c r e m e n t a l r e s i s t a n c e o f t h e s e m i - c o n d u c t o r s p e c i m e n a s a f u n c t i o n o f I 2 67 Dependence o f F ( r ) . o n r S t e a d y s t a t 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 s e m i c o n d u c t o r s p e c i m e n s h o w i n g NR V a r i a t i o n o f Ra l_R( 0 0 ) + R ( o ) J ( s e m i c o n d u c t o r s p e c i m e n ) on I 2 Dependence o f X ( w ) , • ,. o n f** 1 f o r t h e H 0 F o s e m i - c o n d u c t o r s p e c i m e n H i g h f r e q u e n c y r e s i s t a n c e as a f u n c t i o n o f f°"^/2 ( s e m i - c o n d u c t o r s p e c i m e n ) V a r i a t i o n o f X ( w ) T . w i t h f r e q u e n c y f o r t h e s e m i c o n d u c t o r s p e c i m e n 2 Dependence o f R(w) o n f L . F . ( s e m i - c o n d u c t o r ) L o c u s o f t h e impedance o f t h e s e m i -c o n d u c t o r s p e c i m e n S t e a d y - s t a t 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 m e t a l s p e c i m e n i n an e v a -c u a t e d e n c l o s u r e S t e a d y - s t a t 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 m e t a l s p e c i m e n u n d e r one a t m o s p h e r e p r e s s u r e V a r i a t i o n o f t h e i n c r e m e n t a l r e s i s t a n c e 2 w i t h I f o r t h e m e t a l s p e c i m e n i n a n e v a c u a t e d e n c l o s u r e - x i -F i g u r e F a c i n g Page 7 . 11b Dependence o f t h e i n c r e m e n t a l 2 r e s i s t a n c e on I f o r t h e m e t a l s p e c i m e n u n d e r one a t m o s p h e r e p r e s s u r e 75 7 . 1 2 a V a r i a t i o n o f X(w) a s a I H . F . I f u n c t i o n o f f " 1 ( m e t a l s p e c i m e n ) 76 - 3 / 2 7 . 1 2 b Dependence o f R ( w ) f l p o n f ( m e t a l s p e c i m e n ) 76 7 . 1 3 a V a r i a t i o n o f l o w f r e q u e n c y r e a c t a n c e as a f u n c t i o n o f f r e q u e n c y f o r t h e m e t a l s p e c i m e n 77 7 . 1 3 b V a r i a t i o n o f l o w f r e q u e n c y a . c . 2 r e s i s t a n c e w i t h f ( m e t a l s p e c i m e n ) 77 7 . 14 L o c u s o f impedance o f t h e m e t a l s p e c i m e n 78 - 1 -CHAPTER I  INTRODUCTION 1.1 O b j e c t a n d Scope o f t h e T h e s i s A n o m a l o u s f e a t u r e s may a p p e a r i n t h e d . c . a n d a . c . c h a r a c t e r i s t i c s o f s e m i - c o n d u c t o r s a n d m e t a l s due t o a n o n -u n i f o r m t e m p e r a t u r e d i s t r i b u t i o n . T h e s e have been o b s e r v e d b y d i r e c t measurement o f 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 s a n d by i m p l i c a t i o n o f t h e a p p e a r a n c e o f o s c i l l a t i o n s i n G e , I n S b a n d S i a t room t e m p e r a t u r e . I n e x p e r i m e n t s on " m a g n e t i c p i n c h " , one must be c a r e f u l t o e l i m i n a t e o r a t l e a s t a c c o u n t f o r t h e r m a l e f f e c t s . " T h e r m a l p i n c h " c a n g i v e r i s e t o a c u r r e n t - d e n s i t y d i s t r i b u t i o n o f p r e c i s e l y t h e same f o r m a s " m a g n e t i c p i n c h " . The p r e s e n t i n v e s t i g a t i o n shows t h a t a n o m a l i e s i n 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 c o n d u c t o r s c a n a p p e a r due t o t h e r m a l e f f e c t s . I t i s p o s s i b l e t h a t o s c i l l a t i o n s o b s e r v e d i n some e x p e r i m e n t s on s e m i - c o n d u c t o r s u s i n g h i g h c u r r e n t - d e n s i t y , o r h i g h e l e c t r i c f i e l d may a t l e a s t i n p a r t be a s c r i b e d t o t h o s e e l e c t r o - t h e r m a l e f f e c t s w h i c h c a n g i v e r i s e t o N e g a t i v e R e s i s t a n c e ( N R ) . The p u r p o s e o f t h e p r e s e n t w o r k i s t o d i s t i n g u i s h t h e r m a l e f f e c t s w h i c h may be p r e s e n t i n e x p e r i m e n t s where t h e s e e f f e c t s a r e u s u a l l y o v e r l o o k e d e . g . e x p e r i m e n t s on " m a g n e t i c p i n c h " . I t a l s o e n a b l e s one ~ 2 -t o d e t e r m i n e t h e c h a r a c t e r o f t h e h e a t f l o w by measurement o f t h e e l e c t r i c a l c h a r a c t e r i s t i c s o f t h e s p e c i m e n . The r e s u l t s o f t h e p r e s e n t i n v e s t i g a t i o n a r e r e l e v a n t t o t h e d e s i g n o f m e t a l a n d s e m i - c o n d u c t o r b o l o m e t e r s f o r m e a s u r i n g i n c i d e n t r a d i a n t p o w e r . Thesemay a l s o be h e l p f u l i n e x p l a i n i n g t h e a n o m a l i e s i n measurements o f e l e c t r i c a l p r o p e r t i e s due t o s e l f - h e a t i n g . The p r e d o m i n a n t c h a r a c t e r o f h e a t - f l o w ( l o n g i t u d i n a l o r r a d i a l ) d e t e r m i n e s t h e v a r i o u s i m p o r t a n t f e a t u r e s o f t h e s y s t e m e . g . t h e l e n g t h o f t h e p a t h i n v o l v e d i n t h e t h e r m a l r e l a x a t i o n t i m e . T h e r e a r e two b a s i c m o d e l s f o r t h e i n v e s -t i g a t i o n o f t h e t h e r m a l e f f e c t s , n a m e l y , ( i ) D i s t r i b u t e d t e m p e r a t u r e m o d e l i . e . t e m p e r a t u r e i s d i s t r i b u t e d e i t h e r l o n g i t u d i n a l l y o r r a d i a l l y d e p e n d i n g upon t h e c h a r a c t e r o f h e a t f l o w , ( i i ) Lumped t e m p e r a t u r e m o d e l i . e . t h e s p e c i m e n c a n be d e s c r i b e d by a s i n g l e t e m p e r a t u r e T a s i n t h e c a s e o f t h e t h e r m i s t o r o r e l e c t r i c b u l b . Some e a r l i e r w o r k s have been c a r r i e d o u t on t h e t u r n -o v e r phenomenon i n t h e r m i s t o r s ( B u r g e s s , 1955 a ) a n d o n t h e a . c . a d m i t t a n c e o f t e m p e r a t u r e - d e p e n d e n t c i r c u i t e l e m e n t s ( B u r g e s s , 1955 b ) . I n t h e s e m o d e l s , t h e d e v i c e was assumed t o be d e s c r i b e d b y a s i n g l e t e m p e r a t u r e T ( l u m p e d t e m p e r -a t u r e m o d e l ) . F i g . 1 . 1a : C u r r e n t - c o n t r o l l e d NR e x h i b i t e d by s e m i - c o n d u c t o r s . F i g . 1 , 1b: V o l t a g e - c o n t r o l l e d NR e x h i b i t e d by m e t a l s - 3 ~ 1 .2 N e g a t i v e R e s i s t a n c e (NR) a n d T h e r m a l I n s t a b i l i t y I n a s e m i - c o n d u c t o r , t h e e f f e c t o f h e a t i n g due t o t h e a p p l i e d power i s t o c a u s e t h e c u r r e n t t o i n c r e a s e more r a p i d l y w i t h i n c r e a s i n g v o l t a g e t h a n i f t h e t e m p e r a t u r e were h e l d c o n -s t a n t . T h i s c a n l e a d t o t h e e f f e c t o f " T u r n o v e r " i n w h i c h t h e s t e a d y s t a t 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 d i s p l a y s a v o l t a g e maximum a n d w i t h a f u r t h e r i n c r e a s e i n p o w e r , t h e s e m i - c o n d u c t o r w i l l d i s p l a y n e g a t i v e r e s i s t a n c e ( N R ) . T h e r e a r e two d i s t i n c t 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 s - t h e i s o t h e r m a l a n d t h e s t e a d y - s t a t e . I n t h e i s o -t h e r m a l c a s e , t h e c u r r e n t a n d t h e v o l t a g e a r e r e l a t e d i n s u c h a way t h a t t h e power d i s s i p a t e d i n t h e d e v i c e i s p r e v e n t e d f r o m p r o d u c i n g a n y r i s e i n t h e t e m p e r a t u r e o f t h e s p e c i m e n . T h i s may a l s o be a c h i e v e d by u s i n g p u l s e d v o l t a g e w i t h p u l s e -d u r a t i o n l e s s t h a n t h e t h e r m a l t i m e - c o n s t a n t o f t h e s p e c i m e n . I n t h e s t e a d y - s t a t e c a s e , s u f f i c i e n t t i m e i s a l l o w e d t o p a s s u n t i l t h e c u r r e n t comes t o i t s s t e a d y - s t a t e v a l u e a n d c o r r e s p o n d i n g l y , t h e t e m p e r a t u r e h a s a t t a i n e d I t s s t e a d y - s t a t e v a l u e . T h e r e a r e two t y p e s o f N R , n a m e l y , ( i ) c u r r e n t - c o n t r o l l e d t y p e o f NR ( F i g . 1 . 1a) ( i i ) v o l t a g e - c o n t r o l l e d t y p e o f NR ( F i g . 1 . 1 b ) . C u r r e n t - c o n t r o l l e d t y p e o f NR i s a l w a y s a s s o c i a t e d w i t h s e m i -c o n d u c t o r s w h i l e m e t a l s i n g e n e r a l e x h i b i t v o l t a g e - c o n t r o l l e d t y p e o f NR due t o t h e r m a l e f f e c t s . T h e r m a l l y p r o d u c e d NR i s c o m m o n l y - a s s o c i a t e d w i t h a r e l a t i v e l y l o n g r e l a x a t i o n t i m e o f t h e o r d e r o f a m i l l i - s e c o n d . I f , h o w e v e r , t h e c u r r e n t - c o n t r o l l i n g r e g i o n i s of s m a l l d i m e n -s i o n s a n d t h e s p e c i m e n h a s a h i g h t h e r m a l d i f f u s i v i t y ( t r u e a t l o w t e m p e r a t u r e ) , , t h e t h e r m a l t i m e " c o n s t a n t c a n e a s i l y be o f t h e o r d e r o f a m i c r o - s e c o n d . ( B u r g e s s , 1960 a ) When a s e r a i - c o n d u c t o r s p e c i m e n i s b i a s e d i n t h e NR r e g i o n o f i t s s t e a d y - s t a t 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 , t h e l o a d l i n e i n t e r s e c t s t h e d . c . c h a r a c t e r i s t i c a t m u l t i p l e p o i n t s The c u r r e n t i s t h e n a m u l t i - v a l u e d f u n c t i o n o f t h e v o l t a g e . T h i s may g i v e r i s e , t o t e m p e r a t u r e i n s t a b i l i t y a s t h e t e m p e r -a t u r e d i s t r i b u t i o n i s r e f l e c t e d 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 . S i m i l a r l y i n m e t a l s , t e m p e r a t u r e i n s t a b i l i t y may a r i s e when t h e e l e c t r i c f i e l d i s a m u l t i - v a l u e d f u n c t i o n o f t h e c u r r e n t ( i n t h e NR r e g i o n o f t h e d . c . c h a r a c t e r i s t i c ) . The o p e r a t i n g p o i n t i s t h e s o l u t i o n o f t h e f o l l o w i n g t h r e e e q u a t i o n s , g i v e n b y , I = F ( V , T ) < 1 » 2 » 1 ) V ' - E - I S ( 1 . 2 . 2 ) T ~ T a 3 ? < I V J ' ( 1 . 2 . 3 ) E q u a t i o n ( 1 . 2 . 1 ) r e f e r s t o t h e s t e a d y - s t a t 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 a n d e x p r e s s i o n ( 1 . 2 . 2 ) i s t h e e q u a t i o n o f t h e l o a d l i n e , E b e i n g t h e a p p l i e d emf a n d S i s t h e r e s i s t a n c e o f t h e e x t e r n a l c i r c u i t i n s e r i e s w i t h t h e s p e c i m e n . - 5 ~ F i n a l l y , e q u a t i o n ( 1 . 2 . 3 ) g i v e s t h e e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i n t h e s p e c i m e n due t o s e l f - h e a t i n g . F o r t h e " l u m p e d t e m p e r a t u r e ' * m o d e l , a n a l y s i s shows t h a t t h e c r i t e r i o n f o r s t a b i l i t y f o r c u r r e n t - c o n t r o l l e d t y p e o f NR i s S > R n a n d f o r v o l t a g e - c o n t r o l l e d t y p e o f N R , i s S < R n , where R^ i s t h e m a g n i t u d e o f t h e d i f f e r e n t i a l r e -s i s t a n c e / d v A a t t h e o p e r a t i n g p o i n t . CHAPTER 2 D . C . PROPERTIES FOR DIFFERENT MODELS OF HEAT—FLOW 2 . 1 L o n g i t u d i n a l H e a t F l o w : E x p o n e n t i a l M o d e l ( S e m i - C o n d u c t o r s and M e t a l s ) The c r i t e r i a f o r t h e a p p e a r a n c e o f NR i n t h e s t e a d y -s t a t 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 s f o r s e m i - c o n d u c t o r s a n d m e t a l s h a v e been t h e s u b j e c t s o f t h e o r e t i c a l i n v e s t i g a t i o n s i n t h i s c h a p t e r . I n t h i s m o d e l , i t i s assumed t h a t t h e h e a t f l o w due t o t h e p a s s a g e o f c u r r e n t i s p u r e l y l o n g i t u d i n a l , t h e r e b e i n g no s u r f a c e l o s s f r o m t h e s p e c i m e n . The f i l a m e n t i s assumed t o be o f l e n g t h 2 L , t h e two ends o f t h e s p e c i m e n b e i n g k e p t f i x e d a t some a m b i e n t t e m p e r a t u r e T . a The dependence o f c o n d u c t i v i t y o n t e m p e r a t u r e i s assumed t o be o f t h e f o r m , where Cj^ " i s t h e a m b i e n t c o n d u c t i v i t y a t t e m p e r a t u r e T , 0 = t - Ta i s t h e e x c e s s t e m p e r a t u r e o f any s e c t i o n o f t h e s p e c i m e n o v e r t h e a m b i e n t , a n d a i s t h e t e m p e r a t u r e c o e f f i c i e n t o f c o n -d u c t i v i t y , d e f i n e d by = fB(&* 0 ( 2 . 1 . 2 ) The d i f f e r e n t i a l e q u a t i o n , g o v e r n i n g t h e o n e - d i m e n s i o n a l f l o w o f h e a t i s g i v e n by where K i s t h e t h e r m a l c o n d u c t i v i t y o f t h e s e p c i m e n , E i s t h e e l e c t r i c f i e l d a n d J i s t h e c u r r e n t - d e n s i t y . Now, i n t h i s m o d e l , t h e c u r r e n t d e n s i t y J i s i n v a r i a n t w i t h r e s p e c t t o t h e d i s t a n c e a l o n g t h e l e n g t h o f t h e s p e c i m e n w h i l e t h e e l e c t r i c f i e l d E i s a f u n c t i o n o f t h e d i s t a n c e x . A s s u m i n g t h e t h e r m a l c o n d u c t i v i t y K o f t h e s p e c i m e n t o be i n d e p e n d e n t o f t h e v a r i a t i o n o f t e m p e r a t u r e a n d u s i n g e q u a t i o n ( 2 . 1 . 1 ) , e q u a t i o n ( 2 . 1 . 3 ) r e d u c e s t o i r +- —<r- e - o • ( 2 . i . 4 ) The s o l u t i o n o f ( 2 . 1 . 4 ) u n d e r t h e b o u n d a r y c o n d i t i o n s ( i ) A t x = O i . e . a t t h e c e n t r e o f t h e s p e c i m e n , d8 - 0 dx ( i i ) A t x = + L i . e . a t t h e two ends o f t h e s p e c i m e n , 0 = 0 i s g i v e n by ( 2 . 1 . 5 ) e « = "n»-)-T = f i « . LS= v y / » " ' i- > (2.1.6) where 6 ( x ) i s t h e e x c e s s t e m p e r a t u r e a t any d i s t a n c e x f r o m t h e c e n t r e o f t h e s p e c i m e n . The 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 i s c a s e i s F i g , 2 , 1 b , D . C . c h a r a c t e r i s t i c f o r m e t a l s h o w i n g v o l t a g e -c o n t r o l l e d t y p e o f N R . „ 8 -g i v e n by 2K ( 2 . 1 . 7 ) where A i s t h e c r o s s - s e c t i o n a l a r e a o f t h e s p e c i m e n . A p l o t o f c u r r e n t a g a i n s t v o l t a g e i n a c c o r d a n c e w i t h e q u a t i o n ( 2 . 1 . 7 ) , ( F i g . 2 . 1 a ) shows " B r e a k d o w n C h a r a c t e r -i s t i c " w i t h no NR, i n w h i c h t h e c u r r e n t goes t o i n f i n i t y c o r r e s p o n d i n g t o t h e v a l u e o f t h e maximum v o l t a g e , g i v e n by ( 2 . 1 . 8 ) The p r e v i o u s a n a l y s i s f o r s e m i - c o n d u c t o r s c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s by c h a n g i n g t h e s i g n o f t h e t e m p e r -a t u r e c o e f f i c i e n t o f c o n d u c t i v i t y i . e . a s s u m i n g t h e dependence o f c o n d u c t i v i t y v a r i a t i o n w i t h t e m p e r a t u r e o f t h e f o r m cne) _ ere" ( 2 . 1 . 9 ) F o r m e t a l s , t h e e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i s g i v e n b y 10) The c h a r a c t e r i s t i c e q u a t i o n f o r t h i s c a s e i s g i v e n by 1 = K ( 2 . 