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Signal and noise characteristics of photovoltaic P-N junction diodes Galbraith, Donald Stewart 1957

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SIGNAL OF  AND  NOISE  PHOTOVOLTAIC  CHARACTERISTICS  P-N  JUNCTION  DIODES  by DONALD STEWART G A L B R A I T H B. A. , University of British Columbia,  1955  A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T T H E REQUIREMENTS  FOR T H E D E G R E E O F  MASTER OF  SCIENCE  in the Department of Physics  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  O F BRITISH C O L U M B I A  December,  1957  OF  i i ABSTRACT  The noise c h a r a c t e r i s t i c s of i d e a l photovoltaic p-n j u n c t i o n diodes a r e d i s c u s s e d and investigated.  The h y p o t h e s i s i s advanced that  the o p e n - c i r c u i t noise f r o m an i l l u m i n a t e d i d e a l diode i s e n t i r e l y due to the shot n o i s e of the v a r i o u s c u r r e n t c o n t r i b u t i o n s .  Theoretical  j u s t i f i c a t i o n f o r this t h e o r y i s developed and the p a r a m e t e r t, the effective noise t e m p e r a t u r e r a t i o , i s introduced.  The p o s s i b l e r e a s o n s  f o r e x c e s s noise i n p-n photo-diodes o b s e r v e d i n e a r l i e r e x p e r i m e n t s are suggested. The dc and ac b e h a v i o r of a r e a l diode c h o s e n to be v e r y n e a r l y i d e a l i n i t s dc c h a r a c t e r i s t i c i s found to be consistent w i t h e x i s t i n g diode theory. are  The v a r i o u s p a r a m e t e r s a p p r o p r i a t e to the d e v i c e  evaluated. E q u i p m e n t f o r noise m e a s u r e m e n t i s s e l e c t e d and a  c o m p a r i s o n technique adopted.  T h i s m e t h o d avoids m a n y of the p o s s i b l e  e r r o r s inherent i n an absolute m e a s u r e m e n t and a l l o w s an equivalent noise r e s i s t a n c e r e s o l u t i o n of about 200 ohms at r o o m t e m p e r a t u r e i n the 200 c s bandwidth m e a s u r e d .  The o p e n - c i r c u i t noise of the s e l e c t e d  diode i s m e a s u r e d at 20 and 30 k c as a f u n c t i o n of i l l u m i n a t i o n and the r e s u l t s i n t e r p r e t e d i n t e r m s of the equivalent r e s i s t a n c e i n t h e r m a l e q u i l i b r i u m w h i c h w o u l d give the same noise.  C o m p a r i s o n of t h i s set  of v a l u e s w i t h the r e a l part of the j u n c t i o n impedance i n e a c h case indicates t h a t t h e £heory advanced i s adequate to p r e d i c t noise under these c i r c u m s t a n c e s . The  s i g n a l - t o - n o i s e r a t i o f o r a photo-diode u s e d as an  o p e n - c i r c u i t r a d i a t i o n detector i s developed, and s e v e r a l r e c o m m e n d a t i o n s  iii  are made r e g a r d i n g the d e s i g n of a photo-diode to d i s p l a y the m o s t f a v o u r a b l e s i g n a l - t o - n o i s e r a t i o under i l l u m i n a t i o n .  In p r e s e n t i n g the  this thesis in partial fulfilment  requirements f o r an advanced degree at the  of  University  o f B r i t i s h Columbia, I agree t h a t  the  L i b r a r y s h a l l make  it  and  study.  f r e e l y available f o r reference  I  further  agree t h a t : p e r m i s s i o n f o r e x t e n s i v e 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  Department or by h i s r e p r e s e n t a t i v e .  Head o f  my  I t i s understood  t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not  Department o f  be allowed without my  P W J ^  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada.  Columbia,  written  permission.  iv TABLE  OF  CONTENTS  Chapter 1  2  3.  4  Page INTRODUCTION  1  1. 1  Review of Previous Work  1  1. 2  Theoretical Considerations  6  1. 3  Objective of Present Work  12  E X P E R I M E N T A L INVESTIGATIONS  13  2. 1  Direct Current Characteristics  13  2.2  Alternating Current Measurements  2. 3  Noise Measurements  ,  17 24  DISCUSSION O F R E S U L T S  30  3. 1  Inferences from DC and AC Measurements  30  3. 2  Implications of Noise Measurements  34  CONCLUSIONS  36  4. 1  General Comments  36  4. 2  Design Recommendations for Radiation Detectors  38  APPENDDC Shift in DC Bias Due to Rectification of A C Signal from Measuring Bridge  42  BIBLIOGRAPHY  44  ACKNOWLEDGMENTS  I w i s h to thank P r o f e s s o r R. E. B u r g e s s f o r h i s guidance throughout the c o u r s e of t h i s i n v e s t i g a t i o n and f o r h i s m a n y valuable comments d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . The w o r k d e s c r i b e d was c a r r i e d out under Defence R e s e a r c h B o a r d grant number" 9512-22. I a l s o w i s h to thank the B o a r d f o r p e r s o n a l f i n a n c i a l a s s i s t a n c e f r o m A p r i l to D e c e m b e r of 1957. I am indebted to M r . D. A. M c C o y f o r h i s a s s i s t a n c e i n the p r e p a r a t i o n of the f i g u r e s .  D. S. G.  F a c i n g Page 1  -200-L  FIGURE  1  1 CHAPTER 1. 1  1  I N T R O D U C T ION  Review of P r e v i o u s Work  B r i e f l y , a photovoltaic p - n junction diode is a s o l i d state device capable of converting incident illumination (of suitable wavelength) into e l e c t r i c a l energy.  Its operation as a c i r c u i t element can best be  outlined by reference to figure 1.  The dark c h a r a c t e r i s t i c plotted here  is s i m i l a r to that of an ideal diode.  Shockley (1949) has d e r i v e d the  following r e l a t i o n between c u r r e n t (I) and voltage (V) for an ideal p - n junction diode; I = I o ( e x p - g - 1),  (1)  where Io = r e v e r s e saturation current q = e l e c t r o n i c charge k = B o l t z m a n n ' s constant T = absolute temperature of junction. T o do so he assumed that the t r a n s i t i o n r e g i o n between n and p regions has negligible width c o m p a r e d to the diffusion lengths of the c a r r i e r s , and that the c u r r e n t s involved are s m a l l (such that the injected c a r r i e r density in either region is s m a l l c o m p a r e d to the density of the c a r r i e r n o r m a l l y present therein). L a t e r C u m m e r o w (1954) considered the case o c c u r r i n g when photons of energy greater than the energy gap of the semiconductor f a l l on or near the junction creating h o l e - e l e c t r o n pairs which,  crossing  the junction u n s y m m e t r i c a l l y because of the electrostatic field existing there, constitute an additional c ur r en t .  He showed that the current  flow in this case is given by I = U e x p ^ - 1) - B L where B L , the photocurrent, is proportional to the incident light  (2)  2  intensity L.'  In other w o r d s , the i l l u m i n a t i o n causes a  proportional  c u r r e n t c o n t r i b u t i o n in the opposite d i r e c t i o n to the f o r w a r d d a r k current.  In effect, then, i l l u m i n a t i o n d r o p s the diode's d a r k  c h a r a c t e r i s t i c by an amount B L ,  as i s indicated by the c u r v e i n f i g u r e 1  f o r an i l l u m i n a t e d device. That p o r t i o n of the c u r v e f a l l i n g i n the f o u r t h quadrant i s of p a r t i c u l a r i n t e r e s t because it i s the p o r t i o n generated by the i l l u m i n a t e d device and a p a s s i v e load alone.  The  point of i n t e r s e c t i o n  of the a p p r o p r i a t e i l l u m i n a t e d c h a r a c t e r i s t i c w i t h a load line d r a w n f r o m the o r i g i n indicates the voltage a c r o s s and c u r r e n t t h r o u g h that load under that i l l u m i n a t i o n and thus shows the power d e l i v e r e d to the load.  F o r e x a m p l e , the diode tested would d e l i v e r about 8. 5 m i c r o w a t t s  (130//A at 65 mv)  to a 500 ohm  load at the i l l u m i n a t i o n l e v e l plotted.  It should be noted that r e a l d e v i c e s (including that f o r w h i c h f i g u r e 1 was  plotted) exhibit v a r i o u s d e p a r t u r e s f r o m i d e a l behavior.  S e v e r a l of these d e p a r t u r e s w i l l be c o n s i d e r e d i n l a t e r sections. A l t h o u g h these diodes are finding n u m e r o u s a p p l i c a t i o n s as r a d i a t i o n d e t e c t o r s or e n e r g y c o n v e r t o r s , r e l a t i v e l y l i t t l e attention has been paid to c e r t a i n aspects of t h e i r p r o p e r t i e s .  In p a r t i c u l a r , no  thorough i n v e s t i g a t i o n of t h e i r noise behavior under i l l u m i n a t i o n has been p e r f o r m e d . S e v e r a l sets of m e a s u r e m e n t s have been made, however. G i a n o l a (1956) investigated s i l i c o n b r o a d a r e a junctions under openc i r c u i t conditions at v a r i o u s incident light i n t e n s i t i e s and i n the f r e q u e n c y range f r o m 20 cs to 4 kc.  H i s r e s u l t s indicate a l / f r e q u e n c y noise  voltage s q u a r e d s p e c t r u m under constant i l l u m i n a t i o n and a v a r y i n g  3  noise magnitude w i t h v a r y i n g i l l u m i n a t i o n .  T h i s v a r i a t i o n of noise  voltage i s quite m a r k e d and p a s s e s through a m a x i m u m w h i c h a p p e a r s r e a s o n a b l y independent of f r e q u e n c y over the range investigated.  He  e x p l a i n e d h i s r e s u l t s on the b a s i s of current-dependent f l u c t u a t i o n s of c u r r e n t . S p e c i f i c a l l y , he a s s u m e s that the m e a n square c u r r e n t f l u c t u a t i o n i s d i r e c t l y p r o p o r t i o n a l to the photocurrent ( B L i n equation 2) and thus to the i l l u m i n a t i o n ; i . e. , ~i*= s q B L A f w h e r e s i s independent of the magnitude of the photocurrent but i s a function of frequency.  