Low noise SQUIDS

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ter verkrijging van de graad van doctor in de technische wetenschappen

aan de Technische Hogeschool Delft

op gezag van de rector magnificus Prof. ir. B.P.Th. Veltman, voor een commissie aangewezen door

het college van dekanen te verdedigen op dinsdag 13 september 1983 te 14.00 uur door V I C T O R J A N D E W A A L natuurkundig ingenieur geboren te Amsterdam

Het o n d e r z o e k i s v e r r i c h t i n samenwerking met D r . I r . T.M. Klapwijk, I r . P. van den Hamer, I r . R. L L u r b a , G.J. van Nieuwenhuyzen, P. S c h r i j n e r en I r . J . J . P . B r u i n e s .

De T e c h n i s c h e Hogeschool D e l f t h e e f t f a c i l i t e i t e n t e r b e s c h i k k i n g g e s t e l d .

De S t i c h t i n g v o o r Fundamenteel Onderzoek d e r M a t e r i e h e e f t h e t p r o j e k t f i n a n c i e e l o n d e r s t e u n d .

6 CONTENTS page I GENERAL INTRODUCTION 9 R e f e r e n c e s 12 I I THEORETICAL BACKGROUND 15 I I . 1 The J o s e p h s o n E f f e c t 15 I I . 2 Fundamentals o f t h e DC SQUID 20 R e f e r e n c e s 24

I I I HIGH PERFORMANCE DC SQUIDS WITH SUBMICRON NIOBIUM JOSEPHSON

JUNCTIONS 27 A b s t r a c t 27 I I I . 1 I n t r o d u c t i o n 28 111.2 D e s i g n c o n s i d e r a t i o n s 29 111.3 F a b r i c a t i o n 36 111.4 J u n c t i o n c h a r a c t e r i z a t i o n 13

111.5 Performance o f t h e SQUID and i n p u t c o i l 46

I I I .5 . 2 Performance o f t h e SQUID 46 I I I . 5 . 2 P e r f o r m a n c e o f t h e c o u p l e d SQUIDs 49 111.6 G r a d i o m e t e r p e r f o r m a n c e 53 111.7 C o n c l u s i o n 55 Appendix A, C a l c u l a t i o n o f t h e g r a d i o m e t e r i n d u c t a n c e . . . . 56 Appendix B, E s t i m a t i o n o f t h e p a r a s i t i c c a p a c i t a n c e 57 R e f e r e n c e s 58

IV SIMULATION AND OPTIMIZATION OF A DC SQUID WITH FINITE

CAPACITANCE 61 A b s t r a c t 61 IV. 1 I n t r o d u c t i o n 62

IV.2 The dc SQUID model 63 IV.3 The o p t i m i z a t i o n method 66 IV.4 I m p l e m e n t a t i o n on a h y b r i d computer 68

IV.6 D i s c u s s i o n R e f e r e n c e s . . . V CONCLUSION . . . R e f e r e n c e s . . . 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 91

I GENERAL 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 s i g n , f a b r i c a t i o n , and l i m i t a t i o n s o f v e r y s e n s i t i v e SQUID ( S u p e r c o n d u c t i n g Quantum I n t e r f e r e n c e D e v i c e ) magnetometers. The SQUID magnetometer i s based on t h e J o s e p h s o n e f f e c t . I n 1962 B.D. J o s e p h s o n (_]_) p r e d i c t e d t h a t a s u p e r c u r r e n t c a n f l o w between two s u p e r c o n d u c t o r s s e p a r a t e d by a v e r y t h i n i n s u l a t i n g b a r r i e r . He showed t h a t t h e maximum s u p e r c u r r e n t , t h e c r i t i c a l c u r r e n t , f l o w i n g t h r o u g h t h e j u n c t i o n depends on t h e m a g n e t i c f i e l d i n s i d e t h e j u n c t i o n . The s i n g l e J o s e p h s o n j u n c t i o n i s n o t a v e r y s e n s i t i v e magnetometer. J a k l e v i c , Lambe, S i l v e r , and M e r c e r e a u ( 2 ) f i r s t c o n s t r u c t e d t h e dc SQUID w h i c h i s a r i n g o u t o f s u p e r c o n d u c t i n g m e t a l i n t e r r u p t e d by two J o s e p h s o n j u n c t i o n s . The c r i t i c a l c u r r e n t o f t h i s d e v i c e depends on t h e m a g n e t i c f l u x e n c l o s e d by t h e s u p e r -c o n d u -c t i n g r i n g . The -c r i t i -c a l -c u r r e n t i s a p e r i o d i -c f u n -c t i o n o f t h e m a g n e t i c f l u x w i t h a p e r i o d o f one f l u x quantum % =h/2e. As t h e f l u x

-15 2 2 quantum 2.07*10 T'm w i t h i n a t y p i c a l a r e a o f 1 mm c o r r e s p o n d s t o

t h e v e r y s m a l l m a g n e t i c f i e l d o f 2 nT, t h e SQUID i s a v e r y s e n s i t i v e i n s t r u m e n t f o r m e a s u r i n g m a g n e t i c f i e l d s .

F o r g a c s and W a r n i c k ( 3 ) f i r s t made a dc SQUID f o r measurement p u r p o s e s . As J o s e p h s o n j u n c t i o n s t h e y used p o i n t c o n t a c t s , w h i c h c o n s i s t o f a n i o b i u m r o d w i t h a s h a r p p o i n t p r e s s e d o n t o a f l a t n i o b i u m p l a t e . However, r e p r o d u c i b l e f a b r i c a t i o n o f t h e two p o i n t c o n t a c t s was d i f f i c u l t and t h e r e l i a b i l i t y was n o t good enough. A l s o t h e i r e l e c t r o n i c r e a d o u t s y s t e m l i m i t e d t h e o v e r a l l p e r f o r m a n c e . A l a r g e improvement i n t h e use o f p o i n t c o n t a c t s f o r d e v i c e s was r e p o r t e d by Zimmerman e t a l . (H), who p r o p o s e d t h e r f SQUID. T h i s i s a s u p e r c o n d u c t i n g r i n g i n t e r r u p t e d by o n l y one J o s e p h s o n j u n c t i o n . The r f SQUID i s o p e r a t e d w i t h a r a d i o f r e q u e n c y b i a s c u r r e n t , g e n e r a l l y s e v e r a l t e n s o f MHz. T h i s r f SQUID became t h e p o p u l a r SQUID f o r p r a c t i c a l a p p l i c a t i o n s . I n t h e 70's t h e f i r s t c o m m e r c i a l SQUID systems became a v a i l a b l e . The p r o b l e m w i t h t h e s e SQUIDs was s t i l l t h e u n r e -l i a b -l e p o i n t c o n t a c t , w h i c h was s e n s i t i v e t o m e c h a n i c a -l v i b r a t i o n s and t h e r m a l change. I n 1975 C l a r k e , Goubau and K e t c h e n (5_) r e p o r t e d a dc SQUID f a b r i c a t e d w i t h t h i n f i l m t e c h n i q u e s . Such SQUIDs p r o v e d

10

m e c h a n i c a l l y s t a b l e and were r e l a t i v e l y r e s i s t a n t t o t h e r m a l changes. The dc SQUID i n d e e d had a b e t t e r s e n s i t i v i t y t h a n t h e c o m m e r c i a l r f SQUID. The s e n s i t i v i t y o f t h e dc SQUID i s l i m i t e d by t h e v a l u e o f t h e i n d u c t a n c e o f t h e SQUID and t h e p a r a s i t i c c a p a c i t a n c e o f t h e j u n c t i o n s (6_,7_), w h i c h must be made as s m a l l as p o s s i b l e . To o b t a i n t h e same s e n s i t i v i t y w i t h t h e r f SQUID one would need, b e s i d e s t h e s m a l l i n d u c t a n c e and c a p a c i t a n c e a l s o an i m p r a c t i c a l h i g h f r e q u e n c y ( o f t h e o r d e r o f s e v e r a l GHz) f o r t h e b i a s i n g and t h e e l e c t r o n i c r e a d o u t system (8 ) .

A number o f r e s e a r c h e r s n o t i c i n g t h e t h e o r e t i c a l p r e d i c t i o n o f a quantum l i m i t t o t h e s e n s i t i v i t y o f SQUIDs have c o n c e n t r a t e d on u l t r a s e n s i t i v e low i n d u c t a n c e dc SQUIDs. On t h e o t h e r hand i n t h i s r e s e a r c h t h e l i n e i s f o l l o w e d t o c o n s t r u c t a dc SQUID o f a p r a c t i c a l i n d u c t a n c e and o u t o f a s t a b l e s u p e r c o n d u c t i n g m a t e r i a l l i k e n i o b i u m . T h i s was made p o s s i b l e by r e c e n t p r o g r e s s i n t h e r e p r o d u c i b l e f a b r i c a t i o n o f s t a b l e J o s e p h s o n j u n c t i o n s ( 9 , 1 0 ) .

