LOW NOISE SQUIDS

LOW NOISE SQUIDS

proefschrift

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

Victor Jan de Waal

natuurkundig ingenieur

geboren te Amsterdam

Dit proefschrift is goedgekeurd door de promotor Prof.dr.ir. J.E. Mooij

Allen die aan het werk beschreven in dit proefschrift hebben bijge-

dragen, bedank ik hartelijk.

Het onderzoek is verricht in samenwerking met Dr-Ir. T.M. Klapwijk,

Ir. P. van den Hamer, Ir. A . LLurba, G.J. van Nieuwenhuyzen,

P. Sohrijner en Ir. J.J.P. Bruines.

De Technische Hogeschool Delft heeft faciliteiten ter beschikking

gesteld.

De Stichting voor Fundamenteel Onderzoek der Materie heeft het projekt

financieel ondersteund.

CONTENTS

page

. . . . . . . . . . . . . . . . . . . . . I GENERAL INTRODUCTION 9

References . . . . . . . . . . . . . . . . . . . . . . . . 12

I1 THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . 15

11.1 The Josephson E f f e c t . . . . . . . . . . . . . . . . . . 15

11.2 Fundamentals o f t h e DC SQUID . . . . . . . . . . . . . . 20

Refe rences . . . . . . . . . . . . . . . . . . . . . . . 24

I11 H I G H PERFORMANCE DC SQUIDS WITH SUBMICRON NIOBIUM JOSEPHSON

JUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 27

A b s t r a c t . . . . . . . . . . . . . . . . . . . . . . . 27

111.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . 28

111.2 Design 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 . . . . . . . . . . . . . . . 43

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

111.5.2 Performance o f t h e SQUID . . . . . . . . . . . . 46

111.5.2 Performance o f t h e coup led SQUIDS . . . . . . . 49

111.6 Gradiometer performance . . . . . . . . . . . . . . . 5 3

111.7 Conclus ion . . . . . . . . . . . . . . . . . . . . . . . 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 . 5 6

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

Refe rences . . . . . . . . . . . . . . . . . . . . . . . . . . 58

I V SIMULATION AND OPTIMIZATION OF A DC SQUID WITH FINITE

CAPACITANCE . . . . . . . . . . . . . . . . . . . . . . . . . 61

A b s t r a c t . . . . . . . . . . . . . . . . . . . . . . . . 61

IV.l I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . 62

I . 2 The d c SQUID model . . . . . . . . . . . . . . . . . . . 6 3

IV.3 The o p t i m i z a t i o n method . . . . . . . . . . . . . . . . 66

I V . 4 Implementa t ion on a h y b r i d computer . . . . . . . . . . 68

IV.5 Noise and o p t i m i z a t i o n o f t h e SQUID . . . . . . . . . . 71

. . . . . . . . . . . . . . . . . . . . . . . IV.6 Discussion 79

References . . . . . . . . . . . . . . . . . . . . . . . . 81

. . . . . . . . . . . . . . . . . . . . . . . . . . V CONCLUSION 83

References . . . . . . . . . . . . . . . . . . . . . . . 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . Samenvatting 88

. . . . . . . . . . . . . . . . . . . . . . . . . Curriculum Vitae 91

I GENERAL INTRODUCTION

This thesis deals with the design, fabrication, and limitations of

very sensitive SQUID (Superconducting Quantum Interference Device)

magnetometers. The SQUID magnetometer is based on the Josephson

effect. In 1962 B.D. Josephson ( 1 ) predicted that a supercurrent can - flow between two superconductors separated by a very thin insulating

barrier. He showed that the maximum supercurrent, the critical

current, flowing through the junction depends on the magnetic field

inside the junction. The single Josephson junction is not a very

sensitive magnetometer. Jaklevic, Lambe, Silver, and Mercereau (2) first constructed the dc SQUID which is a ring out of superconducting

metal interrupted by two Josephson junctions. The critical current of

this device depends on the magnetic flux enclosed by the super-

conducting ring. The critical current is a periodic function of the

magnetic flux with a period of one flux quantum a, =h/Ze. As the flux

quantum 2.07.10-'~ ~ . m ~ within a typical area of 1 mm2 corresponds to

the very small magnetic field of 2 nT, the SQUID is a very sensitive

instrument for measuring magnetic fields.

Forgacs and Warnick (3) first made a dc SQUID for measurement - purposes. As Josephson junctions they used point contacts, which

consist of a niobium rod with a sharp point pressed onto a flat

niobium plate. However, reproducible fabrication of the two point

contacts was difficult and the reliability was not good enough. Also

their electronic readout system limited the overall performance. A

large improvement in the use of point contacts for devices was

reported by Zimmerman et al. (4), who proposed the rf SQUID. This is a

superconducting ring interrupted by only one Josephson junction. The

rf SQUID is operated with a radio frequency bias current, generally

several tens of MHz. This rf SQUID became the popular SQUID for

practical applications. In the 70's the first commercial SQUID systems

became available. The problem with these SQUIDs was still the unre-

liable point contact, which was sensitive to mechanical vibrations and

thermal change. In 1975 Clarke, Goubau and Ketchen (5) reported a dc - SQUID fabricated with thin film techniques. Such SQUIDs proved

mechanically stable and were relatively resistant to thermal changes.

The dc SQUID indeed had a better sensitivity than the commercial rf

SQUID. The sensitivity of the dc SQUID is limited by the value of the

inductance of the SQUID and the parasitic capacitance of the junctions

( 6 , 7 ) , - - which must be made as small as possible. To obtain the same

sensitivity with the rf SQUID one would need, besides the small

inductance and capacitance also an impractical high frequency (of the

order of several G H z ) for the biasing and the electronic readout

system ( 8 ) . - A number of researchers noticing the theoretical prediction of a

quantum limit to the sensitivity of SQUIDs have concentrated on

ultrasensitive low inductance dc SQUIDs. On the other hand in this

research the line is followed to construct a dc SQUID of a practical

inductance and out of a stable superconducting material like niobium.

This was made possible by recent progress in the reproducible

fabrication of stable Josephson junctions (9,lO). - - SQUIDS offer the possibility to measure low frequency magnetic

-14 fields as small as 10 T and voltages of 10-l2 V. The aim of the

research described in this thesis is the construction of a practical

low noise SQUID. In the early days SQUIDs were used primarily for

measurements on objects at low temperature. Nowadays SQUID systems are

used for a variety of applications. For instance magnetic fields

generated by the human body are studied as well as magnetic suscepti-

bilities and magnetic moments of materials. SQUID systems are used for

measurements of magnetic signals from the earth's crust. The sources

for most of these measurements are outside the helium cryostat. The

cryogenic system used for cooling the SQUID is different for all of

these applications. But the SQUID sensor is mostly the same. It

consists of the SQUID itself with an input coil to couple the signal

into the SQUID. The input terminals of the SQUID can be connected to

different kinds of circuits which pick up the signal from the object

of measurement. In many cases the pick-up circuit is a signal coil

consisting of a superconducting wire wound in a configuration adapted

to the signal to be measured, for instance a spatial gradient of the

magnetic field. Together with the input coil it forms a completely

superconducting circuit, which keeps the enclosed flux constant and

hence transports the magnetic field from signal coil to the SQUID. The

overall sensitivity of the system is mostly limited by the noise of

the SQUID and the loss between SQUID and input coil. Therefore the

development of a low noise SQUID should concentrate on a combination

of a SQUID with an input coil. Reviews of many applications of SQUIDs

are given in Refs.11 and 12. - - The theory of Josephson junctions is often reduced to a simplified

description with the RSJ (Resistively Shunted Junction) model (13,14). -- For some types of junctions this model gives a good quantitative

description. Tesche and Clarke (6) - used a computer model to study the behavior of the dc SQUID. Their calculations give a good estimate of

the noise of a SQUID and are useful for optimization of SQUID para-

meters. They showed that the noise of the dc SQUID could be reduced by

reducing the inductance of the SQUID and the capacitance of the

junctions. In their numeric calculation the parasitic capacitance of

the junctions was not taken into account. The capacitance can have a

large influence on the behavior of the SQUID. A too large capacitance

results in a hysteretic I-F characteristic ( 13,111). This hysteresis -- can make stable biasing of the SQUID impossible. The capacitance also

produces a resonance with the SQUID inductance (15). This largely - influences the transfer function of the SQUID. So influence of the

capacitance on the noise performance of dc SQUIDs is expected. This is

supported by measurements with real dc SQUIDs reported in this thesis

and by others (9). A theory of a SQUID with capacitors is clearly - needed.

Chapter I1 gives a brief introduction to the theory of Josephson

junctions and SQUIDs. Knowledge of basic theory of superconductivity

(16) - is assumed. Chapter I11 will be published in the Journal of Low Temperature Physics (c). It reports on the dc SQUID developed in our

laboratory. A very low noise niobium SQUID is described. It is

fabricated with ultra small niobium junctions with an overlapping area 2 smaller than 1 pm . The junctions are formed according to a recipe

from Daalmans (10). - The photolithographic technique developed for the

fabrication of the SQUIDs is described. Also complete systems con-

s i s t i n g of 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 de-

s c r i b e d . I n t h i s c h a p t e r a n i n t e g r a t e d sys tem 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 is p r e s e n t e d . T h i s d e v i c e is

u s e f u l f o r b iomedica l a p p l i c a t i o n s . Chapter I V 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 Josephson 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 (c,s)- It 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 compl ica ted . It c o n s i s t s of two

coup led 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

independen 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 . With a h y b r i d computer t h e n o i s e o f

t h e sys tem is c a l c u l a t e d and t h e optimum paramete r s o f t h e SQUID a r e

found. Chapter 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 .

Also 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 .

References

B.D. Josephson , Phys .Le t t . l , 251 - (1962)

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 . 1 2 , 1 5 9 - (1964)

P.L. Forgacs and A. Warnick, Rev.Sci.Instrurn.38,214 - (1967)

J . E . Zimmerman, P. Thiene, and J.T. Harding, J.Appl.Phys.41,1572 -

(1970)

J. C l a r k e , W.M. Goubau, and M.B. Ketchen, J.Low Temp.Phys.25,99 - (1976)

C.D. Tesche and J . C l a r k e , J.Low Temp.Phys.29,301 - (1977)

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)

J. ~ u r k i j a r v i and W.W. Webb, Proc. Appl. S u p e r c o n d u c t i v i t y

Conf. Annapol is , IEEE, New York, 1972, p. 581

R.F. Voss, R.B. La lbowi tz , S . I . Ra ide r , and J . C l a r k e , J.App1.

Phys.51,2306 (1980)

10. G.M. Daalmans, Superconduc t ing Quantum I n t e r f e r e n c e Devices and

T h e i r A p p l i c a t i o n s , H.D. Hahlbohm and H . Lubbig e d s . , Wal ter de

G r u y t e r , B e r l i n 1980, p. 399

1 1 . Superconduc t ing Quantum I n t e r f e r e n c e Devices and T h e i r Appl ica-

t i o n s , H.D. Hahlbohm and H . Lubbig e d s . , Wal ter 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 Superconduc t ive 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 eds . , 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 .Appl.Phys.39,3113 - (1968)

14. W.C. S t e w a r t , Appl .Phys .Let t . l2 ,277 - (1968)

15. S.M. F a r i s and E.A. Valsamakis, 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 , McGraw-Hill, New

York, 1975

17. V . J . d e 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 . d e Waal, P. Schr i jner , and R. LLurba, s u b m i t t e d t o J.Low Temp.

I1 THEORETICAL BACKGROUND

1 1 . 1 The Josephson Effect

The widely used theory of Josephson junctions applies to a tunnel

junction between two superconducting metals. The two superconducting

metals are separated by a very thin insulating film. The insulator is

often the natural oxide of the metal with a thickness of 1 to 5 nm.

The junction is called a tunnel junction if the electrons going across

the junction really have to tunnel through a potential barrier,

because no electron states exist inside the barrier. Josephson (1-3) - - showed theoretically using the microscopic theory of superconduc-

tivity, that a supercurrent can flow through a tunnel junction. He

derived the equations for a junction biased with a constant voltage.

The current I flowing through the junction obeys the Josephson

equations:

where V is the voltage across the junction, & is the maximum super-

current through the junction, and cp is the gauge invariant phase

difference between the quantum states of the two superconductors. Only

small junctions are considered, which have dimensions smaller than the

Josephson penetration depth

where A is the area of the junction and d is the effective thickness

of the junction including the penetration depths of both super-

conductors. It is also assumed that the magnetic flux applied to the

junction is small compared to the flux quantum $ =h/2e=2.07-10-~~ Wb.

The a, and a terms are the quasiparticle current and the quasi- 1

particle-pair interference current respectively. They depend on the

v o l t a g e a c r o s s t h e j u n c t i o n , t h e t e m p e r a t u r e and t h e energy g a p of 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 t h e Josephson e q u a t i o n s i s t h e

c u r r e n t o s c i l l a t i n g a t t h e Josephson f requency 2eiiIh. 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 mic roscop ic t h e o r y of 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 superconduc to r s is

where A(T) i s t h e t e m p e r a t u r e dependent ene rgy gap of t h e supercon-

d u c t o r and 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 Josephson j u n c t i o n is t h e weak l i n k , which

c o n s i s t s of two s u p e r c o n d u c t o r s weakly coupled 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 ind 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 . Exper imenta l ly i t h a s been

well 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 Josephson 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 of a t u n n e l j u n c t i o n

t h e g e n e r a l form of Eq. 11.1 remains v a l i d , a l t h o u g h t h e phase depen-

dence c a n be n o n - s i n u s o i d a l . Also t h e 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. Although

Eq. 11.4 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 , t h e same o r d e r

o f magni tude o f t h e maximum L , R p roduc t i s reached w i t h any k i n d of

weak l i n k o r 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 of Josephson t u n n e l

j u n c t i o n s i s from Werthamer ( 6 ) and Larkin and Ovchinnikov ( 7 ) . T h e i r - - t h e o r y t r e a t s t h e g e n e r a l c a s e o f a t ime dependent v o l t a g e a c r o s s t h e

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 are 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 o r a v o l t a g e changing 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 Josephson 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 Josephson e q u a t i o n s 11.1 and 11.2. However, i n many e x p e r i -

men ta l s i t u a t i o n s one h a s 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 requency and i t s

harmonics . C a l c u l a t i o n s (8-10) - - from t h e Werthamer t h e o r y y i e l d I-!

c u r v e s a s shown i n F ig . 11.1. A s f a r a s I know t h e s e I-v c u r v e s have

/ /

I I 0.5 1.0 I.!

VOLTAGE e i / 2 ~ ( 0 )

Fig. II. 1

1-7 curve of a noiseless junction from the Werthamer

theory at T=0.5 Tc with

constant current bias. From

Zorin and Likharev (9). -

never been observed experimentally. This is due to the capacitance

always present in parallel with the junction. This capacitance shunts

the high frequency Josephson oscillations and often causes the

junction to behave like a voltage biased junction. To avoid this

effect the current 2nI , -2~/@o through the capacitor at the largest

Josephson frequencies must be made small compared to the maximum

supercurrent I,. If one assumes a dielectric constant of the junction

barrier material of 10 and a barrier thickness of 2 nm, the super- 2

current density needed becomes lo9 A/m . In practice it is probably hard to realize a high quality tunnel barrier with such a very high

current density.

If the junction is shunted with a capacitor or a resistor, the

Werthamer theory reduces to the Josephson equations 11.1 and 11.2. In

most practical circumstances one of these conditions is satisfied.

Fig. 11.2 shows the schematic of a commonly used model for a Josephson

junction. It consists of a Josephson element shunted with a resistor

and a capacitor ( 1 1 1 2 - - The equations describing the model are

Fig. 11.2

Schemat ic 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 (RSJ)

model w i t h a c a p a c i t o r

t.' OO 0.5 1.0 1.5

VOLTAGE V/I, R

Fig. II .3

I-? c u r v e s f o r 8 =0, 1, 2, C

and 4. From Mccumber ( 1 1 1 . -

where C i s t h e c a p a c i t a n c e of t h e j u n c t i o n and R is t h e s h u n t r e s i s -

t a n c e . I n comparison w i t h Eqs. 11.1 and 11.2 t h e c o s 9 term i s o m i t t e d

and I,, is t a k e n independen t of 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 lowed i f t h e s h u n t r e s i s t o r R is 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 of

t h e t u n n e l b a r r i e r . F ig . 11.3 shows I-v c u r v e s (y is t h e mean

v o l t a g e , ave raged o v e r many Josephson c y c l e s ) f o r v a r i o u s v a l u e s o f

t h e pa ramete r B c

$ i s t h e f l u x quantum h / 2 e . For t h e c a s e 8 :0 one can f i n d t h e C

s o l u t i o n o f Eqs. 11.5 and 11.6 a n a l y t i c a l l y :

m e junctions with a Bc larger than 1 have a hysteretic I-v curve. In the case of a very large capacitance the voltage will be nearly

constant. Then, according to Eqs. 11.5 and 11.6, for nonzero voltage

the mean current through the Josephson element will be zero and the

I-ii curve is that of the resistor only. So far noiseless junctions

were considered. In a real junction there is a thermal noise current

associated with the quasiparticle current. In the RSJ model this noise

is introduced with a Johnson noise current source with spectral

density 4kTIR in parallel with the shunt resistor. This can be

accounted for by an extra term in Eq. 11.5. Calculations of the 1-7 curves of a RSJ junction can be carried out by direct numerical

solution of the Eqs. 11.5 and 11.6 including the noise term (13). - Other techniques have been used also (14-16). Fig. 11.4 shows that the - - noise produces a rounding of the I-v curve near I . The noise

rounding depends on the dimensionless parameter

,

1.5 -

/ /

0 / I I I 0 0.5 1.0 1.5

VOLTAGE ~/I,R

Fig. 11.4

Noise rounded 1-7 curves for r=0.01, 0.05, and 0.2 and

0 = I . From Voss ( 1 3 ) . C -

The physical explanation is that the noise fluctuations switch the

junction between the voltage carrying state and the zero voltage

state. The curve in Fig. 11.4 shows the mean voltage. This behavior is

described by the thermal activation model (17-19). - - So far it was assumed that the spectrum of the noise current was

white. For very high frequencies it is necessary to use the complete

expression of the noise current in the junction including zero point

fluctuations (20) -

Because the noise at the Josephson frequency or its first few harmon-

ics is mixed down to low frequencies due to the nonlinear behavior of

the junction, the quantum fluctuations can produce an excess noise at

low frequency. Also this quantum noise can cause a noise rounding in

the I-V curve (20). -

11.2 Fundamentals of the dc SQUID

A combination of one or more rings of superconducting material

interrupted by one or more Josephson junctions is called a Super-

conducting Quantum Interference Device (SQUID) or interferometer. The

subject of this thesis is the dc SQUID containing two junctions (21). - To get a qualitative understanding of the device the junction model

described above is used. Both junctions obey the equations 11.5 and

11.6. The phase of the superconducting state must be single valued,

which leads to (22) -

where n is an integer, A(r) is the magnetic vector potential, and p 1 and q are the phase differences across the junctions. The integral in