1 . 1 1 ) - 9 ~ T h i s m o d e l p r e d i c t s NR ( F i g . 2 . 1 b ) o f t h e v o l t a g e -c o n t r o l l e d t y p e and t h e v o l t a g e a n d c u r r e n t a t t u r n o v e r a r e r e s p e c t i v e l y g i v e n by Hi-V T = 4 . 3 ( 2 . 1 . 1 2 ) I = 0 . 9 4 a/_J< \ V where A i s t h e c r o s s - s e c t i o n a l a r e a o f t h e s p e c i m e n o = 2.3 • , ( 2 . 1 . 1 3 ) The " T u r n o v e r P o w e r " i s g i v e n by P - 4 . ^ ^ — . ( 2 . 1 . 1 4 ) A l s o , t h e e x c e s s t e m p e r a t u r e a t t h e c e n t r e a t T u r n o v e r i s g i v e n by e (o) - J U 2 . ( 2 . 1 . 1 5 ) T u r n o v e r a 2 . 2 L o n g i t u d i n a l H e a t F l o w : L i n e a r M o d e l ( S e m i - C o n d u c t o r s and M e t a l s ) T h i s m o d e l i s t h e same a s t h e " L o n g i t u d i n a l H e a t F l o w ( e x p o n e n t i a l ) " m o d e l e x c e p t i n g t h e f a c t t h a t i n t h i s m o d e l , t h e t e m p e r a t u r e dependence o f r e s i s t i v i t y i s assumed t o be o f t h e f o r m fHff) = y£ O - C e ) > ( 2 . 2 . 1 ) -10-w h e r e , ^){&) i s t h e r e s i s t i v i t y a t any e x c e s s t e m p e r a t u r e 6 , l^a, i s t h e a m b i e n t r e s i s t i v i t y o f t h e s p e c i m e n , C i s t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y , a n d 9 = T-T_ i s t h e e x c e s s t e m p e r a t u r e o v e r t h e Or a m b i e n t T . a The d i f f e r e n t i a l e q u a t i o n f o r one d i m e n s i o n a l f l o w o f h e a t i s g i v e n by ( s t e a d y - s t a t e ) i ( K i l ) V T " ( | - C 0 > = ° • < 2 - 2 - 2 ) A s s u m i n g K , t h e t h e r m a l 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 t o be i n d e p e n d e n t o f t h e v a r i a t i o n o f t h e t e m p e r a t u r e t h e r e q u i r e d d i f f e r e n t i a l e q u a t i o n t o be s o l v e d i s g i v e n by K ^ +/g T " - ( | - C 9 ) - O • ( 2 . 2 . 3 ) The s o l u t i o n o f ( 2 . 2 . 3 ) u n d e r t h e b o u n d a r y c o n d i t i o n s ( 2 . 1 . 5 ) [ i s g i v e n by c o s k x ( ^ ) ' x ( 2 . 2 . 4 ) The s t e a d y - s t a t 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 i s g i v e n b y , V - 2 ( ^ L ) " X f o , k T ( - ^ - ) " L - ( 2 . 2 . 5 ) - 1 1 -The 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 does n o t p r e d i c t n e g a t i v e r e s i s t a n c e a n d t h e c u r r e n t goes t o i n f i n i t y (Breakdown c h a r a c t e r i s t i c ) , c o r r e s p o n d i n g t o t h e v a l u e o f t h e maximum v o l t a g e g i v e n by ( 2 . 2 . 6 ) The above a n a l y s i s c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s by c h a n g i n g t h e s i g n o f t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y i . e . a s s u m i n g t h e r e s i s t i v i t y v a r i a t i o n w i t h temper* a t u r e t o be o f t h e f o r m , peg) - /&(i + ee) ( 2 . 2 . 7 ) where t h e s y m b o l s have t h e i r u s u a l s i g n i f i c a n c e . A s s u m i n g K t o be c o n s t a n t w i t h r e s p e c t t o t h e v a r i a t i o n o f t e m p e r a t u r e , t h e one d i m e n s i o n a l h e a t f l o w e q u a t i o n i s g i v e n by ( 2 . 2 . 8 ) The s o l u t i o n o f ( 2 . 2 . 8 ) u n d e r t h e b o u n d a r y c o n d i t i o n ( 2 . 1 . 5 ) i s g i v e n by Bi^-Titi-x: - £ The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c i s g i v e n by ctOS ( 2 . 2 . 9 ) - 1 2 -T h i s m o d e l does n o t e x h i b i t n e g a t i v e - r e s i s t a n c e a n d t h e maximum v a l u e o f t h e c u r r e n t i s g i v e n by •a / k_v>-where A i s t h e c r o s s - s e c t i o n o f t h e s p e c i m e n . l -1 {AT ' 2 . 3 . R a d i a l H e a t F l o w : E x p o n e n t i a l M o d e l ( S e m i - C o n d u c t o r s and M e t a l s ) I n t h i s m o d e l , t h e h e a t f l o w due t o t h e p a s s a g e o f c u r r e n t i s assumed t o be p u r e l y r a d i a l w i t h no s u r f a c e h e a t l o s s . The s u r f a c e o f t h e c y l i n d r i c a l s p e c i m e n o f r a d i u s r o i s assumed t o be k e p t f i x e d a t an a m b i e n t t e m p e r a t u r e T • a The c u r r e n t - d e n s i t y J i s a f u n c t i o n o f t h e r a d i a l d i s t a n c e r w h e r e a s t h e e l e c t r i c f i e l d E i s i n v a r i a n t w i t h r e s p e c t t o r . The c o n d u c t i v i t y dependence o n t e m p e r a t u r e i s assumed to be o f t h e f o r m ~ ^ I ^ 6 ' ( 2 . 3 . 1 . ) where 8 ( r ) = T ( r ) - T i s t h e e x c e s s t e m p e r a t u r e o v e r t h e a a m b i e n t and a = d ( I n (T ) i s t h e t e m p e r a t u r e c o e f f i c i e n t o f oTe c o n d u c t i v i t y . - 1 3 -A s s u m i n g t h e t h e r m a l 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 (K) t o be i n v a r i a n t w i t h r e s p e c t t o t h e t e m p e r a t u r e , t h e o n e -d i m e n s i o n a l d i f f e r e n t i a l e q u a t i o n f o r r a d i a l h e a t f l o w i s g i v e n by 4 - f- - 0 9 r ^ ' r 9r K The b o u n d a r y c o n d i t i o n s e m p l o y e d i n t h i s m o d e l a r e ( i ) A t r = O, d 9 - O d r ( i i ) A t r = r Q , 9 =. O The e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i s g i v e n by ( 2 . 3 . 2 ) ( 2 . 3 . 3 ) ( 2 . 3 . 4 ) Thus f r o m ( 2 . 3 . 1 ) a n d ( 2 . 3 . 4 ) , t h e c o n d u c t i v i t y d i s t r i b u t i o n i s g i v e n by F 7 r K V C -2. ( 2 . 3 . 5 ) A l s o , T V - 4 ( 2 . 3 . 6 ) Turnover s i n c e ( I E ) i > u r n o v e r — 4 v K a n d $*(°) i s t n e c o n d u c t i v i t y ~~ a a t t h e a x i s . The r i s e o f t e m p e r a t u r e w i t h 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 t h e e l e c t r i c a l c o n d u c t i v i t y t h e r e b y c a u s i n g a maximum c o n c e n t r a t i o n o f c u r r e n t - d e n s i t y a l o n g t h e a x i s . T h i s e f f e c t may be t e r m e d " T h e r m a l P i n c h " a n d i n some e x p e r i m e n t a l c a s e s , i t i s more i m p o r t a n t t h a n t h e " M a g n e t i c P i n c h " . The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c i s g i v e n by V ~ ~ ~ O L r v T ? > ( 2 . 3 . 7 ) where 2 L i s t h e l e n g t h o f t h e s p e c i m e n . N e g a t i v e r e s i s t a n c e o c c u r s i f t h e c u r r e n t e x c e e d s t h e t u r n o v e r v a l u e , IT - TTQ^^y^ > ( 2 . 3 . 8 a ) a t w h i c h t h e power i s P - ( 2 . 3 . 8 b ) a n d t h e e l e c t r i c f i e l d i s a t i t s maximum p o s s i b l e v a l u e £ r = 7. T ^ - l • (2-3-8c) ( 2 . 3 . 9 ) T h u s , K " z ' R b e i n g t h e " T u r n o v e r r e s i s t a n c e " . T F o r h i g h v a l u e s o f t h e c u r r e n t , t h e v o l t a g e d e c r e a s e s w i t h i n c r e a s i n g c u r r e n t , a c c o r d i n g t o ( 2 . 3 . 1 0 ) =•15— s o t h a t t h e power P a p p r o a c h e s a f i n i t e maximum g i v e n by where P T i s t h e " t u r n o v e r power**. A t t u r n o v e r , t h e t e m p e r a t u r e r i s e a t t h e a x i s i s Ted) ( 2 . 3 . 1 2 ) A l s o , t h e e x c e s s t e m p e r a t u r e a t t h e a x i s i n t e r m s o f power i s g i v e n by ( 2 . 3 . 1 3 ) 9(o) - - ^ . ^ ( j - J E w h e r e , O ^ P ^ 2 P T . The a n a l y s i s f o r t h e c a s e o f s e m i - c o n d u c t o r s c a n be c a r r i e d 1 o v e r t o t h e c a s e o f m e t a l s b y c h a n g i n g t h e s i g n o f t h e t e m p e r a t u r e c o e f f i c i e n t o f c o n d u c t i v i t y . Thus f o r m e t a l s , t h e e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i s g i v e n by G e o ^ \ Jb*. ( 2 . 3 . 1 4 ) The r e s „ t i v i t y d i s t r i b u t i o n i n t h i s c a s e i s g i v e n by - /S. i "2-UK ( 2 . 3 . 1 5 ) A c c o r d i n g t o ( 2 . 3 . 1 5 ) , one f a c e s a r e g i o n o f h i g h e r r e s i s t i v i t y on a p p r o a c h i n g t h e a x i s . Thus most o f t h e c u r r e n t - f l o w i s - 1 6 -c o n f i n e d w i t h i n a s m a l l d e p t h f r o m t h e s u r f a c e , g i v i n g r i s e t o a " T h e r m a l S k i n E f f e c t . " T h i s m o d e l does n o t p r e d i c t NR f o r m e t a l a n d t h e d . c . c h a r a c t e r i s t i c i s g i v e n by \ ( 2 , 3 . 1 6 ) The v o l t a g e goes t o i n f i n i t y when t h e c u r r e n t a p p r o a c h e s t h e maximum v a l u e I 7-( 2 . 3 . 1 7 ) 2 . 4 . R a d i a l H e a t F l o w : L i n e a r M o d e l ( S e m i - C o n d u c t o r s a n d M e t a l s ) I t i s assumed i n t h i s m o d e l t h a t t h e h e a t f l o w due t o t h e p a s s a g e o f c u r r e n t i s p u r e l y r a d i a l a n d t h e t e m p e r a t u r e dependence o f c o n d u c t i v i t y i s assumed t o be o f t h e f o r m , ( 2 . 4 . 1 ) where ( T ( 6 ) i s t h e c o n d u c t i v i t y o f t h e f i l a m e n t a t t h e e x c e s s t e m p e r a t u r e 9 , b i s t h e t e m p e r a t u r e c o e f f i c i e n t o f c o n d u c t i v i t y a n d 8 = T - T a i s t h e e x c e s s t e m p e r a t u r e o v e r t h e a m b i e n t t e m p e r a t u r e T . The d i f f e r e n t i a l e q u a t i o n f o r one d i m e n s i o n a l h e a t f l o w - 1 7 -( a s s u m i n g K t o be c o n s t a n t ) i s g i v e n b y r a r k (i + be) ~ o ( 2 . 4 . 2 ) The e x c e s s t e m p e r a t u r e © a t t h e d i s t a n c e r f r o m t h e c e n t r e u n d e r t h e b o u n d a r y c o n d i t i o n s ( 2 . 3 . 3 ) , i s g i v e n by ( 2 . 4 . 3 ) where J a n d J- a r e t h e B e s s e l f u n c t i o n s o f f i r s t k i n d a n d o f o 1 , o r d e r z e r o a n d one r e s p e c t i v e l y a n d r 0 b e i n g t h e r a d i u s o f t h e s p e c i m e n . The c u r r e n t - v o l t a g e r e l a t i o n s h i p i s g i v e n by ( 2 . 4 . 4 ) From t h e p r o p e r t y o f t h e B e s s e l f u n c t i o n s , i t i s s e e n f r o m e q u a t i o n ( 2 . 4 . 4 ) t h a t 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 i s o f t h e " B r e a k d o w n " t y p e w i t h o u t n e g a t i v e r e s i s t a n c e i n w h i c h t h e c u r r e n t goes t o i n f i n i t y when t h e v o l t a g e r e a c h e s i t s maximum v a l u e V m , g i v e n by ( 2 . 4 . 5 ) - 1 8 -2 L b e i n g t h e l e n g t h o f t h e s p e c i m e n . The f o r e g o i n g a n a l y s i s f o r s e m i - c o n d u c t o r s c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s by c h a n g i n g t h e s i g n o f t h e t e m p e r a t u r e c o e f f i c i e n t o f c o n d u c t i v i t y ( b ) . The e x c e s s t e m p e r a t u r e d i s t r i b u t i o n f o r m e t a l s i s g i v e n b y I. OWN:) ( 2 . 4 . 6 ) where L i s t h e m o d i f i e d B e s s e l f u n c t i o n o f f i r s t k i n d . o The c u r r e n t - v o l t a g e r e l a t i o n s h i p i s g i v e n by 1=. i i r r . ^ j /J£fiY^X,(EWr.) ( 2 . 4 . 7 ) where I a n d In a r e " m o d i f i e d B e s s e l f u n c t i o n s " o f t h e f i r s t O X k i n d and o f o r d e r z e r o and one r e s p e c t i v e l y . From t h e p r o p e r t i e s o f t h e " m o d i f i e d B e s s e l f u n c t i o n " , i t i s known t h a t =-! \ a s x oo . The LV* j 1 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 i s m o d e l does n o t e x h i b i t NR b u t t h e v o l t a g e goes t o i n f i n i t y c o r r e s p o n d i n g t o t h e maximum v a l u e o f t h e c u r r e n t , g i v e n by ( 2 . 4 . 8 ) The r e l a t i v e m a g n i t u d e o f r a d i a l and l o n g i t u d i n a l h e a t f l o w depends on t h e r a t i o o f t h e t h e r m a l c o n d u c t i v i t i e s o f t h e - 1 9 -s p e c i m e n a n d t h e a m b i e n t s u r r o u n d i n g t h e s p e c i m e n and a l s o on t h e g e o m e t r y o f t h e s y s t e m . I f a c y l i n d r i c a l s p e c i m e n o f l e n g t h 2 L a n d r a d i u s r Q and o f t h e r m a l c o n d u c t i v i t y K be s u r -r o u n d e d by a medium o f t h e r m a l c o n d u c t i v i t y K q > h a v i n g a f i x e d t e m p e r a t u r e on t h e s u r f a c e o f r a d i u s R . t h e r a t i o o f t h e r a d i a l o t o t h e l o n g i t u d i n a l h e a t f l o w i s g i v e n by ^ B u r g e s s (1960)^] - UCoHiU - | ( 2 . 4 . 9 ) where The s u r f a c e c o n d u c t i v i t y X i s t h e n g i v e n by \ - 2 > < 8 „ • ( 2 . 4 . 1 0 ) 2 . 5 X - K M o d e l ; L o n g i t u d i n a l H e a t F l o w w i t h S u r f a c e H e a t L o s s ( S e m i - C o n d u c t o r s and M e t a l s ) . The d i s t r i b u t i o n s o f p o t e n t i a l d i f f e r e n c e and e x c e s s t e m p e r a t u r e a l o n g t h i n c u r r e n t - c a r r y i n g c o n d u c t o r s ( s e m i -c o n d u c t o r s a n d m e t a l s ) have been t h e s u b j e c t o f t h e o r e t i c a l i n v e s t i g a t i o n s i n t h i s s e c t i o n . T h i s m o d e l assumes l o n g i t u d i n a l h e a t f l o w w i t h s u r f a c e h e a t l o s s p r o p o r t i o n a l t o t h e e x c e s s t e m p e r a t u r e o f any s e c t i o n o f t h e s p e c i m e n o v e r t h e a m b i e n t . The s p e c i m e n c h o s e n i s t h i n enough s o t h a t e a c h s e c t i o n o f t h e s p e c i m e n i s i s o t h e r m a l . =•20-The s p e c i m e n i s o f l e n g t h 2 L and e a c h end i s m a i n t a i n e d a t some f i x e d a m b i e n t t e m p e r a t u r e T * - a The d i f f e r e n t i a l e q u a t i o n , c o n t r o l l i n g t h e one d i m e n -s i o n a l f l o w o f h e a t i n t h e s t e a d y s t a t e due t o t h e p a s s a g e o f c u r r e n t , i s g i v e n b y , ^ . ( K j £ ) _ ^ e + E T + n T g = 0 , ( 2 . 5 . x , where K i s t h e t h e r m a l 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 , X i s t h e s u r f a c e thermal) c o n d u c t i v i t y , 9 = T - T a i s t h e e x c e s s t e m p e r a t u r e o v e r t h e a m b i e n t , E i s t h e e l e c t r i c f i e l d , J i s t h e c u r r e n t - d e n s i t y , and n i s t h e Thomson c o e f f i c i e n t . The p r e s e n c e o f t h e "Thomson h e a t " t e r m c a u s e s t h e maximum o f t h e t e m p e r a t u r e d i s t r i b u t i o n t o s h i f t f r o m t h e c e n t r e of t h e s p e c i m e n - t h e d i r e c t i o n o f t h e s h i f t d e p e n d i n g on t h e d i r e c t i o n o f t h e c u r r e n t . The e f f e c t o f t h i s t e r m becomes n e g l i g i b l e compared t o o t h e r t e r m s i n e q u a t i o n ( 2 . 5 . 1 ) p r o v i d e d . v\//-> 2-K V K J where 2 L i s t h e l e n g t h o f t h e s p e c i m e n . T h u s , a s s u m i n g t h e t h e r m a l 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 t o be i n v a r i a n t w i t h r e s p e c t t o t h e t e m p e r a t u r e v a r i a t i o n a n d n e g l e c t i n g t h e e f f e c t t o t h e "Thomson h e a t " t e r m , t h e r e q u i r e d d i f f e r e n t i a l e q u a t i o n c o n t r o l l i n g t h e f l o w o f h e a t i s g i v e n by - 2 1 -X ~ i - ^ 9 t E T O ( 2 . 5 . 2 ) The t e m p e r a t u r e dependence o f r e s i s t i v i t y i s assumed t o be o f t h e f o r m , ( 2 . 