The m e a s u r e d m e a n square voltage f l u c t u a t i o n ,  however, i s the product of t h i s c u r r e n t f l u c t u a t i o n w i t h the square of the d y n a m i c impedance of the junction. T h i s l a t t e r t e r m d e c r e a s e s w i t h i n c r e a s i n g i l l u m i n a t i o n (under o p e n - c i r c u i t conditions) so the m a x i m u m i n the noise voltage v e r s u s i l l u m i n a t i o n plot i s q u a n t i t a t i v e l y reasonable, Gianola's a s s u m p t i o n f o r the f o r m of the c u r r e n t n o i s e i s r a t h e r suggestive of the shot noise equation "F=2qlAf. Any attempt to apply t h i s equation d i r e c t l y to a p-n j u n c t i o n , however, is c o m p l i c a t e d by the n e c e s s i t y of s o r t i n g out the c o n t r i b u t i o n s of the v a r i o u s c a r r i e r s to the c u r r e n t and, a s s u m i n g t h e i r i n d i v i d u a l m o t i o n s to be e s s e n t i a l l y u n c o r r e l a t e d , adding up t h e i r c o n t r i b u t i o n s to the noise.  Gianola's diode e x h i b i t e d c u r r e n t noise of the o r d e r of 1 0  4  g r e a t e r than that w h i c h would r e s u l t f r o m shot noise i n a v a c u u m diode c a r r y i n g a c u r r e n t equal to the p-n diode's photocurrent.  4  The objection m i g h t be r a i s e d that the b r o a d a r e a j u n c t i o n s used i n t h i s i n v e s t i g a t i o n are n o t o r i o u s l y poor diodes, e s p e c i a l l y i n that t h e i r r e v e r s e dc c h a r a c t e r i s t i c s indicate a r a t h e r low shunt resistance.  T h i s being the case, the p r e c e d i n g study was not a c t u a l l y  made under o p e n - c i r c u i t conditions, s i n c e r e l a t i v e l y heavy c u r r e n t s could flow i n t e r n a l l y .  s  To i n c o r p o r a t e the n o n - i d e a l behavior of the device studied, G i a n o l a f i t t e d its c h a r a c t e r i s t i c s to those p r e d i c t e d by the Bethe m o d e l ( T o r r e y and W h i t m e r , 1948).  The equation f o r the dc c h a r a c t e r i s t i c s  of this m o d e l d i f f e r s f r o m that of Shockley's (equation 1) by r e p l a c i n g the latter's I w i t h a t e r m Ioexp(-yS-£^), 0  where 0^8^1/2. Although there i s  a t h e o r e t i c a l b a s i s f o r i t s a p p l i c a t i o n to point-contact diodes, the equation can be,applied to j u n c t i o n diodes only on e m p i r i c a l grounds.  The  effect of the e x t r a f a c t o r i s to lower the c h a r a c t e r i s t i c f r o m the i d e a l (Shockley) curve f o r both f o r w a r d and r e v e r s e b i a s , v i z ;  Poor  r e v e r s e diode behavior, however, can g e n e r a l l y be e x p l a i n e d  in t e r m s of low shunt r e s i s t a n c e , while r e d u c e d f o r w a r d c u r r e n t i n d i c a t e s s e r i e s r e s i s t a n c e or h i g h c a r r i e r i n j e c t i o n (violating Shockley's assumption).  Since the f o r w a r d and r e v e r s e n o n - i d e a l c h a r a c t e r i s t i c s  stem f r o m different causes, i t i s unreasonable to expect an a d d i t i o n a l t e r m i n v o l v i n g only one e x t r a p a r a m e t e r to provide m o r e than a rough,  empirical correction. A n o t h e r study by P e a r s o n , M o n t g o m e r y , and F e l d m a n n (1955) points up one of the dangers encountered i n t h i s f i e l d .  Their  m e a s u r e m e n t s i n d i c a t e that a n e a r l y i d e a l s i l i c o n p-n diode i l l u m i n a t e d and r e v e r s e b i a s e d to give a c u r r e n t of the o r d e r of 1 0 t i m e s I e x h i b i t s 4  0  p r e c i s e l y shot noise down to f a i r l y low f r e q u e n c i e s (80 cs). When the r e v e r s e s a t u r a t i o n c u r r e n t i s r a i s e d by a f a c t o r of about 100 by exposing the device to v e r y h u m i d a i r , the noise behavior changes m a r k e d l y ; e x h i b i t i n g a l / f r e q u e n c y c u r r e n t - s q u a r e d s p e c t r u m w i t h a 100 cs value about 10 above shot noise. e  T h e s e r e s u l t s indicate (as indeed the authors  concluded) that e x c e s s noise i n s e m i c o n d u c t o r d e v i c e s can be a s t r o n g l y surface-dependent p r o p e r t y . N o i s e m e a s u r e m e n t s have been made on an InSb photovoltaic c e l l ( M i t c h e l l , G o l d b e r g , and K u r n i c h , 1955) under illumination.  A l/frequency noise voltage-squared spectrum i s reported 3  below 2 kc w i t h s u b s t a n t i a l l y white noise (mean s q u a r e d noise voltage constant w i t h frequency) f r o m 2 kc up to about 200 kc.  Once again,  however, the t a s k of a s c r i b i n g the noise g e n e r a t i o n to a s i m p l e m e c h a n i s m i s c o m p l i c a t e d by the n o n - i d e a l nature of the d e v i c e w h i c h , a c c o r d i n g to the authors, 'exhibited . . .  no noticeable r e c t i f i c a t i o n . *  6  1. 2  Theoretical Considerations  At present, t h e r e i s no a l l - e m b r a c i n g t h e o r y f o r the n o i s e b e h a v i o r of an i d e a l p-n photo-diode although s e v e r a l r e a s o n a b l e c o n j e c t u r e s m a y be made.  O b v i o u s l y , Nyquist's t h e o r e m m u s t apply to  an o p e n - c i r c u i t e d diode i n the d a r k when i n t h e r m a l e q u i l i b r i u m whether the noise i s c o n s i d e r e d f r o m the t h e r m o d y n a m i c o r the c o r p u s c u l a r point of view. T o i l l u s t r a t e t h i s , c o n s i d e r an i d e a l S h o c k l e y diode i n thermal equilibrium.  Its c h a r a c t e r i s t i c s a r e then g i v e n by equation 1  w h i c h m a y be i n t e r p r e t e d as p r e d i c t i n g a c u r r e n t L>expS^ i d i r e c t i o n and I i n the r e v e r s e d i r e c t i o n . 0  n  the f o r w a r d  If we a s s u m e these c u r r e n t s  to be m i c r o s c o p i c a l l y u n c o r r e l a t e d (or m o r e e x a c t l y , if we a s s u m e the c a r r i e r t r a n s i t s c o m p r i s i n g these c u r r e n t s o c c u r independently and r a n d o m l y at m e a n r a t e s - ^ e x p ^ ^ a n d - ^ r e s p e c t i v e l y ) we m a y w r i t e the shot noise equation as 2 q l = 2 q l ( e x p - g ^ + 1). 0  Since we have postulated t h e r m a l e q u i l i b r i u m , V m u s t be z e r o , so 4ql . 0  D i f f e r e n t i a t i n g equation 1 w i t h r e s p e c t to V and setting V equal to z e r o g i v e s f o r the low frequency o p e n - c i r c u i t conductance G  =-§V=-§Y"»  or q l = 0  kTG.  W i t h t h i s s u b s t i t u t i o n , the shot noise equation b e c o m e s l  *  Af  4kTG  w h i c h i s Nyquist's t h e r m o d y n a m i c r e s u l t .  Thus the t h e r m o d y n a m i c and  c o r p u s c u l a r approaches have been shown to b e c o m p a t i b l e i n t h e r m a l (  e q u i l i b r i u m under the a s s u m p t i o n s used.  F a c i n g Page 7  FIGURE  2 i  ELECTROSTATIC  P O T E N T I A L AND QUASI-FERMI  IN A N I L L U M I N A T E D  P+-N  JUNCTION  n V  vi  LEVELS  7  Under i l l u m i n a t i o n , however, the s i t u a t i o n could be m o r e complicated. model.  L e t us f i r s t d e s c r i b e i n g e n e r a l t e r m s the u s u a l photo-diode  F i g u r e 2 shows the i n t e r n a l d i s t r i b u t i o n of the e l e c t r o s t a t i c  potential ijj and the q u a s i - F e r m i l e v e l s <p and <p (Shockley, 1949) f o r a p  o n e - d i m e n s i o n a l model.  n  Here V and I, the voltage a c r o s s and c u r r e n t  through the diode, a r e the same as those appearing i n equation 2. - F o r s i m p l i c i t y , the diode shown i s p -n, r a t h e r than p-n, because under t h i s +  condition v i r t u a l l y a l l the c u r r e n t i n the n r e g i o n at the j u n c t i o n i s due to hole diffusion. F o r s m a l l injected c a r r i e r d e n s i t i e s , the r e v e r s e c u r r e n t a r i s e s f r o m holes diffusing f r o m the n to the p r e g i o n at a rate +  independent of the voltage a c r o s s the j u n c t i o n but p r o p o r t i o n a l to the number of holes a v a i l a b l e f o r such diffusion.  Thus the r e v e r s e c u r r e n t  is I + B L , where I , the m a c r o s c o p i c d a r k s a t u r a t e d r e v e r s e c u r r e n t , Q  c  c o r r e s p o n d s to the t h e r m a l l y generated holes and i s p r o p o r t i o n a l to the e q u i l i b r i u m d e n s i t y of holes i n the n r e g i o n p ; being equal i n fact n  (Shockley, 1949), to Aqp -f^-, where A i s the a r e a of the j u n c t i o n and n  Lip  Dp and L|6 are the d i f f u s i o n constant and d i f f u s i o n length r e s p e c t i v e l y f o r holes i n the n r e g i o n .  The B L t e r m accounts f o r the e x t r a l i g h t -  c r e a t e d holes a v a i l a b l e f o r d i f f u s i o n . The f o r w a r d c u r r e n t , on the other hand, a r i s e s f r o m those holes i n the p r e g i o n having enough e n e r g y to surmount the j u n c t i o n +  " potential b a r r i e r .  T h i s b a r r i e r i s l o w e r e d f r o m its e q u i l i b r i u m height  be a f u n c t i o n of V and i s , i n fact, I exp-^> 0  It should be noted that the p r e c e d i n g d i s c u s s i o n - subject  I  8  to the r e s t r i c t i v e a s s u m p t i o n of low i n j e c t e d c a r r i e r d e n s i t y - i s quite independent of the cause of the voltage a c r o s s the diode; t h i s voltage m a y be a p p l i e d by an e x t e r n a l b a t t e r y , m a y be the o p e n - c i r c u i t photovoltage of the diode, o r m a y be the photovoltage appearing  across  an a r b i t r a r y load. If we assume, as i n the e q u i l i b r i u m case, that the c o n t r i b u t i o n s to the t o t a l c u r r e n t a r e i n d i v i d u a l l y u n c o r r e l a t e d , we m a y w r i t e the shot noise equation f o r the i l l u m i n a t e d case a s = 2 q ( I e x p ^ + I.+ B L ) . 0  We s h a l l now introduce a new v a r i a b l e t, the effective noise t e m p e r a t u r e r a t i o , w h i c h we define as the r a t i o of the m e a n s q u a r e d c u r r e n t o r voltage noise i n a p a r t i c u l a r case to that given by Nyquist's t h e o r e m a p p l i e d to the conductance or r e s i s t a n c e . Thus f o r c u r r e n t noise 4kTG * D i f f e r e n t i a t i n g equation 2 w i t h r e s p e c t to V gives f o r the low frequency diode conductance under i l l u m i n a t i o n G  and hence  »  =4v="k% Pk T" k T < + *• + ex  t  l  =  1  B L  )'  [ l + (1 + ^ " ) e x p ( - ^ ) ] .  <  3 )  (4)  To investigate the o p e n - c i r c u i t case, I = 0 i s s u b s t i t u t e d into equation 2 to give exp - so that  k T  t  yi +  [ l + exp  u  V  >].  