2 Eq. 11-12 is the magnetic flux O through the SQUID ring. Eq. 11.12 t

combined w i t h Eqs. 11.5 and 11.6 d e s c r i b e s t h e d c 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 of 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 is t h e

same as t h e one of a s i n g l e j u n c t i o n shown i n F ig . 11.3. One c a n

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 , Ic o f

t h e SQUID becomes

where Qo i s t h e f l u x quantum h12e. If one i n t r o d u c e s a n i n d u c t a n c e i n

t h e SQUID r i n g t h e modula t ion dep th of t h e c r i t i c a l c u r r e n t i s

reduced . A more complete a n a l y s i s was g i v e n by De Waele and D e Bruyn

Ouboter ( 2 3 , 2 4 ) - - and by F u l t o n e t a l . (5). Numerical c a l c u l a t i o n s of

Tesche and Cla rke ( 2 6 ) and Bru ines e t a l . ( 2 7 ) a r e shown i n F ig . I I . 5 a - - and b. For u s e as a measur ing d e v i c e t h e SQUID is o p e r a t e d w i t h a

Fig . 11.5

C h a r a c t e r i s t i c s of a d c S Q U I D wi th 21,L/QO =1 , pc=O, and r=0.05

a c c o r d i n g t o Tesche and Cla rke ( 2 6 ) . - ( a ) I-V c u r v e s f o r Q =O and O =Qo / 2 .

a a ( b ) V-o c u r v e s f o r c o n s t a n t b i a s c u r r e n t I/Io ~ 1 . 2 , 1.6, 2, and 3 .

a

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 is a p e r i o d i c

f u n c t i o n o f t h e magne 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 of m e r i t of a SQUID used a s magnetometer i s

t h e e n e r g y r e s o l u t i o n ( 2 8 , 2 9 ) , which is d e f i n e d by --

where 5 6 0 ) i s t h e low f requency 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 is t h e i n d u c t a n c e of 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

M i s t h e mutual i n d u c t a n c e between SQUID and c o i l and LC 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 , i f t h e SQUID r i n g is open. Tesche and

Cla rke (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 of a SQUID is - ( 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))

where C i s t h e c a p a c i t a n c e o f t h e Josephson j u n c t i o n s . Th i s r e s u l t

shows t h a t t h e n o i s e o f a d c SQUID can be made low by choos ing a small

i n d u c t a n c e o r a s m a l l c a p a c i t a n c e . However, Tesche and Cla rke 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 of E q . I I . 1 6 w a s 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 t h e 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

t h e 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-T c u r v e ( s e e Sec. 11 .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 c a n r e s u l t - i n I-T c u r v e s l i k e F ig . 11.6. I n Ch. I V c a l c u l a t i o n s o f t h e n o i s e of

a d c 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 .

For v e r y low v a l u e s o f t h e energy r e s o l u t i o n , n e a r P l a n c k ' s

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 Josephson f r e q u e n c y

c a n be above t h e w h i t e n o i s e p a r t o f t h e the rmal n o i s e spect rum (31) 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. 11.1). There i s e v i d e n c e

o f macroscopic quantum p r o c e s s e s , i n which 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 ene rgy (g) . These e f f e c t s c a n

l e a d t o a n i n c r e a s e d v o l t a g e noise . I n t h e l i t e r a t u r e t h e r e is d i s c u s -

s i o n ( 3 1 , 3 3 , 3 4 ) abou t t h e p r e s e n c e o f a quantum l i m i t of 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 d c SQUID is 0.5h (35) . - Because t h e r e s o l u t i o n of 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 is

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

expec ted 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 .

Bes ides t h e whi te n o i s e d e s c r i b e d above, i n any SQUID a low

f r e q u e n c y l / f n o i s e i s presen t . For most p r a c t i c a l measurement sys tems

t h e l / 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. Th i s llf 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 which 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 of t h e l / f n o i s e p robab ly l i es i n t h e

Josephson junc t ions . The n o i s e might 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 (2). Tesche (31) 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 parameters . Up t o now no s a t i s f a c t o r y

/ /

I I S Q U I D w i t h 21, L/@, = 1 and Bc=l

0 I 0 0.5 1.0 1.5 2.0 c a l c u l a t e d w i t h t h e model

VOLTAGE ~/ I ,R d e s c r i b e d i n Ch. IV

theory is available to understand or predict the l/f noise of

Josephson junction devices.

References

1. B.D. Josephson, Phys.Lett.l,251 - (1962)

2. B.D. Josephson, Rev.Mod.Phys.36,46 - (1964)

3. B.D. Josephson, Adv.Phys.l4,419 - (1965)

4. V. Ambegaokar and A. Baratoff, Phys.Rev.Lett.l0,486 - (1963),

Err. Phys.Rev.Lett.l1,104 - (1963)

5. P.W. Anderson and A.H. Dayem, Phys.Rev.Lett.l3,195 (1964) - 6. N.R. Werthamer, Phys.Rev.147,255 - (1966)

7. A.I. Larkin and Yu.N. Ovchinnikov, Sov.Phys.JETP24,1035 - (1967)

8. D.G. Mc Donald, E.G. Johnson, and R.E. Harris, Phys.Rev.B13,1028 - (1976)

9. A.B. Zorin and K.K. Likharev, Sov.J.Low Temp.Phys.3,70 - (1977)

10. W.A. Schlup, J.Phys. Colloque C6, - 39,565 (1978)

11. D.E. McCumbeP, J.Appl.Phys.39,3113 - (1968)

12. W.C. Stewart, Appl.Phys.Lett.l2,277 (1968) - 13. R.F. Voss, J.LOW Temp.Phys.42,151 (1981) - 14. V. Ambegaokar and B.I. Halperin, Phys.Rev.~ett.22,1364 - (1969)

15. J. ~urkijarvi and V. Ambegaokar, Phys-Lett. - 1A,314 - (1970)

16. K. Yoshida, J.Appl.Phys.53,7471 - (1982)

17. J. ~urkijarvi, ~hys.Rev.B6,832 - (1972)

18. T.A. Fulton, IEEE Trans.Mag.l1,749 (1975) - 19. C.D. Tesche, J.Low Temp.Phys.44, 119 (1981) - 20. R.H. Koch, D.J. van Harlingen, and J. Clarke, Phys.Rev.Lett.45, -

26 (1980)

21. R.C. Jaklevic, J. Lambe, A.H. Silver and J.E. Mercereau, Phys.Rev.

Lett.12,159 - (1964)

22. J.E. Zimmerman and A.H. Silver, Phys.Rev.141,367 - (1966)

23. A.Th.A.M. de Waele and R. de Bruyn Ouboter, Physica - 42,225 (1969)

24. A.Th.A.M. de Waele and R. de Bruyn Ouboter, Physica - 42,626 (1969)

25. T.A. Fulton, L.N. Dunkleberger, and R.C. Dynes, Phys.Rev.B6, - 855

(1972)

26. C.D. Tesche and J. Clarke, J.Low Temp.Phys.29,301 - (1977)

27. J.J.P. Bruines, V.J. de Waal, and J.E. Mooij, J.Low Temp.Phys.46, - 383 (1982)

28. V. Radhakrishnan and V.L. Newhouse, J.Appl.Phys.42, - 129 (1971) 29. J.H. Claassen, J.Appl.Phys.5, 2268 (1975)

30. S.M. Faris and E.A. Valsamakis, J.Appl.Phys.52,915 - (1981)

31. R.H. Koch, D.J. van Harlingen, and J. Clarke, Appl.Phys.Lett.38, - 380 (1981)

32. W. den Boer and R. de Bruyn Ouboter, Physica98B,185 - (1980)

33. R.F. Voss, Appl.Phys.Lett.38,182 - (1981)

34. C.D. Tesche, Appl.Phys.Lett.41,490 - (1982)

35. D.J. van Harlingen, R.H. Koch, and J. Clarke, Appl.Phys.Lett.41, - 197 (1982)

36. J. Clarke and G. Hawkins, IEEE Trans.~agn.MAG-11,841 (1975)

37. C.D. Tesche, Appl.Phys.Lett.41,99 - (1982)

111 HIGH PERFORMANCE DC SQUIDS WITH SUBMICRON NIOBIUM

JOSEPHSON JUNCTIONS

A b s t r a c t

We r e p o r t on t h e f a b r i c a t i o n and performance o f low n o i s e a l l -

niobium t h i n f i l m p l a n a r d c SQUIDs w i t h submicron Josephson j u n c t i o n s .

The j u n c t i o n s are evapora ted o b l i q u e l y th rough a m e t a l shadow evapo-

r a t i o n mask, which is made u s i n g o p t i c a l l i t h o g r a p h y w i t h 0.5 pm

t o l e r a n c e . The Josephson j u n c t i o n b a r r i e r is 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 subsequen t o x i d a t i o n i n a glow d i s c h a r g e .

The j u n c t i o n pa ramete r s c a n be reproduced w i t h i n a f a c t o r of 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 a b o u t 3 P A and t h e

r e s i s t a n c e s a r e abou t 100 C2. With SQUIDs hav ing a n i n d u c t a n c e o f 1 nH

t h e v o l t a g e modula t ion is a t l e a s t 60 vV. An i n t r i n s i c energy r e s o l u -

t i o n o f 4 . 1 0 - ~ ~ J/Hz h a s been reached . The SQUIDs a r e coup led 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 niobium 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 .

I n b o t h c a s e s t h e c o i l is g l u e d o n t o 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. Refe r red 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

b e s t coup led energy r e s o l u t i o n ach ieved is 1 . 2 . 1 0 - ~ ~ J /Hz measured i n

a f l u x locked l o o p a t f r e q u e n c i e s above 10 Hz. A s 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 a n 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 is s u i t e d f o r making c i r c u i t s

w i t h SQUID and pick-up 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 h a s a s i z e of 12 mm -17 m m , is s i m p l e t o f a b r i c a t e and

is s u i t a b l e f o r b iomedica l a p p l i c a t i o n s .

111.1 Introduction

The last decade SQUIDs (superconducting quantum interference

devices) have become widely used measuring instruments for small

magnetic fields and many other kinds of small signals. In 1964

Jaklevic et al. (1) constructed the first dc SQUID consisting of a - superconducting ring with two Josephson junctions. Nowadays the most

often used type is the rf SQUID, which consists of a superconducting

ring containing one Josephson junction, generally a point contact. The

rf SQUIDS (2), - biased with a frequency of 20 MHz or larger, have an -28

energy resolution of 10 J/Hz. To obtain a better resolution with

this system it is necessary to use higher frequencies with more

complicated electronics. The last few years research on dc SQUIDs has

been revitalized by the work of Clarke, Goubau and Ketchen ( 3 ) . - The

noise of this type is limited by the capacitance of the junctions ( 4 ) . - Although for research point contacts have been used (51, for optimum reliability it is advantageous to use thin film junctions. A first

step towards a reliable low noise dc SQUID was the cylindrical thin

film niobium-lead dc SQUID of Clarke, Coubau and Ketchen ( 3 ) with a - wire wound input coil. An improvement can be reached with SQUIDs on a

flat substrate with a spiral input coil. A flat substrate facilitates

the use of standard thin film techniques for producing ultra small

junctions with a high I , R product. Jaycox and Ketchen (6,7) - - and Cromar and Carelli (8) made such low noise dc SQUIDs with coupling coil using - lead alloy Josephson junctions. For practical use a disadvantage of

lead alloy is the sensitivity to thermal shock and the poor chemical

resistance. In this respect refractory metal junctions are more

favourable. Several authors reported Josephson junctions of niobium

suited for this application (9,lO). This paper deals with a low noise - - niobium dc SQUID with wire wound as well as thin film input coils.

A different approach for a practical device is the fabrication of

systems with SQUID and pickup coil on a single substrate. Advantages

are the compactness and the possibility of precisely balancing

gradiometers. Ketchen et al. ( 1 1 ) - reported a first order niobium lead gradiometer with a SQUID on one flat substrate. Here we describe a

compact first order gradiometer designed to be an integral part of t.he

SQUID itself. Sec. 111.2 explains the design criteria of thin film

SQUID circuits. We argue, that with small junctions a low noise SQUID

can be made without the necessity to reduce the SQUID inductance to

very small values. Sec. 111.3 contains the fabrication method of the

Josephson junctions and the thin film coils. The properties of the

junctions are dealt with in Sec. 111.4. We describe the noise perfor-

mance of the SQUIDs and the properties of the coils in Sec.III.5.

Sec.III.6 contains the experimental results with the gradiometers. In

Sec.III.7 we discuss possible improvements and give a summary. Part of

this work (12-15) was reported on before. - -

111.2 Design Considerations

For magnetometer and gradiometer applications the important figure

of merit of a SQUID is the energy resolution ( 1 6 ) , which is defined by -

where S 0 ) is the r.m.s. flux noise of the SQUID, L is the SQUID & inductance and k is the coupling coefficient between SQUID and input

coil. The coupling coefficient is defined by

where L is the inductance of the input coil and M is the mutual C

inductance. Theoretical calculations of the energy resolution of dc

SQUIDS have been performed by Tesche and Clarke ( 4 ) . - Assuming a

resistively shunted junction model with thermal noise generated in the

shunt resistor, they predict an optimum energy resolution of

if the parameter B = 21,L/Qo has the optimum value

where Qo is the flux quantum. The numerical factor in Eq. 111.3 was

corrected according to Bruines et al. (17). If the SQUID is operated - in a flux locked loop, the energy resolution is slightly deteriorated.

For the theoretical model with a sine wave modulation signal this

amounts to a factor of 1.6 (4). I17 this model the parallel capacitance - of the junctions is neglected. Real junctions always have a capaci-

tance, which can cause hysteresis in the I-V curve of the junctions.

To make the junctions non-hysteretic the McCumber parameter (18) 2 B =2aI., R C/Qo must be

C

Tesche and Clarke argue that the energy resolution is optimum if 6 is C

about 1. Substituting this into Eq. 111.3 yields

Eq. 111.6 shows, that SQUIDS with a low energy resolution require a

small SQUID inductance or a small junction capacitance. Small induc-

tances make it difficult to couple flux efficiently into the SQUID.

However, the capacitance of the Josephson junctions can be made small

without similar problems. For instance a SQUID in a flux locked loop

with L=1 nH and C:l.lO -1 4 F should have an energy resolution of

4.6.10-~~ J/Hz, if the critical current and the resistance have the

optimum values of 1 P A and 180 n. Niobium is suitable as material for the Josephson junctions,

because of its high critical temperature, the large I,, R product that

can be reached and the long term stability of the parameters of

niobium tunnel junctions. A disadvantage of niobium junctions is the

large relative dielectric constant of its oxide, which is 30 (19). The - -2

specific capacitance of niobium oxide junctions is 0.13 F-m . An

-14 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 a n a r e a

2 smaller t h a n 0.1 Mm . 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 ime d e t e r m i n e s t h e

j u n c t i o n pa ramete 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 t h e s i l i c o n t h i c k n e s s c a n be 10 t i m e s a s l a r g e a s 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 a s small a s

t h a t o f t h e niobium 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 o u r 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

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 . 1 0 - l ~ m2 t u r n

o u t t o have a l m o s t t h e r i g h t & R p roduc 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 u s e of a n 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 of t h e s e

j u n c t i o n s is s o 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 . I n Appendix A t h i s c a p a c i t a n c e is shown t o be a t l e a s t

5-10- l5 F p e r j u n c t i o n , even i f t h e width o f t h e s t r i p s connec ted t o

t h e j u n c t i o n is reduced t o 10 pm.

For most SQUID a p p l i c a t i o n s a n i n p u t c o i l w i t h a n i n d u c t a n c e n e a r

1 pH i s needed. This c o i l must be coupled t o t h e SQUID a s t i g h t l y a s

p o s s i b l e . Genera l ly a c o u p l i n g c o e f f i c i e n t k2 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 wire wound c o i l s . For t h i n f i l m SQUIDs, which 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 o u t o f bulk niobium, 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 . With a c o i l , u s i n g l i n e w i d t h s of 10 pm o r

s m a l l e r , one c a n make many t u r n s on a small area. Promis ing r e s u l t s

were r e p o r t e d by Jaycox and Ketchen (6.1) and by Cromar and

C a r e l l i ( 8 ) . - The c i r c u i t des igned by Jaycox and Ketchen c o n s i s t s of 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 a n i n d u c t a n c e o f 1 nH, t h e s i z e s which w e

c a n a l l o w a r e 10 t i m e s a s l a r g e a s t h e s i z e s of t h e 100 pH sys tem.

There fo re i t is n o t n e c e s s a r y t o p u t t h e SQUID and t h e c o i l on t h e

same s u b s t r a t e . Th i s e n a b l e s u s even, i f necessary , t o u s e w i r e wound

c o i l s . I n p r a c t i c e , p a r t o f t h e SQUID loop 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

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 (20): -

where L . 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 n e a r t h e J

j u n c t i o n s which is n o t coupled 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 ground p l a n e i n s t e a d of t h e SQUID r i n g . For a c o i l

ve ry n e a r t h e SQUID t h e i n d u c t a n c e L is t h e i n d u c t a n c e o f a s t r i p l i n e s

w i t h t h e same l e n g t h a s t h e i n p u t c o i l . Th i s i n d u c t a n c e c a n 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 u s e i s abou t 2 nH. The i n d u c t a n c e of 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 Jaycox and Ketchen (5) . 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 of o u r

c i r c u l a r SQUID w i t h I . D . 1 .4 mm and O.D. 3.4 mrn t o be 1.9 nH. I f we

p u t 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 of

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 pm, which 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 mutual induc-

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 of 70 nH and a n i n p u t induc-

t a n c e o f 0.8 pH. 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 between t h e

s u b s t r a t e s a c c o r d i n g t o Eqs. 111.7-9 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 . Usua l ly it is 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 such 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 imensions o f t h e SQUID. Th i s problem c a n 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 t h e c o i l can be used t o

c o n c e n t r a t e t h e magnet ic f l u x i n s i d e t h e SQUID r i n g . Two h a l f c y l i n -

d e r s o f niobium 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 pm

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 t h e s u p e r c o n d u c t i n g w i r e is

wound. F ig . 111 .1 shows t h e c o n f i g u r a t i o n . Because t h e 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 is s m a l l , t h e 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 t h e c y l i n d e r . Th 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

I I I I

I I Fig. III.1

Construction of an input # L --I-

/-- I . \ coil consisting of wire wound I -- \ - - -< 'I- --=-. around a split niobium core

SQUID. This niobium cylinder does not short the SQUID loop, but only

reduces the inductance of the SQUID. The coupling constant of this

configuration depends on the distance between the film and the

cylinder, the distance between the two halves of the cylinder and the

thickness of the insulation of the wire used. The largest loss is

produced by the leakage between the wire and the cylinder. This

contribution can be estimated analogous to the spiral input coil. The

SQUID is used as described above, input inductances of the order of

several hundreds of nanohenrys can be reached with a 4 mm long

cylinder.

