SUPERCONDUCTORS OUT OF THERMAL EQUILIBRIUM

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SUPERCONDUCTORS OUT OF THERMAL EQUILIBRIUM

A. Schmid

To cite this version:

A. Schmid. SUPERCONDUCTORS OUT OF THERMAL EQUILIBRIUM. Journal de Physique

Colloques, 1978, 39 (C6), pp.C6-1360-C6-1367. �10.1051/jphyscol:19786572�. �jpa-00218063�

JOURNAL DE PHYSIQUE

Colloq~e C6, supplkment au no 8, Tome 39, aotit 1978, page ~ 6 - 1 3 6 0

SUPERCONDUCTORS OUT O F THERMAL E Q U I L I B R I U M

A. Schmid

I n s t i t u t fiir Theorie der Kondensierten Materie, University o f KarEsruhe, F. R. G.

R6sumd.- 1 . Les v a r i a b l e s fondamentales d ' u n d t a t supraconducteur s o n t l e p a r a m s t r e d ' o r d r e e t l a f o n c t i o n d e d i s t r i b u t i o n d e q u a s i - p a r t i c u l e s . En g d n d r a l , c e s v a r i a b l e s d o i v e n t S t r e ddtermin6es c o m e s o l u t i o n s d e l ' d q u a t i o n BCS de "gap" e t d e l ' b q u a t i o n de Boltzmann. 2. Des p r o c e s s u s d e r e l a - x a t i o n QlCmentaires s o n t e x p l i q u d s e t i l l u s t r d s e n d i s c u t a n t d e s e x p 6 r i e n c e s c r u c i a l e s avec d e s su- p r a c o n d u c t e u r s , a u s s i b i e n q u ' a v e c l V 3 H e s u p e r f l u i d e . 3 . MGme dans d e s c a s oii d e s d t a t s 21 quasi-par- c i t u l e s s o n t ma1 d d f i n i s l ' d q u a t i o n de Boltzmann p e u t i t r e c o n s t r u i t e , a v e c , cependant, une s i g n i - f i c a t i o n d i f f 6 r e n t e d e s q u a s i - p a r t i c u l e s . On montre comment l ' d q u a t i o n d e Ginzburg-Landau ddpendant du temps e s t a f f e c t d e p a r l a f o n c t i o n d e d i s t r i b u t i o n d e s q u a s i - p a r t i c u l e s . 4. On e x p l i q u e l e s pro- p r i d t 6 s d e s c e n t r e s d e g l i s s e m e n t d e p h a s e . On p r d s e n t e une t h e o r i e ddmontrant q u ' u n f i l a m e n t porteur d e c o u r a n t p e u t s u b i r d e s t r a n s i t i o n s d e phase d i s s i p a t i v e s , p l u s p a r t i c u l i s r e m e n t dans un S t a t os- c i l l a t o i r e s t a b l e . 5. La s u p r a c o n d u c t i v i t d s t i m u l g e p a r r a d i a t i o n e s t examinde e t l ' o n a j o u t e r a une d i s c u s s i o n s u r l a ~ o s s i b i l i t d de t r a n s i t i o n s d e phase d i s s i p a t i v e s .

A b s t r a c t . - 1 . The r e l e v a n t v a r i a b l e s of a s u p e r c o n d u c t i n g s t a t e a r e t h e o r d e r parameter and t h e qua- s i p a r t i c l e d i s t r i b u t i o n f u n c t i o n . I n g e n e r a l , t h e s e v a r i a b l e s have t o b e determined a s s o l u t i o n s of t h e BCS gap e q u a t i o n and a Boltzmann e q u a t i o n . 2 . B a s i c r e l a x a t i o n p r o c e s s e s a r e e x p l a i n e d and il- l u s t r a t e d by d i s c u s s i n g b a s i c e x p e r i m e n t s w i t h s u p e r c o n d u c t o r s a s w e l l a s w i t h s u p e r f l u i d 3 ~ e . 3 . Even i n c a s e s where q u a s i p a r t i c l e s t a t e s a r e i l l - d e f i n e d , a Boltzmann e q u a t i o n can b e c o n s t r u c t e d , however, w i t h a d i f f e r e n t meaning of t h e q u a s i p a r t i c l e s . I t i s shown how t h e time dependentGinzburg- Landau e q u a t i o n i s a f f e c t e d by 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 f u n c t i o n . 4. The p r o p e r t i e s of phase s l i p c e n t e r s a r e e x p l a i n e d phenomenologically. A t h e o r y i s p r e s e n t e d which shows t h a t a c u r r e n t - c a r r y i n g f i l a m e n t may undergo d i s s i p a t i v e phase t r a n s i t i o n s , p a r t i c u l a r l y i n t o a s t a b l e o s c i l l a t o r y state. 5 . The b a s i c f a c t s of r a d i a t i o n s t i m u l a t e d s u p e r c o n d u c t i v i t y a r e examined and a d i s c u s s i o n i s added on t h e p o s s i b i l i t y of d i s s i p a t i v e phase t r a n s i t i o n .

