NON-EQUILIBRIUM EFFECTS IN CONSTRICTED SUPERCONDUCTORS

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NON-EQUILIBRIUM EFFECTS IN CONSTRICTED SUPERCONDUCTORS

P. Lindelof

To cite this version:

P. Lindelof. NON-EQUILIBRIUM EFFECTS IN CONSTRICTED SUPERCONDUCTORS. Jour-

nal de Physique Colloques, 1978, 39 (C6), pp.C6-1411-C6-1420. �10.1051/jphyscol:19786581�. �jpa-

00218073�

JOURNAL DE PHYSIQUE

Colloque C6, suppliment au no 8, Tome

39, aoat

1978, page C6-1411

NON-EQUILIBRIUM EFFECTS IN CONSTRICTED SUPERCONDUCTORS

P.E. Lindelof,

Physics Laboratory I , H.C. Qrsted Institute, University of Copenhagen, Denmark

Rdsum6.- Des effets varids sont observds sur un supraconducteur hors-dquilibre quand un voltage est appliqud aux bornes d'un conducteur de dimensions restreintes (longueur a, largeur w

,

dpaisseur d).

La petite dimension est choisie infdrieure B la longueur de cohdrence 5 pour dviter les instabilitds.

Nous discutons des effets observds dans un systsme oii deux supraconducteurs massifs sont relids par un conducteur de petites dimensions soit d, w<c ("strips"), soit d ,

R<c

(pont large) soit enfin d,w,R<C (micropont).

Abstract.- Various non-equilibrium effects as a result of a voltage can be observed in superconduc- tors where either the length 2 , the width w or the thickness d is constricted. The dimensions of the constriction is normally chosen to be smaller than the coherence length 5 in order to avoid instabi- lities. We review some effects found in systems where two large superconductors are connected by a constriction with d,w<< (strips), d,R<< (wide bridges) and d,w,R<E (microbridges)

.

1. SUPERCONDUCTORS OUT OF THERMAL EQUILIBRIUM.- Over the last few years the understanding and classifica- tion of non-equilibrium effects in superconductors have improved considerably / l / . In order to characte- rize the non-equilibrium situation the distribution of the excitations out of the superconducting ground state must be known. If there exist an equal number of hole-like and electron-like excitations (evendis- tribution), then the energy gap Aand consequentlythe excitation spectrum can be deduced from the self- consistent BCS gap-equation

A = A J

A

( 1-2n(c))dE ( 1 ) A

kP-2

Here n(~) is the distribution function of the exci- tations, wD is the Debye frequency and A is the BCS coupling parameter. An even non-equilibrium distri- bution will appear if for instance high frequency photons or phonons are injected into the superconduc-

tor. It will relax towards the thermal distribution, given by the lattice temperature, by the inelastic scattering time T or the recombination time T

.

(The phonons must be considered separatly). A non- equilibrium situation can also arise if the overall energy gap is suddenly changed by a supercurrent

(or magnetic field). The relaxation time which de- termines how fast this can happen (the excitation distribution must change) is given by the gap rela- xation time which close to T has the form T~ % 3rE / c t , where t is the reduced temperature t = TITc.

Injection of electrons into a superconductor will create a charge imbalance among the excitations. In a spatial homogeneous situation such an imbalance will relax with a characteristic time which close

to T~ is T~ 2 T ~ /

41-t.

Because of the characteristic form of the

spectrum in superconductors, frequencies above 2A/H will be damped in an anomalous way. This is often

represented as a gap relaxation time H/2A.

In many cases of experimental interest there is a spatial orland time variation of the odd and even non-equilibrium distribution of the excitation as well as of the modulus and argument of the com- plex order parameter. Such a situation is difficult to handle in the general case where two Boltzmann equations and a generalized Ginzburg-Landau equation, all coupled, must be solved simultaneously. Because the elastic scattering time T is normally very short (10-13s), these equations have diffusion character.

For a structure with length L and with rigid boun- daries for the order parameter and the excitation distribution function (often used for microbridges), any relaxation process will happen with at least the characteristic time T = R2/D, where D = vgr.

a

2. THE QUASIPARTICLE DIFFUSION LENGTH.

-

In their s tu- dies of superconducting-normal boundaries Pippard et al. /2/ introduced the concept of charge imbalan-

- -

ce in a superconductor and discovered a "third"

length in superconductors namely the quasiparticle diffusion length. In their approach they wrote up a Boltzmann equation for each of the four branches of the excitation spectrum taken at one particular ener- gy. By a symmetry argument and by addition and sub- traction of these equations they arrived at equa- tions for the quasiparticle electric current J and

*

the charge imbalance Q* integrated over all energies :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786581

JOURNAL DE PHYSIQUE

dQk- J

*

^{1 dJ*}

v

F d X V ~ T v F d t

Here we have i n c l u d e d a p o s s i b l e time v a r i a t i o n i n t h e Boltzrnann e q u a t i o n s . I / T i s t h e t o t a l s c a t t e r i n g r a t e , which e s s e n t i a l l y i s t h e t r a n s p o r t r e l a x a t i o n r a t e i n t h e normal s t a t e . Combining e q u a t i o n ( 2 ) and ( 3 ) g i v e s

and a s i m i l a r e q u a t i o n f o r J

* .

^{A i s} t h e q u a s i p a r t i -

Q

c l e d i f f u s i o n l e n g t h f o r c h a r g e i m b a l a n c e A ⁼

1

Q

(v2.cr

) r .

A s i m i l a r d i f f u s i o n l e n g t h f o r a n unchar- F

Q

ged ( e v e n ) n o n - e q u i l i b r i u m e x i s t : A = ( v 2 r ^.')T E F E Normally r i s much l o n g e r t h a n r , and t h e c h a r a c t e -

Q

r i s t i c f r e q u e n c i e s o f t h e v a r i a t i o n of Q* much lower t h a n t h e t r a n s p o r t r e l a x a t i o n r a t e . Then we c a n ne- g l e c t t h e l a s t two t e r m s i n e q . ( 4 ) . The r e s u l t i n g d i f f u s i o n e q u a t i o n i s of some i n t e r e s t . A d . c . o r

law f r e q u e n c y q u a s i p a r t i c l e c u r r e n t g e n e r a t e d l o c a l - l y i n a s u p e r c o n d u c t o r ( o r a t a SN boundary) d e c a y o v e r l e n g t h s c a l e d e t e r m i n e d by A The q u a s i p a r t i c l e

Q'

p o t e n t i a l p and t h e c h a r g e i m b a l a n c e d e c a y o v e r t h e same d i s t a n c e :

where Q i s a c h a r g e imbalance a t x = 0 .

