RADIATION DETECTION WITH
POINT-CONTACT JOSEPHSON JUNCTIONS
o o
U J U I
RADIATION DETECTION WITH
POINT-CONTACT JOSEPHSON JUNCTIONS
T E R V E R K R I J G I N G V A N D E G R A A D V A N D O C T O R I N D E T E C H N I S C H E W E T E N S C H A P P E N A A N D E T E C H N I S C H E H O G E S C H O O L D E L F T , O P G E Z A G V A N D E R E C T O R M A G N I F I C U S P R O F . I R . B . P . T H . V E L T M A N , V O O R E E N C O M M I S S I E A A N G E W E Z E N D O O R H E T C O L L E G E V A N D E K A N E N , T E V E R D E D I G E N O P D I N S D A G 2 2 S E P T E M B E R 1 9 8 1 T E I A . 0 0 U U R P R O E F S C H R I F T D O O R
TIEMEN POORTER
N A T U U R K U N D I G I N G E N I E U R G E B O R E N T E D E L F T i BIBLIOTHEEK T U Delft P 1673 5123 C 496508P R O F . D R . I R . J . E . M O O I J
I . INTRODUCTION 6
I I . THEORETICAL BACKGROUND 8 11.1. The ac Josephson e f f e c t 8
11.2. The R e s i s t i v e l y Shunted J u n c t i o n Model 10 11.3. C a p a c i t i v e S h u n t i n g o f t h e RSJ Model 13 11.4. J u n c t i o n Impedance 18 11.5. F l u c t u a t i o n s 19 11.6. D e t e c t i o n o f E l e c t r o m a g n e t i c R a d i a t i o n 21 11.7. Wide-Band D e t e c t i o n o f R a d i a t i o n 22 11.8. H e t e r o d y n e D e t e c t i o n w i t h E x t e r n a l L0 25
I I I . WIDE-BAND DETECTION; EXPERIMENTAL TECHNIQUE 30
I I I . 1. M i c h e l s o n I n t e r f e r o m e t e r 30 I I I . 2 . Wide-Band D e t e c t i o n 32
SUBMILLIMETRE AND MILLIMETRE RESPONSE OF JOSEPHSON JUNCTIONS MOUNTED
IN A WAVEGUIDE 34 I n f r a r e d P h y s i c s , J9.317 ( 1 9 7 9 ) . 1. I n t r o d u c t i o n 34 2. D e t e c t o r C o n f i g u r a t i o n 34 3. J u n c t i o n C h a r a c t e r i s t i c s 35 4. S p e c t r a l Response Measurements 36 5. Comparison w i t h T h e o r y 40 6. A b s o l u t e Response Measurements 42 7. C o n c l u s i o n 43
THE PLASMA RESONANCE IN THE RESPONSE AND IN THE RF IMPEDANCE OF A
CAPA-CITIVELY SHUNTED JOSEPHSON JUNCTION IN THE PRESENCE OF THERMAL NOISE. 45 J o u r n a l o f A p p l i e d P h y s i c s , 5_1_, 6305 ( 1 9 8 0 ) .
I . I n t r o d u c t i o n 45 I I . The A n a l o g 45 I I I . PLL A n a l o g w i t h o u t a C a p a c i t i v e Shunt 47
IV. C a p a c i t i v e S h u n t i n g i n t h e P r e s e n c e o f Thermal N o i s e 49
V. The Plasma Resonance 51
IV. HETERODYNE DETECTION 59 I V . 1 . The H e t e r o d y n e R e c e i v e r 59
I V . 2 . H e t e r o d y n e D e t e c t i o n 61
JOSEPHSON HETERODYNE DETECTION AT HIGH BACKGROUND LEVELS 65 S u b m i t t e d t o J o u r n a l o f A p p l i e d P h y s i c s I . I n t r o d u c t i o n 65 I I . The M i x e r 67 I I I . The R e c e i v e r 68 IV. J u n c t i o n C h a r a c t e r i s t i c s 70 V. D e t e r m i n a t i o n o f t h e M i x e r and R e c e i v e r P e r f o r m a n c e 74 V I . Non-Heterodyne Response o f t h e M i x e r 75 V I I . H e t e r o d y n e Response 77 V I I I . C o n c l u s i o n 81 V. CONCLUSION 84 S a m e n v a t t i n g C u r r i c u l u m v i t a e
CHAPTER I
INTRODUCTION
T h i s t h e s i s d e a l s w i t h t h e d e t e c t i o n o f e l e c t r o m a g n e t i c r a d i a t i o n w i t h a J o s e p h s o n j u n c t i o n . A Josephson j u n c t i o n i s a weak l i n k between two s u p e r c o n -d u c t i n g e l e c t r o -d e s . I t e x h i b i t s a number o f i n t r i g u i n g p r o p e r t i e s w h i c h make i t v e r y s u i t a b l e as a d e t e c t o r f o r h i g h f r e q u e n c y e l e c t r o m a g n e t i c f i e l d s . The f r e q u e n c y r a n g e f o r w h i c h i t i s most s u i t e d i s 10-1000 GHz ( t h i s c o r r e s p o n d s t o a w a v e l e n g t h o f 0.3-30 mm).
D e t e c t i o n can o c c u r i n two d i f f e r e n t manners: 1. a s a wide-band i n c o h e r e n t d e t e c t o r w h i c h c o n v e r t s s i g n a l power t o a v o l t a g e a c r o s s t h e j u n c t i o n . 2. as a c o h e r e n t h e t e r o d y n e d e t e c t o r ( m i x e r ) w h i c h f r e q u e n c y c o n v e r t s s i g n a l power a t a h i g h f r e q u e n c y t o a much l o w e r f r e q u e n c y , w h i l e r e t a i n i n g t h e s p e c t r a l i n f o r m a t i o n p r e s e n t i n t h e s i g n a l . B o t h t y p e s o f d e t e c t o r a p p l i c a t i o n s have been i n v e s t i g a t e d e x p e r i m e n t a l l y i n t h i s t h e s i s u s i n g s m a l l p o i n t c o n t a c t J o s e p h s o n j u n c t i o n s w h i c h a r e mounted i n a waveguide t o o p t i m i z e t h e c o u p l i n g o f r a d i a t i o n t o t h e j u n c t i o n . The r e s u l t i s t h a t i n b o t h a p p l i c a t i o n s a v e r y s e n s i t i v e d e t e c t o r i s o b t a i n e d t h a t i s optimum i n t h e f r e q u e n c y r a n g e o f 170-220 GHz w h i c h a p p r o x i m a t e l y 2 c o r r e s p o n d s t o t h e s i n g l e mode f r e q u e n c y r a n g e f o r t h e 0.5*1.0 mm w a v e g u i d e . However i t i s found e x p e r i m e n t a l l y t h a t d i f f e r e n t t y p e s o f p o i n t c o n t a c t s w i t h d i f f e r e n t c h a r a c t e r i s t i c s a r e optimum f o r each a p p l i c a t i o n . F o r w i d e -band d e t e c t i o n p u r p o s e s i t i s no drawback when t h e r e e x i s t s a c a p a c i t a n c e o f
-15
t h e o r d e r o f ( 1 1 0 ) * 1 0 f a r a d i n t h e j u n c t i o n . The h i g h n o r m a l i z e d 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 c u r r e n t v o l t a g e c h a r a c t e r i s t i c i n t h a t c a s e i n -c r e a s e s t h e a m p l i t u d e o f t h e r e s p o n s e , and i n a d d i t i o n t h e r e s p o n s e i s enhanced by a plasma t y p e o f r e s o n a n c e . The e x i s t e n c e o f t h i s plasma r e s o -nance b o t h i n t h e r f impedance o f t h e j u n c t i o n and i n t h e wide-band r e s p o n s e , f o r a n o n - z e r o a v e r a g e v o l t a g e a c r o s s t h e j u n c t i o n , w i l l be d e m o n s t r a t e d i n t h i s t h e s i s . F o r h e t e r o d y n e d e t e c t i o n however i t i s v e r y i m p o r t a n t t o use J o s e p h s o n j u n c t i o n s t h a t p o s s e s s a s h u n t c a p a c i t a n c e t h a t i s as s m a l l as p o s s i b l e . The h e t e r o d y n e r e s p o n s e d e c r e a s e s f o r an i n c r e a s i n g c a p a c i t i v e s h u n t and i n t h i s c a s e t h e r e i s no c o m p e n s a t i n g mechanism p r e s e n t . I t i s t h e r e f o r e i m p o r t a n t t o use p o i n t c o n t a c t s w i t h a v e r y s m a l l c o n t a c t i n g a r e a . T h i s can be r e a l i z e d by u s i n g p o i n t c o n t a c t s formed w i t h v e r y t h i n ( t y p i c a l l y 3-5 pm) s h a r p w h i s k e r s
w i t h a l o w c o n t a c t p r e s s u r e o b t a i n e d t h r o u g h a r e l a t i v e l y l a r g e s p r i n g w h i c h i s compressed o n l y about 10 pm. The j u n c t i o n s t h a t can be f a b r i c a t e d i n t h i s manner have c o n t a c t d i m e n s i o n s s m a l l e r than one m i c r o n and e x h i b i t good h e t e r o d y n e d e t e c t i o n p r o p e r t i e s . They a r e n o t d i s t u r b e d by a s t r o n g non-h e t e r o d y n e r e s p o n s e w non-h i c non-h can be t non-h e c a s e f o r p o i n t c o n t a c t s w i t non-h more r i g i d w h i s k e r s .
The e s s e n c e o f t h i s t h e s i s i s c o n t a i n e d i n t h r e e p a p e r s , two o f w h i c h have been p u b l i s h e d a l r e a d y . The f i n a l paper has been s u b m i t t e d f o r p u b l i c a t i o n i n J o u r n a l o f A p p l i e d P h y s i c s .
Chapter two i s a t h e o r e t i c a l background o f t h e phenomena t h a t o c c u r i n Josephson j u n c t i o n s . I t g i v e s t h e i n f o r m a t i o n n e c e s s a r y t o u n d e r s t a n d t h e t o -p i c s d i s c u s s e d i n t h e -p a -p e r s .