SQUIDs o f f e r t h e p o s s i b i l i t y t o measure low f r e q u e n c y m a g n e t i c -14 -12 f i e l d s as s m a l l as 10 T and v o l t a g e s o f 10 V. The aim o f t h e r e s e a r c h d e s c r i b e d i n t h i s t h e s i s i s t h e c o n s t r u c t i o n o f a p r a c t i c a l low n o i s e SQUID. I n t h e e a r l y days SQUIDs were used p r i m a r i l y f o r measurements on o b j e c t s a t low t e m p e r a t u r e . Nowadays SQUID s y s t e m s a r e used f o r a v a r i e t y o f a p p l i c a t i o n s . F o r i n s t a n c e m a g n e t i c f i e l d s g e n e r a t e d by t h e human body a r e s t u d i e d as w e l l as m a g n e t i c s u s c e p t i -b i l i t i e s and m a g n e t i c moments o f m a t e r i a l s . SQUID systems a r e used f o r measurements o f m a g n e t i c s i g n a l s from t h e e a r t h ' s c r u s t . The s o u r c e s f o r most o f t h e s e measurements a r e o u t s i d e t h e h e l i u m c r y o s t a t . The c r y o g e n i c system used f o r c o o l i n g t h e SQUID i s d i f f e r e n t f o r a l l o f t h e s e a p p l i c a t i o n s . But t h e SQUID s e n s o r i s m o s t l y t h e same. I t c o n s i s t s o f t h e SQUID i t s e l f w i t h an i n p u t c o i l t o c o u p l e t h e s i g n a l i n t o t h e SQUID. The i n p u t t e r m i n a l s o f t h e SQUID can be c o n n e c t e d t o d i f f e r e n t k i n d s o f c i r c u i t s w h i c h p i c k up t h e s i g n a l from t h e o b j e c t o f measurement. I n many c a s e s t h e p i c k - u p c i r c u i t i s a s i g n a l c o i l c o n s i s t i n g o f a s u p e r c o n d u c t i n g w i r e wound i n a c o n f i g u r a t i o n a d a p t e d t o t h e s i g n a l t o be measured, f o r i n s t a n c e a s p a t i a l g r a d i e n t o f t h e m a g n e t i c f i e l d . T o g e t h e r w i t h t h e i n p u t c o i l i t forms a c o m p l e t e l y

s u p e r c o n d u c t i n g c i r c u i t , w h i c h keeps t h e e n c l o s e d f l u x c o n s t a n t and hence t r a n s p o r t s t h e m a g n e t i c f i e l d from s i g n a l c o i l t o t h e SQUID. The o v e r a l l s e n s i t i v i t y o f t h e s y s t e m i s m o s t l y l i m i t e d by t h e n o i s e o f the SQUID and t h e l o s s between SQUID and i n p u t c o i l . T h e r e f o r e t h e development o f a low n o i s e SQUID s h o u l d c o n c e n t r a t e on a c o m b i n a t i o n o f a SQUID w i t h an i n p u t c o i l . Reviews o f many a p p l i c a t i o n s o f SQUIDs a r e g i v e n i n Refs.J_l_ and 1 2 .

The t h e o r y o f J o s e p h s o n j u n c t i o n s i s o f t e n r e d u c e d t o a s i m p l i f i e d d e s c r i p t i o n w i t h t h e RSJ ( R e s i s t i v e l y Shunted J u n c t i o n ) model ( 1 3 , 1 4 ) . For some t y p e s o f j u n c t i o n s t h i s model g i v e s a good q u a n t i t a t i v e d e s c r i p t i o n . Tesche and C l a r k e (6) used a computer model t o s t u d y t h e b e h a v i o r o f t h e dc SQUID. T h e i r c a l c u l a t i o n s g i v e a good e s t i m a t e o f the n o i s e o f a SQUID and a r e u s e f u l f o r o p t i m i z a t i o n o f SQUID p a r a -m e t e r s . They showed t h a t t h e n o i s e o f t h e dc SQUID c o u l d be r e d u c e d by r e d u c i n g t h e i n d u c t a n c e o f t h e SQUID and t h e c a p a c i t a n c e o f t h e j u n c t i o n s . I n t h e i r n u m e r i c c a l c u l a t i o n t h e p a r a s i t i c c a p a c i t a n c e o f the j u n c t i o n s was n o t t a k e n i n t o a c c o u n t . The c a p a c i t a n c e c a n have a l a r g e i n f l u e n c e on t h e b e h a v i o r o f t h e SQUID. A t o o l a r g e c a p a c i t a n c e r e s u l t s i n a h y s t e r e t i c I-V c h a r a c t e r i s t i c ( 1 3 , 1 4 ) . T h i s h y s t e r e s i s can make s t a b l e b i a s i n g o f t h e SQUID i m p o s s i b l e . The c a p a c i t a n c e a l s o p r o d u c e s a r e s o n a n c e w i t h t h e SQUID i n d u c t a n c e ( 1 5 ) - T h i s l a r g e l y i n f l u e n c e s t h e t r a n s f e r f u n c t i o n o f t h e SQUID. So i n f l u e n c e o f t h e c a p a c i t a n c e on t h e n o i s e p e r f o r m a n c e o f dc SQUIDs i s e x p e c t e d . T h i s i s s u p p o r t e d by measurements w i t h r e a l dc SQUIDs r e p o r t e d i n t h i s t h e s i s and by o t h e r s ( 9 ) . A t h e o r y o f a SQUID w i t h c a p a c i t o r s i s c l e a r l y needed. C h a p t e r I I g i v e s a b r i e f i n t r o d u c t i o n t o t h e t h e o r y o f J o s e p h s o n j u n c t i o n s and SQUIDs. Knowledge o f b a s i c t h e o r y o f s u p e r c o n d u c t i v i t y

(16) i s assumed. C h a p t e r I I I w i l l be p u b l i s h e d i n t h e J o u r n a l o f Low Temperature P h y s i c s (J_7_). I t r e p o r t s on t h e dc SQUID d e v e l o p e d i n o u r l a b o r a t o r y . A v e r y low n o i s e n i o b i u m SQUID i s d e s c r i b e d . I t i s f a b r i c a t e d w i t h u l t r a s m a l l n i o b i u m j u n c t i o n s w i t h an o v e r l a p p i n g a r e a 2 s m a l l e r t h a n 1 um . The j u n c t i o n s a r e formed a c c o r d i n g t o a r e c i p e from Daalmans ( 1 0 ) . The p h o t o l i t h o g r a p h i c t e c h n i q u e d e v e l o p e d f o r t h e f a b r i c a t i o n o f t h e SQUIDs i s d e s c r i b e d . A l s o c o m p l e t e systems c o n

-12 s i s t i n g o f SQUID w i t h w i r e wound o r t h i n f i l m i n p u t c o i l a r e d e -s c r i b e d . I n t h i -s c h a p t e r an i n t e g r a t e d -s y -s t e m w i t h SQUID and a f i r -s t o r d e r g r a d i o m e t e r on a s i n g l e s u b s t r a t e i s p r e s e n t e d . T h i s d e v i c e i s u s e f u l f o r b i o m e d i c a l a p p l i c a t i o n s . C h a p t e r IV d e a l s w i t h c a l c u l a t i o n s o f t h e r e s o l u t i o n o f a dc SQUID c o n t a i n i n g i d e a l J o s e p h s o n j u n c t i o n s a c c o r d i n g t o t h e RSJ model i n c l u d i n g a p a r a s i t i c c a p a c i t a n c e ( J _ 3 , 1 4 ) . I t was 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 t h e J o u r n a l o f Low Temperature P h y s i c s ( 1 8 ) . The model used i s r a t h e r c o m p l i c a t e d . I t c o n s i s t s o f two c o u p l e d second o r d e r n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s i n c l u d i n g two i n d e p e n d e n t n o i s e s o u r c e s . An a n a l o g computer i s v e r y s u i t a b l e f o r s o l v i n g t h i s t y p e o f e q u a t i o n s . W i t h a h y b r i d computer t h e n o i s e o f t h e system i s c a l c u l a t e d and t h e optimum p a r a m e t e r s o f t h e SQUID a r e f o u n d . C h a p t e r V g i v e s a c o n c l u s i o n on t h e u s e f u l n e s s o f t h e f a b r i -c a t e d SQUIDs based on e x p e r i e n -c e w i t h them i n p r a -c t i -c a l s i t u a t i o n s . A l s o t h e i m p l i c a t i o n s o f t h e c a l c u l a t i o n s w i t h r e g a r d t o t h e p e r f o r -mance o f t h e SQUIDs f a b r i c a t e d a r e d i s c u s s e d .

Ref erences

1 . B.D. J o s e p h s o n , Phys . L e t t2 5 1 ( 1 9 6 2 )

2 . R.C. J a k l e v i c , J . Lambe, A.H. S i l v e r and J.E. Mercereau, Phys.Rev. L e t t.]2_,159 (1961)

3 . P.L. F o r g a c s and A. W a r n i c k , Rev . S c i . Instrum.38_, 214 ( 1 9 6 7 ) 4. J . E . Zimmerman, P. T h i e n e , and J.T. H a r d i n g , J .Appl .Phys .4j_, 1572

( 1 9 7 0 )

5 . J . C l a r k e , W.M. Goubau, and M.B. Ketchen, J.Low T e m p . P h y s . 2 5 , 9 9 ( 1 9 7 6 )

6 . C D . Tesche and J . C l a r k e , J.Low T e m p . P h y s.29, 301 ( 1 9 7 7 )

7 . J . J . P . B r u i n e s , V . J . de Waal, and J.E. Mooij, J.Low Temp.Phys.46, 383 ( 1 9 8 2 )

8 . J . K u r k i j a r v i and W.W. Webb, P r o c . A p p l . S u p e r c o n d u c t i v i t y Conf. A n n a p o l i s , IEEE, New York, 1972, p. 581

9 . R.F. Voss, R.B. L a i b o w i t z , S . I . R a i d e r , and J . C l a r k e , J . A p p l .

10. G.M. Daalmans, S u p e r c o n d u c t i n g Quantum I n t e r f e r e n c e D e v i c e s and T h e i r A p p l i c a t i o n s , H.D. Hahlbohm and H. L u b b i g e d s . , W a l t e r de G r u y t e r , B e r l i n 1980, p. 399

11. S u p e r c o n d u c t i n g Quantum I n t e r f e r e n c e D e v i c e s and T h e i r A p p l i c a -t i o n s , H.D. Hahlbohm and H. L u b b i g eds., W a l -t e r de G r u y -t e r , B e r l i n 1980

12. F u t u r e Trends i n S u p e r c o n d u c t i v e E l e c t r o n i c s , B.S. Deaver, C M . F a l c o , J.H. H a r r i s , S.A. Wolf e d s . , American I n s t i t u t e o f P h y s i c s , New York 1978

13. D.E. McCumber, J . A p p l . P h y s . 3 9 ,3113 (1968) 1.4. W.C S t e w a r t , A p p l . Phys. L e t t .]2_, 277 (1968)

15. S.M. F a r i s and E.A. V a l s a m a k i s , J.Appl.Phys.52,915 (1981) 16. M. Tinkham, I n t r o d u c t i o n t o S u p e r c o n d u c t i v i t y , M c G r a w - H i l l , New