Another method of coupling signals into the SQUID is the integra-

tion of the SQUID and the pick-up loop, for instance a gradiometer, on

a single substrate. Such a design has the advantage, that the complete

superconducting part of the system is concentrated on a small chip.

Then the space consuming superconducting wires, connections and

screenings can be eliminated. This design was first used by Ketchen,

Clarke, Goubau and Donaldson ( 1 1 - ). They designed a large pick-up loop.

Part of the SQUID loop formed a part of the pick-up loop. Because the

inductance of the SQUID loop is smaller than the inductance of the

pick-up loop, a coupling loss must be accepted. The sensitivity of

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 -

me te r and t h e r e s o l u t i o n of 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 is 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 which is s e n s i t i v e

enough f o r b iomedica l a p p l i c a t i o n s . The d e s i g n might a l s o be u s e f u l i n

a n a r r a y o f SQUIDS.

Fig . II1.2

E q u i v a l e n t c i r c u i t o f a SQUID d i r e c t l y

coup led t o t h e p ick-up c o i l

Fig . 111.2 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 .

L i s t h e i n d u c t a n c e o f t h e pick-up l o o p and is much l a r g e r t h a n t h e 1

SQUID i n d u c t a n c e Ls. We n e g l e c t t h e i r mutua 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 of t h e s i g n a l which i s s e n s e d by t h e i n d u c t o r Ls. Then

it f o l l o w s f o r t h e f l u x n o i s e S 0 ) i n t h e pick-up l o o p from Eq. 111.6 6

This r e s u l t i m p l i e s , t h a t Ls shou ld be made as l a r g e a s p o s s i b l e .

However, f o r l a r g e v a l u e s of Ls Eqs. 111.6 and 111.10 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

o u t t h e f l u x dependence o f t h e SQUID ( 2 2 ) . Th i s w i l l become a s e r i o u s - problem if

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 induc-

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 robab ly be n e a r 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 . It c o n s i s t s of two j u n c t i o n s i n s e r i e s

connec ted 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 ig . 111.3 shows photographs

of t h e g r a d i o m e t e r and t h e p a r t of i t n e a r t h e j u n c t i o n s . The induc-

t a n c e s e e n by t h e j u n c t i o n s is determined mainly by t h e i n d u c t a n c e s of

( a ) Photograph of a g r a d i o -

meter. The l o n g e r s i d e h a s a

l e n g t h o f 16.5 mm.

( 6 ) D e t a i l of t h e g r a d i o m e t e r

wi th t h e j u n c t i o n s

t h e s m a l l e s t r i n g s . The s i g n a l is sensed mainly by t h e l a r g e l o o p s . I n

t h e d e s i g n we avo id 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 r educe 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 of t h e d e v i c e is 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 ize o f t h e loop

o f 1 m m , which l i m i t s t h e SQUID i n d u c t a n c e t o a b o u t 2 nH o r l a r g e r .

Near t h e middle o f t h e g rad iomete r t h e wid th o f t h e g r a d i o m e t e r is

reduced . Th i s r educes 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

region is not so important. This was calculated by Pegrum and

Donaldson (23). Advantages of the present design are the larger - sensitvity of a parallel gradiometer and the simplicity of the

fabrication. In principle the dimensions can be controlled with an

accuracy of the order of 1 pm. If one uses an optically flat sub-

strate, the deviation perpendicular to the surface can be of the same

order. So the balance of thin film gradiometers of 10 mm sizes can be

100 ppm. A disadvantage of the parallel circuit is the closed super-

conducting loop. If the gradiometer is moved in a magnetic field, or

if the magnetic field Changes, large currents can flow in the super-

conducting strips and possibly drive the film normal. This effect

limits the use of these gradiometers to applications with relatively

small changes of the magnetic fields and gradients. The inductance

seen by the junctions is an important parameter in the design of the

circuit. The calculation of the induction is described in Appendix A .

Our gradiometer was designed to have an inductance of 2.7 nH.

111.3 Fabrication

If one wants to take full advantage of the high critical tempera-

ture of niobium, it is necessary to make good quality niobium. In an

ordinary high vacuum system this is done by heating the substrate to

400 "C. It is difficult to combine this heating with a lithographic

procedure for miniaturization, because resists can not tolerate these

temperatures. Daalmans and Zwier (24) developed a method to pattern - submicron niobium Josephson junctions with thin film metal offset

masks, generated with electron beam lithography. These masks can

withstand temperatures larger than 300 "C. The junctions are evaporated

obliquely. The complete pattern with the tunnel junctions is fabri-

cated in one evaporation run and one lithographic step. They fabri-

cated the masks, which consist of chromium and niobium, with electron

lithography. However, the linewidth of 1 pm needed for this process

can also be reached with photolithography. Because of the greater

flexibility of our photolithographic equipment we prefer to make the

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 is 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. Th i s mask c o n s i s t s of 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 m m ) 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 a n o p t i c a l p r o j e c t i o n

sys tem c o n t a i n i n g a microscope 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 a s

a n a n t i r e f l e c t i o n l a y e r on t h e chromium (z), 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 . Af terwards t h i s p a t t e r n i s deve loped , t h e

s i l i c o n i s plasmaetched 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 pm 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 pm chromium f i l m is 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

is 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 is 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 s p a c e between t h e mask and t h e

s u b s t r a t e . The sample is exposed th rough 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. Af te 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 are s e l e c t e d . The y i e l d i s more t h a n 70%. I n 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 pm niobium f i l m

is 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 of

t h e niobium is performed. The chromium is 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 cer ium n i t r a t e . The e t c h i s s t o p p e d when i t h a s

passed 0.5 pm below t h e edge o f t h e niobium f i l m . A t p l a c e s where t h e

niobium is narrower t h a n 1 pm a f r e e hang ing niobium b r i d g e is formed.

T h i s s t r u c t u r e s e r v e s a s 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 sys tem w i t h a 10 kW

e l e c t r o n gun. Between t h e 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 t h e vacuum

system is n o t opened. F ig . 1 1 1 . 4 shows a s c h e m a t i c of t h e c o n f i g u r a -

t i o n . The niobium is e v a p o r a t e d a t a r a t e o f 10 nmls. The p r e s s u r e

d u r i n g e v a p o r a t i o n i s 5 . 1 0 - ~ Pa. During 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

is h e l d a t a n 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 T. This first

niobium 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 t h e n e x t f i l m s under 45 d e g r e e s from t h e o p p o s i t e s i d e .

,NIOBIUM BRIDGE

SECOND NIOBIUM FILM

SILICON FILM

Fig. III.4

Cross-section of a junction with evaporation mask directly after

evaporation

The substrate is cooled to 15 C. A silicon film of 20 nm is deposited.

A dc oxygen glow discharge is applied at a pressure of 5 Pa. The

electrode of the discharge is at the negative side with a voltage of

3kV. The junction barrier is presumably formed by niobium oxide in the

pinholes in the silicon (9). Cooling the substrate during and after the oxidation is necessary to prevent decomposition of this oxide

barrier. The niobium counterelectrode of the junction is evaporated to

a thickness of 300 nm at a substrate temperature of 15 C. To facili-

tate contacting the niobium film, a final 7 nm gold film is evapo-

rated. This gold prevents oxidation of the top niobium film. With a

chromium etch bath of several hours the pattern is lifted-off.

During fabrication the junctions are shunted by the film itself to

prevent burn out of the junctions by electrostatic discharges. After

fabrication this shunt is scratched away. As substrate n-type silicon

of 1 il-cm is used. At room temperature the junctions remain shunted by

the substrate with 50 to 200 il per junction depending on the geometry.

Yet sometimes the junctions can be destroyed by electrostatic dis-

charges. If the junctions are handled with grounded tools only, this

possibility is reduced. The film is contacted with indium press

contacts or by bonding ultrasonically aluminium wire. Pictures of

junctions are shown in Fig. III.5a and b. From the various pictures we

made with optical and electron microscopes we estimate a mean 2

overlapping area of the junction of 0.2 pm . With the fabrication

method used it is not possible to make larger junctions, because the

chromium film thickness, which must be adapted in that case, can not

Fig. III.5

Scanning e l e c t r o n mic rographs of a s m a l l ( a ) a n d a l a r g e ( b l Josephson

j u n c t i o n . The s u b s t r a t e s w e r e t i l t e d o v e r 4 0 degrees . The l e n g t h of

t h e w h i t e b a r c o r r e s p o n d s t o 1 pm.

be made t h i c k e r t h a n 0.6 pm w i t h o u t a f f e c t i n g t h e m e t a l l u r g i c a l

p r o p e r t i e s of t h e f i l m . With t h e same p r o c e s s a l s o t h e SQUID r i n g is

f a b r i c a t e d . F i g s . 111.6 and 111.7 show photographs of d c SQUIDs.

For t h e d e s i g n o f a t h i n f i l m i n p u t c o i l , we dec ided a g a i n s t

e v a p o r a t i n g t h e c o i l d i r e c t l y on t o p of t h e SQUID. The complete

p a t t e r n would i n v o l v e 9 l a y e r s . I n o r d e r t o r educe t h e f a i l u r e r a t e we

f a b r i c a t e d bo th SQUID and c o i l on s e p a r a t e s u b s t r a t e s and g l u e d them

t o g e t h e r a f t e r w a r d s . Whenever p o s s i b l e we used t h e same t e c h n i q u e s

a p p l i e d t o f a b r i c a t e t h e SQUIDs themse lves .

The c o i l ( F i g . 111.8) c o n s i s t s o f a 20 t u r n c o n c e n t r i c s p i r a l w i t h

a n o u t e r d i a m e t e r o f 2.8 mm. The conduc to r s a s wel l as t h e i n t e r m e d i -

a t e s p a c i n g a r e 10 pm wide. Because bo th s o l d e r pads a r e l o c a t e d

o u t s i d e t h e s p i r a l , one i s f o r c e d t o l e a d t h e c u r r e n t from t h e i n n e r

end of t h e s p i r a l back t o t h e o u t s i d e u s i n g a n i n s u l a t e d c r o s s - o v e r .

Because t h e whole f l u x t r a n s f o r m e r c i r c u i t must be 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 , s p e c i a l a t t e n t i o n must be p a i d t o a v o i d i n g t h e f o r m a t i o n

o f a n o x i d e l a y e r between t h e c ross -over and t h e c o i l i t s e l f , when t h e

c o i l i s removed from t h e e v a p o r a t i o n chamber. Th i s problem was s o l v e d

by e v a p o r a t i n g a t h i n g o l d l a y e r ( 7 nm) d i r e c t l y on t o p o f t h e f i r s t

niobium l a y e r t h a t i s t h i c k enough t o a v o i d t h e o x i d a t i o n o f t h e whole

F i g . I I I .6

P h o t o g r a p h o f t he 1 n H d c SQUID. T h e o u t e r d i a m e t e r o f the r i n g

is 450 v m .

s u r f a c e of t h e niobium b u t t h i n enough t o become s u p e r c o n d u c t i n g due

t o t h e p r o x i m i t y e f f e c t . The same t e c h n i q u e was used t o a v o i d a n o x i d e

f i l m on t h e c o n t a c t pads .

We c h o s e s i l i c o n a s s u b s t r a t e m a t e r i a l f o r t h e c o i l t o match t h e

t h e r m a l expans ion o f t h e SQUID s u b s t r a t e . The s p i r a l and t h e s o l d e r

c o n t a c t s were p a t t e r n e d w i t h a p h o t o l i t h o g r a p h i c l i f t - o f f s t e p . We

covered t h e s i l i c o n w i t h a n adhes ion a g e n t (HMDS p r i m e r , Kodak) b e f o r e

a p p l y i n g t h e f o t o r e s i s t . To make s u r e t h a t subsequen t l a y e r s w i l l

c o v e r t h e v a r i o u s s t e p s each l a y e r was made somewhat t h i c k e r t h a n t h e

p r e v i o u s one . The f i r s t Nb l a y e r was 80 nm t h i c k . S i l i c o n was evapo-

r a t e d t h r o u g h a new p h o t o r e s i s t mask t o make a n i n s u l a t i n g b r i d g e by

which t o l e a d t h e c r o s s o v e r a c r o s s t h e s p i r a l . Although we used a 10

nm s i l i c o n l a y e r we s t i l l had t r o u b l e i n g e t t i n g i t t o c o v e r a l l 40

s t e p s i n t h e s p i r a l r e l i a b l y . Evapora t ion o f t h e s i l i c o n a t two

d i f f e r e n t a n g l e s i n two l a y e r s o f 65 nm each s o l v e d t h i s problem.

Because two s i l i c o n l a y e r s were evapora ted th rough a t h i c k (5 pm) mask

o f p h o t o r e s i s t , t h e edges o f t h e s i l i c o n p a t t e r n i t s e l f end i n two

s m a l l e r s t e p s r a t h e r t h a n one b i g one. T h i s a g a i n f a c i l i t a t e s t h e s t e p

c o v e r a g e i n t h e f o l l o w i n g l a y e r s . The dumbbell-shaped c r o s s - o v e r was

a l s o e v a p o r a t e d i n two l a y e r s of 80 nm each a t a n a n g l e t o a l l t h e

s t e p s . It i s a l s o p a t t e r n e d w i t h a l i f t - o f f p r o c e s s . The r e s i s t a n c e o f 2

t h e i n s u l a t i n g s i l i c o n a t 4.2 K was found t o be l a r g e r t h a n 10 C2-m . A

f i n a l i n s u l a t i n g l a y e r of 200 nm s i l i c o n was a p p l i e d t o c o v e r t h e

whole p a t t e r n e x c e p t t h e s o l d e r pads . F ig . 111.8 shows a photograph of

a c o i l .

A t t h e edges o f t h e s u b s t r a t e we i n c l u d e d a s i m p l e c a l i b r a t i o n

p a t t e r n c o n s i s t i n g o f a l o n g , wide s t r i p t h a t i s e v a p o r a t e d s i m u l t a -

n e o u s l y w i t h t h e s p i r a l . By measur ing t h e r a t i o o f i t s r e s i s t a n c e t o

t h a t o f t h e s p i r a l we c o u l d q u i c k l y check i f t h e s p i r a l c o n t a i n e d any

e l e c t r i c a l s h o r t s . T h i s was measured a t a t e m p e r a t u r e o f 10 K t o

e l i m i n a t e t h e conduc t ion th rough t h e s u b s t r a t e .

F i g . III.7

P h o t o g r a p h o f the 2 n H d c SQUID. T h e o u t e r d i a m e t e r o f t h e r i n g i s

3 . 4 mm.

F i g . III.8

P h o t o g r a p h o f the e v a p o r a t e d 20 t u r n n i o b i u m i n p u t c o i l . T h e o u t e r

d i a m e t e r i s 2 .8 mm.

111.4 Junction characterization

The c o n t r o l l a b i l i t y o f t h e r e s i s t a n c e and t h e c r i t i c a l c u r r e n t o f

t h e Josephson j u n c t i o n s i s i m p o r t a n t f o r t h e f a b r i c a t i o n o f SQUIDS.

These p a r a m e t e r s are c o n t r o l l e d w i t h t h e o x i d a t i o n t i m e . We are a b l e

t o r e p r o d u c e t h e c r i t i c a l c u r r e n t s and t h e r e s i s t a n c e s w i t h i n a r a n g e

of a f a c t o r of 2 even i n d i f f e r e n t e v a p o r a t i o n r u n s . T h i s i s s u r -

p r i s i n g because t h e r e p r o d u c i b i l i t y o f t h e d imens ions of t h e evapo-

r a t i o n masks i s i n s u f f i c i e n t t o g e t such a c o n s t a n t o v e r l a p p i n g a r e a .

The r e s o l u t i o n o f t h e f a b r i c a t i o n equipment i s c e r t a i n l y n o t b e t t e r

t h a n 0.2 pm. With a n e l e c t r o n microscope w e a r e a b l e t o s e e t h a t t h e

areas o f t h e j u n c t i o n s d i f f e r by a f a c t o r o f a b o u t 5 . F ig . 111.5 shows

SEM micrographs o f a small and a l a r g e j u n c t i o n which had comparable

c r i t i c a l c u r r e n t s and r e s i s t a n c e s . The numer ica l d a t a seem t o f o l l o w a

l i n e a r r e l a t i o n s h i p between t h e c r i t i c a l c u r r e n t and t h e j u n c t i o n

dimensions . The j u n c t i o n pa ramete r s a l s o depend on t h e e l e c t r i c a l

r e s i s t a n c e from t h e niobium f i l m t o ground d u r i n g t h e f a b r i c a t i o n .

Normally w e u s e n-type s i l i c o n a s s u b s t r a t e m a t e r i a l . T h i s s u b s t r a t e

p r o v i d e s a r e s i s t a n c e o f a b o u t 500 between t h e niobium f i l m and

ground. T h i s r e s i s t a n c e i s due t o t h e c o n t a c t between t h e niobium and

t h e s i l i c o n . I n t h i s case we make j u n c t i o n s o f 1.5 u A and 200 n. I f w e

make- t h e r e s i s t a n c e t o ground s e v e r a l k n , t h e j u n c t i o n I,,R p r o d u c t c a n

become 700 pV - a l s o f o r a j u n c t i o n c r i t i ca l c u r r e n t o f a b o u t 2 PA. If

t h e niobium f i l m is connec ted d i r e c t l y t o ground we a r e n o t a b l e t o

make r e p r o d u c i b l y t h e same k i n d o f 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 s

become 30 PA o r l a r g e r . During t h e o x i d a t i o n t h e s u b s t r a t e i s connec-

t e d t o g round , which is t h e anode o f t h e d i s c h a r g e . The dependence of

t h e j u n c t i o n c h a r a c t e r i s t i c s on t h e r e s i s t a n c e t o ground is presumably

r e l a t e d t o t h e v o l t a g e o f t h e f i l m d u r i n g o x i d a t i o n which i n f l u e n c e s

t h e d i s c h a r g e n e a r t h e niobium f i l m .

