SQUID '85 Superconducting Quantum Interference Devices and their Applications: Proceedings of the Third International Conference on Superconducting Quantum Devices, Berlin (West), June 25-28, 1985 9783110862393, 9783110103304

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SQUID '85 Superconducting Quantum Interference Devices and their Applications

SQUID '85 Superconducting Quantum Interference Devices and their Applications Proceedings of the Third International Conference on Superconducting Quantum Devices Berlin (West), June 25-28,1985 Editors H. D. Hahlbohm · H.Lübbig

W DE G Walter de Gruyter · Berlin · New York 1985

Editors Hans-Dieter Hahlbohm, Prof. Dr. Heinz Lübbig, Dr.-ing. Physikalisch-Technische Bundesanstalt Institut Berlin Abbestraße 2-12 D-1000 Berlin 10

Library of Congress Cataloging in Publication Data

International Conference on Superconducting Quantum Devices (3rd : 1985 : Berlin, Germany), SQUID '85, superconducting quantum interference devices and their applications. Bibliographie: p. Includes indexes. 1. Superconducting quantum interference devices—Congresses. I. Hahlbohm, H. D„ 1930. II. Lübbig, H„ 1932. III. Title. TK7872.S8I57 1985 621.39 85-29189 ISBN 0-89925-144-7 (U.S.)

CIP-Kurztitelaufnahme der Deutschen Bibliothek

SQUID... : superconducting quantum interference devices and their applications ; proceedings of the ... Internat. Conference on Superconducting Quantum Devices. Berlin ; New York : de Gruyter ISSN 0720-7964 NE: International Conference on Superconducting Quantum Devices 3.1985. Berlin (West), June 25-28,1985. ISBN 3-11-010330-3

Copyright © 1985 by Walter de Gruyter & Co., Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm or any other means nor transmitted nor translated into a machine language without written permission from the publisher. Printing: Gerike GmbH, Berlin. - Binding: Dieter Mikolai, Berlin. Printed in Germany.

PREFACE

Rarely has a new physical phenomenon given rise to such strong expectations, and resulted in such strong frustrations, as has the Josephson effect. It is also rare that a new phenomenon has had such a wide influence on physical research. Nobody, of course, can predict future developments, but nearly all the participants in this conference felt at the end that the active history of the Josephson effect and of SQUID technology has by no means reached its final state. From the beginning it has been the goal of IC SQUID to observe, and make available a critical analysis of the current situation of SQUID physics and technology. This 3rd edition of the IC SQUID proceedings reflects the research done over the last 5 years. Reading the 20 invited review lectures, the 16 invited supplementary lectures and the 89 contributed papers, it is clear that there will have to be future volumes. Biomagnetic and HF applications in combination with new materials and improved fabrication techniques, together with new concepts of barrier physics - quantum fluctuations in macroscopic superconducting systems and the quantum interference in normal metal loops - will make necessary a continuing analysis in the future. Special progress may be expected from the future development of ultrasmall systems, three-terminal non-equilibrium devices, and fluxon dynamics in long Josephson junctions. Nevertheless, the constraints in this field have become much more difficult and tedious, which also makes the editors' task more difficult. Being obliged to publish these results as quickly as possible, it was unavoidable that we had to make certain cuts. These concern mainly a comprehensive and detailed classification system. We have therefore been forced to make decisions which might be considered rather arbitrary in certain cases. Our thanks go to all the authors who have enabled us to produce this volume

VI The c o n f e r e n c e w a s p r o p o s e d a n d e n c o u r a g e d b y the E u r o p e a n

Physical

Society. The p u b l i c a t i o n of the p r o c e e d i n g s is a n a p p r o p r i a t e o c c a s i o n to a c k n o w l e d g e o n c e a g a i n the g e n e r o u s s u p p o r t of the c o n f e r e n c e the P r e s i d e n t of the D e u t s c h e r B u n d e s t a g , the B e r l i n S e n a t e , P r e s i d e n t of the P h y s i k a l i s c h - T e c h n i s c h e B u n d e s a n s t a l t , V a c u u m s c h m e l z e G m b H , S . H . E . G m b H , a n d the D e u t s c h e

by the

Siemens AG,

Physikalische

Gesellschaft. It s h o u l d also be e m p h a s i z e d t h a t w i t h o u t the cheerful a n d a c t i v e p a r t i c i p a t i o n of the m e m b e r s of the B e r l i n I n s t i t u t e of the k a l i s c h - T e c h n i s c h e B u n d e s a n s t a l t in all a s p e c t s of the

Physi-

conference,

the m e e t i n g w o u l d n o t h a v e b e e n p o s s i b l e . O u r thanks go to G . S a u e r b r e y , h e a d of the B e r l i n I n s t i t u t e , a n d to all its m e m b e r s . We are also g r e a t l y i n d e b t e d to the loyalty a n d p a t i e n t a s s i s t a n c e of F r a u Α . L o c h m a n n a n d F r a u G. W i l l e . We are

secretarial much

o b l i g e d to F r a u R. F ö h n for the o r g a n i z a t i o n of this v o l u m e , as w e l l as for p e r f o r m i n g the task of r e a d e r w i t h g r e a t We are g r a t e f u l

care.

to the d e G r u y t e r P u b l i s h i n g C o m p a n y for its v a l u -

able c o o p e r a t i o n in p r e p a r i n g these

B e r l i n , A u g u s t 1985

proceedings.

H.-D.

Hahlbohm

H. L ü b b i g

CONTENTS

JUNCTION AND DEVICE PHYSICS Physics of tunneling barriers T.M. Klapwijk

1

Tunnelspectroscopy of inhomogeneous potential barriers showing resonances exemplified for Nb2Û^ J. Halbritter

31

Non-thermal noise in ultrasmall area tunnel junction dc SQUIDS R.T. Wakai and D.J. Van Harlingen

43

Microscopic theory of the Josephson effect in the SNINS and SNIS tunnel junction V.F. Lukichev, A.A. Golubov and M.Yu Kupriyanov

49

Stationary properties of Josephson SN-N-NS microbridges V.F. Lukichev, M.Yu. Kupriyanov and A.A. Orlikovsky

....

55

Alteration of magnetic field dependence for voltage biased SNS Josephson junctions K.D. Bures and K. Baberschke

59

Behaviour of aluminium microbridges irradiated with microwaves S. Wang and P.E. Lindelof

65

Automation of numerical analysis of circuits with Josephson tunnel junctions V.K. Semenov, A.A. Odintsov and A.B. Zorin

71

On a few-degree-of-freedom-model of the Josephson tunneling junction G. Brunk

77

Direct numerical determination of stationary rotational states in Josephson junctions with arbitrary current phase relation G. Brunk, H. Lübbig and Ch. Zurbrügg

83

Analytical calculation of the inductance of Josephson junctions J. Buechler and M. Rieger

89

Influence of the Abricosov's vortices on superconducting tunnel junctions properties A.A. Golubov and M.Yu. Kupriyanov

95

Vili I-V c u r v e s of R F - d r i v e n J o s e p h s o n j u n c t i o n s in the low f r e q u e n c y regime C. V a n n e s t e a n d C.C. Chi

101

R e c o n s t r u c t i o n of the J o s e p h s o n j u n c t i o n p a r a m e t e r s its t r e s h o l d c u r v e s

via 1°7

A. B a r o n e , F. E s p o s i t o , V . K . S e m e n o v and R. V a g l i o P r o p e r t i e s of the l e a d - a l l o y J o s e p h s o n tunnel

junctions

V . Y u . K i s t e n e v , L.S. K u z m i n , A.G. O d i n t s o v , E.A. a n d I.Yu. S y r o m y a t n i k o v

Polunin

113

R e s o n a n t m o d e s in l i g h t - s e n s i s i v e dc S Q U I D s M . Russo a n d G. P a t e r n o

119

R e s o n a n t m o d e s in r e f r a c t o r y b a s e l a y e r d.c.

SQUIDs

G. P a t e r n o , A . M . C u c o l o , G. M o d e s t i n o and R. V a g l i o Q u a n t i z e d flux 0 Q in D C - S Q U I D w i t h

....

I25

LIQ>kT) decreases with increasing kT due to

decreasing with U - see

Fig.4. 2.5 Summary on tunnel channels

In summary,

the

logarithmic

derivative

g (U) = dln(I/U)/dll depending on the

nonlinear growth rate of I(U) only, shows extrema, where I(U), i.e. the transmission coefficients T 2

changes rapidely as shown in Figs. 2-4.

3

These ex-

trema are more pronounced for larger barrier thicknesses, because the angular lobe of tunneling are more

electrons

pronounced

becomes more narrow with

for lower

temperatures

bœause

diminishes. Incoherent, activated tunneling j from

and

jR

by

g H (eU>>kT) decreasing

•gì(eU>>kT) are solely functions of U.

with

increasing 2d and they

the

temperature smearing

("hopping conduction") differs temperature whereas

gD

and

37

Fig. 5: Absolute value (Δ) of the logarithmic derivative g(U) = dln(I/U)/dU as a function of voltage U for a Nb-Nb^O^-Pb tunnel junction. The oxide has been grown by plasma oxidation up to a thickness of about 2d = 5 nm. The narrow symmetric peaks at U = ±15 mV are explained (fig. 2) by R via states 50 mV) is due to a transition layer of lowered barrier at the Nb-Nb„0_ interface, see insert in fig. 6.

With this logarithmic derivative g(U,T) the different tunnel channels can be separated and 2d, φ(χ), φ*(χ) and η^ίχ,ε^) can be evaluated. This and the interaction of tunneling electrons with internal degrees of freedom of the 5 6 barrier yield the following classification of tunnel channels:

IeUI > eV = φ: barrier plateau; JeUI > 0.1eV = φ*: channels in the barrier plateau; I eU I < 50 meV > 2E · resonant tunneling via localized A

I(eU>2 Ej) by 10 - 90Ϊ (GZBA). In contrast to j*

'

states enhancing

D

exp(-t yielding a higher φ* density (Fig. 6). Adjacent (< 5-10 nm) to Nb the V q reduction concentrates on n^ (Ε>Ε ρ -Ε 1 [ η 3 β θ ) - states with E. the energy gain by image charges, image

40 The Nb rich defects, namely

φ* channels and V^s causing resonant

tunneling,

account at 50mV for more than 50? of the tunnel current - see Table 1 . Thus for

Nb-Nbo0_ all the tunnel anomalies summarized in Part 2.5 are fairly 6 strong. For example, the strong GZBA at E^ " E image Œ 1 ^ e r d i s d u e t o t h e onset of n. at E -E. due to the removal of V 's above this energy, the IL F image o superconductor anomalies are due to η^(Δχ < 0.5 nm), the 1/f noise is due to states n. above Ε -E. and the spin flip ZBA is due to localized pairs in K Í F image the midth of the barrier.

3.3 Relation between structure and tunneling

The

above

discussed

tunnelanomalies

of

Nb^O^

are

due

to

φ*

and

being 8rowth

volume defects, which are due to the nonlocal stoichiometry, the

by 0 diffusion and the {NbOg' block nucleation straining the interface. Slowly grown

Nb 0 contains less strain and less V 's yielding a better junction 0 6 10 3 2quality. ' In NbN-oxide Ν -ions on 0 sites destabilize V defects and 6

{MbO/·

0

11

block formation. Thus NbN oxide stays thinner (< 1.5 nm) and con6 12 tains no V q defects yielding better quality tunnel barriers. Because of a similar nonlocal stoichiometry due to ·Μ0,· blocks, Ta„0_ and Ti0_ show anomatJ ¿ lies similar to Nb^O^.

In contrast to the above transition metal oxides of the IV and V column, Al^O^ is

locally

stoichiometric.

At

the

surface

of

A1_0

dangling

bonds

exist 13

pinning E„ 2eV below E„ causing resonant tunneling and tunnel anomalies. Γ Γ These

surface

defects

are

saturated

by

OH-groups

explaining

the

superior

tunnel barrier quality of "dirty, amorphous" A ^ O ^ .

Artificial,

amorphous barriers 19 = 10 /cm 3 eV.

contain

e.g., a-Si n^

a high

density of localized

states,

These states are as dangling bonds immobile

and thus η (χ,E) = const holds. Such states cause various tunnel anomalies, 14 where η.(x,E) » const allows a measurement of the phonon density of states 15 p(e) [Eq.(2)], as has been carried out, e.g., for a-Ge. the defects cannot be changed

Whereas in oxides

in a controlled way, in a-Si n^ is reduced by

3 hydrogénation to about 10 1 7 /eVcm at Ep. Thus GZBA and ZBA are reduced 114 in 5 line with the above theory. In a-Si and a-Ge the large n^ and the small

ε ^ 30 yV

an enhancement

of

Ic

is

observed.

Nonuniform Current Density for V¿ c =0

Recently

there

has been the report

about

the observation

of an in situ

electron spin resonance of a voltage biased Nb AuGd Nb/Pb Josephson Junction where the Junction is used as a microwave generator and detector as well /l/. The disadvantage of the junctions used in these experiments is the point contact method and its almost unknown local geometry. Here we present the magnetic field dependence of the supercurrent for the junctions used in /l/. The method to deduce the local geometry from the diffraction pattern is well known /2/. Figure la shows the experiment with field periods of SS(S)G and 4.S(3)G, Fig. lb the simulation. We had to extend the simple current density curves to the case of independent current pathes with different densities as is shown in Fig.lc. The used simulation is given by:

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

60

s l n [ (• t •i 1s / L' ) ( * / t •oT ) ]

sin[ (nl/L) (•/•o) ]+^ n- -i1 the

i s * t e P' b u t ^ enhancement of the c r i t i c a l c u r r e n t . C i s with the microwave power a d j u s t e d to g i v e the ] m i f9est ™ v e induced s t e p . v=70 MHz.

c r i t i c a l c u r r e n t and the Step height are enhanced by roughly

the

the same amount

relative

At

low f r e q u e n c i e s

the critical current

the m i c r o w a v e - i n d u c e d

to the p r e d i c t i o n of the RSJ m o d e l .

first

cannot be e n h a n c e d ,

step can be o r d e r s of m a g n i t u d e s

ger than the p r e d i c t i o n s of the RSJ m o d e l . The lowest g i v e s the largest d e v i a t i o n quencies

whereas

from the RSJ m o d e l ; at the lowest

it b e c o m e s a c o n s t a n t

(s 1 00) times the RSJ

lar-

temperature fre-

prediction.

68 Discussion Microwaves can have a stimulating influence on the critical current of superconducting microbridges. Two mechanisms are known, which explains such behaviour. Eliashberg

(5) showed that the energy gap

of a homogeneous superconducting film can be enhanced by the nonequilibrium, which is created among the quasiparticles by a high frequency microwave field. Fig.3 shows how the critical current and the first microwave-induced step are enhanced relative to the RSJ prediction at high frequencies. The step is particularly much enhanced close to Τ , where heating effects are negligible. We have also seen enhancement of the energy gap in the subharmonic energy gap structure in the same frequency and voltage regime. However the subharmonic energy gap structure is often difficult to follow when irradiated by microwaves, because the simultaneous appearance of the induced step structure. Thus we feel confident that the enhancement seen in the high frequency/voltage regime is a result of the Eliashberg mechanism. The other mechanism for enhancement of the critical current in superconducting microbridges is due to Aslama-

Fig.3. The maximum microwave-enhanced critical current I /I (lower part) and the maximum first microwave-induced ste¡3m c I. /I plotted versus the normalized frequency. Three temperi tu rë s have been plotted and many different frequencies ranging from 10 MHz to 12 GHz have been used. The solid lines are the predictions of the resistively shunted junction (RSJ) model. Fig.1 corresponds to a normalized frequency of 4 and Fig.2 corresponds to a normalized frequency of 0.005.

69 zov and Larkin (6) and connected with the shoulder structure in the current-voltage characteristic mentioned above. However they find that the microwave-enhancement only appears, when the microbridge has a size comparable to the coherence length, and we should therefore not expect to see this type of enhancement in our microbridges. At low frequencies/voltages we observe enhanced microwaveinduced steps, but never any enhancement of the critical current as illustrated in Fig.3. The enhanced microwave-induced first step at low frequencies is a result of eq.(1) and (3), since 6ls is dissipative here and can be approximated by an effective resistance R eff =17ñ/(2ex E I c ) (2,3). Literally this means that the RSJ prediction shown in Fig.3 is valid here, but with R^ replaced by R e f f For the lowest temperature points in Fig.3 R^ is 100 times larger than Rgff calculated with τ Ε =10 ns. This brings the points at the lowest frequencies in Fig.3 into agreement with an RSJ model with as the determining resistance. The other temperatures follow a similar trend, yet the values of R e f£ are not quite as low, so the displacements of the RSJ curves are not as large. At frequencies where eq.(1) and (4) applies, our experimental points for the maximum step goes parallel to the RSJ prediction, but with a slightly higher value for the maximum step. According to eq.(4) we should expect a 33% increase here and according to our experimentally measured shoulder structure we should rather expect a 60% increase (2). This indicate a defect in eq.(4) with q.j=-jsin-t+At

numerical integration, and moreover, a large number of previous values of

of φ

should be remembered. Time truncation of the kernels (4) or adiabatic expansion of the junction voltage (5) can give only slight progress but not solve the problem.

The key point of our computational method is to approximate the kernels by finite Dirichlet series (6) Μ Ν Σ Σ a Tlm expr ( rρ τ) , „ mn n m=0 n=l

J(τ) = Re

where J(τ)equal s either

J

q(T)

=

J

(T) q

"

(yl/e

with

Re ρ

η

< 0 ,

(2)

J (τ) or Ρ V

δ

'(τ)

(3)

'

Substitution of the expansions (2) into Eq.(J_) yields a series of the integrals of the type F(t)

=

/ d t ' f(t') exp (p(t-t')}

.

(4)

— oo

Instead of direct integration of the new value of

F

Eq.(4) at each time step, one can calculate

using its previous value

F(t+At) = exp(pAt) F(t) + AF, where

AF =

as follows

t+At / dt'f(t')exp{p(t+At-t')} t

.

(5)

One can see that such calculation can be directly embeded into any finite difference scheme. Therefore, quality of this economical method is determined by accuracy of the expansion

of kernels

J

(τ)

(2). This expansion corresponds

Ρ

of the kernels (which are

essentially the complex amplitudes of the tunneling current (2,3)) by rational algebraic functions (7). The approximation can be fulfilled using any standard procedure which minimizes the root-mean-square deviation. Figure 1 shows that already for M=0 and N=3 the approximated amplitudes (solid lines) are in a good agreement with the original functions Iq.p (ω) (dashed lines) , taking into account the finite width 25 of the Riedel peak (8).

73

Fig.l. Components of complex a m p l i t u d e s I (ω) a s f u n c t i o n s of t h e v o l t a g e V = H ω/e f o r a symmetrical ( Δ | = Δ 2 = Δ ) J o l à p h s o n t u n n e l j u n c t i o n . Dashed l i n e s , o r i g i n a l f u n c t i o n s ( t h e BCS a p p r o x i m a t i o n w i t h t h e L o r e n t z b r o a d e n i n g of t h e R i e d e l p e a k ) . S o l i d l i n e s , t h e D i r i c h l e t a p p r o x i m a t i o n (2) of t h e f u n c t i o n s w i t h M = 0 and Ν = 3 . 1 = 2 4 / ( e R j j ) .

Applications

New v e r s i o n of o u r p r o g r a m COMPASS, s u p p l i e d w i t h t h e d e v e l o p e d a l g o r i t h m , u s e d t o a n a l y z e of v a r i o u s J o s e p h s o n - j u n c t i o n c i r c u i t s . As a f i r s t

was

example,

F i g . 2 shows t h e c a l c u l a t e d de I - V c u r v e s of a s i n g l e t u n n e l j u n c t i o n w i t h a =

ical value

Sc

Ό > under i n f r a r e d i r r a d i a t i o n with frequency

where

= 2Δ/Χ. One c a n s e e a number of t h e v e r t i c a l

ω = 1.4

ω

Shapiro s t e p s due to

s u p e r c u r r e n t and smoothed Dayem-Martin s t e p s d u e t o q u a s i p a r t i c l e

current.

Second, we h a v e a n a l y z e d p e r f o r m a n c e of some d i g i t a l

c i r c u i t s of t h e

proposed

( 9 ) . F i g u r e 3a d e p i c t s

R e s i s t i v e S i n g l e F l u x Quantum (RSFQ) l o g i c

a neuristor-type pulse F(t),

is realized Rj

if

R

T c = Σττ/ω^^ = h / ( 2 A ) of t h e

RSFQ l o g i c .

t h a t one-way t r a n s m i s s i o n n e c e s s a r y f o r t h e

i s of t h e o r d e r of

of t h e j u n c t i o n s .

recently

t r a n s m i s s i o n l i n e , w h i c h c o n s i s t of t h e p u l s e

a s e r i e s of f o u r i d e n t i c a l b u f f e r s t a g e s and t h e i n d u c t i v e l o a d

F i g u r e 3b i l l u s t r a t e s

R^

rather

source

L.

RSFQ l o g i c

t h a n of t h e p r e g a p

One c a n s e e a l s o t h a t o n l y t h e minimum t i m e existing

typ-

2^>

resistance

constant

i n s u p e r c o n d u c t i v i t y l i m i t s t h e minimum t i m e d e l a y

74

βν/2Δ Fig.2. de I-V curves of a symmetrical tunnel junction fed with the current I + I cosüJt for several values of the ac current amplitude.

Discussion

The Dirichlet series expansion of the Werthamer-theory kernels has proved to be a very memory-consuming and rapid method of numerical simulation of the Josephson junction circuits. The simulation of typical circuits using our algorithm is only twice slower than that based on the fenomenological RSJ or RSJN models.

75

For example, c a l c u l a t i o n of

the waveforms shown in F i g . 3 takes 20 seconds of

p r o c e s s o r time of an a v e r a g e - s p e e d

( M MIPS)

the

computer.

Acknowledgements

We wish to thank K.K. Likharev f o r f r u i t f u l d i s c u s s i o n s and U. T i t z

for useful

collaboration.

References

1. Semenov, V . K . , V . P . Z a v a l e e v . 2. Werthamer, N.R.

1984. A p p i . Supercond. Conf. R e p o r t s .

1966. Phys. Rev.

3. L a r k i n , A . I . , Yu.N. Ovchinnikov. Sov. Phvs. - JETP. 24, 1035] . 4 . Galey, R . I .

147, 255. 1966. Zh. Exp. T e o r . F i z . _51_» 1535 [ 1967 .

1981. J . A p p i . Phys. 52 ,

5. De L u s t r a c , Α . , P. C r o z a t , R. Adde. 6. Schlup, W.A.

1411.

1983. IEEE Trans. Magn. MAG-19,

1221.