5 . 3 ) where,, ^ i s t h e a m b i e n t r e s i s t i v i t y , a i s t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y and 0 i s t h e e x c e s s t e m p e r a t u r e o v e r t h e a m b i e n t . The b o u n d a r y c o n d i t i o n s e m p l o y e d i n t h i s X-K model a r e g i v e n by A t x = 0 , d 0 — 0 dx A t x = £ L , 0 zz. 0 ( 2 . 5 . 4 ) The e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i s g i v e n by w h e r e , r K K The s t e a d y - s t a t e d . c . c h a r a c t e r i s t i c i s g i v e n b y , ( 2 . 5 . 5 a ) ( 2 . 5 . 5 b ) R + ( | C _ ] M i k . ( 2 . 5 . 6 ) R i (5LV The d i f f e r e n t i a l r e s i s t a n c e i . e . a . c . r e s i s t a n c e a t z e r o F i g . 2 . 2 Dependence o f S^L on r . 0.45-1 F i g . 2 . 3 V a r i a t i o n o f Rj. as a f u n c t i o n o f r . - 2 2 -f r e q u e n c y i s g i v e n b y Rio) ) T h i s m o d e l p r e d i c t s NR p r o v i d e d cSL^3 and r ^ O. A s s u m i n g tasjk 2-,-L 2r | > STL <<L\ ( v a l i d f o r £ T L ^ 3 ) a n d r » C S T L")V Seek"1" S T U , t h e v a l u e o f ( $ U ) a t t u r n -o v e r i s g i v e n by t h e c u b i c e q u a t i o n C S T L ) 3 - 3 ( ^ r L ) ^ - 2r(hTL) - O ( 2 . 5 . 8 ) w h e r e , S^L - L. » Jt b e i n & t h e t u r n o v e r c u r r e n t - d e n s i t y . i . e . w h e r e , CCS ( 2 . 5 . 9 ) <J> = Cos -I r ( 2 . 5 . 1 0 ) ( i * ¥ ) 3 / : The dependence o f ST L on r i s shown i n F i g . 2 . 2 . The c u r r e n t and v o l t a g e a t t u r n - o v e r a r e t h e n g i v e n by ( 2 . 5 . 1 1 ) [ i + z ( i + ¥ r c o s i j ( 2 . 5 . 1 2 ) F i g . 2 . 4 b D . C . c h a r a c t e r i s t i c s f o r s e m i - c o n d u c t o r s w i t h f i x e d K and v a r i a b l e \ . =23-( 2 . 5 . 1 3 ) A l s o , The dependence o f on t h e p a r a m e t e r r i n a c c o r d a n c e w i t h I N a. e q u a t i o n ( 2 . 5 . 1 3 ) i s i l l u s t r a t e d i n F i g . 2 . 3 . I t i s s e e n f r o m t h e above e q u a t i o n t h a t i < | < I • ( 2 . 5 . 1 4 , When K » 0 , t h e " d i s t r i b u t e d t e m p e r a t u r e m o d e l " p a s s e s o v e r t o t h e c a s e o f t h e " l u m p e d t e m p e r a t u r e m o d e l " and t h e v o l t a g e a n d c u r r e n t a t t u r n o v e r f o r t h i s c a s e , a r e g i v e n by a n d A I 1 \"*-L =- A ( T ^ - ) !> ( 2 . 5 . 1 6 ) ( 2 . 5 . 1 7 ) s o t h a t P T | ~ 1 when r — » 0 , t h i s m o d e l does n o t p r e d i c t N R . A g a i n when r—•*-<=*', I T — © e , w h i c h a l s o shows t h a t NR i s n o t a t t a i n a b l e i n t h i s c a s e . The s t e a d y s t a t 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 s f o r c o n s t a n t X and v a r i a b l e K and f o r f i x e d K and v a r i a b l e X a r e shown i n F i g . 2 . 4 a and F i g . 2 . 4 b r e s p e c t i v e l y . The s u r f a c e l o s s p a r a m e t e r X c a n be e x p e r i m e n t a l l y d e t e r m i n e d i n t h e f o l l o w i n g way. A s s u m i n g t h a t X ^ T * " -24= ( v a l i d f o r l o w v a l u e s o f t h e c u r r e n t ) , t h e d e c r e r o e n t a l d . c . r e s i s t a n c e i s g i v e n by •P _ 0 2 Thus f r o m t h e s l o p e o f t h e p l o t o f — — a g a i n s t I , r c a n be e x p e r i m e n t a l l y d e t e r m i n e d and t h u s X c a n be e v a l u a t e d u s i n g e q u a t i o n ( 2 . 5 . 5 b ) . The e x p e r i m e n t a l v a l u e o f X t h u s o b t a i n e d c a n be compared w i t h t h e t h e o r e t i c a l v a l u e o f X . A s s u m i n g N e w t o n ' s l a w o f c o o l i n g t o be v a l i d and n e g l e c t i n g t h e c o n v e c t i o n l o s s compared t o t h e r a d i a t i o n l o s s ( v a l i d i n an e v a c u a t e d e n c l o s u r e ) ] , t h e e x p r e s s i o n f o r X i s g i v e n by -X-- ^ ^ £ - ( 2 . 5 . 1 9 ) w h e r e , <T = 5 . 8 x 1 0 ~ 1 2 w a t t s cm"* 2 d e g " 4 i s t h e S t e f a n ' s c o n s t a n t € i s t h e e m i s s i v i t y and p t h e p e r i m e t e r o f t h e s p e c i m e n w i t h c r o s s - s e c t i o n a l a r e a A . I f , h o w e v e r , t h e c o n v e c t i o n l o s s i s n o t n e g l i g i b l e compared t o t h e r a d i a t i o n l o s s , X i s t h e n g i v e n by 3 = 4 £ £ £ f e f H£ (2.5.20) A A where H i s t h e h e a t - t r a n s f e r c o e f f i c i e n t . The f o r e g o i n g a n a l y s i s f o r s e m i - c o n d u c t o r s c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s b y c h a n g i n g t h e s i g n o f t h e -25-t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y . The r e s i s t i v i t y v a r i -a t i o n w i t h t e m p e r a t u r e i s a s s u m e d , f o r m e t a l s , t o be o f t h e f o r m - /2 ( I f * 0 ) > (2.5.21) where t h e s y m b o l s h a v e t h e i r u s u a l s i g n i f i c a n c e . T h e r e a r e two c a s e s i n t h i s m o d e l d e p e n d i n g on t h e v a l u e o f X . C a s e I . "X > afk ^ The e x c e s s t e m p e r a t u r e d i s t r i b u t i o n f o r t h i s c a s e i s g i v e n by where ^_ ^ - fl, ^ 3""*" K The d . c . c h a r a c t e r i s t i c f o r t h i s c a s e i s g i v e n by T h i s m o d e l does n o t e x h i b i t NR. C a s e I I . "X < dfl 3"^ T h i s c a s e i s a p p l i c a b l e f o r h i g h e r v a l u e s o f t h e c u r r e n t . The e x c e s s t e m p e r a t u r e a t a d i s t a n c e x f r o m t h e c e n t r e o f t h e s p e c i m e n i s g i v e n by where S^- ^ ^ ' ^ . K B e c a u s e o f t h e n a t u r e o f t h e s e c a n t f u n c t i o n , SL i n e q u a t i o n (2.5.24) i s bounded i . e . SL ^ J -v t e r i s t i c s f o r m e t a l s w i t h m e t a l s w i t h f i x e d \ a n d c o n s t a n t K a n d v a r i a b l e V . v a r i a b l e K . - 2 6 -The d . c . c h a r a c t e r i s t i c f o r t h i s m o d e l i s g i v e n by ^ - V 1 CSOy SL ( 6 U V ' ( 2 . 5 . 2 5 ) I n t h i s c a s e a l s o , 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 does n o t e x h i b i t NR a n d t h e v o l t a g e goes t o i n f i n i t y c o r r e s p o n d i n g t o t h e maximum v a l u e o f t h e c u r r e n t , g i v e n by The v a r i a t i o n o f " H S l a n d <5""*", where T ( o ) i s t h e t e m p e r a t u r e a t t h e c e n t r e o f t h e s p e c i m e n , w i t h c u r r e n t - d e n s i t y a r e shown i n F i g . 2 . 5 . and F i g . 2 . 6 . A l s o shown i n F i g . 2 . 7 a a n d F i g . 2 . 7 b a r e 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 s f o r f i x e d K and v a r i a b l e X a n d f o r c o n s t a n t X a n d v a r i a b l e K . - 2 7 -CHAPTER 3  A . C . PROPERTIES OF X - K MODEL 3 . 1 . A . C . T h e r m a l T h e o r y f o r S e m i - C o n d u c t o r s The p r e s e n t a n a l y s i s i s c o n c e r n e d w i t h t h e a . c . impedance o f a t e m p e r a t u r e - d e p e n d e n t c o n d u c t o r when a s m a l l a . c . v o l t a g e i s a p p l i e d s u p e r i m p o s e d on t h e d . c . b i a s w h i c h d e t e r m i n e s t h e o p e r a t i n g c o n d i t i o n o f t h e s p e c i m e n . T h i s i s t h e a . c . a n a l y s i s o f t h e d . c . X - K m o d e l o u t l i n e d i n C h a p t e r 2 . A s e x p e c t e d , t h e p r e s e n c e o f t h e r m a l i n e r t i a , i . e . l a g o f t e m p e r a t u r e r i s e b e h i n d t h e v a r i a t i o n o f t h e a p p l i e d power g i v e s r i s e t o r e a c t i v e impedance w h i c h may be e i t h e r i n d u c t i v e ( s e m i - c o n d u c t o r s ) o r c a p a c i t i v e ( m e t a l s ) . The v o l t a g e , c u r r e n t a n d e x c e s s t e m p e r a t u r e a r e a l l assumed t o have s t e a d y components w i t h r e l a t i v e l y s m a l l s u p e r -i m p o s e d a . c . c o m p o n e n t s , g i v e n by 6 ( x , t ) « © s ( x ) + 9^ ( x ) e x p i u t E ( x , t ) = E s ( x ) + E j ( x ) e x p i w t J ( t ) => J + J , e x p i U t s 1 ( 3 . 1 . 1 ) where ©_ « © . E , « E o a n d J . << J • X 8 JL S X S The d i f f e r e n t i a l e q u a t i o n c o n t r o l l i n g t h e o n e - d i m e n s i o n a l f l o w o f h e a t i s g i v e n by - 2 8 -( 3 . 1 . 2 ) where t h e new p a r a m e t e r c i s t h e h e a t c a p a c i t y p e r u n i t vo lume a n d K i s assumed t o i n d e p e n d e n t o f t h e v a r i a t i o n o f t e m p e r -a t u r e . On s u b s t i t u t i n g f r o m e q u a t i o n ( 3 . 1 . 1 ) i n t o e q u a t i o n ( 3 . 1 . 2 ) , one g e t s t h e d i f f e r e n t i a l e q u a t i o n f o r t h e a . c . c a s e , ^ £ * T . LCAC 9 , -O • ( 3 . 1 . 3 ) The t e m p e r a t u r e dependence o f r e s i s t i v i t y i s assumed t o be o f t h e f o r m f(B) = /g(i-^e) • W . I . * ) The b o u n d a r y c o n d i t i o n s e m p l o y e d i n t h i s A . C . \K m o d e l a r e A t x • 0 i . e . a t t h e c e n t r e , d © i _ _ - s dx = ' A t x = +^  L i . e . a t t h e two e n d s , ( 3 . 1 . 5 ) From e q u a t i o n s ( 3 . 1 . 1 ) , E i s g i v e n by ( a p p l y i n g Ohm's l a w ) , 1 E i = / g [ 3", CI ^ 9 S ) -h T 3 cx 0,"]]. ( 3 . 1 . 6 ) A g a i n by e q u a t i o n ( 2 . 5 . 5 ) where CesU SL K - 2 9 -T h u s , 1 V X + * £ 3 ^ V cosfc£L!/i- ' ( 3 . 1 . 7 ) H e n c e , t h e r e q u i r e d d i f f e r e n t i a l e q u a t i o n t o be s o l v e d t o d e t e r m i n e t h e a . c . e x c e s s t e m p e r a t u r e d i s t r i b u t i o n i s + 2 ccsU £ x - O ( 3 . 1 . 8 ) The s o l u t i o n o f ( 3 . 1 . 8 ) u n d e r t h e b o u n d a r y c o n d i t i o n s ( 3 . 1 . 5 ) i s g i v e n by cesh SL ( 3 . 1 . 9 ) where Z5 K The a . c . impedance a t any a n g u l a r f r e q u e n c y u i s g i v e n b y , 3 ( 3 . 1 . 1 0 ) -30-Then t h e a . c . r e s i s t a n c e and r e a c t a n c e a t any a n g u l a r f r e q u e n c y (j c a n be e x p r e s s e d a s 4. 2a|£."5^~A [^c ( * f tV i2y - - y s*V> / a x ) + ft + q f t 3 / X * 5 c 4 2 x t-y s ^ y ) ^ (3.1.11) a n d ?ec ~ a^j23/){a+^|»V)\^] C^ V/1) ( c a s k ^ c + c o S ^ ) ) { C * f a p . T j T ffcoej 1-} « ) c O + ^ ) * L ; (3.1.12) R(00) R(cu) -R(o) F i g . 3 . 1 : Dependence o f A . C . r e s i s t a n c e on f r e q u e n c y f o r s e m i - c o n d u c t o r b i a s e d i n t h e NR r e g i o n o f i t s d . c . c h a r a c t e r i s t i c . LnX(cu) *-l_ncu F i g . 3 . 2 : Dependence o f X ( u ) o n f r e q u e n c y o n a l o g - l o g p l o t . - 3 1 -where E x p r e s s i o n ( 3 . 1 . 1 1 ) r e d u c e s t o e q u a t i o n ( 2 . 5 . 7 ) w h i c h i s t h e d i f f e r e n t i a l r e s i s t a n c e o f t h e s p e c i m e n / d v \ i n t h e l i m i t V d i y o f ui t e n d i n g t o z e r o . A l s o f r o m e q u a t i o n ( 3 . 1 . 1 2 ) i t i s s e e n t h a t X ( w ) — 0 as u—»0. T h i s i s e x p e c t e d p h y s i c a l l y , a s a t z e r o f r e q u e n c y t h e t e m p e r a t u r e i s a b l e t o a t t a i n i t s s t e a d y v a l u e a t e a c h i n s t a n t i n a c c o r d a n c e w i t h t h e a p p l i e d Power I V . A s w oo , equa^Jb i o n ( 3 . 1 . 1 1 ) r e d u c e s t o w h i c h i s t h e d . c . r e s i s t a n c e o f t h e s p e c i m e n . A t v e r y h i g h f r e q u e n c y , t h e power i s v a r y i n g s o r a p i d l y t h a t t h e t e m p e r -a t u r e r e m a i n s c o n s t a n t a t t h e v a l u e d e t e r m i n e d by t h e a v e r a g e a p p l i e d p o w e r . A l s o , t h e r e a c t a n c e goes t o z e r o a s u—^ oc. The v a r i a t i o n s o f t h e a . c . r e s i s t a n c e a n d r e a c t a n c e b i a s e d i n t h e n e g a t i v e r e s i s t a n c e r e g i o n w i t h f r e q u e n c y a r e shown i n F i g . 3 . 1 and F i g . 3 . 2 . F o r t h i s c a s e ( s e m i - c o n d u c t o r s ) , t h e d ^ c . r e s i s t a n c e o f t h e s p e c i m e n i s a l w a y s g r e a t e r t h a n t h e d i f f e r e n t i a l r e s i s t a n c e , s i n c e =32= SL J ( 3 . 1 . 1 5 ) w h i c h i s a l w a y s ^ 0 , s i n c e , SL s^eJi^SL < te+J^SL < SL. E q u a t i o n s ( 3 . 1 . 1 1 ) and ( 3 . 1 . 1 2 ) a r e s o i n v o l v e d t h a t no u s e f u l i n f o r m a t i o n c a n be o b t a i n e d f r o m t h e i r b e h a v i o u r f o r t h e g e n e r a l c a s e . Hence t h e i r b e h a v i o u r a t b o t h l o w a n d h i g h f r e q u e n c y h a s been t h e s u b j e c t o f f u r t h e r i n v e s t i g a t i o n . A t l o w f r e q u e n c y s u c h t h a t w C « •^-d^'S^' ) > t h e a . c . r e s i s t a n c e i s g i v e n by b L *" 3 ( 3 . 1 . 1 6 ) where R ( o ) i s t h e d i f f e r e n t i a l r e s i s t a n c e o f t h e s p e c i m e n c o r r e s p o n d i n g t o t h e d . c . b i a s g i v e n by e q u a t i o n ( 2 . 5 . 7 ) . I f t h e s p e c i m e n i s b i a s e d i n t h e NR r e g i o n o f i t s d . c . c h a r a c t e r -- 3 3 -i s t i c , t h e l o w f r e q u e n c y a . c . c h a r a c t e r i s t i c p r e d i c t s NR. The r e a c t a n c e a t l o w f r e q u e n c y i s g i v e n by ( 3 - M 7 ) A t h i g h f r e q u e n c y a n d t h e a . c . r e s i s -t a n c e i s g i v e n b y where i t was assumed t h a t x ^ y ^> i . E q u a t i o n ( 3 . 1 . 1 8 ) e n a b l e s one t o d e t e r m i n e t h e t h e r m a l 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 by measurements o f i t s a . c . r e -s i s t a n c e a t h i g h f r e q u e n c i e s . A l s o t h e s e r i e s r e a c t a n c e a t h i g h f r e q u e n c i e s i s g i v e n b y The h e a t c a p a c i t y p e r u n i t vo lume o f t h e s p e c i m e n ( c ) c o r r e s -p o n d i n g t o t h e d . c . b i a s c a n be e x p e r i m e n t a l l y d e t e r m i n e d b y m e a s u r i n g t h e r e a c t a n c e o f t h e s p e c i m e n a t h i g h f r e q u e n c y , a n d u s i n g e q u a t i o n ( 3 . 1 . 1 9 ) . On e x a m i n i n g e q u a t i o n ( 3 . 1 . 1 2 ) , one s e e s t h a t i t i s a l g e b r a i c a l l y i n t r a c t a b l e t o see where t h e maximum o f X ( w ) l i e s . On a l o g - l o g p l o t o f X ( w ) a g a i n s t w , i t i s s e e n t h a t a t l o w - 3 4 -f r e q u e n c i e s , t h e r e a c t a n c e t a k e s o f f l i n e a r l y w i t h a s l o p e +1 and a t h i g h f r e q u e n c i e s t h e s l o p e i s - 1 . ( F i g . 3 . 2 ) The c r o s s - o v e r f r e q u e n c y i s e s t i m a t e d r o u g h l y ( u s i n g e q u a t i o n s ( 3 . 1 . 1 7 ) and ( 3 . 1 . 1 9 ) ) t o be z ^ SL ( 3 . 1 . 2 0 ) A l s o . X ( w ) i s a p p r o x i m a t e l y g i v e n by max. 4 ( 3 . 1 . 2 1 ) I n t e r m s o f t h e e q u i v a l e n t c i r c u i t o f F i g . 3 . 3 , e q u a t i o n ( 3 . 1 . 2 0 ) a n d ( 3 . 1 . 2 1 ) c a n be r e p r e s e n t e d by max X ( w ) max 2 ir L \ R 2 ( 3 . 1 . 2 2 ) w h e r e , » R ( o ) a n d ( 3 . 1 . 2 3 ) The e f f e c t i v e t h e r m a l t i m e c o n s t a n t T J i s t h e n g i v e n b y ( 3 . 1 . 2 4 ) 3 . 2 The a n a l y s i s o f S e c t i o n 3 . 1 f o r s e m i - c o n d u c t o r s c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s by c h a n g i n g t h e s i g n o f t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y i . e . a s s u m i n g t h e t e m p e r a t u r e dependence o f r e s i s t i v i t y o f t h e f o r m ( 3 . 