Y o r k , 1975

17. V . J . de Waal, T.M. Klapwijk, and P. van den Hamer, t o be p u b l i s h e d i n J.Low Temp.Phys.

18. V . J . de Waal, P. S c h r i j n e r , and R. L L u r b a , s u b m i t t e d t o J.Low Temp. Phys.

II THEORETICAL BACKGROUND

II.1 The Josephson Effect

The w i d e l y used t h e o r y o f J o s e p h s o n j u n c t i o n s a p p l i e s t o a t u n n e l j u n c t i o n between two s u p e r c o n d u c t i n g m e t a l s . The two s u p e r c o n d u c t i n g m e t a l s a r e s e p a r a t e d by a v e r y t h i n i n s u l a t i n g f i l m . The i n s u l a t o r i s o f t e n t h e n a t u r a l o x i d e o f t h e m e t a l w i t h a t h i c k n e s s o f 1 t o 5 nm. The j u n c t i o n i s c a l l e d a t u n n e l j u n c t i o n i f t h e e l e c t r o n s g o i n g a c r o s s the j u n c t i o n r e a l l y have t o t u n n e l t h r o u g h a p o t e n t i a l b a r r i e r , because no e l e c t r o n s t a t e s e x i s t i n s i d e t h e b a r r i e r . J o s e p h s o n ( 1 - 3 ) showed t h e o r e t i c a l l y u s i n g t h e m i c r o s c o p i c t h e o r y o f s u p e r c o n d u c -t i v i -t y , -t h a -t a s u p e r c u r r e n -t c a n f l o w -t h r o u g h a -t u n n e l j u n c -t i o n . He d e r i v e d t h e e q u a t i o n s f o r a j u n c t i o n b i a s e d w i t h a c o n s t a n t v o l t a g e . The c u r r e n t I f l o w i n g t h r o u g h t h e j u n c t i o n obeys t h e J o s e p h s o n e q u a t i o n s : I = Lj - s i n <p + (o0 + 'cos <p)V v = 2e d t where V i s 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 , L, i s t h e maximum s u p e r -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 <p i s t h e gauge i n v a r i a n t phase d i f f e r e n c e between t h e quantum s t a t e s o f t h e two s u p e r c o n d u c t o r s . Only s m a l l j u n c t i o n s a r e c o n s i d e r e d , w h i c h have d i m e n s i o n s s m a l l e r t h a n t h e Josephson p e n e t r a t i o n d e p t h X7 = (- — ) ( I I .3 ) J 2itUo lo d where A i s t h e a r e a o f t h e j u n c t i o n and d i s t h e e f f e c t i v e t h i c k n e s s o f t h e j u n c t i o n i n c l u d i n g t h e p e n e t r a t i o n d e p t h s o f b o t h s u p e r -c o n d u -c t o r s . I t i s a l s o assumed t h a t t h e m a g n e t i -c f l u x a p p l i e d t o t h e - 1 5

j u n c t i o n i s s m a l l compared t o t h e f l u x quantum 00 =h/2e=2.07'10 Wb.

The a0 and a terms a r e 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 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 r e s -p e c t i v e l y . They de-pend on t h e ( 1 1 .1 )

16

v o l t a g e a c r o s s the j u n c t i o n , the t e m p e r a t u r e and t h e energy gap o f t h e s u p e r c o n d u c t o r s and a r e i n v e r s e l y p r o p o r t i o n a l t o t h e normal j u n c t i o n r e s i s t a n c e . An i m p o r t a n t f e a t u r e o f the Josephson e q u a t i o n s i s the c u r r e n t o s c i l l a t i n g a t the J o s e p h s o n f r e q u e n c y 2eV/h. Ambegaokar and B a r a t o f f (4_) c a l c u l a t e d t h e maximum s u p e r c u r r e n t o f a t u n n e l j u n c t i o n u s i n g t h e m i c r o s c o p i c t h e o r y o f s u p e r c o n d u c t i v i t y . The r e s u l t f o r a j u n c t i o n between two i d e n t i c a l s u p e r c o n d u c t o r s i s

T nA(T) , , M T ) , ITT H I

Jo - ^ p - t a n h C ^ ) ( I I . 4 )

where A(T) i s t h e t e m p e r a t u r e dependent energy gap o f t h e s u p e r c o n -d u c t o r an-d R i s t h e r e s i s t a n c e o f t h e j u n c t i o n i n t h e normal s t a t e .

Another t y p e o f J o s e p h s o n j u n c t i o n i s the weak l i n k , w h i c h c o n s i s t s o f two s u p e r c o n d u c t o r s w e a k l y c o u p l e d by a c o n d u c t i n g c h a n n e l . An example o f t h i s k i n d i s t h e m i c r o b r i d g e ( 5 ) b e i n g a c o n s t r i c t i o n i n a s u p e r c o n d u c t i n g f i l m . E x p e r i m e n t a l l y i t has been w e l l e s t a b l i s h e d t h a t weak l i n k s a l s o d i s p l a y t h e J o s e p h s o n e f f e c t s . For t h e more g e n e r a l c a s e o f a weak l i n k i n s t e a d o f a t u n n e l j u n c t i o n the g e n e r a l form o f Eq. I I . 1 remains v a l i d , a l t h o u g h the phase dence can be n o n - s i n u s o i d a l . A l s o the v o l t a g e and t e m p e r a t u r e depen-dence o f t h e c r i t i c a l c u r r e n t and t h e c o n d u c t i v i t i e s change. A l t h o u g h Eq. I I .4 o n l y a p p l i e s t o a Josephson t u n n e l j u n c t i o n , the same o r d e r o f magnitude o f t h e maximum I, R p r o d u c t i s r e a c h e d w i t h any k i n d o f weak l i n k or j u n c t i o n . The most e x t e n s i v e t h e o r y o f J o s e p h s o n t u n n e l j u n c t i o n s i s from Werthamer ( 6 ) and L a r k i n and O v c h i n n i k o v ( 7 ) . T h e i r t h e o r y t r e a t s the g e n e r a l c a s e o f a t i m e dependent v o l t a g e a c r o s s the j u n c t i o n . The e q u a t i o n s r e s u l t i n g from t h e c a l c u l a t i o n a r e r a t h e r c o m p l i c a t e d . For c o n s t a n t v o l t a g e or a v o l t a g e c h a n g i n g w i t h a f r e q u e n c y much s m a l l e r t h a n t h e J o s e p h s o n f r e q u e n c y t h e r e s u l t r e d u c e s t o t h e J o s e p h s o n e q u a t i o n s I I . 1 and I I . 2 . However, i n many e x p e r i -m e n t a l s i t u a t i o n s one has a j u n c t i o n b i a s e d w i t h a c o n s t a n t c u r r e n t . Then t h e v o l t a g e o s c i l l a t e s w i t h t h e Josephson f r e q u e n c y and i t s h a r m o n i c s . C a l c u l a t i o n s ( 8 - 1 0 ) from t h e Werthamer t h e o r y y i e l d I-V c u r v e s as shown i n F i g . I I . 1. As f a r as I know these I-V c u r v e s have

Fig. II.1

I-V curve of a noiseless junction from the Werthamer theory at T=0.5 with constant current bias. From Zorin and Likharev (9). "0 0.5 1.0 1.5

VOLTAGE ev/2A(0)

never been o b s e r v e d e x p e r i m e n t a l l y . T h i s i s due t o t h e c a p a c i t a n c e a l w a y s p r e s e n t i n p a r a l l e l w i t h t h e j u n c t i o n . T h i s c a p a c i t a n c e s h u n t s the h i g h f r e q u e n c y J o s e p h s o n o s c i l l a t i o n s and o f t e n c a u s e s t h e j u n c t i o n t o behave l i k e a v o l t a g e b i a s e d j u n c t i o n . To a v o i d t h i s

2 -2

e f f e c t t h e c u r r e n t 2itLj a0 C/ 00 t h r o u g h t h e c a p a c i t o r a t t h e l a r g e s t Josephson f r e q u e n c i e s must be made s m a l l compared t o t h e maximum s u p e r c u r r e n t I, . I f one assumes a d i e l e c t r i c c o n s t a n t o f t h e j u n c t i o n b a r r i e r m a t e r i a l o f 10 and a b a r r i e r t h i c k n e s s o f 2 nm, t h e s u p e r

-9 2

c u r r e n t d e n s i t y needed becomes 10 A/m . I n p r a c t i c e i t i s p r o b a b l y h a r d t o r e a l i z e a h i g h q u a l i t y t u n n e l b a r r i e r w i t h such a v e r y h i g h c u r r e n t d e n s i t y .

I f t h e j u n c t i o n i s s h u n t e d w i t h a c a p a c i t o r o r a r e s i s t o r , t h e Werthamer t h e o r y r e d u c e s t o t h e J o s e p h s o n e q u a t i o n s I I . 1 and I I . 2 . I n most p r a c t i c a l c i r c u m s t a n c e s one o f t h e s e c o n d i t i o n s i s s a t i s f i e d . F i g . I I . 2 shows t h e s c h e m a t i c o f a commonly used model f o r a Josephson j u n c t i o n . I t c o n s i s t s o f a J o s e p h s o n element shunted w i t h a r e s i s t o r and a c a p a c i t o r ( 1 1 , 1 2 ) . The e q u a t i o n s d e s c r i b i n g t h e model a r e

V riV = 1, - s i n 9 + - + C — R dq> : 2e'dT (11.5) (11.6)

18

V O L T A G E v / l „ R

Fig. II.2 Fig. II.3

Schematic of the Resistively I-V curves for B =0, 1, 2, c

Shunted Junction (RSJ) and 4. From McCumber (11). model with a capacitor

where C i s t h e c a p a c i t a n c e o f t h e j u n c t i o n and R i s t h e s h u n t r e s i s -t a n c e . I n c o m p a r i s o n w i -t h Eqs. I I . 1 and I I . 2 -t h e cos &l-t;p -term i s o m i -t -t e d and I„ i s t a k e n i n d e p e n d e n t o f t h e v o l t a g e . These a p p r o x i m a t i o n s a r e a l l o w e d i f t h e s h u n t r e s i s t o r R i s much s m a l l e r t h a n t h e r e s i s t a n c e o f the t u n n e l b a r r i e r . F i g . I I . 3 shows I-V c u r v e s (V i s t h e mean v o l t a g e , a v e r a g e d over many J o s e p h s o n c y c l e s ) f o r v a r i o u s v a l u e s o f the parameter B

c 2nIo R2C

v — % — ( I I-7 )

% i s t h e f l u x quantum h/2e. For t h e c a s e B =0 one can f i n d t h e c

V= R ( I2- l o2)1 ( I > \ ) ( I I . 8 )

V = 0 (|JJ<I„ ) ( I I . 9 )