1978. J . Math. Phys. J_9, 2469.

7. S i m i l a r approximation was used in analog s i m u l a t i o n s by 1982. J . A p p i . Phys. 53, 7458. 8. Z o r i n , A . B . , 629.

LE-28.

K.K. L i k h a r e v ,

S . I . Turovets.

Yablonski,

D.G.

1983. IEEE Trans. Magn. MAG-19,

9. Mukhanov, O . A . , V.K. Semenov. 1985. P r e p r i n t . Department of S t a t e U n i v e r s i t y . March, No. 9.

P h v s i c s , Moscow

ON A FEW-DEGREE-OF-FREEDOM-MODEL OF THE JOSEPHSON TUNNELING JUNCTION

G. BRUNK Technische Universität Berlin, Straße des 17. Juni 135, Sekr. C 8, 1000 Berlin 12

Abstract :

The dynamics of an ideal SIS Josephson tunnel junction

is studied on the base of the Werthamer model. It is the aim of the paper to present the simplest extension of the conventional resistively shunted junction model possible to include 1) the dispersive properties, and 2) the non-linear Self-coupling of the quantum phase. This is done by an appropriate circuit design.

The constitutive description of the superconducting JOSEPHSON tunnel junction has been given by WERTHAMER /1/. In the time domain this model is represented by a non-linear

integro-differential

equation as follows, cf. /2/, /3/: i(t)

*

- Co Λ - f -

with

u+

[

u

-

( N

:= sin ^

Here denote i the current,

+

*

'c

5 i n {

-

u

+

u_ :=

f

( N

--*

" J j

cos ^

( 1 )

(2)

the quantum mechanical phase diffe-

rence, u + and u_ the components of a dynamical state vector, I the critical current, C Q the geometrical capacitance. The two kernels or relaxation functions N + determine the dissipative behaviour of the system /2/. The complex structure of the current phase relation (CPR) and the intricate definition of the relaxation functions do not make popular the use of the WERTHAMER model among experimentalists and engineers. In this contribution a few-degree-of-freedom model is stated. The first aim is to reduce the enormous display of the numerical

eva-

luation of the integro-d»fferential equation. Our second object is to present an intuitive picture of the macroscopic junction dyna-

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

78 raies by means of an electrical network. At the third place we make visible the energy associated with internal degrees of freedom. So the energy balance becomes clearer and considerations on stability may be made easier. At first we utilize the special dependence on the state vector u + which characterizes the CPR as a member of a wider class of constitutive relations /4/. From Eqs. (2) we get sin γ u_ û +

= -

2 u_ u +

(3.1)

u + u_

= (j) /2

u + u_

= (j?/2

(3.2)

and (3.3)

Inseration in the CPR, Eq. (1), yields i (t)

=

- u_ i +

+

u + i_

with the transformed "currents" i i+ =

given by the linear operators

:= C o | Π +

i_ = Y _ [ u J : =

(4)

•

2 Icu+ • N+*¿+

C Q |· tl_ + Ν û _

(5.1) (5.2)

From the voltage v

-

(6)

-

and the CPR, Eq. (4), we deduce the power Ρ

! 4u-

=

-

I

i+

- éu+1 j +

i_)

(7)

v_

(8)

By the definitions i Κ

-

v+,

we have Ρ

=

v+ i+

+

v_ i_

(9)

and v+ =

- u_ ν,

v_ = u + ν .

(10)

The relations, Eqs (4) and (10), render to represent the CPR by the network given in Fig. 1, consisting of ideal transducers with transmission ratios -1 : u_ and

1 : u + modelling the non-line-

arities and linear circuits Y + incorporating the properties of the kernels N. .

79

-Vu. Y*

Y. Figi According to the WERTHAMER theory the FOURIER transforms N + of the relaxation functions must be represented, numerically /5/, /6/ or they must be represented by a rational function.

In the second

case a simulation by means of electronics is possible /7/. Starting from a mathematical point of view, such an approximation can be executed by fitting methods in the frequency domain /7/ or in the time domain, cf. /8/. Here we look at the physical

properties

of circuit elements and ask how to simulate the essential marks of N +

(CO)

by simple circuits. If we demand further to minimize

the number of elements or equivalently the degrees of freedom we find the networks as designed in Fig. 2. In terms of flux coordinates φ ^ C

=

wi th k :- v k

Τγ [ φ ! ~T[Φ. C

2

+

-

A ^ J

TT.

Φ2~]+ * 2

(magnetic)

the total energy is

τ [Φ\ - Φ; ΎΊΓ Φ: ΤΊΖΨ: 1

1

c

1

(11)

80 the rate of d i s s i p a t i o n

is (12)

and the s u p p l i e d p o w e r is by Eqs. Ρ

=

ν

(3.2),

i - I [Φ + f

Kr y

y.

~

r

(6) and

(8)

- Φ- Φ'+] i •

(13)

fi --C.1

II·

(Β)

0

ft

Τ

Ν

«

Fig 2 Here denote

by

φ^,

Φ

β

,

Φ ç

internal

degrees of freedom, φ

φAι + := -he S +

are the state c o o r d i n a t e s

+

defined (14)

r e s t r i c t e d by the

Φ! - Φ ! - ( V « 0

.

constraint (15)

81

The Y

circuit consists of the L C rplasma oscillator connected + o o in parallel with the low frequency L^R^ damping element and the high frequency C^R^ damping branch. The main feature of the Y_ circuit with respect to damping is the strong high-pass behaviour of the C2L2R^-connection. The parameters L Q , L^, L^, R^, C^ and C2 depend on the temperature according to the temperature dependence of I

and the kernels N+ (Fig. 3). The resistance R^ is

nearly constant, whereas the temperature dependence of C^, C2 and L L? is essentially controlled by the gap frequency (jJ . Only the Δ inductance L^ depends strongly on the temperature as the band width of the low frequency damping does so; here we use the limiting frequency cj defined by —e ~ ) := VRpg · 1

82 References : 1. W E R T H A M E R ,

N.R.:

P h y s . Rev. j_47 (1966), 255 - 263 2. BRUNK, G.:

J. N o n - E q u i l i b .

T h e r m o d y n . _5 (1980), 339 - 360

3. W LÜBBIG, In SQUID' a l t e r deH.:Gruyter, B e r l i80, n / N e w York

1980

4. BRUNK, G.: In Recent D e v e l o p m e n t s in N o n e q u i l i b r i u m d y n a m i c s , L e c t u r e Notes in Physics, 199, p. 446, Springer, B e r l i n ... New York 1984 5. SCHLUP, W.: Tables of F u n c t i o n s e n t e r i n g the R e l a t i o n ... . IBM Zürich, 1972

Thermo-

Current-Phase

6. ZURBRÜGG, C h . / G R I G O R I E F F , R.D.: B e r i c h t über das D F G - F o r s c h u n g s v o r h a b e n "Numerik der D i s p e r s i o n s r e l a t i o n e n des JOSEPHSON-TunneIkontaktes (WERTHAMER-Theorie)". TU B e r l i n 1982 - 1983. 7. J A B L O N S K I ,

D.G.:

J. A p p . . Phys. 53

8. ERNE, S . Ν . / L Ü B B I G , Η.:

(1982),

7458 - 7463

J. Appi. Phys. 51 ( 1 9 8 0 a ) ,

4927 - 4929

DIRECT NUMERICAL DETERMINATION OF STATIONARY ROTATIONAL STATES IN JOSEPHSON JUNCTIONS WITH ARBITRARY CURRENT PHASE RELATION

G. Brunk, Ch. Zurbrügg Technische Universität Berlin, Staße des 17. Juni 135, Sekr. C 8 H. Lübbig Physikalisch-Technische Bundesanstalt, Institut Berlin, Abbestr. 2-12 1000 Berlin 10

Abstract A procedure is reported to calculate directly the periodic states of a differential equation or an integrodifferential equation by successice numerical integration. By this procedure the current-voltage characteristic of the JOSEPHSON tunnel junction described by the WERTHAMER model is determined. The calculation yields the stable as well as the unstable branches near the gap frequency. The minimum low voltage current a w has been determined for the WERTHAMER model. This minimum current is much smaller compared with the zero voltage hysteresis parameter of the RSJ-model (Stuart-McCumber criterium) since the damping decreases significantly for fundamental frequencies in the sub gap regime.

1. Problem We treat functional equations: ψ = F [t, ψ ί , ¿ t ].

(1 )

Here the functional F describes an arbitrary current phase relation such as for tunnel junctions, microbridges or any weak links. E.g., in the case of the WERTHAMER model, cf /1/, /2/, the functional F[ . 1 has the structure 1

F [t, ψ., ψ.] := — Sca2

[-2σω - σψ] - sin (2mt + ψ(ί))

(2)

- U_ 0 /°°M + (t) Û + ( t - τ)άτ + U + 0 /°°M_(t) Û_(t - τ )dx

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

84 with u + := sin {Ψ/2), u_ := cos («72) and the quantum phase i>(t) = 2ωΐ + ψ(ί). Here the time is p normalized to the intrinsic gap response time, Κ/2δ; ß Lr := (2e/H) I C (T) C Rj^

denote the McCumber damping parameter, σ = (2/π) cth (δ/2 kT)

the normalized conductivity, iO := IUL n r / I PL( T ) the normalized control current, ω the real DC voltage (I c critical current, C capacitance, R^

and Vpç = —

normal state resistance, M + see below). The advantages of the method used are the following - no transition process leading to the rotational state, i.e. ψ(t) = ψ(ί + π/ω), must be determined - no solution of a complicated system of algebraic equations, as described in /3/, is necessary.

2. Mathematical

procedure

Here a generalization of SAMOILENKOS 1 /4/ method is outlined. For functional differential equations Ψ = F [t, ψ ί , ψ ί ]

ψ ί (τ) := ψ(t + τ), τ

(-Γ, 0),

(3)

r may be infinite with a T-periodic functional F, i.e. F [t + Τ, ψ ΐ

+ T,

ψί

+ T]

= F [t, ψ ΐ 5 ψ1_] for all T-periodic ψ,

T-periodic solutions are determined by the following iteration scheme in the space of T-periodic functions: ^m + 1

=

^o

+

2 ^ ^

^m'

w

^ere ^

ls

a

linear operator on

(4)

the T-periodic functions given by: t Lf(t) = r ( f ( s ) - f)ds, c

1 f = γ1

C+T

c

/ f(s)ds.

By this iteration we find (in case of convergence) a solution of the equation 2 ψ(ί) = ψ 0 + L F [t, ψ ΐ , ψ ί ]. Differentiating this relation twice we get ψ(ΐ) = F [t, ψ ί , Ψ ί ] - μ,

μ = F [t, ψ ^

¿ t ].

Sufficient conditions for the convergence of the iteration are: (i)

The nonlinear operator F is bounded: for

U(t)

sa

ψ( t)

s b

(5)

85 there is a constant M such that F (t, ψ ΐ , ψ ί ) S M

(ii)

The operator F is Lipschitz F (t,

continuous:

Ψ 2 ί ) - F (t, ψ ^ , ijj1t) S K 1

Ψ2(ΐ) - ψ ^ ΐ )

+ K„

(iii)

Ψ2(ί) - ψ1(t)

The constants T, a, b, M, K^, Κ 2 are connected by the following inequalities:

T2M

< a,

- TM s b, 6

- (I K. + - K j 2 2 3

< 1.

3. Calculations

In order to evaluate the WERTHAMER functional

F(t, ψ^., ijjt) for given ψ^ it is

necessary to calculate numerically the integrals

o

/ Μ+

where M ±

(τ) u +

(t - τ) di

and

o

/ M_ (τ) u_ (t - τ) d i ,

denote the gap relaxation functions, cf /1/, /2/. Cutting off in the

time domain by a finite value L of the upper bound involves extremely long computation time and gives poor results since the 4 TT

kernel M_ decays only like

sin t . t

As in our procedure the rotation number ω is given, it is possible to cut off in the frequency domain by replacing u + and u_ by finite Fourier sums. In the case of the WERTHAMER functional we get the current i 2 i

= ßj-σ μ

where

μ = F(t, ψ^.,

The dissipated energie per period is given by D = / Τ F(t, ψ + , ψ.) ψ(ΐ) dt Τ = TTO U t U)

by

86 Using Fourier series for u + and u_ we get ϋ = - σ Σ G ω n=o +

(ηω)

(a^ + b^) + — Σ π n ω n=1

G •

{nu>)

+ db,

n

n

where G + , G_ are the real parts of the Fourier transforms of M + , M_ and a n , b( c n , d n the Fourier coefficients of u + and u_.

4. Results By our procedure we have for example calculated the current-voltage characteristics for the WERTHAMER functional for various values of β^ (Figure a). As the striking property we find the logarithmic singularity of the Fourier transforms of the integral kernels reproduced. The second essential difference between the RSJ-curve and the WERTHAMER curve is the strong lowering of the tunnel characteristic in the sub gap regime. Looking for the reasons of the effect we have tested the dissipation in the junction as distributed to the harmonics of the state variables u + , u_, (cf table). In the RSJ-model the dissipation of the first harmonic predominates while in the present model the dissipation is mainly controlled by the 3rt* harmonic and the 1s1* is undamped (T = 0) or at least weaker damped at intermediate temperatures (Figure b). The rd magnitude of the dissipation of the 3 harmonic is found to be equal in the different cases (RSJ, Τ = 0, T/T^ = 0.72, T^ denotes the transition temperature). In the subgap domain ω < 1 there is a minimum in the current-voltage characteristics due to the WERTHAMER model. As we assume the decreasing branch of the characteristics to be unstable this minimum current a,, w is taken as analog of the Stuart-McCumber parameter a. The result is a strongly diminished minimum a

as shown in Figure c. ω = 0.77

Parts of the dissipated energie in the η th harmonics of u + , u_ η WERTHAMER

1 no damping

3

5

7

88.6 %

10.5 %

0.8 %

30.8 %

62.3 %

6.7 %

0.2 %

79.4 %

18.7 %

1.8 %

0.1 %

Τ = 0 WERTHAMER T/T c = 0.72 RSJ

87

minimum current a

JOSEPHSON JUNCTION at T=0

0.4

0.6

Of ßc

1.4

1.2

1.0

0.8

as function

VOLTAGE ZA

.

b)

τ I

1.5

Tc

rlÚj2

= 0.72

0.66

1J0

" X " 0.5 -

0, Ό

.

05

1

1.0

ι

1

1.5

ω Δ / Δ

•

1

2.0

1st

Figure a)

GH

-j- 0.16

\

0

2.5 3rd

â

3.0

—*

0.01

3.5 5th

— —

ι

4.0 "0

0.5" 1.0 1st

1.5

2.0 2.5 3rd

30

3.5 5th

4.0

ω Δ / Δ ο — —

The current-voltage characteristics of an ideal tunnel junction due to the Werthamer theory;

b)

the conductivities G+/GfJ , G_/G N ( GfJ normal state conductivity) and the spectral distribution of the square rates | Ci | ßc = 1 ;

c)

the low-voltage minimum current as function of ß^.

resp.,

88 The main conclusion here is that the RSJ model is an appropriate one only in the case that the significant Fourier components (here the first harmonic) are above the energy gap.

References 1. Lübbig, H., Proc. SQUID '80, Hahlbohm, H.-D., and H. Liibbig (eds.). 1, (1980). 2. Brunk, G., J. Non-Equilib. Thermodyn. 5, 339, (1980). 3. McDonald, D.E., E.G. Johnson, and R.E. Harris. Phys. Rev. Β J_3> 1028, (1976). 4. Samoilenko, A.M., and N.I. Ronto. Moscow 1979. Numeri cal-analytic methods of investigating periodic solutions, Mir publishers. 5. Zurbrügg, Ch., to be published. 6. Stewart, W.C., Appi. Phys. Lett. Y¿, 277, (1968). 7. McCumber, D.E., J. Appi. Phys. 39, 3113, (1968).

ANALYTICAL CALCULATION OF THE

J. Buechler, M.

INDUCTANCE OF JOSEPHSON

JUNCTIONS

Rieger

L e h r s t u h l für H o c h f r e q u e n z t e c h n i k d e r T e c h n i s c h e n M ü n c h e n , A r c i s s t r a s s e 2 1 , D - 8 0 0 0 M ü n c h e n 2, F R G

Universität

Abstract Analytical ductance

expressions

for the

Fourier

coefficients

and o f L _ 1 ( t ) o f a c u r r e n t - c o n t r o l l e d

L(t)

of the

in-

Josephson

j u n c t i o n are c a l c u l a t e d

b a s e d o n the R S J - m o d e l . W i t h t h e s e

efficients

a linearized

r e p r e s e n t a t i o n o f the n o n l i n e a r

inductance

is p o s s i b l e

parametric

a m p l i f i e r s m a y be

co-

Josephson

a n d the c o n v e r s i o n m a t r i x of m i x e r s

and

calculated.

I n t r o d u c i ion A Josephson therefore

junction

as the n o n l i n e a r

lifiers. The

small

Josephson element paper well The

is a n o n l i n e a r

analytical as o f L _ 1 ( t )

signal

element

solutions

conversion matrix

ding networks, whereas

in m i x e r s

frequency

is d e s c r i b e d are

inductance

and m a y be

applied

and p a r a m e t r i c

amp-

conversion behaviour

of

the

by the c o n v e r s i o n m a t r i x .

In

this

for the c o n v e r s i o n m a t r i c e s

o f L(t)

as

presented. of L(t)

is n e e d e d

for

low

impedance

the c o n v e r s i o n m a t r i x o f L _ 1 ( t )

the c a s e of h i g h i m p e d a n c e

embedding

embed-

is u s e d

in

networks.

Results The

equivalent

lations

circuit which

is s h o w n

by' the d c - c u r r e n t

in F i g .

is t a k e n as the b a s i s

1. T h e J o s e p h s o n

I η . W i t h the n o r m a l i z e d

junction

for the is

calcu-

controlled

values[1]

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

90 2eRI o

í

τ = ω t g

' g =

max

the d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g the e q u i v a l e n t c i r c u i t =

%

s i n

dip 3T

+

^

.

(1)

Fig. 1: E q u i v a l e n t circuit of a c u r r e n t - c o n t r o l l e d J o s e p h s o n t ion The s o l u t i o n of this d i f f e r e n t i a l e q u a t i o n for a Q > 1 φ(τ)

= 2 arctan

w i t h the n o r m a l i z e d Ω = /α

1+Ω

tan

junc

is given by

Ωτ

T

(2)

•TFTl

frequency (3)

o

The n o n l i n e a r J o s e p h s o n

inductance

is given by

[2]

L L(T) =

is

(4)

cosipCr )

with 2e I

From eqs.

(2) and

LC-O

The

= L

inductance

c

(4) we

obtain

cosñT + flsin.^T+Q2 + 1 ίΡΈοΒΩτ-ΩΒίηΩτ

L ( t ) m a y be e x p r e s s e d by the F o u r i e r

•Σ

LCt) = >

L

m

e

jmC

CS)

series (6)

91 w i t h the

coefficients

¿I 2 ΤΓ

m where

LCt)

dr

e

(7)

ξ = Ωτ.

T h e e v a l u a t i o n of eq.

(7) r e s u l t s

,

11

— Ω lei m

L

C

Ω ΙεΙ

m = 0 ,

in

, α > ' o

m S 2 , even, α >1 ' ' ο

, m S 1 , odd, α >1 m- 1 ' ' ' ο

w i t h ε = 1 + jΩ. F i g . 2 s h o w s the a b s o l u t e v a l u e s c u l a t e d f r o m eq. (8).

of the F o u r i e r

coefficients

m = even

.01 Fig.

2: A b s o l u t e v a l u e

1.03

1.1

o f the F o u r i e r

1.3

a

coefficients

L

cal-

92

The coefficients of the Fourier series of 1/L(t) are given by 1/LCT) =Y2 lm e^ ' where

(9)

2n

= i / LlW e"jm?

*

dû)

The following table gives the analytical solution of this integral for m = 0 ... 3. Im

o+ , ·J -4α -ο +4«ο3Ω+5α 2ο -3α Ω-1 . , „ ^ g Ω ο 2 — τ ^ —ν — ?— * — ϊ > Tor 10< α„ S ο aO„ L 2 α 2 α „ Ω 1 C 0 0 6 On ·μ η 2 ) -16α 3 +16α35Ω+28α„"-20α 3Ω-13α 2+5α t Ω+11 4 2Ω+Μ1-Ω ) 0 ο ο _ 0 0 0 fnr 1or < η o/uoA ( λ^ ) , where A is the junction size) the critical currents showed good diffraction patterns I c (Η), an evidence of a high uniformity of the tunnel barrier. These patterns make it possible to determine the net penetration depth of the electrodes as (230±30) nm. Using the literature value λ^ =150 nm for PbAuIn alloy (2), we can restore λ^ of the Pb Sb Q 0 1 7 alloy films as (80±30) nm. Well-pronounced Fiske steps arise at the junction I-V curves in magnetic fields of the same order (H>H^). Voltage positions of these steps imply the junction specific capacitance to equal C/S= 2 (3.7 ± 0.6) uF/cm ; note that this value is in agreement with measured earlier (5) for the same oxide by another method. Interaction with Microstrip Resonator The most interesting features, however, could be observed in weak magnetic fields, H < »•

• · ·

• . •X

ε

—· • _

· •

•

.·· ·\ =2 n

·

'·../ y

· ·.·

V

• •

—

··..·· η = 3

V

·

··..

'Γ.4.·"

· /

· ·.

n=1

9

ι

ι

-1

0

1 1

Be (a-u.)—

Fig. 5 - Magnetic field dependence of the first four steps amplitudes. The experimental data refer to the interferometer in the same conditions of Fig. 2. the computation a value for the coupling parameter Ζ = 0.77 determined from the I ^ / I j ratio has been used. The case of resonances in interferometers with a non-negligible considered in Fig. 5.

β

value is

The experimental magnetic field dependences of the am-

124

plitude of the first four resonances a r e shown. sample of F i g . 2.

The data

refer

to the same

The asymmetry of the threshold c u r v e is also clearly exhi-

bited in the magnetic pattern of the f i r s t and second s i n g u l a r i t y .

At p r e s e n t

no theoretical approach to describe this experimental situation is available. In conclusion, the magnetic field dependence of the critical c u r r e n t and of the resonant modes in light-sensitive interferometers have been investigated. For small

β

v a l u e s , a comparison between experiments and a simple theoretical

approach based on a non-uniform c u r r e n t density profile gives a satisfactory agreement. For l a r g e r

β

values, the interferential behavior of the critical

c u r r e n t is well accounted in the framework of available theories; as far as concerns the c u r r e n t singularities f u r t h e r work is in p r o g r e s s .

References 1.