Daalmans (9) a r g u e s , t h a t t h e niobium o x i d e n e a r t h e p i n h o l e s i n - t h e s i l i c o n p robab ly a c t s a s t h e j u n c t i o n b a r r i e r . The f a b r i c a t i o n of

t u n n e l j u n c t i o n s w i t h a r t i f i c i a l b a r r i e r s l i k e s i l i c o n h a s a lways been

d i f f i c u l t because o f p i n h o l e s . For t h i s r e a s o n i t is c o n s i d e r e d t o be

v e r y d i f f i c u l t t o make a t h i n s i l i c o n f i l m w i t h o u t p i n h o l e s . T h i s

e x p l a n a t i o n is s u p p o r t e d by t h e dependence of t h e j u n c t i o n p a r a m e t e r s

on t h e o x i d a t i o n time. Longer o x i d a t i o n t ime y i e l d s a larger r e s i s -

t a n c e . Fur thermore t h e r e s i s t a n c e i s n o t s t r o n g l y dependent on t h e

s i l i c o n t h i c k n e s s . Thus f a r we have n o t found any i n d i c a t i o n o f t h e

number and t h e s i z e of t h e p i n h o l e s . Another p o s s i b i l i t y might be t h a t

t h e c u r r e n t i s f l o w i n g n e a r t h e edge o f t h e j u n c t i o n . Th i s would c a u s e

a c r i t i c a l c u r r e n t p r o p o r t i o n a l t o t h e l i n e a r d imensions of t h e

j u n c t i o n s i n s t e a d of t h e a r e a , which would e x p l a i n t h e good r e p r o -

d u c i b i l i t y o f t h e j u n c t i o n p a r a m e t e r s . Daalmans a l r e a d y o b s e r v e d , t h a t

t h e j u n c t i o n s have a c o n s i d e r a b l e e x c e s s c u r r e n t a t h i g h v o l t a g e s . The

I-V c u r v e s o f o u r j u n c t i o n s w i t h a t h i c k s i l i c o n f i l m resemble t h e

c u r v e s o f Daalmans, i f t h e y have a h i g h I, R p r o d u c t ( o f t h e o r d e r o f

1 mV). According t o Blonder e t a l . (26) t h e e x c e s s c u r r e n t i s due t o a

v e r y h i g h c r i t i c a l c u r r e n t d e n s i t y , c l o s e t o t h e c u r r e n t d e n s i t y o f -2

t h e p u r e m e t a l . I f we assume a c u r r e n t d e n s i t y o f 1 0 ' ' A-m and a 2

c r i t i c a l c u r r e n t o f 2 V A t h e e f f e c t i v e j u n c t i o n area i s 20 nm , which

is o n l y 1 - 1 0 - ~ t i m e s t h e t o t a l o v e r l a p p i n g a r e a o f t h e j u n c t i o n . Th i s

s u p p o r t s t h e h y p o t h e s i s of Daalmans, t h a t t h e c u r r e n t is f l o w i n g

th rough t h e p i n h o l e s i n t h e s i l i c o n .

The niobium j u n c t i o n s c a n be d e s t r o y e d by i n c r e a s i n g t h e tempera-

t u r e t o a b o u t 100 T. Probably t h i s i s due t o t h e n a t u r e o f t h e o x i d e

o f t h e j u n c t i o n b a r r i e r . There is e v i d e n c e , t h a t t h e o x i d a t i o n o f

niobium b e g i n s w i t h t h e f o r m a t i o n o f l e s s s t a b l e o x i d e s ( 2 7 ) l i k e NbO - and Nb02. These o x i d e s have m e t a l l i c p r o p e r t i e s . Because t h e j u n c t i o n s

w i t h a large c r i t i c a l c u r r e n t need l o c a l l y a v e r y h i g h c u r r e n t

d e n s i t y , o u r b a r r i e r p robab ly w i l l c o n t a i n s u c h less s t a b l e o x i d e s .

The j u n c t i o n s can a l s o be d e s t r o y e d by e l e c t r o s t a t i c d i s c h a r g e s .

G e n e r a l l y t h e j u n c t i o n s c a n be burned o u t by a d c v o l t a g e of t h e o r d e r

of 1 V. We t h i n k t h i s i s due t o h e a t i n g of t h e j u n c t i o n t o above

100 T. I f we assume a l o c a l h e a t i n g i n t h e p l a n e o f t h e j u n c t i o n

b a r r i e r and a c o n d u c t i o n th rough niobium c u b e s of s i z e 0.2 pm a t bo th

s i d e s w i t h a the rmal c o n d u c t i v i t y of 50 W / m . K , we g e t a v o l t a g e

n e c e s s a r y t o h e a t a j u n c t i o n of 200 0 from 4 K t o 400 K of 1 V . T h i s

t e m p e r a t u r e is reached a f t e r a b o u t 1 n s . These f i g u r e s show, t h a t a

p o s s i b l e e x p l a n a t i o n of t h e burn-outs cou ld be a n i n c r e a s e d d i f f u s i o n

due t o a h i g h t e m p e r a t u r e a t t h e b a r r i e r .

The j u n c t i o n s a r e v e r y s t a b l e w i t h r e s p e c t t o t h e r m a l c y c l i n g and

s t o r a g e o v e r l o n g p e r i o d s . We never obse rved a change o f t h e I - V

c h a r a c t e r i s t i c a f t e r s e v e r a l t i m e s c o o l i n g from room t e m p e r a t u r e t o

4 K. Also measurements a f t e r a n i n t e r v a l o f one y e a r d i d n o t show any

change.

The s t a n d a r d p rocedure i n f a b r i c a t i n g SQUIDs i s t o c o n t r o l t h e

b a r r i e r t h i c k n e s s t o g e t t h e r i g h t c r i t i c a l c u r r e n t . The r e s i s t a n c e is

reduced t o t h e p r o p e r v a l u e by a s h u n t of g o l d o r a n a l l o y ( 3 , 6 , 8 ) t o - - - p r e v e n t h y s t e r e s i s . For our j u n c t i o n s we u s e a d i f f e r e n t method t o f i x

t h e j u n c t i o n r e s i s t a n c e . The I , ,R p r o d u c t o f t h e j u n c t i o n s depends on

t h e t h i c k n e s s o f t h e s i l i c o n f i l m and on t h e o x i d a t i o n p a r a m e t e r s . A s

shown by Daalmans ( 9 ) t h e j u n c t i o n s w i t h a t h i n s i l i c o n f i l m ( 2 nm) - have t h e r e l a t i v e l y l a r g e b R p r o d u c t of 1 m V o r even more. These

j u n c t i o n s g e n e r a l l y a r e h y s t e r e t i c a t 4.2 K , which r e n d e r s them

u n s u i t a b l e f o r a p p l i c a t i o n i n SQUIDs w i t h o u t t h e u s e o f a n e x t e r n a l

s h u n t . The j u n c t i o n s we u s e f o r SQUID a p p l i c a t i o n s , w i t h I,,-1.5 yA and

R=200 n have a n o n - h y s t e r e t i c I - V c u r v e a t 4.2 K . These j u n c t i o n s w i t h

a 20 nrn s i l i c o n f i l m have a n I,,R p roduc t o f 300 t o 700 pV. The

j u n c t i o n s w i t h a l a r g e r r e s i s t a n c e t e n d t o have a s m a l l e r I, R p r o d u c t .

The c a p a c i t a n c e of t h e s e j u n c t i o n s i s e s t i m a t e d a t 1 . 1 0 - ~ ~ F from

r e s o n a n c e s t r u c t u r e i n t h e I - V c u r v e s o f t h e SQUIDs (28), a s d i s c u s s e d - i n s e c t i o n V . C a l c u l a t i n g t h e c a p a c i t a n c e from t h e geometry of t h e

j u n c t i o n , we g e t a t l e a s t 5 . 1 0 - ~ ~ F f o r t h e c a p a c i t a n c e o f t h e banks

a t t a c h e d t o t h e j u n c t i o n ( s e e Appendix B) and 1.10 -I5 F f o r t h e

c a p a c i t a n c e of t h e o v e r l a p p i n g p a r t of t h e j u n c t i o n s e p a r a t e d by t h e 2

s i l i c o n assuming a j u n c t i o n a r e a of 0.2 ym . We c a n n o t e s t i m a t e t h e

c a p a c i t a n c e o f t h e niobium o x i d e p a r t of t h e j u n c t i o n . However, if w e

assume t h a t t h e p i n h o l e s i n t h e s i l i c o n t a k e a 20% p a r t o f t h e

j u n c t i o n a r e a , which seems r a t h e r l a r g e , w e g e t a c a p a c i t a n c e of

5.10-l5 F f o r t h e niobium o x i d e . These f i g u r e s a g r e e i n o r d e r o f

magni tude w i t h t h e c a p a c i t a n c e o f abou t I - I O - ~ ~ F e s t i m a t e d from t h e

measurements. I f t h e t e m p e r a t u r e is reduced t o 1 K t h e c r i t i c a l

c u r r e n t i n c r e a s e s w i t h a f a c t o r o f two. H y s t e r e s i s a p p e a r s between 3

and 4 K due t o t h e l a r g e r c r i t i c a l c u r r e n t and t h e s m a l l e r n o i s e

2 rounding. At 4.2 K the junctions are hysteretic if I,, R reaches 5 2 10 pA.n . With the estimated capacitance of 1-10 -14

F this means,

that the junctions become hysteretic for 0 =3. This is consistent C

with the calculation of Voss (30) for a single junction with a shunt - capacitor in the presence of thermal noise. It is also consistent with

the experimental observations of Voss et a 1 1 0 who found no - hysteresis at values of 0 up to 5 in the case of junctions with small

C

critical current.

111.5 Performance of SQUID and input coil

111.5.1 Performance of the SQUID

The SQUIDs are mounted in a superconducting lead shielded environ-

ment in a vacuum can or in a helium bath at 4.2 K. We investigated

SQUIDs of about 4 nH, 1 nH (Fig. 111.6) and 2 nH (Fig. 111.7). The

resistance is typically 60 to 150 n, the critical current is 2 to 5 pA and the I, R product is 0.3 to 0.4 mV. As stated before, critical

currents and resistances of the SQUID can be reproduced in different

evaporation runs within a range of a factor of 2. This makes it likely

that the asymmetries in the critical current and the resistance are

small also. Figs. 111.9 and 10 show the I-V curves of a 5 and a 1 nH

dc SQUID for different values of the flux inside the ring. The

vol tage V ( pV)

Fig. 111.9

I - V curves of a li nH dc S Q U I D . The

two curves were measured with 0.5 Q

difference in flux in the S Q U I D .

0 0 u 200 LOO 600

v o l t a g e V(pV)

F i g . 111 .10

I-V c u r v e s o f a 1 nH d c SQUID.

The t w o c u r v e s were measured

w i t h 0 . 5 O,, d i f f e r e n c e i n f l u x

i n t h e SQUID.

applied f lux drQo)

F i g . I I I . 1 1

V-0 c u r v e s o f t h e 1 nH d c SQUID a t v a r i o u s v a l u e s o f t h e b i a s a

c u r r e n t . In t h e f i g u r e t h e parame te r i s t h e b i a s c u r r e n t i n ud.

dependence o f t h e v o l t a g e on t h e f l u x of t h e 1 nH SQUID i s shown i n

Fig . 111.11. The symmetry o f t h e c u r v e s i n d i c a t e s a good symmetry o f

t h e i n d u c t a n c e and t h e j u n c t i o n pa ramete r s ( 4 ) . The v o l t a g e modula t ion - of t h e $ nH SQUIDs i s 50 t o 100 pV. The 1 nH SQUIDs have a modula t ion

o f a b o u t 50 pV and t h e l a r g e i n d u c t a n c e SQUIDs 5 t o 10 pV. The I - V

c u r v e s a lways show resonances ( 28 ) due t o t h e c a p a c i t a n c e of t h e

j u n c t i o n s and t h e i n d u c t a n c e o f t h e r i n g . The Q o f t h i s r e s o n a n t

c i r c u i t i s e s t i m a t e d t o be between 1 and 6 f o r v a r i o u s SQUIDs. As t h e

Q becomes l a r g e f o r t h e s m a l l i n d u c t a n c e SQUIDs w i t h l a r g e r c a p a c i -

t a n c e , t h e s t r u c t u r e i n t h e I - V c u r v e s becomes more pronounced. From

t h e r e s o n a n c e f requency we can de te rmine t h e r a t i o C/L, assuming t h a t

both j u n c t i o n s have t h e same c a p a c i t a n c e . With a n e s t i m a t e d i n d u c t a n c e

o f 1 nH o f t h e SQUID o f F ig . 111.10 we get a j u n c t i o n c a p a c i t a n c e o f

l . l ~ - ~ ~ t o 2.10-l4 F. SQUIDS w i t h t h e s e pa ramete r s a r e never h y s t e r -

e t i c . The 4 nH SQUIDs c o n t a i n j u n c t i o n s w i t h l a r g e r banks up t o 0.3 mm -14

wide. 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 was e s t i m a t e d a t 3-10 t o

4 - 1 0 - ~ ~ F l e a d i n g t o a Bc o f a b o u t 3 t o 6 . These SQUIDs were sometimes

h y s t e r e t i c .

For n o i s e measurements t h e SQUIDs a r e s c r e e n e d by a superconduc-

t i n g niobium c y l i n d e r i n s i d e a vacuum chamber w i t h hel ium exchange

g a s . The SQUID is r e a d o u t w i t h a s t a n d a r d f l u x l o c k e d l o o p w i t h a

100 kHz modula t ion s i g n a l ( 3 ) . The impedance o f t h e SQUID is matched - w i t h a r e s o n a n t c i r c u i t w i t h Q = 5 t o 10 t o a p r e a m p l i f i e r w i t h a n

optimum s o u r c e impedance o f 100 k n and a n o i s e t e m p e r a t u r e o f 1 K.

Before c l o s i n g t h e f l u x l o c k e d loop t h e b i a s c u r r e n t , t h e d c f l u x and

t h e modula t ion s i g n a l a r e a d j u s t e d t o g e t maximum s i g n a l from t h e

r e s o n a n t c i r c u i t . G e n e r a l l y t h i s p rocedure y i e l d s t h e l o w e s t n o i s e .

The n o i s e o f t h e SQUID is measured a t t h e o u t p u t o f t h e f l u x locked

l o o p . For t h e SQUID of F i g s . 111.10 and 111.11 we measured a n o i s e o f

6 . 1 0 - ~ a, -HZ-' a t 4.2 K. T h i s c o r r e s p o n d s t o a n i n t r i n s i c energy

r e s o l u t i o n o f 8 . 1 0 - ~ ~ JIHz, which is t h e l o w e s t f i g u r e we ach ieved a t

t h i s t e m p e r a t u r e . Genera l ly we cou ld n o t improve t h e n o i s e performance

by a d j u s t i n g t h e b i a s c u r r e n t o r t h e a m p l i t u d e of t h e modula t ion

s i g n a l . We a l s o o b s e r v e no d i f f e r e n c e i n t h e performance between

l o c k i n g t h e SQUID i n a maximum o r a minimum o f t h e c r i t i c a l c u r r e n t .

The model o f a dc SQUID (4 ,171 mentioned above g i v e s a n energy - - r e s o l u t i o n o f JIHz f o r a SQUID w i t h R=150 0 and L-1 nH i n a

f l u x locked l o o p . Probably t h e SQUID n o i s e i n c r e a s e s due t o o u r large

Bc. When we measured t h e same SQUID a t 5 K w e found a n o i s e of

4.10-32 JIHz. A t t h i s t e m p e r a t u r e t h e c r i t i c a l c u r r e n t of t h e SQUID

was 3 P A . The s m a l l e r Bc p robab ly produced t h e smaller SQUID n o i s e .

Th i s SQUID had a reduced width of t h e s t r i p s l e a d i n g t o t h e j u n c t i o n s .

With SQUIDs hav ing wider s t r i p s , many o f them hav ing a l s o a s m a l l e r

induc tance ,we never found a n energy r e s o l u t i o n lower t h a n

2 . 1 0 - ~ ' JIHz. Those SQUIDs were h y s t e r e t i c more o f t e n , which agrees

w i t h a l a r g e r j u n c t i o n c a p a c i t a n c e . The n o i s e was measured a t f r equen-

c i e s from 20 Hz t o 5 kHz w i t h a 3 Hz wide f i l t e r a t t h e o u t p u t o f t h e

f l u x locked l o o p . The n o i s e spect rum g e n e r a l l y was w h i t e . Sometimes a

l l f component was found a t f r e q u e n c i e s below 1 kHz. The l a r g e s t l / f

n o i s e component e v e r measured was 3 . 1 0 - ~ ~ ( 1 ~ z / f ) JIHz. I n most c a s e s

we measured, t h a t t h e l / f component, i f p r e s e n t , was a t least below

4 . 1 0 - ~ ~ ( 1 ~ z / f ) J/Hz.