The j u n c t i o n s w i t h a Bc l a r g e r t h a n 1 have a h y s t e r e t i c I-V c u r v e . I n

t h e c a s e o f a v e r y l a r g e c a p a c i t a n c e t h e v o l t a g e w i l l be n e a r l y c o n s t a n t . Then, a c c o r d i n g t o Eqs. I I .5 and I I . 6 , f o r nonzero v o l t a g e t h e mean c u r r e n t t h r o u g h t h e Josephson element w i l l be z e r o and t h e I-V c u r v e i s t h a t o f t h e r e s i s t o r o n l y . So f a r n o i s e l e s s j u n c t i o n s were c o n s i d e r e d . I n a r e a l j u n c t i o n t h e r e i s a t h e r m a l n o i s e c u r r e n t 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 c u r r e n t . I n t h e RSJ model t h i s n o i s e i s i n t r o d u c e d w i t h a Johnson n o i s e c u r r e n t s o u r c e w i t h s p e c t r a l d e n s i t y 4kT/R i n p a r a l l e l w i t h t h e shunt r e s i s t o r . T h i s can be a c c o u n t e d f o r by an e x t r a term i n Eq. I I . 5 . C a l c u l a t i o n s o f t h e I-V c u r v e s o f a RSJ j u n c t i o n c a n be c a r r i e d o u t by d i r e c t n u m e r i c a l s o l u t i o n o f t h e Eqs. I I . 5 and I I . 6 i n c l u d i n g t h e n o i s e term ( 1 3 ) . Other t e c h n i q u e s have been u s e d a l s o ( 1 4 - 1 6 ) . F i g . I I . 4 shows t h a t t h e n o i s e p r o d u c e s a r o u n d i n g o f t h e I-V c u r v e near I = Io - The n o i s e r o u n d i n g depends on t h e d i m e n s i o n l e s s parameter

r= 2li£l (11.10) I> %

2 0

The p h y s i c a l e x p l a n a t i o n i s t h a t the n o i s e f l u c t u a t i o n s s w i t c h t h e j u n c t i o n between the v o l t a g e c a r r y i n g s t a t e and the z e r o v o l t a g e s t a t e . The c u r v e i n F i g . I I . 4 shows t h e mean v o l t a g e . T h i s b e h a v i o r i s d e s c r i b e d by t h e t h e r m a l a c t i v a t i o n model (17-19) .

So f a r i t was assumed t h a t the s p e c t r u m o f the n o i s e c u r r e n t was w h i t e . For v e r y h i g h f r e q u e n c i e s i t i s n e c e s s a r y t o use t h e c o m p l e t e e x p r e s s i o n o f 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 i n c l u d i n g z e r o p o i n t f l u c t u a t i o n s (20)

S ( f ) = ^ - c o t h ( | ^ r ) (11.11)

Because t h e n o i s e a t t h e J o s e p h s o n f r e q u e n c y or i t s f i r s t few harmon-i c s harmon-i s mharmon-ixed down t o low f r e q u e n c harmon-i e s due t o t h e n o n l harmon-i n e a r b e h a v harmon-i o r o f t h e j u n c t i o n , t h e quantum f l u c t u a t i o n s can produce an e x c e s s n o i s e a t low f r e q u e n c y . A l s o t h i s quantum n o i s e can cause a n o i s e r o u n d i n g i n t h e I-V c u r v e (20_).

I I .2 Fundamentals of the dc SQUID

A c o m b i n a t i o n o f one or more r i n g s o f s u p e r c o n d u c t i n g m a t e r i a l i n t e r r u p t e d by one o r more J o s e p h s o n j u n c t i o n s i s c a l l e d a Super-c o n d u Super-c t i n g Quantum I n t e r f e r e n Super-c e D e v i Super-c e (SQUID) or i n t e r f e r o m e t e r . The s u b j e c t o f t h i s t h e s i s i s the dc SQUID c o n t a i n i n g two j u n c t i o n s ( 2 1 ) . To g e t a q u a l i t a t i v e u n d e r s t a n d i n g o f t h e d e v i c e t h e j u n c t i o n model d e s c r i b e d above i s u s e d . Both j u n c t i o n s obey t h e e q u a t i o n s I I . 5 and I I . 6 . The phase o f t h e s u p e r c o n d u c t i n g s t a t e must be s i n g l e v a l u e d , w h i c h l e a d s t o (22)

' r ( p2+ | ^ ' ^A(r),dl= 2 n n d i . 1 2 )

where n i s an i n t e g e r , A(r) i s t h e m a g n e t i c v e c t o r p o t e n t i a l , and <p and a r e the phase d i f f e r e n c e s a c r o s s the j u n c t i o n s . The i n t e g r a l i n

combined w i t h Eqs. I I .5 and I I . 6 d e s c r i b e s t h e dc SQUID. I f t h e f l u x i n s i d e t h e SQUID r i n g i s z e r o , t h e e q u a t i o n s a r e e q u i v a l e n t w i t h t h e e q u a t i o n s o f a s i n g l e RSJ model j u n c t i o n . Then t h e I-V c u r v e i s t h e same a s t h e one o f a s i n g l e j u n c t i o n shown i n F i g . I I .3 . One can e a s i l y d e r i v e t h a t t h e maximum s u p e r c u r r e n t , c r i t i c a l c u r r e n t , I o f the SQUID becomes

i t *

Ic= 2\ | c o s (-^) | ( 1 1 . 1 3 )

where % i s t h e f l u x quantum h/ 2 e . I f one i n t r o d u c e s an i n d u c t a n c e i n the SQUID r i n g t h e m o d u l a t i o n d e p t h o f t h e c r i t i c a l c u r r e n t i s r e d u c e d . A more c o m p l e t e a n a l y s i s was g i v e n by De Waele and De Bruyn Ouboter (2_3,2_4) and by F u l t o n e t a l . {25). N u m e r i c a l c a l c u l a t i o n s o f Tesche and C l a r k e ( 2 6 ) and B r u i n e s e t a l . (27) a r e shown i n F i g . I I . 5 a and b . F o r use as a m e a s u r i n g d e v i c e t h e SQUID i s o p e r a t e d w i t h a

Fig. II. 5

Characteristics of a dc SQUID with 2I0L/%=1, $c=0, and T=0.05

according to Tesche and Clarke (26). (a) I-V curves for 0 =0 and $ = 4„ / 2 .

a a

22

c o n s t a n t b i a s c u r r e n t . Then t h e v o l t a g e a c r o s s t h e SQUID i s a p e r i o d i c f u n c t i o n o f t h e m a g n e t i c f l u x ( F i g . I I . 5 b ) .

The i m p o r t a n t f i g u r e o f m e r i t o f a SQUID used as magnetometer i s t h e energy r e s o l u t i o n ( 2 8 , 2 9 ) , w h i c h i s d e f i n e d by

S( 0 )

e= j (11.14)

2Lk

where S^O) i s t h e low f r e q u e n c y f l u x n o i s e power s p e c t r a l d e n s i t y o f t h e d e v i c e , L i s t h e i n d u c t a n c e o f t h e SQUID and k i s t h e c o u p l i n g c o n s t a n t between SQUID and i n p u t c o i l d e f i n e d by

(11.15)

M i s t h e m u t u a l i n d u c t a n c e between SQUID and c o i l and L i s t h e c

i n d u c t a n c e o f t h e i n p u t c o i l , i f t h e SQUID r i n g i s open. Tesche and C l a r k e (26) a r g u e t h a t t h e optimum energy r e s o l u t i o n o f a SQUID i s ( w i t h a c o r r e c t i o n o f B r u i n e s e t a l . (21))

e=16kT-(LC)^ (11.16)

where C i s t h e c a p a c i t a n c e o f t h e J o s e p h s o n j u n c t i o n s . T h i s r e s u l t shows t h a t t h e n o i s e o f a dc SQUID c a n be made low by c h o o s i n g a s m a l l i n d u c t a n c e or a s m a l l c a p a c i t a n c e . However, Tesche and C l a r k e d i d n o t i n c l u d e t h e c a p a c i t a n c e o f t h e j u n c t i o n s i n t h e i r computer c a l c u -l a t i o n s . The r e s u -l t o f Eq. 11.16 was o b t a i n e d , assuming t h a t Bc v a l u e s

o f 0 o r 1 y i e l d the same energy r e s o l u t i o n and t h a t t h e v a l u e o f 1 i s the optimum. The c a p a c i t a n c e c a n have a l a r g e i n f l u e n c e on t h e b e h a v i o r o f t h e SQUID, l i k e a h y s t e r e t i c I-V c u r v e (see S e c . I I . 1) and a r e s o n a n c e w i t h t h e i n d u c t o r o f t h e SQUID ( 3 0 ) , which can r e s u l t i n I-V c u r v e s l i k e F i g . I I . 6 . I n Ch. IV c a l c u l a t i o n s o f t h e n o i s e o f a dc SQUID w i t h c a p a c i t o r s a r e p r e s e n t e d .

Fig. II.6

/ I-V curves of a noiseless dc

0 V.

0 0.5 1.0 1.5

V O L T A G E v / l0R 2.0

SQUID with 2IaL/%=1 and B^=7

calculated with the model described in Ch. IV

c o n s t a n t h a l s o quantum e f f e c t s p l a y a r o l e . The J o s e p h s o n f r e q u e n c y can be above t h e w h i t e n o i s e p a r t o f t h e t h e r m a l n o i s e s p e c t r u m ( 3 1 ) a n a l o g t o t h e c a s e o f a s i n g l e j u n c t i o n (Sec. I I . 1 ). There i s e v i d e n c e o f m a c r o s c o p i c quantum p r o c e s s e s , i n w h i c h t h e SQUID can t u n n e l between l o c a l minima i n t h e p o t e n t i a l energy (32)- These e f f e c t s c a n l e a d t o an i n c r e a s e d v o l t a g e n o i s e . I n t h e l i t e r a t u r e t h e r e i s d i s c u s -s i o n ( 3 1 , 3 3 , 3 4 ) about t h e p r e s e n c e o f a quantum l i m i t o f t h e energy r e s o l u t i o n . The l o w e s t measured r e s o l u t i o n o f a dc SQUID i s 0. 5h ( 3 5 ) • Because t h e r e s o l u t i o n o f t h e SQUIDs c o n s i d e r e d i n t h i s t h e s i s i s s t i l l f a r above t h e quantum l i m i t , t h e s i m p l e j u n c t i o n models a r e e x p e c t e d t o g i v e a r e a s o n a b l e e s t i m a t e .