2 . 1 ) w h i c h i s o b e y e d b y m e t a l s f o r a w i d e r a n g e o f t e m p e r a t u r e s . The e x c e s s a . c . t e m p e r a t u r e - d i s t r i b u t i o n f o r m e t a l s i s g i v e n b y I > O J F i g . 3 . 4 V a r i a t i o n o f a . c . r e s i s t a n c e w i t h f r e q u e n c y f o r m e t a l s . L.F. >Lncj H.F F i g . 3 . 5 . V a r i a t i o n o f r e a c t a n c e ( c a p a c i t i v e ) w i t h f r e q u e n c y on a l o g - l o g p l o t f o r m e t a l s . ! - 3 6 -( 3 . 2 . 2 ) ( 3 . 2 . 3 ) where The a . c . r e s i s t a n c e a n d r e a c t a n c e f o r t h i s c a s e c a n be o b t a i n e d f r o m e q u a t i o n s ( 3 . 1 . 1 1 ) a n d ( 3 . 1 . 1 2 ) by r e p l a c i n g a by - a ( F i g . 3 . 4 a n d F i g . 3 . 5 ) . A t l o w f r e q u e n c y ooc « a,fi/5^~-~\ , a n d t h e a . c . r e s i s t a n c e i s g i v e n by "> £ f f J ^ S L - is kt+i£L BL fc*>) J < M g y g T / y a ) O v ( 3 . 2 . 4 ) A l s o , t h e r e a c t a n c e a t l o w f r e q u e n c y i s g i v e n by - 3 7 -SL / J ( 3 . 2 . 5 J When "X «. 9 t h i s i s s i m P l l f i e d t o ( 3 . 2 . 6 ) A t h i g h f r e q u e n c i e s s u c h t h a t *->C c ^ ^ s — 7v a n d y L- | , t h e a . c . r e s i s t a n c e i s g i v e n by fiVco) ~ i ? ^ ) -V- ^ • ( 3 . 2 . 7 ) The t h e r m a l c o n d u c t i v i t y K o f t h e s p e c i m e n c a n be e x p e r i -m e n t a l l y d e t e r m i n e d by measurements o f t h e h i g h f r e q u e n c y a . c . r e s i s t a n c e . The r e a c t a n c e a t h i g h f r e q u e n c i e s i s g i v e n by a i "1 X* ' ( 3 . 2 . 8 ) E x p r e s s i o n ( 3 . 2 . 8 ) c a n be u s e d t o d e t e r m i n e t h e h e a t c a p a c i t y p e r u n i t vo lume ( c ) o f t h e s p e c i m e n f r o m h i g h f r e q u e n c y r e a c t a n c e m e a s u r e m e n t s . The e x p r e s s i o n f o r t h e c r o s s - o v e r f r e q u e n c y and X ( w ) . may be o b t a i n e d f o r : m e t a l s f r o m e q u a t i o n ( 3 . 1 . 2 1 ) and max. ( 3 . 1 . 2 2 ) b y c h a n g i n g t h e s i g n o f a . =38-CHAPTER 4 STEP-FUNCTION BEHAVIOUR OF X - K MODEL 4 . 1 T h e r m a l T i m e - C o n s t a n t as a F u n c t i o n o f C u r r e n t a n d  E x t e r n a l C i r c u i t (Lumped M o d e l . S e m i - C o n d u c t o r s a n d M e t a l s ) . I n t h i s m o d e l , i t i s assumed t h a t t h e s p e c i m e n c a n be c h a r a c t e r i z e d b y a f i x e d t e m p e r a t u r e T i . e . t h e t e m p e r a t u r e i s u n i f o r m t h r o u g h o u t t h e s p e c i m e n . I f t h e n o n - s t e a d y s t a t e c h a r a c t e r i s t i c i s c o n s i d e r e d , i t i s e v i d e n t t h a t t h e i n p u t e l e c t r i c a l power must e q u a l t h e sum o f t h e d i s s i p a t e d power p l u s t h e r a t e a t w h i c h t h e r m a l e n e r g y i s b e i n g s u p p l i e d t o t h e s p e c i m e n . I f t h e r a t e o f h e a t l o s s i s d e t e r m i n e d b y t h e i n s t a n t a n e o u s e x c e s s t e m p e r a t u r e 6 o v e r t h e a m b i e n t , t h e b a l a n c e o f power c o n d i t i o n may be e x p r e s s e d as T V = + ( 4 . 1 . D w h e r e , g i s t h e t h e r m a l c o n d u c t a n c e between t h e d e v i c e a n d t h e a m b i e n t , c i s t h e h e a t c a p a c i t y o f t h e s p e c i m e n . I f a power V 0 I Q = g & Q h a s been a p p l i e d t o t h e s p e c i m e n f o r s u c h a t i m e t h a t a l l t r a n s i e n t s h a v e d i e d down and t h e n t h e s p e c i m e n i s o p e n - c i r c u i t e d , t h e e x c e s s t e m p e r a t u r e © w i l l be g i v e n b y s i =io+r, (t) ••0 • e V= V 0 +V, (tj F i g . 4 . 1 C i r c u i t f o r t h e r m a l s t e p - f u n c t i o n a n a l y s i s . =39-9= 9.e ( 4 . 1 . 2 ) The q u a n t i t y " £ " i s c a l l e d t h e " i n t r i n s i c t h e r m a l t i m e g c o n s t a n t " o f t h e s p e c i m e n and w i l l be d e n o t e d by C&• When t h e w o r k i n g p o i n t o f t h e s p e c i m e n i s c h o s e n s o t h a t t h e power d i s s i p a t i o n i n i t r a i s e s i t s t e m p e r a t u r e above t h a t o f t h e a m b i e n t , t h e n i n g e n e r a l t h e r a t e o f h e a t l o s s i s n o t c h a r a c t e r i z e d by t h e " i n t r i n s i c t h e r m a l t i m e c o n s t a n t " " e f f e c t i v e t h e r m a l t i m e c o n s t a n t " *2~ *• T h i s i s b e c a u s e c h a n g i n g i n s t a n t a n e o u s power d i s s i p a t i o n d u r i n g t e m p e r a t u r e change ( f o r s e m i - c o n d u c t o r s ) c a n e i t h e r a i d o r i n h i b i t t h e r a t e o f l o s s o f h e a t a c c o r d i n g a s t h e e l e c t r i c a l f e e d a p p r o a c h e s c o n s t a n t v o l t a g e o r c o n s t a n t c u r r e n t c o n d i t i o n . We c o n s i d e r t h e a p p l i c a t i o n o f a s m a l l i n c r e m e n t a l s t e p f u n c t i o n o f v o l t a g e e t o a s e r i e s c o m b i n a t i o n o f t h e s p e c i -men and r e s i s t a n c e S s u p e r i m p o s e d o n a d i r e c t v o l t a g e E 0 w h i c h h a s b e e n a p p l i e d f o r a t i m e s u f f i c i e n t l y l o n g f o r a s t e a d y s t a t e c o n d i t i o n t o have been r e a c h e d . ( F i g . 4 . 1 ) t0 b u t b y a m o d i f i e d t h e r m a l t i m e c o n s t a n t , c a l l e d t h e Now, f o r t £ O , a r _ i w ( 4 . 1 . 3 ) a t ~ " s vt n o -where , I - I + I x ( t ) V - V + V± ( t ) V , ( 4 . 1 . 4 ) a n d X l « I o , V L « V o . F o r s e m i - c o n d u c t o r s , t h e t e m p e r a t u r e dependence o f r e s i s t a n c e i s assumed t o be o f t h e f o r m , R « R (1 - a ©) ( 4 . 1 . 5 ) a. • a b e i n g t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t a n c e . Thus U s i n g e q u a t i o n ( 4 . 1 . 6 ) , e q u a t i o n ( 4 . 1 . 1 ) c a n be w r i t t e n a s at Thus t h e e f f e c t i v e t h e r m a l t i m e c o n s t a n t f o r s e m i -c o n d u c t o r s i s g i v e n b y where ^© = £ i s t h e i n t r i n s i c t h e r m a l t i m e c o n s t a n t o f t h e S s p e c i m e n . T h i s e f f e c t i v e t h e r m a l t i m e c o n s t a n t t may be u s e d a s a c r i t e r i o n o f t h e r m a l i n s t a b i l i t y i n t h e s e n s e t h a t when i s p o s i t i v e , a l l t h e t r a n s i e n t s d i e down a f t e r a s u f f i -c i e n t l y l o n g t i m e and t h e s y s t e m i s s t a b l e . B u t a n e g a t i v e v a l u e o f 2" may l e a d t o t h e r m a l i n s t a b i l i t y . - 4 1 -Thus f o r s e m i - c o n d u c t o r s when S R, t h e r e i s no t h e r m a l i n s t a b i l i t y whereas f o r S < R , t h e r m a l i n s t a b i l i t y may a r i s e p r o v i d e d f^f * f K ^ f - (-.1..) S i n c e i n t h e NR r e g i o n o f a s e m i - c o n d u c t o r where t h e l o a d l i n e i n t e r s e c t s 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 a t m u l t i p l e p o i n t s , t h e c u r r e n t i s a m u l t i - v a l u e d f u n c t i o n o f t h e v o l t a g e a n d hence t e m p e r a t u r e i n s t a b i l i t y may a r i s e a s t h e t e m p e r a t u r e d i s t r i b u t i o n i s r e f l e c t e d i n t h e I - V c h a r a c t e r i s t i c . When t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y i s z e r o , a s e x p e c t e d s i n c e t h e r m a l f e e d b a c k : i s a b s e n t i n t h i s c a s e . T h e r m a l f e e d b a c k i s due t o t h e f a c t t h a t t h e power s u p p l i e d t o t h e s p e c i m e n i s v a r y i n g w i t h t i m e b e c a u s e o f t h e change o f r e s i s t a n c e o f t h e s p e c i m e n . The t h e r m a l f e e d b a c k i s a l s o a b s e n t when t h e e x t e r n a l r e s i s t a n c e i n s e r i e s w i t h t h e s p e c i m e n e q u a l s t h e r e s i s t a n c e o f t h e s p e c i m e n ( c o n s t a n t power s o u r c e ) . I n t h i s c a s e , m a x i -mum power i s s u p p l i e d t o t h e s p e c i m e n a n d i s c o n s t a n t as c a n be s e e n f r o m t h e e q u a t i o n ( 4 . 1 i l 0 ) Any d e c r e a s e o f V i n P = I V i n c r e a s e s t h e c u r r e n t I t e n d i n g t o k e e p t h e power P. c o n s t a n t . I t i s a l s o s e e n f r o m e q u a t i o n ( 4 . 1 . 8 ) t h a t f o r s e m i -=42-c o n d u c t o r t h e e f f e c t i v e t h e r m a l t i m e c o n s t a n t i s s m a l l e r t h a n t h e i n t r i n s i c t h e r m a l t i m e c o n s t a n t "fo f o r a c o n s t a n t c u r r e n t s o u r c e whereas f o r a c o n s t a n t v o l t a g e s o u r c e , i t i s g r e a t e r t h a n *c"l • The f o r e g o i n g a n a l y s i s f o r s e m i - c o n d u c t 0 1 6 c a n be c a r r i e d o v e r t o t h e c a s e o f m e t a l s by c h a n g i n g t h e s i g n o f t h e t e m p e r -a t u r e c o e f f i c i e n t o f r e s i s t i v i t y . The e f f e c t i v e t h e r m a l t i m e c o n s t a n t f o r m e t a l s i s g i v e n by Z ~ | , £ g » T - E - ~ ' ( 4 - i a i ) When S ;> R, t e m p e r a t u r e i n s t a b i l i t y may a r i s e i n m e t a l s p r o v i d e d ~ ~ a ! > a ( 4 . 1 . 1 2 ) F o r S < R , t h e r e c a n n o t be any t h e r m a l i n s t a b i l i t y . F o r a c o n s t a n t - c u r r e n t s o u r c e (S ^> R ) , f >Ti w h e r e a s f o r a c o n -s t a n t - v o l t a g e s o u r c e ( S « R ) , t<T0 . A g a i n f o r c o n s t a n t power s o u r c e (S = R ) , 'ZT^'To due * ° * n e a b s e n c e o f t h e phenomenon o f t h e r m a l f e e d b a c k . 4 . 2 T h e r m a l Time C o n s t a n t a s a F u n c t i o n o f t h e B i a s i n g C u r r e n t a n d E x t e r n a l C i r c u i t ( D i s t r i b u t e d M o d e l ; S e m i - C o n d u c t o r s a n d M e t a l s ) I n t h e c a s e o f d i s t r i b u t e d m o d e l , t h e d i f f e r e n t i a l e q u a t i o n f o r one d i m e n s i o n a l h e a t f l o w f o r t h e u n s t e a d y =43-s t a t e c a s e i s g i v e n by ( a s s u m i n g K t o be c o n s t a n t ) -70 ^ c | | - , ( 4 . 2 . D where c i s t h e h e a t c a p a c i t y p e r u n i t vo lume o f t h e s p e c i m e n . The p r e s e n t a n a l y s i s i s c o n c e r n e d w i t h t h e " e f f e c t i v e t h e r m a l t i m e c o n s t a n t " o f a t e m p e r a t u r e d e p e n d e n t s p e c i m e n ( s e m i - c o n d u c t o r a n d m e t a l ) when a s m a l l i n c r e m e n t a l s t e p -f u n c t i o n o f v o l t a g e e i s a p p l i e d t o a s e r i e s c o m b i n a t i o n o f t h e s p e c i m e n a n d r e s i s t a n c e S , s u p e r i m p o s e d on a d i r e c t v o l t a g e E g . Now, f o r t £ 0 , 2t ~ - AS H " "AS 7>t ( 4 t 2 * 2 ) w h e r e , J - J 0 + J , ( t ) ^ V = V g + V1 ( t ) y ( 4 t 2 . 3 ) a n d J x ( t ) < < J g ; V x ( t ) « Vg The t e m p e r a t u r e dependence o f r e s i s t i v i t y i s assumed t o be o f t h e f o r m a b e i n g t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y . - 4 4 -Thus 9 E _ T - A S ( 4 - 2 - 5 ) Vi-SZ ' * 9 t F o r c o n s t a n t c u r r e n t (S"5>R)-, . t h e s o l u t i o n o f e q u a t i o n ( 4 . 2 . 1 ) u n d e r t h e b o u n d a r y c o n d i t i o n s ( 2 . 5 . 4 ) i s g i v e n by ^ C a r s l a w and J a e g e r ; 1959^] oo .1. ( 4 . 2 . 6 ) where n » 0 , 1, 2 , e t c . . B = / 6 ^ T ^ . ( 4 . 2 . 7 ) a n d - > + a f t T ^ . From e q u a t i o n ( 4 . 2 . 6 ) , t h e t h e r m a l t i m e c o n s t a n t "t^ c o r r e s p o n d i n g t o t h e n t h mode i s g i v e n by 4 e ( 4 . 2 . 8 ) - 4 5 -1 B Q i s t h e l a r g e s t o f t h e B n f s a n d XQ i s t h e l a r g e s t o f 2"n ' s . Hence 6Q i s t h e d o m i n a n t component f o r a l l t i m e , An i n s p e c t i o n o f e q u a t i o n ( 4 . 2 . 8 ) shows t h a t a l l 2Tn's a r e p o s i t i v e . The t h e r m a l t i m e c o n s t a n t f o r m e t a l s u n d e r c o n s t a n t -c u r r e n t c o n d i t i o n i s g i v e n by T- = >-a/a.:r+cw.ri£ (4-2,9) where a i s t h e p o s i t i v e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y . Now, a c c o r d i n g t o t h e X - K m o d e l ( see a l s o F i g . 4 . 2 ) , t h e c u r r e n t - d e n s i t y J i s bounded by Q / 1 T ^ ( 4 . 2 . 1 0 ) Thus a l l T n * s a r e p o s i t i v e i m p l y i n g t h a t t h e r m a l i n -s t a b i l i t y c a n n o t a r i s e i n m e t a l s f o r c o n s t a n t c u r r e n t . The c r i t e r i o n f o r s t a b i l i t y c a n a l s o be d e r i v e d f r o m t h e t r a n s i e n t a n a l y s i s o f t h e e q u i v a l e n t c i r c u i t . F o r " l u m p e d t e m p e r a t u r e m o d e l " , t h e e q u i v a l e n t c i r c u i t f o r s e m i - c o n d u c t o r s c a n be r e p r e s e n t e d by a s e r i e s c o m b i n a t i o n o f t h e d i f f e r e n t i a l r e s i s t a n c e a t t n e ° P e r a t i n g p o i n t w i t h a p a r a l l e l c o m b i n a t i o n o f a r e s i s t a n c e R 2 a n d a n i n d u c t a n c e L . ( B u r g e s s , 1 9 5 5 b ) . A s s u m i n g a s m a l l - 4 6 -c u r r e n t p e r t u r b a t i o n o f t h e f o r m i - 10Q , t h e c o n s t a n c y o f t h e s o u r c e emf r e q u i r e s t h a t S + m.\ + - o , c 4 . , . u , S b e i n g t h e r e s i s t a n c e o f t h e s o u r c e . Thus b = - ^ 4 - ^ ) ( 4 . 2 . 1 2 ) F o r s t a b i l i t y , p must be n e g a t i v e w h i c h r e q u i r e s t h a t S + g ) > o S i n c e f o r " l u m p e d t e m p e r a t u r e m o d e l " , ~ ~ ) - / ? 1 •> ( 4 . 2 . 1 4 ) i t c a n be s e e n t h a t t h i s s t a b i l i t y c r i t e r i o n i s i d e n t i c a l w i t h e q u a t i o n ( 4 . 1 . 9 ) . The c r i t e r i o n o f s t a b i l i t y f o r m e t a l s i s g i v e n by F i g . 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 s o f s e m i - c o n d u c t o r s as a f u n c t i o n o f \ . - 4 7 -One must d i s c u s s s t a b i l i t y i n t e r m s o f t h e s o u r c e r e s i s t a n c e a s i t depends on t h e e x t e r n a l c i r c u i t c o n -d i t i o n s . I n a s e m i - c o n d u c t o r i f t h e r e e x i s t s a c o n s t a n t -v o l t a g e s o u r c e l a r g e r t h a n t h e " t u r n o v e r " o r a s y m p t o t i c v o l t a g e , t h e s y s t e m w i l l be t h e r m a l l y u n s t a b l e . T h i s c a n be shown by a c o n s i d e r a t i o n o f t h e b a l a n c e o f power e q u a t i o n ( 4 . 1 . 1 ) i n t h e f o l l o w i n g way. The t e m p o r a l dependence o f t h e e x c e s s t e m p e r a t u r e i s g i v e n by -It X 0 ( t ) ^ - e c V(\)e^ d t 9 ( 4 . 2 . 