111.5.2 Performance of t h e coup led SQUIDs

S e v e r a l t h i n f i l m c o i l s o f t h e t y p e o f F ig . 111.8 as d e s c r i b e d

above were made and t e s t e d . A t room t e m p e r a t u r e t h e c o i l s were s h o r t e d

by t h e s u b s t r a t e t o a b o u t 10 n. A t a t e m p e r a t u r e of a b o u t 10 K we

found a r e s i s t a n c e o f t h e c o i l o f abou t 10 k n . T h i s r e s i s t a n c e c o u l d

be compared w i t h t h e r e s i s t a n c e o f a s t r i p on t h e same s u b s t r a t e t o

check f o r s h o r t s a t t h e c r o s s - o v e r . A t 4.2 K t h e i n d u c t a n c e of t h e

c o i l s was 1.5 pH, which a l s o i s a t e s t f o r s h o r t s . A v e r y i m p o r t a n t

p r o p e r t y o f such c o i l s i s t h e r e s i s t a n c e o f t h e c o i l a t 4.2 K . A s t h e

two niobium f i l m s a r e c o n t a c t e d th rough a g o l d f i l m , which i s n o t

s u p e r c o n d u c t i n g a p r i o r i , t h e r e s i s t a n c e must be measured. I f t h e g o l d

f i l m would remain normal , t h e r e s i s t a n c e would be I O - ' ~ . By c o n n e c t i n g

t h e c o i l t o t h e i n p u t c i r c u i t of a SQUID, making t h e c o n t a c t t o t h e

c o i l w i t h niobium s c r e w s , a n d a p p l y i n g a magne t i c f i e l d t o t h e c o i l we

c o u l d measure t h e decay t i m e of t h i s c i r c u i t . We found, t h a t t h e

r e s i s t a n c e , if p r e s e n t , was a t l e a s t s m a l l e r t h a n 4 - 1 0 - ' ~ n . A s t h i s

r e s i s t a n c e i s much s m a l l e r t h a n t h e expec ted r e s i s t a n c e o f t h e g o l d

f i l m i n t h e normal s t a t e , we c o n c l u d e , t h a t t h e g o l d f i l m becomes

s u p e r c o n d u c t i n g due t o t h e p rox imi ty e f f e c t . The c r i t i c a l c u r r e n t o f

t h e c o i l s is a t l e a s t 10 m A . These c o i l s have proven t o be v e r y

r o b u s t . We never obse rved any damage due t o t h e r m a l c y c l i n g , t o

c l e a n i n g t h e c o i l w i t h a t i s s u e o r t o s o l d e r i n g t o t h e c o n t a c t pads .

F ig . 111.7 shows a photograph of t h e SQUID des igned f o r c o u p l i n g t o

t h e c o i l s . Using t h e marks on t h e s u b s t r a t e o f t h e c o i l t h e SQUID and

t h e c o i l a r e p o s i t i o n e d o p p o s i t e t o each o t h e r w i t h a n a c c u r a c y o f

0.1 mm. The r i n g i s broad enough t o coup le t o t h e 20 t u r n c o i l and

s t i l l t o a l l o w a n e r r o r i n t h e p o s i t i o n i n g o f 0 .3 mm. The s u b s t r a t e s

a r e e l e c t r i c a l l y i n s u l a t e d from each o t h e r w i t h a 7 pm t h i c k p o l y e s t e r

f o i l . Then t h e two s u b s t r a t e s a r e g l u e d t o g e t h e r w i t h a n epoxy r e s i n .

I f n e c e s s a r y t h e SQUID and t h e c o i l can e a s i l y be removed from each

o t h e r and be r e u s e d . F i g . 111.12 shows I - V c u r v e s of t h i s

c o n f i g u r a t i o n . The most i m p o r t a n t parameter t o measure i s t h e c o u p l i n g

c o e f f i c i e n t k2 between c o i l and SQUID. Of one o f t h e s e d e v i c e s we

measured a mutual i n d u c t a n c e of 38 nH. The i n p u t i n d u c t a n c e c a n be

measured w i t h c o n v e n t i o n a l room t e m p e r a t u r e e l e c t r o n i c s . I f t h e b i a s

c u r r e n t o f t h e SQUID is f a r above t h e c r i t i c a l c u r r e n t we measure t h e

i n p u t i n d u c t a n c e L . I n t h i s c a s e t h e impedance of t h e Josephson C

j u n c t i o n s i s much l a r g e r t h a n t h e impedance o f t h e SQUID r i n g a t t h e

f r e q u e n c y of t h e measurement, and hence t h e SQUID w i l l behave a s a n

open c i r c u i t . The measured i n d u c t a n c e is 1.2 pH. For z e r o b i a s c u r r e n t

t h e SQUID i n d u c t a n c e i s s h o r t e d by t h e Josephson j u n c t i o n s i f t h e

Josephson i n d u c t a n c e h/4nel , i s much 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 SQUID r i n g . For t h e SQUID w i t h a c r i t i c a l c u r r e n t o f 50 pA we

c a l c u l a t e a Josephson i n d u c t a n c e o f 0.04 nH, which is much 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 r i n g . I n t h i s c a s e we measure a n i n p u t

i n d u c t a n c e o f 0.62 pH. It i s e a s i l y shown t h a t t h e r a t i o o f t h e s e two 2

i n d u c t a n c e s i s 1-k . From t h e s e r e s u l t s we c a n c a l c u l a t e t h e SQUID

i n d u c t a n c e of 2 .3 nH and a c o u p l i n g e f f i c i e n c y k2 of 0.50. T h i s 2 . 3 nH

i s t h e most a c c u r a t e v a l u e o f t h e SQUID i n d u c t a n c e we can g i v e . It i s

i n good agreement w i t h t h e v a l u e e s t i m a t e d i n Sec. 111.2. We e s t i m a t e

a c c o r d i n g t o Eq. 111.9 t h a t t h e c o u p l i n g e f f i c i e n c y i s t h e p r o d u c t o f

a f a c t o r o f 0.8 due t o t h e l o s s a t t h e o u t e r p a r t o f t h e SQUID r i n g

n e a r t h e j u n c t i o n s and a f a c t o r of 0.6 due t o t h e d i s t a n c e between

c o i l and SQUID. The i n p u t i n d u c t a n c e 1.2 nH is l a r g e r t h a n c a l c u l a t e d

i n S e c . I I I . 2 . This i s p robab ly due t o a larger d i s t a n c e between t h e

two s u b s t r a t e s , which c a u s e s t h e s t r i p l i n e i n d u c t a n c e L t o i n c r e a s e S

t o 0.4 pH. However, t h e c o u p l i n g e f f i c i e n c y is r e l a t i v e l y good and

o n l y l i t t l e c a n be g a i n e d by improving t h e c o u p l i n g .

These SQUIDS were t e s t e d w i t h t h e c o i l s coup led t o them. The

p r o p e r t i e s o f t h e SQUID s t r o n g l y depend on t h e l o a d o f t h e i n p u t c o i l .

Fig. 1 1 1 . 1 2

I - V c u r v e s o f a 2 nH d c S Q U I D

c o u p l e d t o a s p i r a l i n p u t

c o i l . The two c u r v e s were

measured wi th 0.5 O0

d i f f e r e n c e i n f l u x i n t h e

S Q U I D .

Fig. I I I . 1 3

( a ) V-@ c u r v e o f a 2 nH d c a

2 - 30n S Q U I D c o u p l e d t o a s p i r a l

> i n p u t c o i l . The i n p u t c o i l i s 0

20 .- l o a d e d w i t h a 1 kn r e s i s t o r . d

0 > The S Q U I D is b i a s e d a t t h e 10

optimum b i a s p o i n t .

( b ) V-@ c u r v e o f t h e 2 nH 0

a

0 1 0 1 S Q U I D c o u p l e d t o a s p i r a l

applied f l u x $ a ( $ o ) i n p u t c o i l .

For i n s t a n c e F i g s . I I I . 1 3 a and I I I . 1 3 b show t h e dependence of t h e

v o l t a g e on t h e f l u x f o r a SQUID w i t h c o i l loaded w i t h a 1 k n r e s i s t o r

o r open. Q u a l i t a t i v e l y t h e V-Q, c u r v e o f a SQUID w i t h c o i l a lways

resembles t h e c u r v e of Fig . I I I . 1 3 b . Probably t h e p a r a s i t i c c a p a c i -

t a n c e s i n t h e c o i l and a t t h e o u t p u t o f t h e c o i l i n f l u e n c e t h e SQUID.

Without t h e c o i l t h e v o l t a g e modula t ion is 5 t o 10 pV b u t i f t h e c o i l

i s coup led t o t h e SQUID t h e modula t ion i n c r e a s e s t o between 10 and -4

25 ~ I V . The f l u x r e s o l u t i o n improves from 1.0.10 $ .HZ-' t o -3 3 . 6 . 1 0 ~ ~ ~ -Hz ' . With t h e measured mutual i n d u c t a n c e o f 39 nH we f i n d

a n o v e r a l l ene rgy r e s o l u t i o n o f 2 . l 0 - ~ ~ J /Hz. The optimum c u r r e n t b i a s

p o i n t i s n o t a lways a t t h e p o i n t o f maximum v o l t a g e modula t ion . Th i s

is i n agreement w i t h t h e c a l c u l a t i o n o f Tesche ( 2 9 ) - c o n c e r n i n g a d c

SQUID w i t h a c a p a c i t a n c e s h u n t i n g t h e SQUID i n d u c t a n c e . If t h e i n p u t

c o i l i s connec ted t o a 1 kn r e s i s t o r t h e o v e r a l l ene rgy r e s o l u t i o n

improves t o l . 2 . 1 0 - ~ ~ JIHz. Th i s i s p robab ly due t o t h e larger

averaged t r a n s f e r f u n c t i o n ( F i g . I I I . 3 a ) . The i n t r i n s i c energy

r e s o l u t i o n is sti l l a f a c t o r o f 10 l a r g e r t h a n t h e r e s o l u t i o n o f t h e

1 nH SQUID. T h i s i s due t o t h e r e l a t i v e l y l a r g e SQUID i n d u c t a n c e of

2 nH, which produces a modula t ion smaller by a f a c t o r o f 5 . Yet no

measurement was performed w i t h a c o i l coup led t o t h e i n p u t c o i l , which

is a c o n f i g u r a t i o n used f o r many a p p l i c a t i o n s . I n most c a s e s t h e

p ickup c o i l w i l l have t h e optimum i n d u c t a n c e ( 3 ) - e q u a l t o t h e induc-

t a n o e o f t h e i n p u t c o i l . Then t h e e f f e c t i v e i n d u c t a n c e s e e n by t h e

j u n c t i o n s w i l l be reduced t o 1.5 nH. For t h i s c a s e w e e x p e c t a b e t t e r

ene rgy r e s o l u t i o n , approach ing t h e r e s o l u t i o n o f t h e 1 nH SQUID.

Bes ides t h e method o f c o u p l i n g t o a t h i n f i l m c o i l , t h e r e is a l s o

t h e p o s s i b i l i t y o f c o u p l i n g t h e s e SQUIDS e f f i c i e n t l y t o a w i r e wound

c o i l a s d e s c r i b e d i n Sec. 111.2. For t h i s purpose w e u s e d t h e same

SQUID d e s i g n a s f o r t h e t h i n f i l m c o i l s . The c o i l s c o n s i s t of 20 t u r n s

o f 0.13 mm t h i c k w i r e on a 4 mm d i a m e t e r niobium c o r e . The mutual

i n d u c t a n c e between c o i l and SQUID is 4 nH. The i n p u t i n d u c t a n c e i s

a b o u t 90 nH. The niobium c o r e 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 from

2 nH t o l e s s t h a n 0.3 nH. The l o w e s t measured f l u x r e s o l u t i o n o f t h e

SQUID i n t h i s c o n f i g u r a t i o n is 1 . 4 . 1 0 - ~ 0 ~ -HZ-'. Th i s c o r r e s p o n d s t o a n

energy r e s o l u t i o n w i t h r e s p e c t t o t h e i n p u t c o i l o f ~ . I o - ~ ~ J/Hz.

Although the input inductance of the coil used is rather small, this

is not a serious limit because this inductance can be made much

larger with a thin wire with more turns. Also at the inner side of the

core a number of turns can be wound. This would have a negligible

effect on the coupling efficiency. The coupling efficiency k2 for this

configuration is 0.6 to 0.7. By changing various dimensions of SQUID

and coil also the coupling efficiency can be increased.

111.6 Gradiometer performance

The gradiometers are measured directly inside the helium bath.

They are electromagnetically shielded by a superconducting lead shield

from disturbances from outside the helium bath. The gradiometers were

made with a critical current of 1 to 10 PA and a resistance of 100 to

300 n. The &,R product is typically 600 pV. Most of the gradiometers

were slightly hysteretic. Biased in a point without hysteresis, they

can still be used. The maximum change in voltage when changing the

magnetic flux in the SQUID is 5 to 10 pV for a SQUID with an I,, R

product of 600 P V . That the signal from the gradiometer is so small is

due to the large inductance (2.7 nH) of the SQUID. An inductance near

3 nH will result in a small voltage modulation due to thermal noise

(22). - To determine the sensitivity a magnetic gradient is applied to

the gradiometer with a quadrupole coil. The mean gradient in the plane

of the gradiometer is known by calculation to better than 10%. We -8 -1 -1

measured a transfer ratio of the gradiometer of 3.5-10 T-m .Q, . The noise of the gradiometer, measured in a flux locked loop, is

typically 1.0.10-~0,, .HZ-'. This flux noise i's 10 times as high as the

noise of a 1 nH SQUID, which is reasonable if we take into account the

voltage modulation which is 10 times smaller than in the case of the -12 -1 -1

SQUID. This implies, that the gradient noise is 3.5'10 T-m .Hz ' . This sensitivity is comparable to the figure of the much larger system

of Ketchen et al. (11) - and to a planar wire wound gradiometer coupled to a conventional SQUID (31). The gradiometers were mounted in a - fiberglass dewar. Inside the dewar a thin metal foil screening was

mounted at a distance of 0.8 mm. Due to the reduction of the induc-

tance of the SQUID by the metal foil the voltage modulation increased

and the flux resolution improved to ~.Io-~@, .HZ-'. With this arrange-

ment magnetocardiograms were made without any adjustment of the

balance of the gradiometer. With the same arrangement we measured the

balance of the gradiometer for perpendicular fields. The sensitivity

for a homogeneous magnetic field from a Helmholtz coil was

7.10-~ TI*,, which corresponds to an intrinsic balance of 300 ppm.

This balance is reasonable in view of the 3 pm resolution of the

fabrication of the large pattern. We expect a better balance for

perpendicular fields as the substrate is flat within about 1 pm. To

improve the balance we used a second SQUID magnetometer. By adding the

two signals from the SQUID and the magnetometer we could improve the

balance by a factor of 15.

As mentioned above sometimes the junctions burn out due to

electrostatic discharges. With the gradiometers we got similar

problems even if we had taken careful precautions. This only happened

if the gradiometer was in the helium for several weeks. We believe

that occasionally the current in one of the strips in the gradiometer

passes the critical currrent. This can be illustrated with a configu-

ration as in Fig. 111.2. We assume L1>>Ls. The current will heat the

film, and the voltage across the normal region becomes larger. This

process will not stop before a large part of the energy of the

inductor coupled to this normal part will be unloaded. If the normal

region is in parallel with the junctions, a voltage of 1 V can develop

across them, which could be enough to burn them out. If the normal

spot is in the inductor L the voltage across the junctions will be s' the largest. We expect that this is a general problem with this

configuration and that it can occur also with larger junctions which

are not so sensitive. In a magnetometer, or any other system which

consists of only one ring, no large energy can be stored because the

flux immediately leaks out at the junctions. A possible solution would

be the incorporation of a constriction in the large conductor L Then 1' the voltage across the junctions is LS/L1 times the voltage across the

normal r e g i o n . An o t h e r s o l u t i o n is t h e use o f a s e r i e s ar rangement of

t h e g r a d i o m e t e r . Then t h e large c i r c u l a t i n g c u r r e n t s a r e avo ided .

111.7 Conclus ion

T h i s paper shows t h a t p r a c t i c a l low n o i s e d c SQUIDs c a n be made

w i t h submicron niobium Josephson j u n c t i o n s . The niobium j u n c t i o n s are

c h e m i c a l l y r e s i s t a n t and s t a b l e d u r i n g the rmal c y c l i n g o r s t o r a g e o v e r

l o n g p e r i o d s . The d e f i n i t i o n o f t h e j u n c t i o n s i z e s i s good enough t o

r e p r o d u c e t h e c r i t i c a l c u r r e n t s and r e s i s t a n c e s w i t h i n a f a c t o r of 2 ,

even though t h e j u n c t i o n dimensions can n o t be reproduced w i t h i n t h i s

r a n g e .

The b e s t measured energy r e s o l u t i o n o f t h e 1 nH SQUID is

J IHz , which is 60 t i m e s P l a n c k ' s c o n s t a n t . Although t h i s

r e s o l u t i o n i s very low f o r a 1 nH SQUID i t i s s t i l l a f a c t o r of 10

h i g h e r t h a n p r e d i c t e d by t h e computer model. I n t h i s model t h e s h u n t

c a p a c i t a n c e of t h e 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 . Probably t h e

n o i s e is l i m i t e d by t h e r e l a t i v e l y l a r g e Bc of t h e Josephson

j u n c t i o n s . Such a n i n c r e a s e o f t h e n o i s e o f t h e j u n c t i o n is a l s o

p r e s e n t i n a s i n g l e j u n c t i o n (30). The SQUIDs w i t h a t h i n f i l m i n p u t c o i l r e a c h e d a n energy r e s o l u -

t i o n o f 1 . 2 - 1 0 - ~ ~ JIHz r e f e r r e d t o t h e i n p u t c o i l . A s far as w e know

t h i s i s t h e b e s t c o u p l e d energy r e s o l u t i o n y e t a c h i e v e d w i t h a n a l l

niobium t h i n f i l m SQUID. With a w i r e wound i n p u t c o i l w e r e a c h e d a

r e s o l u t i o n o f ~ . I o - ~ ~ JIHz. Th i s r e s u l t shows t h a t i t i s p o s s i b l e t o

c o u p l e a low n o i s e t h i n f i l m SQUID e f f i c i e n t l y t o a w i r e wound c o i l .

These f i g u r e s were measured w i t h r e s p e c t t o t h e i n p u t c o i l u s i n g a

f l u x locked l o o p , which is t h e r e l e v a n t measurement c o n d i t i o n f o r

p r a c t i c a l a p p l i c a t i o n s . For both t y p e s o f c o n f i g u r a t i o n w e e x p e c t

f u r t h e r improvement i f t h e i n p u t c o i l i s coupled t o a n i n d u c t i v e

pickup c o i l due t o a r e d u c t i o n of t h e e f f e c t i v e i n d u c t a n c e of t h e

SQUID.

The r e s u l t s w i t h t h e t h i n f i l m g r a d i o m e t e r show t h e p o s s i b i l i t y t o

make ve ry s e n s i t i v e d e v i c e s u s i n g o n l y a s m a l l t o t a l a r e a . T h i s i s

interesting for applications where a limited area is available or for

use in an array of gradiometers and SQUIDS. The sensitivity of

3.10-l2 Tam -1 .Hz - ' is good enough for magnetocardiography and many

other applications. It is comparable to the resolution of other much

larger systems. Yet improvement of the sensitivity is possible.

Because of the low noise and the good stability of the junctions

the system is promising for practical applications. As the energy

resolution referred to the input coil is much larger than the figure

for a single SQUID and the theoretical limits, we still expect large

improvements to be possible. Without essentially changing the design

of the devices they can be made very small for application in a closed

cycle refrigerator.