B e s i d e s t h e w h i t e n o i s e d e s c r i b e d above, i n any SQUID a l o w f r e q u e n c y 1 / f n o i s e i s p r e s e n t . For most p r a c t i c a l measurement systems the 1 / f n o i s e i s i m p o r t a n t a t f r e q u e n c i e s below 1 Hz. T h i s 1 / f n o i s e i s a s e r i o u s l i m i t a t i o n i n s i t u a t i o n s i n w h i c h a good l o n g term s t a b i l i t y i s needed. The o r i g i n o f t h e 1 / f n o i s e p r o b a b l y l i e s i n t h e J o s e p h s o n j u n c t i o n s . The n o i s e m i g h t be caused by t e m p e r a t u r e f l u c t u a -t i o n s (3_6). Tesche ( 3 7 ) s t u d i e d t h e SQUID f l u c t u a t i o n s assuming f l u c t u a t i o n o f t h e j u n c t i o n p a r a m e t e r s . Up t o now no s a t i s f a c t o r y

24

t h e o r y i s a v a i l a b l e t o u n d e r s t a n d o r p r e d i c t t h e 1/f n o i s e o f J o s e p h s o n j u n c t i o n d e v i c e s .

References

1. B.D. J o s e p h s o n , Phys . L e t t .1_, 251 (1962) 2. B.D. J o s e p h s o n , Rev. Mod . Phys . 36_, 46 (1964) 3. B.D. J o s e p h s o n , Adv. Phys ._14_, 41 9 (1965)

4 . V. Ambegaokar and A. B a r a t o f f , Phys . Rev. L e t t .J_0,486 ( 1 9 6 3 ) , E r r . Phys. Rev. L e t t . Vl_, 104 ( 1 9 6 3 )

5. P.W. Anderson and A.H. Dayem, Phys .Rev . L e t t .J_3,195 ( 1 9 6 4 ) 6. N.R. Werthamer, Phys • Rev•147,255 (1966)

7. A . I . L a r k i n and Yu.N. O v c h i n n i k o v , Sov. Phys .JETP24_, 1 035 (1967) 8. D.G. Mc D o n a l d , E.G. J o h n s o n , and R.E. H a r r i s , Phys.Rev.BI3,1028

(1976)

9. A.B. Z o r i n and K.K. L i k h a r e v , Sov.J.Low Temp. Phys .3_, 70 (1977) 10. W.A. S c h l u p , J . P h y s . C o l l o q u e C6, 39_,565 ( 1 9 7 8 )

11. D.E. McCumber, J . A p p l . Phys . 39_, 31 1 3 (1968) 12. W.C. S t e w a r t , A p p l . P h y s . L e t t .12,277 ( 1 9 6 8 ) 13- R.F. Voss, J.Low Temp.Phys.42,151 (1981)

14. V. Ambegaokar and B . I . H a l p e r i n , Phys.Rev.Lett.22_, 1364 (1969) 15. J . Kurkijärvi and V. Ambegaokar, P h y s . L e t t . 1 A,314 (1970) 16. K. Y o s h i d a , J . A p p l . Phys . 53_, 7471 (1982)

1 7 . J . Kurkijärvi, Phys.Rev.B6,832 (1972) 18. T.A. F u l t o n , IEEE T r a n s .Mag. Vt_, 749 (1975) 19. C D . Tesche, J.Low Temp. Phys .44, 1 19 (1981 )

20. R.H. Koch, D.J. van H a r l i n g e n , and J . C l a r k e , P h y s • R e v . L e t t . 4 5 , 26 (1980)

2 1 . R.C J a k l e v i c , J . Lambe, A.H. S i l v e r and J . E . M e r c e r e a u , Phys.Rev. Lett-22_,159 (1964)

22. J.E. Zimmerman and A.H. S i l v e r , Phys.Rev.141,367 (1966)

2 3 . A.Th.A.M. de Waele and R. de Bruyn Ouboter, P h y s i c a 42,225 (1969) 24. A.Th.A.M. de Waele and R. de Bruyn O u b o t e r , P h y s i c a 42_,626 (1969) 25. T.A. F u l t o n , L.N. D u n k l e b e r g e r , and R.C. Dynes, Phys.Rev.B6, 855

(1972)

2 6 . C D . Tesche and J . C l a r k e , J.Low Temp. Phys .29_, 301 ( 1977) 27. J . J . P . B r u i n e s , V . J . de Waal, and J . E . Mooij, J.Low Temp. Phys .46,

383 (1982)

28. V. R a d h a k r i s h n a n and V.L. Newhouse , J . A p p l . Phys . 42_, 129 (1971 ) 29. J-H. C l a a s s e n , J . A p p l . Phys . 46_, 2268 ( 1975)

3 0 . S.M. F a r i s and E.A. V a l s a m a k i s , J.Appl.Phys.52,915 (1981) 31. R.H. Koch, D.J. van H a r l i n g e n , and J . C l a r k e , A p p l . P h y s • L e t t . 3 8 ,

380 (1981)

32. W. den Boer and R. de Bruyn O u b o t e r , P h y s i c a 9 8 B , 1 8 5 (1980) 33. R.F. Voss, Appl.Phys.Lett.38_, 182 ( 1981 )

34. C D . Tesche, A p p l . Phys . L e t t .4l_, 490 (1982)

35. D.J. van H a r l i n g e n , R.H. Koch, and J . C l a r k e , A p p l . P h y s . L e t t . 4 1 , 197 (1982)

36. J . C l a r k e and G. Hawkins, IEEE Trans.Magn.MAG-11,841 (1975) 37. C D . Tesche, A p p l . Phys . L e t t .41, 99 (1982)

I I I HIGH PERFORMANCE DC SQUIDS WITH SUBMICRON NIOBIUM JOSEPHSON JUNCTIONS

Abstract

We r e p o r t on t h e f a b r i c a t i o n and p e r f o r m a n c e o f l o w n o i s e a l l -n i o b i u m t h i -n f i l m p l a -n a r dc SQUIDs w i t h s u b m i c r o -n J o s e p h s o -n j u -n c t i o -n s . The j u n c t i o n s a r e e v a p o r a t e d o b l i q u e l y t h r o u g h a m e t a l shadow evapo-r a t i o n mask, w h i c h i s made u s i n g o p t i c a l l i t h o g evapo-r a p h y w i t h 0.5 uevapo-rn t o l e r a n c e . The J o s e p h s o n j u n c t i o n b a r r i e r i s formed by e v a p o r a t i n g a t h i n s i l i c o n f i l m and w i t h a s u b s e q u e n t o x i d a t i o n i n a g l o w d i s c h a r g e . The j u n c t i o n p a r a m e t e r s c a n be r e p r o d u c e d w i t h i n a f a c t o r o f 2. T y p i c a l c r i t i c a l c u r r e n t s o f t h e SQUIDs a r e about 3 uA and t h e r e s i s t a n c e s a r e about 100 fi. W i t h SQUIDs h a v i n g an i n d u c t a n c e o f 1 nH the v o l t a g e m o d u l a t i o n i s a t l e a s t 60 uV. An i n t r i n s i c energy r e s o l u

--32

t i o n o f 4 - 1 0 J/Hz has been r e a c h e d . The SQUIDs a r e c o u p l e d t o w i r e wound i n p u t c o i l s o r t o t h i n f i l m i n p u t c o i l s . The t h i n f i l m i n p u t c o i l c o n s i s t s o f a n i o b i u m s p i r a l o f 20 t u r n s on a s e p a r a t e s u b s t r a t e . In b o t h c a s e s t h e c o i l i s g l u e d onto a 2 nH SQUID w i t h a c o u p l i n g e f f i c i e n c y o f a t l e a s t 0.5. R e f e r r e d t o t h e t h i n f i l m i n p u t c o i l t h e - 3 0 b e s t c o u p l e d energy r e s o l u t i o n a c h i e v e d i s 1.2*10 J/Hz measured i n a f l u x l o c k e d l o o p a t f r e q u e n c i e s above 10 Hz. As f a r a s we know t h i s i s t h e b e s t f i g u r e a c h i e v e d w i t h an a l l r e f r a c t o r y m e t a l t h i n f i l m SQUID. The f a b r i c a t i o n t e c h n i q u e used i s s u i t e d f o r making c i r c u i t s w i t h SQUID and p i c k - u p c o i l on t h e same s u b s t r a t e . We d e s c r i b e a compact p l a n a r f i r s t o r d e r g r a d i o m e t e r i n t e g r a t e d w i t h a SQUID on a

-12 -1 s i n g l e s u b s t r a t e . The g r a d i e n t n o i s e o f t h i s d e v i c e i s 3*10 T*m The g r a d i o m e t e r has a s i z e o f 12 mm *17 mm, i s s i m p l e t o f a b r i c a t e and i s s u i t a b l e f o r b i o m e d i c a l a p p l i c a t i o n s .

28

III.1 Introduction

The l a s t decade SQUIDs ( s u p e r c o n d u c t i n g quantum i n t e r f e r e n c e d e v i c e s ) have become w i d e l y used m e a s u r i n g i n s t r u m e n t s f o r s m a l l m a g n e t i c f i e l d s and many o t h e r k i n d s o f s m a l l s i g n a l s . In 1964 J a k l e v i c e t a l . (1) c o n s t r u c t e d the f i r s t dc SQUID c o n s i s t i n g o f a s u p e r c o n d u c t i n g r i n g w i t h two J o s e p h s o n j u n c t i o n s . Nowadays t h e most o f t e n used t y p e i s t h e r f SQUID, w h i c h c o n s i s t s o f a s u p e r c o n d u c t i n g r i n g c o n t a i n i n g one J o s e p h s o n j u n c t i o n , g e n e r a l l y a p o i n t c o n t a c t . The r f SQUIDs ( 2 ) , b i a s e d w i t h a f r e q u e n c y o f 20 MHz o r l a r g e r , have an