1 6 ) w h e r e , p ( t ) i s t h e i n s t a n t a n e o u s p o w e r . H e n c e , Thus as l o n g as y Q , t h e t e m p e r a t u r e w i l l i n c r e a s e i n d e f i n i t e l y w i t h t i m e g i v i n g r i s e t o t h e r m a l i n s t a b i l i t y . - 4 8 -I n a m e t a l e x h i b i t i n g NR, f o r c u r r e n t l a r g e r t h a n t h e " t u r n o v e r c u r r e n t " , t h e same argument c a n be a p p l i e d a n d t h e r m a l i n s t a b i l i t y seems t o a r i s e . S i m i l a r l y i n a m e t a l w h i c h d i s p l a y s c u r r e n t t u r n -o v e r o r a c u r r e n t a s y m p t o t e , c o n n e c t i o n t o a s o u r c e o f c u r r e n t g r e a t e r t h a n t h e s e u p p e r bounds r e s u l t s i n t h e r m a l i n s t a b i l i t y . Specimen too D. PG. { f t Vertical — « DUAL BEAM SCOPE DIFFERENTIAL INPUTS Horizontal Fig. 5.1 PULSE MEASUREMENT OF CURRENT - VOLTAGE CHARACTERIC OF THE SPECIMEN. -49= CHAPTER 5 EXPERIMENTAL TECHNIQUES : SEMI-CONDUCTORS 5 . 1 . P r e p a r a t i o n o f • t h e S p e c i m e n a n d , d . c . Measurements S i n c e i n t r i n s i c s e m i - c o n d u c t o r s a r e v e r y s e n s i t i v e t o t e m p e r a t u r e c h a n g e , t h i n f i l a m e n t s o f i n t r i n s i c germanium were c h o s e n f o r e x p e r i m e n t a l i n v e s t i g a t i o n s i n an a t t e m p t t o v e r i f y t h e n \ - K " t h e o r y i n s e m i - c o n d u c t o r s ; E a r l i e r i n t h e r e s e a r c h programme, " t w o t e r m i n a l " s p e c i m e n s o f germanium were t r i e d and 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 were made by " a l l o y i n g t e c h n i q u e " . Ohmic c o n t a c t s were n e c e s s a r y f o r a v o i d i n g 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 e f f e c t s . An Ohmic c o n t a c t c a n be d e f i n e d t o be one w h i c h d e l i v e r s c u r r e n t t o t h e d e v i c e w i t h o u t e n t e r i n g i n a n a c t i v e p r o c e s s . The " P u l s e m e t h o d " ( F i g . 5 . 1 ) was u s e d a s a r o u g h c r i t e r i o n f o r t h e s e l e c t i o n o f t h e s p e c i m e n . By u s i n g p u l s e s o f d u r a t i o n much s h o r t e r t h a n t h e t h e r m a l t i m e c o n -s t a n t o f t h e s p e c i m e n , t h e I - V c h a r a c t e r i s t i c was t r a c e d o u t on t h e T e k t r o n i x (Type 502) d u a l beam o s c i l l o s c o p e b y v a r y i n g t h e o u t p u t o f 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 (Type B 7 B ) . F o r ohmic c o n t a c t s , t h e I - V c h a r a c t e r i s t i c w o u l d be l i n e a r f o r b o t h t h e d i r e c t i o n s o f t h e c u r r e n t . Spec imens - S O -s e l e c t e d by t h i s method were s u b j e c t e d t o more t e s t s e l a b o -r a t e d l a t e r . C o n s i d e r a b l e s u c c e s s was o b t a i n e d i n m a k i n g ohmic c o n t a c t s w i t h germanium up t o 20-n>cm r e s i s t i v i t y , by u s i n g a n t i m o n y - d o p e d ( 0 . 6 % ) g o l d w i r e t o n - t y p e Ge and p u r e g o l d o r Cu w i r e t o p - t y p e G e . B u t a l l t h e s e a l l o y i n g t e c h n i q u e s d i d n o t s u c c e e d when germanium o f h i g h e r r e s i s -t i v i t y was t r i e d . A f t e r t r y i n g a l l o y i n g c o n t a c t s w i t h d i f f e r e n t m e t a l s u s i n g d i f f e r e n t r e c i p e ( T u r n e r , 1 9 5 9 ) , t h i s " t w o - t e r m i n a l * * s p e c i m e n method was abandoned i n f a v o u r o f t h e " f o u r t e r m i n a l " m e t h o d . The s p e c i m e n r e p o r t e d i n t h i s t h e s i s was made f r o m a b o u l e o f i n t r i n s i c 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 saw. T h i s s l i c e was a b r a d e d i n r o u g h and t h e n 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 i s p r o c e s s was r e p e a t e d , u s i n g e a c h t i m e f i n e r and f i n e r c a r b o r u n d u m powder u n t i l a v e r y t h i n w a f e r was o b t a i n e d . T h i s was 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 measured on 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 were 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 ) , g i v i n g =» 4 5 . 3 J f t . c m . S p e c i m e n s were p r e p a r e d o u t o f t h i s w a f e r b y s l i c i n g i t i n t o a number o f r e c t a n g u l a r s p e c i m e n s u n d e r t h e w i r e - s a w . T h e s e f i n e s p e c i m e n s were t h e n g e n t l y a b r a d e d w i t h f i n e c a r b o r u n d u m powder and c l e a n e d - 5 1 -i n d e - i o n i z e d w a t e r . The s p e c i m e n was t h e n p l a c e d i n HgOg o (30%) s o l u t i o n a n d p l a c e d i n o v e n a t 68 C f o r a c o u p l e o f m i n u t e s . I t was t h e n r i n s e d w i t h d e - i o n i z e d w a t e r a n d e t c h e d i n C P ^ s o l u t i o n a n d t h e n 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 h a d d r i e d on 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 two ends o f t h e f i l a m e n t ( c u r r e n t t e r m i n a l s ) . The 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 mm. d i a m e t e r and 3 mm. l e n g t h and 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 above t h e e u t e c t i c t e m p e r a t u r e ( 3 6 5 ° C ) o f t h e t h r e e - p h a s e G e - A u - S b s y s t e m . M i c r o - m a n i p u l a t o r s were u s e d t o b r i n g t h e g o l d ( 0 . 6 % Sb) w i r e i n c o n t a c t w i t h t h e germanium f i l a m e n t . P r o v i d e d t h a t t h e w i r e and t h e germanium s u r f a c e were k e p t s c r u p u l o u s l y c l e a n by 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 t h e germanium 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 . The h e a t i n g c o i l was t h e n s l o w l y c o o l e d t o t h e room t e m p e r a t u r e . B o t h t h e c u r r e n t - c o n t a c t s were made i n t h i s way . V o l t a g e p r o b e c o n t a c t s were made by d i r e c t s o l d e r -i n g o f f i n e a n t i m o n y - d o p e d g o l d w i r e t o t h e s p e c i m e n u s i n g Wood 's m e t a l and p u r e Z n C l ' s o l u t i o n as f l u x . A C u - c o n s t a n t a n t h e r m o - c o u p l e was s o l d e r e d t o one end o f t h e s p e c i m e n t o c h e c k t h e c o n s t a n c y o f t h e a m b i e n t t e m p e r -F i g . 5 . 2 : B r i d g e f o r m e a s u r i n g t h e l i f e - t i m e o f t h e m i n o r i t y c a r r i e r s . - 5 2 -a t u r e a t t h e e n d . S h o r t e n i n g o f t h e l i f e - t i m e o f t h e m i n o r i t y - c a r r i e r s was a c h i e v e d b y . a b r a d i n g t h e s u r f a c e s o as t o i n c r e a s e t h e r e c o m b i n a t i o n o f h o l e s and e l e c t r o n s i n t h e f i l a m e n t . The two ends o f t h e f i l a m e n t were c h o s e n t o be o f s u c h l e n g t h t h a t t h e m i n o r i t y c a r r i e r s i n j e c t e d a t 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 w o u l d r e c o m b i n e b e f o r e e n t e r i n g t h e b u l k o f t h e s p e c i m e n . I n o r d e r t o a c h i e v e t h i s , t h e l i f e - t i m e was made as s m a l l a s p o s s i b l e . A l s o , f o r t h e a p p l i c a b i l i t y o f t h e " X - K " m o d e l , t h e l i f e - t i m e o f t h e c a r r i e r s s h o u l d be n e g l i g i b l e compared t o t h e t h e r m a l t i m e c o n s t a n t o f t h e f i l a m e n t . The l i f e - t i m e o f t h e m i n o r i t y c a r r i e r s was measured by t h e b r i d g e - c i r c u i t o f F i g . 5 . 2 . The f i l a m e n t c o n s t i -t u t e s one arm o f t h e b r i d g e , t h e o t h e r s were made o f o r d i n a r y l i n e a r r e s i s t o r s . A l l t h e components o f t h e b r i d g e were p r o p e r l y s h i e l d e d a n d g r o u n d e d t o a v o i d 60 c p s i n t e r f e r e n c e . The r e s i s t a n c e R i n s e r i e s w i t h t h e s p e c i m e n i s X l a r g e compared t o r e s i s t a n c e o f t h e f i l a m e n t . A f t e r t h e a p p l i c a t i o n o f t h e p u l s e , t h e e x p o n e n t i a l v o l t a g e d e c a y a c r o s s t h e s p e c i m e n was s i m u l a t e d by t h e e x p o n e n t i a l a c r o s s t h e R-C c o m b i n a t i o n . B a l a n c e was o b t a i n e d by a d j u s t i n g R a n d C s e p a r a t e l y s o t h a t b a l a n c e was o b t a i n e d t h r o u g h o u t t h e d u r a t i o n o f t h e p u l s e a s was shown b y F i g . 5 . 3 : B r i d g e - c i r c u i t f o r m e a s u r i n g t h e d . c . r e s i s t a n c e o f t h e f i l a m e n t a s a f u n c t i o n o f c u r r e n t o r t e m p e r a t u r e - 5 3 -a p p e a r a n c e o f a s t r a i g h t h o r i z o n t a l l i n e on T e k t r o n i x D u a l beam Type 502 d i f f e r e n t i a l CRO. The l i f e - t i m e o f t h e m i n o r i t y c a r r i e r s i s t h e n g i v e n by T-RC ( 5 . 1 . D P u l s e s o f one m i c r o s e c o n d d u r a t i o n w i t h a r e p e t i t i o n r a t e o f 3 0 p . p . s . f r o m a 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 B7B) were u s e d . B a l a n c e was o b t a i n e d f o r R «= 1025 JTL. a n d C = 0 . 0 0 4 2 f i F . "2T = 4 . 3 j i s e c . T a k i n g t h e maximum v a l u e o f t h e d . c . a p p l i e d i . e . - 1 £ = 3 0 v . c m t h e d i s t a n c e t h r o u g h w h i c h t h e i n j e c t e d c a r r i e r s were swept by t h e d . c . f i e l d b e f o r e r e c o m b i n a t i o n i s 4 . 8 mm ^ jie = 3 . 9 x 1 0 3 c n ^ s e c " 1 v o l t - 1 ; S m i t h ( 1 9 5 7 ) \ • The two ends o f t h e s p e c i m e n were c h o s e n t o be 9 mm. l o n g e a c h s o t h a t t h e b u l k was f r e e f r o m c a r r i e r s i n j e c t e d a t t h e c o n t a c t s . The d i m e n s i o n s o f t h e s p e c i m e n were measured on 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 a n d a r e g i v e n b e l o w : T o t a l l e n g t h o f t h e f i l a m e n t = 2 . 3 0 cm L e n g t h o f t h e c e n t r a l p o r t i o n ( 2 L ) = 0 . 5 0 cm B r e a d t h - 0 . 1 1 cm and w i d t h = 0 . 0 2 cm. The r e s i s t a n c e o f t h e s p e c i m e n was m e a s u r e d a s a f u n c t i o n o f t h e c u r r e n t by u s i n g t h e b r i d g e - c i r c u i t o f F i g . 5 . 3 when t h e s p e c i m e n was k e p t immersed 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) i n an a t t e m p t t o \ 10 0 0 0 H —o- o o 9 000-8 0 0 0 -7000-— r - f — < -S Q Q O I (mo) R(FL) 3 Fig. 5.4 D.C. FORWARD AND REVERSE CHARACTERISTICS OF THE SEMI-CONDUCTOR SPECIMEN IMMERSED IN A TEMPERATURE BATH. -54= c h e c k w h e t h e r t h e s p e c i m e n was f r e e f r o m c o n t a c t e f f e c t s . The t e m p e r a t u r e o f t h e b a t h was a r b i t r a r i l y k e p t c o n s t a n t t h e f i l a m e n t a n d e n d " l w r e s p e c t i v e l y * A s t h e r e s i s t a n c e c a n be measured a c c u r a t e l y , t h i s method was u s e d a s a c r i t e r i o n f o r t h e s e l e c t i o n o f t h e s p e c i m e n . The dependence o f r e s i s t a n c e on c u r r e n t f o r b o t h t h e d i r e c t i o n s o f t h e c u r r e n t i s shown i n F i g . 5 . 4 . The c o n s t a n c y o f t h e r e s i s t a n c e o f t h e f i l a m e n t o v e r t h e v a l u e o f t h e c u r r e n t u s e d shows t h a t t h e s p e c i m e n i s f r e e f r o m i n j e c t e d c a r r i e r s . The s l i g h t d e c r e a s e o f t h e r e s i s t a n c e a t h i g h e r v a l u e o f t h e c u r r e n t may be due t o s e l f - h e a t i n g o f t h e s p e c i m e n . A l s o , no a p p r e c i a b l e change o f t h e r e -s i s t a n c e was o b s e r v e d on r e v e r s i n g t h e d i r e c t i o n o f t h e c u r r e n t . The r e s i s t a n c e o f t h e s p e c i m e n was t h e n measured a s a f u n c t i o n o f t h e b a t h - t e m p e r a t u r e o v e r t h e r a n g e between 20°C t o 80°C t o d e t e r m i n e t h e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y o f t h e s p e c i m e n . A v e r y l o w v a l u e o f t h e c u r r e n t (5 jiamp) was u s e d . To e n s u r e t h a t t h e r e was no h e a t i n g due t o t h e c u r r e n t , t h e c u r r e n t was d o u b l e d a t b a -l a n c e a n d t h e b a l a n c e was f o u n d t o be unchanged;. The o a t 2 0 . 3 C . The b a l a n c e e q u a t i o n s f o r t h e b r i d g e a r e 10 000 8 000 (deg-'A) — i 1 1 1 3Jx|0~5 32XI0-5 33XI0"5 34xl0~5 35x10 -5 F i g . 5 . 6 P l o t o f L n R a g a i n s t T _ 1 . - 5 5 -dependence o f r e s i s t a n c e on t e m p e r a t u r e i s i l l u s t r a t e d i n F i g . 5 . 5 . The p l o t o f L n R a g a i n s t T ( d a t a o b t a i n e d f r o m F i g . 5 . 5 ) was f o u n d t o be l i n e a r w i t h i n 5% between 295°K and 330°K. The c h o i c e o f germanium w h i c h i s i n t r i n s i c a t a n d above room t e m p e r a t u r e p r o v i d e s a r e l a t i v e l y l a r g e t e m -p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y a n d y e t p e r m i t s a s a t i s -f a c t o r y w i d e r a n g e o f t e m p e r a t u r e o v e r w h i c h t h e e x p o n e n t i a l c o n d u c t i v i t y m o d e l i s an a c c u r a t e r e p r e s e n t a t i o n . The t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y was d e t e r m i n e d t o be a = - 0 . 0 4 1 d e g ^ C a s a g a i n s t t h e p r e d i c t e d v a l u e o f - 0 . 0 4 8 d e g _ i C . F i g . 5 . 6 i l l u s t r a t e s t h e dependence o f L n R on T " 1 w h i c h i s a s t r a i g h t l i n e w i t h a s l o p e o f b = 4480°K. T h i s e x p e r i m e n t a l v a l u e o f b c a n be compared w i t h t h e t h e o r e t i c a l v a l u e o f b = 4533°K f o r germanium w h i c h i s i n good agreement w i t h t h e e x p e r i m e n t a l v a l u e . I n t h e above c a l c u l a t i o n , Ee = 0 . 7 8 2 ev ( M a c F a r l a n e e t a l , 1957) a n d 5 o K - 1 . 3 8 x 1 0 " 1 6 e r g s d e g " - C . I n o r d e r t o k e e p t h e two ends c o n s t a n t , f o u r f l a t p i e c e s o f b r a s s p l a t e were p l a c e d i n c l o s e t h e r m a l c o n t a c t w i t h t h e e n d s , t h e i n t e r v e n i n g s p a c e between t h e s e m i -c o n d u c t o r ( e n d s ) a n d b r a s s p l a t e s b e i n g c o v e r e d by t h i n m i c a (1 m i l ) s h e e t s . The m i c a s h e e t s a c t e d a s a good e l e c t r i c a l i n s u l a t o r between t h e m e t a l and t h e s e m i -V to pump V/////A 4 M i c a Constantan S p e c i m e n 1 V////A vacustat Brass Plai;e •Copper to potentiometer D.C. Fig, 5 . 7 : Circuit for current-voltage measurements. -56= c o n d u c t o r w h i l e p r o v i d i n g a good t h e r m a l p a t h . The e n t i r e a s s e m b l y was t h e n p l a c e d i n a g l a s s - t u b e . The p r e s s u r e i n -s i d e t h e g l a s s t u b e was k e p t c o n s t a n t (0.8mm of Hg) d u r i n g t h e e x p e r i m e n t . A b i g b l a c k c a r d b o a r d box was u s e d a s a c o v e r t o c u t o f f a l l l i g h t f r o m t h e s p e c i m e n . The c i r c u i t o f F i g . 5 . 