H.H. Zappe and B . S . L a n d m a n . , 1978., J . Appi. P h y s . 49, 4149

2.

D.B. Tuckerman and J . H . Magerlein. 1980. Appi. P h y s . Lett. 37, 241.

3.

G. Paterno, A.M. Cucolo and G. Modestino. 1985. J . A p p i . P h y s . 57, 1680

4.

A. Barone and M. Russo. 1983. in "Advanced in Superconductivity" (B. Deaver and J . Ruvalds e d s . ) Plenum, p . 197.

5.

A. Barone, G. Paterno, M. Russo and R. Vaglio. 1977. P h y s . Status Solidi A41, 393.

6.

A.M. Cucolo and G. Paterno. 1983. Cryogenics 23, 639.

7.

Won Tien Tsang and T . van Duzer. 1975.J. A p p i . P h y s . 46, 4573

8.

A. Barone and G. Paterno. 1982. "Physics and Applications of the Josephson Effect" Wiley, New York, Chap. 12.

9.

A. M. Cucolo, M. R. Saggese, and G. Paterno. 1983. P h y s . 51, 102.

10.

A. M. Cucolo and G. Paterno. 964.

11.

M. Russo and R. Vaglio.

12.

G. Paterno, A.M. Cucolo, R. Vaglio and G. Modestino (this conference, p. 1 1 5 .

J

1983. IEEE T r a n s . Mag

Low Temp MAG-19

1978. P h y s . Rev. B_17, 2171.

RESONANT MODES IN REFRACTORY BASELAYER d.c. SQUIDs

G. Paterno Associazione EURATOM-ENEA sulla Fusione, C.R.E. Frascati, C.P. 65 - 00044 Frascati, Rome (Italy) A.M. Cucolo, G. Modestino*, R. Vaglio Dipartimento di Fisica, University of Salerno, 84100 Salerno, (Italy)

Introduction Due to the nonlinear interaction of the ac Josephson current with the resonant circuit formed by the loop inductance L and the junction capacitance C, resonances can occur in the current voltage characteristics of a dc SQUID (1,2,3). They appear as current singularities at

voltages integer multiples of V r =

jLC/2,

where φο is the flux quantum. In this paper we report on the magnetic

field dependence of the current amplitude of the resonance

induced singularities. The samples investigated were two junction inteferometers

(L>ie /b9) = L c (1-K2·) [1+K* (l+ficosef' ]

Lm =

(6)

For the case with the weak link broken (p-» 0): L m = L J 1 - K 2 (1-M, /M 0 ) ] For

(7)

an ideal toroidal SQUID the spatial distributions

screening and

the

and coil screening currents are equal.

flux

Thus M 0 / M , = 1

broken link inductance given by (7) becomes

space inductance L e ,

of

the

free

as is to be expected for an ideal toroidal

coil.

The Mutual Inductance Of A Two Hole SQUID We

will

give

an

example

of

a

calculation

of

the

mutual

inductance for a particular SQUID using the above ideas. For

a coil of Ν turns,

symmetric

short in comparison with the hole of

two hole SQUID in which it has been

placed,

we

a can

derive an approximate expression for M. We confine our attention

145 here

to

length

a

single layer coil,

lc

and

cross

a uniformly wound

sectional area A c . >

2d}, where

the

1>, and

of

coil

is

d^

are

sufficiently

short

respectively

the length and diameter of the hole] one can

that ju = NAc/A^.

[(lv, -l c )

When

solenoid

show

The cross sectional area of the hole is Av,. We

assume that the long solenoid formula will give us a good enough approximate expression for the flux screening inductance of hole: L), =

μ0Α),/1>,. For a symmetrical two hole SQUID

the

the flux

screening inductance of the SQUID L,= L^/2. Thus, for the mutual inductance M = juL, we find: M = ju0NAc/21k

(8)

This is exactly the empirical relation for M which was found Gif fard

et

al

( 3 ) to give agreement (within 10%)

with

by their

measured values of M for a number of coils.

Conclusions By

dividing the currents in the SQUID into flux

coil

screening

and

screening currents it is possible to define inductances in

a consistent way. The 'mutual inductance' M of the SQUID cannot, in

general,

through flux. of

a

be defined as the ratio between the magnetic

well-defined surface and the current

We defined M =

the

with L, the flux screening

SQUID and μ a parameter related to the

current.

This

response

of

producing

coil

flux the

inductance screening

yields the usual flux periodicity Αφ = φα . a SQUID to a current through a coil coupled to

can be analysed using the concepts described in this paper.

The it

146 Acknowledgments We thank Dr.

C.N.

Guy for many fruitful discussions. This work

has been supported by a grant from the S.E.R.C.

References 1.

C.N. Guy, J.G. Park. 1984. J.Phys.D 17, 871.

2.

J.G. Park, A.T. Cayless, P.A.J, de Groot. to be published in J.Phys.D.

3.

R.P.

Giffard,

Phys. 6, 533.

R.A.

Webb, J.C. Wheatley. 1972. J.Low Temp.

Nb-Si-Nb TUNNIÎL JUNCTION RF SQUID

S.Q. Xue, W.C. Qiao, L.X. Xiao National Institute of Metrology, He Pin Li, Peking, China G.Q. Cui, X.F. Meng, J.Ζ. Li, X.W. Wu, T.P. Zhang Peking University, Peking, China

ABSTRACT Λ new RP SQUID using a Nb-Si-Nb tunnel junction and the fabrication of small area tunnel junction with a Si barrier layer are described. This kind of junction shows very good reliability in mechanics and temperature cycling. The energy resolution of the RF SQUID is

1.9 x 1 0 2 R j/Hz.

INTRODUCTION There are three kinds of junctions used in RF SQUID: point contact, thin film microbridge and tunnel junction. Point contact junctions are easy to fabricate but not good enough in mechanics and temperature cycling. Thin film microbridges have good stability, but their operating temperature range is too small. Of three kinds of junctions thin film tunnel junctions are of the best stability and reliability, and they are ideal for RF SQUID. Of various tunnel junctions, all-Nb junctions have tho best stability, but junctions with oxidized Nb barrier have hysteretic I-V curve because of large dielectric constant of N b o 0 c 1 ) ' ( £ - 29.5 ) , which is not suitable for RF SQUID. Because Si 2) barriers are of smaller dielectric constant (t - 9.3 ) , which can be made into a high quality tunnel junction, they have been used instead of Nb^O^. It is possible to make junctions either with hysteretic I-V curve or without it. The latter is suitable for RF SQUIDs. JUNCTION FAiiRIOATIO?) The Nb thin films were deposited on glass substrate bv electron beam evaporation with a pressure of 5 χ 1 0 - 6 Torr, at the rate

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

148 of 100 A/sec.

The thickness of Nb films was 2000-3000 Â. The

critical temperature

Τ

c

of the Nb film varied with the substrate J3) ' if the Τ , was sub

temperature Τ . The Τ was 8.R-9.0 Κ or 9.2 Κ s e 100 'C or higher than 250 "C respectively.

After deposition of the base layer, a thin Hi coating was immediately evaporated on Nb film, without breaking down the vacuum condition, at a rate of 2-4 A/sec with the substrate e temperature β

around 200 C. The thickness of Si coating was 40-60 A. The base pattern was formed on Si coated Nb film by photolithography and plasma etching using AZ-1350J photoresist. To make the upper Nb pattern the lift-off technique was used. During the time of deposition of the upper pattern, the temperature

of

the substrate must be kept not higher than 150° C because of the photoresist. Pig. 1

shows the patterns of the device. A scann-

ing electron micrograph of the junction 2 is shown in Fig. 2. The area of the junction is about 0.3/¿m . ANALYSES OP Si TUNNEL BARRIERS 0 Thin Si films of 40 A thickness, which were deposited on monocrystal substrates of NaCl and then taken off, have been investigated using TKM. The evaporating conditions,such as

deposi-

tion rate, substrate temperature, etc. were the same as that for deposition on Nb films. Results showed that Si films were noncrystalline and continuous, and no pinholes were found. The composition and structure of oxidized Si barriers, which were oxidized in room air for more than 24 hours with a thickness of 40 Â, was analysed by XPS. Results indicated that there existed SÌO2 at the top of the barrier, Si only at the bottom (no 3ÌO2 was detected) and a thin transitive layer of SiO ( 2 > x>0 ) in between. PROPERTIES OF JUNCTIONS

ρ The I-V curve of the junction with area of 2/I characteristic at ηΦο consists of a OUT IN XDC series of roughly triangular features, which decrease in amplitude with

155 increasing Ij^, in which the positive step slopes (figure 2) have become strongly negative. [ηφο 4b.

Applying an additional half quantum of external

DC

flux

(η + ί)Φο] splits and inverts these triangles, as can be seen in figure The positive step slope features which are associated with

performance

AC

lower

biased SQUID magnetometers can be recovered from these 430 MHz

Ij!^ characteristics in two distinct ways. Either the SQUID 0UT v e r s u s magnetometer system bandwidth can be reduced or, alternatively, noise can be V

injected from an external source, via the tank circuit, into the weak link ring.

The outcome of using either technique, or a combination of both, is

that the slope of the SQUID features in

V "OUT

I. IN

can be altered at

will and in the limit of small bandwidth and large noise amplitude the features can be made to disappear altogether.

The effects of noise and bandwidth in an

AC biased SQUID magnetometer can be seen very clearly in the examples provided

OUT 'OUT

'IN arbitrary units

arbitrary

units

Figure S.

in figures 5a and 5b, respectively.

In figure 5a we show the effect of

increasing UHF flux noise, band-stopped within the 3dB bandwidth of the tank circuit, on a section of the in phase r ' system bandwidth here is 10 MHz.

ν„ Μ _ OUT

versus

IT>, IN

characteristic.

The

The same smearing out of the characteristic

occurs over a large frequency range for the injected noise, from a few MHz to a

156 few GHz.

In figure 5b we demonstrate the effect of decreasing system band-

width on a section of the characteristic.

In the upper characteristic the

system bandwidth has been reduced to 100 kHz, in the lower to 10 kHz.

From the above arguments we conclude that the positive step slope in AC biased SQUID magnetometer

(flux mode) characteristics is an artifact created by

limitations in the performance of the magnetometer electronics systems and, by implication, that the quasiclassical model

[equations

(1) and (2)] does not

provide a satisfactory description of an AC-biased SQUID magnetometer.

We

note that essentially identical noise and bandwidth dependent behaviour is observed in charge mode AC biased SQUID voltmeters

(6,7,10).

References

1.

Leggett, A. J. : Proc. VI Int. Conf. on Noise in Physical Systems (N.B.S. Special Pub. 614, CODE : X5NB SAB - Library of Congress Cat. Card. No. 81-600084), 335.

2.

Solymar, L. 1972. In : Superconductive Tunnelling and Applications. Chapman and Hall, Chapter 14.

3.

McCumber, D. E. 1968. : J. Appi. Phys. 39, 3113.

4.

Josephson, B. D. 1962. : Phys. Letts,

5.

de Gennes, P. G. 1966. In Superconductivity of Metals and Alloys.

6.

Prance, R. J., T. D. Clark, J. E. Mutton, H. Prance, T. P. Spiller, R. Nest 1985. : Phys. Letts. 107A, 133.

7.

Prance, H.,R. J. Prance, J. E. Mutton, T. P. Spiller, T. D. Clark, R. Nest. 1985: Quantum Duality in SQUID Rings, to be published in Festkörperprobleme Vol. XXV.

8.

Kurkij ärvi, J. 1973 : J. Appi. Phys. £4, 3729.

9.

Long, A. P., T. D. Clark, R. J. Prance. 1980 : Rev. Sei. Instrum.

10.

251. Benjamin.

8.

Prance, R. J., T. D. Clark, H. Prance, T. P. Spiller, J. E. Mutton, Noise and Bandwidth Limitations in AC Biased SQUID Systems, in preparation.

THE ASYMMETRY EFFECT IN DC-SQUID

Xu, Fengzhi Institute of Physics, Academia Sinica, Beijing, China '¿eng, Pei Wei Chang Chun College of Geology, Chang Chun, China.

As we know in a double junction DC-SQUID the SQUID'S critical current is a period function of magnetic field.

As is well

known, periods depending on the junction area A and the loop area S can be observed, viz.

Η χ = > A, thus H 2 >> Hj_, when the external field is changed the critical current of the junctions is changed by an amount which we define as 01(Hi).

Fig. 1 I-H curve

In a real DC-SQUID, it is difficult to make a symmetrical double junction. The assymmetry of the inductance can cause the amplitude of the voltage modulation to decrease for some biasing currents, but it does not cause amy change of the field periods. In this paper we shall mainly discuss the current asymmetry which is of two kinds.

The first, due to l c i(0)

We call statical asymmetry.

IC2(0).

Its treatment is complicated,

but unimportant here, and for simplicity we ignore it.

By

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

158 setting

l c i ( 0 ) - I C 2 ( 0 ) - 1/2 I c ( 0 ) .

The second

aeymmetry

is Ι ι ( Φ θ , ν ) =5^ l2(

- - ñ ¡ -

H

i

A t t h e same t i m e , t h e S Q U I D v o l t a g e c h a n g e s f r o m t h e m i n i m u m

159 to m a x i m u m t h e n r e t u r n s to t h e m i n i m u m , period. origin.

i.e. c h a n g e s

a

B u t t h e S Q U I D s t a t e a t H^ p o i n t is d i f f e r e n t w i t h In other w o r d s , w h e n the c u r r e n t 01(H) c h a n g e s by

01(Hi), the SQUID'S state changes W e f i r s t d e f i n e other p o i n t s

once.

(see Fig. 3) for

convenience.

Imagine integers mj, n\2 etc., a n d consider s u c c e s s i v e

field

i n t e r v a l s H = 0 to H = Hj./mj., H = H i / m i to H = Η χ ( 1 - 1/mj) l/m2 + H i / m i = H ^ / M 2 w h e r e I/M2 = 1/mi + 1/π>2 - l/mim2,

W h e n the f i e l d is i n c r e a s e d f r o m zero, t h e c u r r e n t of j u n c t i o n is c h a n g e d .

e t c

the

T h e c h a n g e i n c l u d e s two p a r t s :

the

f i r s t is c a u s e d b y the field a n d is p r o p o r t i o n a l to 0 1 ( H ) , w h i l e the s e c o n d is c a u s e d b y t h e a s y m m e t r y a n d is p r o p o r t i o n a l to ΔΙ(Η,).

W h e n t h e s u m m a t i o n of t h e t w o

is e q u a l to 01 (Hj.),. t h e s t a t e of t h e S Q U I D

parts

is c h a n g e d .

We

s u g g e s t t h a t t h i s o c c u r s a t t h e p o i n t Η - H j / m i so t h a t Η

ΔΙ(ϊφ + 01 Now suppose

Η

=

(2)

that Η ΔΙ(^)

Combining equation

- (η - 1) § i ( H 1 )

(3) w i t h e q u a t i o n

=

Htilñ

(2), w e h a v e

1^(0)

(3)

160

tm^

k

^

- η

Thus as Η - Hj/n there is a state change.

The current of

the junction is changed from 01(H^) to 1/n 61(Hi).

The

voltage of the SQUID V S 1 - (1 - ¿)V o

(5)

where v 0 is the amplitude of the SQUID's voltage. We defined ΔΙ(Η) as a current difference between the junctions in the different states.

At Η - H^/mi the state

change occurs, so that ΔΙ discharges.

When the field is

increased from H^/m^ to l/m2 ( 1 - l/Mj_)H]_.

ΔΙ(Η) increases

from zero to ΔI ( Ι/Π12 ( 1- l/Mj^) Ηχ) , i.e. Hi/m^ is its new origin.

But 01(H) is increased from 6I(Hj/mi) to 0I(l/m2(l-

Ι/Μ^Ηχ).

AI

When

2' ΔI( jj-) changes in direction. 3 3 These functions ought to be changed.

When

1 —

161 W e n o w d i s c u s s t h e m o d u l a t i n g a m p l i t u d e of t h e s m a l l oscillation.

This

T h e d e p t h of m o d u l a t i o n

and

is t h u s V r j w h e r e ,

V„. = V„. - V. = a.V RU S3 3 30

(8)

a. = (1 3

(9)

H

Conclusion:

periodic

is o b t a i n e d b y s u b t r a c t i n g Vj f r o m V g j .

1

M j

- sin

M.

1

if ΔΙ (^ ) = ¿ ( n - 1 ) 6 1 ( H 1 ) , t h u s t h e former

H^ is d i v i d e d

into η s m a l l e r p e r i o d s .

The oscillation

period peaks

are at η

—---. 2 η

T h e a mrp l i t u d e are a . V . J3 0

The current flows through the superconducting A I L l flux

loop

producing

if

AIL. i m

Φ o

v(10)

'

t h e n a m H ^ p e r i o d c a n b e o b s e r v e d , w h e r e m is a n a r b i t r a r y number.ΔΙ flows through the 3unction producing AIL^ flux

and

if AIL. = - Φ A q o

k(11)

'

t h e n a q H ¡ p e r i o d c a n b e o b s e r v e d , w h e r e q is a n a r b i t r a r y number.

A d d i n g H ^ a n d H2, w e

periods:

H 0 = H j / n , Ηχ - 0/S, H2 = 0/A'

qHj_.

in p r i n c i p l e o b s e r v e H

3

=

five

mHl» H4 =

U s u a l l y m a n d q are v e r y large n u m b e r s .

We now use the asymmetry effect theory to consider data from a n e x p e r i m e n t w h i c h w a s p e r f o r m e d six y e a r s

ago(2).

162

In o u r e x p e r i m e n t w e o b s e r v e d H1

=

1.4 χ

10~ '*G and H 3

4 periods, viz. H

1 χ

ο

3

= 1.8 χ

1 0 ~ G . T h e s e are in

10-5

υ

.

good

a g r e e m e n t w i t h o u r t h e o r y . W e did n o t g i v e a d e t a i l e d

discus-

s i o n for lack of s p a c e . W e w i l l u s e E q . ( 8 ) and

(9) t o

calcu-

l a t e the a m p l i t u d e of the s m a l l e r o s c i l l a t i o n .

H^

divides

into

r e s p o n d s t o the l o w e s t v o l t a g e y definition

period

14 s m a l l e r p e r i o d s , η = 14. T h e p o i n t A it is the p o i n t

is

cor-

1/m^

in

our

(see F i g . 5) .

VD1 = RI

(1 - -- - sin - ) V - 0.70 η n o

T h e d i s t a n c e of A B is t h e v o l t a g e V q A

= 0.62V , so V„, = 0 . 6 2 V , and c o El o g i v e n in T a b l e 1.

V

o

of t h e S Q U I D . A B = V ,

so on. T h e s e v a l u e s

are

V „ ./V Rr °

0., 70

0..43

0..21

-0..01

-0., 14

-0..26

V

0.. 62

0..35

0.. 28

-0..02

-0..21

-0.. 27

0..08

0..08

0,.01

0..07

./V ej o

(v„.-v . )/v R3 β]'' o

-0..07

0,.01

Table 1 T h e a m p l i t u d e of s m a l l e r o s c i l l a t i o n V „ . are c a l c u l a t e d v a l u e s , V . are m e a s u r e d v a l u e s . Κ] e] The results

of T a b l e

an a d e q u a t e

account

structure

in our

1 show that the asymmetry theory of t h e p o s i t i o n

< 2,

This

the H ^ p e r i o d

period,

is a g e n e r a l J o s e p h s o n

are s y m m e t r i c a l ,

is n o t d i v i d e d . η

is an

but at t h e former

even where η

ΔΙ

hyperfine

ideal Josephson

period

period

The H^ effect.

into a r e g u l a r

there

is a h i g h e r

is d i v i d e d

is not an

effect.

= 0, η = 1.

is n o t d i v i d e d

W h e n η > 2, t h e former periods,

s i z e of t h e

data.

This a s y m m e t r y effect two junctions

and

gives

integer.

into η

If

the

period If 1
10 GHZ) to achieve sufficient high emitted power. In noise thermometry the need for a

radiofrequency pump-signal causes technical

problems due to the

large volume of the SQUID circuit / tank circuit combination. As a proposal to overcome these difficulties we studied the possibility to combine the advantages of DC-SQUIDs and common R-SQUIDs to a "Resistive DCSQUID": a superconducting ring interrupted by two Josephson junctions and a resistance. In Fig. 1 a lumped circuit model of the Resistive DC-SQUID is shown. The two Josephson junctions are represented by the capacitance-free Resistive-ShuntedJunction (RSJ) model with critical currents resistances R ^

and I ^ and quasiparticle

and R ^ . L a n d R are the loop inductance

and the resistance

resp. The circuit is biased by two independent dc-currents: the main bias current (1 + α) I ß (distributed into the paths I ß and αΙ β ) and the resistor bias current

which flows through the resistive part and the parallel-

connected junctions. The appropriate choice of the main bias-current causes the junction voltages to oscillate in the microwave region whereas the voltage drop across the small resistance R produces a low frequency component for the junction voltages by interference of the two slightly different high-frequency signals. So the

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

166 (

'(Uo)lB i Fig. 1 Lumped c i r c u i t model of the R e s i s t i v e DC-SQUID

c i r c u i t w i l l behave l i k e an usual DC-SQUID exposed to a linear increasing magnetic f l u x . Consequently the amplitude of the low-frequency component of V(t) i s of the same order as the voltage modulation in an usual DC-SQUID (pV-region) and the frequency f ^ i s related to the dc-voltage < V^ > across R by the relationship < V

Here

=

R

> =

%

f

L

1)

i s the fluxquantum.

To prove t h i s behaviour numerical c a l c u l a t i o n s have been performed on the basis of the model shown in F i g . 1 and a practical r e a l i z a t i o n of the c i r c u i t has been tested experimentally.