28

energy r e s o l u t i o n o f 10 J/Hz. To o b t a i n a b e t t e r r e s o l u t i o n w i t h t h i s system i t i s n e c e s s a r y t o use h i g h e r f r e q u e n c i e s w i t h more c o m p l i c a t e d e l e c t r o n i c s . The l a s t few y e a r s r e s e a r c h on dc SQUIDs has been r e v i t a l i z e d by t h e work o f C l a r k e , Goubau and Ketchen ( 3 ) . The n o i s e o f t h i s t y p e i s l i m i t e d by t h e c a p a c i t a n c e o f t h e j u n c t i o n s ( 4 ) . A l t h o u g h f o r r e s e a r c h p o i n t c o n t a c t s have been used ( 5 ) , f o r optimum r e l i a b i l i t y i t i s a d v a n t a g e o u s t o use t h i n f i l m j u n c t i o n s . A f i r s t s t e p towards a r e l i a b l e low n o i s e dc SQUID was t h e c y l i n d r i c a l t h i n f i l m n i o b i u m - l e a d dc SQUID o f C l a r k e , Goubau and K e t c h e n (3) w i t h a w i r e wound i n p u t c o i l . An improvement can be r e a c h e d w i t h SQUIDs on a f l a t s u b s t r a t e w i t h a s p i r a l i n p u t c o i l . A f l a t s u b s t r a t e f a c i l i t a t e s t h e use o f s t a n d a r d t h i n f i l m t e c h n i q u e s f o r p r o d u c i n g u l t r a s m a l l j u n c t i o n s w i t h a h i g h I„ R p r o d u c t . J a y c o x and K e t c h e n (6,7) and Cromar and C a r e l l i (8) made s u c h low n o i s e dc SQUIDs w i t h c o u p l i n g c o i l u s i n g l e a d a l l o y J o s e p h s o n j u n c t i o n s . For p r a c t i c a l use a d i s a d v a n t a g e o f l e a d a l l o y i s t h e s e n s i t i v i t y t o t h e r m a l shock and t h e poor c h e m i c a l r e s i s t a n c e . In t h i s r e s p e c t r e f r a c t o r y m e t a l j u n c t i o n s a r e more f a v o u r a b l e . S e v e r a l a u t h o r s r e p o r t e d J o s e p h s o n j u n c t i o n s o f n i o b i u m s u i t e d f o r t h i s a p p l i c a t i o n ( 9 , 1 0 ) . T h i s paper d e a l s w i t h a low n o i s e n i o b i u m dc SQUID w i t h w i r e wound as w e l l as t h i n f i l m i n p u t c o i l s . A d i f f e r e n t a p p r o a c h f o r a p r a c t i c a l d e v i c e i s the f a b r i c a t i o n o f s y s t e m s w i t h SQUID and p i c k u p c o i l on a s i n g l e s u b s t r a t e . Advantages a r e t h e compactness and the p o s s i b i l i t y o f p r e c i s e l y b a l a n c i n g g r a d i o m e t e r s . Ketchen e t a l . (11) r e p o r t e d a f i r s t o r d e r n i o b i u m l e a d g r a d i o m e t e r w i t h a SQUID on one f l a t s u b s t r a t e . Here we d e s c r i b e a

compact f i r s t o r d e r g r a d i o m e t e r d e s i g n e d t o be an i n t e g r a l p a r t o f t h e SQUID i t s e l f . S e c . I I I . 2 e x p l a i n s t h e d e s i g n c r i t e r i a o f t h i n f i l m SQUID c i r c u i t s . We a r g u e , t h a t w i t h s m a l l j u n c t i o n s a low n o i s e SQUID can be made w i t h o u t t h e n e c e s s i t y t o r e d u c e t h e SQUID i n d u c t a n c e t o v e r y s m a l l v a l u e s . S e c . I I I . 3 c o n t a i n s t h e f a b r i c a t i o n method o f t h e J o s e p h s o n j u n c t i o n s and t h e t h i n f i l m c o i l s . The p r o p e r t i e s o f t h e j u n c t i o n s a r e d e a l t w i t h i n S e c . I I I . 4 . We d e s c r i b e t h e n o i s e p e r f o r -mance o f t h e SQUIDs and t h e p r o p e r t i e s o f t h e c o i l s i n Sec. I I I . 5 . S e c . I I I . 6 c o n t a i n s t h e e x p e r i m e n t a l r e s u l t s w i t h t h e g r a d i o m e t e r s . I n S e c . I I I . 7 we d i s c u s s p o s s i b l e improvements and g i v e a summary. P a r t o f t h i s work (12-15) was r e p o r t e d on b e f o r e .

I I I .2 Design Considerations

For magnetometer and g r a d i o m e t e r a p p l i c a t i o n s t h e i m p o r t a n t f i g u r e o f m e r i t o f a SQUID i s t h e e n e r g y r e s o l u t i o n (J_6_), w h i c h i s d e f i n e d by

E = S£0)/(2Lk2) ( I I I . 1 )

where S|0) i s t h e r.m.s. f l u x n o i s e o f t h e SQUID, L i s t h e SQUID i n d u c t a n c e and k i s t h e c o u p l i n g c o e f f i c i e n t between SQUID and i n p u t c o i l . The c o u p l i n g c o e f f i c i e n t i s d e f i n e d by

( I I I . 2 )

where L i s t h e i n d u c t a n c e o f t h e i n p u t c o i l and M i s t h e m u t u a l c

i n d u c t a n c e . T h e o r e t i c a l c a l c u l a t i o n s o f t h e energy r e s o l u t i o n o f dc SQUIDs have been p e r f o r m e d by Tesche and C l a r k e (4_) . Assuming a r e s i s t i v e l y s h u n t e d j u n c t i o n model w i t h t h e r m a l n o i s e g e n e r a t e d i n t h e shunt r e s i s t o r , t h e y p r e d i c t an optimum energy r e s o l u t i o n o f

30

i f t h e parameter 8 = 21, L/$0 has t h e optimum v a l u e

Z\ L/% = 1 ( I I I . 4 )

where % i s t h e f l u x quantum. The n u m e r i c a l f a c t o r i n Eq. I I I .3 was c o r r e c t e d a c c o r d i n g t o B r u i n e s e t a l . (J_J_) . I f t h e SQUID i s o p e r a t e d i n a f l u x l o c k e d l o o p , t h e energy r e s o l u t i o n i s s l i g h t l y d e t e r i o r a t e d . For t h e t h e o r e t i c a l model w i t h a s i n e wave m o d u l a t i o n s i g n a l t h i s amounts t o a f a c t o r o f 1.6 ( 4 ) . In t h i s model t h e p a r a l l e l c a p a c i t a n c e o f t h e j u n c t i o n s i s n e g l e c t e d . R e a l j u n c t i o n s a l w a y s have a c a p a c i -t a n c e , w h i c h c a n c a u s e h y s -t e r e s i s i n -t h e I-V c u r v e o f -t h e j u n c -t i o n s . To make t h e j u n c t i o n s n o n - h y s t e r e t i c t h e McCumber parameter ( 1 8 )

2

Bc= 2 i t l0 R Cl% must be

211a R2C 7 *0 £ 1 ( I I I . 5 )

Tesche and C l a r k e a r g u e t h a t t h e e n e r g y r e s o l u t i o n i s optimum i f (5^ i s about 1. S u b s t i t u t i n g t h i s i n t o Eq. I I I .3 y i e l d s

E s; l 6 kDT V ( L C ) 7 k2 ( I I I . 6 )

Eq. I I I . 6 shows, t h a t SQUIDs w i t h a low e n e r g y r e s o l u t i o n r e q u i r e a s m a l l SQUID i n d u c t a n c e o r a s m a l l j u n c t i o n c a p a c i t a n c e . S m a l l i n d u c -t a n c e s make i -t d i f f i c u l -t -t o c o u p l e f l u x e f f i c i e n -t l y i n -t o -t h e SQUID. However, t h e c a p a c i t a n c e o f t h e J o s e p h s o n j u n c t i o n s c a n be made s m a l l w i t h o u t s i m i l a r p r o b l e m s . F o r i n s t a n c e a SQUID i n a f l u x l o c k e d l o o p

-14

w i t h L = 1 nH and C = 1 » 1 0 F s h o u l d have an energy r e s o l u t i o n o f - 3 3

4.6*10 J/Hz, i f t h e c r i t i c a l c u r r e n t and t h e r e s i s t a n c e have t h e optimum v a l u e s o f 1 uA and 180 Q.

Niobium i s s u i t a b l e as m a t e r i a l f o r t h e J o s e p h s o n j u n c t i o n s , because o f i t s h i g h c r i t i c a l t e m p e r a t u r e , t h e l a r g e I j R p r o d u c t t h a t can be r e a c h e d and t h e l o n g term s t a b i l i t y o f t h e p a r a m e t e r s o f n i o b i u m t u n n e l j u n c t i o n s . A d i s a d v a n t a g e o f n i o b i u m j u n c t i o n s i s t h e l a r g e r e l a t i v e d i e l e c t r i c c o n s t a n t o f i t s o x i d e , w h i c h i s 30 ( 1 9 ) . The

_2 s p e c i f i c c a p a c i t a n c e o f n i o b i u m o x i d e j u n c t i o n s i s 0 . 1 3 F*m . An

o x i d e j u n c t i o n o f t h i s m a t e r i a l w i t h C=1»10 F must have an a r e a 2

s m a l l e r t h a n 0.1 vim . The j u n c t i o n s i n t r o d u c e d by Daalmans (9) have s u c h a v e r y s m a l l c a p a c i t a n c e . He f i r s t e v a p o r a t e d a s i l i c o n f i l m and o x i d i z e d t h e f i l m a f t e r w a r d s . The o x i d a t i o n t i m e d e t e r m i n e s t h e j u n c t i o n p a r a m e t e r s . The c a p a c i t a n c e o f t h e s e j u n c t i o n s i s s m a l l e r because the s i l i c o n t h i c k n e s s can be 10 t i m e s as l a r g e as t h e o x i d e t h i c k n e s s and because t h e d i e l e c t r i c c o n s t a n t i s 3 t i m e s as s m a l l as t h a t o f t h e n i o b i u m o x i d e . The j u n c t i o n s a r e s u i t a b l e f o r our a p p l i -c a t i o n f o r two r e a s o n s . The s m a l l j u n -c t i o n -c a p a -c i t a n -c e p r o v i d e s a good

-13 2 energy r e s o l u t i o n . Moreover, j u n c t i o n s w i t h an a r e a o f 2"10 m t u r n o u t t o have a l m o s t t h e r i g h t I,, R p r o d u c t f o r n o n - h y s t e r e t i c o p e r a t i o n w i t h o u t t h e use o f an e x t e r n a l s h u n t . The c a p a c i t a n c e o f t h e s e j u n c t i o n s i s so s m a l l t h a t a l s o t h e p a r a s i t i c c a p a c i t a n c e o f t h e l e a d s t o t h e j u n c t i o n s can be a l a r g e p a r t o f t h e t o t a l j u n c t i o n c a p a c i -t a n c e . In Appendix A -t h i s c a p a c i -t a n c e i s shown -t o be a -t l e a s -t -15 5*10 F per j u n c t i o n , even i f t h e w i d t h o f t h e s t r i p s c o n n e c t e d t o the j u n c t i o n i s r e d u c e d t o 10 um.