Appendix A

Calculation of the gradiometer inductance

The gradiometer contains a configuration of n connected straight

segments. To calculate the inductance seen by the junctions we first

determine the self-inductances L. and the mutual inductances M. of J ~k

the segments. For the mutual inductances we use the formula for

straight thin filaments ( 3 2 ) . For the self-inductance the formula for - a strip with uniform current flow is used. It is also possible to use

a correction for a superconducting strip (23). - The flux Q through any 1

loop composed of the straight segments is

where 1 is the number of the loop, n and m are the numbers of the

segments. u is + I depending on the direction of the current if nl -

segment n is part of loop 1, and zero otherwise. The same equation

applies to the flux d seen by the junctions. The flux through any 1

closed loop is zero. The total current into an intersection of

segments must be zero:

If an input current I is chosen, the other currents are calculated by

solving equations A1 and A2. Then one of the equations A1 yields the

flux B The inductance to be calculated is is B /I. 1 ' 1

Appendix B

Estimation of the parasitic capacitance

For very small Josephson junctions the parasitic capacitance due

to the electric field outside the oxide barrier can be a large part of

the total capacitance. Generally the situation is very complicated

because the leads to the junctions also have an inductance which

changes and reduces the influence of the capacitance behind it. We

assume that this circuit can be replaced by a single capacitance. The

capacitance which is seen by the junction in series with an inductance

larger than h/(4neL,) (the Josephson inductance) will not contribute

to the parasitic capacitance. In our case with junctions of 1.5 V A

this inductance is 0.2 nH. To estimate the order of magnitude of this

contribution to the total capacitance we made 100 times magnified

scale models of parts of the banks attached to the junctions out of

copper foil. Fig. 111.14 shows the configurations used. The estimated

inductances of the configurations of Fig. III.lQa,b,c and d are

respectively 0.2, 0.2, 0.3 and 0.4 nH and the measured capacitances

are 0.5, 1.1, 0.9 and 1.6 fF. The capacitances are not strongly

dependent on the length of the strip if it is much larger than its

width. The influence of the dielectric constant of the substrate is

simple to calculate because the charge is concentrated exactly on the

plane between the halfspaces with dielectric constants E of the 1

silicon and E of the free space. In each halfspace the field will be 2

the same as if the dielectric constant has the same value in the whole

space. This field satisfies the boundary conditions. So the parasitic

capacitance contains the factor (E +E )/2. For a silicon substrate 1 2

with a relative dielectric constant of 12 the capacitances must be

Four different geometries of which the capacitances were measured

using scale models.

multiplied by 6.5. The figures show that on a silicon substrate the

parasitic capacitance of banks without a constriction will be about

I - I o - ~ ~ F and the capacitance of the banks with constriction will be

about 5.10 -15 F. We do not consider narrower strips, because that

would produce a considerable increase of the inductance of the SQUID

loop.

References

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J.E. Zimmerman, P.Thiene and J.T.Harding, J.Appl.Phys.41,1572 - (1970)

J. Clarke, W.M. Goubau and M.B. Ketchen, J. Low Temp. Phys.25,99 -

(1976)

C.D. Tesche and J. Clarke, J.Low Temp.Phys.29,301 - (1977)

A.Th.A.M. de Waele and R. de Bruyn Ouboter, Phys.42,225 - (1969)

J.M. Jaycox and M.B. Ketchen, IEEE Trans-Magn. MAG-17,400 (1981 )

M.B. Ketchen and J.M. Jaycox, Appl.Phys.Lett.40,736 - (1982)

M.W. Cromar and P. Carelli, Appl.Phys.Lett.28,723 - (1981)

G.M. Daalmans, Superconducting Quantum Interference Devices and

Their Applications, H.D. Hahlbohm and H. Lubbig eds., Walter de

Gruyter, Berlin (1980), p. 399

10. R.F. Voss, R.B. Laibowitz, S.I. Raider and J. Clarke, J.App1.

Phys.51,2306 - (1980)

11. M.B. Ketchen, W.M. Goubau, J. Clarke and G.B. Donaldson, J.Appl.

Phys.49,4111 - (1978)

12. V.J. de Waal, P. van den Hamer and J.E. Mooij, Superconducting

Quantum Interference Devices and their Applications, H.D.Hahlbohm,

H.Lubbig eds., Walter de Gruyter, Berlin (1980), p.391

13. V.J. de Waal, P. van den Hamer and J.E. Mooij and T.M. Klapwijk,

IEEE Trans.Magn. MAG-17, 858 (1981) - 14. V.J. de Waal and T.M. Klapwijk, Appl.~hys.Lett.41,669 (1982) - 15. V.J. de Waal, P. van den Hamer and T.M. Klapwijk, Appl-Phys-Lett.

42,389 (1983) - 16. V. Radhadkrishnan and V.L. Newhouse, J.Appl.Phys.42,129 (1971) - 17. J.J.P.Bruines, V.J. de Waal and J.E. Mooij, J.Low Temp.Phys.46,383 -

(1982)

18. D.E. McCumber, J.Appl.Phys.39,3113(1968) - 19. J.H.Magerlein, IEEE Trans.Magn. MAG-17, 286( 1981 )

20. M.B. Ketchen, IEEE Trans.Magn. MAG-17,387 (1981)

21. W.H. Chang, J.Appl.Phys.50,8129 (1979) - 22. M. Tinkham, Introduction to superconductivity, McGraw-Hill, New

York (1975), p.212

23. C.M. Pegrum and G.B. Donaldson, Superconducting Quantum

Interference Devices and their Applications, W. de Gruyter, Berlin

(1980), H.D. Hahlbohm and H. Lubbig eds., p.535

24. G.M. Daalmans and J. Zwier, Future Trends in Superconductive

Electronics, Charlottesville (1978), B.S. Deaver, C.M. Falco,

J.H. Harris, S.A. Wolf eds.

25. H.A.M. van den Berg and J.B. van Staden, J.Appl.Phys.50,1212(1979) - 26. G.E. Blonder, M. Tinkham and T.M.Klapwijk, Phys.Rev.B25,4515 - (1982)

27. I. Lindau and W.E. Spicer, J.Appl.Phys.45,3720 - (1974)

28. S.M. Faris and E.A. Valsamakis, J.Appl.Phys.52,915 (1981) - 29. C.D. Tesche, J.Low Ternp.Phys.47,385 - (1982)

30. R.F. Voss , J.Low Temp. Phys .42,15 - 1 (1 98 1 )

31. D.Cohen, Proc. XI1 International Conference on Medical and

B i o l o g i c a l Engineer ing , Je rusa lem, I s r a e l , August 19-24, 1977,

p.15

32. W. Grover , Induc tance C a l c u l a t i o n s , Dover i n c . , New York (1946)

IV SIMULATION AND OPTWZATION OF A DC SQUID WITH FINITE CAPACITANCE

A b s t r a c t

T h i s paper d e a l s w i t h t h e c a l c u l a t i o n s of t h e n o i s e and t h e

o p t i m i z a t i o n o f t h e energy r e s o l u t i o n o f a dc SQUID w i t h f i n i t e

j u n c t i o n c a p a c i t a n c e . Up t o now n o i s e c a l c u l a t i o n s o f d c SQUIDS were

performed u s i n g a model w i t h o u t p a r a s i t i c c a p a c i t a n c e s a c r o s s t h e

Josephson j u n c t i o n s . A s t h e c a p a c i t a n c e s l i m i t t h e performance o f t h e

SQUID, f o r a good o p t i m i z a t i o n one must t a k e them i n t o a c c o u n t . The

model c o n s i s t s o f two coup led non l i n e a r second o r d e r d i f f e r e n t i a l

e q u a t i o n s . The e q u a t i o n s a r e v e r y s u i t a b l e f o r s i m u l a t i o n w i t h a n

a n a l o g c i r c u i t . We implemented t h e model on a h y b r i d computer. The

n o i s e s p e c t r u m from t h e model i s c a l c u l a t e d w i t h a f a s t f o u r i e r

t r a n s f o r m . A c a l c u l a t i o n o f t h e energy r e s o l u t i o n f o r one set o f

p a r a m e t e r s t a k e s abou t 6 minu tes o f computer t ime . D e t a i l e d r e s u l t s o f

t h e o p t i m i z a t i o n a r e g i v e n f o r p r o d u c t s o f i n d u c t a n c e and t e m p e r a t u r e

o f LTz1.2 nH.K and 5 nH.K. Within a r a n g e of B and B c between 1 and 2 ,

which is optimum, t h e energy r e s o l u t i o n is n e a r l y independen t of t h e s e

v a r i a b l e s . I n t h i s r e g i o n t h e energy r e s o l u t i o n i s n e a r t h e v a l u e

c a l c u l a t e d w i t h o u t p a r a s i t i c c a p a c i t a n c e s . R e s u l t s o f t h e o p t i m i z e d

energy r e s o l u t i o n a r e g i v e n as a f u n c t i o n o f LT between 1.2 nH.K and

10 nH.K.

IV.1 Introduction

A d c SQUID (Superconduc t ing Quantum I n t e r f e r e n c e Device) is a v e r y

s e n s i t i v e magnet ic f i e l d measur ing i n s t r u m e n t . It 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 i n t e r r u p t e d by two Josephson j u n c t i o n s . The l as t

t e n y e a r s c o n s i d e r a b l e e f f o r t ( 2 - ,2 -' -9 3 - 4) h a s been devo ted t o making a

p r a c t i c a l l y u s e f u l d c SQUID magnetometer. The s e n s i t i v i t y of t h e s e

SQUIDs is l i m i t e d by the rmal n o i s e from t h e Josephson j u n c t i o n s . For

c a l c u l a t i o n o f t h e n o i s e t h e Josephson j u n c t i o n s a r e r e p r e s e n t e d by

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(5 -? - 6 ) . The n o i s e o f d c

SQUIDS was approximated by Cla rke , Ketchen and Goubau ( 1 - ) and Tinkham

(7). - Tesche and Cla rke (8 ) - c a l c u l a t e d t h e n o i s e o f a d c SQUID w i t h

i d e a l RSJ j u n c t i o n s ( 5 , c ) . - A f t e r t h e i n t r o d u c t i o n o f a c o r r e c t i o n by

Bru ines e t a l . (9) t h e r e s u l t s were i n r e a s o n a b l e agreement w i t h t h e - measurements. However, s o f a r it was assumed t h a t 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 was n e g l i g i b l e . For p r a c t i c a l SQUIDs t h i s is n o t a lways

t h e case 4 . - The c a p a c i t a n c e can 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 of j u n c t i o n s and SQUIDs. For i n s t a n c e i t can produce h y s t e r e -

sis i n t h e I-F c u r v e ( 6 ) and can c a u s e a resonance w i t h t h e SQUID - i n d u c t a n c e (10) . - To p r e v e n t h y s t e r e s i s o f t e n a r e s i s t i v e s h u n t is

a p p l i e d . It is d o u b t f u l i f t h e optimum paramete r s o f t h e SQUID a r e i n

t h e r e g i o n of n e g l i g i b l e c a p a c i t a n c e .

The d c SQUID w i t h c a p a c i t o r s is d e s c r i b e d by two coup led second

o r d e r d i f f e r e n t i a l e q u a t i o n s c o n t a i n i n g two independen t n o i s e t e rms .

To s o l v e t h e s e e q u a t i o n s w i t h t h e d i g i t a l computer would consume a

v e r y l a r g e amount of computa t ion t ime . However, because t h e e q u a t i o n s

of Josephson c i r c u i t s c o n s i s t of d i f f e r e n t i a l e q u a t i o n s w i t h o n l y one

n o n l i n e a r term, t h e y a r e v e r y s u i t a b l e f o r s i m u l a t i o n w i t h a n a n a l o g

computer. The t ime used f o r s i m u l a t i o n o f t h i s t y p e o f d i f f e r e n t i a l

e q u a t i o n s w i t h a n a n a l o g machine is i n p r i n c i p l e n o t dependent on t h e

o r d e r o f t h e d i f f e r e n t i a l e q u a t i o n . I n t h e l i t e r a t u r e mechan ica l

(5,c) and e l e c t r o n i c (13-21) -- a n a l o g u e s o f j u n c t i o n s were d e s c r i b e d .

Although t h e s i n g l e Josephson j u n c t i o n h a s been s t u d i e d o f t e n w i t h

a n a l o g s , o n l y few c a l c u l a t i o n s were performed on t h e d c SQUID. Analog

s i m u l a t i o n s of t h e d c SQUID w i t h c a p a c i t a n c e have been performed w i t h

a mechanical analog by Zimmerman and Sullivan (22) and with an - electronic analog by Tuckerman (23) - and Henry and Prober (24). The - analog simulations up to now concerned noise free dc SQUIDS. This

paper describes calculations of the sensitivity and an optimization of

a dc SQUID containing junctions with capacitance and with thermal

noise using a hybrid computer.

In Sec. IV.2 we shortly describe the model used. Sec. IV.3 shows

the method of optimization. In Sec. IV.4 we present the method of

calculation with an Applied Dynamics 4 analog computer. The sine

generators, needed for simulation of the junctions, were present in

the computer and proved to be suitable for our application. Sec. IV.5

shows some characteristics of the model and compares results with

other calculations. We give results of the noise calcuLations and

optimization and analyze the influence of small changes of the

parameters from the optimum point. We show results of the optimized

energy resolution as a function of inductance and temperature.

Sec. IV.6 summarizes the results with this model and compares them

with other calculations and measurements. A design procedure of SQUID

systems is presented.

IV.2 me dc SQUID model

The model (Fig. IV.l) consists of two Josephson elements (25-27) -- with critical current I, each shunted by a resistor R and a capacitor

C (5,6) and connected through an inductor L. We consider the case - - of a symmetrical SQUID, because small asymmetries of critical current

or resistance were shown by Tesche and Clarke to have a minor influ-

ence on the characteristics (8). The SQUID is biased with a constant - current I. The noise is introduced with two independent noise current

sources I and I in parallel with the junctions. 'he power spectral n 1 n2

densities S f) and S f) of the noise currents are 1'; 14

Fig. IV. 1

Schematic of the dc SQUID model with finite capacitance

where k is Boltzmann's constant, T is the temperature, and R is the

junction resistance.The currents through the two branches of the loop

are

1 - 0 1 1 nl dV1 I - I 'sin(6 ) + V / R + I + C.- dt

and

2 0 2 2 n2 dt dV2 I z I -sin(6 ) + V / R + I + C.-

where V1 and V are the voltages across the junctions. The phases 6 2 1 and 6* are related to the voltages by the Josephson equations

d6 /dt = (4nelh)Vl 1

and

d6 2 /dt : (4ne/h)V2

The phase difference is determined by the total flux Qt (28)

The t o t a l f l u x i s

where Oa i s t h e e x t e r n a l l y a p p l i e d f l u x and J is t h e c u r r e n t c i r c u -

l a t i n g th rough t h e SQUID l o o p

The t o t a l v o l t a g e a c r o s s t h e SQUID is

and

A mutual i n d u c t a n c e p r e s e n t between t h e two h a l v e s o f t h e r i n g d o e s

n o t a p p e a r i n t h e s e e q u a t i o n s , which was shown i n Ref.8. - To g e t a

d i m e n s i o n l e s s s e t of e q u a t i o n s w e u s e t h e commonly used u n i t s

$ 1 (ZnI, R) f o r t ime , I, f o r c u r r e n t , I, R f o r v o l t a g e , and 0, f o r f l u x .

For t h e d i m e n s i o n l e s s q u a n t i t i e s lower c a s e c h a r a c t e r s a r e used . The 2

d i m e n s i o n l e s s pa ramete r s used a r e B=2I, L/a0 , 6,=2nI, R C/@, and

= 2 n k T l 0 ) . Then t h e s e t o f d i m e n s i o n l e s s e q u a t i o n s becomes

For o p t i m i z a t i o n o f t h e SQUID w e u s e t h e energy r e s o l u t i o n E

( 2 9 , 3 0 ) , which i 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 of a S Q U I D f o r - - magnetometer a p p l i c a t i o n s

I n t h i s e q u a t i o n S f ) i s t h e e f f e c t i v e 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 d e f i n e d by 6

where S J f ) i s t h e v o l t a g e n o i s e power s p e c t r a l d e n s i t y and is t h e

a v e r a g e v o l t a g e .

IV.3 The optimization method

The e n e r g y r e s o l u t i o n is a f u n c t i o n o f t h e s e v e n independen t

v a r i a b l e s of t h e model I,, R , C , L , T, I , and Qa. When minimizing t h e

energy r e s o l u t i o n one i s l i m i t e d by t h e v a l u e s o f t h e s e p a r a m e t e r s

which can be reached i n p r a c t i c e . For i n s t a n c e , i f t h e t e m p e r a t u r e

c o u l d be chosen a r b i t r a r y low, t h e r e would be no l i m i t a t i o n on t h e

a c h i e v a b l e energy r e s o l u t i o n .

For t h e & .R p r o d u c t t h e r e i s a maximum (31) - depending on t h e

m a t e r i a l , t h e j u n c t i o n t y p e and t h e t e m p e r a t u r e . With t h e s h u n t e d

j u n c t i o n s commonly used f o r d c SQUIDS t h e I,, - R p r o d u c t i s f a r from t h e

maximum. Hence we assume i n t h i s p a p e r , t h a t I, and R a r e c o m p l e t e l y

independen t . The r e s u l t s of Tesche and Cla rke a l r e a d y i n d i c a t e , t h a t ,

i f we o p t i m i z e w i t h r e s p e c t t o R , I , , I , and Qa, f o r each combina t ion

o f t h e o t h e r v a r i a b l e s t h e energy r e s o l u t i o n n e a r t h e optimum i s a

monotonously i n c r e a s i n g f u n c t i o n o f L , C and T. So L , C , and T s h o u l d

a lways be made a s small a s p o s s i b l e . Techno log ica l l i m i t s o f min ia -

t u r i z a t i o n o f t h e j u n c t i o n impose r e s t r i c t i o n s on t h e c a p a c i t a n c e C .

The t e m p e r a t u r e is g e n e r a l l y a l s o l i m i t e d by t e c h n o l o g i c a l p o s s i b i l i -

t i e s . Reduction o f t h e i n d u c t a n c e i s l i m i t e d . because i t must be

p o s s i b l e t o c o u p l e a c o i l t o t h e SQUID e f f i c i e n t l y . Reduction o f t h e

i n d u c t a n c e below a c e r t a i n v a l u e w i l l o f t e n reduce t h e c o u p l i n g

e f f i c i e n c y and hence t h e o v e r a l l performance. So t h e optimum v a l u e o f

t h e i n d u c t a n c e depends both on t h e method of c o u p l i n g and on t h e

r e s u l t s o f t h e model c a l c u l a t i o n s .