Calculations Applying K i r c h h o f f ' s law and the Josephson r e l a t i o n s to the c i r c u i t ( F i g . 1) the R e s i s t i v e DC-SQUID can be described by a system of three dimensionless differential

equations:

167

J ^ - 2 T T Í

Ai

dr

=

[ W - £ - ¿ ( ? J ]

Λ

2ΤΤβ

Ldr

_ i&l

dt- J

-Vf /

c

Here the following3 normalizations have been used: η = I r /Ι Γ γ = R/R L2 Li , qi iR/iCi, ξ = iB/iCi, δ = y i C i LI r / Φ , τ = t R /L, ε Li o qi cp^ and J ( Ψ 2 '' "

norma

l i z e d t° the corresponding critical currents - and

the RqI c "products are assumed to be equal for both junctions. The systems of equations 2), 3), 4) has been solved numerically using an Adams iteration procedure and the low frequency component of Φι and φ 2 resp. have been evaluated for different parameter sets. One selected result, the dependence of the normalized low frequency on the normalized resistor-current, is shown in Fig. 2. Fig. 2:

0.5

Dependence of the normalized frequency f. of the low-frequency component V.(t) of V(t) on the normalized bias current I... The dashed line indicates the R(R + R ) Q2* qi dc-impedance ρ = R + R + R qi q2

0.4 0.3 0.2

O

0.1 0

Parameters : 2πβ = 0.8, c 0.1

0.3

0.2

OA

= 0.002 1, η = 1, ξ = 1.1

0.5

A f RIC L For sufficient high bias currents the impedance of the Resistive DC SQUID aproaches the constant value of R II (R

+ R ) (dashed line in Fig.2). In the qa qi region Ι^/Ις > 0.4 the deviations from linearity are less then 1 %. Low bias

currents may be preferable to increase the signal amplitude V^(t). Therefore on the basis of the first results of the calculations careful choice of the parameters for an optimal compromise between accuracy (e.g. in noise-thermometer applications) and LF-voltage resolution will be necessary.

Experiments A Resistive DC-SQUID was prepared in planar Niobium technology. The configuration, which may be seen schematically in the lower l e f t corner of Fig. 3, was patterned using optical lithography and l i f t - o f f technique. The two Josephson junctions are part of the single Nb layer. They are variable-thickness microbridges situated on a step edge in the s i l i c o n wafer. The c r i t i c a l currents have been adjusted

by pulse heating to about 20 μΑ, the quasiparticle resistance R^

i s about 0.15 Ω. To prevent oxydation the whole Nb-structure i s coated with a 0.05 pm copper layer . The Cu-layer enables ultrasonic bonding not only of the leads to the terminals but also of a blob of 25 μ Aluminium bond-wire across a small gap in the Nb-ring to realize the r e s i s t i v e part R. For t h i s type of -4 r e s i s t o r s values in the order of 10

Ω have been measured. I f used for noise

thermometry, t h i s configuration seems to be suitable for direct preparation on a copper block to achieve good thermal contact with the objects to be temperature measured. The Resistive DC-SQUID has been tested in the c i r c u i t shown in Fig. 3.

'bH^HI-

JL irfc -EU Í R :

Í#t Fig. 3:

TP

ELECTR. SHIELD

OSCILLOSCOPE

V(t)

LEAD SHIELD H E 4 BATH

Test c i r c u i t for the Resistive DC-SQUID.

COUNTER

169 The bias currents Ig and

are generated by two separate battery-powered

circuits containing filters to prevent high-frequency interferences. The lowfrequency component V L (t) of the junction voltage V(t) is defected with a high-gain amplifier in a conveniently restricted frequency band, displayed on an oscilloscope and its frequency is counted. Shielding of the sample against electrical and magnetical fields has been provided with superconducting lead shields. In Fig. 4. a result of the measurements is presented.

35-

Fig. 4: Measured dependence of the frequency f. on the resistor bias current I N . The dashed line indicates the effective zero-line, ξ « 1.

30·-

100

200 fL

300

(kHz)

— .

In each case two values of the bias current

generate voltage drops across R

of equal intensity but opposite signs. To each of them corresponds the same frequency value f^. We suppose that the zero shift on the

- axis is caused

by thermoelectric and leakage currents. Nevertheless the linear dependence of f^ on I

is demonstrated. Using the relation Δ f. Ο Δ I Ν

for the determination of the impedance we found R = 1.1 · 10

5)

Ω in good

agreement with the resistance value taken from direct DC-voltage-current measurements at the sample. This result and an observed amplitude of the signal V L (t) in the microvolt range confirm the predicted behaviour of the Resistive DC SQUID in the first tests. Further experiments are in progress to optimize the parameter sets, to reduce the influences of amplifier noise, leakage currents etc. and to investigate the deviations from linearity.

170 Conclusion A superconducting r i n g interrupted by two Josephson junctions and a r e s i s t i v e segment ( R e s i s t i v e DC-SQUID) has been described and tested with a lumpedc i r c u i t model and by experimental r e a l i z a t i o n . The c i r c u i t proved to be a p p l i cable as a voltage-to-frequency converter with high output signal at i n t e r mediate frequencies. The f i r s t r e s u l t s are very promising for the future development of the R e s i s t i v e DC-SQUID as e.g. a high performance voltage-controlled o s c i l l a t o r for d r i v i n g cryoelectronic c i r c u i t s or as a accurate low-temperature noise thermometer suitable for good thermal contact to the objects to be measured.

Acknowledgements The authors wish to thank Dr. K. D. Klein for the design of the planar configuration and for providing the chromium mask and Mrs. M. Peters for her technical assistance during the microfabrication.

References 1.

Zimmerman, J . E . , J.A. Cowen, H. S i l v e r . 1966. Appi. Phys. Lett. 9, 353-355.

2.

Erné, S.N.. 1978. In: Proc. 5th I n t . Conf. on Noise in Physical Systems (D. Wolf, Ed.). Springer, Berlin-Heidelberg-New York.

3.

Hoffmann, Α., Β. Buchhol ζ. 1984. J. Phys. E.: S c i . Instrum. JT_, 1035.

THREE TERMINAL NON-EQUILIBRIUM SUPERCONDUCTING DEVICES

R.A. Buhrman School of Applied and Engineering Physics Cornell University Ithaca, New York 11853

I.

Introduction

For a number of years there has been a continuing interest in possible development of a three terminal superconducting device that would exhibit electrical characteristics fundamentally similar to those of a bipolar or field effect transistor.

The recent cessation of several of the major development efforts

in high speed Josephson digital electronics has dramatically heightened the appreciation of the desirability of attaining this goal.

But if such a device

is to eventually prove competitive with semiconductor technology and is to be an improvement over existing Josephson technology it must meet some very stringent performance standards.

Beyond the attainment of a significant power gain

or equivalently the capability of supporting a large device fan-out, these requirements include, arguably, the capability of attaining a loaded switching speed < 100 psec., a power dissipation level < 10 yW suitable for high density applications, and, to be compatible with superconducting stripline impedances and with the above criteria,, the ability to operate on voltage levels, £ 1 0 mV.

These are clearly a very challenging set of requirements.

A number of different three terminal device concepts have been proposed over the years (1).

One class of such devices which has received considerable

attention recently is those whose operation is based on the suppression of the superconducting state in a selected region of a superconducting thin film by creation of non-equilibrium population of quasiparticles.

It is the purpose of

this paper to review some of the recent work on this type of device and to discuss the prospects of non-equilibrium superconductivity eventually yielding a device which is capable of meeting the above goals.

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

172 II.

CLINK

The concept of a non-equilibrium superconducting device was apparently first proposed by Volkov (2) who suggested that a superconducting thin film strip can be made to exhibit Josephson behavior if superconductivity in a localized region of the film were suppressed by laser excitation of quasiparticles in that region.

Subsequently Wong and co-workers (3) demonstrated that good

Josephson-like behavior could in fact be produced in thin film microstrip by Injecting quasiparticles with a tunnel junction into a narrow section of the microstrip, by heating a narrow region with a superimposed thin film heater, or by the imposition of a focused laser beam onto the film.

The major objective of this work was to develop a means of producing a controllable superconducting weak link; hence the name CLINK that was coined for this device.

But, focusing on the quasiparticle injection version, this device con-

cept also had some possibilities for serving as a three terminal switching device with power amplification.

This switching operation could be carried out

by first supercurrent biasing the microstrip with a current lb < I c the critical current of the microstrip.

where

Upon injection of an ade-

quate quasiparticle current into a small region of the microstrip the critical current of that region would be reduced below the bias level l¡j.

That region

would then abruptly switch to the normal state resulting in a localized hot spot at the injected region.

Owing to the poor film-substrate thermal contact

and the one-dimensional geometry of the microstrip, the hot spot would propagate along the entire length of the microstrip, driving it normal, and yielding an output voltage It,Rm where R m is the normal state resistance of the microstrip.

Examining the results of Wong et .al. we see that power gain was in

fact obtained in this mode of device operation.

Indeed as Gallagher (1) has

pointed out this particular device is one of the very few three terminal superconducting devices ever to show power gain.

But while it was not measured, it is reasonable to expect that a CLINK device would be rather slow.

Its turn-on and turn-off time would be set by the rate

of thermal transport down a long microstrip which in these experiments was 1.6 mm long. competitive.

Thus as a digital device, this original CLINK structure is not

173 III.

Qui teron

Subsequent t o the I n i t i a l work of Wong e t . a l . , F a r i s ( Ό proposed a somewhat d i f f e r e n t type of gap suppression device f o r a p p l i c a t i o n in d i g i t a l l o g i c c i r cuits.

This device which was named the Quiteron, and which was subsequently

r e a l i z e d and t e s t e d by Gallagher and co-workers ( 5 ) , d i f f e r e d from the previous work in t h a t i t used a second tunnel j u n c t i o n c a l l e d the acceptor t o monitor and respond t o the suppression of the superconducting energy gap in a t h i n f i l m region which was subjected t o q u a s i p a r t i c l e i n j e c t i o n by the o t h e r , i n j e c t o r , junction. The geometry of the Quiteron i s t h a t which has been used e x t e n s i v e l y over the years in non-equilibrium superconductivity experiments ( 6 ) .

As sketched in

F i g . 1 i t c o n s i s t s of two, stacked t h i n f i l m tunnel j u n c t i o n s with the shared, middle e l e c t r o d e being the s i t e where the gap suppression occurs.

The b a s i s of

the Quiteron i s the use of the n o n - l i n e a r i t y of the acceptor I-V c h a r a c t e r i s t i c to provide the device response.

Thus in the off s t a t e , with no q u a s i p a r t i c l e

current from the i n j e c t o r j u n c t i o n , i f the acceptor i s biased a t a voltage l e v e l eV < Δ 2 (ο) + Δ3 l i t t l e c u r r e n t flow w i l l occur (in the i d e a l case the subgap current would be z e r o ) .

Upon s u f f i c i e n t q u a s i p a r t i c l e i n j e c t i o n via the

i n j e c t o r j u n c t i o n , gap suppression in f i l m 2 w i l l r e s u l t in the i n i t i a t i o n of acceptor current once Δ2 < eV a c c - Δ 3 .

The output power swing t h a t can be

achieved with the Quiteron depends of course on the n a t u r e of the load. 2

Assum-

2

ing a matched l i n e a r load, AP aco « A 3 / e R a c c f o r complete gap suppression.

But if as i n d i c a t e d in the f i g u r e a non-linear load i s used, such as

t h a t provided by the input of a following Quiteron, then the maximum power swing i s l a r g e r , AP a c c = (Δ 2 (o) + A 3 ) 2 / e 2 R a 0 C , and can occur f o r only a p a r t i a l suppression of Δ 2 —a considerable advantage.

The Quiteron, which has been r e a l i z e d with power gain, has a number of a t t r i butes t h a t at f i r s t make i t seem very a t t r a c t i v e as an a l t e r n a t i v e to convent i o n a l Josephson c i r c u i t s , p a r t i c u l a r l y i t s i n v e r t i n g nature and supposed t r a n sistor-like qualities.

However subsequent experiments and analyses have r e -

vealed major d i f f i c u l t i e s with t h i s device concept.

The major concerns a r e

with the lack of i s o l a t i o n between input and o u t p u t , the l a t c h i n g operating with power gain and, p a r t i c u l a r l y , the slow switching speed.

Figure 1.

(1) Schematic representation of the Quiteron geometry.

(ii) The I-V characteristic of the acceptor junction for the case of no injector junction current (line a) and sufficient injector current for a ~ 30? reduction of Δ 2 , the superconducting energy gap of the shared electrode (line b). Note the negative resistance portion of the uninjected I-V characteristic. This feature is necessary for device gain. Line c represents the nonlinear load line which would be provided by the input of a following Quiteron.

175 Isolation Both the lack of isolation and the latching behavior of the Quiteron are consequences of the fundamental fact that there is no basic separation between the effect of the control signal and the output signal, on the shared electrode. Thus current from either the injector junction and acceptor junction will suppress the gap of the shared electrode with the degree of gap suppression being determined basically only by the amount of power supplied by the quasiparticles.

Thus if the device is to have significant power gain the acceptor junc-

tion once switched to the high current state must deliver more power (quasiparticle current) to the shared electrode than the injector.

Hence once the gap

is suppressed, it will stay suppressed until both the injector and acceptor currents are removed, i.e. the device will latch.

One result of this fact is

that if the device is to exhibit gain it is necessary that the acceptor junction I-V characteristic exhibit negative resistance at e V a c c = Δ 2 + Δ 3 , as indicated in Fig. 1.

If, alternatively, the device is operated in a mode where

the acceptor power is less than the injector, i.e. with loss, then it will not latch.

The Quiteron then is much closer to a dual of the Josephson device than

to a transistor since it is voltage biased below a critical point (Δ2(ο) + Δ 3 ), where low quasiparticle (subgap) current flows.

Once Δ 2 is reduced to less

than eV a c c - Δ 3 a large quasiparticle current flow commences, which further suppresses A 2 and ensures that Δ 2 stays suppressed even if the injector power is removed.

This fundamental point can be further substantiated by examining a circuit analysis of the Quiteron by Frank (7) who showed that if the fan-out of a Quiteron circuit is greater than unity it will indeed latch-up.

Frank also showed

that with the use of the nonlinear load non-latching operation with unity fanout was possible, but this of course is of little practical value. As Gallagher has demonstrated (1), the equivalency of the effect of injector and acceptor power on the shared electrode also results in essentially no isolation between input and output.

Thus even if the latching nature of the Quit-

eron were acceptable, the isolation problem seems quite forbidding for circuit applications. Hunt et. al. (8) have pointed out that this latter problem can be greatly alleviated, albeit at a cost, through replacement of the SIS injector junction with a NIS injector.

As has been verified experimentally (8) such an Injector is

176 essentially just as efficient in gap suppression as is an SIS junction.

Alter-

natively a higher resistance SIS injector junction could be used such that it was always biased above

+ Δ 2 , perhaps in the off condition at eVj n j > Δ, -

+ Δ 2 , in the on condition at eVj n j » 2 ^

+ Δ 2 ), the factor of four change in

Injector power being such as to bring about the necessary gap suppression from an initially only partially suppressed state.

In either case the input resis-

tance of the device will appear to be essentially ohmic, with changes in Δ 2 having little effect on the input current.

The cost of this adaptation is that

the maximum output power swing is diminished and a complete gap suppression is required to accomplish this maximum power swing.

Thus the conditions necessary

to achieve power gain become more stringent, with it being necessary either to decrease the specific resistance of the acceptor junction substantially or to very significantly thin the shared electrode.

Note that in the experimental

7

device (1) the specific resistance was 1CT fi cm 2 and the shared electrode thickness was d = 30 nm thick.

Switching speed To be competitive any alternative logic device must be very fast, be it nonlatching or latching.

It Is this aspect that is probably the most serious flaw

of the Quiteron, as presently developed.

The two basic times in the Quiteron

are (1) the time required to create a quasiparticle population in the shared electrode sufficient to suppress the gap by the required amount, i.e. the turnon-time, and (2) the time required for the excess quasiparticle population to decay once the injector and acceptor currents are reduced—the turn-off-time.

In the ideal case, the minimum turn-on time, as discussed by Gallagher, can be simply determined from a Rothwaf-Taylor analysis of the injection process. Assuming the injection voltage Is eV i n j = Δ, + Δ 2 , quasiparticles are injected at low energy and inelastic scattering Is not an important, time consuming step In building up the non-equilibrium quasiparticle population Nq p ·

Then,

since Δ = Δ(ο) (1 - 2.0n) where η = Νς ρ /4Ν(ο)Δ(ο) and is the normalized quasiparticle density and N(o) is the electron density of states, the time required for the necessary gap suppression is 4t

.

2 eN(o) Vol I, . inj

6Δ

.

2 eN(o)d J, . inj

6Δ

(1)

This assumes that the mean effective quasiparticle lifetime (recombination time) is greater than St.

For a sufficiently thin middle electrode and high

177 injector current density

the turn-on-time can indeed be relatively short,

6t » 100 psec. Unfortunately, as Gallagher has reported (1,9), the realized Quiteron is less than ideal.

In particular, the measured turn on response of this device, as

shown in Fig. 2, has both a fast component < 1 nsec, due to the quasiparticle injection time and a much slower response extending over many nanoseconds. Gallagher has interpreted this latter, slow response as being due to a slow heating of the top, third electrode, which is relatively poorly cooled by the liquid helium in which the device is immersed.

The interpretation is supported

by the fact that the Quiteron response varied greatly when the cooling medium was altered, from He gas to He liquid and from normal liquid He to superfluid. Thus the device exhibited a rapid gap suppression in the thin middle electrode and then a slower gap suppression in the thicker top electrode as it gradually heated up.

The turn-off-time of the conventional Quiteron device is determined by the rate at which the Injected quasiparticles, from both the Injector and acceptor junctions, can be removed from the middle (and top) electrode.

For the practical

case where high T 0 superconductors with fast quasiparticle recombination times TR are used, the rate limiting step in this process is escape of the

TIME (ns) Figure 2.

Measured change in voltage of an acceptor junction of a Quiteron subjected to a step injector current. (Figure reproduced from ref. 1 with permission of the author.)

178 recombination phonons out of the superconducting f i l m .

This i s a r e s u l t of the

f a c t that the pairbreaking time i g f o r such superconductors i s also short. Hence until the phonon escapes there i s a high p r o b a b i l i t y of phonon trapping and regeneration of q u a s i p a r t i c l e s in the material. quasiparticle r e l a x a t i o n time t e f f

As a r e s u l t , the e f f e c t i v e

in high T 0 superconductors tends to be

determined by the much slower phonon escape time than by q u a s i p a r t i c l e recombination ( 1 0 ) . τ

es

The phonon escape time of a superconductor f i l m of thickness d i s -

Ü-1 ην

(2)

s where v s i s the phonon v e l o c i t y and η i s the phonon transmlssivity at the film-substrate i n t e r f a c e .

Measured values of η range from < 0 . 1 t o 0.5.

Thus

only i f the material i s s u f f i c i e n t l y thin (d < 10 nm) and a c o u s t i c a l l y well coupled to a substrate can we expect t e f f

< 100 psec.

Note that since, in

the actual Quiteron both the thin middle f i l m and thick top electrode are heated, the r e s u l t would be expected to be a very slow t u r n - o f f - t i m e > 1 nsec. This was in f a c t observed.

The combination of a slow turn-on and even slower

t u r n - o f f time of the Quiteron device makes i t noncompetitive to conventional Josephson devices, despite the achievement of power gain.

IV.

A l t e r n a t i v e Edge Junctions Device Geometry

In p a r a l l e l to the Quiteron work Hunt and co-workers (8,11) have been i n v e s t i gating an a l t e r n a t i v e type of gap suppression structure with the Intent of addressing the c r i t i c a l problem of switching speed.

As noted above the time

scale of a gap suppression device I s set by the e f f e c t i v e r a t e of r e l a x a t i o n of the non-equilibrium q u a s i p a r t i c l e population.

I f , as in the Quiteron,

this

r e l a x a t i o n Is to proceed by emission of recombination phonons and subsequent phonon escape, i t i s essential that the i n j e c t e d superconductor both have a f a s t i n t r i n s i c recombination r a t e and be s u f f i c i e n t l y thin and a c o u s t i c a l l y well coupled to i t s substrate on the f i l m so that reabsorption of the emitted phonons i s u n l i k e l y .

The a l t e r n a t i v e i s f o r r e l a x a t i o n to proceed via quasi-

p a r t i c l e d i f f u s i o n from the i n j e c t e d region to an adjacent, w e l l cooled superconducting e l e c t r o d e .

For non-equilibrium r e l a x a t i o n to proceed predominantly

by d i f f u s i o n , a l l dimensions of the i n j e c t e d region should be somewhat smaller than the basic q u a s i p a r t i c l e I n e l a s t i c s c a t t e r i n g length.

In a d d i t i o n ,

i d e a l l y should be a three dimensional fan-out of the superconductor

there

electrode

179 from the small, 1 μπι, I n j e c t e d r e g i o n .

In other words, the s t r u c t u r e should

resemble t h a t of a "well-cooled" quasi-three-dimensional microbridge ( 1 2 ) . The s t r u c t u r e employed by Hunt and co-workers in t h e i r non-equilibrium device experiments was not i d e a l but did explore non-equilibrium response in the r e gime where q u a s i p a r t i c l e d i f f u s i o n plays an e s s e n t i a l r o l e in determining the o v e r a l l device response. in F i g . 3.

A schematic r e p r e s e n t a t i o n of Hunt's device i s shown

I t c o n s i s t s of a p a i r of tunnel j u n c t i o n s formed on the edges of a

superconductorr-insulator-superconductor t h i n f i l m s t a c k .

The shared e l e c t r o d e

I s a narrow t h i n f i l m m i c r o s t r u c t u r e which fans out i n t o two dimensions a s h o r t d i s t a n c e , < 1 um, from the bottom j u n c t i o n .

The e s s e n t i a l aspect of t h i s geom-

e t r y i s t h a t the two superconducting f i l m s on which the I n j e c t o r and acceptor j u n c t i o n s are formed serve as two dimensional "heat" sinks f o r the non-equilibrium q u a s i p a r t l c l e s and phonons generated at the tunnel j u n c t i o n s .

But the

removal of nonequilibrium q u a s i p a r t l c l e s from the shared e l e c t r o d e i s l i m i t e d by one dimensional d i f f u s i o n down the narrow m i c r o s t r i p to i t s two dimensional fan-out r e g i o n . shared

Thus, as demonstrated e x p e r i m e n t a l l y , the energy gap in t h e

e l e c t r o d e can be suppressed completely by q u a s i p a r t i c l e i n j e c t i o n

Common electrode

Figure 3.

Schematic r e p r e s e n t a t i o n of the double edge j u n c t i o n s t r u c t u r e used in nonequilibrium gap suppression experiments.

180 without creating a significant heating effect in the injector or acceptor

elec-

t r o d e — a significant improvement over the conventional Quiteron structure.

The response time of the energy gap in the shared electrode to a change in quasiparticle injection rate is largely determined, as indicated above, by quasiparticle diffusion to the fanout region, assuming that this distance d < A e f f where X e ff is the effective quasiparticle recombination length, including phonon trapping effects.

This response time is then approximately

2

T

d

d Γ~ν

=

r^

(3)

g where 8, is the electron elastic mean free path and v g the group velocity of the quasiparticles, which we will approximate with vp.

Note that for d = 1

μπι, td < 100 ps for J, > 10 nm.

Under a steady state injection current Ij n j and with V j n j » Δ 2 + Δ 3 the density of non-equilibrium quasiparticles at the injecting junction in the counter-electrode film can be estimated as

Ν

-

IlnJ

J, or Ν

xd/2eAfd

lnj

qP

-

Jlnj

1

.