The o p e n - c i r c u i t noise t e m p e r a t u r e r a t i o , then, i s t-c=i(i + i ) =l that i s , the open-rcircuit noise of a n i l l u m i n a t e d i d e a l diode i s , under  9  these a s s u m p t i o n s , equilibrium.  that expected f r o m the same impedance i n t h e r m a l  If c u r r e n t f l o w s , however, i n c r e a s e d noise m a y be expected  s i n c e then V < V f o r a given  BL  o c  Subject to our a s s u m p t i o n s ,  then,we have shown that although  an o p e n - c i r c u i t e d i l l u m i n a t e d i d e a l diode e x h i b i t s t h e r m a l noise only, use of the diode to d e l i v e r c u r r e n t to a l o a d (or to accept c u r r e n t f r o m a s o u r c e ) w i l l i n c r e a s e the noise. device w i l l next be c o n s i d e r e d .  The effect of c u r r e n t flow w i t h i n the  T h i s s i t u a t i o n c o u l d a r i s e i n an i d e a l  diode if the i l l u m i n a t i o n w e r e not p e r f e c t l y u n i f o r m .  L e t us suppose,  for e x a m p l e and as a r o u g h a p p r o x i m a t i o n , that h o l e - e l e c t r o n p a i r s a r e being c r e a t e d by light u n i f o r m l y i n p a r t of the diode (to w h i c h we m a y thus apply equation 2), but a r e being p r o d u c e d only t h e r m a l l y i n the r e s t of it (to w h i c h we m a y apply equation 1).  T h e n we m a y r e p r e s e n t the  s y s t e m by two diodes i n p a r a l l e l : Diode 1: I,= I o , ( e x p ^  _ 1  ) "  Diode 2: I = I ( e x p - j ^ - 1) z  0 2  where diodes 1 and 2 r e p r e s e n t the i l l u m i n a t e d and d a r k p o r t i o n s r e s p e c t i v e l y of the complete device. p a r t s of the diode to be at the same  We s h a l l a l s o a s s u m e the two temperature.  P r o c e e d i n g i n the s a m e way as b e f o r e , we w r i t e f o r the shot noise d e n s i t y - £ = 2q(Lwexp^-+ I ,+ B L ) kT 0  B  L  10  and  iif  ~M~  =  qV  2q(I 2exp Q  k T  + I0z) .  = 2q[(I„,+ L ^ e x p - ^ + I 0 , + I0z+ B L J  Hence  The t o t a l c u r r e n t through the composite diode i s  Uz) - B L  I = (Io,+ M e x p ^ -  so that i t s low frequency conductance i s _ dl_ _q_ . _q_V ~ dV ~ k T ^ *«/ PkT ' G  Then  I o l +  t =  ex  4kTG  =*[ < T3T;»^-S-'/1+  1+  Since I c = I01+ Ioz. the noise t e m p e r a t u r e r a t i o i s unchanged by the nonu n i f o r m i t y of the i l l u m i n a t i o n . qVoc  so that  In p a r t i c u l a r , i n the o p e n - c i r c u i t c o n d i t i o n  , ,  B L  toc= 1.  Thus the i n t e r n a l c u r r e n t p r o d u c e d i n the diode by n o n - u n i f o r m i l l u m i n a t i o n does not cause an a d d i t i o n to the m e a s u r a b l e noise.  This  r e s u l t s i m p l i f i e s the e x p e r i m e n t a l techniques, s i n c e no s p e c i a l attention need be p a i d to the homogeneity of the incident light beam o r to the d e t a i l e d g e o m e t r y of the junction. The objection m i g h t r e a s o n a b l y be r a i s e d that the holee l e c t r o n p a i r s produced by the light a r e being c r e a t e d w i t h e n e r g i e s i n m a n y cages c o n s i d e r a b l y i n e x c e s s of the t h e r m a l e n e r g y of the c r y s t a l l a t t i c e , and it would seem p o s s i b l e that p r i o r to t h e r m a l i z a t i o n these p a i r s could lead to a d d i t i o n a l noise.  Davydov (Tauc, 1957) has shown,  however, that t h i s t h e r m a l i z a t i o n takes place i n a t i m e v e r y s h o r t c o m p a r e d w i t h the l i f e t i m e s of the m i n o r i t y c a r r i e r s : r e p r e s e n t a t i v e f i g u r e s being 10""seconds f o r the t h e r m a l i z a t i o n p r o c e s s and lO'^seconds  f o r a t y p i c a l l i f e t i m e . T h i s i s f u r t h e r support f o r the t h e o r y that the noise under i l l u m i n a t i o n should be p r e d i c t e d - at l e a s t a p p r o x i m a t e l y by Nyquist's e x p r e s s i o n s , s i n c e the p h o t o c a r r i e r s w i l l have e n e r g i e s c o r r e s p o n d i n g to the l a t t i c e t e m p e r a t u r e except f o r a v e r y s h o r t i n i t i a l p e r i o d of elevated energy.  12 1. 3  Objective of P r e s e n t Work.  The objective of the p r e s e n t w o r k i s to provide an e x p e r i m e n t a l b a s i s f o r a noise generation m o d e l w h i c h , i t i s hoped, w i l l p r e d i c t the o p e n - c i r c u i t noise voltage appearing at the t e r m i n a l s of an i l l u m i n a t e d photo-diode i n t e r m s of the m e a s u r a b l e p a r a m e t e r s of that diode.  In p a r t i c u l a r , i t i s p r o p o s e d to investigate the  p o s s i b i l i t y of p r e d i c t i n g this noise voltage on a 'quasi-Nyquist' b a s i s ; that i s , can the noise of an i l l u m i n a t e d device s t i l l be a s c r i b e d to the N y q u i s t - p r e d i c t e d t h e r m a l noise of the r e a l part of the j u n c t i o n impedance at the same f r e q u e n c y (as the noise m e a s u r e m e n t i s m a d e ) and under the same i l l u m i n a t i o n ? T o t h i s end, a photo-diode has been s e l e c t e d whose d i r e c t c u r r e n t c h a r a c t e r i s t i c s appear n e a r l y i d e a l .  Various measurements  a r e d e s c r i b e d f r o m w h i c h the p a r a m e t e r s c h a r a c t e r i z i n g the device can be i n f e r r e d .  T h e s e include d i r e c t c u r r e n t m e a s u r e m e n t s of s h o r t - c i r c u i t  c u r r e n t and o p e n - c i r c u i t voltage under v a r y i n g i l l u m i n a t i o n , c u r r e n t and voltage under constant i l l u m i n a t i o n , and impedance at f r e q u e n c i e s f r o m 1 to 100 kc under v a r y i n g i l l u m i n a t i o n .  The diode's noise has  been m e a s u r e d ( l i m i t e d to f r e q u e n c i e s f r o m 20 to 30 kc by equipment and techniques a v a i l a b l e ) as a f u n c t i o n of i l l u m i n a t i o n and i n t e r m s of equivalent noise r e s i s t a n c e .  13 CHAPTER 2. 1  2  -  E X P E R I M E N T A L  INVESTIGATIONS  Direct Current Characteristics v  Photo-diodes a v a i l a b l e f o r s e l e c t i o n w e r e types 5C and 1N188A grown g e r m a n i u m p-n junctions m a n u f a c t u r e d by C l e v i t e Transistor Products.  The d i r e c t c u r r e n t c h a r a c t e r i s t i c s of two of the  f o r m e r and five of the l a t t e r w e r e m e a s u r e d both i n the d a r k and under constant i l l u m i n a t i o n , and one of the f o r m e r was s e l e c t e d f o r f u r t h e r i n v e s t i g a t i o n because of i t s n e a r - i d e a l p e r f o r m a n c e .  The e x p e r i m e n t a l  setup was as follows? 2kn  A/VWW VAA/  V ^ / T  1  VacuomvoI+m<z  The r e s u l t s of these m e a s u r e m e n t s w i t h the s e l e c t e d diode a r e those shown i n f i g u r e 1. The s a t u r a t i o n evidenced by the r e v e r s e c h a r a c t e r i s t i c i n d i c a t e s c o n s i s t e n c y w i t h equations 1 and 2 f o r the d a r k and light c h a r a c t e r i s t i c s r e s p e c t i v e l y . F u r t h e r evidence f o r the a p p l i c a b i l i t y of these equations is obtained by c o n s i d e r i n g the p h o t o c h a r a c t e r i s t i c equation.  Examining  equation 2 under s h o r t - c i r c u i t conditions (V = 0) gives Isc= - B L  (5)  and under o p e n - c i r c u i t conditions (I = 0) gives 0 = I (exp^f0  1) - B L .  FIGURE  Facing Page 14  3  FORWARD DARK, FORWARD I L L U M I N A T E D , AND  PHOTOCHARACTERISTICS O F S E L E C T E D DIODE A  a  5 --  A  F o r w a r d licjhf c h a r a c h z r i s l i c  a  Forward  °.  d a r k charctc+<zrbiic  Pho4"ocriaracf<2ris+ic  4 --  3 --  2 --  -"Slope = 2>75  vol+5  - 1  (-ft- = 39. G voi-hr' a l 2 0 ° C )  0 . 2  Volts -I --  -—1~O.OZQju A  a  14  C o m b i n i n g these gives the p h o t o c h a r a c t e r i s t i c equation f o r the m o d e l : 11^1= I„(exp-g^- 1) or  (6)  ln( |lsc| + I ) = j ^ V ^ + ln(U) . Q  T h i s equation i s of the same f o r m as the f o r w a r d d a r k c h a r a c t e r i s t i c . T h u s a plot of l n ( | l s c | + Io) v e r s u s Voo should be a s t r a i g h t line of slope •r^L-if t h i s m o d e l i s v a l i d . good agreement.  F i g u r e 3 i s s u c h a plot and suggests r a t h e r  It should be noted that the ordinate a x i s i n t e r c e p t of  the line obtained f r o m the high c u r r e n t r e a d i n g s (such that |l |»I ) gives sc  0  an a p p r o x i m a t i o n to ln(I„); using t h i s value f o r I enables the plot of Q  ln(|lsc|+  Io) to be completed at low c u r r e n t s .  ,  The m e a s u r e m e n t s f o r the p h o t o c h a r a c t e r i s t i c plot w e r e obtained u s i n g a h e a v i l y shunted m i r r o r - t y p e g a l v a n o m e t e r f o r I , a s c  potentiometer f o r Voc, and a 150 watt incandescent p r o j e c t i o n l a m p f o r a light source.  V a r i a t i o n of light i n t e n s i t y was p r o v i d e d f o r by running  the l a m p f r o m 60 cs power obtained through a v a r i a b l e a u t o t r a n s f o r m e r . The o p t i c a l a r r a n g e m e n t c o n s i s t e d of a single c o n v e r g i n g lens of about 20 cm f o c a l length to concentrate the light somewhat and a b l o c k of c l e a r l u c i t e about 1 i n c h t h i c k to m i n i m i z e heating effects.  The l u c i t e  is a good a b s o r b e r of those wavelengths g r e a t e r than that c o r r e s p o n d i n g to the energy gap i n g e r m a n i u m w h i c h would produce only heat i n the diode. The slope of the p h o t o c h a r a c t e r i s t i c line i n f i g u r e 3 i s about 37. 5 volt"', w h e r e a s -j^-for r o o m t e m p e r a t u r e (20°C) i s 39. 6 volt" ; 1  so the r e s u l t s a r e i n reasonable agreement.  The o r d i n a t e a x i s i n t e r c e p t  leads to an e s t i m a t e that I is about 0. 3juA f o r t h i s diode. Q  the same o r d e r as the r e v e r s e s a t u r a t i o n c u r r e n t noted.  T h i s i s of The d i f f e r e n c e  at -2. 0 v o l t s b i a s would l e a d to a value of [5 of about 0. 005 if the Bethe  15  e x p r e s s i o n (page 4) w e r e f i t t e d there.  