It might p e r h a p s b e n e c e s s a r y t o e x p l a i n why some p e o p l e

-

i n c l u d i n g myself

-

t h i n k t h a t i t i s a n i n t e r e s t i n g s u b j e c t t o s t u d y t h e b e h a v i o u r of super- c o n d u c t o r s o u t of t h e r m a l e q u i l i b r i u m . I t i s c e r t a i n - l y t r u e t h a t , s i n c e t h e d i s c o v e r y of t h e t h e o r y of s u p e r c o n d u c t i v i t y twenty y e a r s ago, t h e r e have been so many i n v e s t i g a t i o n s on t h e r e s p o n s e of a super- conductor t o p e r t u r b a t i o n s which means n o t h i n g more t h a n i n v e s t i g a t i o n s on i t s b e h a v i o u r when d r i v e n out of thermal e q u i l i b r i u m . I s h a l l mention o n l y a few t h i n g s s u c h a s e l e c t r o m a g n e t i c r e s p o n s e , a t t e n u a t i o n of u l t r a s o u n d and n u c l e a r s p i n r e l a x a t i o n . You may remember t h a t t h e q u a n t i t a t i v e e x p l a n a t i o n of t h e s e phenomena had been a major s u c c e s s a t t h e b e g i n n i n g of t h e epoch of t h e BCS t h e o r y . A s h o r t t i m e l a t e r , a g r e a t i n t e r e s t i n t h e Ginzburg-Landau t h e o r y a r o s e , s i n c e t h i s t h e o r y was a b l e t o e x p l a i n , f o r i n s t a n c e , t h e r a t h e r i n v o l v e d magnetic s t r u c t u r e of supercon- d u c t o r s . ~ £ t h e second t y p e . At t h a t t i m e , Gorkov was a b l e t o show t h a t t h e Ginzburg-Landau e q u a t i o n s were a consequence of t h e BCS t h e o r y . Almost every- one seemed t o be convinced t h e n , t h a t t h e t h e o r y of s u p e r c o n d u c t i v i t y could b e e x p r e s s e d i n r e l a t i o n s i n v o l v i n g o n l y t h e o r d e r p a r a m e t e r .

The r a t h e r l a r g e a c t i v i t y of r e s e a r c h / I / i n

t h e f i e l d of what i s nowadays c a l l e d "nonequilibrium s u p e r c o n d u c t i v i t y " h a s shown t h a t t h i s c o n v i c t i o n h a s been premature.

1 . BCS GAP EQUATION AND BOLTZMANN EQUATION.- An im-

p o r t a n t l e s s o n t h a t should b e l e a r n t from t h e r e - s u l t s of t h e c u r r e n t r e s e a r c h i s , t h a t t h e BCS theo- r y

-

and I wish t o i n c l u d e under t h i s name a l l t h e subsequent r e f i n e m e n t s

-

p r o v i d e s u s w i t h t h e p r o p e r t o o l f o r t h e d e s c r i p t i o n of t h e s u p e r c o n d u c t i n g s t a - t e even i f i t i s f a r away from thermal e q u i l i b r i u m . There i s a s e t of r e l e v a n t v a r i a b l e s , which i n c l u d e s t h e o r d e r parameter (wave f u n c t i o n of t h e Cooper p a i r s a l i a s energy gap) on t h e one s i d e and t h e q u a s i p a r t i c l e s and t h e i r d i s t r i b u t i o n f u n c t i o n n on

P t h e o t h e r . There i s a c o n n e c t i o n between b o t h t y p e s of v a r i a b l e s which i s t h e BCS gap e q u a t i o n (A =

1 ~ 1 )

It i s i m p o r t a n t now t o r e a l i z e t h a t t h e r a t e 1 / . r E , a t which t h e q u a s i p a r t i c l e s approach t h e r m a l e q u i I i - brium w i t h r e s p e c t t o e n e r g y , i s r a t h e r slow a s com- pared w i t h t h e gap f r e q u e n c y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786572