A,

2e

A

Td

AlnJ

(4)

2e A f d

.

d

»

(5)

IP

m

I v„ F

where A m is the cross sectional area of the counter-electrode microstrip and A

inj

the

cross sectional area of the injecting junction.

reduce Δ to 0.5Δ 0 an injection current Jj.nj

J* inj

-

Ν(ο)Δ 0 ev

^ϋ— F A inj

is

Thus in order to

required where

ί

(6)

d

Note that for a 100 nm counter-electrode Nb film with short mean free path if

I « 10 nm and with A m = A ^ n j we have J i n j _ 10 7 A/cm 2 . Am/A¿nj

If the ratio of

Is reduced by thinning the counter electrode film to 30 nm while *

increasing the injector electrode to » 300 nm J i n j could be reduced to ~ 10 6 A/cm 2 .

Further reductions in J^nj are possible by increasing d or decreasing

S, but only at the expense of directly Increasing the device response time above 100 pseo.

These large values of critical injection current, which are in very

good numerical accord with experimental results, show that junctions with very low specific assistance RA « 1 0 _ β Ω cm 2 are required in order to have a fast, gap suppression, gap recovery process.

Indeed the balance between the

181 necessity of having both a fast quasiparticle recovery rate and a large excess quasiparticle density directly mandates a very high injection current.

Since

in order for a gap suppression device to exhibit gain J a c c must be larger than Jinj> assuming equal area junctions, it is also required that the acceptor junction have an even lower specific resistance.

While these constraints

might be altered by the use of a different geometry the necessity for high injection current density levels is fundamental. Very low specific resistance tunnel junctions can and have been fabricated (8).

However it is a universal observation that low specific resistance (or

equivalently high critical current density) junctions have increased subgap currents, with this leakage current rising rapidly for RA > 1CT 7 Ώ cm 2

(8,13).

As pointed out by Hunt et.al. (8) a result of this leakage current is that tunnel junctions with a fast quasiparticle response time do not exhibit the negative resistance feature at eV = Δ 2 + Δ 3 necessary for the Qulteron mode of device operation.

In other words, high J c tunnel junctions with rapid quasi-

particle removal by diffusion do not exhibit a sharp enough turn on of quasiparticle current flow when V a c c is swept through ώ 2 + Δ 3 to allow the attainment of significant power gain.

Thus while a device such as that studied by

Hunt et.al. can have a reasonably fast response time, large signal power gain is not possible unless this response time is lengthened to an unacceptable value, > 1 nsec.

This is in accord with the experimental results.

Unless some

means are developed for producing very low specific resistance junctions with low subgap leakage, or a superconductor with a greatly reduced carrier density (see next section) can be employed as the shared electrode, the practical value of a gap suppression quasiparticle device appears to be low.

V.

Non-Equilibrium Josephson Device

A straightforward variation on the Qulteron which appears to have some potential is to use the suppression of the critical supercurrent of the acceptor junction, instead of the increase in quasiparticle current, as the output signal from gap suppression.

While such a device is not inverting it does avoid

the problem the Quiteron has with leakage in high current density junctions. The device would be operated by biasing the acceptor junction with a supercurrent If, < I 0 and then switching the junction to the ohmic line of an SIN junction by quasiparticle injection from the Injector junction, which for best isolation should be a NI(S/N) junction, that is the injector electrode should

182 be a normal metal.

A superconductor i n j e c t o r could be used however.

As I s the

Qulteron and conventional Josephson devices, t h i s device i s latching—once the shared e l e c t r o d e i s driven normal i t w i l l stay normal u n t i l

i s removed.

Indeed the device o p e r a t i o n i s r a t h e r s i m i l a r t o a conventional Josephson device with the d i f f e r e n c e being t h a t the switching i s accomplished here by gap suppression in the shared e l e c t r o d e via q u a s i p a r t i c l e i n j e c t i o n r a t h e r than by the a p p l i c a t i o n of a magnetic f i e l d or by d i r e c t c u r r e n t i n j e c t i o n t o the junction.

As a r e s u l t t h i s device i s r e f e r r e d to as JONES f o r JOsephson with

Normal Electron Switching. The advantage of the JONES device over conventional Josephson l o g i c element i s t h a t i t i s compact, r e q u i r i n g no i n d u c t i v e loops as do some Josephson c i r c u i t s , and has the p o s s i b i l i t y of s i g n i f i c a n t power gain, where the power gain i s here defined to be t h e r a t i o of t h e i n j e c t e d power necessary to completely suppress Δ in the shared e l e c t r o d e divided i n t o the output power obtained when the acceptor current i s biased s a f e l y below I 0 of the unperturbed a c c e p t o r .

This

power gain i s obtained a t the expense of decreasing device speed s i g n i f i c a n t l y below t h a t of conventional Josephson devices. A schematic r e p r e s e n t a t i o n of one version of a JONES device i s shown in Fig. 4. As can be seen i t i s q u i t e s i m i l a r to the o r i g i n a l double edge j u n c t i o n s t r u c t u r e of Hunt e t . a l . except t h a t the areas of the acceptor and i n j e c t o r j u n c t i o n s have been a d j u s t e d t o reduce the current density requirements f o r the acceptor j u n c t i o n .

In a preliminary experiment Hunt has f a b r i c a t e d and t e s t e d

such a device, using Nb I n j e c t o r and d e t e c t o r e l e c t r o d e s and a t h i n « 50 nm AÎ, microbridge of 0.5 μιη width and - 1.0 vim length as the shared e l e c t r o d e .

AS,

was chosen so t h a t the necessary i n j e c t i o n c u r r e n t l e v e l s would be lower than t h a t necessary f o r a higher T c m a t e r i a l .

The acceptor j u n c t i o n area was

approximately twice the i n j e c t o r a r e a . I-V c h a r a c t e r i s t i c s f o r the i n j e c t o r and acceptor j u n c t i o n s are shown in F i g . 5 measured a t « 0.5K.

As can be seen both j u n c t i o n s e x h i b i t a c r i t i c a l s u p e r c u r -

r e n t of the l e v e l expected f o r a Nb-A5, tunnel j u n c t i o n with the measured normal state resistance. a SIN j u n c t i o n .

But once I c i s exceeded the I-V c h a r a c t e r i s t i c i s t h a t of Thus the e f f e c t of the q u a s i p a r t i c l e i n j e c t i o n current t h a t

flows once I c i s exceeded i s to suppress A^j, to zero; only when I j n j i s reduced below a c r i t i c a l value does a supercurrent biased SIS j u n c t i o n .

recover and the device switch back to The p o s s i b i l i t i e s of such a device f o r

switching with power gain i s i l l u s t r a t e d by Fig. 5 where the I-V of t h e

183

»50nm 2 - D or 3 - D I Fonout of Electrode Injector — Acceptor

Figure U.

XX χ χ x x V x x x x x x x x

(Ì) Schematic representation of a JONES device.

(ii) I-V characteristic of the acceptor junction of a JONES device. The dotted line is the quasiparticle curve that would result if there were no gap suppression in the shared electrode. The solid line represents the I-V characteristic the device will exhibit, assuming complete gap suppression whenever there is a normal current flowing through the acceptor. The dashed line represents a linear load line with A being the bias point in the off state, Β the operating point is the switched state.

184

Figure 5.

Measured I-V c h a r a c t e r i s t i c s of a prototype Nb-AH-Nb JONES device. The v e r t i c a l s c a l e i s 100 μΑ/div; the h o r i z o n t a l s c a l e i s 0.5 mV/div. (The s l i g h t slope on the supercurrent portion of the I-V c h a r a c t e r i s t i c s i s done to a small contact r e s i s t a n c e . ) (i)

The acceptor j u n c t i o n , with no i n j e c t o r c u r r e n t .

(ii)

The i n j e c t o r j u n c t i o n with no acceptor c u r r e n t .

( i l l ) The acceptor I-V c h a r a c t e r i s t i c with 65 μΑ of injector current bias.

185 acceptor junction i s shown f o r the case of an i n j e c t o r q u a s i p a r t i c l e current of 60 μΑ.

Since t h i s i n j e c t o r current i s nearly 4 times smaller than I 0 of the

unperturbed acceptor junction the device gain p o s s i b i l i t i e s are apparent. Of course, f o r the JONES device to be p r a c t i c a l this type of r e s u l t must be obtained with a higher T c material, perhaps with Nb or NbN.

In p r i n c i p l e

t h i s simply requires that J c of the i n j e c t o r and acceptor junctions be raised to the appropriate l e v e l s .

But while the JONES device concept avoids the prob-

lem of high subgap currents, the problems of manufacturability and s t a b i l i t y of the required very high current density, J c > 106 A/cm2, junctions are l i k e l y to be s i g n i f i c a n t . An a l t e r n a t i v e approach would be to employ a material for the counterelectrode which has an unusually low density of s t a t e s f o r a superconductor. BaPb(ι_ x )Bi x 0 3

i s one such material that has received considerable atten-

tion l a t e l y f o r which theory and experiment (15,16) suggest that N(0) "" 1021 cm3 e V _ 1 .

Since the q u a s i p a r t i c l e density necessary f o r gap suppression in

t h i s material should scale with N ( 0 ) , i t should be possible to f a b r i c a t e a JONES device in which the c r i t i c a l current density of the acceptor junction could be ~ 105 A/cm2 while maintaining a quasiparticle d i f f u s i o n limited switching speed < 100 pS.

VI.

Quasiparticle I n j e c t e d Microstrip

Another and more a t t r a c t i v e device p o s s i b i l i t y can be seen by f i r s t noting that the r o l e of the acceptor junction in the JONES device i s r e a l l y that of an impedance transformer.

In other words the device i s biased by a supercurrent

l b i n j e c t e d into the microstrip by the acceptor junction, and hence we must have I b < I c acceptor

and

^

< ic.m·

Here

pairing current of the microstrip ( 1 7 ) .

:c

,m ^

^e

c r i t i c a l de

Upon application of the q u a s i p a r t i c l e

i n j e c t i o n current the microstrip becomes normal r e s u l t i n g in an output voltage V0 = Ijj ( R j > a c c microstrip.

+

Rm'

where Rm i s the normal state resistance of the

Up u n t i l now i t has been assumed that Rm < Rj

V - I. R. < o b J,acc ~

* 2e

aco

and hence (7)

Δ?

But i f Rj i s reduced to zero, i . e .

if

a S-S contact i s formed between the

acceptor electrode and the microstrlpe then

can

be

increased to

186

1(3 < I C i I I 1 and the output voltage then becomes V0 = Ib^m·

Thus the bias

current and output voltage are determined by the geometry of the mlcrostrip rather than by the impedance of the tunnel junction.

With t h i s modification we

have gone nearly f u l l c i r c l e back to an optimized version of the o r i g i n a l , t r o l l a b l e link

con-

concept.

A possible version of this device structure i s shown schematically in F i g . 6. I t consists of a thin, ~ 30 nm, microstripe of ~ 1 um χ 1 μπι dimensions formed on the edge of a normal metal f i l m .

The normal metal f i l m

junction contact constitutes the i n j e c t o r junction.

microstrip tunnel

Preferably the microstrip

should terminate at both ends in a quasi-three dimensional fan-out, thus having the geometry of a r e l a t i v e l y long variable thickness microbridge.

This w i l l

allow the greatest l e v e l of bias current to be used without the power l e v e l of the device when the microstrip i s in the normal s t a t e , being so high as to also drive the electrodes normal.

The device parameters for this type of controlled link can be estimated as f o l lows.

Assuming a Nb microstrip of the above dimensions with electron mean f r e e

path I - 100 A, we have Rm -1Ω and t^ - 1 0 0 psec, and the required quasip a r t i c l e current for s i g n i f i c a n t gap suppression i s I j n j eVinj

"

~ 3 mA (assuming

¿W· This in turn requires that J c , i n j ~ 3 x 105 A/cm2 assuming

a 1 ym χ 1 μπι i n j e c t o r area.

The microstrip can be biased with a current up to

the l i m i t set by the c r i t i c a l current density of the microbridge.

Superconductor S

Microbridge

I /¿m χ I μτη χ 3 0 nm

¡>>>»»»»¡»>i»¡>n

Ν

I s e

nmdmnf Injector

Figure 6.

Junction

Schematic representation of a proposed q u a s i p a r t i c l e i n j e c t e d microstrip device For best performance S' should be a low c a r r i e r density superconductor

Assuming

187 J

c,m > 5 χ 10' A/cm2 (17), a bias current of 10 mA should be acceptable

y i e l d i n g I^ = 10 mA.

The r e s u l t i s then a device with both a s i g n i f i c a n t

current and power gain.

Again we point out that t h i s device gain has been

obtained by s i g n i f i c a n t l y slowing down the response time of the device compared t o t h a t of an ideal high current density tunnel j u n c t i o n or microbridge. The most obvious d i f f i c u l t y with t h i s c o n t r o l l e d l i n k device i s i t s high l e v e l of power d i s s i p a t i o n , = 100 pW, when the microbridge i s in the normal s t a t e . This problem i s f a i r l y fundamental (since t h i s power l e v e l would r e s u l t in s i g n i f i c a n t heating in both the i n j e c t o r and acceptor e l e c t r o d e s and in the f a n - o u t region of the shared e l e c t r o d e .

This power level also exceeds the

l e v e l considered a p p r o p r i a t e for high density cryogenic c i r c u i t s . )

The opera-

t i n g power l e v e l can be reduced by narrowing the width of m i c r o s t r i p but a linewidth of 0.1 μΓπ would be required t o reduce the operating power to a more acceptable 10 pW while maintaining device g a i n .

While such linewidths are

c e r t a i n l y a t t a i n a b l e t h i s requirement would make the device too d i f f i c u l t f o r present day LSI manufacturing. material.

The a l t e r n a t i v e i s to change the m i c r o s t r i p

One p o s s i b i l i t y could be to use a lower T c m a t e r i a l (but s t i l l

with T c > 4.2K) so t h a t the necessary q u a s i p a r t i c l e i n j e c t i o n current would be reduced, as would the maximum bias current

Another and more a t t r a c -

t i v e p o s s i b i l i t y i s , again, t o use a superconducting m a t e r i a l with a low dens i t y of s t a t e s .

If we again consider BaPb(i_ x )Bi x 0 3 as an example, we see

t h a t since both J j . n j

and

sllou d

l

s c a l e with N(0) such a device with the

dimensions of F i g . 6 should be operable with an i n j e c t o r j u n c t i o n current dens i t y of = 3 χ IO1* A/cm 2 , which i s r o u t i n e l y a t t a i n a b l e with Nb base e l e c t r o d e s , and with Ij) of » 1 mA, r e s u l t i n g in an o n - s t a t e power d i s s i p a t i o n l e v e l of 10 μΚ.

This device looks very a t t r a c t i v e , and appears to be c l e a r l y the most

promising of the various non equilibrium devices considered h e r e . VI.

Summary

In t h i s paper I have reviewed various non-equilibrium t h r e e terminal superconducting devices.

The Quiteron, while i n i t i a l l y promising, has major f a i l i n g s

t h a t make I t uncompetitive as a high speed t h r e e terminal device.

By examining

a l t e r n a t i v e s to the Quiteron s t r u c t u r e we come to the conclusion t h a t a q u a s i p a r t i c l e I n j e c t e d m i c r o s t r i p device somewhat s i m i l a r in concept t o the o r i g i n a l CLINK device of Wong e t . a l . has considerable promise, provided t h a t a low c a r r i e r density superconductor i s employed as the m i c r o s t r i p m a t e r i a l . l a s t proviso appears to be e s s e n t i a l .

This

188

This d e v i c e has an a c c e p t a b l e l e v e l

of power d i s s i p a t i o n ,

appears capable of

high s w i t c h i n g speed and i t o f f e r s s i g n i f i c a n t current and power g a i n . ital

circuit

margins. fast,

it

The disadvantages are t h a t i t

i s a l a t c h i n g d e v i c e and t h a t ,

i s much slower than a simple Josephson j u n c t i o n of

density.

In

dig-

terms the d e v i c e i s capable of a l a r g e f a n - o u t with good o p e r a t i n g

equivalent

On the plus s i d e the d e v i c e i s compact, y e t manufacturable,

r e q u i r i n g no g r e a t e r

level

current perhaps

of l i t h o g r a p h y c a p a b i l i t y than that which has a l -

ready been developed f o r present-day d i g i t a l Josephson c i r c u i t s . f u l production w i l l

while

Its

success-

r e q u i r e e s t a b l i s h i n g the f e a s i b i l i t y of using a low

d e n s i t y superconductor f o r the i n j e c t e d m i c r o s t r i p .

carrier

At l e a s t one m a t e r i a l

can-

d i d a t e e x i s t s and the performance e s t i m a t e s f o r t h i s d e v i c e are s u f f i c i e n t l y promising t o j u s t i f y a s i g n i f i c a n t l e v e l on t h i s p a r t i c u l a r

non-equilibrium

of

continued r e s e a r c h and development

device.

Acknowledgments

In w r i t i n g t h i s paper I have b e n e f i t e d from very h e l p f u l and extended

discus-

sions with W.J. G a l l a g h e r ,

as w e l l

S.I.

Raider,

as from the f i n a n c i a l support of

B.D. Hunt and R.P. R o b e r t a z z l

the O f f i c e of Naval

Research.

Referenees 1. W.J. G a l l a g h e r ,

IEEE Trans, on Magn., MAG-21, 709 ( 1 9 8 5 ) .

2. A.F. Volkov, Zh. Eksp. T e o r . F i z . 811 ( 1 9 7 1 ) ] .

60, 1500 ( 1 9 7 1 ) .

[ S o v . Phys.,

JETP 33,

3. T.W. Wong, J . T . C . Yeh, and D.N. Longenberg, Phys. Rev. L e t t . 37, 150 ( 1 9 7 6 ) ; T.W. Wong, J . T . C . Yeh and D.N. Longenberg, IEEE Trans, on Magn., MAG-1 3, 743 ( 1 977). 4. S.M. F a r i s , U.S. Patent 1,331,158 ( f i l e d

6/6/80).

5. S.M. F a r i s , S . I . R a i d e r , W.J. Gallagher and R.E. Drake, IEEE Trans, on Magn., MAGr-19, 1293 ( 1 9 8 3 ) ; W.J. Gallagher and S . I . Raider p r i v a t e communication. 6. Nonequlllbrlum S u p e r c o n d u c t i v i t y , Phonons and K a p i t z o Boundaries, e d . Κ. Gray, New York:Plenum P r e s s , 1981. 7. D.J. Frank, J. Appi. Phys. 56, 2553 ( 1 9 8 1 ) . 8. B.D. Hunt, R.P. R o b e r t a z z i 717 ( 1 9 8 5 ) .

and R.A. Buhrman, IEEE T r a n s , on Magn., MAG-21,

189 9. W.J. G a l l a g h e r , P r o c . 17th I n t . Conf. on Low Temp. P h y s . , E l s e v i e r S c i e n c e P u b l i s h e r s B.V. 1981, p . 1 9 1 . 10. C.C. Chi, M.M.T. Loy and D.C. Cronemyer, Phys. Rev. B23, 121 ( 1 9 8 1 ) . 11. B.D. Hunt and R.A. Buhrman, IEEE T r a n s , on Magn., MA0-19, 1155 ( 1 9 8 3 ) . 12. M. Tlnkham, M. Octavio and W.J. Skocpol, J . Appi. Phys. 48, 1311 ( 1 9 7 7 ) . 13. S . I . R a i d e r , IEEE T r a n s , on Magn. MAG-21, 110 ( 1 9 8 5 ) . 14. B.D. Hunt,

unpublished.

15. B. B a t l o g g , Physica 126B, 275 ( 1 9 8 1 ) . 16. L.F. M a t t h e l s s and D.R. Hamann, Phys. Rev. B28, 4227 ( 1 9 8 3 ) . 17. 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 , p p . 1 1 6 - 1 2 0 , McGraw H i l l , New York ( 1 9 7 5 ) .

QUITERON FOR LOGIC APPLICATIONS

I. Iguchi and H. Kashimura Institute of Materials Science, University of Tsukuba Sakura-mura, Ibaraki 3 05, Japan

In 1983, Faris (1,2) devised a superconducting three terminal device named "Quiteron" using nonequilibrium phenomena due to tunnel injection of quasiparticles.

This device composed of the

Nb-I-Nb-I-PblnAu double-tunnel-junction structure proved to have the promising features of switching time less than 300ps, power gain of 2 and current gain of 8.

Later Frank (3) studied the be-

havior of quiteron and discussed its applicability to logic circuits .

It is the purpose of this paper to demonstrate the digital application of quiterons, especially the operation of logic gate circuits (NAND, NOR) utilizing two quiterons.

Fig. 1 shows the stru-

cture of a quiteron and the proposed circuits for the NAND and NOR gates.

The operational principles are shown in Fig. 2.

The simu-

lated I-V characteristics were calculated using the tunnel formula with 20% gap reduction and ΙΟμν gap smearing.

For the NAND gate,

the detectors of two quiterons are connected in series and the output voltage becomes tv;ice that of a single quiteron.

ig ) INJ.

Ig

JL

A
/K), t h e

tunneling

and

tunneling

which admit an adequate

and

remains

master

an especcially

of i n t e r e s t a r e n o t v e r y

large:

equasimple

hf > Θ /τ·]ΐ)· As

fluctuations

the

vanishing,

G /G,p

_ ^SET^O

I-V

l ( V ) = G,pV),

noise.

when G g

out" and

remains

one

is g r o w i n g

the usual

t h a t of t h e d c c u r r e n t : ckets

growth.

classical

be

spectrum

so t h a t

in-

for to

f o r 1·^

distinguished.

conductance,

the S E T s i so t h a t a p i c t u r e o f the B l o c h

(2,3)

G

is c o m p l e t e l y

valid.

202 Discussion Conditions lations

( 1_0) , ( 1_1_) of e x p e r i m e n t a l

are

close

tailed analysis generally tion

of

values

those

According

the b o t h

effects

realization

of t h e S E T

of the B l o c h o s c i l l a t i o n s ,

of t h e p r o b l e m

valid.

able

to

carried

out earlier

to t h i s a n a l y s i s ,

is q u i t e p o s s i b l e

(2,3)

small

de-

remains

experimental

for

oscil-

so t h a t a

but

observaattain-

of C, Τ a n d G . s

If t h e s e

experiments

practical ble.

In, p a r t i c u l a r ,

an e x t e r n a l tions. tage ple

confirm

applications

steps

one

standard

I-V curve, the S E T

of the d c

current,

on t h e B l o c h o s c i l l a t i o n s

the former

standard

so t h a t

might

fabricate

i t c o u l d be o p e r a b l e junctions

dition

( 1_0) , w h i c h r e q u i r e s

Useful

discussions

lin,

V.V.