As t h e f r e q u e n c y w of t h e q u a s i p a r t i c l e c u r - r e n t i n c r e a s e s , t h e second t e r m on t h e l e f t come i n t o p l a y and s h o r t e n t h e e f f e c t i v e d e c a y l e n g t h and g i v e a p h a s e s h i f t of t h e q u a s i p a r t i c l e c u r r e n t a s a f u n c t i o n of t h e d i s t a n c e from t h e q u a s i p a r t i c l e cur- r e n t s o u r c e . When w~ >> l t h e second t e r m on t h e

Q

l e f t i n e q . ( 4 ) d o m i n a t e s o v e r t h e f i r s t and f o r a harmonic v a r i a t i o n o f t h e q u a s i p a r t i c l e c u r r e n t s o u r - c e t h e s o l u t i o n i s :

Q" = Qo e (6)

/%

- r e p r e s e n t s t h e s k i n d e p t h f o r c h a r g e i m b a l a n c e i n a s u p e r c o n d u c t o r .

3 . THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS.- Up t o now o n l y two a p p r o a c h e s h a v e been c h o s e n i n o r d e r t o g e t n u m e r i c a l r e s c l t s f o r s u p e r c o n d u c t o r s w i t h a v a r i a t i o n of a n o n - e q u i l i b r i u m d i s t r i b u t i o n i n s p a c e and time. One i s t o l e t t h e o r d e r p a r a m e t e r b e a c o n s t a n t i n s p a c e and s o l v e t h e Boltzmann e q u a t i o n f o r t h e q u a s i p a r t i c l e s ' , which we mentioned i n t h e p r e v i o u s s e c t i o n . The o t h e r i s t o l e t t h e q u a s i p a r -

t i c l e d i s t r i b u t i o n b e r u l e d by t h e v a r i a t i o n of t h e o r d e r p a r a m e t e r , w i t h t h e r e s u l t t h a t a time-depen- d e n t Ginzburg-Landau e q u a t i o n .

The s i m p l e s t time-dependent e x t e n s i o n of t h e Ginzburg-Landau e q u a t i o n s h a s t h e form / 3 /

a

^{2 i u + 2 i e + 2}

+

7 )

^{J, =} ( I - [ $ I 2 ) $ + c 2 ( v

-

= A ) $ (7)

where J, i s t h e r e d u c e d o r d e r p a r a m e t e r ,

2

t h e v e c t o r p o t e n t i a l , 5 t h e c o h e r e n c e l e n g t h , T t h e r e l a x a t i o n t i m e , 11 i s t h e e l e c t r o c h e m i c a l p o t e n t i a l f o r t h e p a i r s and q u a s i p a r t i c l e s . J i s t h e p a i r c u r r e n t den- -+

s i t y and Y o i s t h e e q u i l i b r i u m v a l u e of t h e o r d e r pa- r a m e t e r . These e q u a t i o n s c a n b e s t r i c t l y d e r i v e d on- l y f o r g a p l e s s s u p e r c o n d u c t o r s

131.

They h a v e , a t l e a s t u n t i l r e c e n t l y 1 4 1 , b e e n t h e o n l y t r a c t a b l e time-dependent Ginzburg-Landau e q u a t i o n s , and t h e y have t h e r e f o r e been w i d e l y u s e d , a l s o o u t s i d e t h e i r regime o f proved v a l i d i t y .

4. PHASE-SLIP CENTERS. - A 1-dimensional w h i s k e r 151 o r m i c r o s t r i p / 6 , 7 / w i t h t r a n s v e r s e d i m e n s i o n s s m a l l e r

t h a n o r of t h e o r d e r of 5 h a s c u r r e n t - v o l t a g e c h a r a c - t e r i s t i c s (IVCs) w i t h a s t e p - l i k e s t r u c t u r e . It was shown by Skocpol

% &

1 6 1 t h a t t h i s s t r u c t u r e i s d u e t o d e s c r e t e a p p e a r a n c e o f r e s i s t i v e r e g i o n s o f l e n g t h A When t h e c r i t i c a l c u r r e n t o f such a one-

Q '

d i m e n s i o n a l s u p e r c o n d u c t o r i s e x c e e d e d , one o r seve- r a l o f t h e s e p h a s e - s l i p c e n t e r s (PSCs) d e v e l o p a t p l a c e s where t h e c r i t i c a l c u r r e n t i s j u s t s l i g h t l y d e p r e s s e d . When a PSC i s formed, t h i s a p p e a r a n t l y r e p e l o t h e r c e n t e r s w i t h i n a d i s t a n c e which presuma- b l y i s some c h a r g e i m b a l a n c e d i f f u s i o n l e n g t h s . The p a i r s a r e o n l y a f f e c t e d i n a c o r e of t h e p h a s e - s l i p c e n t e r which h a s t h e s i z e of a c o h e r e n c e l e n g t h . Here t h e ' o r d e r p a r a m e t e r p h a s e $ p e r i o d i c a l l y s l i ~ s

2 eii by ZTI w i t h t h e J o s e p h s o n f r e q u e n c y

*

d t ⁼

^-. M

^{T h i s}

p i c t u r e h a s been confirmed i n t h e r e c e n t e x p e r i m e n t s by Dolan and J a c k e l 171.