C h a p t e r t h r e e i s an i n t r o d u c t i o n s p e c i f i c t o t h e f i r s t paper w h i c h t r e a t s t h e wide-band d e t e c t i o n p r o p e r t i e s o f p o i n t - c o n t a c t J o s e p h s o n j u n c t i o n s mount-ed i n a w a v e g u i d e . The i n t r o d u c t i o n g i v e s some a d d i t i o n a l i n f o r m a t i o n about t h e method o f s p e c t r a l a n a l y s i s o f t h e r e s p o n s e and about t h e d e t e c t i o n c i r -c u i t i n v o l v e d .
The second paper c o n t a i n s an e x t e n s i v e s t u d y o f t h e phenomena t h a t o c c u r when a Josephson j u n c t i o n i s shunted by a c a p a c i t a n c e . I t i s n o t p r o v i d e d w i t h a s e p a r a t e i n t r o d u c t i o n . The main r e s u l t i s t h a t i t i s d e m o n s t r a t e d w i t h a p h a s e - l o c k e d - l o o p a n a l o g o f t h e R e s i s t i v e l y Shunted J u n c t i o n model o f a Josephson j u n c t i o n t h a t , i n c a s e o f a s u i t a b l e c a p a c i t i v e s h u n t , an enhance-ment o f t h e wide-band r e s p o n s e o c c u r s a t an a t t e m p t f r e q u e n c y . T h i s a t t e m p t f r e q u e n c y i s r e l a t e d t o t h e plasma r e s o n a n c e f r e q u e n c y t h a t o c c u r s when t h e j u n c t i o n i s b i a s e d i n t h e s u p e r c u r r e n t . C h a p t e r f o u r i s an i n t r o d u c t i o n t o t h e f i n a l paper w h i c h c o n c e r n s t h e ap-p l i c a t i o n o f our s m a l l ap-p o i n t c o n t a c t s as a m i x i n g e l e m e n t f o r a h e t e r o d y n e d e t e c t i o n r e c e i v e r . T h i s c h a p t e r g i v e s some a d d i t i o n a l i n f o r m a t i o n about t h e c o n s t r u c t i o n o f t h e r e c e i v e r and a l s o an i n t r o d u c t i o n t o two d i f f e r e n t methods o f d e t e r m i n i n g t h e p e r f o r m a n c e o f a m i x i n g e l e m e n t . In t h e paper i t i s shown t h a t t h e l o w - p r e s s u r e j u n c t i o n s have a low m i x e r n o i s e t e m p e r a t u r e and a com-p a r a t i v e l y l a r g e c o n v e r s i o n e f f i c i e n c y i n t h e f r e q u e n c y r a n g e o f 170-220 GHz, even w i t h a r e l a t i v e l y h i g h t h e r m a l background r a d i a t i o n l e v e l .
C h a p t e r f i v e summarizes t h e r e s u l t s on h e t e r o d y n e d e t e c t i o n and g i v e s recommendations f o r f u r t h e r improvements.
CHAPTER I I
THEORETICAL BACKGROUND
In t h i s c h a p t e r a t h e o r e t i c a l background w i l l be l a i d f o r t h e u n d e r s t a n d -i n g o f t h e phenomena t h a t have been s t u d -i e d -i n t h -i s t h e s -i s . They -i n v o l v e t h e i n t e r a c t i o n o f e l e c t r o m a g n e t i c waves w i t h weak l i n k s between s u p e r c o n d u c t o r s .
i Such weak l i n k s a r e c a l l e d J o s e p h s o n j u n c t i o n s a f t e r B.D.Josephson . He d e s c r i b e d t h e i r p r o p e r t i e s on t h e b a s i s o f t h e quantum t h e o r y o f e l e c t r o n t u n -n e l i -n g t h r o u g h a t h i -n o x i d e b a r r i e r betwee-n two s u p e r c o -n d u c t i -n g e l e c t r o d e s . The t u n n e l i n g c u r r e n t t h r o u g h such a j u n c t i o n can be d i v i d e d i n t o s i n g l e e l e c -t r o n o r q u a s i - p a r -t i c l e -t u n n e l i n g and J o s e p h s o n -t u n n e l i n g o f e l e c -t r o n p a i r s . J o s e p h s o n showed t h a t c o r r e l a t e d " s u p e r c o n d u c t i n g " p a i r s o f e l e c t r o n s w i l l t u n n e l a c r o s s t h e b a r r i e r w i t h o u t a v o l t a g e d r o p . T h i s n o n - d i s s i p a t i v e c u r r e n t i s v e r y s e n s i t i v e t o e l e c t r o m a g n e t i c f i e l d s and can t h e r e f o r e be used as a d e t e c t o r o f e l e c t r o m a g n e t i c r a d i a t i o n . The i n t e r a c t i o n o f t h e J o s e p h s o n j u n c t i o n w i t h p u r e l y m a g n e t i c f i e l d s w i l l be o m i t t e d h e r e because i t i s o u t s i d e t h e scope o f t h i s t h e s i s . Reviews on h i g h f r e q u e n c y p r o p e r t i e s o f J o s e p h
-2 3 son j u n c t i o n s have been g i v e n by R i c h a r d s and by Adde and V e r n e t . More
gen-e r a l i n f o r m a t i o n on s u p gen-e r c o n d u c t i v i t y and t h gen-e J o s gen-e p h s o n gen-e f f gen-e c t s c a n bgen-e found 4 5
i n books by Tinkham and by Solymar .
I I . 1 . The ac J o s e p h s o n E f f e c t
When a s u p e r c o n d u c t o r i s c o o l e d below i t s t r a n s i t i o n t e m p e r a t u r e T , a second o r d e r phase t r a n s i t i o n t a k e s p l a c e and a new t y p e o f o r d e r i n g o c c u r s . The m e t a l behaves as though i t c o n t a i n s two f l u i d s , a n o r m a l f l u i d o f e l e c t r o n - l i k e e x c i t a t i o n s ( t h e q u a s i - p a r t i c l e s ) and a s u p e r - f l u i d o r quantumf l u i d w h i c h i s i n some r e s p e c t a Bose c o n d e n s a t e o quantumf e l e c t r o n p a i r s . The c o n -d e n s a t e c a r r i e s no e n t r o p y ; e l e c t r o n s c a n e n t e r i t , i n p a i r s , a t a l w a y s t h e same e n e r g y . The s u p e r c o n d u c t i n g s t a t e i s d e s c r i b e d by a complex o r d e r
parame-i y *
t e r VQeJ , where W = n t h e p a i r d e n s i t y and X i s t h e phase f a c t o r w h i c h i s a f u n c t i o n o f b o t h t i m e and p o s i t i o n . F o r d e c r e a s i n g t e m p e r a t u r e T, n ^ i n -c r e a s e s s t e a d i l y from z e r o a t Tq t o a s a t u r a t i o n v a l u e f o r T< TQ/2. When two s u p e r c o n d u c t o r s a r e w e a k l y l i n k e d , s u c h t h a t each c a n s t i l l be r e g a r d e d as i n s t a t i c e q u i l i b r i u m , f u l l c o h e r e n c e between t h e phase f a c t o r s on e a c h s i d e o f t h e b a r r i e r i s p r e v e n t e d . A n o n - d i s s i p a t i v e p a i r c u r r e n t c a n t h e n f l o w a c r o s s
t h e b a r r i e r . I t depends on t h e phase d i f f e r e n c e 0 between t h e two phase f a c -t o r s and i s g i v e n by^ I = Ics i n 0 , ( 2 . 1 ) where Iq j_s tn e m a xim u m z e r o v o l t a g e c u r r e n t t h a t c a n be c a r r i e d by t h e j u n c -t i o n . As was f i r s -t shown b y A n d e r s o n ^ a c o u p l i n g e n e r g y E = - h 7 2 e( Icc o s 0 + Ib0 ) ( 2 . 2 ) i s a s s o c i a t e d w i t h t h e phase d i f f e r e n c e 0. Here If e i s t h e b i a s c u r r e n t t h r o u g h
t h e j u n c t i o n and 2TTH i s P l a n c k s c o n s t a n t . When I ^ > I t h e phase d i f f e r e n c e i s no l o n g e r c o n s t a n t (no p h a s e - l o c k ) , b u t a f u n c t i o n o f t h e f i n i t e p o t e n t i a l d r o p V t h a t now o c c u r s a c r o s s t h e b a r r i e r . The quantum t h e o r y o f tunneling''''^ t h e n p r e d i c t s a t i m e dependence o f t h e phase d i f f e r e n c e hd0/dt = 2eV . ( 2 . 3 ) I f t h e v o l t a g e i s c o n s t a n t i n t i m e V ( t ) = Vb, i n t e g r a t i o n o f Eq.(2.3) g i v e s 0= 2 e Vbt / R + 0Q . E q u a t i o n (2.1) then p r e d i c t s an ac c u r r e n t t h r o u g h t h e b a r r i e r w i t h f r e q u e n c y v= 2 e Vb/ h , c o r r e s p o n d i n g t o 484 Ghz/mV. T h i s c a n be t h o u g h t o f as t h e e m i s s i o n o f p h o t o n s w i t h e n e r g y hv w h i c h e q u a l s t h e change i n t h e pot e n pot i a l e n e r g y 2eV pot h a pot o c c u r s when one C o o p e r p a i r pot u n n e l s a c r o s s pot h e b a r -r i e -r . T h i s ac c u -r -r e n t h a s no a v e -r a g e component (T=0; t h e b a -r d e n o t e s t i m e I Ic -F i g . I I - 1 . I-V c u r v e of a J o s e p h s o n t u n n e l j u n c t i o n f o r T c l o s e to z e r o . 2A/e a v e r a g i n g ) so t h a t t h e t i m e a v e r a g e d c u r r e n t - v o l t a g e c u r v e ( I - V c u r v e ) o f t h e i d e a l J o s e p h s o n element a t a t e m p e r a t u r e c l o s e t o z e r o i s a c u r r e n t s p i k e w i t h a m p l i t u d e I a t Vbr O and 1-0 f o r V^tO. However when V approaches 2A/e, where
2A i s t h e f u l l s u p e r c o n d u c t i n g gap v a l u e , s i n g l e e l e c t r o n s s t a r t t o t u n n e l t h r o u g h t h e b a r r i e r , r e p r e s e n t i n g a h i g h l y n o n - l i n e a r r e s i s t a n c e p a r a l l e l t o t h e J o s e p h s o n e l e m e n t . F o r V> 2A/e t h e v o l t a g e dependence o f t h i s q u a s i
-p a r t i c l e c u r r e n t c o r r e s -p o n d s t o a l i n e a r r e s i s t a n c e . The r e s u l t i n g I-V c u r v e i s shown i n F i g . I I - 1 . The c o n n e c t i o n between t h e two p a r t s o f t h e I-V c u r v e depends on t h e s o u r c e impedance o f t h e b i a s u n i t .