Schmidt

with C M 0 areas

Braginskiy,

Zorin

this

are

F to

I.A.

multifunto

similarity,

superconductiif

satisfy

of o r d e r

one the

con-

100 n m .

Devyatov,

gratefully

vol-

a

in d e s i g n

a t r o o m -1t e7m p e r a t u r e s

junction

with V.B. and A.B.

to d e v e l o p

the

by

in

by i n t e r v a l s

(2). D e s p i t e to

possi-

oscilla-

result

similar

bound

several

locked

and Bloch will

separated

is n o t n e c e s s a r i l y

tunnel

be p h a s e

locking

quite

(5),

can b e c o m e

oscillations

that based vity,

can

j u s t as J o s e p h s o n

this phase

can u s e

of t h i s w o r k

oscillations

oscillations

signal,

to Eq.(1^3)

in t h e j u n c t i o n

to e f . T h u s

damental

the S E T

microwave

According

predictions

of the n o v e l

V.V.

Migu-

achnowledged.

References 1.

Larkin, A.I., K.K. B + C 126, ¿ U .

2.

Likharev, K.K., A.B. Zorin. 1985. I E E E T r a n s . M a g n . 21_ (in p r e s s ) .

3.

A v e r i n , D . V . , A . B . Z o r i n , K . K . Li_kharev. Fiz. 88, 697 [Sov. P h y s . - J E T P ] .

Λ.

Cohen,M.H., 8, 316.

5.

A f t e r this w o r k h a d been c o m p l e t e d , we l e a r n e d a b o u t the p a p e r B e n - J a c o b , Ε.. , Y . G e f e n , 1 9 8 5 . P h y s . L e t t . 1 0 8 A, 2 8 9 , w h e r e t h e p o s s i b i l i t y o f t h e S E T o s c i l l a t i o n s is m e n t i o n e d a l t h o u g h no c h a r a c t e r i s t i c s of this effect are calculated.

L.M.

Likharev,

Yu.N.

Falicov, J.C.

Ovchinnikov. J. L o w T e m p .

Phillips.

1985. 1962.

198 4.

Physica

Phys. ¿9, Zh.

Eksp.

Phys.

347; Teor.

Rev.

Lett.

CHARGE OSCILLATIONS AND ZENER NOISE DUE TO INTERBAND TRANSITIONS IN SMALL JUNCTIONS

E. Ben-Jacob Department of P h y s i c s , The U n i v e r s i t y of Michigan Ann Arbor, Michigan 48109 Y. Gefen* Department of Physics and Astronomy, T e l - A v i v Tel-Aviv, Israel

University

K. Mullen Department of P h y s i c s , The U n i v e r s i t y of Michigan Ann Arbor, Michigan 48109 Z. Schuss Department o f Applied Mathematics, T e l - A v i v Tel-Aviv, Israel

1.

University

Introduction

One of the most e x c i t i n g t h i n g s we learned about in the Third

International

Conference "SQUID 85" i s that the recent advances in f a b r i c a t i o n techniques and the present state of cryogenics brought us to a new domain of p h y s i c s . 1 i s the "Mesophysics" domain, where we are dealing with ultrasmall are neither microscopic nor macroscopic, but somewhere in between. expected that new e f f e c t s and new phenomena w i l l

This

systems that It

is

be encountered which w i l l

widen our understanding of physics as a whole. As we have learned, four new main "mesophysics" e f f e c t s have already been predicted.1

We refer t o :

(a) macroscopic quantum e f f e c t s predicted in

small capacitance Josephson j u n c t i o n s and SQUIDS, (b) h/e o s c i l l a t i o n s

in

small normal r i n g s of a s i z e such that they are l a r g e r than the e l a s t i c s c a t t e r i n g length but s h o r t e r than the i n e l a s t i c s c a t t e r i n g l e n g t h , (c) charge

S Q U I D ' 8 5 - S u p e r c o n d u c t i n g Q u a n t u m Interference Devices a n d their Applications © 1985 Walter d e Gruyter & Co., Berlin · N e w York - Printed in Germany.

204 oscillations

in small

capacitance Josephson j u n c t i o n s and normal

tunnel

j u n c t i o n s such that the c h a r g i n g energy of t r a n s f e r r i n g one charge i s the dominant energy, and (d) shot noise e f f e c t s in the d i s s i p a t i v e s t a t e s of small capacitance elements.

simplest1

The f i r s t two p r e d i c t i o n s y i e l d the

e x p l a n a t i o n s to some o b s e r v a t i o n s and there are claims that some experimental r e s u l t s can be explained based on the t h i r d e f f e c t , 2 but the r e s u l t s are controversial.

still

There are no d i r e c t experiments yet to v e r i f y the f o u r t h

effect. In t h i s paper we d i s c u s s charge o s c i l l a t i o n s . that the energy spectrum of a small external

I t has been a r g u e d 2 " 5

capacitance Josephson j u n c t i o n d r i v e n by

f i e l d e x h i b i t s bands s t r u c t u r e s i m i l a r to the extended zone scheme of

an e l e c t r o n in a l a t t i c e .

The analog of the l a t t i c e momentum i s the external

d r i v i n g f i e l d , so when the f i e l d i s increased l i n e a r l y in time i t leads to oscillations

in energy, and consequently to o s c i l l a t i o n s

in voltage

v o l t a g e i s the d e r i v a t i v e of the energy with respect to the d r i v i n g We would l i k e to emphasize that though Ref. 2-4 and Ref. 5 give p r e d i c t i o n f o r the Josephson j u n c t i o n s , they take d i f f e r e n t approaches.

field). similar

physical

To make the d i f f e r e n c e between the two approaches more o b v i o u s ,

one should c o n s i d e r a normal tunnel Clark et a l . , 2

junction.

According to the approach of

Ivanchencko and Z i l b e r m a n 3 , and Likharev and A v e r i n 4 there w i l l

be no charge o s c i l l a t i o n s

in a normal tunnel

junction.

Ben-Jacob and Gefen 5 does p r e d i c t charge o s c i l l a t i o n s junctions.

(the

These charge o s c i l l a t i o n s

The approach of in normal

in a normal tunnel j u n c t i o n are a l s o

p r e d i c t e d in a recent p r e p r i n t by Averin and L i k h a r e v . 6 suggested by Schon. 7 Josephson j u n c t i o n s

tunnel

Additional

support

is

In t h i s paper we d i s c u s s charge o s c i l l a t i o n both in ( S e c t i o n 2) and in normal tunnel

junctions

t a k i n g an approach c l o s e to that of Ben-Jacob and Gefen.

( S e c t i o n 3)

In Section 4 we

d i s c u s s interband t r a n s i t i o n s due to Zener t u n n e l i n g w i t h i n the framework of a master equation approach.

We show that the Zener t u n n e l i n g leads to a new type

205 of noise that has long time correlations and that does not satisfy the fluctuation-dissipation

relations.

point out possible experimental

Finally in Section 5 we give a summary and

verification of our predictions and discuss

the effect of quasi-particle tunneling on charge oscillations in Josephson junctions.

2.

Charge Oscillations in Small

Josephson

Junction

The Hamiltonian of the junction, neglecting quasi-particles tunneling, is given by H = Hc + HT

(2.1)

where H c is the charging Hamiltonian and Ηγ is the pair tunneling

Hamiltonian.

In the absence of Ηγ the eigenstates of H c are determined by the number of pairs, n, charging the capacitor.

So, in terms of the number operator ή, the

Hamiltonian H c is given by

The corresponding charging energy is given by

Επ

= = E c n 2 «n,m

Here we consider the case of an externally classical

(2.3)

(a c number ) biased

field defined so that (2.4)

Hc = Ec(" * lex'2· We discuss the realization of q e x below.

The energy as function of q e x for

various values of η is shown in Fig. 2.1. Next we include Ηγ.

We use the phenomenological

definition of the tunneling

1 Hamiltonian given by,8

1 = | E j ( 2 6 n > m - m-l

/. ,, (4 15 ·

Next we assume q e x increases slowly in time, namely Ij/2e ( E j / E c ) ^ " 1 )

lex «

(4.2)

Under t h i s condition the p r o b a b i l i t y Pm,m+1 of Zener tunneling from the m

th

band to the m+1 band i s given by P

m,m + 1

s

e

*Pt-

> 2kßT. p r i n c i p l e i f we can get down to capacitances as low as 10*18 charge o s c i l l a t i o n s

We

normal

Thus, in

can observe

at room temperature provided the r e s i s t a n c e i s not too

l a r g e , so that fi/RC »

2kgT.

We conclude by p o i n t i n g out that the p r e d i c t i o n of the charge

oscillation

(which i s a s i n g l e charge e f f e c t ) makes i t p o s s i b l e to invent many new l o g i c a l elements. level

These elements w i l l

be very f a s t , w i l l

operate on a s i n g l e

and in p r i n c i p l e can operate at room temperatures.

charge

217 Acknowledgements One of us ( E . B - J ) would l i k e to thank Dr. and his many useful

H. Lubbig for h i s

hospitality

c o n v e r s a t i o n s d u r i n g the "SQUID 85" conference.

We would

l i k e to thank M. B ü t t i k e r , T.D. C l a r k , J. K u r k i j a r v i , G. Schön, T.P. and R.A. Webb f o r many useful d i s c u s s i o n s d u r i n g the "SQUID 85"

Spiller

conference.

References * permanent a d d r e s s : The Weitmann I n s t i t u t e of S c i e n c e , Rehovot, (1)

The proceedings of the "SQUID 85" conference edited by H. H. and H. LÜbbig ( i n p r e s s ) and references

(2)

Israel. Hahlbohm

therein.

A. Widom, G. M e g a l o u d i s , T.D. C l a r k , H. Prance, R.J. Prance, J . and Math. 15 (1982) 3877.

Phys.

R . J . Prance, J . E . Motton, H. Prance,

T.D. C l a r k , A. Widom and G. M e g a l a n d i s : Helv. Phys. Acta _56 ( 1 9 8 3 ) . T.P. Spi 11 e r , J . E .

Prance, H. Prance, R . J . Prance, T.D. Clark and

R. Nest, "Quantum Mechanical Weak Link C o n s t r u c t i o n R i n g , "

Flux Band Dynamics of a Superconducting in

preparation.

T.P. Spi l i e r , J . E . Mutton, H. Prance, R . J . Prance and T.D. C l a r k , preprint (3)

submitted to "SQUID 85" Conference, B e r l i n , June,

1985,ρ.3^1

Yu. M. Ivanchenko and L.A. Zilberman, Zh. Eksp. Teor. F i z . _55 (1968) 2395 [JETP 28 (1969)

1272].

(4)

K.K. Likharev and A.R. A v e r i n , p r e p r i n t No. 7 (1984).

(5)

E. Ben-Jacob, Y. Gefen, Phys. L e t t . 108A (1985)

(6)

A.R. A v e r i n and K.K. L i k h a r e v , p r e p r i n t submitted to "SQUID 85"

289.

Conference, B e r l i n , June, 1985, p* - i l f . (7)

G. Schön, p r e p r i n t submitted to "SQUID 85" Conference, B e r l i n , June, 1985, p . 1 5 A >

(8)

D . J . S c a l a p i n o in " T u n n e l i n g in S o l i d s " , E. B u r s t e i n and S. Lundquist e d s . , Plenum, N.Y.

(1968).

Κ. M u l l e n , Ε. Ben-Jacob,

(unpublished).

E. Ben-Jacob, E. M o t t o l a , G. Schön, Phys. Rev. L e t t . 51 (1983) "Mathematics of C l a s s i c a l

2064.

and Quantum P h y s i c s " by F.W. Byron, J r . and

R.W. F u l l e r , Addi son-Wesley P u b l i s h i n g Company

(1969).

The treatment presented here n e g l e c t s quantum mechanical among v a r i o u s t r a n s i t i o n a m p l i t u d e s , and rather uses

interference

transition

p r o b a b i l i t i e s , i n c o r p o r a t e d in a master e q u a t i o n , to d e s c r i b e the time e v o l u t i o n of the system.

We present t h i s treatment here only

to suggest that the t r a n s i t i o n s , and t h e r e f o r e the have an i n t r i g u i n g mechanical

spectral

character.

fluctuations,

A more complete quantum

treatment i s needed to e s t a b l i s h the c o r r e c t

of the system. Y. Gefen, E. Ben-Jacob, A.O. C a l d e i r a

(to be p u b l i s h e d ) .

behavior

THE LIFETIMES OF NON-EQUILIBRIUM D I S S I P A T I VE STEADY STATES

E. Ben-Jacob Department of P h y s i c s , The U n i v e r s i t y of Michigan Ann A r b o r , Michigan 48109 D . J . Bergman School of P h y s i c s and Astronomy, T e l - A v i v Ramat-Aviv, I s r a e l 69978

University

B . J . Matkowsky Department of E n g i n e e r i n g Sciences and Applied Mathematics The Technological I n s t i t u t e , Northwestern U n i v e r s i t y Evanston, I l l i n o i s 60 201 Z. Schuss School of Mathematical S c i e n c e s , T e l - A v i v Ramat-Aviv, I s r a e l 69978

1.

University

Introduction

Noise induced t r a n s i t i o n s of the h y s t e r e t i c Josephson j u n c t i o n from the zero v o l t a g e s t a t e i n t o the non-zero v o l t a g e s t a t e has been t h o r o u g h l y in the l i t e r a t u r e

( 1 ) , in a wide temperature range, i n c l u d i n g very low

temperature range, ( 2 - 4 ) . classical interest

This problem was s t u d i e d in the context of the

a c t i v a t i o n process of a p a r t i c l e

in a p o t e n t i a l

i s the problem of noise induced t r a n s i t i o n s

well

(5).

in the opposite

from the non-zero v o l t a g e s t a t e i n t o the zero v o l t a g e s t a t e ( 6 , 7 ) . of t h i s p r e s e n t a t i o n

i s to summarize our recent a n a l y t i c a l

l a t t e r problem of t r a n s i t i o n s . transitions

investigated

from the

Of equal direction The purpose

r e s u l t s on the

In c o n t r a s t to the e x p e r i m e n t a l l y well

zero v o l t a g e s t a t e , the t r a n s i t i o n s

studied

in the opposite

d i r e c t i o n were not well s t u d i e d e x p e r i m e n t a l l y and have not been compared to theoretical

predictions.

I t i s our aim to encourage e x p e r i m e n t a l i s t s to c a r r y

SQUID '85 - Superconducting Quantum Interference Devices and their Applications © 1985 Walter de Gruyter & Co., Berlin · New York - Printed in Germany.

220 out measurements of this transition process by providing a concise summary of our theoretical

results.

In Section 2 we describe the transitions from the non-zero voltage stage of a single Josephson junction due to the thermal

noise.

We give the

transition rate as a function of the junctions' parameters.

In Section 3 we

calculate the lifetime of the zero-voltage step and the first harmonic step of a junction with induced microwave radiation

(8).

Finally, in Section 4, we

calculate the lifetime of the non-zero voltage state at low temperatures where shot noise effects dominate those of thermal

noise.

In this range of

temperatures we use the master equation to describe the junction's noisy dynamics

2.

(9).

The Lifetime of the Voltage State of a Hysteretic Josephson Junction

In this section we review the effect of thermal of a current driven Josephson junction. junction assuming the RSJ (10,11) model. described in dimensional

noise on the voltage state

First we review the dynamics of the The dynamics of the junction is

units, by the phenomenological

equation

.dV

(2.1)

where C, R and Ij are the capacitance, resistence and critical current of the junction, respectively.

Is = 0 .

(2.15)

The Boltzmann d i s t r i b u t i o n P

B - e-E/T ,

(2a

*

When t h i s approximation

i s i n a p p l i c a b l e , ω 5 θ ρ may be found by numerical solution of the n o i s e l e s s equation of motion. We note that there i s usually two s o l u t i o n s to (3.14), the plus sign leading to

ω sep

and the minus sign leading to ω sep =

ω

.

s

π [I+Ji

ω

·

(3,16)

When the fluctuating frequency ω+Δω reaches either one of these values, a t r a n s i t i o n takes place to a new steady s t a t e , the general character of which can. be discerned by considering a different type of experiment, where the external d r i v i n g frequency i t s e l f i s varied.

In that case, when ω i s decreased

down to ω + , the system jumps to a higher step ( i . e . , some subharmonic n/m>l), while when ω i s increased up to ω_, the system jumps to a lower subharmonic s t e p - - u s u a l l y to the zero-voltage step.

We thus conclude, somewhat

p a r a d o x i c a l l y , that an exit by a Δω>0 f l u c t u a t i o n will take the system to a lower s t e p - - i n our case (where we assume that there i s no overlapping low-lying subharmonic steps) to the zero-voltage s t e p — while an exit by a Δω5)

= coth(eV/2k B T).

Next, we identify /N(E)dEe(r - t ) as the mean net current I across the junction, so that

I = /e(r - I)N(E)dE) =

(4.6)

/N(E)dEe(r - I )/coth(eV/2kBT) = V/R where N(E) is the density of states per unit energy and R is the normal Ohmic resistance of the junction.

Thus

J ( r + Ζ)N(E)dE = (V/eR)coth(eV/2kBT). We consider the case that r and i

(4.7)

can be replaced by their mean values (with

respect to N(E)) and consequently we obtain* 4

= V/eR[l - exp(-eV/k B T)]

r

= V/eR[exp(-eV/kBT) - 1]

(4.8)

*If e/C i s not negligible relative to V one may attempt to refine (4.4) by including the change in Ep due to the transfer of a single charge. Such refinements were at one time suspected to be important in the discussion of the zero voltage state. However we have found that the resulting corrections are of no practical importance.

237 Thus the dymanics Eqs. (4.1) and (4.2) in the presence of quasi-particle tunneling are replaced by Eq. (4.2) by the stochastic equation

(4.9)

V(t+dt) = V(t) - (Ij/C)sinfl(t)dt + (Ioc/C)dt + o(dt)

(4.9)

with probability [l-(r - t ) d t ] + o(dt) V(t+dt) = V(t) - (Ij/C)sin6dt + (Ioc/C)dt + 0(dt) ± e/C with probabilities respectively.

rdt or i d t + 0(dt)

(4.10)

We set

p(0,V,t) = Pr[9(t) = θ , V(t) = V| Θ (0) = θ 0 , V(0) = V 0 ] The transition probability density p(9,V,t) satisfies the forward equation, or the master

Kolmogorov

equation = -(ZeV/h)

r(V-e/C,0)p(V-e/C,6,t)

- [I d c/C -

Ij sine]

+

+ * ( V+e/C,e ,t)p(V+e/C,e ,t) -

[r(V,8) + £(V,8)]p(V,9,t) with the initial

(4.11)

(4.12)

conditions

p(e,V,t) + p ( 9 = 9 0 , V = V 0 )

as t • 0.

(4.13)

To be consistent with the notations of Ref. 5 we introduce the dimensionless variables T-

2 e k B T/tfIj, t +

twj, E -

E/Ej

(4.14)

.

(4.15)

and p(e,V,t) -

P(0,9,t) .

eV=-fiUJ6/2,

V/RIj = G0

We obtain

In these dimensionless units the elementary change in charge is given by q = ewj/Ij = f1uij/2Ej

(4.16)

Thus q is the ratio of the zero point energy of the junction to the barrier height.

It is also the inverse of the number of quasi-particles

across the junction in one period 2n/uj.

transferred

Note that q is also the parameter

which determines the range where MQT becomes important, that is, for q that is not too small MQT can be appreciable

(2).

238 Our analysis concerns the case where q is small, so we can solve the master equation asymptotically.

In the chosen units we have £ = exp(-qe/T),

(r + I

)/( r - I ) = coth(qê/2T), q(r - t ) = G9

(4.17)

r = G6/q[l - exp(-q9/T)] I Now the master equation

= G8/q[exp(q6/T) - 1] .

(4.12) takes the form | P . i J P _ ( I - s 1 n e ) | j

+

(4.18) r(6, e - q ) p ( 6 , θ-q.t) +

[Γ(Θ,Θ)

-

In summary, the master equation (1)

l (θ , é+q)p (Θ ,θ+q.t) -

4 (θ ,θ )]P (θ ,θ ,t )

(4.18) holds under the following

The dynamics of θ obeys the continuous classical

assumptions

Josephson

relations

(4.1) and (4.3) between jumps.

(2)

The quasi-particle tunneling is elastic and is controlled by the

instantaneous

Fermi occupation probability of single particle states on the

two sides of the junction. (3)

The tunneling matrix elements and the density of states are

essentially energy (4)

independent.

The individual

quasi-particle tunneling is uncorrelated.

First we consider the averaged dynamics of the "stochastic" obtained by averaging Eqs. (4.9) and (4.10) over all jumps.

This also

corresponds to retaining only the first term in the Kramer-Moyal expansion of the master equation jP 3 t

by Eq. (4.17). (2.3).

=

U>

(4.18).

series

We obtain a damped Liouville equation

+-q»[(r:l)p] 3θ

junction

= u +

G L Í Í P l

{4

.19)

3Θ

Eq. (4.19) is Liouville's equation for the damped dynamics

Hence, on the average, the motion is bistable as described in Ref. 7.

239 In particular there is a non-equilibrium steady state solution whose phase space trajectory is given by (2.11). We employ again the WKB method, as in Section 2, to construct an asymptotic solution of the master equation. θ = I/G + (G/I)cos9 + We write the eikonal equation in the form IJW 30

+

(l-sin-Gé) 3Θ

Q¡ { [ ^ 3Θ

exp(- ™)-exp(-q6/T) 3Θ

+

[ e x p ( ^ - qê/T) • 3Θ

l/[l-exp(-q^-/T)] } = 0 3Θ

(4.20)

- cosh(qè/2T)]/sinh(qê/2T)

(4.21)

or alternatively, in the form LW + G9[cosh(q6/2T - — ) 3Θ

Eqs. (4.20) (4.21) are the generalization of the eikonal equation of Section

2. The constant W contours are the steady state trajectories of the equation θ + g(e,K) + sine = I

(4.22)

where g(é ,K) is a non-linear dissipative force given by g(e.K) = -(GT/2qK)[cosh(q6(1-2K)/T)-cosh(qê/T)]/sinh(qê/T)

(4.23)

Where Κ is a parameter. In the limit K->0 (i.e., on S 0 ) we have g(θ,K)->G6 These contours are given by θ = + (l/)cose + 0(G 2 )

(4.24)

= -(T/2qK)log(l-2qIK/GT)

(4.25)

with

( denotes averaging over constant W contours).