7 was u s e d t o measure 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 s p e c i m e n . The v o l -t a g e a c r o s s t h e s p e c i m e n was m e a s u r e d by a d . c . VTVM ( H e w l e t t P a c k a r d M o d e l 412 A ) . The p r o c e d u r e f o l l o w e d i n p l o t t i n g t h e p o i n t - b y - p o i n t I - V c h a r a c t e r i s t i c 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 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 , 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 h a d been t a k e n , t h e p o i n t s o b t a i n e d were 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 u r v e h a d n o t c h a n g e d . 5 . 2 A . C . M e a s u r e m e n t s The s m a l l s i g n a l A . C . impedance o f t h e s p e c i m e n , s u p e r i m p o s e d on t h e a p p l i e d d . c . was m e a s u r e d . a s a f u n c t i o n o f t h e f r e q u e n c y o v e r t h e r a n g e between 0 . 1 0 c p s t o 5 , 0 0 0 c p s w i t h t h e b r i d g e c i r c u i t o f F i g . 5 . 8 . The a . c . v o l t a g e u s e d was v e r y s m a l l compared t o t h e d . c . b i a s . The d i r e c t c u r r e n t t h r o u g h t h e s p e c i m e n was k e p t c o n s t a n t a t 1=3.50 mA t h r o u g h o u t t h e a . c . m e a s u r e m e n t s . D.C. 500V Signal Generator t o pump M i c a , to potentiometer > Y////M <H END "I" [ E N D ¥ H b r a s s p l a t e e c i m e n ^ t o V a c u s t a t R s, ww—(T)-" Diff. CRO. d* if Diff. CRO b A V \ r b F i g . 5 . 8 : A . C . B r i d g e c i r c u i t f o r m e a s u r i n g t h e impedance o f t h e s p e c i m e n ( S e m i - C o n d u c t o r ) a s a f u n c t i o n o f f r e q u e n c y . - 5 7 -The f i v e - s t a g e f i l t e r - c i r c u i t u s e d i n t h e b r i d g e -c i r c u i t was n e c e s s a r y t o s u p p r e s s t h e l o w f r e q u e n c y n o i s e a n d 60 c p s i n t e r f e r e n c e c o m i n g f r o m t h e D . C . power s u p p l y ( M o d e l 4 0 6 ) . I n o r d e r t o e l i m i n a t e t h e 60 c p s i n t e r f e r e n c e , e a c h component o f t h e b r i d g e - c i r c u i t was p r o p e r l y s h i e l d e d a n d g r o u n d e d . F o r h i g h f r e q u e n c y m e a s u r e m e n t s , H e w l e t t - P a c k a r d A u d i o O s c i l l a t o r ( M o d e l 200 C D) was u s e d a s t h e A . C . s o u r c e w h i l e f o r f r e q u e n c i e s l e s s t h a n 50 c p s U l t r a Low F r e q u e n c y O s c i l l a t o r ( M o d e l 4 0 0 C) was u s e d . T e k t r o n i x t y p e 502 D u a l - b e a m o s c i l l o s c o p e was u s e d as a n u l l d e t e c t o r . The method o f e l l i p s e was u s e d a s a n u l l d e t e c t i o n , t h e o u t -p u t f r o m t h e s i g n a l g e n e r a t o r b e i n g a p p l i e d t o t h e h o r i z o n t a l sweep o f t h e CRO w h i l e t h e o u t p u t between a and b a n d c a n d d ( F i g . 5 . 8 ) were p u t on t h e two v e r t i c a l d . c . d i f f e r e n t i a l i n p u t s . A t b a l a n c e , t h e e l l i p s e d e g e n e r a t e s i n t o a h o r i -z o n t a l s t r a i g h t l i n e . F o r v e r y l o w f r e q u e n c i e s ( f ^ 1 c p s ) , H o n e y w e l l Type 104 W l - G g a l v a n o m e t e r was u s e d a s a n u l l - d e t e c t o r . The r e s i s t a n c e R' (WW) u s e d i n s e r i e s w i t h t h e s p e c i m e n was v e r y l a r g e compared t o t h e e n d r e s i s t a n c e . C o r r e c t i o n s were made f o r n e g l e c t i n g t h e r e s i s t a n c e o f t h e end o f t h e s p e c i m e n and t h e c o r r e c t e d v a l u e o f t h e r e s i s t a n c e i n s e r i e s w i t h t h e f i l a m e n t i s d e n o t e d by R . ^Specimen K^) y&-P.G. Rt Fig. 5 , 9 : Bridge for measuring the effeotive thermal time constant of the filament. 5 8 -When n u l l i s o b t a i n e d on b o t h t h e d e t e c t o r s , t h e b a l a n c e e q u a t i o n s f o r t h e b r i d g e a r e g i v e n by R ( u ) - b Rs L = R s . b c x L ( 5 . 2 . 1 ) X ( * i ) o R b u c , s 1 where t h e s p e c i m e n i s assumed t o be r e p r e s e n t e d as a c o m b i -n a t i o n o f a . c . r e s i s t a n c e R(w) i n s e r i e s w i t h an i n d u c t i v e r e a c t a n c e X (<•/)• 5 . 3 Measurement o f t h e E f f e c t i v e T h e r m a l T ime C o n s t a n t o f t h e F i l a m e n t The e f f e c t i v e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n was m e a s u r e d b y u s i n g t h e c i r c u i t o f F i g . 5 . 9 . The o u t p u t o f t h e d . c . power s u p p l y was a d j u s t e d s o t h a t t h e c u r r e n t t h r o u g h t h e s p e c i m e n was k e p t c o n s t a n t a t I q = 3 . 5 0 mA. S u p e r i m p o s e d on t h e s t e a d y d . c . b i a s , p u l s e s o f 60 m i l l i s e c d u r a t i o n were a p p l i e d f r o m a U n i t P u l s e r (Type 1 2 1 7 - A ) . The p u l s e a p p l i e d ( 1 . 0 v o l t ) was v e r y s m a l l c o m p a r e d t o t h e d . c . b i a s (50 v ) . The e x p o n e n t i a l d e c a y o f v o l t a g e a c r o s s t h e s p e c i m e n was s i m u l a t e d b y a d j u s t i n g R a n d C s e p a r a t e l y . The b a l a n c e was i n d i c a t e d b y t h e a p p e a r a n c e o f a s t r a i g h t h o r i z o n t a l l i n e on t h e T e k t r o n i x D u a l Beam (Type 502) CRO. - 5 9 -B a l a n c e was o b t a i n e d f o r R =» 10 KSL , C =» 4 . 0 5 \iFf R = 1 KSl , R = 1 K i t , R = 1 0 K X L a n d R = 1 K _Q_ , 2 3 1 s g i v i n g "2" = 4 0 . 5 m i l l i s e c o n d s . The p u l s e r e p e t i t i o n r a t e u s e d was 15 p . p . s . - . 6 0 -CHAPTER 6  EXPERIMENTAL TECHNIQUES (METALS) 6 . 1 . S e l e c t i o n o f S p e c i m e n a n d d . c . M e a s u r e m e n t s E a r l i e r i n t h e r e s e a r c h programme, d . c . e x p e r i m e n t s were c a r r i e d o u t b y u s i n g t u n g s t e n , a luminum a n d p u r e i n d i u m w i r e s as s p e c i m e n s . A s m e t a l s have v e r y l o w r e s i s t i v i t i e s a t room t e m p e r a t u r e , i t was d i f f i c u l t t o measure t h e change i n r e s i s t a n c e due t o c u r r e n t . M o r e o v e r , f o r t h i c k m e t a l w i r e s (1 m i l ) , t h e r e a c t a n c e maximum i s i n t h e n e i g h b o u r -h o o d o f 10 1 c p s where i t i s d i f f i c u l t t o make a . c . m e a s u r e m e n t s . A l s o , i n o r d e r t o be c o m p a t i b l e w i t h t h e X - K m o d e l , a n i s o t h e r m a l c r o s s - s e c t i o n was e s s e n t i a l n e c e s s i t a t i n g t h e u s e o f a t h i n s p e c i m e n . A l l t h e s e d i f f i c u l t i e s were overcome by t a k i n g - 4 p l a t i n u m w i r e o f 0 . 9 3 x 10 cm r a d i u s , u s e d i n m a k i n g " L i t t l e F u s e " . The two ends o f t h e s p e c i m e n were s o l d e r e d t o two b i g c o p p e r r o d s w h i c h a c t e d as h e a t s i n k s i n o r d e r t o k e e p t h e a m b i e n t f i x e d d u r i n g a n y e x p e r i m e n t . T h e s e two c o p p e r b a r s were a g a i n a t t a c h e d t o two b i g b r a s s p l a t e s . The e n t i r e a s s e m b l y was p l a c e d i n a g l a s s t u b e , s o t h a t t h e a s s e m b l y c o u l d be e v a c u a t e d down t o a p r e s s u r e o f 1 0 ~ 3 mm o f H g . S p e c i m e n F i g . 6 . 1 . : B r i d g e f o r m e a s u r i n g t h e d . c . r e s i s t a n c e o f t h e s p e c i m e n a s a f u n c t i o n o f c u r r e n t o r t e m p e r a t u r e . t o p u m p i / K t o V a c u s t a t F i g . 6 . 2 C i r c u i t f o r c u r r e n t - v o l t a g e measurement - 6 1 -The following procedure was adopted i n selecting the specimen. The wire contained i n a glass tube was viewed under the microscope (magnification 200) while the currenst through the specimenwas varied. Some specimens were found to have abrupt constrictions at discrete points along the length of the specimen and these points were viewed as bright spots under the microscope when.the current was high. These specimens were rejected. The dimensions of the specimen reported i n t h i s thesis were determined i n the following way. The specimen was made up of platinum. The wire was viewed under a high power projection micro-scope (Mag. 2000) and the image was projected on a ground glass plate, the diameter of the specimen being measured at f i v e d i f f e r e n t points along i t s length. The mean value of the diameter of the specimen was found to be - 4 1.86 x 10 cm. The length of the specimen was determined by measurement of the ambient resistance of the specimen for very low value of the current and was found to be 0.22 cm. The b r i d g e - c i r c u i t of F i g . 6.1 was used to measure the resistance of the specimen as a function of temperature. In the balanced condition, the balance equation i S given by (6.1.1) •=•62— I n t h e p r e s e n t e x p e r i m e n t , R =» R = 1 KSl, s o t h a t R=R_. 1 2 3 The d - c c u r r e n t was s u p p l i e d by a b a t t e r y i n s e r i e s w i t h a h i g h r h e o s t a t s o t h a t v e r y l o w c u r r e n t was a l l o w e d t© p a s s ( 5 f iamp.) t h r o u g h t h e s p e c i m e n t© measure t h e r e -s i s t a n c e . The s p e c i m e n was p l a c e d , f o r t h e t e m p e r a t u r e r a n g e between 20°C a n d 1 5 0 ° C , i n s i d e t h e 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 1 3 4 ) . H o n e y w e l l Type 104 Wl - G was u s e d a s a n u l l - d e t e c t o r . F o r t e m p e r a t u r e r a n g e between 150°C t o 3 7 0 ° C , a f u r n a c e was made by u s i n g n i c h r o m e w i r e o n an a s b e s t o s s h e e t , w r a p p e d a r o u n d a u n i f o r m g l a s s t u b e a n d c u r r e n t • f rom a d . c . power s u p p l y was u s e d f o r h e a t i n g t h e f u r -n a c e . . A C u - c o n s t a n t a n t h e r m o - c o u p l e was i n s e r t e d i n -s i d e t h e t u b e by t h e s i d e o f t h e s p e c i m e n , t o measure t h e t e m p e r a t u r e o f t h e f u r n a c e . S u f f i c i e n t t i m e (3 t o 4 h o u r s ) was a l l o w e d t o p a s s , s o t h a t t h e s p e c i m e n was a t t h e same t e m p e r a t u r e as t h a t o f t h e f u r n a c e f o r e a c h r e a d i n g . I n t h i s w a y , r e a d i n g s were t a k e n u p t o t h e o i t e m p e r a t u r e 370 C . E a c h o f t h e s e r e a d i n g s was c h e c k e d by d e c r e a s i n g t h e t e m p e r a t u r e i n o r d e r t o e n s u r e t h e r e p r o -d u c i b i l i t y o f t h e r e s u l t s . The dependence o f r e s i s t a n c e o f t h e s p e c i m e n o n t e m p e r a t u r e i s shown i n F i g . 6 . 3 . The r e s i s t a n c e o f t h e s p e c i m e n was f o u n d t o v a r y l i n e a r l y w i t h t h e e x c e s s t e m p e r -o a t u r e up t o 320 C . The v a l u e o f t h e t e m p e r a t u r e c o e f f i c i e n t 5x|02-6 (deg c) Fig. 6.3 VARIATION OF RESISTANCE OF THE SPECIMEN AS A FUNCTION OF EXCESS TEMPERATURE. "•63 s" of r e s i s t i v i t y was found to be 3.8 x 10 deg C. This compares favourably with the value of a =3.9 x 10~3 •^ X deg C (platinum) obtained by Stauss (Stauss, 1941). The specimen was then placed inside a metal-glass sealed tube and evacuated to 10 mm of Hg. The c i r c u i t of F i g . 6.2 was used to investigate the steady-state current-voltage c h a r a c t e r i s t i c . The d.c. current was supplied from a battery i n series with rheostats which permitted a range of current from 5 jiA to 3.5 mA. The voltage across the specimen was measured by a d.c. VTVM (Hewlett Packard Model 412A); The procedure followed i n p l o t t i n g the point-by point I-V curve was the same as that for semi-conductors. After the maximum reading had been taken the points so obtained were checked as the voltage was reduced, to ensure that the c h a r a c t e r i s t i c had not changed. The —3 pressure within the enclosure was kept constant at 10 mm of Hg during the experiments. The resistance of the specimen was measured as a function of the current with the bridge c i r c u i t of F i g . 6.1 over the range of current from 30 \iA to 2.52 mA. to pump S i g n a l G e n e r a t o r e - » t o V a c u s t a t H o r i z o n t a l ojf " CRO D.C. Rs ^ F i g . 6 . 4 : A . C . B r i d g e f o r m e a s u r i n g t h e s m a l l s i g n a l a . c . impedance o f t h e s p e c i m e n ( m e t a l ) a s a f u n c t i o n o f f r e q u e n c y . -64-6*2 The small signal.a.c. impedance of the specimen was measured using the bridge c i r c u i t of F i g . 6.4 over the frequency range between 0.20 cps to 5000 cps, while keeping the d.c. bias through the specimen f i x e d at I <=» 1.50 mA. A l l the resistance and capacitance boxes were properly shielded and grounded. The output between a and b (Fig. 6.4) was placed on the d.c. d i f f e r e n t i a l ( v e r t i c a l ) input of DUHMont Type 411 cathode ray oscilloscope while the out-put from the a.c. si g n a l generator (Shasta Wide Rang® O s c i l l a t o r , Type Model 301A) was d i r e c t l y applied to the horizontal sweep of the CRO. The balance was indicated by a st r a i g h t horizontal l i n e . bias. For low frequency, U l t r a Low Frequency O s c i l l a t o r (Model 400 C) was used as a si g n a l generator and the CRO was replaced by Honeywell Type 104 Wl - G galvanometer. specimen to be represented by a series combination of a.c. resistance R(w) with a capacitive reactance X(w), are given by, The a.c. applied was small compared to the d.c. The balance equations of the bridge, assuming the (6.2.1) F i g . 6 . 5 : B r i d g e f o r m e a s u r i n g t h e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n ( M e t a l ) . - 6 5 -2 Now, when (wC R ) < ^ 1, t h e b a l a n c e e q u a t i o n s s t o R(oS> ~ K x ^ • ( 6 . 2 . 2 ) 6 . 3 . Measurement o f t h e E f f e c t i v e T h e r m a l Time C o n s t a n t o f t h e S p e c i m e n The t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n was m e a s u r e d b y u s i n g t h e B r i d g e - c i r c u i t o f F i g . 6 . 5 . The two r a t i o - a r m s R a n d R were p r e c i s i o n c a r b o n r e s i s -1 . 2 t o r s . To r e d u c e t h e 60 c p s i n t e r f e r e n c e , e a c h component o f t h e b r i d g e was s h i e l d e d a n d g r o u n d e d . When R = R , t h e r e s i s t a n c e arm a e q u a l s t h e a m b i e n t r e s i s t a n c e o f t h e s p e c i m e n . R e q u a l s t h e i n c r e a s e o f t h e r e s i s t a n c e o f t h e s p e c i m e n . F o r m e a s u r i n g t h e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n , t h e o u t p u t o f t h e D . C . Power S u p p l y ( M o d e l 4 0 6 ) was a d j u s t e d s o t h a t t h e c u r r e n t t h r o u g h t h e s p e c i -men was Jjy Q • =» 1 . 5 0 mA. Now, s u p e r i m p o s e d on t h i s d . c . b i a s , p u l s e o f t i m e - d u r a t i o n 3 0 m i l l i s e c f r o m a 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 B7B) was a p p l i e d . The p u l s e a p p l i e d ( 0 . 