The Bethe equation was thus not  considered further. F i g u r e 3 shows a l s o the data of f i g u r e 1 r e p l o t t e d w i t h s e m i l o g a r i t h m i c c o - o r d i n a t e s ( I h a s been added to e a c h s c  i l l u m i n a t e d c u r r e n t reading to enable i t s f o r m to be c o m p a r e d w i t h the others' m o r e e a s i l y ) .  In g e n e r a l both these sets of data agree f a i r l y  w e l l w i t h the p h o t o c h a r a c t e r i s t i c , although both f a l l somewhat below i t . T h i s behavior can be e x p l a i n e d by a s s u m i n g the e x i s t e n c e of s e r i e s r e s i s t a n c e w i t h i n the diode.  C o n s i d e r the s i m p l e equivalent c i r c u i t :  Although V ' i s c l e a r l y the quantity u s e d i n the b a s i c diode equations, V i s the quantity a c t u a l l y m e a s u r e d .  However V ' = V - I R so we m a y  r e w r i t e equation 1 i n t e r m s of the m e a s u r a b l e voltage V as I -Uexp[aiV^l]-  u  and f o r the p u r p o s e s of f i g u r e 3 as ln(|l| + 1„) = - £ T ( V  -  IR)  +  ln(I ) . 0  Substituting the known and p r e v i o u s l y c a l c u l a t e d v a l u e s and u s i n g an e x p e r i m e n t a l point w h i c h appears r e p r e s e n t a t i v e of the f o r w a r d d a r k line (point at 0. 2 volt) i n d i c a t e s that the c u r v a t u r e of f i g u r e 3 i s consistent w i t h a s e r i e s r e s i s t a n c e somewhat g r e a t e r than 50 ohms.  The f o r w a r d  d a r k c h a r a c t e r i s t i c c o r r e c t e d on the a s s u m p t i o n of 60 ohms s e r i e s r e s i s t a n c e i s plotted i n f i g u r e 4. The l i n e a r i t y i m p r o v e m e n t made by this c o r r e c t i o n i s quite apparent.  The slope of the l i n e shown i s about  36 volt" c o m p a r e d with-^p = 39. 6 volt" . 1  1  The fact that the s l o p e s of the l i n e s plotted ( p a r t i c u l a r l y the last-mentioned) are not quite equal to-j^jr m a y indicate a d e p a r t u r e f r o m the behavior p r e d i c t e d by Shockley, although not n e c e s s a r i l y a departure f r o m g e n e r a l i d e a l b e h a v i o r , of w h i c h Shockley's m o d e l i s a s p e c i a l case (for s m a l l injected c a r r i e r d e n s i t i e s ) .  T h i s point w i l l  be d i s c u s s e d i n connection w i t h the i n t e r p r e t a t i o n of c e r t a i n of the alternating current measurements. No attempt was made i n these o r l a t e r e x p e r i m e n t s to m e a s u r e the absolute magnitude of the light i n t e n s i t y incident o n the junction.  Such f a c t o r s as c e l l o r i e n t a t i o n and j u n c t i o n g e o m e t r y would i  m a k e this m e a s u r e m e n t of l i t t l e m e a n i n g i n any case.  The s i g n i f i c a n t  quantity, i n the light of the p r e v i o u s l y d e r i v e d r e l a t i o n Isc= - B L is the s h o r t - c i r c u i t c u r r e n t through the diode.  This i s always a  m e a s u r e of the rate of p r o d u c t i o n of photo-electron-hole p a i r s , and avoids a l l d i f f i c u l t i e s i n v o l v e d i n m e a s u r i n g the s p e c t r a l d i s t r i b u t i o n of the light and i n c o n s i d e r i n g the i n t e r n a l s t r u c t u r e of the device.  17 2. 2  Alternating Current Measurements  F o l l o w i n g the s u c c e s s i n r e l a t i n g the d i r e c t c u r r e n t c h a r a c t e r i s t i c s to a s i m p l e m o d e l , it was hoped that the a l t e r n a t i n g c u r r e n t behavior c o u l d i n s o m e f a s h i o n a l s o be e x p l a i n e d i n t e r m s of a s i m p l e p i c t u r e . A r e a s o n a b l e equivalent c i r c u i t f o r a r e a l diode at some f i x e d b i a s i s :  c-  where R and C m a k e up the j u n c t i o n impedance and g e o m e t r i c shunt capacitance of the device and R s r e p r e s e n t s the bulk r e s i s t a n c e of the s e m i c o n d u c t o r together w i t h contact and l e a d r e s i s t a n c e s .  Suppose  we m e a s u r e t h i s admittance ( s t i l l at a f i x e d b i a s ) as a f u n c t i o n of frequency and i n t e r m s of i t s p a r a l l e l components. _  1  Y  X  Then  1 + JcoCR Z ~ R + R ( l + jcbCR) 3  _ R + Rs +UJ*C RR + 2  3  i^CR  (R s + R)* + (ooCRRs)2 (R s + R)  Rs Rs _(R + R) + ( w C R R ) T ( s+ R)* + (wCRRsf' S  z  s  R  T h e n if we adopt the notation Y = G + jS . , .., we m a y identify  „ 1 G =  and  S=  R F (Rs + R) l f S [ ( R 5 + R ) * + (^ ooCRRs)5  (R s + R f + (coCRRaf *  If we now l e t Rm =— and Cm be the p a r a l l e l components a c t u a l l y m e a s u r e d ,  F I - G U R E  5 *1 f»  o 0 oo  era E X P E R I M E N T A L O F  A R R A N G E M E N T D A R K  A C  F O R  M E A S U R E M E N T  A D M I T T A N C E  » '  °°  18  we see that  S = toCm  and t h e r e f o r e  ( R + R f + (ooCRRsf = 7 ^ . J  s  Substitution of this into the e x p r e s s i o n f o r G gives J_  R m=  G  «-» - UAVS - ±_Mi±*L S-T3 Uxvi VsV) Urn. =  Cm  Hence if we plot the m e a s u r e d conductance G (the r e c i p r o c a l of the m e a s u r e d p a r a l l e l r e s i s t a n c e component of the admittance) against the m e a s u r e d p a r a l l e l capacitance, we should obtain a s t r a i g h t line i n t e r s e c t i n g the ordinate a x i s a t — . T h i s line i s generated by v a r y i n g  Rs the frequency of m e a s u r e m e n t . The e x p e r i m e n t a l a r r a n g e m e n t f o r the d a r k admittance m e a s u r e m e n t i s shown i n f i g u r e 5. P o t e n t i o m e t e r  1 i s u s e d to set the  ac s i g n a l l e v e l a c r o s s the diode to a s u f f i c i e n t l y low l e v e l to a v o i d d i s p l a c e m e n t of the operating voltage (at the m e a s u r e d c u r r e n t ) and e x c e s s i v e h a r m o n i c d i s t o r t i o n due to the diode's n o n - l i n e a r i t y . Potentiometer  2 i s used to set the b i a s l e v e l .  B i a s c u r r e n t r a t h e r than  voltage i s m e a s u r e d since i t i s g e n e r a l l y the m o r e s e n s i t i v e p a r a m e t e r . The output i s o l a t i o n t r a n s f o r m e r i s o r i e n t e d f o r m i n i m u m 60 cs m a g n e t i c pickup f r o m the m a n y s o u r c e s i n the l a b o r a t o r y .  F i n a l balance,  obtained  by adjustment of the R ^ a n d Cm e l e m e n t s , i s d i s p l a y e d as an e l l i p t i c a l p a t t e r n on an o s c i l l o s c o p e to d i s c r i m i n a t e against h a r m o n i c s , n o i s e , and hum. To check the o p e r a t i o n of the b r i d g e , a d u m m y diode was p r e p a r e d a s a test c i r c u i t . ,  It c o n s i s t e d of a 0. 06 ^ F c a p a c i t o r and a  470 ohm r e s i s t o r i n p a r a l l e l r e p r e s e n t i n g the j u n c t i o n impedance, both i n s e r i e s w i t h a 100 ohm r e s i s t o r r e p r e s e n t i n g the s e r i e s r e s i s t a n c e . The admittance of this d u m m y was m e a s u r e d f r o m 5 to 100 k c and the  FIGURE  Facing Page 19  6  M E A S U R E D C O N D U C T A N C E VERSUS MEASURED P A R A L L E L  CAPACITANCE  FOR DUMMY AND R E A L  -H  O  1  1  1  1  0.2  O.I  C  m  (yuF)  1  DIODE  —| 0.3  1  \ 0.4-  19  r e s u l t s plotted as suggested above - see f i g u r e 6.  The ordinate a x i s  i n t e r c e p t of the best s t r a i g h t line t h r o u g h the points c o n f i r m s e x a c t l y (within the l i m i t of a c c u r a c y of the graph) the value of s e r i e s r e s i s t a n c e , used.  :  A l s o plotted on this g r a p h i s a t y p i c a l set of points obtained f r o m m e a s u r e m e n t of the d a r k diode admittance under f o r w a r d bias.  The pronounced c u r v a t u r e evident here was noted at a l l b i a s e s  i n v e s t i g a t e d (+20 to +150 uA). /  It m a y t h e r e f o r e be c o n c l u d e d that the  s i m p l e equivalent c i r c u i t suggested cannot be a p p l i e d to the device. T h i s m e t h o d a l s o e l i m i n a t e s f r o m c o n s i d e r a t i o n the two slightly m o r e complicated resistance-capacitance networks depicted be lows  The f i r s t of these leads to a s t r a i g h t line d i s p l a c e d v e r t i c a l l y f r o m that of the s i m p l e case by the added shunt conductance-^-, w h i l e the second leads to a s t r a i g h t line d i s p l a c e d h o r i z o n t a l l y by the added c a p a c i t a n c e C'. N e i t h e r of these n e t w o r k s , t h e r e f o r e , a r e a p p l i c a b l e as equivalent circuits. Since we hope to r e l a t e m e a s u r e d diode n o i s e to m e a s u r e d diode conductance, the l a c k of a s i m p l e m o d e l on w h i c h to base the l a t t e r is r e g r e t t a b l e but by no m e a n s s e r i o u s .  Because the diode admittance  can r e a d i l y be m e a s u r e d under conditions to be encountered, no f u r t h e r effort was d i r e c t e d to finding a m o d e l by w h i c h it m i g h t be p r e d i c t e d .  FIGURE  7 *1 $» o  E X P E R I M E N T A L ARRANGEMENT  FOR M E A S U R E M E N T  O F ILLUMINATED AC A D M I T T A N C E  CO  ( * >  00  IN OPEN-CIRCUIT  CONDITION  ft o  Sig  A-C. Iin<z  The measurement of diode admittance under illumination • is undertaken in very much the same manner.  The effective illumination  at the junction is determined by measuring the diode's short-circuit current (see equation 5).  Then the device is switched into the bridge  circuit in series with a large capacitance to ensure open-circuit conditions.  A similar capacitance in the variable arm of the bridge  minimizes error due to the added element. The requirement of open-circuit measurement, however, introduces an interesting complication.  The rest of the bridge may be  considered to be an alternating current source, so the diode-capacitor system may be thought of as a rectifier-filter combinations  A net dc bias will therefore appear across the diode in these ' circumstances even without illumination.  Hence the photo-bias  supposedly set by adjusting the illumination for a given short-circuit current will be disturbed when alternating current from the bridge is applied.  It is shown in the appendix that this additional bias ( A V ) is  given by  where V,sinwt is the applied ac voltage.  