240 In Fig. 4.1 we show the constant W contours for high (T>>q) and low (T0 (I c = critical current of junction,

= flux quantum) which may typically be of the order of

the thermal energy of a single atom at room temperature!

(2) As we shall see

below, an essential condition for the observation of the effects of quantum superposition of states of a macroscopic variable is that the dissipation associated with the classical motion of such a variable should be sufficiently small. Again Josephson devices score:

for an ideal tunnel-oxide junction the dissipa-

tion should be proportional to the number of normal quasiparticles present, and hence totally negligible at temperatures far below the bulk transition temperature, and while in any real junction there are almost certainly other parasitic sources of dissipation, they can often by made very small indeed, corresponding to effective shunting resistances of the order of a Μ Λ .

(3) Josephson devices also have

the advantage that one can vary at least some of the parameters (eg external current or flux) in a controlled way and, even more importantly, determine all

245 the experimental parameters (in principle!) from experiments conducted entirely in the classical regime and subject to classical interpretation.

Thus, in

principle at least, one should be able to make predictions of the effect of quantum superposition in which there are no adjustable parameters.

This feature

should be particularly important in the event that the experimental behaviour fails to agree with the theoretical expectations (cf below).

It is convenient to divide the relevant existing and planned experiments on superconducting devices into two main classes, depending on whether the primary aim is to verify (or not) that the application of quantum mechanics to a macroscopic variable gives results consistent with experiment, or to exclude alternative objective descriptions of reality at the macroscopic level. discussed further below.)

(The distinction is

In the first category fall the various experiments [4]

which have been done on 'macroscopic quantum tunnelling'(MQT)

(escape from a

metastable potential well of a macroscopic variable such as the phase difference across a Josephson junction), the beautiful experiments just reported by the Berkeley group [5] on energy level quantisation, and the various experiments done by the Sussex and Kharkov groups [6] in which a SQUID ring is coupled to a tuned tank circuit.

While the last-mentioned group of experiments is indirect

and their interpretation typically involves a number of subsidiary assumptions, the experiments on MQT and on energy level quantisation seem to involve a minimum of such interpretative assumptions and to constitute strong evidence that at this level the motion of a macroscopic variable is indeed well described by quantum mechanics.

The question of how far they are direct evidence for the validity of

the concept of superposition of macroscopically distinct states is a delicate one (cf ref [3] > sections 4 and 5); in view of the development in the next paragraph I will not attempt to discuss it explicitly here.

Ideally, in any case, one would like to verify not merely that quantum mechanics gives a correct account of the behaviour of a macroscopic object, but that alternative theories must give an incorrect one, in particular, that under appropriate circumstances the attribution of objective macroscopic properties to such an object must be incompatible with experiment.

It may help here to draw an

analogy with the history of work on the EPR paradox.

In this case the original

paper of Einstein et al [7] demonstrated that in certain circumstances it is impossible.within the quantum formalism, to attribute objective properties to microscopic objects even when they are physically isolated (provided, that is, that one wishes to maintain local causality).

Since the relevant experiments,

such as they were, were consistent with the predictions of the quantum formalism,

246 many people were thereby convinced that local objectivity was indeed not a property of the world at the microscopic level (call this stage 1).

However, any-

one who found this rather radical notion repugnant could take refuge in the fact that the experiments themselves in no way refuted the concept of local objectivity; it was only their interpretation in quantum-mechanical terms which did so.

It

was therefore rightly regarded as a major development in the study of the paradox when nearly thirty years later Bell [8] showed that under certain circumstances an experimental result which agreed with the quantum prediction must automatically be incompatible with local objectivity irrespective of the formalism within which it is interpreted, and when subsequently experiments were actually carried out in this regime and gave the quantum-mechanically predicted results (stage 2). In the parallel case of the quantum measurement paradox I believe that we can now say with some confidence that the experiments mentioned above have taken convincingly to stage 1:

us

can we now proceed to stage 2?

To discuss this, let us focus on the experiment which seems to be the most suitable candidate for a stage 2 role, namely the 'macroscopic quantum coherence (MQC) experiment discussed in this session by Claudia Tesche [9].

In such an experi-

ment one would take a SQUID ring (ie a bulk superconducting ring interrupted by a Josephson junction) and bias it with a suitable external flux in such a way that the potential energy, expressed as a function of flux or equivalently of circulating current I, has two degenerate minima corresponding to current +I Q . the magnitude I

Typically

of the circulating current would be a fraction of the critical

current, hence perhaps of the order of a few ^.A.

Thus, the states of clock-

wise- and counterclockwise-circulating current may reasonably be called 'macroscopically 1 distinct.

However, quantitative calculations show (cf ref [10])

that the rate of tunnelling Δ

through the barrier between those two configurations

need not be negligibly small.

Straightforward application of quantum mechanics to this situation 4 neglecting dissipation, yields the results that (a) Any observation of the value of the circulating current will yield either the result +I Q or the result -I

(with a

small spread about each value due to zero-point fluctuations) (b) if the current is known to be +1

at any time tj, then the probability P(t 2 : t ^

of finding it

to be +I Q at some subsequent time t 2 (the system being in the meantime undisturbed) will be given by the expression

P(t 2 : t ^

= Í O + cosAítg-t,))

(1)

247 In the following discussion I shall assume that the experiment confirms prediction (a) - were this not to be the case, it would imply that there is something seriously wrong with our model - and concentrate on the verification, or other wise, of prediction (b).

Suppose that it is possible to attain the parameter regime necessary for this MQC experiment, and that it is carried out. physics community as a whole? rately.

Namely:

What is the likely reaction of the

I am afraid that I can predict it only too accu-

If the experiment comes out in favour of the theoretical pre-

diction, eqn (1), we shall be told that the result is trivial; whereas, if it. doesn'tjWe shall be told that we were naive to have expected it in the first place!

(Perhaps, in fairness, I should add that the groups who do the telling

in the two cases will not have a totally overlapping membership.)

Why should

this be so?

The attitude of those who find the result trivial will be that 'everyone knows' that quantum mechanics describes the physical world, and hence it is no surprise to find that it works in this situation too.

At a slightly more sophisticated

level, it will be argued that doubts about the applicability of quantum mechanics to the notion of a macroscopic variable have already been laid to rest by existing experiments on MQT and on energy quantization and that the MQC experiment adds nothing that is qualitatively new.

As a matter of fact, I believe

that it almost certainly does add something even at 'stage Γ

of the argument

(cf ref [3],section 5) but in any case that is not the main issue:

the whole

point of the MQC experiment is that, if it comes out in agreement with the quantum prediction, it thereby provides not only evidence for quantum mechanics but also evidence against objectivity (realism) at the macroscopic level.

To

be more precise, let us define a 'macro-realistic' theory as one which incorporates the following two postulates: (1)

A macroscopic object which has available to it two macroscopically distinct states must at all times be in one or other of these states, irrespective of whether or not it is observed.

(2)

It is in principle possible to ascertain the state without any radical effect on the subsequent behaviour of the system.

With this definition it can be shown that no macrorealistic theory can reproduce, for all values of t^ and tg, the correlations given by the quantum-mechanical expression (1).

The details of the argument, and some necessary cautions, are

given in ref [11].

Thus, if the MQC experiment can be carried out, for appro-

priate values of the times in question, and the result agrees with (1), we shall

248 have brought the quantum

measurement paradox to a stage of development

comparable

to that of the EPR paradox.

Suppose on the other hand that the result of the experiment comes out against the prediction (1), in fact Derhaps is qualitatively

different.

W h y , in this

event, will a large subsection of the physics community tell us that we were naive to expect the result (1) or anything

like it in the first place?

The

reason is a very old argument in the quantum theory of measurement, which goes crudely as follows (cf ref [12]): quantum-mechanical

To obtain the characteristic

(oscillatory)

behaviour (1) we must require that the wave function at an

intermediate time be of the linear superposition form

4

= a(t)$+ + b(t)i

(2)

where +, - refer to the two macroscopically different states, with I = +1 respectively.

However, it is argued, any macroscopic system will

interact

strongly and irreversibly with its environment, and as a result the wave fucntion of the

'universe' (which originally is just % Q [ a ( t ) i + + b(t)í_], where χ

ο

is

some environment wave function) will very rapidly be converted into the form

*un = a(t)*A + b(t)X-$where the environment wave functions gonal.

and %

(3)

are very nearly mutually

ortho-

Once this has happened, it is argued, the evolution will be quite diff-

erent from that given by eqn ( 1 )^and in fact in most cases P ( t 2 : t ^ the form i(1 + exp macro-realistic

- t^))

will be of

(a form which is always compatible with

theories).

To meet this objection one needs to construct models of the relevant devices which incorporate as fully as possible the complex and irreversible interactions which they undergo with their environment (the latter including the normal

component,

the radiation field, the phonons, nuclear spins and much else) and to demonstrate explicitly that even for those models the behaviour (1), or something close to it to violatemacro-realism,

is still obtained.

of this approach have been pursued:

sufficiently

To date two main variants

one, exemplified by the paper of Eckern et

al [13], is to take a very specific model, that of an ideal tunnel-oxide

junction

described by the standard tunnelling Hamiltonian, and carry out calculations as rigorously as possible on this basis. quantum-mechanical

If one is quite confident

that the

results will be obtained in the experiment, this is an

249 entirely satisfactory procedure.

However, in the event that the experiment shows

a quite different type of behaviour from (1), say an exponential anyone

tries

relaxation,and

to claim this as evidence for the breakdown of the quantum formalism

at the macrolevel, it will immediately be pointed out that there are all sorts of noise and dissipation mechanisms in such devices which are not accounted for in the tunnelling Hamiltonian, and it will be suggested that those are responsible for the qualitative discrepancy from (1).

It is precisely in order to combat

this kind of anticipated objection that the author and his coworkers have concentrated on a rather different approach, which tries to relate the

'dephasing 1

effect

of the environment as rigorously and model-independently as possible to properties of the system which can be observed in purely classical experiments, in particular to the classical dissipation:

see for example refs [14] and [10],

While the

arguments developed to date in this connection are certainly not free of

loop-

holes, one's confidence in them is increased by the fact that they do seem to be able to relate the MQT data to classically observaole parameters in a not entirely trivial way.

Thus, to treat a failure of an MQC experiment (conducted in a suitable

parameter regime, of course) to satisfy the prediction (1) or something like it as prima facie evidence for a possible breakdown of quantum mechanics at this level is less question-begging than it might seem at first sight.

If the MQC experiment is attempted, it may of course be found that one cannot in practice attain the parameter regime which is necessary for (1), or something close to it, to be the correct quantum mechanical prediction.

This would of

course be disappointing, but there should be (indeed already has been) a lot of spin-off on both the experimental and theoretical sides.

Assuming that the para-

meter regime is accessible, either outcome to the experiment would be highly significant:

If the quantum-rrechanical

results are obtained, then (with the

reservations indicated in ref [11]) the idea of realism or objectivity even at the macro-level

is dead:

while if the quantum-mechanical

results are not

obtained, then (after a good deal of closing of loopholes, obviously) we have a real possibility of concluding that the quantum formalism fails at the macroscopic level.

With a little optimism, we may hope that by the next IC SQUID we shall

know which of these radically different conclusions about the macro-world correct!

This work was supported by the National Science Foundation through grant no DMR 83-15550.

is

250 References

[1]

E.Schrodinger, Die Naturwissenschaften 23 ,807

(1935)

[2]

A.J.Leggett, Cont Physics 25

[3]

A. J. Leggett, Prog.Theor. Phys.supplement no. 59 , 80

[4]

R.de Bruyn Ouboter, Physica 126B , 423 (1983) and references cited therein; S.Washburn et al, Phys Rev Letters 54 , 2712 (1985); D.B.Schwartz et al, these proceedings, p.

[5]

J.Clarke et al, these proceedings, p. 31f.

[6]

R,J.Prance et al, Phys.Letters 107A , 133 (1985) and earlier references cited therein; I.M.Dmitrenko, G.M.Tsoi and V. Ν .Shnyrkov, Fiz Nizkh Temp 10, 211 (1984); (Translation: Sov Journ of Low Temp Phys _10 , 111 ( 1 9 8 4 a n d earlier work cited therein

[7]

A.Einstein et al, Phys.Rev.47 ,777

[8]

J.S.Bell, Physics (USA) _1_ ,195

[9]

C.D.Tesche, these proceedings

[10]

A.J. Leggett in Percolation, localisation and superconductivity, ed.A.M. Goldman and S.A.Wolf, NATO Advanced Study Institute vol 109 (Plenum, New York 1984)

[11]

A.J.Leggett and Anupam Garg, Phys.Rev. Letters 54 , 857

[12]

R.H.Kraichnan, Phys Rev Letters 5 4 , 2 7 2 3 A.J. Leggett and Anupam Garg, ibi kT) the quantum mechanical nature of those degrees of freedom which are responsible for the noise is important and the noise - in contrast to what the classical result (3) or (4) suggests remains finite even at T=0. The noise may be due either to a shunt resistor or due to quasiparticle tunneling across the junction. In the following section, I will describe appropriate modifications of the Langevin equation and give the limits where such a description is sufficient. On the other hand, for sufficiently small capacitance (equivalent to the mass of a particle) quantum mechanical effects associated with the phase degree of freedom φ may show up and, for example, quantum tunneling in the phase space may occur. Since ψ describes the state of a macroscopic object -the Josephson junction- these quantum effects carry the label "macroscopic". As long as damping effects can be ignored, all these quantum effects follow from a simple Hamiltonian. The theoretical problem is to justify that φ indeed can be considered a quantum mechanical variable and to account for the damping. In Section III a fully quantum mechanical description including the effect of dissipation due to quasiparticle tunneling will be derived and com-

253 pared to models describing the dissipation of a shunt resistor. In Section IV, the effect of damping on quantum effects will be analyzed in several examples.

II.

Some Simple Concepts

1) Gaussian quantum noise Even at very low temperatures, because of the quantum mechanical nature of the resistor degrees of freedom, the noise remains finite If the shunt resistor is an ideal linear element, with a response characterized by a frequency independent resistance R ^ (in other words, if the damping is proportional to the velocity) then the fluctuation-dissipation theorem in its general quantum mechanical form requires a power spectrum (1 ) Vco, = £ c o t h | £

.

(5,

At large frequencies tî|u)|>kT this spectrum increases proportional to I co I - up to a cutoff , which any realistic resistor must have - and thus differs from the classical form (4). This difference is difficult to observe directly, since even moderately low temperatures correspond to high frequencies, e.g. T=1 Κ is equivalent to ü)=kT/fií»1C>^ sec However, if the resistor is embedded in a nonlinear circuit, e.g. in parallel to a Josephson junction, the resulting frequency mixing makes the nonclassical part of Sjioo) observable (see Koch et.al. (2)). Under certain conditions (see Sect. IV) it is consistent to take into account the quantum effects associated with the noise source while still treating the phase degree of freedom as classical. (Quantum effects associated with the phase will be discussed belcw. ) In those cases the dynamics of the RSJ is described by a quantum Langevin equation (2-5) which has the classical form (1), however, the power spectrum of the noise has the quantum form (5). From the general theory we learn that the quantum Langevin equation is sufficient if the potential is linear or harmonic, or if the quantum effects related to the anharmonicities can be ignored in comparison to the effects of the noise. This last condition is satisfied if the damping is strong (3), i.e. if

254

-fi — » sh

e R

1

or

e R

li — sh

»

2p 2 = f sRh , C c I fi

3

c

.

(6)

It should be noted that expectation values derived from the quantum Langevin equation correspond to a specific ordering of the equivalent quantum mechanical operators tions in

,

(7)

—CO

where the kernel current

I

(V)

(Xj(t)

dependent step at V/I^

for

eV=2A

eV>> 2Δ-

α Ι (ω) = - ¿

is related to the well-known quasiparticle

for constant voltage

V

-which has a temperature

(in a symmetric junction) and is

I

qp^ V ^ =

by I

(V=ìio/e)

.

(8)

255 The associated noise is shot noise. In the classical limit, the power spectrum of a constant voltage biased junction is 3 ι ( ω ) = e I g p (V) coth For voltages

.

(9)

eV>kT, the fluctuations are proportional to the mean

current itself, which is characteristic for shot noise. (Also it is interesting to see that the quantization of the charge to a similar structure as the quantum mechanical

ft

e

leads

does.)

In general, the current noise can be expressed as (7) I(t) = ζ 1 (t). cos Here, ξ^

and

ξ2

+ ξ 2 (t). sin î^t).

_

(10)

are two independent Gaussian random variables

with correlation functions = δ.. 2e 2 Ka R (t-t1 ) 1

(11)

J

and

aR

in its general quantum form, consistent with the fluctua-

tion-dissipation theorem, is ccr(cj)

= i oij (ω) coth ^

.

(12)

If the junction is biased with a constant voltage

V

the power

spectrum resulting from (10)-(12) takes the well-known form of Dahm et.al. (8) S x (ω) = l Σ e Iqp(V±fico/e) coth

,

which reduces to (9) in the classical limit

(13)

-fioj>1 the dynamics of a current biased normal junction follows from the Langevin equation e

κ

θ

= I(t)

.

(19)

[If we included an extra factor 2 in the definition of Θ in Eq. (17), the analogy between normal and superconducting junction would be even more striking. However, this appears unnatural. Rather, we note that in a superconductor both the phase φ of the order parameter (the Cooper pairs) as well as the voltage V and its in-

257 tegral

Θ

are relevant degrees of freedom. The quasiparticle tun-

neling and noise depend on the latter phase relation links the two phases as

Θ . However, Josephson 1 s

φ=2Θ, with the factor

2

being

related to the Cooper pair.] Finally, we comment on a heuristic model for shot noise, put forward by Ben-Jacob, Bergman, Matkowsky and Schuss (12). Motivated by the constant voltage result (9) they suggested a Langevin description where the noise in general is given by = Sïlti. coth ^ ¿ i i ó(t-t')

.

(20)

However, this noise spectrum in general violates the fluctuationdissipation theorem and, as is shown in Ref. 11, in no limit produces correct results, except in the trivial limit

eV(t)0) the smearing is negligibly small and hence the Langevin equation is completely sufficient; however, also the noise is classical. More generally, we can compare the smearing with the uncertainty due to the stochastic fields ξ^ and · F o r strong damping and thus strong noise, the latter is more important. Quantitatively the restrictions are given in Eq. (15). In the case of ohmic damping and Gaussian noise higher order terms in

χ

arise solely due to anharmonicities of the potential

υ(φ),

and weaker restrictions given in Eq. (6) are sufficient (3). Beyond that, if the potential is linear or harmonic, the quantum Langevin equation is exact. Notice, the quantum Langevin equations, although their form is that of a classical equation of motion, account for all quantum effects in a harmonic system. Only those quantum effects relying on anharmonicities, for example tunneling through potential barriers, are not properly accounted for. For weak damping, we can proceed using different approximations to be discussed in the following paragraph. The comparison of numerical solutions of the Langevin equation with shot noise and the weak damping expansions shows that the shot noise quantum Langevin equation indeed fails for weak damping if the capacitance and temperature are small

CkT/e2

for

[see Eq. (43)] - which in an expansion

can be truncated to low order terms in

I

and

α(τ) .

For example, the probability distribution for the values of the charge at one fixed time - e.g.

τ=0

- is

P(Q) = JdQ δ (Q (0) -Q) ρ {Q ( τ ) }

.

(52)

Alternatively, we may remain in the phase representation. E.g. p(Q) = JdAcp exp (i ^

Δφ) Ζ[Δφ]

(53)

where φ 0 +Δφ Γ 0Φ e" S[tp]/tt «>ο

Ζ [ΔΦ] = JAP

.

(54)

The equivalence of (52) and (53) was shown in Ref. 11. Similarly, expectation values of

Q

also follow from derivatives of

Ζ[Δφ] ,

e.g. < Q 2 >

= - IF§R

DIS)^

Ζ[Δ

^ΙΔΦ=Ο

(55)

·

In order to demonstrate and investigate the difference between shot noise and Gaussian quantum noise of an equivalent resistor, we considered in Refs. 10 and 11 a normal tunnel junction, i.e. ignored any potential (Ic=0). We found explicit analytic results both in lowest nontrivial order in the expansion in classical approximation for

*ti/RCkT«1

and in semi-

CkT/e2>>1. Both limits are contained

in the following result for the second moment (11) •fiß

= g

+ e2

J dx α ( τ ) β ( τ ) [

... E

"

N M

-1]

(56)

o where

G

is the 2nd moment with Gaussian noise

-2α(ω ) „ = CkT Σ — v ft - 2 ω 2 -2α(ω )

(57)

and β(τ) = 2 kT

ft e

„ Γ γ · . 12 Σ ω 2 / ft C- 2 ω 2 -2α(ω ν ) v*0

h(τ) = ^ Σ λ ( 1-cos ν co τ)/ñ !2ων"2α(ων) •η ν*0

(1-cos ω^τ) (58)

277 A plot of

(in the limit Δ=0) is given in Fig. 4 (as the solid

line) and compared to both the classical result

c =CkT

(dotted

2

line) and the Gaussian result

„ (dashed line). Whereas the Cj quantum mechanical results can exceed the classical value considerably depending on the strength of

1/Rj^ , the difference between

quantum shot noise and quantum Gaussian noise is small, except for very small values of

CkT/e2 .

C kT/e 2 Fig. 4. Charge fluctuations in a junction (see text). The parameters are (a) -ft/e2R=0.25, (b) fi/e*R=0.05 . The cutoff is chosen as Ω RC=200 . c If the noise is shot noise the distribution function for charge fluctuations is not of a Gaussian form. But numerically, the difference to the distribution for Gaussian fluctuations is small again. (See Ref. 11 for plots and analytic results.) Finally, the power spectrum of charge fluctuations in a tunnel junction, to lowest order in the tunneling rate, i.e. for

-fi/RCkT«1 ,

is (10,11) 1 1 S_(ω) = ω}z ù} υ 0

00

J dQ -«χ. °

2 1 exp(—Q /2CkT) 4 i Σ l e iq p ° ε=±1 σ=±1

[ (6E® +σ*ω)/β] υ ο

χ n(6E¿ +σίίω) o

(59)

278 ±1 where η(E) is the Bose function, Z Q =/dQ o exp(-Q^/2CkT) and 6Eg = 2 (Q0±e) /2C-Q^/2C is the energy difference before and after a quasiparticle tunneling process, which increases or decreases the initial charge Q q . The interpretation of (59) is the following: The system can be in different states characterized by the charge Q q , each weighted by the appropriate Boltzmann factor. (This simple form is a consequence of our restriction to lowest order in 1/R.) The current noise is a sum from different events increasing or decreasing the charge (e=±1) and depends on the corresponding energy difference

±1

6EQ

. Only for a constant voltage bias we have

6E

±1

=±eV

and we recov§r what is equivalent to the result (13) of Dahm et al. ±1

(8). However, for small significantly from

C , the energy difference

6EQ

differs

eV(Q Q ) . Quantum mechanics introduces as usual

a frequency shift by

±ήω .