To what e x t e n t t h e a . c . J o s e p h s o n e f f e c t can b e d e t e c t e d i n a true one-dimensional p h a s e - s l i p c e n t e r i s n o t c l e a r . Both r e f e r e n c e s 161 and / 7 / r e p o r t s on microwave induced s t e p s i n p h a s e - s l i p c e n t e r s , b u t

i n b o t h c a s e s a n o t c h was made i n t h e t r i p i n o r d e r t o provoke t h e p h a s e - s l i p c e n t e r . I n f a c t t h e r e c e n t c a l c u l a t i o n s / 4 / i n d i c a t e t h a t an i s o l a t e d PSC, i n a homogeneous one-dimensional s u p e r c o n d u c t o r can e x i s t o n l y i n t h e c u r r e n t r e g i o n below t h e c r i t i c a l c u r r e n t

i . e . , i n t h e h y s t e r e t i c r e g i o n . They found t h a t a p e r i o d i c a r r a y of PSCs c a n e x i s t , w i t h a n e x c e s s

s u p e r c u r r e n t s i m i l a r t o t h e c r i t i c a l c u r r e n t and w i t h a s p a t i a l e x t e n s i o n of A a s found experimen-

Q

t a l l y / 5 , 6 , 7 / . They a l s o p r e d i c t e d t h a t t h e s e PSCs e x h i b i t a n o s c i l l a t o r y b e h a v i o u r w i t h t h e Josephson f r e q u e n c y . Presumably such o s c i l l a t i o n s would b e m o d i f i e d when w > l / r a s i n E q . ( 6 ) , and g r a d u a l l y

Q

d e c a y when w>l/rE. However, a more p r e c i s e p i c t u r e of t h e non-equilibrium s t a t e of a PSC, i n p a r t i c u - l a r t h e t r a n s i e n t s f o l l o w i n g t h e p h a s e - s l i p p r o c e s s and i t s r e l a t i o n t o t h e c h a r a c t e r i s t i c t i m e s i n t h e s u p e r c o n d u c t o r , i s s t i l l a m a t t e r of c o n s i d e r a b l e i n t e r e s t .

5.WIDE BRIDGES WITH FLUX-FLOW.- Superconducting f i l m s much w i d e r t h a n t h e coherence l e n g t h h a s a n o n n e g l i g i b l e s e l f - m a g n e t i c f i e l d from t h e a p p l i e d cur- r e n t . T h i s means t h a t t h e v e c t o r p o t e n t i a l i n e q . ( 7 ) and (8) p l a y s a n i m p o r t a n t r o l e . A t h i n f i l m i s nor- m a l l y a t y p e I1 superconductor a s t h e p e n e t r a t i o n d e p t h i s g i v e n by i 2 / d , where d i s t h e f i l m - t h i c k - n e s s and A i s t h e b u l k p e n e t r a t i o n d e p t h . An e s t i - mate of t h e IVCs o f a wide b r i d g e must t h e r e f o r e b e based on t h e p i n n i n g and v i s c o s i t y of f l u x - l i n e s i n t h e f i l m . The f i r s t c o n s t r i c t i o n s c o n s i d e r e d / 8 / were presumably of t h i s t y p e . It was found t h a t t h e motion of f l u x - l i n e s p e r p e n d i c u l a r t o t h e c u r r e n t

(and d r i v e n by t h e L o r e n t z f o r c e ) could be synchro- n i z e d by a r a d i o - f r e q u e n c y s i g n a l i n c l o s e analogy t o t h e Josephson e f f e c t . Such e f f e c t s h a s been one of t h e consequences of l a t e r t h e o r e t i c a l work / 9 , l O f . The t h e o r e t i c a l p r e d i c t i o n s / 9 , 1 0 / a r e i n some con- t r a d i c t i o n . I n p a r t i c u l a r t h e b e h a v i o u r a t low v o l - t a g e s were Ref. / 9 / p r e d i c t s a low dynamic r e s i s t a n - c e and Ref. / l o / a h i g h dynamic r e s i s t a n c e . E x p e r i - ments /11,12/ on wide b r i d g e s have n o t been s u f f i - c i e n t l y c l e a r c u t a s t o s o r t o u t t h e t h e o r e t i c a l pre- d i c t i o n s . It i s t o be expected t h a t t h e t h e o r y f o r

che v i s c o s i t y of moving v o r t i c e s may have t o be r e - Fined i n l i g h t o f t h e r e c e n t p r o g r e s s of non-equili- brium s u p e r c o n d u c t i v i t y 1131.

6. MICROBRIDGES (CTBs AND VTBs).-In t h e n e x t few sec- t i o n s we s h a l l d e s c r i b e some e f f e c t s i n m i c r o b r i d g e

n e s s b r i d g e s (CTBs). I n t h e f i r s t experiment / 8 / on s u p e r c o n d u c t i n g b r i d g e s t h e s e were t r u l y two-dimen-

s i o n a l , a s t h e y were made u s i n g a m e t a l mask. By t h e i n t r o d u c t i o n o f t h e c r o s s - s c r a t c h method /14,15/

(Fig. 1) which i s now w i d e l y u s e d , t h i s s i m p l i c i t y was l o s t s i n c e t h e d e p t h o f t h e second s c r a t c h i s n o t ' w e l l c o n t r o l l e d . I n subsequent experiments / I 6 1 on m i c r o b r i d g e s o f aluminium i t was found t h a t a . c . Josephson e f f e c t was h a r d l y v i s i b l e i n CTBs b u t o n l y i n VTBs. There i s e s s e n t i a l l y two r e a s o n s why VTBs show a s u p e r i o r Josephson e f f e c t . 1) -1 l o c a l i z e s t h e v a r i a t i o n of t h e o r d e r parameter 1171 and g i v e r i g i d bounds f o r t h e n o n - e q u i l i b r i u m r e g i o n and 2) i t i n -

c r e a s e s t h e h e a t conductance from t h e b r i d g e , i . e . , k e e p s t h e b r i d g e c o l d 1181. I n c a s e s where t h e cen- t r a l b r i d g e r e g i o n i s t h r e e - d i m e n s i o n a l i t i s , how- e v e r , w e l l t o remember t h a t o n l y a f i l m - t h i c k n e s s - d i s t a n c e fsom t h e m i d d l e of t h e c o n s t r i c t i o n t h e geometry i s two-dimensional. On t h e s c a l e of t h e c h a r a c t e r i s t i c l e n g t h s i n t h e s u p e r c o n d u c t o r t h e c o n s t r i c t i o n w i l l i n many c a s e s b e two-dimensional.

True VTB i n terms of a l l c h a r a c t e r i s t i c l e n g t h s h a s h a r d l y been made. Very t h i c k f i l m s must be used.

Clean p o i n t c o n t a c t s /19/ on t h e o t h e r hand a r e t r u e VTBs.