It is interesting to note that this  bias shift is independent of the level of illumination. The effect of this shift in dc bias is also investigated in the appendix and it is shown that the error in the measured conductance  Facing Page 21  F I G U R E  8  I  21  from this cause will be less than 1% if V, is less than about 5 mv. For this reason the signal generator output is kept as low and the detector sensitivity raised as high as practicable (limited by noise and hum) so that the ac signal across the diode can be kept sufficiently low. Illumination is provided by a low-voltage lamp run from a variable dc supply to avoid ripple fluctuations in the light intensity.  The experimental  arrangement is shown in figure 7. Since we are now to be concerned with frequency-dependent quantities, we shall designate the diode conductance by the symbol G(io) and thus its low frequency value (G in earlier sections) by G(0). The open-circuit low frequency conductance (which we shall call Go(0) ) is obtained from equation 3 (page 8) by setting 1 = 0. G  / A . -(°)  =  qlo k f e  X p  Thus;  qVoc k f  ^(io+M. Therefore successive plots of G0(us) versus the short-circuit current should approach, as the frequency is decreased, a straight line of slope q -pjr and abscissa axis intercept -1^.  Figure 8 is such a plot.  Since the 1 and 2 kc plots on this graph are virtually indistinguishable, their slope should be a good approximation to the zero frequency value.  The slope of the 1 kc line shown in figure 8 is about  35 volt"1, compared with the calculated value for-^r of 39. 6 volf'at room temperature.  It might be noted, however, that the value found is in  good agreement with the value of 36 volt"1 obtained from consideration of the forward dark characteristic (page 15). The scales of the quantities plotted in figure 8 make any accurate estimate of I „ f r o m the axis crossing quite impossible.  The  Facing Page 22  FIGURE  9  22  f i g u r e does, n e v e r t h e l e s s , suggest our p r e v i o u s e s t i m a t e of 0. 3yuA i s not at a l l unreasonable. The v a r i a t i o n w i t h f r e q u e n c y of the diode's conductance m a k e s p o s s i b l e a check on the a p p l i c a b i l i t y of Shockley's c a r r i e r d i f f u s i o n t h e o r y and, s h o u l d t h i s t h e o r y be v a l i d , p e r m i t s an e s t i m a t e of the m i n o r i t y c a r r i e r l i f e t i m e X .  Shockley's t h e o r y p r e d i c t s that  Y(u>) = G(0)(1 + juszf. . Q,(UJ) = Gj(0)Re(l + jcuTjk  Therefore  1 Wl  UJ^T  + 2  (6)  at any f i x e d b i a s o r l e v e l of i l l u m i n a t i o n (which d e t e r m i n e Gj(0) ). Now  G«,(a>) i s a f u n c t i o n both of f r e q u e n c y and of i l l u m i n a t i o n  (as m e a s u r e d by I ) . We have shown that f o r low f r e q u e n c i e s , however, Go(0) . Io * c * * independent of i l l u m i n a t i o n ( t h e o r e t i c a l l y equal sc  + Uscl  s  a  0 1 1 8  2 1 1 1  to -j^L ), and f i g u r e 8 i n d i c a t e s that at higher f r e q u e n c i e s t h i s p r o p o r t i o n a l i t y i s m a i n t a i n e d , although the p r o p o r t i o n a l i t y constant i s a f u n c t i o n of frequency. Shockley's t h e o r y m a y  be u s e d to p r e d i c t the v a r i a t i o n of  t h i s p r o p o r t i o n a l i t y constant (which i s , at a g i v e n f r e q u e n c y , the slope of the a p p r o p r i a t e line i n f i g u r e 8) by d i v i d i n g both s i d e s of equation 6 by ( I + |lscj). 0  Therefore:  CUto)  r  0(0)  1 +Ml  Io + | M Io+ UflclL  + uf-T* 'A  2  The v a l u e s of the s l o p e s of the l i n e s i n f i g u r e 8 a r e plotted against f r e q u e n c y i n f i g u r e 9, as i s a l i n e d e r i v e d f r o m Shockley's t h e o r y a s s u m i n g a~C of iOjUsec. U s i n g t h i s value f o r the l i f e t i m e , the c o r r e c t i o n t e r m i n equation 6 i s about 1. 005 at 1 k c , i n d i c a t i n g that the 1 kc m e a s u r e m e n t s  23  a r e a good a p p r o x i m a t i o n to the z e r o f r e q u e n c y c h a r a c t e r i s t i c s .  The  value of toU i t s e l f v a r i e s f r o m about 0. 2 at 1 kc to about 20 at 100 kc. The good a g r e e m e n t between e x p e r i m e n t and Shockley's t h e o r y o v e r t h i s wide range of toC i s s t r o n g evidence i n f a v o u r of i t s a p p l i c a b i l i t y . Thus the a l t e r n a t i n g c u r r e n t m e a s u r e m e n t s l e n d c o n f i r m a t i o n to the v a l i d i t y of Shockley's d i f f u s i o n t h e o r y a p p l i e d to t h i s case, and a l s o indicate that the low f r e q u e n c y b e h a v i o r of the diode (uJC<C<l) can be p r e d i c t e d by d i f f e r e n t i a t i o n of equation 2 under the a p p r o p r i a t e conditions.  The m e a s u r e d slope of the low f r e q u e n c y  conductance v e r s u s s h o r t - c i r c u i t c u r r e n t (figure 8) does not agree w i t h the t h e o r e t i c a l l y p r e d i c t e d value of-^.  T h i s r e s u l t c o n f i r m s the  i n d i c a t i o n s i n c e r t a i n of the dc m e a s u r e m e n t s (page 16)that the diode i s i n some r e s p e c t n o n - i d e a l .  24 2. 3  Noise Measurements  The m e a s u r e m e n t of noise w i t h i n some bandwidth i n v o l v e s * e s s e n t i a l l y , s e l e c t i o n of that p a s s band by a suitable f i l t e r and d e t e c t i o n of the average s i g n a l t h e r e i n b y a suitable detector.  In p r a c t i c e , s m a l l  noise s i g n a l s m u s t be a m p l i f i e d (before o r after f r e q u e n c y s e l e c t i o n ) and hence some method m u s t be a v a i l a b l e by w h i c h the noise of i n t e r e s t m a y be r e c o g n i z e d against the b a c k g r o u n d of a m p l i f i e r noise. In g e n e r a l , t h i s m e a n s that the a m p l i f i e r noise w i l l l i m i t the r e s o l u t i o n of the s o u r c e noise m e a s u r e m e n t s by c o m p l e t e l y o v e r s h a d o w i n g v e r y s m a l l changes. The average s i g n a l voltage to be expected a f t e r d e t e c t i o n is a f u n c t i o n of the bandwidth i n w h i c h i t i s m e a s u r e d ; being s t r i c t l y p r o p o r t i o n a l if the detector f o l l o w s a square law and if the n o i s e i s white, i . e. constant i n m e a n square value w i t h r e s p e c t to frequency, as i s t h e r m a l noise.  The v a r i a n c e of the detected output w i l l v a r y i n some  i n v e r s e f a s h i o n w i t h the bandwidth so that the w i d e r the bandwidth the l a r g e r and m o r e n e a r l y constant w i l l be the detected s i g n a l . F i l t e r i n g m a y be e m p l o y e d to reduce v a r i a t i o n s i n the output s i g n a l , but it m u s t be u s e d w i t h c a r e i f there i s any s p u r i o u s i n t e r f e r e n c e present i n o r p i c k e d up by the m e a s u r i n g set, s i n c e long time-constant f i l t e r i n g w i l l s m o o t h these out and add t h e i r c o n t r i b u t i o n s to the r e c o r d e d l e v e l w i t h no i n d i c a t i o n of t h e i r t r a n s i e n t nature. S e v e r a l f a c t o r s g o v e r n the choice of m e a s u r e m e n t i  frequency.  T o m e a s u r e what m i g h t be c a l l e d j u n c t i o n n o i s e a s opposed  to e x c e s s surface-dependent n o i s e , low f r e q u e n c i e s should be avoided. O n the other hand, m e a s u r e m e n t at high f r e q u e n c i e s i s made d i f f i c u l t  25  by the effect of shunt capacitance i n the a m p l i f i e r input c i r c u i t . E q u i p m e n t a v a i l a b l e f o r f i l t e r i n g and d e t e c t i o n i n c l u d e d a G e n e r a l R a d i o type 736-A wave a n a l y z e r c o v e r i n g 20 c s t o 16 k c w i t h a bandwidth of about 4 c s , and a S i e r r a type 121 a n a l y z e r c o v e r i n g 15 to 500 k c w i t h a bandwidth of about 200 cs.  A d d i t i o n a l f i l t e r i n g was  added to the detector stages of both a n a l y z e r s i n o r d e r t o reduce the fluctuations of output voltage. B e c a u s e of i t s m u c h n a r r o w e r s a m p l i n g band, the lower frequency a n a l y z e r r e q u i r e d a m u c h longer t i m e - c o n s t a n t f i l t e r f o l l o w i n g the detector to reduce these f l u c t u a t i o n s to a u s e f u l degree.  However,  i n t e r m i t t e n t and p r o b a b l y r a n d o m i n t e r f e r e n c e f r o m f l u o r e s c e n t l a m p s t a r t e r s and e l e c t r i c m o t o r s i n the v i c i n i t y was noted at a m e a n rate c o m p a r a b l e w i t h the r e c i p r o c a l of t h i s n e c e s s a r y t i m e constant.  These  i n t e r f e r i n g b u r s t s w e r e , t h e r e f o r e , a v e r a g e d along w i t h the noise r e c o r d e d by t h i s a n a l y z e r .  The s h o r t e r t i m e constant needed f o r  reasonable r e s u l t s w i t h the higher frequency a n a l y z e r left the i n t e r f e r e n c e a l m o s t intact (that i s , it appeared s t i l l as b u r s t s at the d e t e c t o r - f i l t e r output) so i t c o u l d be i g n o r e d when i n t e r p r e t i n g the r e s u l t s . r e a s o n the S i e r r a equipment was u s e d e x c l u s i v e l y .  F o r this  The a d v e r s e effect  of input shunt capacitance at higher f r e q u e n c i e s i n d i c a t e d that the lower p a r t of this instrument's range would give the m o s t a c c u r a t e measurements.  T h e s e m e a s u r e m e n t s w e r e , t h e r e f o r e , made at 20 and  30 kc. The wave a n a l y z e r i t s e l f has i n s u f f i c i e n t gain to enable »  diode noise to be m e a s u r e d d i r e c t l y , so two p r e a m p l i f i e r s a r e c a s c a d e d ahead of i t . The f i r s t of these i s a s i n g l e tube a m p l i f i e r w i t h a voltage gain of about 2. 6 w h i c h i s designed w i t h as low an input capacitance as  Facing Page 26  . FIGURE  10  USE O F RESISTANCE BOX T H E R M A L NOISE TO D E T E R M I N E NOISE BANDWIDTH  T  I  i  2  1  3  i  4-  i  S  CKIL)  i  (o  :  r  i  7  26  practicable.  Its gain v a r i e s s l o w l y w i t h t i m e over a range of a b o u t - 1 0 % .  The second unit i s a T e c h n o l o g y decade a m p l i f i e r w i t h a s t a b i l i z e d gain of 1000.  The power inputs to both units are regulated. B e c a u s e of the v a r i a t i o n i n detector output, whose m o d i f i e d  f i l t e r s y s t e m has a t i m e constant of about 0. 