The e x i s t e n c e o f t h e ac Josephson c u r r e n t can be d e m o n s t r a t e d by a p p l y i n g a v o l t a g e V ( t ) = Vb+ Vgc o s u )st t o a j u n c t i o n . Then Eq.(2.3) y i e l d s
0 = ubt + a s i n "st + 0Q . (2.4)
where 0^= 2eVb/R and a= 2 e Vs/ R «s ( t h e r e d u c e d s i g n a l a m p l i t u d e ) . W i t h Eq.(2.1)
we o b t a i n
I ( t ) = Ics i n ( u >bt + a s i n u jgt + 0Q) , (2.5)
w h i c h can be expanded i n t o F o u r i e r components r e s u l t i n g i n 00 I ( t ) = lQJZ Jn( a ) s i n d ^ t + nu)gt+ 0Q) , (2.6) n=-<x> where Jn a r e B e s s e l s f u n c t i o n s o f t h e f i r s t k i n d o f o r d e r n. F o r a l l v a l u e s o f f o r w h i c h nwg= 0, I ( t ) c o n t a i n s dc components I = IcJn( a ) s i n 0o , (2.7)
w h i c h a r e dc b e a t s between t h e ac J o s e p h s o n c u r r e n t w i t h 0^= 2eVb/R and t h e
i n t e r n a l l y g e n e r a t e d n^ harmonic o f O Js. F o r t h e i d e a l J o s e p h s o n e l e m e n t t h e s e
dc b e a t s a r e v i s i b l e on t h e I-V c u r v e as c u r r e n t s p i k e s a t v o l t a g e s V= nhtog/2e; n= 1,2,.. . For n=a=0, Eq.(2.7) r e d u c e s t o E q . ( 2 . 1 ) . F o r n= 0,
0O= V 2 ( f o r w h i c h t h e maximum s u p e r c u r r e n t o c c u r s ) and a/0, Eq.(2.7) g i v e s
t h e r e d u c t i o n o f t h e dc z e r o v o l t a g e c u r r e n t and f o r n/0, a/0 and 0Q= IT/2 E q . ( 2 . 7 ) g i v e s t h e c u r r e n t r a n g e o f t h e d i f f e r e n t dc b e a t s . T h i s e f f e c t was
Q
f i r s t measured by S h a p i r o i n 1963. The microwave r a d i a t i o n t h a t i s e m i t t e d by a J o s e p h s o n j u n c t i o n a t a f r e q u e n c y v ^ r 2 e Vb/ h was f i r s t measured i n 1965 by
Yanson e t a l . ^ .
I I . 2 . The R e s i s t i v e l y Shunted J u n c t i o n Model
The v o l t a g e b i a s e d t u n n e l j u n c t i o n o u t l i n e d i n t h e p r e v i o u s p a r a g r a p h does n o t i n c l u d e many o f t h e i m p o r t a n t f e a t u r e s o f J o s e p h s o n p o i n t c o n t a c t s w h i c h a r e most f r e q u e n t l y used f o r d e t e c t i o n p u r p o s e s . A l t h o u g h p o i n t c o n t a c t s ex-h i b i t gap s t u c t u r e a t V= 2A/e wex-hen t ex-h e y a r e s u f f i c i e n t l y ex-h i g ex-h - o ex-h m i c , t ex-h e y a l s o show a s t r o n g n o n - t h e r m a l q u a s i - p a r t i c l e c u r r e n t f o r V< 2A/e w h i c h i s p r o b a b l y 1 o c a r r i e d by m i c r o b r i d g e s between t h e two e l e c t r o d e s P o i n t c o n t a c t s a r e p r o b a b l y b e s t d e s c r i b e d by t h e R e s i s t i v e l y Shunted J u n c t i o n (RSJ) model t h a t c o n s i s t s o f an i d e a l J o s e p h s o n element w i t h a s u p e r -c u r r e n t Ics i n 0 i n p a r a l l e l w i t h a v o l t a g e i n d e p e n d e n t r e s i s t a n c e R. T h i s model
was f i r s t proposed by Aslamazov e t a l .10 i n 1969 f o r a b r i d g e t y p e Josephson j u n c t i o n . F o r a t e m p e r a t u r e c l o s e t o Tc t h e t o t a l c u r r e n t t h r o u g h s u c h a b r i d g e i s g i v e n by K t ) = V ( t ) / R + Ics i n [ ( 2 e / R ) [ V ( t ) d t ] . ( 2 . 8 ) A c t u a l l y a t h e o r e t i c a l c a l c u l a t i o n1 o f t h e t o t a l c u r r e n t t h r o u g h a j u n c -t i o n based on -t h e -t u n n e l i n g H a m i l -t o n i a n y i e l d s a s l i g h -t l y d i f f e r e n -t r e s u l -t . I -t i s found t h a t Eq(2.8) s h o u l d c o n t a i n s t i l l a n o t h e r term [ V ( t ) / R ] a c o s 0 w h i c h d e s c r i b e s t h e i n t e r f e r e n c e between t h e q u a s i - p a r t i c l e c u r r e n t and t h e phase dependent p a i r c u r r e n t ( t h e q u a s i - p a r t i c l e - p a i r i n t e r f e r e n c e o r " c o s 0 " t e r m ) . However t h i s c o s 0 term i s e x p e c t e d n o t t o i n f l u e n c e t h e b e h a v i o r o f a J o s e p h
-11
son j u n c t i o n s i g n i f i c a n t l y , c e r t a i n l y n o t when t h e shunt c a p a c i t a n c e i n t h e 12
j u n c t i o n i s s m a l l ( s e e p a r . I I 3 ) . T h e r e f o r e t h i s q u a s i p a r t i c l e p a i r i n -t e r f e r e n c e c u r r e n -t i s n e g l e c -t e d .
A t i m e dependent s o l u t i o n o f E q . ( 2 . 8 ) , when a f i x e d c u r r e n t i>1 i s im-posed i s10'13
v ( t ) = ( i2- 1 ) / [ i - c o s ( a i0t ( i2- 1 )1 / 2) ] , ( 2 . 9 )
where v= V / IQR and i = I / I a r e t h e r e d u c e d v o l t a g e and c u r r e n t r e s p e c t i v e l y
and where uQ= 2 e IcR / n . When Eq.(2.9) i s t i m e averaged we o b t a i n t h e h y p e r b o l i c
I-V c u r v e f o r t h e RSJ m o d e l , ( s e e F i g . I I . 2 ) g i v e n b y
v = 0 f o r |T|< 1 ; v = ( ! / ] i | ) ( I2- 1 )1 / 2 f o r |I|> 1 . (2.10)
When t h e j u n c t i o n i s c u r r e n t b i a s e d as i s u s u a l i n p r a c t i c e ( i . e . a h i g h e x t e r n a l s o u r c e impedance much l a r g e r t h a n t h e j u n c t i o n impedance Z ) , an ac v o l t a g e i s g e n e r a t e d a c r o s s t h e p a r a l l e l r e s i s t a n c e R by t h e ac Josephson c u r r e n t s . From E q s . ( 2 . 1 ) and ( 2 . 3 ) we s e e t h a t t h e r e s i s t i v e f e e d b a c k t h a t t h e n e x i s t s p r o d u c e s a s t r o n g harmonic, d i s t o r t i o n o f t h e J o s e p h s o n o s c i l l a t i o n and i s r e s p o n s i b l e f o r i t s s p e c t r u m r i c h i n h a r m o n i c components and a l s o f o r t h e shape o f t h e IV c u r v e . T h i s i s i l l u s t r a t e d i n F i g . 1 1 2 . The r e d u c e d v o l -t a g e a c r o s s -t h e j u n c -t i o n c o n s i s -t s o f a dc v o l -t a g e componen-t v= i and a s e r i e s o f n a r r o w p e r i o d i c v o l t a g e p u l s e s t h a t r e p r e s e n t s u c h an a v e r a g e v o l t a g e t h a t t h e IV c u r v e i s g i v e n by E q . ( 2 . 1 0 ) . F o r i n c r e a s i n g T t h e ac v o l t a g e i s g e t -t i n g more and more s i n u s o i d a l and -t h e a v e r a g e v o l -t a g e i s a s y m p -t o -t i c a l l y ap-p r o a c h i n g v= T.
The e f f e c t o f ac c u r r e n t s i n j e c t e d i n t o t h e j u n c t i o n i s now d i f f e r e n t from t h e c a s e o f an i d e a l v o l t a g e d r i v e n t u n n e l j u n c t i o n . I n s t e a d o f c u r r e n t s p i k e s , t h e I-V c u r v e o f an i r r a d i a t e d j u n c t i o n shows c o n s t a n t v o l t a g e s t e p s a t v o l t a g e s v= nu s/wg , o r v= nfi where Q i s t h e reduced s i g n a l f r e q u e n c y d e
F i g . I I 2 . Time e v o l u t i o n of the v o l -tage i n a c u r r e n t b i a s e d j u n c t i o n o b e y i n g the RSJ m o d e l , showing t h e s t r o n g harmonic d i s t o r t i o n r e l a t e d to the r e s i s t i v e f e e d b a c k . To the l e f t the dc I - V c u r v e i s shown ( t a k e n from R e f . 1 4 ) . f i n e d by 0= to /u - Rio / 2 e l R. U n f o r t u n a t e l y t h i s s i t u a t i o n cannot be d e a l t 0 w i t h a n a l y t i c a l l y . However a n u m e r i c a l d e r i v a t i o n o f t h e e f f e c t s 15,16 showed t h a t f o r v » 1 and ft»1 t h e s i z e o f t h e s t e p s i s r a t h e r w e l l d e s c r i b e d by t h e B e s s e l f u n c t i o n i n E q . ( 2 . 7 ) . F o r JK1 t h e s t e p s i z e s show l a r g e d e v i a t i o n s from t h i s e q u a t i o n . An example o f t h e i n f l u e n c e o f r a d i a t i o n on t h e I-V c u r v e i s g i v e n i n F i g . I I - 3 .
I
»- voltage
F i g . I I - 3 . I - V c u r v e s f o r a p o i n t - c o n t a c t w i t h o u t r f b i a s ( l e f t ) and f o r i n c r e a s i n g v a l u e s of r f power ( r i g h t ) .