I n f a b r i c a t i o n t h e v a r i a b l e s L, C and T c a n be c o n t r o l l e d v e r y

p r e c i s e l y . They a r e chosen i n t h e d e s i g n o f t h e sys tem. The v a r i a b l e s

I,, and R a r e o f t e n l e s s p r e c i s e l y c o n t r o l l e d i n t h e f a b r i c a t i o n

p r o c e s s . Often t h e r e a r e d e v i a t i o n s from t h e optimum v a l u e s . It is

a l s o p o s s i b l e t o change t h e i r v a l u e s d u r i n g t h e f a b r i c a t i o n w i t h o u t

change o f t h e geometry o f t h e d e s i g n . The v a l u e s of I and @ a r e a a d j u s t e d d u r i n g o p e r a t i o n o f t h e SQUID. The most h e l p f u l f i g u r e s f o r

d e s i g n i n g t h e SQUID a r e t h e energy r e s o l u t i o n op t imized w i t h r e s p e c t

t o I,, , R , I and 9 as a f u n c t i o n o f T, L , and C. a

Thus w e o p t i m i z e w i t h r e s p e c t t o I,,, R , ma and I, k e e p i n g t h e

v a l u e s o f L , C and T f i x e d . To f i n d t h e f o r m u l a t i o n i n d i m e n s i o n l e s s

v a r i a b l e s w e u s e t h e set o f seven independen t v a r i a b l e s C, L , 0 , B c ,

r , i , and q . The energy r e s o l u t i o n becomes a

For o p t i m i z a t i o n we u s e t h e d i m e n s i o n l e s s r e s o l u t i o n E

2 4 where we removed from E t h e f a c t o r 0, - ( C / L ) , which is c o n s t a n t ,

because C and L remain f i x e d . Now w e have t o o p t i m i z e w i t h r e s p e c t t o

t h e d i m e n s i o n l e s s v a r i a b l e s 0 , B c , i , and . Because a

t h e c o n s t a n t t e m p e r a t u r e and i n d u c t a n c e l e a d t o t h e c o n s t r a i n t Br=con-

s t a n t . T h i s q u a n t i t y Br is p r o p o r t i o n a l t o t h e r a t i o o f t h e t h e r m a l

energy kT and the energy of one flux quantum in the inductor. It is a

measure for the noise rounding of the Y-aa curve (7). A typical value - is 0.17 for a 1 nH SQUID at a temperature of 4.2 K.

IV.4 Implementation on a hybrid computer

The equations IV.11,12,13 and 14 of the SQUID model were simulated

using a hybrid computer consisting of an Applied Dynamics 4 analog

computer and a PDP 11/45 digital computer. The analog machine contains

summers, amplifiers, integrators, coefficient units, switches,compar-

ators, A-D converters and several nonlinear elements. The analog

computer is controlled by a PDP 11/45 digital computer. The digital

computer can set the value of the coefficients and the initial condi-

tions of the integrators, read the A-D converters and control the

operation of the analog machine. This system is very versatile and

suited for our application. Fig. IV.2 shows the configuration used for

the simulation of the SQUID. A survey of hybrid computation techniques

is given in Ref. 32 and 33. - - Fig. IV. 3 shows the basic circuit of the dc SQUID simulator. It

contains two branches, each representing a junction. They are con-

nected via the circulating current J. The integrator time constant,

which determines the duration of a time unit of the dimensionless

model, is 0.5 ms. To generate the sin(6) terms in equations IV. 12 and

13 we used the available forward rate resolvers. This device computes

the sine of the integral of its input, making use of the property

sin(6)=sin(6+2nn). The value of is kept inside the region (-n , n ) by

shifting the sign of the integrator input each time the boundary [gI=n is reached. Because 6 itself is not available at an output of this

component we use a separate integrator to compute 61-62. However,

differences in offset between the resolvers introduce an error in the

model which is equivalent with a linearly increasing applied flux.

This error is partly compensated with coefficient "a" in Fig. IV. 3,

which is adjusted as accurately as possible to get the same offset in

both branches of the SQUID. Due to asymmetry still present the

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Fig. I V . 3

Schemat ic of t h e d c S Q U I D a n a l o g s i m u l a t o r

d u r a t i o n o f a r u n i s l i m i t e d t o 1000 t o 2000 time u n i t s . T h i s same

problem was n o t i c e d i n Refs. 23 and 24. These workers avo ided t h e

problem by d e s i g n i n g a s e p a r a t e s i n e g e n e r a t o r , and s u b t r a c t i n g t h e

phases d i r e c t l y a f t e r t h e i n t e g r a t o r s . Th i s s o l u t i o n would be more

d i f f i c u l t w i t h o u r machine. For t h e j u n c t i o n s two independen t n o i s e

s o u r c e s a r e needed. These a r e made by r e c o r d i n g s u c c e s s i v e l y t h e n o i s e

from a random n o i s e g e n e r a t o r on two d i f f e r e n t t r a c k s of a t a p e

r e c o r d e r . The bandwidth o f t h e n o i s e is 3 kHz, which is i n dimension-

less model u n i t s 1.5. I n most c a s e s t h i s i s a t l e a s t a f a c t o r o f 5

above t h e Josephson f requency . The c o n f i g u r a t i o n used s l i g h t l y d i f f e r s

from F ig . IV.3 a s t o s c a l i n g f a c t o r s t o a v o i d o v e r l o a d and keep t h e

s i g n a l large enough f o r optimum use of t h e dynamic r a n g e of t h e a n a l o g

COmpOnents. The computer contains lights which warn for overload

situations.

The integrators start with initial condition zero. First the

system runs for 250 time units in order to eliminate transients of the

model. The duration of this stage must be much larger than the

duration of a Josephson cycle, which is 2x1;. Then the measurements

on the model are started. The longest lasting calculation we can make

is 1000 to 2000 time units due to the asymmetries mentioned above.

Generally we make runs between 800 and 1300 time units, which last

0.4 to 0.65 s real time. The speed of the calculation is limited by

the resolvers, which become less accurate for faster calculations. The

fastest speed achieved is 2000 dimensionless time units per second.

The calculation is repeated 20 to 200 times, depending on the quantity

we measure. The results are averaged.

IV.5 Noise and optimization of the SQUID

The SQUID voltage ; is measured by the digital computer via an

A-D converter from an integrator which averages the voltage v. For

voltage measurements we calculate during 500 to 1000 time units. The

calculation is repeated at least 60 times and the results are aver-

aged. The total duration of a voltage calculation is at least 45000

time units, which is 23 s real time.

To test the model we computed i-; and v-q characteristics with a

the SQUID model. For several sets of parameters these were compared

with the calculations of a single junction of Voss (z) and with a

SQUID without capacitance of Tesche and Clarke ( 8 ) . This comparison is - possible in the cases 0.0 or B -0 respectively. We found an agreement

C

with those calculations to within 5 %. Fig. IV.4 shows i-7 curves

from our calculation for 6 - 1 , l'z0.05 and t3 = I -6 and y-qa curves for C

the same set of parameters. The value of Bc at which the i-v becomes

hysteretic is always larger than 1 for D 0 . 0 1 . We examined the

hysteresis of a noiseless SQUID for 8.1. Fig. IV.5 shows the value of

B at which the voltage becomes double valued as a function of bias C

Vol tage V

F i g . IV.4

( a ) i-Y c u r v e s of a SQUID

w i t h 0=1, 0 =1.6, 0r=0.05 and C

p =O and 0.5 a Ib) v-cpa c u r v e s of a SQUID

with 0=1, 0 =1.6, 0r=0.05 a n d C

and b i a s c u r r e n t s of 1.2,

1.5, 1.8, a n d 2.2

0 0.25 RS

opplied flux +a

current i

Fig. IV.5

0 at which hysteresis C

becomes visible for a

noiseless SQUID for

q =0, 0.25 , and 0.5, a and 8=1, as a function

I of current.

current. As expected for a noiseless SQUID we obtain the same value as

for a single junction ( 6 ) . - In the region i"1 .5 which is the region

where the SQUID is biased optimally, no hysteresis is observable for

values of Bc up to 2, even for a noiseless SQUID. Hence SQUID3 can be stably biased, and are useful as a measuring device for values of 0

C

much larger than 1.

To measure the SQUID voltage noise power spectral density we use

two different techniques. The first method makes use of an analog band

pass filter with center frequency 5 . 1 0 - ~ and 4 and an analog

squarer. The output of the squarer is integrated to get the voltage

noise power within a certain bandwidth. A disadvantage of this method

is that the bandwidth over which is measured is only a part of the

n o i s e spec t rum. An o t h e r method u s e s a F a s t F o u r i e r Transform by t h e

d i g i t a l computer. The v o l t a g e i s f i r s t f i l t e r e d w i t h a low p a s s f i l t e r

t o p r e v e n t a l i a s i n g from f r e q u e n c i e s h i g h e r t h a n one h a l v e t h e sample

f requency. Then t h e s i g n a l i s sampled each 1 m s . A F a s t F o u r i e r

Transform is performed on 512 samples . Th i s p r o c e s s i s r e p e a t e d 120

times. The n o i s e s p e c t r a a r e averaged . The l o w e s t f r equency we c a n

measure depends on t h e maximum d u r a t i o n o f a c a l c u l a t i o n . Most o f t h e

c a l c u l a t i o n s were r e s t r i c t e d t o f r e q u e n c i e s above 1 . 1 0 - ~ , a l t h o u g h -4

sometimes we measured from 5.10 t o check f o r n o i s e a t ve ry low

f r e q u e n c i e s . For t h e low f requency s p e c t r a l d e n s i t y we u s e a n a v e r a g e

o f t h e measured v a l u e s a t t h e f l a t p a r t of t h e low f requency spect rum.

Fig. IV.6 shows t h e v o l t a g e n o i s e power spect rum of a SQUID w i t h

i - 1 . 5 , Bl':0.05, ~ 0 . 3 , 6.1, and 6 - 1 . 6 . The n o i s e power measured v i a a c

t h e a n a l o g method i s g e n e r a l l y e q u a l t o t h e r e s u l t w i t h t h e d i g i t a l

method w i t h i n 10% . However, t h e accuracy o f t h e low f requency v o l t a g e

s p e c t r a l d e n s i t y w i t h t h e FFT method i s b e t t e r , because t h e lat ter

method y i e l d s d a t a from a l a r g e r bandwidth. The f l a t p a r t o f t h i s

spect rum c a n be averaged . For l a r g e 6 and f o r P O , i n which c a s e s t h e

SQUID e q u a t i o n s reduce t o t h e e q u a t i o n s of a s i n g l e j u n c t i o n , t h e

s p e c t r a were i n r e a s o n a b l e agreement w i t h t h e s p e c t r a g i v e n by Voss

( 3 4 ) . For l a r g e v a l u e s o f 6 o f t h e o r d e r of 3 o r l a r g e r , we a l s o - c '

reduced frequency f

Fig. IV.6

Voltage n o i s e spect rum of a SQUID w i t h B Y = 0.05, 8=1, 0 =1.6, i = l . 6 C

and ip =0.3 a

found an increased low frequency noise.

For the calculation of the energy resolution we need the transfer

function 87/8p of the SQUID. The transfer function is found by a

measuring the voltage for p +0.04 and q -0.04. This calculation a a

extends over a total period of 150.000 time units. Sometimes we get

voltages as small as 0.01 for applied flux of 0.25 or smaller and

currents of 1.5 and smaller. As the duration of the period of the

Josephson oscillation is 2 x / 7 , the number of periods over which is

averaged becomes small for small voltages. In our calculations we

generally restrict the calculations to a mean voltage above 0.1.

The resolution is calculated using Eq. IV. 18. The duration of a

calculation of the resolution with the FFT method is about 6 minutes.

We optimized the resolution with respect to 8, Bc, i, and p for the a 6I' values 0.05 and 0 2. The results are shown in table IV. 1. An

attempt to optimize for Br-0.01 did not succeed because of an in-

creased noise at frequencies below for 6 >1.5. We are able to C

optimize for frequencies above but as the frequency relevant for

a practical dc SQUID is often much smaller than our figures for

6l':O. 01 would not be practically relevant. Fig. IV. 7 gives the

transfer function 8 7 1 8 ~ the low frequency voltage noise power a'

spectral density S$O) and the resolution E as a function of i and p a' Below i z 1 . 5 the dependence on the bias current is only weak. We are

not able to extend our calculations to below i-1.4 due to the large

Josephson periods for small voltages and hence the long calculations

needed. Probably for this reason we do not find a minimum as was found

by Tesche and Clarke (8). Fig. IV. 7 shows, that there is at least a -

Table IV. 1

Parameters of an optimized dc SQUID

1.0 1 . 2 1 . 4 1.6 1 . 8 2 . 0 0 0.1 0 . 2 0 .3 0 . L 0 . 5

b i a s cu r ren t i a p p l i e d f l u x 9,

Fig. IV. 7

Fig. IV.7 ( left p a g e )

T r a n s f e r f u n c t i o n av/apa, v o l t a g e n o i s e S(O), and r e s o l u t i o n E a s a v

f u n c t i o n o f c u r r e n t i and a p p l i e d f l u x p (Br=0.05, @ = I , B =1.6 a c

i n d i c a t e d w i t h + and 6r=0.2, b 1 . 5 , 6 =1 i n d i c a t e d w i t h 01 C

( a ) T r a n s f e r f u n c t i o n av /aq a s a f u n c t i o n of i w i t h c o n s t a n t q =0.3 a a

( b ) T r a n s f e r f u n c t i o n aG/aq a s a f u n c t i o n o f p with c o n s t a n t i = 1 . 5 a a

(cl Voltage n o i s e S(OI a s a f u n c t i o n of i w i t h c o n s t a n t p =0.3 v a

( d ) Voltage n o i s e a s a f u n c t i o n o f p w i t h c o n s t a n t i = 1 . 5 a

(e) R e s o l u t i o n E a s a f u n c t i o n of i wi th c o n s t a n t p =0.3 a

( f ) Reso lu t ion a s a f u n c t i o n o f q w i t h c o n s t a n t i = 1 . 5 a

10% r e g i o n o f t h e pa ramete r s i and q n e a r t h e optimum f o r which t h e a

r e s o l u t i o n does n o t s i g n i f i c a n t l y change. So t h e a d j u s t m e n t o f t h e

p a r a m e t e r s i s n o t v e r y c r i t i c a l .

The pa ramete r s 6 , and B e , which cor respond t o t h e c r i t i c a l c u r r e n t

I, and t h e r e s i s t a n c e A o f t h e j u n c t i o n s , are t h e pa ramete r s which a r e

c o n t r o l l e d d u r i n g f a b r i c a t i o n . A s t h e s e p a r a m e t e r s a f t e r f a b r i c a t i o n

of t h e SQUID o f t e n d i f f e r from t h e o p t i m a l l y des igned v a l u e s , i t is

i n t e r e s t i n g t o know t h e dependence o f t h e r e s o l u t i o n on B and 6 C.

However, when t h e SQUID is i n s t a l l e d t h e c u r r e n t and a p p l i e d f l u x a r e

a d j u s t e d t o g e t t h e b e s t n o i s e performance. So i f 6 o r Bc a r e chosen

non-optimal, t h e f i g u r e o f p r a c t i c a l s i g n i f i c a n c e is t h e r e s o l u t i o n

o p t i m i z e d w i t h r e s p e c t t o t h e pa ramete r s i and p For t h i s r e a s o n we a'

d e f i n e E(Br,B,B ) as t h e r e s o l u t i o n op t imized i n each p o i n t w i t h C

r e s p e c t t o i and p I n Fig. IV. 8 we show t h e dependence o f t h e a'

r e s o l u t i o n E(61',6,Bc) on 0 and Bc. Fig. IV.8 shows t h a t t h e r e s o l u t i o n

is independen t o f 6 and PC between 1 and 2. For Bc v a l u e s above 2 no

v a l u e s o f t h e low f requency v o l t a g e s p e c t r a l d e n s i t y c a n be o b t a i n e d ,

because i n t h i s r e g i o n a n i n c r e a s e d low f requency n o i s e i s obse rved , -3 which d o e s n o t s a t u r a t e above t h e s m a l l e s t f r equency observed o f 10 .

For B v a l u e s above 2 o n l y a v e r y s low i n c r e a s e of t h e r e s o l u t i o n is

observed.

For v a l u e s of 6 n e a r 3 o r l a r g e r t h e low f requency n o i s e can C

degrade t h e performance w i t h a f a c t o r o f 10 o r more. For i n s t a n c e a

Fig. IV.8

( a ) Reso lu t ion opt imized with r e s p e c t t o i and rp a s a f unc t i on o f 8. a

1OE is i n d i c a t e d with + f o r 8r=0.05 and 0 =1.6 and E is i n d i c a t e d C

with o f o r B r d . 2 and 0 =1 . C

(b) Resolu t ion op t imized wi th r e s p e c t t o i and cp a s a f u n c t i o n o f 0 . a c

IOE is i n d i c a t e d wi th r f o r pr= 0.05 and & = I , and E is i n d i c a t e d

wi th o f o r 8l'=0.2 and 8=1.5.

Fig. I V . 9

Reso lu t ion op t imized wi th r e s p e c t

0 0.1 0.2 0.3 0.4 t o i, pa, 0 and B a s a f u n c t i o n C

P r

SQUID wi th 0 =3, 8.3 and r.0. 17, which a r e t h e parameters o f t h e C

niobium SQUID r e p o r t e d i n Ref. 35 and Ch. 111, has a r e s o l u t i o n E>1 a t

a f requency of and probably l a r g e r a t lower f requenc ies . This i s

i n r e a sonab l e agreement wi th t h e r e s u l t measured ~ = 8 . 1 0 - ~ ' JIHz (2,

Ch. 111) from which we estimate Ez2.

For design of SQUIDS it is often interesting to know the optimum

energy resolution of a SQUID as a function of the design variables L,

C, and T, assuming that the other parameters are optimized. If the

energy resolution must be minimized, the quantities L, C, and T sould

be made as small as possible. Fig. IV.9 shows the resolution E(0r)

optimized with respect to the other parameters as a function of Br. With Fig. IV. 9 and Eqs. IV. 17 and IV. 18 one can calculate the optimum

energy resolution of a SQUID, to make an optimum design of any SQUID

configuration.