*

1

MICROMETER

c o n s t r i c t i o n s , which most p r o b a b l y a r e r e l a t e d t o

non-equilibrium phenomena. F i r s t however I should Fig. 1 : C r o s s - s c r a t c h e d m i c r o b r i d g e 1151. (Scanning e l e c t r o n m i c r o s c o p e ) .

l i k e t o p u t a few remaks on t h e d i f f e r e n c e between

v a r i a b l e t h i c k n e s s b r i d g e s (VTBs) and c o n s t a n t t h i c k - 7.THE DAYEv-WYATT In the early experiments

C6-1414 ^JOURNAL DE PHYSIQUE

18,201 t h e microbridges were exposed t o a h i g h f r e - quency e l e c t r o m a g n e t i c f i e l d . Two e f f e c t s were s t u - d i e d namely t h e i n d i r e c t v e r i f i c a t i o n of the_ a.c.

Josephson e f f e c t by watching t h e s t e p s and secondly a p e c u l i a r phenomenon, t h e Dayem-Wyatt e f f e c t , where t h e c r i t i c a l c u r r e n t was enhanced by t h e microwave f i e l d . Whereas t h e b r i d g e s s t u d i e d were on t h e ver- ge of showing t h e a.c. Josephson e f f e c t , t h e Dayem- Wyatt e f f e c t was q u i t e dramatic. It seems now near- l y a decade l a t e r , t h a t t h e geometry used by t h e s e a u t h o r s (CTB a few pm wide and l e s s than I pm long) was optimum i n order t o s e e t h e e f f e c t i n t i n o r indium. Smaller cross-scratched (VTB) b r i d g e s a few t e n t h s of a m i n s i z e have a much s m a l l e r Dayem- Wyatt e f f e c t 1141 which on t h e o t h e r hand appear a t much lower temperatures (Fig. 2 ) . Another d i f f e r e n c e

F i g . 2 : Dayem-Wyatt e f f e c t a t d i f f e r e n t temperatu- r e s and frequency f o r an indium b r i d g e w i t h dimen- s i o n s a s i n f i g u r e 1 .

i s t h a t t h e enhancement i s not seen f o r f r e q u e n c i e s v > 2 ~ / h i n c r o s s scratched b r i d g e s , whereas enhance- ments of T a r e observed i n t h e l a r g e r b r i d g e s . E l i a s h b e r g 1211 published an e x p l a n a t i o n of t h i s e f f e c t which suggested t h a t t h e observed phenomenon was much more g e n e r a l than the f i r s t experiments 1201 did suggest. He simply showed t h a t r e p l a c i n g t h e Fermi-Dirac d i s t r i b u t i o n i n E q . ( l ) by a non-equi- 1ibri:im d i s t r i b u t i o n where q u a s i p a r t i c l e s a r e pushed up i n energy (by a microwave f i e l d ) lead t o a solu- t i o n f o r A l a r g e r than t h e thermodynamical e q u i l i - brium gap. Subsequent i n v e s t i g a t i o n s /22/ have i n f a c t supported h i s I d e a s . I n p a r t i c u l a r t h o s e expe- riments which were not r e l a t e d t o c o n s t r i c t i o n s , were extremely convincing / 2 3 / . I r o n i c a l l y enough

t h e e x p l a n a t i o n of t h e Dayem-Wyatt e f f e c t i n cons- t r i c t i o n s i s s t i l l a m a t t e r of d i s p u t e , although t h e

main consensus i s t h a t t h e fundamental physics i s t h e same.

When t h e c o n s t r i c t i o n becomes s m a l l e r than t h e coherence l e n g t h , i t has t h e c h a r a c t e r i s t i c f e a t u r e s of t h e Josephson e f f e c t . A t f i n i t e v o l t a g e s

V

t h e r e i s a n o s c i l l a t i n g p o r e n t i a l a c r o s s t h e b r i d g e s w i t h t h e frequency v =

-

2eV and harmonics. This p o t e n t i a l

h

i s connected w i t h an o s c i l l a t i n g s u p e r c u r r e n t and normal c u r r e n t through t h e b r i d g e i n a n t i p h a s e . It i s q u i t e n a t u r a l t o expect t h a t t h i s o s c i l l a t i o n g i v e s r i s e t o a Dayem-Wyatt e f f e c t a s w e l l . This i d e a was suggested i n Ref. /24/ and h a s been persued i n a number of papers s i n c e then.

8. THE SHOULDER (FOOT), THE HYSTERESIS AND THE EXCESS CURRENT.- Superconducting c o n s t r i c t i o n s , which i n a c o n s i d e r a b l e temperature regime below T have dimen- s i o n s s m a l l e r than t h e coherence l e n g t h , a r e u s u a l l y compared w i t h t h e r e s i s t i v e l y shunted (RSJ) model

where $ i s t h e phase d i f f e r e n c e a c r o s s t h e b r i d g e s , I i s t h e c r i t i c a l c u r r e n t and R t h e b r i d g e r e s i s - t a n c e . The I V C according t o t h i s model and with I a s t h e c o n t r o l l e d parameter, has t h e form

which i s not q u i t e i n accordance with t h e experimen- t a l r e s u l t s . The RSJ model was t h e o r e t i c a l l y e x p l a i - ned by Azlamasov and L a r k i n , based on Ginzburg Lan- dau theory. Considerable e f f o r t has s i n c e been d i - r e c t e d towards g e n e r a l i z i n g t h e GL e q u a t i o n i n o r d e r t o e x p l a i n t h e d e v i a r i o n s from RSJ model.

The main d e v i a t i o n of t h e experiments from t h e I V C of t h e RSJ model i s t h e o b s e r v a t i o n of an excess c u r r e n t a t a l l v o l t a g e s and a l l temperatures r e l a - t i v e t o what t h e RSJ model p r e d i c t s . F i g u r e 3 shows t h e development of t h e I V C s of a small indium b r i d g e a s t h e temperature i s decreased. S l i g h t l y below T

(3.431 K) t h e I V C has n e a r l y the f o m of eq. (I 1) where t h e r e s i s t a n c e of t h e b r i d g e a s expected, i s

s m a l l e r t h a n t h e r e s i s t a n c e of t h e t o t a l f i l m i n t h e normal s t a t e . However, t h e r e i s one s i g n i f i c a n t d i f - f e r e n c e a s t h e assymptote of t h e I V C a t high v o l t a - ges do not e x t r a p o l a t e through t h e o r i g o . There appears t o be an excess c u r r e n t a t high v o l t a g e s which i s of t h e order Io. A t lower temperatures

(3.342 K , 3.267 K , 2.794 K ) a shoulder appears i n

IVCs a t low v o l t a g e s (20 U V f o r indium). A t s t i l l

7 I

I n m~crobr~dge

-

CI Voltage V (mV)