The b e h a v i o r o f t h e RSJ model can be s t u d i e d w i t h t h e h e l p o f l o w f r e q u e n -17 18
cy e l e c t r o n i c a n a l o g s ' w i t h w h i c h t h e n u m e r i c a l c a l c u l a t i o n s can be e a s i l y r e p r o d u c e d and w i t h w h i c h many o t h e r a s p e c t s o f t h e RSJ model c a n be s t u -d i e -d1 9'2 0.
The c u r r e n t r e d i s t r i b u t i o n p r o c e s s i n t h e j u n c t i o n , between t h e n o r m a l and the s u p e r c o n d u c t i n g components i s r e l a t e d w i t h a t i m e - c o n s t a n t g i v e n b y ( o ^ )- 1 = [ 2 e IcR / R ]- 1. (2.11) 14 I t can be shown t h a t t h e j u n c t i o n s e n s i t i v i t y t o e x t e r n a l r a d i a t i o n w i s P s o f t e n p r o p o r t i o n a l t o (a^/u)s) . T h e r e f o r e good h i g h f r e q u e n c y p e r f o r m a n c e o f J o s e p h s o n j u n c t i o n s r e q u i r e s l a r g e v a l u e s o f t h e p r o d u c t I R. C a l c u l a t i o n s
from t h e m i c r o s c o p i c t h e o r y o f t u n n e l j u n c t i o n s show t h a t f o r T « T t h e max-c imum v a l u e o f t h i s p r o d u c t i s g i v e n by ICR = TrA/2e . (2.12) 21 F o r s t r o n g c o u p l i n g s u p e r c o n d u c t o r s w h i c h a r e most u s e f u l f o r d e t e c t o r ap-p l i c a t i o n s A=2kTc. T h i s means t h a t t h e r e i s a l i n e a r dependence between t h e
t r a n s i t i o n t e m p e r a t u r e Tq o f a s u p e r c o n d u c t o r and t h e i n t e r n a l t i m e - c o n s t a n t .
To have a good h i g h f r e q u e n c y d e t e c t o r i t i s t h e r e f o r e common t o use Nb as an e l e c t r o d e m a t e r i a l . T h i s i s a l s o t h e c a s e i n our e x p e r i m e n t s . F o r Nb TQ= 9.25 K which i s the h i g h e s t v a l u e f o r the s u p e r c o n d u c t i n g e l e m e n t s . The
r e s u l t i n g maximum p o s s i b l e v a l u e o f O>Q/2TT a t a t e m p e r a t u r e o f 4-5 K i s about 1000 GHz. F o r optimum c o u p l i n g o f r a d i a t i o n t o a j u n c t i o n and because o f t h e s m a l l c a p a c i t a n c e s i n t h e j u n c t i o n , p o i n t c o n t a c t s a r e most w i d e l y used as
22
h i g h f r e q u e n c y d e t e c t o r s . As has been shown by Zimmerman Eq.(2.12) a p p e a r s 23 t o be v a l i d a l s o f o r p o i n t c o n t a c t s . In a d d i t i o n McDonald e t a l . have c a l -c u l a t e d t h e I-V -c u r v e o f an i d e a l -c u r r e n t b i a s e d J o s e p h s o n t u n n e l j u n -c t i o n , showing t h a t i t c a r r i e s a n o n - z e r o a v e r a g e c u r r e n t f o r v o l t a g e s b e l o w t h e gap v o l t a g e , s i m i l a r t o p o i n t c o n t a c t s . In our e x p e r i m e n t s we use Nb p o i n t c o n -t a c -t s o f w h i c h -t h e c o n -t a c -t s u r f a c e s ( w h i c h a r e s m a l l e r -t h a n 1 pm a c r o s s ) are p r e p a r e d by e l e c t r o c h e m i c a l p o l i s h i n g and c h e m i c a l e t c h i n g . At 4.5-5 K j u n c -t i o n s can be made i n -t h i s way, w i -t h I R p r o d u c -t s u p -t o -t h e -t h e o r e -t i c a l maximum o f 2.4 mV f o r R20 ohm. F o r t h e optimum r e s i s t a n c e r a n g e o f 4080 ohm, t y p i -c a l 1 8 p r o d u -c t s range from 1/2-3/4 o f t h i s maximum v a l u e . T h i s i s l a r g e enough t h a t around 200 GHz (where we i n t e n d t o work) t h e r e d u c e d s i g n a l f r e
-19
quency 0 i s 0.25-0.4; low enough t o g i v e optimum h e t e r o d y n e d e t e c t i o n . The I-V c u r v e s o f our j u n c t i o n s show c l e a r gap s t r u c t u r e , a l s o a t 2A/2e and 2A/3e. At 4.5-5 K t h e gap v o l t a g e 2A/e= 2.8-3.0 mV f o r our j u n c t i o n s , w h i c h i s i n good agreement w i t h t h e o r y .
J J .3. C a p a c i t i v e S h u n t i n g o f t h e RSJ Model
The e f f e c t o f a c a p a c i t i v e shunt o f t h e RSJ model ( s e e F i g . I I - 4 ) i s t h a t t h e ac J o s e p h s o n c u r r e n t s a r e s h u n t e d , e s p e c i a l l y t h e i r h i g h e r h a r m o n i c s . The
24
r e s u l t o f t h i s s h u n t i n g , on t h e I-V c u r v e , was i n v e s t i g a t e d by S t e w a r t and 25
McCumber . S i n c e t h e h i g h e r h a r m o n i c s o f t h e ac c u r r e n t s a r e more s h u n t e d , t h e ac v o l t a g e becomes much more s i n u s o i d a l and t h e r e f o r e does c o n t r i b u t e l e s s t o t h e a v e r a g e v o l t a g e a c r o s s t h e j u n c t i o n . When t h e c a p a c i t i v e s h u n t i s so l a r g e t h a t i t s r e d u c e d v a l u e 3Q = 2 e IcR2C / R > 1, t h e ac v o l t a g e i s i n f l u e n c e d
F i g . I I - 4 . E q u i v a l e n t c i r c u i t o f an i d e a l c u r r e n t b i a s e d J o s e p h s o n j u n c t i o n , shunted by a v o l t a g e i n d e p e n d -ent r e s i s t a n c e and by a c a p a c i t a n c e .
s u c h t h a t t h e I-V c u r v e becomes h y s t e r e t i c . T h i s i s i l l u s t r a t e d i n F i g . I I - 5 . P a r t a shows t h e RSJ model w i t h i t s h y p e r b o l i c IV c u r v e and t h e n o n -s i n u -s o i d a l ac v o l t a g e w i t h c o n -s t a n t a m p l i t u d e . P a r t b i -s f o r a c a p a c i t i v e s h u n t such t h a t Bc= 2.5. The I-V c u r v e shows a h y s t e r e s i s and the ac v o l t a g e
i s more s i n u s o i d a l , w h i l e the a m p l i t u d e d e c r e a s e s w i t h i n c r e a s i n g f r e q u e n c y . When d e c r e a s i n g t h e b i a s c u r r e n t , t h e a v e r a g e v o l t a g e r e t u r n s t o z e r o when t h e ac v o l t a g e component f i r s t c r o s s e s V=0. I t i s c l e a r t h a t f o r i n c r e a s i n g c a p a -c i t i v e s h u n t i n g t h i s w i l l happen a t a l o w e r v a l u e o f t h e b i a s -c u r r e n t so t h a t 24 25 t h e h y s t e r e s i s i n c r e a s e s ' . These e f f e c t s can be v e r y w e l l d e m o n s t r a t e d w i t h t h e h e l p o f a s i m p l e m e c h a n i c a l a n a l o g f i r s t d e r c r i b e d by A n d e r s o n ^ . The c o u p l i n g e n e r g y a s s o c i a t e d w i t h t h e phase d i f f e r e n c e 0 a c r o s s t h e j u n c t i o n i s g i v e n by E q . ( 2 . 2 ) . T h i s c a n be n o r m a l i z e d t o E'= - i 0 - c o s 0 , (2.13) w h i c h i s a l s o t h e p o t e n t i a l e n e r g y l e v e l o f a t i l t e d w a s h i n g b o a r d t h a t i s shown i n F i g . I I 6 . The n o r m a l i z e d b i a s c u r r e n t i t h r o u g h t h e j u n c t i o n c o r r e s -ponds t o t h e t i l t o f t h e w a s h i n g - b o a r d and t h e p h a s e - d i f f e r e n c e 0 c o r r e s p o n d s t o t h e h o r i z o n t a l p o s i t i o n on t h e w a s h i n g - b o a r d . The t o t a l c u r r e n t t h r o u g h a j u n c t i o n w i t h o u t a c a p a c i t i v e s h u n t i s g i v e n by E q . ( 2 . 8 ) . T h i s e q u a t i o n can be r e d u c e d t o v ( t ) = i - s i n 0 , (2.14) f o r a c o n s t a n t b i a s c u r r e n t . W i t h a r e d u c e d t i m e T=t^t and Eq(2.3) one o b t a i n s
d0/dx = i - s i n 0 . (2.15) T h i s does n o t o n l y d e s c r i b e t h e b e h a v i o r o f 0 as a f u n c t i o n o f i b u t i t i s
a l s o t h e e q u a t i o n o f m o t i o n o f a p a r t i c l e t h a t moves under t h e i n f l u e n c e o f g r a v i t y i n a v i s c o u s medium i n t h e p o t e n t i a l e n e r g y g i v e n by E q . ( 2 . 1 3 ) . The l e f t s i d e o f Eq.(2.15) d e n o t e s t h e damping f o r c e due t o t h e v i s c o u s medium. I n t h i s s i t u a t i o n t h e p a r t i c l e h a s w e i g h t b u t i t s i n e r t i a i s n e g l e c t e d . T h i s
F i g . 1 1 - 3 . E f f e c t o f a c a p a c i t i v e shunt on the v o l t a g e p u l s e s and t h e r e b y on the I - V c u r v e , (a) shows the I - V c u r v e and the b e h a v i o r of the ac v o l t a g e f o r the RSJ m o d e l ; (b) g i v e s the same i n f o r m a t i o n f o r 3c= 2 . 5 . The e n v e l o p e of the ac v o l t a g e i s c l e a r l y shown.
means t h a t g r a v i t y c a n e x e r t a f o r c e on t h e p a r t i c l e so t h a t i t c a n move down t h e w a s h i n g b o a r d when t h a t i s t i l t e d s u f f i c i e n t l y ; |i|>1. However t h e p a r t i -c l e -c a n n o t d e v e l o p a momentum o r k i n e t i -c e n e r g y t h a t -c a n be used t o keep i t g o i n g when t h e b o a r d i s a g a i n t i l t e d s u c h t h a t l o c a l l y dE/d0> 0 ( | i | < 1 ) . The phase s l i p d0/dx i s i n t h i s c a s e p r o p o r t i o n a l w i t h dE/d0. I t i s g r e a t e s t when 0=-TT/2 and s m a l l e s t when 0= TT/2. W i t h Eq.(2.2) t h e r e s u l t i n g d0/dx c o r r e s p o n d s t o Eq.(2.9) and t h e r e b y t o t h e n o n - s i n u s o i d a l v o l t a g e p u l s e s i n F i g s . I I - 2 and I I - 5 .