8 seconds, the output i s p e r m a n e n t l y r e c o r d e d as a f u n c t i o n of t i m e on a s e l f - b a l a n c i n g r e c o r d i n g potentiometer.  E x a m i n a t i o n of the r e c o r d f r o m this i n s t r u m e n t o v e r  a s a m p l i n g t i m e of a m i n u t e enables a good e s t i m a t e to be m a d e of the a v e r a g e r e a d i n g and a l s o r e v e a l s any m a r k e d gain d r i f t i n any of the equipment. A s a test of the noise m e a s u r i n g set and of m e a s u r e m e n t techniques, the noise output of a r e s i s t a n c e box was m e a s u r e d .  The  box  c o n s i s t s of w i r e - w o u n d r e s i s t o r s w h i c h should be quite f r e e of any but t h e r m a l noise.  In any case, the box was m e a s u r e d under o p e n - c i r c u i t  conditions and so could not generate any cur rent-de pendent noise. The r e s u l t s of one s u c h t r i a l (20 kc) a r e shown i n f i g u r e 10. Now  "e* = 4 k T R A f  and t h e r e f o r e  Af  "5* 4kTR  e B u t i s the slope of the line d r a w n i n f i g u r e 10 c o r r e c t e d f o r the gain of the a m p l i f i e r (2580 i n t h i s case), so that z  ~& "R and thus  =  2. 32 x 10"" , (2580)* volt/ohm,  Af = 210  1 4 V  sec . -1  T h i s value i s i n good a g r e e m e n t w i t h the value of ± 100 cs (to -3db s p e c i f i e d by the m a n u f a c t u r e r .  points)  The magnitude of the point s c a t t e r  r e f l e c t e d onto the R a x i s suggests that the p r e c i s i o n of m e a s u r e m e n t c o r r e s p o n d s to about 200 ohms.  27  When the dc power to the i l l u m i n a t i n g l a m p i s i n t e r r u p t e d the light output f a l l s off w i t h a m e a s u r e d decay t i m e of about 0. 3 seconds. T h i s i n d i c a t e s that the t h e r m a l i n e r t i a of the l a m p w i l l p r o h i b i t i l l u m i n a t i o n fluctuations o c c u r r i n g at f r e q u e n c i e s higher than s e v e r a l c y c l e s per second (oJC f o r the l a m p i s g r e a t e r than 1 f o r f>0. 5 c s ) so that no photovoltaic noise should be i n t r o d u c e d d i r e c t l y b y fluctuations of the light i n t e n s i t y i n the frequency range of i n t e r e s t . It i s to be expected that the v a r i o u s a m p l i f i e r gains (except, perhaps, that of the decade a m p l i f i e r ) and the bandwidth of the wave a n a l y z e r w i l l d r i f t over a p e r i o d of time.  This makes a  s u b s t i t u t i o n technique f o r noise m e a s u r e m e n t p a r t i c u l a r l y d e s i r a b l e , s i n c e w i t h such a m e t h o d the m e a s u r e m e n t i s s t r i c t l y c o m p a r a t i v e and i s independent of the absolute v a l u e s of gain and bandwidth i n v o l v e d . Its s u c c e s s depends on the s u b s t i t u t i o n being made i n a t i m e s h o r t c o m p a r e d to any d r i f t t i m e , but t h i s c r i t e r i o n can be met. The choice of a noise s t a n d a r d f o r s u b s t i t u t i o n i s e a s i l y made i n t h i s case.  A s a t u r a t e d noise diode i s prone t o develop f l i c k e r  n o i s e , it r e q u i r e s c a r e f u l l y f i l t e r e d power s u p p l i e s , and it depends o n an e x t e r n a l m e t e r f o r a c c u r a c y .  A r e s i s t a n c e box, on the other hand,  r e q u i r e s only s h i e l d i n g and, a s s u m i n g w i r e - w o u n d r e s i s t o r s and low-noise s w i t c h e s , r e p r e s e n t s a v e r y stable and a c c u r a t e t h e r m a l noise s o u r c e since both the v a l u e s of r e s i s t a n c e and absolute t e m p e r a t u r e m a y r e a d i l y be known to a few p a r t s i n 10 . e  A t high f r e q u e n c i e s the shunt capacitance  of the decade box w i l l v a r y a c c o r d i n g to the decade s w i t c h e s i n use, but this d i f f i c u l t y w a s not encountered to any m a r k e d degree throughout t h i s investigation.  Facing Page 28  FIGURE  11  COMPOSITE E X P E R I M E N T A L A R R A N G E M E N T FOR NOISE AND A D M I T T A N C E M E A S U R E M E N T UNDER  ILLUMINATION  Switch I  Bztrord <zr  28  Once again the effective i l l u m i n a t i o n at the j u n c t i o n i s m e a s u r e d i n t e r m s of the s h o r t - c i r c u i t c u r r e n t . To a v o i d the effect of e r r o r between the i l l u m i n a t i o n l e v e l s set f o r these noise m e a s u r e m e n t s and those u s e d f o r the ac admittance m e a s u r e m e n t s , the admittance i s m e a s u r e d a g a i n d i r e c t l y f o l l o w i n g the noise m e a s u r e m e n t and thus under n e a r l y e x a c t l y s i m i l a r conditions. The composite e x p e r i m e n t a l a r r a n g e m e n t i s shown i n f i g u r e 11 and the e x p e r i m e n t a l p r o c e d u r e i s as f o l l o w s : 1.  The wave a n a l y z e r i s set to the a p p r o p r i a t e f r e q u e n c y  and,  w i t h s w i t c h 1 i n the I s c p o s i t i o n , the light i n t e n s i t y i s set a p p r o p r i a t e l y  i  by o b s e r v a t i o n of the magnitude of I s c . 2.  The s i g n a l generator i s set to the wave a n a l y z e r f r e q u e n c y (there i s s u f f i c i e n t leakage between the c i r c u i t s that a d i r e c t connection i s u n n e c e s s a r y f o r this adjustment), s w i t c h 1 p l a c e d i n the Yoc p o s i t i o n , and the b r i d g e balanced w i t h the Rm  and Cm  elements  by o b s e r v i n g the e l l i p t i c a l p a t t e r n on the o s c i l l o s c o p e . 3.  The s i g n a l generator i s t u r n e d off (to prevent i n t e r f e r e n c e ) , s w i t c h 1 t u r n e d to the N p o s i t i o n and, w i t h s w i t c h 2 as shown i n f i g u r e 11, a r e c o r d i n g i s m a d e of the wave a n a l y z e r output.  4.  S w i t c h 2 i s t h r o w n to place the s t a n d a r d r e s i s t a n c e box  R  N  in the noise set input c i r c u i t and the value of R N i s adjusted to b r i n g the r e c o r d e r to the same average balance p o s i t i o n as d i d the diode's noise.  R  N  , R  Now  > and Cm are r e c o r d e d .  m  the effective noise t e m p e r a t u r e  ratio  ~eiF 1  =  4kTR Af  w h e r e R i s the r e a l part of the j u n c t i o n impedance under the conditions of  ing Page 29  FIGURE  12  NOISE T E M P E R A T U R E VERSUS  2  RATIO  ILLUMINATION  T  "3I o p c of Least-Sc] u a nzs Line = O.OOS yuA" 1  0  5  o  o  20  IO  Isc  CyuA)  x  20  kc  °  30  kc  30  Illumination  4o  m e a s u r e m e n t , and"e?"is the m e a s u r e d m e a n s q u a r e d noise voltage. But  eft = 4 k T R A f ,  and hence  t = j[~= j ^ ( l + urC Rm) .  N  m  The r e s u l t s of the v a r i o u s m e a s u r e m e n t s made a r e shown i n f i g u r e 12. It i s d i f f i c u l t to a r r i v e at an a c c u r a t e e s t i m a t e of the possible e r r o r i n the values f o r t because of the m a n y f a c t o r s involved. However, R  m  can be m e a s u r e d w i t h the bridge d e s c r i b e d to w i t h i n 3 %  and Cm to w i t h i n 5 % o r 50 pF, w h i c h e v e r i s greater. be e s t i m a t e d to w i t h i n 10%.  R  N  can generally  C o u p l e d w i t h these e r r o r s a r e s e v e r a l  m o r e subtle ones; namely, i n a c c u r a c y i n the value u s e d f o r the frequency ( w h i c h appears i n the f i n a l e x p r e s s i o n f o r t above), d r i f t i n i l l u m i n a t i o n , gain, and bandwidth between the v a r i o u s phases of one m e a s u r e m e n t , and o c c a s i o n a l p i c k - u p of a p e r i o d i c i n t e r f e r e n c e by the noise set (such i n t e r f e r e n c e was s o m e t i m e s o b s e r v a b l e on an o s c i l l o s c o p e m o n i t o r i n g the input to the wave a n a l y z e r ) .  A r e a s o n a b l e estimate, of the t o t a l  p o s s i b l e e r r o r in t taking these f a c t o r s into account would be about ± 1 5 % . T h i s f i g u r e i s i n reasonable a g r e e m e n t w i t h the point s c a t t e r evident i n f i g u r e 12.  30 CHAPTER 3. 1  3  -  DISCUSSION  Inferences f r o m DC  and A C  OF  RESULTS  Measurements  B o t h the dc and the ac m e a s u r e m e n t s indicate a s l i g h t departure i n the diode c h a r a c t e r i s t i c s f r o m those p r e d i c t e d by Shockley. S p e c i f i c a l l y , h i s m o d e l p r e d i c t s a c u r r e n t dependence on voltage of the f o r m ( e x p g ^ - 1), w h i c h leads a l s o to the low f r e q u e n c y o p e n - c i r c u i t conductance dependence (page 21) cao)  =-j^(I.+ | I * | ) .  Our m e a s u r e m e n t s c o n f i r m the g e n e r a l f o r m s of both e x p r e s s i o n s (that i s , i n the f o r w a r d d i r e c t i o 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 found to be exponential and the o p e n - c i r c u i t low f r e q u e n c y conductance is p r o p o r t i o n a l to the i l l u m i n a t i o n and thus to the s h o r t - c i r c u i t c u r r e n t ) ^ but they indicate that Shockley's c o n s t a n t - ^ r i  n  e i t h e r c a s e should be  r e p l a c e d e m p i r i c a l l y by a somewhat s m a l l e r value; about 0. 9 5 - ^ f o r the q p h o t o c h a r a c t e r i s t i c plot, 0. 91 f o r the c o r r e c t e d f o r w a r d d a r k q c h a r a c t e r i s t i c plot, and 0. 89"pp" *  o r  *  o w  f r e q u e n c y conductance plot.  T h i s d i s c r e p a n c y i s evidence that Shockley's a s s u m p t i o n s a r e not e n t i r e l y v a l i d i n t h i s case.  In p a r t i c u l a r , h i s r e s u l t f o l l o w s f r o m  the r e s u l t that the hole c o n c e n t r a t i o n at the n side of the j u n c t i o n t r a n s i t i o n i  r e g i o n p(x ) i s given by Ttn  p(x „) = pnexp-g^  (7)  T  where p  n  i s the e q u i l i b r i u m c o n c e n t r a t i o n of holes i n the n r e g i o n (and  thus a l s o at t h i s plane) and V i s the voltage a c r o s s the junction.  Used  as a boundary c o n d i t i o n f o r Shockley's s o l u t i o n of the c o n t i n u i t y equation i n the n r e g i o n , t h i s e x p r e s s i o n leads to a d i f f u s i o n hole c u r r e n t p r o p o r t i o n a l to (expgrjr - 1).  