0 2

01

L 6 8 ,

0 , I I I I I / I I I I I _

6 2b ^'

^{40 60 80 100 120 1LO 160}

Voltage V [pV)

Fig. 3 : Current-voltage c h a r a c t e r i s t i c s a t a num- ber of d i f f e r e n t temperatures f o r an indium b r i d g e s i m i l a r i n s i z e t o t h a t shown i n f i g u r e 1.

lower temperatures t h e r e i s a l a r g e d i s c o n t i n u o u s jump f i r s t on t h e top of t h e shoulder (and t h e sub- harmonic energy gap s t r u c t u r e ) (2.794 K) and back a t a d i f f e r e n t v a l u e of t h e c u r r e n t ( h y s t e r e s i s ) ( l . 8 K ) . How t h e s e t r e n d s i n t h e development of t h e IVCs a s a f u n c t i o n of temperature v a r i e s w i t h t h e b r i d g e s i z e and geometry i s o n l y vaguely known. Q u a l i t a t i - v e l y it seems t h a t t h e shoulder appears a t lower vol-

t a g e s and c l o s e r t o T f o r l a r g e r b r i d g e s . The ex- c e s s c u r r e n t a t high v o l t a g e s s c a l e s w i t h t h e c r i t i - c a l c u r r e n t . The h y s t e r e s i s mentioned above g e t s s m a l l e r a t t h e lowest temperature when t h e b r i d g e i s s m a l l e r .

Since t h e GL e q u a t i o n s have been so success- f u l , g i v i n g t h e fundamental b a s i s of t h e RSJ model 1171 i t i s n a t u r a l t o expect t h a t t h e a d d i t i o n a l f e a t u r e s seen experimentally might be obtained by u s i n g t h e TDGL eq. (7) w i t h eq. ( 8 ) . Eq. ( 7 ) has i n f a c t been thoroughly i n v e s t i g a t e d b o t h i n t h e vol- t a g e c o n t r o l l e d case and t h e c u r r e n t c o n t r o l l e d case.

The c u r r e n t c o n t r o l l e d case was i n v e s t i g a t e d i n r e f . 1251. The average c u r r e n t a l o n g t h e b r i d g e was t a - ken t o be t h e sum of t h e s u p e r c u r r e n t eq.(8) and a normal Ohmic c u r r e n t . An e x c e s s s u p e r c u r r e n t 0.75 I.

was found a t high v o l t a g e s independent of t h e r e l a - x a t i o n time used. This model f a i l e d t o show any shoulder b u t i t gave h y s t e r e s i s a t l a r g e c r i t i c a l c u r r e n t . The TDGL i n t h e v o l t a g e c o n t r o l l e d c a s e i s a n a l y t i c a l l y s o l v a b l e and an e x a c t c u r r e n t p h a s e re- l a t i o n a t low v o l t a g e s can b e determined 1261. I n r e f e r e n c e 1261 t h i s s t a t i c c u r r e n t phase r e l a t i o n was used i n t h e r e s i s t i v e l y shunted model with c u r r e n t c o n t r o l and gave s u r p r i s i n g l y good agreement w i t h experiment i f a r a t h e r long r e l a x a t i o n time was used.

There i s , however, no j u s t i f i c a t i o n i n t h e d e r i v a - t i o n of t h e TDGL f o r having such l a r g e r e l a x a t i o n times (but t h e e q u a t i o n has s o f a r no j u s t i f i c a t i o n f o r superconductors w i t h energy gap anyway). F u r t h e r - more t h e r e was no j u s t i f i c a t i o n f o r u s i n g t h e s t a t i c CPR i n a c u r r e n t c o n t r o l l e d c a s e ( i t appears t o be v a l i d only i n a r e g i o n of s o l u t i o n s where t h e shoul- der does n o t appear i n t h e c a l c u l a t i o n s ) .

Recently t h e r e h a s been a t t e m p t s t o e x p l a i n t h e shoulder by going beyond TDGL 127-301. Golub 1301 has d e r i v e d a TDGL v a l i d f o r microbridges shor- t e r than

5

w i t h a timedependent term of t h e form

a

^{2 i ~}

+ -) + T

+.

In what e s s e n t i a l l y corresponds t o v o l t a g e c o n t r o l Golub d e r i v e a low- v o l t a g e expansion of a s t a t i c c u r r e n t phase r e l a t i o n w i t h an e x c e s s c u r r e n t a s f o r t h e u s u a l TDGL. Unfor-

t u n a t e l y i t i s n o t c l e a r whether t h i s s o l u t i o n w i l l be v a l i d i n a c u r r e n t c o n t r o l l e d c a s e .

Another approach has been taken by Aslamazov and Larkin /29/. They base t h e i r c a l c u l a t i o n s on the i d e a t h a t t h e normal component of t h e c u r r e n t i n c o n s t r i c t i o n s c r e a t e a q u a s i p a r t i c l e non-equilibrium which enhances t h e energy gap i n t h e c o n s t r i c t e d r e - gion by t h e E l i a s h b e r g 1211 mechanism. They do n o t s e p a r a t e l y c o n s i d e r t h e e f f e c t of t h e d . c . c u r r e n t and t h e a.c. c u r r e n t because they assume t h a t

f

i s s o l a r g e , t h a t t h e frequency i s much h i g h e r t h a n t h e i n v e r s e i n e l a s t i c r e l a x a t i o n time. The q u a s i p a r - t i c l e s a r e a c c e l e r a t e d by t h e e l e c t r i c f i e l d b u t can only g e t r i d of t h e i r excess energy by i n e l a s t i c s c a t t e r i n g p r o c e s s e s . Due t o t h e Andre'ev r e f l e c t i o n s a t t h e ends of t h e c o n s t r i c t i o n (when t h e gap i n t h e c o n s t r i c t i o n i s suppressed) t h e s e q u a s i p a r t i c l e s cannot d i f f u s e o u t of t h e c o n t r i c t e d r e g i o n , and a non-equilibrium d i s t r i b u t i o n i n t h e b r i d g e r e s u l t s . This then enhances t h e gap a s mentioned, and g i v e s r i s e t o t h e shoulder. The model proposed by Aslama- zov and Larkin a p p l i e s only i n a v e r y r e s t r i c t e d

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t e m p e r a t u r e r a n g e where : ~(TY-~ < Q < E(T).