When one adds a c a p a c i t i v e s h u n t t o t h e RSJ m o d e l , an e n e r g y component 2 CV /2 i s added t o t h e H a m i l t o n i a n t h a t d e s c r i b e s t h e s y s t e m . I n t h e w a s h i n g -b o a r d a n a l o g t h i s c o r r e s p o n d s t o a k i n e t i c e n e r g y t e r m caused -b y t h e i n e r t i a o f t h e now m a s s i v e p a r t i c l e . W i t h t h e c a p a c i t i v e s h u n t t h e t o t a l c u r r e n t t h r o u g h t h e j u n c t i o n i s g i v e n b y I = Ics i n 0 + V ( t ) / R + CdV/dt , (2.16)
* 0
F i g . I I - 6 . The washing board a n a l o g ; E ' i s the p o t e n t i a l energy g i v e n by E q . ( 2 . 1 3 ) . The t i l t o f the board c o r r e s p o n d s to the dc c u r r e n t i n j e c t e d i n t o the j u n c t i o n . F o r |i|<1 t h e r e are l o c a l minima i n the p o t e n t i a l energy ( a ) ; f o r |i|>1 these minima d i s a p p e a r ( b ) .
w h i c h c a n b e r e d u c e d t o
i = s i n 0 + d0/dx+ Bcd20/ d T2 . ( 2 . 1 7 )
w h i c h i s a l s o t h e e q u a t i o n o f m o t i o n o f t h e m a s s i v e p a r t i c l e i n t h e p o t e n t i a l e n e r g y o f E q . ( 2 . 1 3 ) . 3 i s a damping p a r a m e t e r t h a t i s d e t e r m i n e d by t h e r a t i o
c -1
o f t h e p a r t i c l e i n e r t i a (C) and t h e v i s c o u s damping (R ) . When gQ>1 t h e
mo-t i o n o f mo-t h e p a r mo-t i c l e i s damped so l i mo-t mo-t l e mo-t h a mo-t , once i mo-t moves down mo-t h e w a s h i n g b o a r d i t w i l l be a b l e t o overcome a c e r t a i n p o t e n t i a l e n e r g y b a r r i e r ( c o u p l i n g e n e r g y o f E q . ( 2 . 2 ) ) . I t w i l l o n l y be r e c a p t u r e d when t h e minimum i n t h e k i n e t -i c e n e r g y , a t t h e l o c a l max-imum -i n t h e p o t e n t -i a l e n e r g y ( t h e e n e r g y b a r r -i e r ) , r e a c h e s z e r o . T h i s r e s u l t s i n a h y s t e r e t i c b e h a v i o r as i n F i g . I I - 5 . The s m a l l e r damping o f t h e m o t i o n o f t h e p a r t i c l e means t h a t i t s m o t i o n w i l l b e -come more and more s i n u s o i d a l w h i l e t h e a m p l i t u d e o f t h e o s c i l l a t i o n d e c r e a s e s b e c a u s e t h e r a t i o o f t h e k i n e t i c e n e r g y o f t h e p a r t i c l e , t o t h e e n e r g y b a r r i e r becomes l a r g e r f o r i n c r e a s i n g 3Q ( s e e a l s o F i g . I I - 5 ) . I t i s e a s y t o see t h a t t h e m a s s i v e p a r t i c l e c a n r e s o n a t e i n t h e p o t e n t i a l w e l l t h a t e x i s t s when |T|<1. I f t h e p a r t i c l e i s s l i g h t l y d i s t u r b e d from i t s e q u i l i b r i u m c o r r e p o n d i n g t o i = s i n 0 , t h e p a r t i c l e w i l l undergo damped o s c i l l a -t i o n s g i v e n by 3cd2<J)/dT2+ d<ydx+ ^ s i n 0 = 0 , (2.18) where 4> i s t h e e x c u r s i o n o f t h e p a r t i c l e w i t h r e s p e c t t o t h e e q u i l i b r i u m p o s i
-t i o n . I f 6c> 1 t h e s e o s c i l l a t i o n s a r e l i g h t l y damped and we n o t i c e t h a t t h e i r
reduced f r e q u e n c y v a r i e s w i t h t h e e q u i l i b r i u m v a l u e o f 0 a c c o r d i n g t o
n_ = (cos0/B )1 / 2 . ( 2 . 1 9 )
P c
I n t h e Josephson j u n c t i o n t h i s r e s o n a n c e c o r r e s p o n d s t o a Josephson plasma i
r e s o n a n c e . In t h e RSJ t a n c e C and an i n d u c t a n c e
1
r e s o n a n c e . In t h e RSJ model the r e s o n a n c e o c c u r s between t h e shunt c a p a c i
LT( 0 ) = Pi/2el c o s 0 , (2.20)
J c
w h i c h r e p r e s e n t s t h e i n d u c t i v e b e h a v i o r o f t h e p a i r c u r r e n t ^ 1= Ics i n 0 , as can
be seen from d l / d t = I ( d 0 / d t ) c o s 0 and d0/dt= 2eV/R. T h i s plasma r e s o n a n c e was c 26
o b s e r v e d e x p e r i m e n t a l l y by Dahm e t a l . . The unreduced plasma r e s o n a n c e f r e -quency i s g i v e n by on = ( 2 e Icc o s 0 / R C )1 / 2 . (2.21) F o r t u n n e l j u n c t i o n s a t y p i c a l v a l u e f o r t h e plasma f r e q u e n c y i s o f t h e o r d e r o f 10 GHz. F o r p o i n t c o n t a c t s w i t h r e s i s t a n c e v a l u e s o f 50-100 ohm and t y p i c a l —1S c a p a c i t a n c e s o f (1-100)*10 f a r a d t h i s c o r r e s p o n d s t o plasma f r e q u e n c i e s from 100-600 GHz a t Bc v a l u e s o f 3-100. When t h e s e j u n c t i o n s a r e exposed t o a
s u f f i c i e n t l y h i g h n o i s e l e v e l , t h e y can s t i l l be used f o r r a d i a t i o n d e t e c t i o n 27
as was p r o p o s e d by T o l n e r . T h i s w i l l be d e m o n s t r a t e d w i t h t h e h e l p o f an 20
e l e c t r o n i c a n a l o g f o r t h e Josephson j u n c t i o n , i n t h i s t h e s i s . In t h a t c a s e t h e r e s p o n s e can be enhanced a t an a t t e m p t f r e q u e n c y r e l a t e d t o t h e plasma r e s o n a n c e f r e q u e n c y . Our p o i n t c o n t a c t s can be made f r e e o f h y s t e r e s i s by ap-p l y i n g a s m a l l c o n t a c t ap-p r e s s u r e . The c o n t a c t a r e a between t h e two e l e c t r o d e s can then be s m a l l e r t h a n one square m i c r o n . F o r R= 100 ohm and I = 15 uA t h e
-15
c a p a c i t a n c e i n t h e c o n t a c t must be s m a l l e r t h a n 2*10 F f o r Bc t o be s m a l l e r
t h a n one.
The o x i d e l a y e r on t h e e l e c t r o d e s i s s i m p l y produced by l e a v i n g them i n a i r a t room t e m p e r a t u r e f o r about 5 min. I t s e s t i m a t e d t h i c k n e s s i s 5
p Q
a n g s t r o m s . The d i e l e c t r i c c o n s t a n t er f o r Niobium o x i d e29 i s rough i y 30.
15
T h i s means t h a t f o r a 2*10 F c a p a c i t a n c e i n t h e j u n c t i o n , t h e a c t u a l con-t a c con-t a r e a i s aboucon-t 4 * 1 0 ~1 5 m2, o r 6*10~8 m i n d i a m e t e r . T h i s i s about 10% o f
t h e c o n t a c t d i a m e t e r e s t i m a t e d from t h e shape o f t h e p o i n t o f t h e w h i s k e r . The d i f f e r e n c e i s caused by t h e i r r e g u l a r i t i e s o f t h e e l e c t r o d e s on a s u b - m i c r o n s c a l e , so t h a t t h e e l e c t r o d e s can t o u c h o n l y on a number o f p l a c e s on t h e ap-p a r e n t c o n t a c t a r e a .