For most SQUID a p p l i c a t i o n s an i n p u t c o i l w i t h an i n d u c t a n c e n e a r 1 uH i s needed. This c o i l must be c o u p l e d t o the SQUID as t i g h t l y as

2

p o s s i b l e . G e n e r a l l y a c o u p l i n g c o e f f i c i e n t k l a r g e r t h a n 0.5 i s r e a c h e d w i t h w i r e wound c o i l s . For t h i n f i l m SQUIDs, w h i c h a r e s m a l l e r t h a n t h e c o n v e n t i o n a l SQUIDs made out o f b u l k n i o b i u m , a s m a l l t h i n f i l m c o i l i s more s u i t a b l e . W i t h a c o i l , u s i n g l i n e w i d t h s o f 10 um or s m a l l e r , one can make many t u r n s on a s m a l l a r e a . P r o m i s i n g r e s u l t s were r e p o r t e d by J a y c o x and Ketchen (6_,7_) and by Cromar and C a r e l l i ( 8 ) . The c i r c u i t d e s i g n e d by J a y c o x and Ketchen c o n s i s t s o f a 100 pH SQUID r i n g w i t h a l a r g e o u t e r d i a m e t e r compared t o t h e i n n e r d i a m e t e r . On t h i s r i n g a s p i r a l i n p u t c o i l o f 10 t o 100 t u r n s was d e p o s i t e d . They showed a c o u p l i n g e f f i c i e n c y o f 0.8 t o 0.9. Because we have s e n s i t i v e SQUIDs w i t h an i n d u c t a n c e o f 1 nH, t h e s i z e s w h i c h we can a l l o w a r e 10 t i m e s as l a r g e as t h e s i z e s o f t h e 100 pH s y s t e m . T h e r e f o r e i t i s not n e c e s s a r y t o put t h e SQUID and the c o i l on the same s u b s t r a t e . T h i s e n a b l e s us even, i f necessary, t o use w i r e wound c o i l s . In p r a c t i c e , p a r t o f the SQUID l o o p c o n t a i n i n g t h e Josephson j u n c t i o n s i s o u t s i d e t h e i n p u t c o i l . Then t h e e l e c t r i c c i r c u i t i s

3 2 d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s ( 2 0 ) : M n ( L - L . ) n (L-L.)+L J s ( 1 - L . / L W I + L / n2( L - L . ) ) ( I I I . 7 ) L, ( I I I . 8 ) i -1 ( I I I .9 ) J

where i s t h e i n d u c t a n c e o f t h e p a r t o f t h e SQUID r i n g near t h e j u n c t i o n s w h i c h i s n o t c o u p l e d t o t h e c o i l , n i s t h e number o f t u r n s and L i s t h e i n d u c t a n c e a s s o c i a t e d w i t h t h e c o i l i f i t would be above

s

a s u p e r c o n d u c t i n g g r o u n d p l a n e i n s t e a d o f the SQUID r i n g . For a c o i l v e r y near t h e SQUID the i n d u c t a n c e i s the i n d u c t a n c e o f a s t r i p l i n e w i t h t h e same l e n g t h as the i n p u t c o i l . T h i s i n d u c t a n c e can be c a l c u l a t e d w i t h t h e a n a l y t i c a l e x p r e s s i o n o f Chang ( 2 1 ) . The SQUID i n d u c t a n c e we use i s about 2 nH. The i n d u c t a n c e o f a wide t h i n s u p e r c o n d u c t i n g s q u a r e r i n g was c a l c u l a t e d by J a y c o x and K e t c h e n ( 6 ) . With t h e a i d o f t h e i r r e s u l t we e s t i m a t e t h e i n d u c t a n c e o f our c i r c u l a r SQUID w i t h I.D. 1.4 mm and O.D. 3.4 mm t o be 1.9 nH. I f we put o n t o t h i s SQUID a 20 t u r n c i r c u l a r c o i l w i t h a mean d i a m e t e r o f 2.4 mm and a l i n e w i d t h o f 10 pm a t a d i s t a n c e o f 10 um, w h i c h seems r e a s o n a b l e f o r two s u b s t r a t e s g l u e d t o g e t h e r , we g e t a m u t u a l i n d u c t a n c e o f 38 nH, a s t r i p l i n e i n d u c t a n c e o f 70 nH and an i n p u t i n d u c -t a n c e o f 0.8 uH. So -t h e c o u p l i n g l o s s due -t o -t h e d i s -t a n c e be-tween -the s u b s t r a t e s a c c o r d i n g t o Eqs. I I I . 7 - 9 i s o n l y 9%.

We a l s o used w i r e wound c o i l s , because t h e y a r e s i m p l e r t o c o n s t r u c t t h a n t h i n f i l m c o i l s . U s u a l l y i t i s d i f f i c u l t t o c o u p l e e f f i c i e n t l y t o s u c h c o i l s , because t h e t h i c k n e s s o f t h e w i r e i s l a r g e compared t o t h e d i m e n s i o n s o f t h e SQUID. T h i s p r o b l e m can be s o l v e d i n t h e f o l l o w i n g way. A s u p e r c o n d u c t i n g c o r e f o r the c o i l can be used t o c o n c e n t r a t e t h e m a g n e t i c f l u x i n s i d e the SQUID r i n g . Two h a l f c y l i n -d e r s o f n i o b i u m a r e g l u e -d t o g e t h e r e l e c t r i c a l l y i s o l a t e -d by a 10 um t h i c k p o l y e s t e r f o i l . Around t h i s c y l i n d e r the s u p e r c o n d u c t i n g w i r e i s wound. F i g . I I I . 1 shows t h e c o n f i g u r a t i o n . Because the d i s t a n c e between t h e two c y l i n d e r h a l v e s i s s m a l l , the f l u x w i l l c o n c e n t r a t e t h r o u g h the c y l i n d e r . T h i s c y l i n d e r i s g l u e d o n t o a c i r c u l a r t h i n f i l m

Fig.

, L _ _ I Construction of an input

coil consisting of wire wound around a split niobium core

SQUID. T h i s n i o b i u m c y l i n d e r does n o t s h o r t t h e SQUID l o o p , b u t o n l y r e d u c e s t h e i n d u c t a n c e o f t h e SQUID. The c o u p l i n g c o n s t a n t o f t h i s c o n f i g u r a t i o n depends on t h e d i s t a n c e between t h e f i l m and t h e c y l i n d e r , t h e d i s t a n c e between t h e two h a l v e s o f t h e c y l i n d e r and t h e t h i c k n e s s o f t h e i n s u l a t i o n o f t h e w i r e u s e d . The l a r g e s t l o s s i s produced by t h e l e a k a g e between t h e w i r e and t h e c y l i n d e r . T h i s c o n t r i b u t i o n can be e s t i m a t e d a n a l o g o u s t o t h e s p i r a l i n p u t c o i l . The SQUID i s used as d e s c r i b e d above, i n p u t i n d u c t a n c e s o f t h e o r d e r o f s e v e r a l hundreds o f nanohenrys c a n be r e a c h e d w i t h a 4 mm l o n g c y l i n d e r .

Another method o f c o u p l i n g s i g n a l s i n t o t h e SQUID i s t h e i n t e g r a -t i o n o f -t h e SQUID and -t h e p i c k - u p l o o p , f o r i n s -t a n c e a g r a d i o m e -t e r , on a s i n g l e s u b s t r a t e . Such a d e s i g n has t h e a d v a n t a g e , t h a t t h e c o m p l e t e s u p e r c o n d u c t i n g p a r t o f t h e system i s c o n c e n t r a t e d on a s m a l l c h i p . Then t h e space consuming s u p e r c o n d u c t i n g w i r e s , c o n n e c t i o n s and s c r e e n i n g s can be e l i m i n a t e d . T h i s d e s i g n was f i r s t used by K e t c h e n , C l a r k e , Goubau and Donaldson (JJ_). They d e s i g n e d a l a r g e p i c k - u p l o o p . P a r t o f t h e SQUID l o o p formed a p a r t o f t h e p i c k - u p l o o p . Because t h e i n d u c t a n c e o f t h e SQUID l o o p i s s m a l l e r t h a n t h e i n d u c t a n c e o f t h e p i c k - u p l o o p , a c o u p l i n g l o s s must be a c c e p t e d . The s e n s i t i v i t y o f

34

t h e s e g r a d i o m e t e r s depends on t h e i r s i z e , t h e geometry o f t h e g r a d i o -meter and t h e r e s o l u t i o n o f t h e SQUID. I f a v e r y s e n s i t i v e SQUID i s u s e d , i t i s p o s s i b l e t o make a s m a l l g r a d i o m e t e r w h i c h i s s e n s i t i v e enough f o r b i o m e d i c a l a p p l i c a t i o n s . The d e s i g n m i g h t a l s o be u s e f u l i n an a r r a y o f SQUIDs.

Fig. III.2

Equivalent circuit of a SQUID directly coupled to the pick-up coil

F i g . I I I . 2 shows t h e e l e c t r i c c i r c u i t o f t h e d e v i c e c o n s i d e r e d , i s t h e i n d u c t a n c e o f t h e p i c k - u p l o o p and i s much l a r g e r t h a n t h e SQUID i n d u c t a n c e Lg. We n e g l e c t t h e i r m u t u a l i n d u c t a n c e . We n e g l e c t

a l s o t h e p a r t o f t h e s i g n a l w h i c h i s sensed by t h e i n d u c t o r . Then i t f o l l o w s f o r t h e f l u x n o i s e S^O) i n t h e p i c k - u p l o o p from Eq. I I I . 6

S(0) = 3 2 kDT Ln 2L ~JC5 ( I I I . 1 0 )

p B i s

T h i s r e s u l t i m p l i e s , t h a t Lg s h o u l d be made as l a r g e as p o s s i b l e .