I n r e c e n t a r t i c l e s 127,281 t h e s h o u l d e r h a s been s t u d i e d i n t h e l i g h t of t h e t h e o r i e s i n r e f e - r e n c e s 1291 and 1301. Octavio

st

1271 a t t e m p t s t o g i v e a q u a l i t a t i v e p h y s i c a l p i c t u r e of t h e i d e a behind t h e two a p p a r e n t l y d i f f e r e n t t h e o r i e s . T h e i r model h a s t h e f o l l o w i n g b a s i c c o n t e n t : As i s w e l l - known t h e s u p e r c u r r e n t has a s i n u s o i d a l dependence on t h e phase. A l s o t h e modulus o f t h e o r d e r parame- t e r ( t h e gap) i n t h e b r i d g e - m i d d l e v a r k e s p e r i o d i c a l - l y w i t h t h e phase such t h a t when t h e gap i s d e c r e a - s i n g t h e s u p e r c u r r e n t i s p o s i t i v e and when t h e gap i s i n c r e a s i n g t h e s u p e r c u r r e n t i s n e g a t i v e . They i n t e r p r e t t h e t h e o r i e s a s g i v i n g a time-lag between t h e gap and t h e s u p e r c u r r e n t . The gap i s enhanced (by t h e E l i a s h b e r g mechanism ?) f o r p o s i t i v e super- c u r r e n t s and s u p p r e s s e d f o r n e g a t i v e s u p e r c u r r e n t s (when t h e gap i n c r e a s e s ) . T h i s a l t o g e t h e r g i v e a r e c t i f i c a t i o n of t h e a . c . s u p e r c u r r e n t enhancing t h e d . c . s u p e r c u r r e n t . T h i s g i v e s r i s e t o t h e s h o u l d e r . They d e m o n s t r a t e t h i s e f f e c t f o r a v o l t a g e b i a s e d s i t u a t i o n . Although a p p e a l i n g , s u c h a q u a l i t a t i v e model does n o t s e p a r a t e t h e d i f f e r e n t t y p e s o f ti'me-

l a g ( r e l a x a t i o n t i m e s ) . And indeed any of t h e r e l a - x a t i o n models proposed a l l g i v e r i s e t o a n e x c e s s c u r r e n t , when v o l t a g e i s t h e c o n t r o l l e d p a r a m e t e r . I t i s i n f a c t e a s y t o show t h a t t h e r e i s no r e a l d i f f e r e n c e between t h e model proposed by Deaver a 1 1311 and t h a t proposed by O c t a v i o 1271 ( i t -

d o e s n o t m a t t e r whether t h e r e i s a time-lag i n t h e supercomponent due t o gap r e l a x a t i o n o r q u a s i p a r t i - c l e r e l a x a t i o n ) . We have s o l v e d t h e e q u a t i o n s e t up by Deaver

5

1311 w i t h c u r r e n t c o n t r o l (on ana-

l o g computer) and we f i n d ( a s f o r t h e TDGL) t h a t t h e r e i s no s h o u l d e r i n t h e IVCs c a l c u l a t e d i n t h i s c a s e . The s h o u l d e r h a s s o f a r been observed i n mi- c r o b r i d g e s made of t i n 115,271, indium 1261, l e a d 1321 and aluminium 1331. From t h e s e p a p e r s i t seems u n l i k e l y t h a t t h e s h o u l d e r i s l i m i t e d t o b r i d g e s w i t h Q %

5 .

It r a t h e r a p p e a r s i n b r i d g e s w i t h a v e r y

l a r g e a . c . Josephson c u r r e n t d e n s i t y . I s does n o t simply seem t o appear a t a v o l t a g e i n v e r s e propor- t i o n a l t o r 127,291 b u t r a t h e r something l i k e

qR .

A phenomenon v e r y a k i n t o t h e s h o u l d e r i s observed i n h i g h q u a l i t y niobium p o i n t c o n t a c t i r r a d i a t e d w i t h s u b - m i l l i m e t e r r a d i a t i o n 134,351.

It i s n a t u r a l t o e x p e c t t h a t t h e s h o u l d e r observed i n t h i s c a s e h a s t h e same o r i g i n a s t h a t observed i n m i c r o b r i d g e s . I n t h i s c o n n e c t i o n we might mention t h a t c o n s t r i c t e d SNS j u n c t i o n s 1361 s i m i l a r l y have a s h o u l d e r .

9. THE JOSEPHSON RADIATION AND MICROWAVE INDUCED STEPS.- The a. c. Josephson e f f e c t can b e e i t h e r d i r e c - t l y d e t e c t e d o r measured i n d i r e c t l y by o b s e r v i n g t h e h e i g h t o f t h e microwave induced s t e p s i n t h e IVC.

The maximum r a d i a t i o n i s u s u a l l y one o r two o r d e r s of magnitude s m a l l e r t h a n p r e d i c t e d by t h e RSJ model, b u t t h i s i s presumably connected w i t h a bad c o u p l i n g of t h e b r i d g e t o t h e d e t e c t o r . The t e m p e r a t u r e de- pendence of t h e i n t e g r a t e d e m i t t e d power 1331 cor r e s p o n d s q u i t e c l o s e l y t o t h e e x p e c t a t i o n of t h e RSJ model, which p r e d i c t s a q u a d r a t i c dependence of t h e power of t h e fundamental Josephson f r e q u e n c y on

(1-t) c l o s e t o T and a s a t u r a t i o n of t h e power when f i w / 2 e ~ I > > I . The e m i t t e d r a d i a t i o n h a s been measured a s a f u n c t i o n of f r e q u e n c y 1371 and h e r e a d e v i a t i o n from t h e RSJ-model i s found a s t h e e m i t t e d power does n o t i n c r e a s e w i t h f r e q u e n c y i n t h e range

2eRIo

w < -

YI .

As a f u r t h e r r e s u l t which must be ex-

p l a i n e d we f i n d t h a t t h e e m i t t e d r a d i a t i o n a t 10GHz from I n i s about two o r d e r s of magnitude l a r g e r t h a n found f o r A 1 b r i d g e s w i t h comparable s u p e r c u r - r e n t and r e s i s t a n c e and mounted i d e n t i c a l .