I I . 4 . J u n c t i o n Impedance
The r f impedance o f a j u n c t i o n w h i c h obeys t h e RSJ model i s a complex f u n c t i o n o f a l l j u n c t i o n p a r a m e t e r s . In t h e s u p e r c u r r e n t ( |T|<1 and 0=0), t h e impedance i s j u s t t h e shunt r e s i s t a n c e R i n p a r a l l e l w i t h t h e i n d u c t i v e e l e -ment t h a t r e p r e s e n t s t h e p a i r c u r r e n t ( g i v e n by E q . ( 2 . 2 0 ) ) . The complex im-pedance o f t h e j u n c t i o n ( n o r m a l i z e d t o R) f o r s m a l l r f c u r r e n t s ds« IG) i s t h e n g i v e n by zrf = Re(Z)/R+ jIm(Z)/R= j f i c o s 0 ) / ( n2+ c o s20 ) . (2.22) F o r |T[->1 (cos0->-0) t h e j u n c t i o n impedance a p p r o a c h e s R f o r a l l f r e q u e n c i e s . When t h e j u n c t i o n i s b i a s e d s u c h t h a t T>1 t h e s i t u a t i o n i s d i f f e r e n t . F o r a dc and r f c u r r e n t b i a s , t h e t o t a l c u r r e n t t h r o u g h t h e j u n c t i o n i s g i v e n by Ib+ Isc o s t ost = V ( t ) / R + IQs i n 0 ( t ) . (2.23) o r i n r e d u c e d u n i t s v ( t ) = 1+ isc o s o )st - s i n 0 . (2.24) When t h e j u n c t i o n i s phase l o c k e d t o t h e i n d u c e d r f c u r r e n t 0 = ust + 0Q . (2.25) F o r 0o=+TT/2, s i n ( o jst + 0o) = +cosoJst so t h a t t h e s e a r e t h e c a s e s where t h e v o l -t a g e a c r o s s -t h e j u n c -t i o n i s i n phase w i -t h -t h e i n d u c e d c u r r e n -t . I n -t h i s c a s e t h e j u n c t i o n impedance i s r e a l . However when -TT/2<0o<TT/2 t h e impedance i s
com-p l e x . From E q s . ( 2 . 2 4 ) and (2.25) i t can be seen t h a t , when t h e j u n c t i o n i s phase l o c k e d t o t h e r f s i g n a l , t h e r f v o l t a g e a c r o s s t h e j u n c t i o n i s a l m o s t i n d e p e n d e n t o f t h e a m p l i t u d e o f t h e i n d u c e d r f c u r r e n t . T h i s means t h a t f o r a s m a l l r f c u r r e n t , w h i c h i s j u s t s u f f i c i e n t t o phase l o c k t h e j u n c t i o n , t h e r f impedance can be v e r y l a r g e . F o r v e r y l a r g e Ig however, t h e m a j o r p a r t o f t h e
r f c u r r e n t must f l o w t h r o u g h t h e r e s s i s t a n c e R as t h e p a i r c u r r e n t i s l i m i t e d t o Ic> T h i s means t h a t z^p becomes more e q u a l t o u n i t y f o r i n c r e a s i n g Is. F o r
i <1 and i n t h e l o w e r p a r t o f t h e i n d u c e d s t e p t h e t h i r d term i n Eq.(2.24) i s 30
l a r g e r t h a n t h e s e c o n d . The impedance i s t h e n n e g a t i v e . T h i s means t h a t t h e 30
j u n c t i o n a c t s as an o s c i l l a t o r g e n e r a t i n g power a t u>s. A u r a c h e r e t a l . h a v e
c a l c u l a t e d t h e r f impedance n u m e r i c a l l y f o r a number o f s i t u a t i o n s , w i t h o u t t h e r m a l n o i s e . One o f t h e i r r e s u l t s i s shown i n F i g . I I 7 f o r a r e d u c e d f r e -quency fl=1.
When t h e j u n c t i o n i s used as a d e t e c t o r , i t s o u t p u t impedance i s q u i t e d i f f e r e n t from t h e r f i n p u t impedance. S i n c e t h e r e d u c e d o u t p u t f r e q u e n c i e s
a r e u s u a l l y s m a l l e r than 0.01, t h e o u t p u t s i g n a l i s a v e r a g e d o v e r many c y c l e s o f t h e J o s e p h s o n o s c i l l a t i o n . I t i s t h e r e f o r e e q u a l t o 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 Rd= d V / d l o f t h e I-V c u r v e .
When t h e j u n c t i o n i s shunted w i t h a c a p a c i t a n c e , t h e r f impedance w i l l show a p l a s m a r e s o n a n c e , as m e n t i o n e d e a r l i e r . T h i s w i l l be i n v e s t i g a t e d i n 20 t h i s t h e s i s i n t h e p r e s e n c e o f t h e r m a l n o i s e and f o r n o n z e r o a v e r a g e v o l -t a g e a c r o s s -t h e j u n c -t i o n , w i -t h an e l e c -t r o n i c a n a l o g . I I . 5 . F l u c t u a t i o n s I n t u n n e l j u n c t i o n s i n t h e l i m i t e V « kDT ( kn i s B o l t z m a n n s c o n s t a n t ) , t h e 31 32
n o i s e can be i n t e r p r e t e dJ 'J as a s h o t n o i s e c o n t r i b u t i o n from t h e Josephson
p a i r c u r r e n t ( w h i c h a r i s e s from t h e c o u p l i n g o f t h e c o h e r e n t l y t u n n e l i n g Coop-er p a i r s t o random f l u c t u a t i o n s due t o t h e b a c k g r o u n d b l a c k b o d y 3 19 r a d i a t i o n ^ ' ) p l u s t h e t h e r m a l ( J o h n s o n ) n o i s e a s s o c i a t e d w i t h t h e q u a s i -p a r t i c l e r e s i s t a n c e . I n c a s e o f -p o i n t c o n t a c t s , t h e t h e r m a l f l u c t u a t i o n s a r e 33
e x p e c t e d t o be d o m i n a n t . T h i s i s i n r e a s o n a b l e agreement w i t h some measure-ments ' , w h i l e o t h e r s '3 f i n d a s i g n i f i c a n t l y h i g h e r n o i s e l e v e l t h a t t h e y i n t e r p r e t e d as a p r e s e n c e o f p a i r f l u c t u a t i o n s . When t h e q u a s i - p a r t i c l e r e s i s t a n c e i s assumed t o be c o n s t a n t , as i n t h e 1 9 RSJ m o d e l , t h e n o i s e c u r r e n t i n t h e j u n c t i o n c a n be r e d u c e d t o 3 In 2( v ) - 2EB/R . (2.26)
where B i s t h e b a n d w i d t h and E i s t h e l a r g e s t o f eV, hv o r 2 kRT . These r e g i m e s
19
can be i d e n t i f i e d as s h o t n o i s e , p h o t o n n o i s e and t h e r m a l n o i s e dominated r e s p e c t i v e l y . I f we l i m i t o u r s e l v e s t o t h e t h e r m a l n o i s e r e g i m e , a r e d u c e d n o i s e parameter T= 2 e kBT / H Ic can be i d e n t i f i e d w h i c h i s e q u a l t o t h e r a t i o o f
t w i c e the t h e r m a l energy kgT and t h e maximum J o s e p h s o n c o u p l i n g e n e r g y HI /e [ f r o m E q . ( 2 . 2 ) ; E(Tr)-E(Q)]. T h i s s i t u a t i o n can a l s o be s t u d i e d i n t h e w a s h i n g - b o a r d a n a l o g . When t h e r e i s no n o i s e p r e s e n t (f=0; as assumed u n t i l now) and |l|<1 i n the s t a t i c s i t u a t i o n , t h e p a r t i c l e i s a t r e s t i n an e q u i l i -b r i u m p o s i t i o n a t a l o c a l minimum o f t h e p o t e n t i a l e n e r g y . When t h e r m a l n o i s e i s p r e s e n t (T/0) the m o t i o n o f t h e p a r t i c l e can be t h o u g h t o f as a Brownian m o t i o n . When t w i c e t h e t h e r m a l e n e r g y kDT i s o f t h e same s i z e as t h e r e s i d u a l
D
e n e r g y b a r r i e r ( t h e c o u p l i n g e n e r g y ) , t h e p a r t i c l e can be e x c i t e d a c r o s s t h e b a r r i e r . When t h e j u n c t i o n i s h e a v i l y damped, t h e p a r t i c l e w i l l be r e c a p t u r e d i n t h e n e x t p o t e n t i a l w e l l , from w h i c h i t can a g a i n be e x c i t e d . The c o n s e -quence o f t h i s b e h a v i o r i s t h a t v o l t a g e p u l s e s a r e p r e s e n t , r e s u l t i n g i n a n o n - z e r o a v e r a g e v o l t a g e , a l s o when | i | < 1 . The I-V c u r v e w i l l show a r o u n d i n g o f f o f t h e sudden v o l t a g e o n s e t a t | i ] = 1 . The r e s u l t i n g I-V c u r v e s f o r t h e
oo RSJ model f o r s e v e r a l v a l u e s o f V have been c a l c u l a t e d by Ambegeokar e t a l .
39
u s i n g t h e E i n s t e i n - F o k k e r - P l a n c k e q u a t i o n . The r e s u l t s a r e v e r y s i m i l a r t o 40
c a l c u l a t i o n s on phase c a p t u r e o f a p h a s e - l o c k e d - l o o p ; as shown by Bak e t 18 a l . a p h a s e - l o c k e d - l o o p i s a p e r f e c t a n a l o g o f t h e RSJ model. The t h e r m a l n o i s e t h e n has t o be s i m u l a t e d by an e x t e r n a l l y i n d u c e d n o i s e c u r r e n t In = ( 4 kBTeB / R )1 / 2 , (1) rms [ f r o m Eq.(2.26) w i t h E= 2 ( k g T ) ] where Tg i s t h e e f f e c t i v e n o i s e t e m p e r a t u r e o f t h e j u n c t i o n . The e f f e c t o f a c a p a c i t i v e s h u n t on t h i s s i t u a t i o n w i l l be s t u -20 d i e d on such an a n a l o g i n t h i s t h e s i s The i n d u c e d c o n s t a n t v o l t a g e s t e p s w i l l a l s o be rounded o f f by t h e r m a l f l u c t u a t i o n s . The c o u p l e d s i g n a l power r e q u i r e d t o l o c k t h e phase d i f f e r e n c e
41
0 t o t h e s i g n a l has been c a l c u l a t e d by Stephen and i s g i v e n by
P > ( 2 e kBTg/ h )2R , (2)
-12
or about 10 W f o r R= 50 ohm and Tg= 5 K e l v i n . WhenT>1, t h e t h e r m a l e n e r g y
i s h i g h enough t h a t even f o r t h e maximum c o u p l i n g e n e r g y ( a t i=0) t h e phase-l o c k e d c o n d i t i o n c a n n o t be m a i n t a i n e d . The s u p e r c u r r e n t w i phase-l phase-l have d i s a p p e a r e d c o m p l e t e l y . F o r an i d e a l Nb p o i n t c o n t a c t a t an e f f e c t i v e t e m p e r a t u r e T -5 K,
¡4 e
I I . 6 . D e t e c t i o n o f E l e c t r o m a g n e t i c R a d i a t i o n
The Josephson j u n c t i o n can be used as a d e t e c t o r o f e l e c t r o m a g n e t i c r a d i a -t i o n i n f i v e d i f f e r e n -t ways, lip a. B o l o m e t e r . The t e m p e r a t u r e dependence o f e i t h e r t h e s u p e r c u r r e n t o r t h e o n s e t o f t h e q u a s i - p a r t i c l e c u r r e n t o f a t u n n e l j u n c t i o n , i s used as a thermometer t o d e t e c t t h e r a d i a t i o n - i n d u c e d t e m p e r a t u r e change o f an a b s o r b i n g e l e m e n t . b. Wide-band d e t e c t o r . The r a d i a t i o n i n d u c e d dependence o f t h e s u p e r c u r r e n t , u s u a l l y o f p o i n t c o n -t a c -t s , i s used as power d e -t e c -t o r . c. H e t e r o d y n e d e t e c t o r w i t h i n t e r n a l L o c a l O s c i l l a t o r ( L O ) ^ .