0 5 v o l t ) was s m a l l compared w i t h t h e d . c . b i a s . The o u t p u t between e a n d f was d i s p l a y e d o n - 6 6 -t h e d i f f e r e n t i a l i n p u t s o f T e k t r o n i x D u a l Beam ( Type 502) CRO. The b r i d g e - b a l a n c e was o b t a i n e d b y a d j u s t i n g t h e p a r a m e t e r s R a n d C o f t h e b r i d g e - c i r c u i t and t h e b a l a n c e was i n d i c a t e d b y t h e a p p e a r a n c e o f a s t r a i g h t h o r i z o n t a l l i n e on t h e CRO a s t h e e x p o n e n t i a l r i s i n g c u r v e due t o h e a t i n g was s i m u l a t e d by t h e e x p o n e n t i a l a c r o s s t h e R-C c o m b i n a t i o n . The e f f e c t i v e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n i s t h e n g i v e n b y T-RC ' (6.3.1) B a l a n c e was o b t a i n e d f o r a = 1880J~L , R «= 450 J 7 - , C = 23.3 j i f , R, = 100 SI, R - ' 1000 _TL and R n = 2 0 0 - H -1 2 ? a n d S = 3 5 0 -i2- , S b e i n g t h e r e s i s t a n c e i n s e r i e s w i t h t h e spec imen. . The e f f e c t i v e t h e r m a l t i m e c o n s t a n t was f o u n d t o be 1 0 . 5 m i l l i s e c . The p u l s e r e p e t i t i o n r a t e u s e d was 3 0 p . p . s . 8CH 0 1 1 1 r -4 8 12 16 r > F i g . 7 . 2 . Dependence o f F ( r ) o n r . - 6 7 -CHAPTER 7 EXPERIMENTAL RESULTS AND INTERPRETATIONS 7 . 1 . S e m i - C o n d u c t o r The s u r f a c e l o s s p a r a m e t e r X f o r t h e s e m i - c o n d u c -t o r s p e c i m e n was e x p e r i m e n t a l l y d e t e r m i n e d f r o m t h e d . c . r e s i s t a n c e measurement a t l o w v a l u e s o f t h e c u r r e n t . From t h e d . c . X - K t h e r m a l t h e o r y , t h e d e c r e m e n t a l d . c . r e -s i s t a n c e f o r l o w v a l u e s o f t h e c u r r e n t 1 ? . - 2 A K r ( l $ ) J ( ' The dependence o f F ( r ) = — ( I — feiLi^L ) on r i s T V N[P / 2 shown i n F i g . 7 . 2 . A p l o t o f (R^— "R)/)^against I i s a 4 - 2 s t r a i g h t l i n e w i t h a s l o p e o f 1 .12 x 10 amps ( F i g . 7 . 1 ) . U s i n g e q u a t i o n ( 7 . 1 . 1 ) , r was e v a l u a t e d t o be 0 . 7 0 , t a k i n g K = 0 . 6 0 w a t t s c m " 1 d e g = 1 C ( S m i t h , 1 9 5 9 ) , 2 L » 0 . 5 0 cm, A = 2 x 1 0 ~ 3 c m 2 , a = 0 . 0 4 1 d e g ^ C a n d R « 9 . 7 5 x 1 0 3 - H - . a - 3 - 1 X was t h u s d e t e r m i n e d t o be 6 . 7 w a t t s cm deg C a u s i n g e q u a t i o n ( 2 . 5 . 5 b ) . T h i s e x p e r i m e n t a l l y d e t e r m i n e d v a l u e o f X may be compared w i t h t h e p r e d i c t e d v a l u e o f X based on S t e f a n ' s r a d i a t i o n l a w . A s s u m i n g t h e c o n v e c t i o n l o s s t o be n e g l i g i b l e c o m p a r e d w i t h t h e r a d i a t i o n l o s s , a n d ( T - T ) <<T a , X due t o r a d i a t i o n i s g i v e n by - 6 8 -Ar - ^ p ( 7 . 1 . 2 ) , where (T" = 5 . 8 x 10 w a t t s cm 2 deg"""* i s t h e S t e f a n ' s c o n s t a n t , £ i s t h e e m i s s i v i t y , i s t h e a m b i e n t t e m p e r a t u r e p i s t h e p e r i m e t e r and A t h e a r e a o f c r o s s * s e c t i o n o f t h e s p e c i m e n . U s i n g e q u a t i o n ( 7 . 1 . 2 ) , X r i s e v a l u a t e d t o be 0 . 3 ™3 —1 w a t t s cm deg C w h i c h i s v e r y s m a l l compared t o t h e e x p e r i m e n t a l l y o b s e r v e d v a l u e o f X «• 0 . 6 1 , S m i t h (1959) . T h i s i n f e r s t h a t c o n v e c t i o n l o s s i s n o t n e g l i -g i b l e c o m p a r e d t o t h e r a d i a t i o n l o s s . The s u r f a c e l o s s p a r a m e t e r due t o c o n v e c t i o n i s g i v e n by X - ^ " . 1 . 3 ) where H i s t h e h e a t t r a n s f e r c o e f f i c i e n t w h i c h has been e s t i m a t e d f o r t h e p r e s e n t e x p e r i m e n t (P = 0 . 8 mm o f H g . ) t o be 2 . 5 x 1 0 = 2 w a t t s cm™2 d e g ~ A C , g i v i n g Xc - 3 . 6 w a t t s cm™3 d e g ™ ^ . ^ F i s h e n d e n and S a u n d e r s , 1950^) . Thus t h e t o t a l s u r f a c e l o s s p a r a m e t e r (X «« X, . + X_) was e s t i -mated t o be 3 . 9 w a t t s cm™3 deg™ AC as compared w i t h t h e —3 —1 e x p e r i m e n t a l l y o b s e r v e d v a l u e o f 6 . 7 w a t t s cm deg A C . I n t h e X - K t h e r m a l t h e o r y t h e " T h o m s o n - h e a t ' ' t e r m was n e g l e c t e d and t h i s a s s u m p t i o n i s v a l i d p r o v i d e d =69-« ( l ± « | - - £ N L • ( 7 . 1 . 4 ) 2.K v K F o r t h e d . c . e x p e r i m e n t , a s s u m p t i o n ( 7 . 1 . 4 ) i s v a l i d f o r =2 J 4 25 amps, cm , i . e . I = 50 roA where t h e Thomson c o -e f f i c i e n t n f o r germanium was t a k e n a s 0 . 1 8 mv deg™ 1 C ^ J o h n s o n a n d L a r k - H o r o v i t z , 1953 ~] . The s t e a d y - s t a t 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 i g . 7 . 3 ) d i s p l a y e d NR o f I > 5 . 0 mA. The p r e d i c t e d v a l u e o f —£- (R,_ b e i n g t h e " t u r n o v e r " r e s i s t a n c e ) i s g i v e n b y ^ _ [ l +z^r)(Ky)'Vs| +4(u|>f] \7.1.5) i i r j i i i cos E x p r e s s i o n ( 7 . 1 . 5 ) g i v e s ; , L i l = 0 . 3 4 ( r = 0 . 7 0 ) a s compared w i t h t h e o b s e r v e d v a l u e o f _ X = 0 . 3 6 . A g a i n , a c c o r d i n g t o t h e \ - K t h e o r y , t h e d . c . r e s i s t a n c e a n d t h e d i f f e r e n t i a l r e s i s t a n c e a r e r e s p e c -t i v e l y g i v e n b y 12. - (5L) a + l l 6 L ( 7 a ' 6 ) a n d 15 V (volts) K H 20 40 I (mA) 60 80 100 F i g . 7 . 3 S t e a d y - s t a t 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 I00~| o f t n e s e m i - c o n d u c t o r s p e c i m e n s h o w i n g NR. o 80H _R(C0) + R{p 40H Ix 10 I2 (amps2 ) 2x10' F i g . 7 . 4 V a r i a t i o n o f ( s e m i - c o n d u c t o r s p e c i m e n ) ' R(oo)+R(o) on I - 7 0 -c 2 y A s s u m i n g t a n h o L ^ 1, s e c h o L <<1 and I ;> one g e t s f r o m e x p r e s s i o n s ( 7 . 1 . 6 ) and ( 7 . 1 . 7 ) , 7-ft. = r H- ( 7 . X . 8 , 'Rcc I"2" 2 o n I i s l i n e a r f o r The dependence o f to®} 4- R/o") I 1 2 mA ( i n t h e NR r e g i o n o f t h e d . c . c h a r a c t e r i s t i c ) 4 —2 w i t h a s l o p e o f 4 . 8 x 10 amps. , a s shown i n F i g . 7 . 4 , 4 —9 compared w i t h t h e p r e d i c t e d v a l u e o f 4 . 2 x 10 amps. ( I ^ 4 0 m A ) . The v a l u e o f r • 1 . 0 o b t a i n e d by e x t r a -p o l a t i o n compares f a v o u r a b l y w i t h r = 0 . 7 d e t e r m i n e d f r o m e q u a t i o n ( 7 . 1 . 1 ) . F o r e x p e r i m e n t s on t h e s e m i - c o n d u c t o r ( g e r m a n i u m ) , t h e d . c . X - K t h e r m a l t h e o r y i s f o u n d t o be v a l i d f o r I ^ 4 0 mA. F o r h i g h e r v a l u e s o f t h e c u r r e n t t h e a s s u m p -t i o n s made i n t h e t h e o r y b r e a k down a s t h e l i n e a r d e p e n -dence o f r e s i s t i v i t y o f t h e s e m i - c o n d u c t o r on t e m p e r a t u r e i s u n t e n a b l e . A l s o f o r h i g h e x c e s s t e m p e r a t u r e compared t o t h e a m b i e n t , N e w t o n ' s l a w o f c o o l i n g becomes i n v a l i d n e c e s s i a t i n g t h e u s e o f S t e f a n ' s r a d i a t i o n l a w w h i c h has t h e e f f e c t o f r e d u c i n g t h e t e m p e r a t u r e g r a d i e n t a l o n g t h e s p e c i m e n . The v a r i a t i o n o f t h e t h e r m a l 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 s h o u l d be t a k e n i n t o a c c o u n t i n t h e t h e o r y a s t h e t h e r m a l c o n d u c t i v i t y o f germanium v a r i e s — l —1 a p p r o x i m a t e l y l i n e a r l y f r o m 0 . 6 0 w a t t s cm deg C a t room t e m p e r a t u r e t o 0 . 5 0 w a t t s cm™1 d e g ^ C a t 95°C ( M c C a r t h y a n d B a l l a r d , 1 9 5 5 ) . 40 s e m i - c o n d u c t o r s p e c i m e n . ( s e m i - c o n d u c t o r s p e c i m e n ) - 7 1 -The s m a l l s i g n a l a . c . impedance o f t h e s p e c i m e n , superimposed on t h e c o n s t a n t d . c . b i a s ( I n „ =» 3 . 5 0 mA) was m e a s u r e d o v e r t h e f r e q u e n c y r a n g e o f 0 . 1 0 c p s t o 5 k c s . The e x c e s s t e m p e r a t u r e a t t h e c e n t r e o f t h e s p e c i m e n was o e s t i m a t e d t o be 7 C c o r r e s p o n d i n g t o I n r = 3 . 5 0 mA. ^ u s i n g e q u a t i o n ( 2 . 5 . 5 ) , t a k i n g S L « 1 . 2 . The maximum v a l u e o f t h e "Thomson h e a t " was e v a l u a t e d i n «.*> t h i s a . c . e x p e r i m e n t t o be 5 x 10 w a t t s as compared '> w i t h t h e l o n g i t u d i n a l a n d J o u l e h e a t o f 2 . 4 x 10™ 1 w a t t s a n d 4 . 5 x 1 0 ~ 2 w a t t s r e s p e c t i v e l y . The h e a t c a p a c i t y p e r u n i t vo lume and t h e t h e r m a l 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 were e x p e r i m e n t a l l y d e t e r m i n e d f r o m h i g h f r e q u e n c y a . c . m e a s u r e m e n t s . A c c o r d i n g t o t h e a . c . X - K t h e r m a l t h e o r y , t h e r e a c t a n c e a t h i g h f r e -q u e n c y i s g i v e n by X(00)HIF - ^ ) ( 7 . 1 . 9 ) The dependence o f X ( w ) H F on f " 1 i s a s t r a i g h t l i n e w i t h a s l o p e o f 5 . 9 2 x 10 H e n r y a s shown i n F i g . 7 . 5 . U s i n g t h i s s l o p e a n d e q u a t i o n ( 7 . 1 . 9 ) , c was e v a l u a t e d t o be 1 .3 J o u l e s cm"" 3 d e g _ 1 C (a f \ J 8 2 = 5 . 6 w a t t s cm"" 3 deg C ) a s c o m p a r e d w i t h t h e l i t e r a t u r e v a l u e o f 1 ,5 J o u l e s c m " 3 d e g ^ C f^ S m i t h , 1959 ~\ . The v a l u e o f K was e x p e r i m e n t a l l y d e t e r m i n e d f r o m t h e p l o t o f R ( « ) _ _ H . F . a g a i n s t f " 3 / 2 ( F i g . 7 . 6 ) w h i c h I s a s t r a i g h t l i n e w i t h 5x|0_l F i g . 7 . 7 V a r i a t i o n o f X ( u ) T _ w i t h f r e q u e n c y f o r t h e s e m i - c o n d u c t o r s p e c i m e n , 1850-1 F i g . 7 . 8 Dependence o f R ( w ) L F on t ( s e m i - c o n d u c t o r s p e c i m e n ) -72-a s l o p e o f 1 .4 x s e c 3 ^ 2 . Now, u s i n g e q u a t i o n - l - 1 ( 3 . 1 . 1 8 ) , K was d e t e r m i n e d t o be 0 . 6 2 w a t t s cm deg C - 1 - I a s compared w i t h t h e v a l u e o f K » 0 . 6 0 watts cm <feg C obtained by M c C a r t h y a n d B a l l a r d ( 1 9 5 5 ) . The h i g h f r e q u e n c y a p p r o x i -m a t i o n i s v a l i d f o r t h i s e x p e r i m e n t f o r f ^> 200 c p s . F i g . 7 . 7 shows t h e dependence o f X ( w ) on f L . F . ( l i n e a r f o r f $; 0 . 5 0 c p s ) . A c c o r d i n g t o t h e t h e r m a l t h e o r y , t h e r e a c t a n c e a t l o w f r e q u e n c y i s g i v e n by K 7 . 1 . 1 0 ) The p r e d i c t e d v a l u e o f t h e p l o t o f X ( w ) T „ a g a i n s t f was c a l c u l a t e d f r o m e q u a t i o n ( 7 . 1 . 1 0 ) t o be 450 F a r a d i n c o m -p a r i s o n t o t h e e x p e r i m e n t a l l y o b s e r v e d v a l u e o f 410 F a r a d ( F i g . 7 . 7 ) . o The dependence o f R ( u ) T _ o n f was o b s e r v e d t o o be l i n e a r ( F i g . 7 . 8 ) w i t h a s l o p e o f 1100 -fU s e c a s c o m -2 p a r e d w i t h t h e t h e o r e t i c a l v a l u e o f 8 4 0 - ^ - s e c c a l c u l a t e d f r o m e q u a t i o n ( 3 . 1 . 1 6 ) o f C h a p t e r 3 . The r e a c t a n c e r e a c h e d i t s maximum v a l u e c o r r e s -p o n d i n g t o f _ _ _ = 4 . 5 c p s a s compared w i t h t h e p r e d i c t e d v a l u e o f f m a x = 3 . 4 c p s , c a l c u l a t e d f r o m t h e e x p r e s s i o n Fig. 7.9 LOCUS OF IMPEDANCE OF THE SEMI-CONDUCTOR SPECIMEN. - 7 3 -^max. T ^ C V Z-'Ra. "(7 .1 .11) A l s o , X ( « ) m a x . was o b s e r v e d t o be 6 1 0 - Q , ( F i g . 7 . 9 ) a s c o m -p a r e d w i t h t h e p r e d i c t e d v a l u e o f 907 - 0 - ^ c a l c u l a t e d f r o m e q u a t i o n ( 3 . 1 . 2 1 ) . The d i s c r e p a n c y b e t w e e n t h e e x p e r i -m e n t a l l y o b s e r v e d a n d t h e t h e o r e t i c a l v a l u e s may be due t o t h e f a c t t h a t r o u g h a p p r o x i m a t i o n s were e m p l o y e d as t h e g e n e r a l e x p r e s s i o n f o r X ( u ) i s a l g e b r a i c a l l y q u i t e i n v o l v e d . The l o c u s o f t h e a . c . impedance ( F i g . 7 . 9 ) was n o r m a l t o t h e r e a l a x i s a t b o t h l o w a n d h i g h f r e q u e n c i e s a s e x p e c t e d f r o m t h e t h e r m a l t h e o r y b u t o v e r t h e i n t e r -m e d i a t e f r e q u e n c y r e g i o n , i t c o u l d be a c c u r a t e l y d e s -c r i b e d by an a r c o f a c i r c l e w i t h t h e c e n t r e b e l o w t h e a x i s . The t h e r m a l t h e o r y o f f e r s no e x p l a n a t i o n t o t h i s " c i r c u l a r a r c " l o c u s f o r t h e i n t e r m e d i a t e f r e q u e n c y r e g i o n . The g e n e r a l e x p r e s s i o n f o r t h e " c i r c u l a r a r c " l o c u s f o r impedance i s g i v e n b y Z(00)~Rt°) - 7 ^ - * ( 7 . 1 . 1 2 ) where n may r a n g e f r o m z e r o t o u n i t y . Z ( u ) i n e q u a t i o n ( 7 . 1 . 1 2 ) i s c a u s a l and c a n t h e r e -1000-800 t V (mV) 600-1 400 200-F i g . 7 . 1 0 a S t e a d y - s t a t e c u r r e n t - v o l t a g e C h a r a c t e r e r i s t i c f o r t h e m e t a l s p e c i m e n i n a n e v a c u a t e d e n c l o s u r e . V (volts) 4-3-A 2H F i g . 7 - 1 0 b . S t e a d y - s t a t 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 m e t a l s p e c i m e n u n d e r one a t m o s p h e r e p r e s s u r e . I (mA) - 7 4 -f o r e r e p r e s e n t a p h y s i c a l l y r e a l l a z a b l e s y s t e m |_MacDonald a n d B r a c h m a n , 1956 . The e f f e c t i v e t h e r m a l t i m e c o n s t a n t o f t h e s p e c i m e n ( X ) was m e a s u r e d a s 40i,5 m i l l i s e c c o r r e s p o n d i n g t o I D c = 3.50 mA. The i n t r i n s i c t h e r m a l t i m e c o n s t a n t ( T0 ) was d e t e r m i n e d t o be 44 m i l l i s e c u s i n g t h e e x p r e s -s i o n -v-< A Cr ( 7 . 1 . 1 3 ) The p r e d i c t e d v a l u e o f , c a l c u l a t e d f r o m t h e e q u a t i o n was f o u n d t o be 40 m i l l i s e c . as compared w i t h t h e e x p e r i -m e n t a l v a l u e o f 44 m i l l i s e c . 7 . 2 M e t a l The s t e a d y s t a t 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 m e t a l s p e c i m e n c o n t a i n e d i n a n e v a c u a t e d e n -—3 c l o s u r e (p « 10 mm o f Hg) i s shown i n F i g . 7 . 1 0 a . The s u r f a c e l o s s p a r a m e t e r A. was e x p e r i m e n t a l l y d e -t e r m i n e d f r o m t h e d . c . r e s i s t a n c e measurement f o r l o w 2 on I ( F i g . 7 . 1 1 a ) was o b s e r v e d t o be l i n e a r f o r I i 0 . 1 0 mA. w i t h a s l o p e o f 6 . 4 x 1 0 5 a m p s " 2 . U s i n g e q u a t i o n ( 7 . 1 . 1 ) , r was d e t e r m i n e d t o be 0 . 0 9 ( a s s u m i n g 8 x i O " 3 -F i g . 7 . 1 1 a V a r i a t i o n o f t h e i n c r e m e n t a l r e s i s t a n c e on I f o r F i g . 7 f l l i b Dependence o f t h e i n c r e m e n t a l r e s i s t a n c e o n I f o r t h e m e t a l s p e c i m e n u n d e r one a t m o s p h e r e p r e s s u r e . =75= K = 0 . 7 2 w a t t s c m " 1 d e g ^ C , ( s m i t h e l s , 1955) f r o m w h i c h X - 3 - 1 was e v a l u a t e d t o be 5 . 5 w a t t s cm deg C . U s i n g e q u a t i o n ( 7 . 1 . 2 ) , X r was c a l c u l a t e d t o be 4 . 8 w a t t s c m " 3 d e g ^ C , t a k i n g £ = 0 . 3 5 ( S m i t h e l s , 1 9 5 5 ) , —4 „ o o p = 5 . 