The microwave induced s t e p s i n t h e IVCs a r e a g a i n i n good agreement w i t h t h e RSJ model i f one p a r t i c u l a r f r e q u e n c y (10 GHz) i s s e l e c t e d and t h e power and t e m p e r a t u r e dependence i s s t u d i e d . The f r e q u e n c y dependence o f t h e maximumstepheight which can b e induced h a s been s t u d i e d / 3 8 / ( P i g . 4 ) . Here c l e a r d e v i a t i o n s from t h e RSJ model i s observed. A t low f r e q u e n c y t h e s t e p h e i g h t s a p p e a r t o o b i g which presumably i s r e l a t e d t o t h e s h o u l d e r , g i v i n g a s m a l l e r e f f e c t i v e 2eRI

/ h

and t h u s l a r g e r s t e p s . A t h i g h f r e q u e n c y a r e l a t i v e l y slow decay of t h e

s t e p h e i g h t i s observed when t h e f r e q u e n c y i s i n c r e a - s e d . A s i z a b l e s t e p c a n b e observed a t 70 GHz ( F i g . 4 ) . No s t e p h a s s o f a r been observed i n m i c r o b r i - dges a t h i g h e r f r e q u e n c i e s /38,39/

10. THE SUBHARYONIC ENERGY GAP STRUCTURE.

-

S t r u c t u r e s i n t h e IVCs o f Josephson t u n n e l j u n c t i o n s , m i c r o b r i - dges and p o i n t c o n t a c t s a t v o l t a g e s c o r r e s p o n d i n g t o s u b m u l t i p l e s o f t h e e n e r g y gap have r e g u l a r l y been r e p o r t e d i n t h e l i t e r a t u r e . F i g u r e 5 shows

dV/dI vs V a t a number of temperatures f o r an indium can be made between t h e odd 2A/2n+l-series and t h e mic-robridge

.

The subharmonic energy gap s t r u c t u r e even 2A/2n-series.

Voltage (10pVJ Div )

Fig. 4 : Microwave induced s t e p s i n t h e current-vol-

tage characteristic at several different frequencies ^{Fig. 5} : D i f f e r e n t i a l r e s i s t a n c e v s v o l t a g e taken f o r an indium microbridge similar to that shown in a t v a r i o u s temperatures. Same b r i d g e a s i n f i g u r e s

£ i n u r e 1. The c h a r a c t e r i s t i c without e x t e r n a l r a d i a - and 3.

t i o n i s shown a t t h e top. The o t h e r c h a r a c t e r i s t i c s

-

a r e taken a t t h e microwave power which produces t h e maximum f i r s t s t e p .

a t 2~15.1 i s q u i t e r e a d i l y seen. Determining what n l a b e l corresponds t o which peaks i n dV/dI i s n o t always easy. I n g e n e r a l , however, our r e s u l t s 1331 a r e c o n s i s t e n t w i t h a s e r i e s where A d e c r e a s e almost l i n e a r l y a s t h e v o l t a g e i s i n c r e a s e d and where t h e e x t r a p o l a t i o n of t h i s dependence t o A = 0 approxi- mately corresponds t o t h e v o l t a g e where t h e a . c . Josephson e f f e c t i s extinguished /39/. This p o i n t of e x t i n c t i o n a l s o approximately corresponds t o t h e broad f e a t u r e i n f i g u r e 6 which may be i n t e r p r e t e d a s t h e v o l t a g e where J o u l e h e a t i n g i n c r e a s e t h e b r i d g e temperature above i t s t r a n s i t i o n temperature.

Close t o t h e t r a n s i t i o n temperature t h e subharmonic

s e r i e s i s a r e g u l a r but not a dominant f e a t u r e i n

-

t h e I V C s . A t low temperatures t h e SGS i s on t h e 1 MICROMETER

o t h e r hand v e r y o f t e n t h e dominating f e a t u r e a t Val- Fig. 6 : Burned o u t c r o s s - s c r a t c h e d rnicrobridge.

t a g e s h i g h e r than the shoulder. Here a d i s t i n c t i o n (scanning e l e c t r o n microscope).

JOURNAL DE PHYSIQUE

I n t h e f i r s t d a t a on SGS i n m i c r o b r i d g e s t h e r e s u l t s were i n t e r p r e t e d a s t h e e n e r g y gap was i n - c r e a s i n g a s t h e v o l t a g e was i n c r e a s e d / 2 4 / ; t h i s h o w - e v e r , seems very r a r e l y /33/ t o be t h e c a s e . The i n - t e r p r e t a t i o n g i v e n i n t h a t paper s u g g e s t e d t h a t t h e SGS appeared a s a consequence of p a i r - b r e a k i n g by t h e photon energy o f t h e Josephson r a d i a t i o n a t a number o f harmonics. Although t h i s e x p l a n a t i o n pro- b a b l y c o n t a i n s t h e e s s e n t i a l i d e a , a more d e t a i l e d p i c t u r e which draw t h e c o n n e c t i o n t o more g e n e r a l i d e a s i n non-equilibrium s u p e r c o n d u a t i v i t y s h o u l d b e persued. The i n t e r e s t i n g o b s e r v a t i o n t h a t t h e odd and even s e r i e s a r e d i f f e r e n t may be of importance h e r e s i n c e i t s u g g e s t s t h a t a normal e l e c t r o n cros- s i n g t h e b r i d g e w i t h o u t s u f f e r i n g i n e l a s t i c c o l l i - s i o n . Another problem worth c o n s i d e r i n g i s t o what e x t e n t t h e p a i r b r e a k i n g happens i n t h e c o n s t r i c t e d r e g i o n o r i n t h e banks l e a d i n g up t o t h e c o n s t r i c - t i o n . The v e r y w e l l d e f i n e d peaks i n dV/dI a t 2 ~ / n observed i n t h e s m a l l e s t b r i d g e s 1331 seem t o i n d i - c a t e t h a t t h e gap i n t h e background and n o t i n t h e b r i d g e (where t h e gap o s c i l l a t e i n time) i s r e l e v a n t f o r t h e SGS. T h i s a g a i n l e a d s t o t h e q u e s t i o n of t h e p r e c i s e p a i r b r e a k i n g mechanism. One p o s s i b i l i t y i s t h a t t h e o s c i l l a t i n g Josephson v o l t a g e modulate t h e d e n s i t y of s t a t e on each s i d e of t h e b r i d g e . ( T h i s i s presumably t h e q u a l i t a t i v e c o n t e n t o f t h e tunne- l l i n g c a l c u l a t i o n on c u r r e n t c o n t r o l l e d Josephson t u n n e l j u n c t i o n s 1401. The non-equilibrium d i s t r i - b u t i o n of q u a s i p a r t i c l e s on each s i d e o f t h e b r i d g e w i l l peak a t e n e r g i e s neV above t h e bottom o f t h e e x c i t a t i o n spectrum. Indeed t h e number o f e x c i t a t i o n s which a r e o u t of t h e e q u i l i b r i u m w i t h i n s a y 5 and which a r e c r e a t e d by t h e q u a s i p a r t i c l e c u r r e n t through t h e b r i d g e can e a s i l y be of t h e same o r d e r a s t h e t o t a l l y t h e r m a l l y e x c i t e d q u a s i p a r t i c l e s . The i n e l a s t i c s c a t t e r i n g r a t e and t h e phonon genera- t i o n i s t h u s s u b s t a n t i a l . Phonons w i t h e n e r g y l a r g e r t h a n 28 w i l l have a v e r y h i g h p a i r b r e a k i n g p r o b a b i - 1 J t y . Phonon g e n e r a t i o n h a s i n d i r e c t l y been observed i n normal m e t a l p o i n t c o n t a c t s 1421. D i r e c t phonon e m i s s i o n a t t h e gap f r e q u e n c y h a s been d e t e c t e d i n c o n n e c t i o n w i t h t h e SGS i n t u n n e l j u n c t i o n s 1411.