The j u n c t i o n i s used a s a m i x e r between t h e i n t e r n a l J o s e p h s o n o s c i l l a t i o n (LO) a t 0^= 2 e Vb/ h and a s i g n a l a t w . The major d i s a d v a n t a g e i s t h e l a r g e r e
-l a t i v e -l i n e w i d t h1 3 o f u^, w h i c h i s ( i n t h e t h e r m a l n o i s e l i m i t ) g i v e n by
Ao^/o^ = 3 kBTe( 2 e / h ) rd 2/ Ic , (2.29)
where r ^ i s t h e n o r m a l i z e d d i f f e r e n t i a l r e s i s t a n c e d v / d i . I n t h i s d e t e c t o r mode, t h e Josephson l i n e w i d t h d e t e r m i n e s t h e l i n e w i d t h o f n a r r o w band m i x i n g components. F o r Tg= 5 K, R= 50 ohm, I s 30 p A and rr f=2 t h e r e l a t i v e l i n e w i d t h
i s 8 . W O- 2, w h i c h i s about 20 GHz f o r a 200 GHz s y s t e m .
34 d. H e t e r o d y n e d e t e c t o r w i t h e x t e r n a l L0 J .
The j u n c t i o n i s used as a m i x e r between an e x t e r n a l l y s u p p l i e d L 0 a t o ^0 and a
s i g n a l a t w . The l i n e w i d t h o f t h e i n t e r m e d i a t e f r e q u e n c y ( I F ) m i x i n g com-ponent a t wI F = I^Q-W I i s independent o f Ao^.
45 e. Q u a s i - p a r t i c l e m i x e r w i t h e x t e r n a l L0
T h i s method o f d e t e c t i o n was d e v e l o p e d o n l y r e c e n t l y . However i t seems t h e most p r o m i s i n g o n e . Here t h e s t r o n g n o n l i n e a r i t y o f t h e I-V c u r v e o f a S u p e r c o n c u c t o r I n s u l a t o r S u p e r c o n d u c t o r ( S I S ) t u n n e l j u n c t i o n a t t h e gap v o l -t a g e 2A/e i s used f o r h i g h f r e q u e n c y m i x i n g . I n -t h i s c a s e -t h e -t u n n e l j u n c -t i o n i s n o t used as a Josephson j u n c t i o n ; sometimes t h e p a i r c u r r e n t i s even r e -moved by a p p l y i n g a m a g n e t i c f i e l d , so t h a t i t c a n n o t i n t e r f e r e . The non-l i n e a r i t y can be so s t r o n g t h a t t h e c non-l a s s i c a non-l m i x e r t h e o r y b r e a k s down and photon a s s i s t e d q u a s i - p a r t i c l e t u n n e l i n g t h e o r y must be used t o u n d e r s t a n d t h e d e v i c e p e r f o r m a n c e ( r e s u l t i n g i n D a y e m - M a r t i n1^ v o l t a g e s t e p s a t v o l t a g e s
i s t w i c e t h e s e p a r a t i o n o f t h e r a d i a t i o n i n d u c e d s t e p s t h a t o c c u r i n Josephson p a i r t u n n e l i n g a t t h e same s i g n a l f r e q u e n c y ) . As a r e s u l t o f t h e quantum
47
t h e o r y , Tucker p r e d i c t e d a p o s s i b l e c o n v e r s i o n g a i n ( t h e a v a i l a b l e power a t t h e I F i s l a r g e r than t h e a v a i l a b l e s i g n a l power) w h i c h was s h o r t l y t h e r e a f t e r found b y Shen e t a l . a t 36 GHz. The g r e a t a d v a n t a g e o f q u a s i - p a r t i c l e m i x e r s over Josephson m i x e r s i s t h a t t h e i r m i x e r n o i s e t e m p e r a t u r e Tm i s c o m p a r a b l e
w i t h t h e quantum n o i s e l i m i t T = F W kc, w h i l e f o r J o s e p h s o n m i x e r s T » nu>/kn
? 3 M IIQ m B' m B
' ' ' . I t i s q u i t e w e l l p o s s i b l e t h a t t h e q u a s i - p a r t i c l e m i x e r s w i l l be b e t t e r t h a n Josephson m i x e r s , a l s o a t f r e q u e n c i e s o f t h e o r d e r o f 200 GHz.
D e t e c t o r modes b. and d. w i l l be i n v e s t i g a t e d i n t h i s t h e s i s and w i l l t h e r e f o r e be examined somewhat c l o s e r i n t h e n e x t two p a r a g r a p h s .
I I . 7 . Wide-Band D e t e c t i o n o f R a d i a t i o n
A s q u a r e - l a w d e t e c t o r i s o b t a i n e d ( t h e o u t p u t v o l t a g e i s p r o p o r t i o n a l t o t h e i n p u t power) when t h e d e f o r m a t i o n o f t h e I-V c u r v e under t h e i n f l u e n c e o f a m p l i t u d e modulated r a d i a t i o n i s u s e d . Each s i g n a l l e v e l c o r r e s p o n d s t o a c e r t a i n maximum s u p e r c u r r e n t and i f t h e j u n c t i o n i s b i a s e d a t a c o n s t a n t c u r r e n t s l i g h t l y l a r g e r than Iq, t h e o u t p u t v o l t a g e f o l l o w s t h e s i g n a l modulaF i g . I I 8 . V i d e o square law d e -t e c -t i o n w i -t h a c u r r e n -t b i a s e d j u n c t i o n . The r e s p o n s e i s the a m p l i t u d e modulated o u t p u t v o l t a g e of the j u n c t i o n t h a t o c c u r s when i t i s a l t e r n a t i n g -l y exposed to two d i f f e r e n t r a d i a t i o n l e v e l s . I bias 0 1 0 voltage t i o n ( s e e F i g . I I - 8 ) . The d e r i v a t i o n o f t h e v o l t a g e r e s p o n s e h a s been g i v e n b y 14 SO K a n t e r e t a l . and L i k h a r e v e t a l . , u s i n g a second o r d e r p e r t u r b a t i o n method based on t h e u n p e r t u r b e d s o l u t i o n o f t h e t i m e e v o l u t i o n o f t h e v o l t a g e as g i v e n b y Aslamazov e t a l . " " \ I n t h e absence o f f l u c t u a t i o n s t h e v o l t a g e r e s p o n s e A V i s g i v e n by
A T = c ia 2/ H X ï2- ie 2r3 / 2 v H 2 - u b 2 ) • (2-30)
There a r e two r e g i o n s o f wide-band d e t e c t i o n c o r r e s p o n d i n g t o t h e i n t e r n a l Josephson f r e q u e n c y l a r g e r than o r s m a l l e r t h a n t h e s i g n a l f r e q u e n c y u s e p a r a t e d by r e s o n a n t d e t e c t i o n when w =1^. The most i m p o r t a n t r e g i o n i s ^ « wg, when T i s s l i g h t l y above IC < W i t h o u t n o i s e rd= Rd/R= dv/dT= T/v= i ( i2- 1 ) ~1^2. E q u a t i o n (2.30) then becomes e q u a l t o AV = ( Is 2/ 4 T ) ( Rd/ f i2) . ( 2 . 3 D In t h i s r e g i m e , t h e c h a r a c t e r i s t i c t i m e t o r e a c h t h e s t e a d y s t a t e Josephson o s c i l l a t i o n i s l a r g e r than the s i g n a l p e r i o d , w h i c h e x p l a i n s t h e d e c r e a s e o f t h e r e s p o n s e w i t h i n c r e a s i n g f r e q u e n c y . F o r wb> > ug t h e r e s p o n s e i s c l a s s i c a l and can be w r i t t e n a s AV = ( I g2/ ^ ) d ^ / d l2 , (2.32) i . e . p r o p o r t i o n a l t o t h e c u r v a t u r e o f t h e I-V c u r v e . F o r w ^ - o ^ t h e r e s p o n s e shows a s i n g u l a r i t y t h a t i s p a r t i a l l y washed o u t when t h e r m a l n o i s e i s p r e s e n t . E q u a t i o n s (2.31) and (2.32) are s t i l l v a l i d i n case o f a l o w l e v e l
;ren .20 50 o f t h e r m a l n o i s e . The r e s p o n s e f o r t h e s e d i f f e r e n t s i t u a t i o n s w i l l be a n a l y z e d w i t h a p h a s e - l o c k e d - l o o p a n a l o g i n t h i s t h e s i s We w i l l r e s t r i c t o u r s e l v e s now t o t h e c a s e to <<w . The r e a l p a r t o f t h e 20 ^0 j u n c t i o n impedance i s i n t h i s c a s e a p p r o x i m a t e l y e q u a l t o R ' . I f t h e ap-p l i e d s i g n a l ap-power i s P and t h e c o u ap-p l i n g e f f i c i e n c y w i t h t h e j u n c t i o n i s C ^ , t h e s i g n a l power i n t h e j u n c t i o n i s C ^ P = Ig 2R / 2 . The v o l t a g e r e s p o n s e i s t h e n g i v e n b y AV/P = Ch fRd/ 2 K 22R . (2.33) The r.m.s. n o i s e v o l t a g e a c r o s s t h e j u n c t i o n i n t h e t h e r m a l n o i s e l i m i t i s 33 51 g i v e n b yJ ' Vn = rd[ ( 2 + I "2) 2 kBT B ' R ]1 / 2, (2.34) rms where B' i s t h e p o s t d e t e c t i o n b a n d w i d t h ( t h e n o i s e b a n d w i d t h a f t e r phase s e n -s i t i v e d e t e c t i o n ) . When B' i -s 1 Hz ( c o r r e -s p o n d i n g t o a t i m e - c o n -s t a n t o f about 0.3 s e c . ) and i = 1 , t h i s r e d u c e s t o Vn = rd( 6 kBT R )1 / 2. (2.35) rms
The N o i s e E q u i v a l e n t Power (NEP) i s t h e n
NEP (1 Hz) r Vn / ( A v / P ) = (21 Q2/^) ( 6 kBT R )1 / 2 ' (2,36)
These c a l c u l a t e d v a l u e s o P v and NEP can be 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 s t h a t w i l l be p r e s e n t e d T l t e r ^2 ( s e e page 4 2 ) . F o r R= 90 ohm, r =2.5,