8 4 x 10 cm a n d A » 2 . 7 2 x 10 cm • The h e a t t r a n s f e r c o e f f i c i e n t H f o r t h i s e x p e r i m e n t was e s t i -•=5 - 2 -1 mated t o be 1 .1 x 10 w a t t s cm deg C f r o m w h i c h 1 was c f o u n d t o be 0 . 2 5 w a t t s c m " 3 d e g ^ C . [ j J s i n g e q u a t i o n ( 7 . 1 . 3 ) } - 3 - 1 Thus t h e p r e d i c t e d v a l u e o f X = 5 . 1 w a t t s cm deg C compares f a v o u r a b l y w i t h t h e e x p e r i m e n t a l l y d e t e r m i n e d v a l u e o f X s 5 . 5 w a t t s c m " 3 d e g ^ C . F o r t h e p u r p o s e o f c o m p a r i s o n , t h e d . c . c h a r a c t e r -i s t i c o f t h e same s p e c i m e n u n d e r one a t m o s p h e r e p r e s s u r e 1 — 2 i s shown i n F i g . 7 . 1 0 b . From t h e s l o p e ( 4 . 4 2 x 1 0 ° amps ) o f t h e p l o t o f — ° " a g a i n s t I ( F i g . 7 . 1 1 b ) r was d e -t e r m i n e d t o be 1 .2 f r o m w h i c h X was e v a l u a t e d t o be 138 —3 — l w a t t s cm deg A C . H f o r t h i s c a s e was e s t i m a t e d t o be 5 . 3 x l o " 3 w a t t s cm"* 2 d e g " * C g i v i n g X c = 113 w a t t s c m " 3 d e g ^ C . The s u r f a c e l o s s p a r a m e t e r X was t h u s e s t i -mated t o be 1 1 7 . 8 w a t t s c m " 3 d e g ^ C a s a g a i n s t t h e e x p e r i -=3 i m e n t a l l y o b s e r v e d v a l u e o f 138 w a t t s cm d e g ~ A C , s h o w i n g t h a t c o n v e c t i o n l o s s p l a y s a p r e d o m i n a n t p a r t i n c o n -t r o l l i n g t h e v a l u e o f X . The l i n e a r dependence o f r e s i s t i v i t y on t e m p e r a t u r e assumed i n t h e d . c . X - K t h e r m a l t h e o r y i s v a l i d when t h e e x c e s s t e m p e r a t u r e d o e s n o t e x c e e d 320°C ( F i g . 6 . 3 ) . F i g . 7 . 1 2 a Dependence o f X ( w ) „ _ as a f u n c t i o n o f H . F . ' f~1 ( m e t a l s p e c i m e n ) 4x10 -4 -4 12x10 f " 1 B c p s ) " 1 ] ^ 20x10 -4 224.401 224.30H 224. Z0-{ 224. ICH 224.00 IxlOH i 1 r 2x|0"5 3xl0" 5 4xl0~5 f " % C(cps)- 3 / 2] 5xl0 - 5 F i g . 7 . 1 2 b Dependence o f R ( « ) H j , on f 3 / 2 ( m e t a l s p e c i m e n ) -76= A l s o t h e a s s u m p t i o n o f i n d e p e n d e n c e o f K w i t h r e s p e c t t o t h e v a r i a t i o n o f t e m p e r a t u r e i s a p p r o x i m a t e l y v a l i d when t h e e x c e s s t e m p e r a t u r e does n o t e x c e e d 250°C a s t h e t h e r m a l c o n d u c t i v i t y o f P l a t i n u m v a r i e s f r o m K => 0 . 6 9 w a t t s e r a " 1 deg-^C a t 0°C t o 0 . 7 8 w a t t s c m " 1 d e g " 1 C a t 250°C ( S m i t h e l s , 1 9 5 5 ) . The s m a l l s i g n a l a . c . impedance o f t h e s p e c i m e n s u p e r i m p o s e d on t h e c o n s t a n t d . c . b i a s ( I D c = 1 . 5 0 mA) was measured o v e r t h e f r e q u e n c y r e g i o n o f 0 . 2 0 c p s t o 5 , 0 0 0 c p s . The e x c e s s t e m p e r a t u r e a t t h e c e n t r e o v e r t h e a m b i e n t f o r I ~ n 1 . 5 0 mA. was e s t i m a t e d t o be 244°C ( SL - i . o ) u s i n g t h e e x p r e s s i o n 0fe) = T W - - £ = ^ { , t ^ . ] J - L _ _ , ] ( 7 . 2 > 1 ) F o r t h i s e x p e r i m e n t t h e maximum v a l u e o f "Thomson h e a t " was c a l c u l a t e d t o be 1 .6 x 1 0 " ^ w a t t s compared w i t h t h e " J o u l e h e a t " o f 4 . 6 x 1 0 " 3 w a t t s . The dependence o f on f " was o b s e r v e d 3 t o be l i n e a r f o r f ^ 700 c p s w i t h a s l o p e o f 4 . 7 x 10 : ( F a r a d ) " " . ( F i g . ( 7 . 1 2 a ) . U s i n g t h e e x p r e s s i o n o f r e -a c t a n c e a t h i g h f r e q u e n c y i . e . - 2 ^ L S . ffc^ , ( 7 . 2 > 2 ) oOC —1 - 3 c was d e t e r m i n e d t o be 2 . 2 J o u l e s d e g C cm compared w i t h t h e l i t e r a t u r e v a l u e o f 2 . 5 J o u l e s c m " 3 d e g ^ C f o r f 2 [(cps)2] - 7 7 -2 —3 —1 p l a t i n u m , ( a ^ a J s = 1 4 4 . 5 w a t t s cm deg C a n d R(<=° ) = 224-T2- ) . The t h e r m a l c o n d u c t i v i t y K o f t h e s p e c i m e n was o b t a i n e d f r o m t h e a . c . r e s i s t a n c e measurements a t h i g h - 3 / 2 f r e q u e n c i e s . A p l o t o f R(w) _ a g a i n s t f was f o u n d H . F . t o be a s t r a i g h t l i n e ( f 700 c p s ) w i t h a s l o p e o f 6 x 1 0 3 - Q - s e c 3 / 2 ( F i g . 7 . 1 2 b ) f r o m w h i c h K was d e t e r m i n e d t o be 0 . 7 8 w a t t s c m " 1 deg'-'-C a s compared w i t h t h e l i t e r -a t u r e v a l u e o f 0 . 7 2 w a t t s c m " 1 d e g ^ C . The v a l u e o f $ L was e x p e r i m e n t a l l y d e t e r m i n e d f r o m t h e s l o p e (10 H e n r y ) o f t h e p l o t o f X ( « ) L . F . a g a i n s t f . ( F i g . 7 . 1 3 a ) w h i c h was l i n e a r f o r f < 2 . 5 c p s . A s s u m i n g X « a y ^ J g ' ( v a l i d f o r t h i s e x p e r i m e n t ) , t h e r e a c t a n c e a t l o w f r e q u e n c y i s g i v e n b y 3lRt~)-fto)} +X&-f ( 7 . 2 . 3 ) f r o m w h i c h S L was e v a l u a t e d t o be 1 . 0 , t a k i n g R ( 0 ) = 3 4 8 SL. 2 The dependence o f R(w ) L F # o n f was o b s e r v e d t o 2 be l i n e a r ( f ^ 2 . 5 c p s ) w i t h a s l o p e o f - 3 . 0 SLsec ( F i g . 7 . 1 3 b ) compared w i t h t h e p r e d i c t e d v a l u e o f - 2 . 2 SL s e c / c a l c u l a t e d f r o m e q u a t i o n ( 3 . 2 . 4 ) , The c r o s s - o v e r f r e q u e n c y ( f ^ - . , ) was c a l c u l a f r o m t h e X - K t h e r m a l t h e o r y b y u s i n g t h e e x p r e s s i o n Fig. 7.14 LOCUS OF IMPEDANCE OF THE METAL SPECIMEN. - 7 8 -max. ire ( 7 . 2 . 4 ) a n d was f o u n d t o be 2 4 . 5 c p s compared w i t h t h e e x p e r i m e n -t a l l y o b s e r v e d v a l u e o f 3 0 c p s . The e x p e r i m e n t a l l y o b s e r v e d v a l u e o f X ( w ) =48-TL max, ( F i g . 7 . 1 4 ) i s s m a l l c o m p a r e d w i t h t h e p r e d i c t e d v a l u e o f X ( w ) max. = 93 SL , c a l c u l a t e d b y u s i n g t h e e x p r e s s i o n , ( 7 . 2 . 5 ) The d i s c r e p a n c y b e t w e e n t h e o b s e r v e d a n d t h e p r e d i c t e d v a l u e may be a t t r i b u t e d t o t h e r o u g h a p p r o x i -m a t i o n s made i n d e r i v i n g e q u a t i o n ( 7 . 2 . 8 ) a s a n e x a c t e x p r e s s i o n o f X(w) max. c o u l d n o t be o b t a i n e d due t o t h e t r a n s c e d e n t a l n a t u r e o f X ( w ) . -79-Th e l o c u s o f t h e a . c . Impedance ( F i g . 7.14) was n o r m a l t o t h e r e a l a x i s i n b o t h l o w and h i g h f r e q u e n c y r e g i o n s a s e x p e c t e d f r o m t h e X - K t h e r m a l t h e o r y b u t o v e r t h e i n t e r m e d i a t e f r e q u e n c y r e g i o n , i t c o u l d be d e s c r i b e d by an a r c o f a c i r c l e w i t h t h e c e n t r e b e l o w t h e r e a l a x i s . T h i s c o u l d n o t be e x p l a i n e d on t h e b a s i s o f t h e a . c . X - K t h e r m a l t h e o r y . The m e a s u r e d v a l u e o f t h e e f f e c t i v e t h e r m a l t i m e c o n s t a n t f o r I D (v ^ 1.50 mA. was o b t a i n e d a s 6.20 m i l l i s e c a s c o m p a r e d w i t h t h e p r e d i c t e d v a l u e o f 8.4 m i l l i s e c , c a l c u l a t e d f r o m t h e e x p r e s s i o n t a k i n g S = 350 SL and R = 224 SL . - 8 0 -CHAPTER 8  CONCLUSIONS The work 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 r m a l t h e o r i e s f r o m " l u m p e d " t o " d i s t r i b u t e d " t e m p e r a t u r e m o d e l s . The c r i t e r i a f o r t h e a p p e a r a n c e o f t h e r m a l b r e a k d o w n a n d N e g a t i v e R e s i s t a n c e (NR) i n t h e d . c . a n d a . c . c h a r a c t e r -i s t i c s o f b o t h m e t a l s a n d s e m i - c o n d u c t o r s have been p r e d i c t e d . The p r e d o m i n a n t c h a r a c t e r o f t h e h e a t f l o w ( l o n g i t u d i n a l o r r a d i a l ) d e t e r m i n e s t h e v a r i o u s i m p o r t a n t f e a t u r e s o f t h e s y s t e m e . g . t h e l e n g t h o f t h e p a t h i n -v o l v e d i n t h e t h e r m a l r e l a x a t i o n t i m e . I t a l s o d e t e r m i n e s t h e f o r m o f t h e s t e a d y - s t a t e c h a r a c t e r i s t i c s . T h e o r e t i -c a l a n a l y s i s shows t h a t a n o n - z e r o s u r f a c e l o s s p a r a -m e t e r X i s e s s e n t i a l f o r t h e a p p e a r a n c e o f NR i n s e m i -2 c o n d u c t o r s . The d i m e n s i o n l e s s p a r a m e t e r r = X L K may be u s e d a s a c r i t e r i o n f o r t h e " l u m p e d " a n d " d i s t r i b u t e d " m o d e l s . F o r t h e lumped m o d e l , r >> 1 w h e r e a s f o r t h e d i s t r i b u t e d m o d e l , r 1. The p r e s e n t a n a l y s i s f o r t h e " d i s t r i b u t e d " t e m p e r a t u r e m o d e l may be c a r r i e d o v e r t o t h e c a s e o f " l u m p e d t e m p e r a t u r e m o d e l " by a s s u m i n g r —»• c*> . E x p e r i m e n t s were p e r f o r m e d i n o r d e r t o c h e c k t h e d . c . and a . c . X - K t h e r m a l t h e o r i e s f o r m e t a l s a n d s e m i -- 8 1 -c o n d u c t o r s . F o r e x p e r i m e n t s o n t h e s e m i - c o n d u c t o r s p e c i -men, t h e t h e o r y i s f o u n d t o be v a l i d f o r c u r r e n t - d e n s i t y •=2 J 6 20 amps, cm . F o r h i g h e r v a l u e s o f t h e c u r r e n t -d e n s i t y , t h e l i n e a r v a r i a t i o n o f r e s i s t i v i t y w i t h t e m p e r a t u r e assumed i n t h e t h e o r y may n o t be v a l i d . N e w t o n ' s l a w o f c o o l i n g , e m p l o y e d i n t h e t h e o r y s h o u l d be r e p l a c e d by S t e f a n ' s r a d i a t i o n l a w . T h i s has t h e e f f e c t o f r e d u c i n g t h e t e m p e r a t u r e g r a d i e n t a l o n g t h e s p e c i m e n . M o r e o v e r t h e "Thomson e f f e c t " c a n no l o n g e r be n e g l e c t e d a n d must be c a r e f u l l y a c c o u n t e d f o r i n e x -p e r i m e n t s u s i n g h i g h c u r r e n t - d e n s i t y . A c o m p a r i s o n b e t w e e n t h e d . c . c h a r a c t e r i s t i c s o f t h e m e t a l s p e c i m e n ( p l a t i n u m ) i n a n e v a c u a t e d e n c l o s u r e (X - 5 . 5 w a t t s cm ° deg C) a n d u n d e r one a t m o s p h e r e p r e s s u r e (X = 113 w a t t s cm deg'-'-C) shows t h a t c o n v e c t i o n l o s s p l a y s a d o m i n a n t r o l e i n c o n t r o l l i n g t h e v a l u e o f t h e s u r f a c e l o s s p a r a m e t e r X . I t s h o u l d be s t r e s s e d t h a t f o r t h e a p p l i c a b i l i t y o f t h e X - K t h e r m a l ^theory, t h e s p e c i m e n s h o u l d be t h i n enough t o e n s u r e an i s o t h e r m a l c r o s s - s e c t i o n . The X - K t h e r m a l t h e o r y f o r t h e m e t a l s p e c i m e n h a s been f o u n d t o be v a l i d f o r J £ 5 x 1 0 4 —2 amps, cm . The e x p e r i m e n t a l l y o b s e r v e d a . c . c h a r a c t e r -i s t i c s a t b o t h h i g h a n d l o w f r e q u e n c i e s a r e i n t e r p r e t e d o n t h e b a s i s o f t h e a . c . X - K t h e r m a l t h e o r y f o r b o t h 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 p e c i m e n s . The - 8 2 -" c i r c u l a r a r c " l o c u s o f t h e i m p e d a n c e s o f t h e m e t a l and t h e s e m i - c o n d u c t o r s p e c i m e n s o b s e r v e d e x p e r i m e n t a l l y c o u l d n o t be e x p l a i n e d on t h e b a s i s o f t h e p r e s e n t a . c . X - K t h e r m a l t h e o r y . 1 The p r e s e n t i n v e s t i g a t i o n e n a b l e s one t o d e t e r m i n e t h e n a t u r e o f h e a t f l o w f r o m measurements o f t h e e l e c -t r i c a l c h a r a c t e r i s t i c s o f t h e s p e c i m e n a n d a l s o t o e v a l u a t e t h e s u r f a c e and l o n g i t u d i n a l h e a t l o s s e x p e r i -m e n t a l l y . A n o m a l o u s e l e c t r i c a l f e a t u r e s o b s e r v e d i n m e t a l s o r s e m i - c o n d u c t o r s may be e x p l a i n e d o n t h e b a s i s o f t h e p r e s e n t w o r k . I t i s a l s o r e l e v a n t t o t h e d e s i g n o f m e t a l o r s e m i - c o n d u c t o r b o l o m e t e r s f o r m e a s u r i n g t h e i n c i d e n t r a d i a n t p o w e r . I n c o n c l u s i o n , f u r t h e r e x p e r i m e n t s s h o u l d be p e r -f o r m e d w i t h s e m i - c o n d u c t o r s f o r r ^ 1 . The t h e o r y f o r t h i s c a s e h a s been w o r k e d o u t i n s e c t i o n 2 . 1 . F u r t h e r measurements s h o u l d a l s o be made by v a r y i n g t h e s u r f a c e l o s s p a r a m e t e r X i n b o t h m e t a l s a n d s e m i - c o n d u c t o r s . F u r t h e r m o r e , t h e e x p o n e n t i a l f o r m o f c o n d u c t i v i t y dependence on t e m p e r a t u r e ( s e m i - c o n d u c t o r s ) , t h e v a r i a t i o n o f t h e t h e r m a l c o n d u c t i v i t y a n d t h e h e a t c a p a c i t y p e r u n i t vo lume o f t h e s p e c i m e n w i t h t e m p e r -a t u r e s h o u l d be t a k e n i n t o a c c o u n t i n t h e \ - K t h e r m a l t h e o r y . BIBLIOGRAPHY B u r g e s s , R . E . P h y s . S o c . P r o c . B , 6 8 , 766 (1955a) B u r g e s s , R . E . P h y s . S o c . P r o c . B , 6 8 , 908 (1955b) B u r g e s s , R . E . C a n . J . P h y s . 3 8 , 369 (1960a) B u r g e s s , R . E . P r o c . o f t h e C o n f e r e n c e on S e m i - C o n d u c t o r P h y s i c s , P r a g u e , 818 (1960b) C a r s l a w , H . S . a n d J a e g e r . J . C o n d u c t i o n o f H e a t i n S o l i d s . O x f o r d U n i v e r s i t y P r e s s ( S e c o n d E d i t i o n ) , L o n d o n (1959) F i s h e n d e n , M. a n d S a u n d e r s , O . A . A n I n t r o d u c t i o n t o H e a t T r a n s f e r . O x f o r d U n i v e r s i t y P r e s s , ( F i r s t E d i t i o n ) , London (1950) J o h n s o n , V . A . and L a r k - H o r o v i t z , K. P h y s . R e v . 9 2 , 226 (1953) L a n d e r , J . J . P h y s . R e v . 7 4 , 479 (1948) M a c D o n a l d , J . R . a n d B r a c k m a n , M . K . R e v . M o d . P h y s . 2 8 . 393 (1956) M a c F a r l a n e , G . G . , M c L e a n , T . P . , Q u a r r i n g t o n , J . E . a n d R o b e r t s , V . P h y s . R e v . 1 0 8 . 1377 (1957) M c C a r t h y , K . A . and B a l l a r d , S . S . P h y s . R e v . 9 9 , 1104 (1955) S m i t h , R . A . S e m i - C o n d u c t o r s . C a m b r i d g e , U n i v e r s i t y P r e s s ( F i r s t E d i t i o n ) L o n d o n (1959) S m i t h e l s , C . J . M e t a l R e f e r e n c e Book , V o l . 2 , ( S e c o n d E d i t i o n ) , B u t t e r w o r t h s S c i e n t i f i c P u b l i c a t i o n s , L o n d o n (1955) S t a u s s , H . E . T e m p e r a t u r e ; I t s Measurements a n d C o n t r o l i n I n d u s t r y . R e i n h o l d P u b l i s h i n g C o r p o r a t i o n , New Y o r k , U w S . A . (1941J T u r n e r , D . R . J . E l e c t r o c h e m i c a l S o c . 1 0 6 . 786 (1959) V a l d e s , L . B . P r o c . I . R . E . 4 2 , 4 2 0 ( 1 9 5 4 ) . 

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