In Ref. 11 I also investigated the charge fluctuations due to shot noise of a system with a harmonic potential. This approximately describes the effect of Josephson coupling as long as the phase fluctuations remain small. Again the difference between shot noise and Gaussian quantum noise remains small. 5) Discrete charge states Instead of the systems which we considered so far, where the junction was assumed to be connected by leads to an external circuit, we may also consider an isolated tunnel junction. Realistic examples are a granular material or a junction in line between two other junctions (34). In this system, the allowed states are characterized by integer charges. Hence the integrations over the charge JdQ o encountered above are to be replaced by sums, e.g. Ζ =

Σ

w ext m=-co ° °

(63)

where Q 0 = Q e x t + m e ®ext "*"s a parameter. In the phase representation, the external charge is accounted for by a phase factor

280

=

Zq

ext

Σ

η

;άφ °

expfi \

fe e

2lm\

'

^ - M Ä ™ Φο

βφ

.-SM/«

while in the charge representation (46) the integration

(64)

/dQ 0

has

to be replaced by a sum

Σ and Q =Q ,+me. For strong coupling to m o ext the outside world we should integrate over all Qex^. a n d recover what we used in the main part of this paper. The concept of an external charge was introduced by Prance et al. (35) and was implicit in an earlier paper by Widom et al. (36). They int ucet suggested that Q e x t = c ®ext c a n ^ ^ ^Y the change of the flux in a (normal or superconducting) ring. Provided that the condition of weak coupling through the leads discussed above applies, an external current can change Q e x t directly as 2 e xt =I ext" (Notice, the change has to be adiabatically slow.) Ben-Jacob and Gefen (37) suggested this as a model for a current source. It should be noted, however, that because of the restrictions mentioned this is not a unique model. E.g. a large inductance SQUID with large trapped flux a so a m ® e x t , such that ^ext^ext^11' ^ °del f° r a current biased junction [see the discussion after Eq. (31)] and is described by a tilted potential U(ip). Also in the work of Likharev et al. (38) it is implied that an external current changes a parameter related to Q e x t · [They arrive at a representation (64) by a different physical picture. They argue that in an extended potential U(cp)~cos φ, with states differing by 2n being distinguishable, the states of the system are extended, coherent Bloch states labelled by a pseudo-wave vector q. The trace over all bands for fixed q takes the form (64) with 3 = Q e x t /2e] There is an important difference between the present work and that of Refs. 35, 36 and 38. In the Hamiltonian formulation of those papers, quasiparticle tunneling could not be accounted for. They describe only Cooper pair tunneling. This corresponds to summing only over charge states Q e x t +m-2e and accordingly in the phase representations the phase factor is exp(iQ gxt /2e* 2πη) and ψ^=φο+2πη .

If quasiparticle tunneling could be completely ignored (but see below) , it is immediately clear that

Zext

and similarly the energy

levels of the system are periodic in ^ext period 2e . Provided that the charging energy for elementary charges is large

281

compared to the tunneling energy ergy for small Q e x t , and at

QGXT Qext

is

=±e

e 2 /2C>hI c /2e , the ground state en-

E Q (Q E X F C ) = Q| X T / 2 C ' i t b

an

is

2e-periodic in

d splitting to the next energy band is

•ftlc/2e . The band picture is shown in Fig. 5a. If

Qex1-

is varied

adiabatically slowly, such that the quasi-static analysis applies, physical observable quantities, like the expectation value of the actual charge

or of the voltage, will oscillate with a frequen-

cy given by the change of f = Q e x t /2e

Qext -

(36,38) (65)

These oscillations are equivalent to the Bloch oscillations in solid state physics (38). For them to exist it is necessary that the system remains in its ground state, which restricts the temperatures to kT

an
1/2) , the properties of the system will oscillate f = Q e xt/ e ·

We should add that shunting the junction by a weak ohmic resistor or closing the junction in a SQUID loop will smear out the features discussed here (for large ical meaning of

Qext

R ^

and large

L ) or destroy the phys-

altogether (for small

Further details will be presented in Ref. 40.

and small

L ).

284 V.

Conclusion

A significant amount of work has been devoted to the analysis of ohmic dissipation and Gaussian noise, both in the classical as well as in the quantum limit, and how they influence macroscopic quantum phenomena. Indeed in many experimental situations, this model is perfectly adequate. On the other hand, there exist other models, which have their physical realization and significance, and it is instructive to analyze them as well. In the classical limit, shot noise has been extensively examined in the past. Here, I discussed the quantum mechanical generalization. The description is derived from the accepted microscopic Hamiltonian of Josephson

junctions,

the dissipation being due to quasiparticle tunneling. The representation in the phase variable is roughly of the same structure as the model for Gaussian noise and ohmic dissipation, but due to the trigonometric phase dependence and the energy gap in the electrodes, is slightly more complicated. The equivalent charge representation displays explicitly the discrete charge transfer.

In spite of the increased complexity, compared to Gaussian-ohmic models, we could still quantitatively analyze our model, applied to different problems. E.g. we estimate the effect of quasiparticle tunneling on macroscopic quantum tunneling or the strength of charge fluctuations. If the capacitance is small, CkT/e 2 1ϊω 0

which

implies

these

inequalities

b a s e d on W K B t y p e

x0

of t h e t r a n s i t i o n 1ίω0/α for α > > 1 .

m u s t be s a t i s f i e d approximations

is

AU>fiω α /d

However,

in o r d e r t h a t t h e

applies.

narrow

N o t e t h a t for

theory strongly

301 damped s y s t e m s the barrier

much lower than

can

for weakly

damped

systems. Fig. 10 shows the rate in the crossover region together

with

the high t e m p e r a t u r e formula [ d a s h e d line]

and the low

rature f o r m u l a function

tempe-

(26) [ d o t t e d

d i s c u s s e d below. The

(16) line]

crossover

(22) smoothly

matches

onto t h e s e f o r m u l a s valid

out-

side the c r o s s o v e r region. As a consequence

of the

theory, properly

experi-

mental data from d i f f e r e n t

x/x0 Fig. 10

crossover

scaled

tems near T 0

The c r o s s o v e r

function

single

For temperatures be 1ow T 0 the action

sys-

should fall on a

line.

(10) has a t h i r d

stationary

t r a j e c t o r y , the s o - c a l l e d bounce s o l u t i o n qg(-r). The vicinity this t r a j e c t o r y

gives the dominant

c o n t r i b u t i o n to the

of

imaginary

part of the free energy, and the decay rate Γ is o b t a i n e d in the form Γ =

f

exp(-S B /fi)

(26)

where Sg is the action of the bounce t r a j e c t o r y . mechanical

prefactor

f

The

quantum

is r e l a t e d to the spectrum of small

t u a t i o n s about the bounce. An explicit a n a l y t i c a l

evaluation

(26) is generally not p o s s i b l e except for t e m p e r a t u r e s 1/2 range

(fiu)0/dAU)

T 0 < T 0 - T

AU *ωβ

damping

and does not d e p e n d on the form of the d e p e n d s on

In the case of a cubic p o t e n t i a l

[η(ω)Ξη]

damping,

Φ 0. It t u r n s out t h a t the e x p o n e n t of the

details

and s t r i c t l y

Ohmic

[4]

α ^

g

,

^

1 / 2 ^ 2 .

(32)

b e c o m e s e x a c t for weak and strong d a m p i n g

and it

fers only by few p e r c e n t damping.

dif-

from the exact slope for i n t e r m e d i a t e 2 Very r e c e n t l y , the Τ - e n h a n c e m e n t of t h e rate w a s o b s e r v e d

for u n d e r d a m p e d

CBJs

[lO] and for o v e r d a m p e d

SQUID r i n g s

[7],

303

Fig. α=0

il

The decay rate as a f u n c t i o n of t e m p e r a t u r e for two values of the damping

a=1

A numerical calculation damping

of the decay rate Γ for a w i d e range of

p a r a m e t e r s has been p e r f o r m e d by Chang

[37] at zero t e m p e r a t u r e

and

Chakravarty

and by G r a b e r t , O l s c h o w s k i ,

and Weiss

[38] at finite T. Fig. 11 r e p o r t s n u m e r i c a l r e s u l t s for a system with a cubic p o t e n t i a l with barrier height AU = 5fiu)0 . The

loga-

rithm of the rate is given as a f u n c t i o n of the i n v e r s e

tempera-

t u r e . In such an A r r h e n i u s plot the c l a s s i c a l r e s u l t is

shown

by a falling

straight line_. The circles show the rate for an un-

damped system. B e c a u s e of quantum e f f e c t s the rate does not crease

continuously

de-

as Τ is lowered but f l a t t e n s off at a value

d e t e r m i n e d by the W K B - d e c a y rate of the g r o u n d state in the met a s t a b l e well. transition

For the u n d a m p e d system t h e r e is a r a t h e r

between the c l a s s i c a l regime of t h e r m a l hopping

the q u a n t u m regime of t u n n e l i n g . The t r i a n g l e s show the rate of a system with d i m e n s i o n l e s s

Ohmic

a=l. The c l a s s i c a l rate is slightly

r e d u c e d since the

a t t e m p t frequency

damping

is d i m i n i s h e d by d a m p i n g .

decay rate, however, pression

sharp

coefficient

The zero

classical temperature

is strongly r e d u c e d by an e x p o n e n t i a l

factor. As a f u n c t i o n of t e m p e r a t u r e the rate now

gradually

changes

tunneling

and t h e r e is a large r e g i o n w h e r e t h e r m a l and

fluctuations temperature

and

decay

from c l a s s i c a l hopping

into pure

quantum

i n t e r p l a y . The v a l u e s of the rate at the are shown by closed

symbols.

suprather

quantum

crossover

304 The theory of MQT rates o u t l i n e d here should readily

be

applicable

to the J o s e p h s o n devices d i s c u s s e d in the p r e v i o u s section. decay of a meta stable state can only be o b s e r v e d with probability on CBJs

reasonable

if the barrier is not too high. T h e r e f o r e ,

[8-10,3l]

The

experiments

observe escape r a t e s for values of the bias

cur-

rent I close to the c r i t i c a l current I c , and fluxoid q u a n t u m s i t i o n s in SQUID rings are o b s e r v e d for applied fluxes the critical

flux

Φ

where the barrier v a n i s h e s

[7,3 θ] . N a t u r a l l y ,

one must make sure that the barrier is still larger than For the range of p a r a m e t e r s of e x p e r i m e n t a l

tran-

Φχ close to fiü¿/d.

i n t e r e s t , the part

the p o t e n t i a l r e l e v a n t for the decay problem

can well be r e p r e -

sented by a cubic. W i t h i n the RSJ model, the J o s e p h s o n device then c h a r a c t e r i z e d

by

three p a r a m e t e r s

of is

Δι) , ü¿= ω^ and a. The re-

sults in this s e c t i o n have been e x p r e s s e d

in t e r m s of t h e s e

para-

m e t e r s and they are r e l a t e d to the familiar j u n c t i o n p a r a m e t e r s Table

in

II.

Table II: R e l a t i o n between MQT p a r a m e t e r s and v a r i a b l e s the RSJ model for a CBJ and an rf SQUID with

used in

β =2π L I / Φ 0 > 1 . L

c

°

AU CBJ

2 5/2 3

M_

I ^3/2

/ ^ ç 1/2 ι Φο0 '

2

rf SQUID

4

37?

(Bl-D

In summary,

1/4

O c x.3/2 LI * (Ti )

u

A 1/4 jZ' c

2RQo0

2 1 / 2 π 1 / 4 (ίΡβ 2 - 1 ) 1 / 8 Φ -Φ Ι_ c χ. 1/4 1/2 ^Φ ' (LC) °

2RQi),

I think that for J o s e p h s o n s y s t e m s the d e p e n d e n c e

MQT r a t e s on damping and t e m p e r a t u r e stood as far as theory

is now r e a s o n a b l y

well

under-

based on the RSJ model is c o n c e r n e d .

I have

i n d i c a t e d that one can easily i n c o r p o r a t e of the j u n c t i o n c o n d u c t i v i t y

a frequency

into the t h e o r y .

dispersion

There are

presently

also a t t e m p t s at c a l c u l a t i n g the decay rate by a r e a l - t i m e lation

of

formu-

[39] w h i c h w o u l d allow for an e x t e n s i o n to extremely

d a m p e d s y s t e m s . A more d e t a i l e d c o m p a r i s o n b e t w e e n theory e x p e r i m e n t will be p o s s i b l e by m e a s u r i n g the j u n c t i o n

under-

and

parameters

305 independently [7],

4.

Work

[ 4 0 ] or u s i n g

in t h i s

TRANSITIONS

For t h e ations

escape

an

undamped

distinct tential posed

υ(Φ)

flux

effects

system

would

Let

asymmetric

proportional

to

are

ΤιΔ 0

Now,

at

large

effectively tunneling

where

=

P^(t)

and

in e i t h e r

of t h e

out

coherent known

well,

σ2/Δ^ 2

+

of t h e

externally Fig.

form

2. T h e

well AU,

energy bias

po-

has

the

thermal

double

and the

conveniently

of bias

various

energy

fio, t h e

where

im-

tunnel

energy

AU a n d fiu)0

well

behaves

dynamics

of

by

(33)

occupation

respectively.

probabilities In t h e

absence

of t h e of

left-

dissipa-

behavior

(Δ2/Δ2)

cos(Abt)

, and where left-hand

superposition

phenomenon

system,

which

(34)

1/2

+ σ )

from the

double

bias

devi-

However,

- PR(t)

oscillatory

=

in

a n d ti Δ 0 , t h e

state

an

height

small

under

part

and the mean

for

is c h a r a c t e r i z e d

Ρ ρ {t ) are t h e

shows

(Δ ο

k g T , fi σ

a biased two

the

well, and

quantitative

of U h a s t h e

as d e p i c t e d

well,

unbiased

with

PL(t)

2

starts

well

undamped

and r i g h t - h a n d

laboratories.

macroscopically

by

part

barrier

P(t)

=

double

of

threaded This

The

hand

Δ^

SQUID Φ0/2.

low t e m p e r a t u r e s

P(t)

junctions

mechanics.

low-energy

the

transitions

P(t)

interference the

to

conditions

- Φ„/2.

tion

where

show

for

scales:

like

tunnel several

leads

Φχ

compared

in

quantum

energy

fiü)0 of e x c i t a t i o n s kgT.

only

arise

us c o n s i d e r

of a h y s t e r e t i c

characteristic

shunted

in p r o g r e s s

of o r d i n a r y

can

of a p p r o x i m a t e l y

splitting

is

dissipation

predictions

new

externally

DOUBLE-WELL

problem

states.

a slightly is

IN A

from the

qualitatively

direction

of t w o

of q u a n t u m

I have

well. energy

The

assumed

oscillation

eigenstates.

coherence.

that the arises

This

system from

is t h e

a

well-

306 In t h e

presence

tunneling be

dissipation

frequency

Debye-Waller can

of

For the

by m e a n s

introduced

the

functional of t h e

influence

functional

environment.

The

been

by

studied

the

method

For

a Brownian

motion

w(q,t)

influence

by

by a

the

mechanical

evolution

functional

[41 ] . In t h i s

et a l .

an the

bath

has

discussion

equation

of

distribution

w(q,t)

written

= /D[qjD[q,]exp(ifi"1{s[q]-S[q']}-fi"1o[q,q'])w(qi>0)

where

the

q ( o)

= q'(o)

of

[43],

deterministic probability

of

heat

extended

approach

by

influence

an O h m i c

P(t)

integral

is m o d i f i e d

frictional

Leggett

'dressed' of

system from

a

is s u p p r e s s e d

[ 1 3 , 4 2 ] . An

obeying

quantum

be

the

by

the

and Vernon

functional

given

particle

which system

undamped

authors

was recently

t may

Feynman

describing

various

(5), t h e

at t i m e

by

damped

of t h e r e a l - t i m e

techniques action

is r e p l a c e d

Δ = Δαexp[-R(α)]

factor.

calculated

Δα

integral

is o v e r

= q^,

q(t)

all

paths

= q'(t)

q(t),

= q^,

q'( τ ),

and w h e r e

(35)

with

q^

is

integrated

over ; t = f dT o

S[q] is t h e

action

. „ {i m q ¿ - V ( q ) }

of t h e

undamped

(36)

particle,

and

t τ φ [ q . q ' ] = J dT J d s ( q ( τ ) - q ' ( τ ) } { Q ( x - s ) q ( s ) - Q * ( x - s ) q ' ( s ) } o o is t h e

influence Q(T)

For [cf.

= η

a double Fig.

2],

P(t)

=

functional in

well, and

[üiüaïïkgT

P(t)

wherein

sinh(Z^lIl)]

w(q,t)

will

-

. C O

Now,

for

(35)

is d o m i n a t e d

+

*

.

i n

(38)

ti be

is g i v e n

o J dq w ( q , t )

(37)

°° Jdq

centered

around

+q0/2

and

-q0/2

by

w(q,t)

.

(39)

0

low t e m p e r a t u r e s

and

small

by m u l t i - i n s t a n t o n

bias the

functional

trajectories.

integral

Grabert

and

307 Weiss

[44] have summed the c o n t r i b u t i o n of these paths in the

dilute bounce gas a p p r o x i m a t i o n

by a m e t h o d due to

[45]. This leads to an integral

representation

Zinn-Justin

for P(t)

can be e v a l u a t e d further in d i f f e r e n t r e g i o n s of the

which

parameter

space. Chakravarty

and Leggett

d i s s i p a t i o n on quantum

[13] have noticed the strong effect coherence.

most r e g i o n s of the p a r a m e t e r space the c o h e r e n c e completely

and the particle t u n n e l s

to the other. transitions

Incoherent

incoherently

is

destroyed

from one well

tunneling means that s u b s e q u e n t

are s t a t i s t i c a l l y

independent.

This behavior

for all systems with a d i m e n s i o n l e s s O h m i c d i s s i p a t i o n α

tunneling is

found

parameter

nq 2 /2rfi o

=

C

of

In fact, it t u r n s out that in

(40)

larger than 1 i r r e s p e c t i v e of the values of kgT and tío, and also for all systems with ot^Cl p r o v i d e d that kgT ciently large.

and/or fio are

Estimates of the lowest t e m p e r a t u r e

and

bias r e q u i r e d are given in Ref. 44. In the r e g i o n of t u n n e l i n g the dilute bounce gas approximation to the integral

for P(t) becomes exact at low t e m p e r a t u r e s .

of t u n n e l i n g t r a n s i t i o n s

incoherent functional The

is d e t e r m i n e d by the t u n n e l i n g

dynamics rates

from the l e f t - h a n d to the r i g h t - h a n d well and vice versa

?

f

. .2 2ïïkBT 1 Δ , Β ,2αC -1 4 ωω~~ ( ' ° ΐίω0

=

=

These r a t e s

e X p (

, τισ , 2FT) V

suffi-

smallest

[46]

I Γ (α + ί1ίσ/2π^Τ) I c Β r

(2ac)

(41)

exp(fio/kgT)f

(42)

have a r a t h e r c o m p l i c a t e d d e p e n d e n c e

bias, and damping,

on

and they are related by d e t a i l e d

The o c c u p a t i o n p r o b a b i l i t i e s e q u a t i o n w h i c h yields

P^(t)

[44,46,47]

temperature, balancing.

and Ρ ρ ( t ) now obey a master

308 P(t)

= ΡM

where Ρ(°°) =

+ [l-P(»)] exp(-rt)

- tanh

(43)

(fia/2kgT) is the e q u i l i b r i u m

value of P,

and w h e r e Γ = î + f . P(œ)

At zero t e m p e r a t u r e , starts out

= 1 and hence P(t)

from the lower well

ticle starting 2 exp(-rt)-l.

Ξ 1 if the

particle

(σ 0 ) we find P(t) = > 1 t h e r e are only

transi-

t i o n s from the upper to the lower w e l l . The t u n n e l i n g rate

Γ =

1 â! 2 ω0

(l£i) 2 a c - l ω0

T(2ac)

for t h e s e t r a n s i t i o n s

[48]

is a n o n a n a l y t i c

f u n c t i o n of the

bias.

In the absence of a bias, we have P(°°) = 0 and P(t) decays nentially:

Γ

P(t) = e x p ( - T t ) . Γ

Λ2

= 4 i° 2

ω

Γ

Γ(

(α

)

The t u n n e l i n g

ïïkDT

„

r ^) 4nr-> 1 +α

2

c

for σ = 0 and Τ = 0 the t u n n e l i n g and Moore

,

on t e m p e r a t u r e .

symmetry

breaking

is a^ < ^

This

[49] and by Bray double

is a t r a n s i t i o n at T=0, a c = l

well leading

for a^ > 1 [49-51] .

Let me now turn to the region of coherent ^ range of p a r a m e t e r s

Particularly,

rate v a n i s h e s for a c > 1.

[50 ] . It means that for the s y m m e t r i c a l

with Ohmic d i s s i p a t i o n there

smaller.

fact was first noticed by C h a k r a v a r t y

to s p o n t a n e o u s

expo-

[13 ]

e

now shows a n o n a n a l y t i c d e p e n d e n c e remarkable

rate

tunneling.

The

relevant

and kgT, fio both of o r d e r tiA or

In this region P(t) may be w r i t t e n

as a sum of three

terms : P(t)

=

P 0 ( t ) + P 1 ( t ) + Δ P(t)

The last term, AP(t), d e s c r i b e s

(46)

c o r r e c t i o n s to the dilute

gas a p p r o x i m a t i o n to the u n d e r l y i n g r e a s o n s to believe that this term

functional

is n e g l i g i b l y

bounce

i n t e g r a l . We small except

have for

309 long times, t >> Ι / Δ α w h i c h considered

here

[43], P ^ t )

zero t e m p e r a t u r e

are i r r e l e v a n t for the

d e s c r i b e s a power law b a c k g r o u n d

[13] which r e s o l v e s into a sum of

at finite T. This term can be c a l c u l a t e d to be of order o c of p a r t i c u l a r

problem

[52], Hence,

explicitly

in the r e g i o n

interest in view of the

is n e g l i g i b l e too. The interesting

at

exponentials and is found

a^

(8)

BT

Μω

»

is

ρ>'

where

pt

is s m a l l

compared

prediction

T

(kBT/AU)î.nat

unity.

In

the

(10)

quantum

regime

at

Τ = 0,

the

is

=

esc

with

=

>%/kB[7.2(1

+ 0.87/QK1

- pq)]

(kßT