I I . HEATING EFFECTS. -Heat i s s u p p l i e d t o t h e b r i d g e r e g i o n from o u r b a t t e r y . I f b i a s e d a t 2 mA and 1 mV a b r i d g e l i k e t h a t shown i n f i g u r e 2 w i l l have a h e a t i n p u t of 10' w/cm3. T h i s h e a t must b e c a r r i e d away by phonons, photons o r e l e c t r o n energy d i f f u - s i o n . By f a r t h e l a r g e s t c o n t r i b u t i o n i s t h e l a s t

one. However, e v e n t u a l l y t h e h e a t must b e t r a n s f e r - r e d from t h e f i l m t o t h e g l a s s s u b s t r a t e . T h i s pro- blem h a s been analyzed by Skocpol e t a l . / 1 8 , 3 9 / i n a number of p a p e r s . They f i n d t h a t whereas i n CTBs t h e h e a t i n g w i l l e x t i n g u i s h t h e Josephson e f f e c t (make t h e b r i d g e normal) a t v o l t a g e which i s around 2A t h e h e a t i n g i s much l e s s i m p o r t a n t i n YTBs. At l a r g e v o l t a g e s a SNS j u n c t i o n r e s u l t s e v e n t u a l l y . Then t h e s u p e r c u r r e n t depends e x p o n e n t i a l l y on t h e w i d t h of t h e normal s e c t i o n . T h i s i s b o r n o u t i n e x p e r i m e n t s 1391.

The problem of g e t t i n g r i d of t h e produced h e a t i s o n e of t h e major problems i n h a n d l i n g t h e s e b r i d g e s . I n p a r t i c u l a r when a c c i d e n t a l e l e c t r i c pul- s e s r e a c h e s t h e b r i d g e t h e r e s u l t may l o o k a s i n f i g u r e 6 .

12.SUMMARY.-~u~erconductingmicrobridges havesome of t h e most pronounced non-equilibrium e f f e c t s i n e x i s - t e n c e . On t h e o t h e r hand t h e s e e f f e c t s p r e s e n t s a n i n t r i g u i n g t h e o r e t i c a l problem of c o n s i d e r a b l e com- p l e x i t y . Even t h e s m a l l e s t m i c r o b r i d g e s d o n o t be- have a s t h e RSJ model p r e d i c t s b u t show an e x c e s s c u r r e n t ( i n s u f f i c i e n t v o l t a g e ) . L a r g e r b r i d g e s have a c h a r a c t e r i s t i c s h o u l d e r i n t h e i r IBC and t h e v e r y r e g u l a r SGS. The power of e m i t t e d Josephson r a d i a - t i o n f a l l o f f a t h i g h f r e q u e n c y and s o does t h e s t e p s t r u c t u r e . I n l a r g e r (CTB) b r i d g e s where t h e Joseph- son e f f e c t i s s m a l l t h e Dayem-Wyatt e f f e c t a p p e a r s . T h i s e f f e c t h a s a l r e a d y had a s t r o n g impact on t h e t h e o r y o f n o n - e q u i l i b r i u m s u p e r c o n d u c t i v i t y . Wide b r i d g e where flux-flow i s dominating r e p r e s e n t s a t y p e of b r i d g e which d e s e r v e s more a t t e n t i o n . The r e s u l t s o b t a i n e d on s u p e r c o n d u c t i n g c o n s t r i c t i o n s should be compared w i t h non-equilibrium e f f e c t s i n o t h e r t y p e s o f weak l i n k s , i n p a r t i c u l a r t h e p r o x i - m i t y t y p e b r i d g e s 1431 which s h a r e s a l o t of i t s p r o p e r t i e s w i t h m i c r o b r i d g e s .

I would b e h i g h l y s u r p r i s e d i f n o t t h e f u r - t h e r e x p e r i m e n t a t i o n w i t h s u p e r c o n d u c t i n g micro- b r i d g e s w i l l show many more i n t e r e s t i n g e f f e c t s . Let me j u s t mention t h e p o s s i b i l i t y of phonon emission and d e t e c t i o n u s i n g such b r i d g e s and t h e e f f e c t s which o c c u r when s e v e r a l m i c r o b r i d g e s a r e brought i n t o c l o s e p r o x i m i t y ( a r r a y s ) . The r e c e n t l y observed 1441 c o l l e c t i v e mode i n s u p e r c o n d u c t o r s m y a l s o have a n o t y e t r e a l i z e d impact on t h e p r o p e r t i e s of micro- b r i d g e s . The v e l o c i t y of t h i s mode i n a l u m i n i u m f i l m s i s found t o be 1 7 km/s ( t 4 0 . 9 9 6 ) . I n a c o n s t r i c t i o n w i t h r i g i d b o u n d a r i e s and II = 0.5 pm t h i s should

give a broad resonance at 1.7 GHz. One of the expe- rimental prerequisits for a theoretical treatment is a knowledge about the size and geometry of the bridges ; such information is difficult to obtain, especially for VTBs. The properties of the thin- film in which the bridges are made is also of si- gnificance. It has a polycrystalline structure, where the arain sizes can easily be similar in size

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