IS'. 6 Discussion

Both the calculated voltages and the noise spectra were compared

with calculations on less complicated circuits (8,34). - - The calcula-

tions show good agreement. The optimum resolution is found to be at B and Bc between 1 and 2, iz1.5 and qa=O. 3, close to the values pre-

dicted by Tesche and Clarke (8). For values of Br smaller than 0. 1 we find according to Fig. IV.9 an optimum resolution of

in agreement with the expectation that at low temperature, where the

noise is expected not to influence the average voltage, the resolution

must be proportional to temperature. Using Eqs. IV. 17-20 we get a

resolution of

This is 3/11 times the result found previously with the calculation

without capacitance (8,9). - - So the introduction of a capacitance up to

0 =I yields a voltage noise somewhat lower than without the capacitor. C

The optimum values of current and applied flux are almost always

i=1. 5 and cp ~0.3. A 0 larger than the optimum value probably causes a a c

s w i t c h i n g between two v o l t a g e s t a t e s o f t h e SQUID. Although t h e SQUID

is n o t b i a s e d i n a p o i n t w i t h two s o l u t i o n s of t h e v o l t a g e ( s e e

F ig . IV.5) , t h e n o i s e c u r r e n t s c a u s e t h e SQUID t o r e a c h t h e r e g i o n

w i t h two s t a t e s . S i m i l a r behav io r was r e p o r t e d by Voss ( 3 4 ) f o r a - s i n g l e j u n c t i o n . The optimum v a l u e s o f Bc and 6 depend on Br. For

l a r g e r 0 r t h e optimum t e n d s t o s h i f t t o a s m a l l e r 6 v a l u e , b u t n o t C

s m a l l e r t h a n 0.8 t o 1 . This c a n be unders tood i n t h e f o l l o w i n g way.

For a SQUID w i t h a l a r g e r n o i s e power t h e r e g i o n o f two s t a t e s i s

r e a c h e d more o f t e n , and hence t h e Bc must be chosen s m a l l e r t o make

t h e r e g i o n of h y s t e r e s i s s m a l l e r and t o push i t f u r t h e r away.

Using o u r numer ica l r e s u l t s one c a n o p t i m i z e t h e d e s i g n of a SQUID

c i r c u i t . F i r s t t h e t e m p e r a t u r e T and t h e c a p a c i t a n c e C must be chosen

a s small a s p o s s i b l e . Then t h e i n d u c t a n c e must be o p t i m i z e d . Because

t h e t e m p e r a t u r e i s c o n s t a n t , t h e p roduc t 0 r is p r o p o r t i o n a l t o L. For

i n s t a n c e f o r a SQUID coupled t o a n i n p u t c o i l one must minimize t h e

energy r e s o l u t i o n r e f e r r e d t o t h e i n p u t c o i l e l k 2 as a f u n c t i o n o f L.

I f t h e c o u p l i n g e f f i c i e n c y k2 a s a f u n c t i o n o f L is known, t h e

r e s o l u t i o n c a n be op t imized u s i n g F ig . IV.9. For t h e c o n f i g u r a t i o n of

a p l a n a r g r a d i o m e t e r d i r e c t l y connec ted t o a SQUID l o o p ( 3 6 , 3 7 , C h . I I I ) - - one must minimize E/L (3 '7) . From Fig . IV.9 one can c a l c u l a t e t h a t E / L - is w i t h i n 10% from t h e optimum w i t h i n t h e range o f t3r between 0 .1 and

0.4, o r L-0.5 t o 2nH f o r T-4.2 K.

D i sadvan tages of o u r a n a l o g t e c h n i q u e o f c a l c u l a t i o n were t h e

r e l a t i v e l y l a r g e amount o f t i m e w e needed t o implement t h e sys tem and

t h e sometimes o c c u r r i n g f a i l u r e s o f components and w i r e s , which make

t h e sys tem l e s s r e l i a b l e t h a n a d i g i t a l computer. Also a s e r i o u s

l i m i t a t i o n o f t h e implementa t ion chosen by us a r e t h e l a r g e i n a c c u r a -

c i e s o f c a l c u l a t i o n s r e q u i r i n g more than 2000 t ime u n i t s . Due t o t h i s

l i m i t a t i o n of t h e d u r a t i o n o f a c a l c u l a t i o n n o i s e a t v e r y low frequen-

c y i s n o t s e e n , which might sometimes be p r e s e n t a t v a l u e s o f Bc

l a r g e r t h a n 1 . An advan tage i s t h e p o s s i b i l i t y t o moni to r e v e r y s i g n a l

needed and t o examine t h e i n f l u e n c e o f change of t h e p a r a m e t e r s d u r i n g

c a l c u l a t i o n . It would be i n t e r e s t i n g t o compare s i m u l a t i o n s of t h i s

model on h y b r i d and d i g i t a l computers .

References

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V CONCLUSION

This research project was started with the aim to make a practical

useful instrument. To meet requirements of stability and sensitivity

the direction towards small all niobium thin film devices was chosen.

Bearing this starting-point in mind the SQUIDs with input coil and the

integrated gradiometer described in the preceding chapters were

developed.

The SQUIDs coupled to thin film coils reached a sensitivity of

2.l0-~~ J/Hz with respect to the input coil. This is the best figure

yet reported for an all niobium thin film device. The noise level of

2.10-30 J/Hz reached with the SQUID and thin film input coil is a

large improvement compared to the resolution of 1.10-~' J/Hz of the rf

SQUID commercially available already for years. The input inductance

is 1 pH, which is a usual value for practical SQUIDs nowadays. Apart

from the low noise and the good coupling, other advantages of this

device are the small size and the rigidness of the system, which makes

instabilities due to vibrating wires impossible.

Although this device completely in thin film is very promising for

practical application, up to now only the simpler system of SQUID with

wire wound input coil was used as measurement instrument. For coupling

to the SQUID with a wire wound input coil a method with a niobium core

was designed and tested. The resolution of this circuit is

3-10-~O J/Hz referred to the input coil. The conditions under which

these devices were used are not discussed in this thesis in detail.

This SQUID with coil was mounted in an electromagnetically shielding

case to get a simply to handle instrument. The SQUID in this

configuration reached the noise level mentioned above and was used for

rock magnetometry and resistance measurements.

The single chip gradiometer reaches a sensitivity of

3.10-l2 T. m-I. HZ-'. Without any external balancing mechanism a balance

of 300 ppm was achieved. The gradiometers were tested in a biomedical

measurement system to perform measurements on the human heart. In this

situation the balance could be improved in the perpendicular direction

by adding the signal from a SQUID magnetometer. As mentioned in

Ch. 111 the performance of this device is comparable to a conventional

system with a wire wound coil. However the size of this device is much

smaller. This type of relatively small gradiometers is expected to

become particularly interesting for applications in large arrays of

SQUIDs for biomedical applications and for use in small cryocoolers

( 1 ) . - A problem with the first order gradiometer is the large outer

loop, which sometimes carries large currents. If the critical current

of the strip is reached, a shift in the signal is observed. It is even

possible that due to the voltage across the strip the junctions are

damaged. This problem might even be present with much larger, less

sensitive junctions. In general I think, that a second order serial

gradiometer ( 2 ) - is the most appropriate solution to overcome these

problems because of the absence of large circulating currents with the

additional advantage of a better rejection of environmental noise.

Although when mounted in the cryostat the sensors proved to be

reliable, during mounting and transport the devices were sometimes

damaged electrically. This burning out of the junctions can be due to

electrostatic discharges or to magnetic fields from the electric mains

sensed by the wires and transferred to the SQUID. The origin of the

problem is most probably the size of the junctions and the character

of the junction barrier. Because the critical current of the SQUID is

imposed by the inductance of the SQUID (see Ch. 11) and the I.,R

product is imposed by the material, which is niobium generally, nearly

all SQUIDs will have the same junction resistance. This causes the

junctions to have about the same dissipation due to signals from

outside, neglecting the protection by a resistive shunt possibly

present. The larger junctions have a larger area for cooling. Also the

geometry of the junction barrier with many small traces through the

silicon film might contribute to the local dissipation near the 2

pinholes. Probably a larger junction area, of about 1 pm , will

prevent such burnouts. Furthermore the larger junctions need a shunt

resistor, which also provides some protection. This can be combined

with a casing for the SQUID which screens from electrical voltages and

magnetic fields as good as possible.

The simulation of the dc S Q U I D without capacitor of Tesche and

Clarke (3) - and Bruines et al. (5) already gave a good estimate of the noise of a S Q U I D system. They assumed that in the optimum case the

junction hysteresis parameter should have the value Bc=l. This

assumption is justified by the results reported in Ch. I V . The

simulation with capacitance presented in Ch. I V yields for B = I about C

314 times the value with the simplified model. The optimum resolution

turns out to be independent of Bc and B for a broad range of

parameters.

For very large values of Bc, of the order of 2.5 or larger, an

increased low frequency noise is observed. Although the noise at very

low frequencies was not calculated, the calculations at reduced

frequency show a reasonable agreement with the observations at

lower frequencies in Ch. 111. The improvement of the sensitivity due

to an increase of the temperature noticed in Ch. I11 is reasonable

because of the accompanying reduction of B whereas the accompanying c'

increase of Br has a minor influence. So the resolution of the niobium S Q U I D is in good agreement with theory. The deviation from the optimum

energy resolution is due to the large Bc, of about 3. Improvement of

the S Q U I D developed is expected, if Bc is reduced. For the optimum

value near B = I an improvement of a factor of about 20 is expected. C

The most proper way to reduce B is to shunt each junction with a C

resistor of about 200 ohm. An other advantage of this shunt will be

the additional protection against externally applied voltages.

During this research a niobium dc S Q U I D with an energy resolution

of ~ . I o - ~ ~ JIHz became commercially available ( 5 ) . - This hybrid S Q U I D

is said to contain thin film junctions, probably with a bulk niobium

toroidal S Q U I D ring, but the properties of the junctions and the

design of the S Q U I D are not known. The size of the junctions is also

unknown, but most probably the junction area will be of the order of 2

several vm , with which size the resolution reported can easily be reached according to theory. Compared to this hybrid S Q U I D an advan-

tage of the S Q U I D with submicron junctions is the potentially low

noise of the order of lom3' J/Hz as measured with the single S Q U I D

(Ch. 1 1 1 ) and yet possibly improvable with an order of magnitude if

the junctions are resistively shunted. Large advantages of thin film

SQUIDs are the good mechanical stability and chemical resistance of

the junctions, much better than of point contacts. As to the noise and

the chemical and mechanical stability the devices are excellent, as

expected for thin film niobium devices. However, a dc SQUID will

always remain more sensitive to burnouts than the standard rf SQUID

because the junction of the latter is shunted by the SQUID ring

itself.

Further research on the connection of superconducting wires to

films, with a solder for instance, is useful for applications of

devices with a thin film input coil. An interesting future direction

is the development of devices with the pick-up circuit on the chip,

such as described in this thesis, or second order serial gradiometers

(Z), - possibly with a SQUID magnetometer for electronically balancing.

This type of relatively small devices is expected to become particu-

larly interesting for biomedical applications and for use in small

refrigerators.

Finally, the calculations of the resolution presented in this

thesis show a remarkable agreement with measurements of both the SQUID

described in this thesis and other SQUIDs. With the aid of the analog

model the prediction of the sensitivity of dc SQUIDs is possible with

a very good accuracy, which is an important aid for the design of very

low noise dc SQUID systems. It is shown, that the fabrication of high

performance niobium devices is not extremely difficult, that without

very sophisticated apparatus the fabrication and use of very small

junctions is possible, but that the vulnerability of submicron devices

is still a very difficult to solve problem. In my opinion a large

improvement has been made compared to the lead alloy junction and

point contact technologies.

References

1. D.B. Sullivan, J.E. Zimmerman, and J.T. Ives, Refrigeration for

Cryogenic Sensors and Electronic systems, J.E.Zimrnerman,

D.B. Sullivan, and S.E. McCarthy eds., NBS special publication

607 (1981)

2. V.J. de Waal, G.J. van Nieuwenhuyzen, and T.M. Klapwijk,

Proceedings of the Applied Superconductivity Conference,

Knoxville, Tennessee, USA, (1982)

3 . C.D. Tesche and J. Clarke, J.Low Temp.Phys.29,301 - (1977)

4. J.J.P.Bruines, V.J. de Waal and J.E. Mooij, J.Low Temp.Phys.46, -

383 (1982)

5. Data from S.H.E. Corporation, San Diego, California

Samenvatting

Dit proefschrift behandelt ontwerp, fabricage en optimalisering

van laagfrequent magnetometers gebaseerd op Josephson juncties.

Gekozen is voor de dc SQUID, het gevoeligste type. Het onderzoek is

gericht op de ontwikkeling van een praktisch bruikbaar en zeer

gevoelig instrument.

Na een inleiding in het vakgebied worden theoretische aspecten van

Josephson juncties en SQUIDs beknopt besproken. Het blijkt dat de

signaal-ruis verhouding van dc SQUIDs verbetert bij verkleining van de

parasitaire capaciteit van de Josephson juncties. Dit wordt bereikt

door miniaturisering van de juncties.

De schakelingen worden gefabriceerd met dunne film techniek. Met

de gebruikte methodes van fotolithografie en schaduwopdampen kunnen

lijnbreedtes van ca. 0,5 pm met een nauwkeurigheid van enkele tienden

pm gerealiseerd worden. Dit maakt het mogelijk de fabricageparameters

goed te beheersen en eventueel ook in grote series te produceren. Alle

circuits zijn gemaakt van de supergeleider niobium, omdat dit materiaal

een hoge kritische temperatuur heeft en relatief stabiele juncties

oplevert. De oppervlakte van de juncties is ongeveer 0,3.0,3 pm2. De

tunnelbarrikre van de Josephson juncties wordt gevormd door over de

niobiumlaag eerst een siliciumlaag op te dampen en dan met een

glimontlading te oxyderen. De capaciteit van de juncties is ongeveer

lo-14 F en de zelfinductie van de SQUID is 1 nH. De juncties hebben

een weerstand van ongeveer 300 fl bij een & R produkt van ca. 300 pV. De

kritische stroom en weerstand zijn reproduceerbaar binnen ongeveer een

factor 2 . Deze juncties zijn geschikt voor een dc SQUID zonder gebruik

van een externe parallelweerstand. De laagst behaalde energieresolutie

gemeten in een flux locked loop is 8.10-~~ JIHz bij een werktemperatuur

van 4,2 K.

SQUIDs met een zelfinductie van 2 nH zijn gekoppeld aan een dunne

film niobium spoel. De laagste energie resolutie gemeten ten opzichte

van de inkoppelspoel is JIHz. Dit is de laagste waarde van een

dunne film niobium systeem bekend uit de literatuur. Ook draadgewik-

kelde spoelen gekoppeld aan dezelfde SQUIDs zijn beproefd. Hiemee is

een resolutie van 3 . 1 0 - ~ ~ J/Hz ten opzichte van de inkoppelspoel

bereikt. Praktisch gebruik van deze SQUIDs wordt nog belemmerd door

soms optredende doorslag van de juncties.

Dunne film technologie is bijzonder geschikt voor het integreren

van het complete circuit van signaalspoel en SQUID op kkn substraat.

Als eerste aanzet hiertoe is een gerntegreerd circuit van SQUID met

eerste orde gradiometer ontwikkeld. Deze gradiometers met afmetingen

van 12'19 mm2 meten een niet-diagonaal component van de gradient van

het magneetveld. De bereikte gevoeligheid is 3.10-~' pm-' .Hz-$.

Tot nu toe uitgevoerde berekeningen aan dc SQUIDs op basis van het

RSJ (Resistively Shunted Junction) model geven een goede kwantitatieve

beschrijving van de SQUID, zolang de hystereseparameter kleiner dan 1

is. Voor grotere waarden moet de junctiecapaciteit bij de berekeningen

in aanmerking genomen worden. Deze configuratie, beschreven door twee

gekoppelde tweede orde niet-lineare differentiaalvergelijkingen, is

gesimuleerd op een hybride rekenmachine. Dit model blijkt een goede

voorspelling van de gevoeligheid van SQUIDs te leveren. Voor kleine

waarden van de capaciteit;, dus een hystereseparameter kleiner dan 1

geven de berekeningen ongeveer dezelfde resultaten als zonder capa-

citeit. Voor het geval waarin de capaciteit niet verwaarloosbaar is

wordt de energieresolutie groter. Dit is ook kwantitatief overeen-

komstig de ervaringen met de in dit proefschrift beschreven SQUID.

Met behulp van deze simulatie is het nu mogelijk de parameters van

de SQUID te optimaliseren. Bij vaste geometrie van de SQUID kunnen de

optimale junctie weerstand en kritische stroom bepaald worden. De

energieresolutie van een optimale SQUID als functie van de zelf-

inductie is berekend.

Dit onderzoek heeft aangetoond dat het mogelijk is een zeer

gevoelige praktisch bruikbare SQUID te maken met submikron junctie-

afmetingen zonder extreem hoge eisen aan de fabricageapparatuur te

stellen. De gebruikte dunne film technologie maakt het mogelijk de

sensoren te miniaturiseren, hetgeen belangrijk is voor toepassingen in

arrays van sensoren of in kleine gesloten koelsystemen. Het model

introduceert de mogelijkheid de gevoeligheid van dc SQUIDs te voorspel-

len en te optimaliseren.

Curriculum vitae

Op 28 juli 1954 is de promovendus geboren te Amsterdam. Van 1966

tot 1972 bezocht hij het Vossius Gymnasium te Amsterdam,waar hij in mei

1972 het examen Gymnasium 0 met goed gevolg aflegde.

In dat jaar is hij begonnen met de studie technische natuurkunde in

Delft. Tijdens de afstudeerfase heeft hij gewerkt in de groep Molecuul-

analyse / Supergeleiding aan microfabricage in dunne films. In 1978

slaagde hij met lof voor ingenieursexamen Technische Natuurkunde. Eind

1978 is hij aangesteld door de Stichting voor Fundamenteel Onderzoek

der Materie om te werken aan de ontwikkeling van dunne film dc SQUID

meetsystemen in de Vakgroep Supergeleiding, Technische Hogeschool

Delft, onder leiding van prof.dr.ir. J.E. Mooij. Hierbij is gewerkt aan

ontwerp en bouw van genoemde systemen en zijn mogelijkheden voor

commerciele exploitatie van deze ontwikkelingen onderzocht.