A s i m i l a r t r e a t m e n t of e l e c t r o n c u r r e n t  31  has a s i m i l a r r e s u l t , so that the t o t a l c u r r e n t a c r o s s the j u n c t i o n should qV be p r o p o r t i o n a l to ( e x p ^ j r - 1). M i s a w a (1955), however, points out that the hole c o n c e n t r a t i o n i n question i s a c t u a l l y g i v e n by P( fn) [P(XT„) + n x  where n  n  n  - p„]  = nfexpgX.  i s the e q u i l i b r i u m c o n c e n t r a t i o n of e l e c t r o n s i n the n r e g i o n .  If the i n j e c t e d c a r r i e r d e n s i t y i s s m a l l (as i n Shockley's t r e a t m e n t ) , pbc ) - p T(  n  i s v e r y s m a l l c o m p a r e d w i t h n , and the equation r e d u c e s to ni jgV sV P( ^) H; PkT P" PkT ' n  4  x  =  e x  Thus Shockley's r e s u l t m a y  =  e x  be seen to be a s p e c i a l case of M i s a w a ' s  equation. F o r the l i m i t i n g case of h i g h l e v e l c a r r i e r i n j e c t i o n , though, p(x ) w i l l be v e r y m u c h g r e a t e r than n Tri  n  - p  n  and the equation b e c o m e s  [p(x-rj] = n * e x p ^ 2  or  p ( x ) = nj e x p | ^ r  (8)  Tn  Thus when the i n j e c t e d c a r r i e r d e n s i t y i s high, t h i s equation leads to qV  a c u r r e n t p r o p o r t i o n a l to exp r o u g h l y to exp 0.  . Since we found a p r o p o r t i o n a l i t y  can a s s u m e our c o n d i t i o n s f a l l between the  two e x t r e m e s , but a r e m u c h m o r e n e a r l y those upon w h i c h Shockley's deductions a r e based.  In other w o r d s , the j u n c t i o n voltage range we  have c o v e r e d i s that f o r w h i c h the c h a r a c t e r i s t i c s a r e changing f r o m those of the l o w - l e v e l to those of the h i g h - l e v e l case.  The  change-over  i s e v i d e n t l y s u f f i c i e n t l y g r a d u a l that our s t r a i g h t line (figure 4) i s a good a p p r o x i m a t i o n to what m u s t a c t u a l l y be a c u r v e . The c o n d i t i o n f o r the onset of h i g h - l e v e l i n j e c t i o n i n the f o r w a r d d i r e c t i o n of a p-n j u n c t i o n i s obtained by c o n s i d e r i n g the state  32  when the hole density at the n-side of the t r a n s i t i o n r e g i o n i s just equal to the e q u i l i b r i u m e l e c t r o n density, that i s ; p(x ) = n . Tn  n  In t h i s case p ( x ) i s given by e i t h e r the low l e v e l (equation 7) o r the Tn  high l e v e l (equation 8) f o r m u l a e above; p ( x ) = n = n; e x p - ^ To  n  qV p exp^r  =  since  PnH  n  n  = ni .  Solving f o r the voltage Vo at w h i c h t h i s change-over takes place we see: q  ni  q  p  n  In 5 o h m - c e n t i m e t e r m a t e r i a l at r o o m t e m p e r a t u r e the f o l l o w i n g a r e r e p r e s e n t a t i v e a p p r o x i m a t e values f o r the v a r i o u s c o n c e n t r a t i o n s : n = 4. 2 x 10  14  n  m = 2. 5 x 10  13  cm".  3  cm;  p = 1. 5 x 10 cm ,z  n  7  5  3  T h e s e values p r e d i c t a change-over voltage of about 0. 14 volt; a value w i t h i n , but t o w a r d the upper end of, the voltage range i n v e s t i g a t e d experimentally. Thus the dc and ac m e a s u r e m e n t s suggest that the diode investigated i s n e a r l y i d e a l , although over the range of f o r w a r d voltage kT investigated (up to 0. 22 volt or about 9"^- a c r o s s the junction) low injected c a r r i e r d e n s i t y c h a r a c t e r i s t i c s ( i . e. Shockley c h a r a c t e r i s t i c s ) a r e being departed f r o m .  T h e y a l s o indicate that the low frequency ac  behavior i s adequately p r e d i c t e d by d i f f e r e n t i a t i o n of the dc c u r r e n t voltage c h a r a c t e r i s t i c .  Although no s i m p l e ac equivalent c i r c u i t f o r the  device i s suggested, j u s t i f i c a t i o n i s found f o r the a p p l i c a t i o n of Shockley's c a r r i e r d i f f u s i o n t h e o r y to a p r e d i c t i o n of the v a r i a t i o n of conductance w i t h f r e q u e n c y o v e r a wide (100;1) range of frequency.  34 3.'2  I m p l i c a t i o n s of N o i s e M e a s u r e m e n t s  The e x p e r i m e n t a l r e s u l t s f o r the noise t e m p e r a t u r e  ratio  t as a function of i l l u m i n a t i o n i n t e r m s of the s h o r t - c i r c u i t c u r r e n t I 5 0 are plotted i n f i g u r e 12 (for both 20 and 30 kc). T h e y d i s p l a y a s c a t t e r ' of the same magnitude as the e s t i m a t e d e r r o r .  The slope of the l e a s t -  squares l i n e through the points i s only 0. 003 ju. A" , a n e g l i g i b l e 1  quantity i n the light of the s c a t t e r .  A noteworthy point i s that the v a l u e s  of t f o r z e r o i l l u m i n a t i o n a r e d i s t r i b u t e d i n m u c h the same way as a r e those under i l l u m i n a t i o n .  If the diode i s i n t h e r m a l e q u i l i b r i u m ,  however, t m u s t be unity, so the e x p e r i m e n t a l points should s c a t t e r about t = l ; i . e. , some should l i e above t h i s value and some below. O n the c o n t r a r y , though, the e x p e r i m e n t a l points r e p r e s e n t i n g r e s u l t s at z e r o i l l u m i n a t i o n a l l l i e above t = l , w i t h an average value of 1. 125. The m e a s u r e m e n t s i n d i c a t e d f o r l»c=0 w e r e made w i t h the diode under n e g l i g i b l e i l l u m i n a t i o n .  The galvanometer u s e d to m e a s u r e  Ischas a m a x i m u m s e n s i t i v i t y of 0. OS^A/mm at a r e s i s t a n c e of 360 ohms. With the diode i n its 'dark' c o n d i t i o n there was no m e a s u r a b l e c u r r e n t reading on t h i s m e t e r .  Since the diode's d a r k o p e n - c i r c u i t r e s i s t a n c e  is m u c h g r e a t e r than 360 ohms, we m a y conclude that the 'dark' s h o r t c i r c u i t c u r r e n t i s l e s s than 0. Ol^uA, w h i c h i s about l / 3 0 of I„.  This  indicates that the diode was i n t h e r m a l e q u i l i b r i u m and hence should have had a noise t e m p e r a t u r e  r a t i o of unity.  The s m a l l d i s c r e p a n c y noted i n the z e r o i l l u m i n a t i o n points of f i g u r e 12, t h e r e f o r e , m u s t be a s c r i b e d to some s y s t e m a t i c e r r o r i n the m e a s u r e m e n t procedure a r i s i n g p r o b a b l y f r o m c a l i b r a t i o n d i s c r e p a n c i e s between the r e f e r e n c e r e s i s t a n c e box and the impedance  bridge at the frequencies used.  If this systematic e r r o r is now  subtracted from the values of t under illumination it is seen that even at the m a x i m u m illumination l e v e l t would not exceed  1.12.  The important feature of the noise m e a s u r e m e n t s is that the o p e n - c i r c u i t noise temperature ratio is v e r y n e a r l y unity and v a r i e s negligibly over a s h o r t - c i r c u i t c u r r e n t range f r o m z e r o to over 130 t i m e s I 0 .  T h i s result is in v e r y good agreement with that  predicted in section 1. 2, and suggests that the simple assumptions made t h e r e i n are quite adequate to p r e d i c t the noise behavior of openc i r c u i t e d ideal photo-diodes when subjected to intense illumination which produces a v e r y large departure from the e q u i l i b r i u m values of the c a r r i e r densities and flows and of the potential distribution. It might be thought that we have c o n s i d e r e d only the noise a r i s i n g in the diode itself and have neglected that inherent in the incident illumination.  T h i s is not the case, however, since i n our t h e o r e t i c a l  d i s c u s s i o n of shot noise in an illuminated diode we wrote (page 8); = 2 q ( I e x p g V r + I0+ 0  BL);  The f i r s t two c u r r e n t t e r m s determine the diode's noise in t h e r m a l e q u i l i b r i u m , while the last ( B L ) accounts for the extra c u r r e n t flow due to illumination.  Thus the p h o t o - p a i r - p r o d u c t i o n takes place at a m e a n  BL rate —^— and the resulting current adds its shot noise contribution to that existing in e q u i l i b r i u m .  In other w o r d s , a p h o t o - h o l e - e l e c t r o n pair is  created only when a photon is incident; thus the photocurrent includes the randomness of the incident illumination.  Facing Page 36  FIGURE  13  PHOTO-DIODE EQUIVALENT CIRCUITS FOR SIGNAL AND NOISE IN OPEN-CIRCUIT O P E R A T I O N  °  i  D . C Cj<znczra+or ( v o l t a g e  o r current^)  Nlois<z Q e n e r c i l o r ( v o l t a a < 3  or current)  36 CHAPTER 4. 1  4  -  CONCLUSIONS  General Comments  The  r e s u l t s of this i n v e s t i g a t i o n s t r o n g l y suggest that the  noise of an o p e n - c i r c u i t e d i d e a l photo-diode m a y be r e g a r d e d as t h e r m a l noise a r i s i n g f r o m random c a r r i e r m o t i o n a c r o s s the junction.  The  a d d i t i o n a l c a r r i e r s produced by the i l l u m i n a t i o n change both the j u n c t i o n impedance and the m e a n - s q u a r e d noise c u r r e n t i n s u c h a f a s h i o n that, p r o v i d i n g the m e a n c u r r e n t i s z e r o ( i . e. o p e n - c i r c u i t conditions), the i  noise r e m a i n s equal to that a s c r i b e d by Nyquist's t h e o r e m to t h e r m a l fluctuations i n the j u n c t i o n impedance.  T h i s i s c o n s i s t e n t w i t h the r a p i d  t h e r m a l i z a t i o n of the a d d i t i o n a l c a r r i e r s by the v e r y n u m e r o u s c o l l i s i o n s w i t h the c r y s t a l l a t t i c e d u r i n g t h e i r l i f e t i m e . In view of these r e s u l t s , the s i m p l e r e p r e s e n t a t i o n s shown i n f i g u r e 13 d e s c r i b e the photovoltaic and noise generation of an i d e a l , o p e n - c i r c u i t e d diode. The fact that no e x c e s s noise (above t h e r m a l ) was noted suggests that the i d e a l photo-diode should be an e x c e l l e n t r a d i a t i o n detector.  The s i g n a l voltage obtained f r o m one of these d e v i c e s i s 'free';  that i s , no c u r r e n t need be s u p p l i e d the device f r o m an e x t e r n a l s o u r c e , as i s n e c e s s a r y to develop a s i g n a l a c r o s s a photoconductive detector. E x c e s s c u r r e n t n o i s e , t h e r e f o r e , does not appear at the photo-diode terminals.  