However, f o r l a r g e v a l u e s o f Eqs. I I I . 6 and I I I . 1 0 a r e no l o n g e r v a l i d . Then t h e t h e r m a l n o i s e o f t h e f l u x i n t h e SQUID r i n g w i l l smear out t h e f l u x dependence o f t h e SQUID ( 2 2 ) . T h i s w i l l become a s e r i o u s problem i f

Ls K $ „2/ 4 kBT ( I I I . 1 1 )

For t h i s c o n f i g u r a t i o n t h e r e a r e no c a l c u l a t i o n s a v a i l a b l e i n t h e l i t e r a t u r e t o f i n d t h e optimum SQUID i n d u c t a n c e . The optimum i n d u c -t a n c e f o r a SQUID o p e r a -t e d a -t 4.2 K w i l l p r o b a b l y be near 2 nH.

We d e s i g n e d a f i r s t o r d e r g r a d i o m e t e r w i t h t h e two l o o p s o f t h e g r a d i o m e t e r i n p a r a l l e l . I t c o n s i s t s o f two j u n c t i o n s i n s e r i e s c o n n e c t e d t o s e v e r a l r i n g s i n p a r a l l e l . F i g . I I I . 3 shows p h o t o g r a p h s o f t h e g r a d i o m e t e r and t h e p a r t o f i t near t h e j u n c t i o n s . The i n d u c -t a n c e seen by -t h e j u n c -t i o n s i s d e -t e r m i n e d m a i n l y by -t h e i n d u c -t a n c e s o f

t h e s m a l l e s t r i n g s . The s i g n a l i s sensed m a i n l y by t h e l a r g e l o o p s . I n the d e s i g n we a v o i d l a r g e a r e a s f i l l e d up w i t h s u p e r c o n d u c t i n g m a t e r i a l t o reduce t h e movement o f f l u x p e n e t r a t i n g t h e f i l m . The SQUID can be b i a s e d by a c o n t a c t i n s i d e t h e r i n g and one a t t h e o u t e r l o o p . The f a b r i c a t i o n o f t h e d e v i c e i s s i m p l e , because t h e r e a r e no c r o s s i n g s o f l i n e s . The c o n t a c t pads demand a minimum s i z e o f t h e l o o p o f 1 mm, w h i c h l i m i t s t h e SQUID i n d u c t a n c e t o about 2 nH o r l a r g e r . Near t h e m i d d l e o f t h e g r a d i o m e t e r t h e w i d t h o f t h e g r a d i o m e t e r i s r e d u c e d . T h i s r e d u c e s t h e i n d u c t a n c e , w h i l e t h e s e n s i t i v i t y i n t h i s

36

r e g i o n i s not so i m p o r t a n t . T h i s was c a l c u l a t e d by Pegrum and Donaldson ( 2 3 ) . Advantages o f the p r e s e n t d e s i g n are the l a r g e r s e n s i t v i t y o f a p a r a l l e l g r a d i o m e t e r and the s i m p l i c i t y o f t h e f a b r i c a t i o n . I n p r i n c i p l e t h e d i m e n s i o n s can be c o n t r o l l e d w i t h an a c c u r a c y o f t h e o r d e r o f 1 \im. I f one uses an o p t i c a l l y f l a t sub-s t r a t e , the d e v i a t i o n p e r p e n d i c u l a r t o t h e sub-s u r f a c e c a n be o f t h e sub-same o r d e r . So t h e b a l a n c e o f t h i n f i l m g r a d i o m e t e r s o f 10 mm s i z e s can be 100 ppm. A d i s a d v a n t a g e o f t h e p a r a l l e l c i r c u i t i s t h e c l o s e d s u p e r -c o n d u -c t i n g l o o p . I f t h e g r a d i o m e t e r i s moved i n a m a g n e t i -c f i e l d , or i f t h e m a g n e t i c f i e l d changes, l a r g e c u r r e n t s can f l o w i n t h e s u p e r -c o n d u -c t i n g s t r i p s and p o s s i b l y d r i v e t h e f i l m n o r m a l . T h i s e f f e -c t l i m i t s the use o f t h e s e g r a d i o m e t e r s t o a p p l i c a t i o n s w i t h r e l a t i v e l y s m a l l changes o f the m a g n e t i c f i e l d s and g r a d i e n t s . The i n d u c t a n c e seen by the j u n c t i o n s i s an i m p o r t a n t parameter i n t h e d e s i g n o f t h e c i r c u i t . The c a l c u l a t i o n o f t h e i n d u c t i o n i s d e s c r i b e d i n Appendix A. Our g r a d i o m e t e r was d e s i g n e d t o have an i n d u c t a n c e o f 2.7 nH.

I I I .3 Fabrication

I f one wants t o t a k e f u l l advantage o f t h e h i g h c r i t i c a l tempera-t u r e o f n i o b i u m , i tempera-t i s n e c e s s a r y tempera-t o make good q u a l i tempera-t y n i o b i u m . I n an o r d i n a r y h i g h vacuum system t h i s i s done by h e a t i n g t h e s u b s t r a t e t o 400 °C. I t i s d i f f i c u l t t o combine t h i s h e a t i n g w i t h a l i t h o g r a p h i c p r o c e d u r e f o r m i n i a t u r i z a t i o n , because r e s i s t s can n o t t o l e r a t e t h e s e t e m p e r a t u r e s . Daalmans and Z w i e r (24) d e v e l o p e d a method t o p a t t e r n s u b m i c r o n n i o b i u m J o s e p h s o n j u n c t i o n s w i t h t h i n f i l m m e t a l o f f s e t masks, g e n e r a t e d w i t h e l e c t r o n beam l i t h o g r a p h y . These masks can w i t h s t a n d t e m p e r a t u r e s l a r g e r t h a n 300 °C. The j u n c t i o n s a r e e v a p o r a t e d o b l i q u e l y . The c o m p l e t e p a t t e r n w i t h t h e t u n n e l j u n c t i o n s i s f a b r i c a t e d i n one e v a p o r a t i o n r u n and one l i t h o g r a p h i c s t e p . They f a b r i -c a t e d t h e masks, w h i -c h -c o n s i s t o f -chromium and n i o b i u m , w i t h e l e -c t r o n l i t h o g r a p h y . However, t h e l i n e w i d t h o f 1 um needed f o r t h i s p r o c e s s can a l s o be r e a c h e d w i t h p h o t o l i t h o g r a p h y . Because o f t h e g r e a t e r f l e x i b i l i t y o f our p h o t o l i t h o g r a p h i c equipment we p r e f e r t o make t h e

m e t a l masks w i t h p h o t o r e s i s t .

The f i r s t s t e p o f t h e f a b r i c a t i o n i s t o make a c o n t a c t mask o f t h e d e s i g n e d SQUID. T h i s mask c o n s i s t s o f a t h i n f l e x i b l e g l a s s s u b s t r a t e (50 mm '50 mm "0.2 mm) w i t h a chromium f i l m o f 80 nm and a s i l i c o n f i l m o f 10nm. On t h i s s u b s t r a t e a f i l m o f p h o t o r e s i s t AZ 1350 i s spun. The p a t t e r n i s p r o j e c t e d on t h e p h o t o r e s i s t w i t h an o p t i c a l p r o j e c t i o n system c o n t a i n i n g a m i c r o s c o p e o b j e c t i v e . The s i l i c o n f i l m s e r v e s as an a n t i r e f l e c t i o n l a y e r on t h e chromium ( 2 5 ) , t o p r e v e n t s t a n d i n g wave e f f e c t s i n t h e r e s i s t f i l m . A f t e r w a r d s t h i s p a t t e r n i s d e v e l o p e d , t h e s i l i c o n i s p l a s m a e t c h e d and t h e chromium i s c h e m i c a l l y e t c h e d . T h i s method a l l o w s p a t t e r n i n g masks w i t h 0.5 vim r e s o l u t i o n .

The shadow e v a p o r a t i o n mask i s made on a s i l i c o n s u b s t r a t e . F i r s t a 0.55 um chromium f i l m i s d e p o s i t e d . A l a y e r o f p h o t o r e s i s t AZ 1350 i s spun o n t o t h e s u b s t r a t e . The c o n t a c t mask i s t i g h t l y p r e s s e d o n t o t h e s i l i c o n s u b s t r a t e by e v a c u a t i n g t h e space between t h e mask and t h e s u b s t r a t e . The sample i s exposed t h r o u g h t h e c o n t a c t mask t o a p a r a l l e l beam from a mercury lamp. A f t e r d e v e l o p i n g t h e s u c c e s f u l p r i n t s a r e s e l e c t e d . The y i e l d i s more t h a n 70%. In t h e e v a p o r a t o r t h e s u b s t r a t e s a r e c l e a n e d w i t h a glow d i s c h a r g e and a 0.1 um n i o b i u m f i l m i s d e p o s i t e d . By d i s s o l v i n g t h e p h o t o r e s i s t w i t h a c e t o n e a l i f t - o f f o f the n i o b i u m i s p e r f o r m e d . The chromium i s c h e m i c a l l y e t c h e d w i t h a s o l u t i o n o f ammonium c e r i u m n i t r a t e . The e t c h i s s t o p p e d when i t has passed 0.5 um below t h e edge o f t h e n i o b i u m f i l m . At p l a c e s where t h e n i o b i u m i s n a r r o w e r t h a n 1 um a f r e e h a n g i n g n i o b i u m b r i d g e i s f o r m e d . T h i s s t r u c t u r e s e r v e s as t h e shadow e v a p o r a t i o n mask f o r t h e j u n c t i o n s and t h e e n t i r e SQUID.

The j u n c t i o n s a r e e v a p o r a t e d i n a h i g h vacuum s y s t e m w i t h a 10 kW e l e c t r o n gun. Between the d i f f e r e n t e v a p o r a t i o n s t e p s the vacuum system i s not opened. F i g . I I I . 4 shows a s c h e m a t i c o f t h e c o n f i g u r a -t i o n . The n i o b i u m i s e v a p o r a -t e d a -t a r a -t e o f 10 nm/s. The p r e s s u r e

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d u r i n g e v a p o r a t i o n i s 5*10 Pa. D u r i n g t h e f i r s t s t a g e t h e s u b s t r a t e i s h e l d a t an a n g l e o f 45 d e g r e e s and h e a t e d t o 250 °C. T h i s f i r s t n i o b i u m f i l m i s made 200 nm t h i c k . Then t h e s u b s t r a t e i s r o t a t e d f o r e v a p o r a t i o n o f the n e x t f i l m s under 45 d e g r e e s from the o p p o s i t e s i d e .