tQ- 10 pk and a s i g n a l f r e q u e n c y o f about 200 GHz, t h e c a l c u l a t e d r.m.s. n o i s e v a l u e i s about 5 * 1 0 ~1 0 V / H z1^2, w h i c h i s e q u a l t o t h e e x p e r i m e n t a l v a l u e . The
c a l c u l a t e d ^ ca lo - 8 * 1 0 ~1 6 W / H z1 / 2 f o r Ch f= 1 . T h i s h a s t o be compared w i t h N E Pe x p " W / H z1 / 2 f o r t h e whole s y s t e m . When t h e e x p e c t e d o p t i c a l
t r a n s m i s s i o n o f about 0.2 i s used t o c a l c u l a t e t h e e x p e r i m e n t a l d e t e c t o r NEP, a v a l u e o f about 1*10~1^ W/Hz1^2 i s found w h i c h i s i n e x c e l l e n t agreement w i t h
the c a l c u l a t e d p e r f o r m a n c e . When t h e RSJ model i s s h u n t e d w i t h a c a p a c i t a n c e s u c h t h a t Bc< 1, t h e wideband r e s p o n s e w i l l d e c r e a s e because t h e e f f e c t i v e a m p l i t u d e o f t h e i n -duced h i g h f r e q u e n c y c u r r e n t s i s d e c r e a s e d . S i n c e 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 20 o f t h e I-V c u r v e i n c r e a s e s i n t h i s c a s e ( s e e p a r . I I . 3 ) , t h e l o s s i n t h e most i m p o r t a n t component o f t h e r e s p o n s e , g i v e n by E q . ( 2 . 3 1 ) , i s somewhat compen-s a t e d . Becaucompen-se o f t h e i n c r e a compen-s e o f Vf i ( p r o p o r t i o n a l w i t h r . ) however, t h e
NEP does n o t b e n e f i t from t h i s c o m p e n s a t i n g mechanism a s can be seen from E q . ( 2 . 3 6 ) . I t w i l l t h e r e f o r e d e t e r i o r a t e f o r i n c r e a s i n g B . When 3 >1 t h e
24 25
n o i s e l e s s I-V c u r v e w i l l become h y s t e r e t i c ' . When a s u f f i c i e n t l y h i g h t h e r m a l n o i s e l e v e l i s p r e s e n t however, t h e j u n c t i o n w i l l be s w i t c h i n g between the two s o l u t i o n s o f Eq.(2.16) c o n t i n u o u s l y . When t h e v o l t a g e i s averaged o v e r a t i m e t h a t i s l o n g e r than t h e maximum l i f e t i m e o f t h e j u n c t i o n i n e a c h
20
s o l u t i o n , t h e I-V c u r v e a p p e a r s c o n t i n u o u s . Wide-band d e t e c t i o n t h e n e x i s t s because t h e a v e r a g e l i f e t i m e o f t h e j u n c t i o n i n t h e two s t a b l e s o l u t i o n s i s i n f l u e n c e d by r a d i a t i o n . T h i s means t h a t t h e a v e r a g e I-V c u r v e i s changed so t h a t a v o l t a g e r e s p o n s e e x i s t s a l s o i n t h i s c a s e . I t w i l l be shown l a t e r w i t h
20
the e l e c t r o n i c a n a l o g t h a t t h i s r e s p o n s e c o n s i s t s f o r an i m p o r t a n t p a r t o f an enhanced e x c i t a t i o n o u t - o f t h e p h a s e - l o c k e d s o l u t i o n i n t h e s u p e r c u r r e n t , a t t h e p l a s m a - r e s o n a n c e r e l a t e d a t t e m p t - f r e q u e n c y . T h i s r e s p o n s e c a n even be g r e a t e r than t h e r e s p o n s e f o r 3c=0. However because o f t h e s w i t c h i n g o f t h e
j u n c t i o n between t h e two s o l u t i o n s , t h e n o i s e l e v e l i n c r e a s e s s t r o n g l y . The NEP o f t h i s d e t e c t i o n mechanism c a n t h e r e f o r e n o t be a s good a s f o r Sc=0.
The wideband r e s p o n s e o f a p o i n t c o n t a c t t h a t i s mounted a c r o s s a r e c -52
t a n g u l a r waveguide w i l l be i n v e s t i g a t e d i n t h i s t h e s i s t o f i n d t h e s p e c t r a l r e s p o n s e o f s u c h a system and t o d e t e r m i n e whether i t c a n be used a s a p r a c t i c a l wideband r e c e i v e r . The r e s u l t o f t h i s e x p e r i m e n t i s t h a t a v e r y s e n s i -t i v e r e c e i v e r i s o b -t a i n e d -t h a -t h a s a r e l a -t i v e l y s m a l l o p -t i c a l b a n d w i d -t h . When
4?
t h i s d e t e c t o r i s compared w i t h a s u p e r c o n d u c t i n g b o l o m e t e r i t i s found t h a t the NEP o f t h e b o l o m e t e r i s s l i g h t l y worse t h a n t h e b e s t NEP o f o u r wide-band d e t e c t o r ^ . The b o l o m e t e r however has t h e advantage o f a more s t a b l e b e h a v i o r t h a n a p o i n t c o n t a c t . Which d e t e c t o r would have t o be chosen depends on t h e
s p e c t r a l i n f o r m a t i o n t h a t one wants t o c o l l e c t . When i t i s j u s t n e c e s s a r y t o d e t e c t as s m a l l as p o s s i b l e a t e m p e r a t u r e d i f f e r e n c e , i t i s o f g r e a t advantage t o use t h e b o l o m e t e r t h a t can have a v e r y l a r g e o p t i c a l b a n d w i d t h . T h i s i s a l s o t r u e f o r v e r y h i g h f r e q u e n c i e s (>300 GHz) s i n c e t h e s e n s i t i v e t y o f t h e b o l o m e t e r i s e s s e n t i a l l y i n d e p e n d e n t o f s i g n a l f r e q u e n c y , w h i l e t h e wide-band Josephson d e t e c t o r s e n s i t i v i t y d e c r e a s e s w i t h i n c r e a s i n g f r e q u e n c y . However when i t i s more i n t e r e s t i n g t o measure a s e l e c t i v e o p t i c a l f r e q u e n c y i n t e r v a l , t h e Josephson wideband d e t e c t o r can be used w i t h advantage because no e x t e r -n a l f i l t e r i -n g has t o be added.
I I . 8 . Heterodyne D e t e c t i o n w i t h E x t e r n a l LP
34
T h i s i s t h e most common mode o f o p e r a t i o n o f a Josephson m i x e r . The 34 4q S3
j u n c t i o n i s u s u a l l y a p o i n t c o n t a c t ' ' b e c a u s e o f t h e good c o u p l i n g and 54
s m a l l j u n c t i o n c a p a c i t a n c e , a l t h o u g h r e c e n t l y Daalmans e t a l . used a vapour d e p o s i t e d s u b - m i c r o n Nb-Nb j u n c t i o n as a m i x e r a t 230 GHz. The j u n c t i o n i s i r r a d i a t e d w i t h a s i g n a l (oi ) and an LO ( ^ Q ) . When t h e s i g n a l power i s much s m a l l e r t h a n t h e LO power ( I « I ^ Q ) and t h e I F f r e q u e n c y ( p a r . I I - 6 d ) i s much s m a l l e r t h a n t h e LO f r e q u e n c y (ojjp« W ^ Q ) , t h e c u r r e n t induced i n t h e j u n c t i o n may be w r i t t e n as IL 0sinaL 0t+ 1s3 i n%t " ( IL 0+ I s c o s V ) sin[,i ot+ t V W 3 1 " ^ 1 - (2-37) T h i s c o r r e s p o n d s t o a phase and a m p l i t u d e m o d u l a t i o n o f t h e L0 a t t h e I F . 55 S i n c e t h e phase m o d u l a t i o n does n o t p l a y an i m p o r t a n t r o l e , t h e n e t e f f e c t i s an a m p l i t u d e m o d u l a t i o n o f I ^0 and t h e r e b y o f t h e s i z e I Q o f t h e z e r o v o l -t a g e s -t e p . T h i s m o d u l a -t i o n can be measured as -t h e v o l -t a g e a c r o s s -t h e j u n c -t i o n at t h e I F . I t i s maximum where R, i s maximum. When I i s r e d u c e d by t h e LO
• 34 power t o about Ic/ 2 ( e . g . s e e F i g . I I - 9 ) I Q i s l i n e a r l y dependent on and
we have a l i n e a r d e t e c t o r t h a t c o n v e r t s s i g n a l power t o I F power. An impor-t a n impor-t f e a impor-t u r e o f impor-t h i s d e impor-t e c impor-t o r i s impor-t h a impor-t impor-t h e m i x i n g componenimpor-t a impor-t impor-t h e I F i s n o impor-t r e l a t e d t o t h e ac Josephson f r e q u e n c y , so t h a t t h e j u n c t i o n can i n p r i n c i p l e be b i a s e d anywhere on t h e I-V c u r v e . The l i n e w i d t h o f t h e f r e q u e n c y c o n v e r t e d s i g n a l i s a l s o n o t r e l a t e d t o t h e Josephson l i n e w i d t h and can be as n a r r o w as t h a t o f t h e c o h e r e n t s o u r c e s t h a t can be used f o r m i x i n g ( s e v e r a l kHz on a s i g n a l f r e q u e n c y o f 200 G H z ) .
The n o i s e o f a h e t e r o d y n e r e c e i v e r i n t h e m i l l i m e t e r wave r e g i o n i s gene r a l l y dominatgened by t h gene r m a l n o i s gene from t h gene m i x gene r o r t h gene I F a m p l i f i gene r . T h gene r gene -f o r e t h e u s e -f u l -f i g u r e o -f m e r i t i s t h e r e c e i v e r n o i s e t e m p e r a t u r e T w h i c h