It should be e m p h a s i z e d , however, that t h i s f r e e d o m f r o m  e x c e s s noise a p p l i e s only under o p e n - c i r c u i t conditions and, as i s i n d i c a t e d by equation 4 (page 8), a d d i t i o n a l c u r r e n t noise w i l l be o b s e r v e d if the diode i s u s e d to d e l i v e r c u r r e n t to a load. A n o t h e r point w o r t h e m p h a s i z i n g i s the c o n d i t i o n a p p l i e d  37  throughout that the diode be ideal.  When a diode i s being chosen f o r  use as an o p e n - c i r c u i t photovoltaic detector, it m i g h t s e e m  reasonable  to use f o r r e s p o n s i v i t y c o n s i d e r a t i o n s i t s o p e n - c i r c u i t voltage v e r s u s i l l u m i n a t i o n c h a r a c t e r i s t i c s and to neglect the f o r m of i t s g e n e r a l current-voltage characteristic.  T h i s i n v e s t i g a t i o n , however, i n d i c a t e s  that i n a d d i t i o n the diode m u s t be i d e a l ( i . e. be c h a r a c t e r i z e d by equation 2, page 1) throughout the o p e n - c i r c u i t voltage range i n question. It may  then be expected to d i s p l a y a higher s i g n a l - t o - n o i s e r a t i o  under a given i l l u m i n a t i o n than a n o n - i d e a l diode of higher absolute responsivity.  The l a t t e r may  have i n t e r n a l shunt paths c a u s i n g a net  flow of c u r r e n t t h r o u g h the j u n c t i o n due to c i r c u l a t i n g flow even though no e x t e r n a l c u r r e n t flows.  The t r e a t m e n t given e a r l i e r (equation 4)  then i n d i c a t e s that the effective noise t e m p e r a t u r e r a t i o w i l l e x c e e d unity, i m p l y i n g a p o o r e r s e n s i t i v i t y .  38 4. 2  Design Recommendations for Radiation Detectors  Using the results of earlier analyses, we can draw several conclusions of value in designing a photo-diode for open-circuit detection application.  '  '  The open-circuit signal voltage appearing across the diode under illumination is v  =  = J  f  l n  <  1 + i  i7)  and the r . m . s. noise voltage appearing with it in a bandwidth A f is: VN = ( 4 k T A f R ( U ) ) i  In the majority of detection applications the frequency range of interest will extend no higher than several kilocycles per second so we need only consider noise in this low frequency band (u5U« 1). open-circuit conditions .  kT  dV  qV«.  R(w) = " a l = - X e x p ( q  =  kT.,  qT ( 0  and hence  [AF,,  In this case and under  VM = 2 k T ^ ^ ( l +  1 +  1ET  -)  B L -.  X")  B L->42 v  Thus we may write the open-circuit signal-to-noise (voltage) ratio as BL Io+ BL" * ln(l +— ) . 4q Af At the limit of sensitivity, B L « I„, so the signal-to-noise ratio becomes 3k  V N " 2(qI 0 Af)^  *  To consider the implications of this equation on detector design we will assume that the diode is illuminated evenly over the entire junction area A so that B can be written in the form bA.  V*  39  A s s u m i n g a p - n s t r u c t u r e , we have (page 7); +  Dp (DP I = Aqpri7 = Aqpntp^ £  e  t  -Up  Vs _ _b_L  so that  VM~  2q pnAf • J  i L p  5  '(10)  E v i d e n t l y then, to s e c u r e the m o s t favourable s i g n a l - t o - n o i s e r a t i o the f o l l o w i n g conditions should be met; 1.  A should be l a r g e (for u n f o c u s s e d r a d i a t i o n )  2.  p should be s m a l l n  3. X p should be l a r g e 4. D p should be s m a l l . The  a r e a A of the j u n c t i o n i s l i m i t e d by the o p t i c a l s y s t e m  u s e d to provide the s i g n a l i l l u m i n a t i o n , the space a v a i l a b l e f o r the detector, and the technique u s e d i n p r o d u c i n g the junction. f a c t o r s l i e beyond the scope of t h i s investigation.  These  It should be noted,  though, that i f a c e r t a i n t o t a l amount of incident r a d i a t i o n i s a v a i l a b l e that i s , i f B L i s f i x e d - the m a x i m u m s i g n a l - t o - n o i s e r a t i o w i l l be attained by f o c u s s i n g t h i s r a d i a t i o n on as s m a l l an a r e a as p o s s i b l e so as to m a k e b L A ^ l a r g e f o r a given value of b L A . The m i n o r i t y c a r r i e r density p i s r e l a t e d to the e n e r g y n  gap of the s e m i c o n d u c t o r Eg by the r e l a t i o n pn=^ex (-^) P  where K i s a constant f o r the p u r p o s e s of t h i s d i s c u s s i o n .  Hence pn  can be r e d u c e d by choosing a m a t e r i a l w i t h a l a r g e e n e r g y gap.  It m u s t  be r e m e m b e r e d , however, that the diode i s photovoltaic only f o r photons of energy g r e a t e r than the energy gap, so that Eg m u s t be l e s s than o r equal to h v f o r the r a d i a t i o n to be detected.  Hence f o r a m o n o c h r o m a t i c  40  r a d i a t i o n detector i t i s advantageous to choose Eg = hv. R e d u c t i o n of p region.  n  i s a l s o effected by heavy doping of the n  T h i s w i l l g e n e r a l l y cause only a n e g l i g i b l e d e c r e a s e i n T , the P  m i n o r i t y c a r r i e r l i f e t i m e and i n D  P  , the d i f f u s i o n constant f o r holes  i n this r e g i o n . A d e c r e a s e i n t e m p e r a t u r e w i l l produce fewer t h e r m a l p a i r s i n the n r e g i o n and thus a c o o l e d detector w i l l d i s p l a y a m o r e favourable s i g n a l - t o - n o i s e r a t i o . Since the d i f f u s i o n constant and the m o b i l i t y of holes i n the n r e g i o n a r e r e l a t e d by the E i n s t e i n equation  Dp m a y  be reduced by choosing a m a t e r i a l i n w h i c h /^pis s m a l l . A t y p i c a l m a x i m u m s e n s i t i v i t y may  of suitable v a l u e s i n equation 9.  be c a l c u l a t e d by use  F o r e x a m p l e , the f o l l o w i n g values  lead to a value for the r e v e r s e s a t u r a t i o n c u r r e n t I  0  A = 0. 005  cm  e q u a l to 0. 3yuA:  2  p n = 3 x 10" c m ~ ( c o r r e s p o n d i n g 3  to 1 ohm-cm n-type  Ge at 300 K. ) Dp = 45 cm sec-»(Ge at 300 2  X  K)  = 30 x 10~ sec (value found e x p e r i m e n t a l l y ) fc  P  The m i n i m u m detectable power, defined as the i n c i d e n t photon e n e r g y in a unit bandwidth w h i c h w i l l give unity s i g n a l - t o - n o i s e r a t i o , i s tr h \& P =fs(BL) „= 2E,(i) £  when the incident r a d i a t i o n i s m o n o c h r o m a t i c and of f r e q u e n c y  For  g e r m a n i u m , E^ = 0. 7 e V c o r r e s p o n d i n g to a r a d i a t i o n wavelength of 1. 8ju. Thus a g e r m a n i u m diode w i t h the above-mentioned r e v e r s e s a t u r a t i o n c u r r e n t of 0. 3 juA has a m i n i m u m detectable power P equal to 3 x 10'^watt.  A n excellent photoconductive c e l l noted i n the l i t e r a t u r e (Smith, Jones, and C h a s m a r , 1957) has a m i n i m u m detectable power, when cooled, -14-  of 6. 4 x 10 watt.  It a p p e a r s , then, that the i d e a l o p e n - c i r c u i t e d p-n  photo-diode can compete f a v o u r a b l y w i t h photoconductive d e v i c e s i n s e n s i t i v i t y and has the added advantage of r e q u i r i n g no e x t e r n a l power supply.  APPENDIX Shift in DC Bias Due to Rectification of AC Signal from Measuring Bridge  Consider the circuit shown below;  V(t) E "5in u)t  (^)  4=c«  Now we know  I = I0(exp|^T " *) " E L  (2)  and we will assume that the voltage across the diode V consists of two parts; i. e.  V = V + V, sinu>t .  Hence  I = Ic exp^(Vo + ' V . 6 i h i o t ) - 1  0  - BL .  If we assume the ac part of the voltage across the diode is small so kT that V, «  , this expression may be expanded to give I = I 0 ( e x p £ £ - 1) - B L kT + loexp^[^sinu>t  +(^J(1 - cos2<ot)  and hence the average current<I> is given by <I> = L ( e x p - g - 1) - B L +  (Lexp^^J  .  Because of the condenser in the circuit, however, the average current must be zero.  Therefore: exp kT  1  +W  = 1+  BL Io  The dq portion V, of the voltage across the diode is made up of the photovoltage due to illumination and the bias shift ( A V ) due to rectification of the ac signal from the bridge.  That is;  43  kT  BL,  V = — i n ( l +— ) + A V . 0  e x p  Hence  kT  =  ^ *  )  P ^kT^  e x  '  kT and i f the bias shift due to r e c t i f i c a t i o n i s s m a l l ( A . V « ), then q exp  k  \i +  T  I o  A  1  +  k  T  )•  B y substitution of t h i s r e s u l t into equation 11 above it m a y r e a d i l y be shown that; AV  ~ " 4kT *  The conductance G a c t u a l l y m e a s u r e d by the b r i d g e i s m  , given by  G  „ m  =  d l t qlo l L ; & e  K  x  Io  kT  p  qVo k f  '  P  kT  = a(0)ex ^ P  -G (0)(l + - ^ ) o  Hence i f  V , = 25 mv,  Gm = 0. 75G„(0)  V, = 10 mv,  G =  0. 96G40)  V, = 5 mv,  G =  0. 99G40).  m  m  kT kT In t h i s l a s t case, VI« and A V = 0. 25 m v w h i c h i s a l s o « — so the q q a s s u m p t i o n s m a d e i n the d e r i v a t i o n a r e v a l i d .  i  44 BIBLIOGRAPHY  Cummerow, R. L . , "Photovoltaic Effect in p-n Junctions", Physical Review, 95, p. 16 (July 1, 1954). Gianola, U. F . > "Photovoltaic Noise in Silicon Broad Area p-n Junctions", Journal of Applied Physics, 27, p. 51 (January, 1956). Misawa, T . , " E m i t t e r Efficiency of Junction T r a n s i s t o r " , Journal of the Physical Society of Japan, 10, p. 362 (May, 1955). Mitchell, G. R. , Goldberg, A. E . , and Kurnich, S. W. , "InSb Photovoltaic C e l l " , Physical Review, 97, p. 239 (January, 1955). Pearson, G. L . , Montgomery, H. C. , and Feldmann, W. L . , "Noise in Silicon p-n Junction Photocells", Journal of Applied Physics, 27, p. 91 (January, 1956). Shockley, W. , " T h e Theory of p-n Junctions in Semiconductors and p-n Junction T r a n s i s t o r s " , Bell System Technical Journal, 28, p. 435 (July, 1949). Smith, R. A . , Jones, F. E . , and Chasmar, R. P. , " T h e Detection and Measurement of Infra-red Radiation", Oxford University Press, 1957. Tauc, J. , "Generation of an emf in Semiconductors with Nonequilibrium Current C a r r i e r Concentrations", Reviews of Modern Physics, 29, p. 308 (July, 1957). Torrey, H. C. , and Whitmer, C. A . , " C r y s t a l Rectifiers", McGraw-Hill Book Company, Inc. , 1948.  

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