(One might s a y , t o some e x t e n t , t h a t t h i s i n e q u a l i t y j u s t means t h a t t h e BCS t h e o r y i s a q u a n t i t a t i v e the- o r y f o r superconductors though s t r i c t l y s p e a k i n g , i t i s t h e i n e q u a l i t y k T << EF). We observe t h a t a t

B c

t h e t r a n s i t i o n temperature t h e r e i s t h e following o r d e r of magnitudes :

Metal A 1 Sn Pb

P l e a s e n o t e t h a t t h e r e l a x a t i o n r a t e 1/.rE ( T ~ : i n e - l a s t i c c o l l i s i o n time) depends on t h e gap a s w e l l a s on t h e energy and temperature i n t h e s e n s e t h a t t h e l a r g e r t h e former, and t h e s m a l l e r t h e l a t t e r quan- t i t i e s , t h e s m a l l e r t h e r e l a x a t i o n r a t e . As a r u l e electron-phonon c o l l i s i o n s c o n t r i b u t e e x c l u s i v e l y t o t h i s r e l a x a t i o n process.

I n thermal e q u i l i b r i u m , 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 f u n c t i o n n i s equal t o a u n i v e r s a l

P

f u n c t i o n , namely t h e Fermi f u n c t i o n

no(E) = [ e x p ( ~ / k B ~ )

+ g-l.

I n a l l o t h e r c a s e s , t h e d i s t r i b u t i o n f u n c t i o n has t o be found by s o l v i n g a Boltzmann e q u a t i o n of t h e form

where ($,A) a r e t h e e l e c t r o m a g n e t i c p o t e n t i a l s . Sin- +

1 .

ce - fi0 = pS may be c a l l e d t h e e l e c t r o c h e m i c a l po- 2

t e n t i a l (per article) of t h e p a i r s , @ h a s t h e mea- n i n g o f a p a i r chemical p o t e n t i a l .

Concluding t h e d i s c u s s i o n of t h e f i r s t p a r t of t h e t h e o r y , I should mention t h a t one h a s t o s o l - ve, i n a d d i t i o n t o t h e BCS gap e q u a t i o n and t h e Boltzmann e q u a t i o n , a l s o Maxwell's e q u a t i o n , which, i n p a r t i c u l a r , a l s o i m p l i e s charge c o n s e r v a t i o n .

2. TWO BASIC RELAXATION PROCESSES.- For i l l u s t r a t i o n , l e t me f i r s t mention t h e i n v e s t i g a t i o n s of C l a r k e 1 3 1 and Tinkham 141 (1972) on a phenomenon f r e q u e n t l y r e f e r r e d t o a s t h e appearance of branch imbalance.

I n t h e experiment (Clarke) a normal c u r r e n t i s i n - j e c t e d ( v i a a t u n n e l i n g j u n c t i o n ) i n t o a supercon- d u c t o r and charge b a l a n c e i s maintained by a super- c u r r e n t outflow. The f i n i t e r a t e of conversion of normal c u r r e n t JN i n t o s u p e r c u r r e n t JS must produce a f i n i t e d i f f e r e n c e i n t h e e l e c t r o c h e m i c a l poten- t i a l s pN and pS of t h e normal and of t h e s u p e r f l u i d components, ( i . e . Cooper p a i r s and q u a s i p a r t i c l e s ) r e s p e c t i v e l y . The measured d i f f e r e n c e i s p r o p o r t i o -

aE a n a E a n

I

^P n a l t o a r e l a x a t i o n time T d e f i n e d a s f o l l o w s

"

a t "p

+[$$-ss

+ I { ~ > = P .

Q

Js

1

The e x p r e s s i o n i n t h e s q u a r e b r a c k e t i s of t h e form ' p - ' ~ = ~ ' 2 e N ( 0 ) ' T ~

'

of a Poisson b r a c k e t and r e p r e s e n t s t h e d r i f t of t h e

and t h i s r e l a x a t i o n time i s found t o a g r e e w e l l with q u a s i p a r t i c l e i n ( r ,p) space. The c o l l i s i o n i n t e g r a l

Tinkham's t h e o r e t i c a l r e s u l t I{n

1

i n c l u d e s i n e l a s t i c c o l l i s i o n s w i t h phonons a s

P

w e l l a s e l a s t i c s c a t t e r i n g a t i m p u r i t i e s . I t can b e A (0)

T~ =

h ( ~ ) .

T ~ ( T ~ )

.

o b t a i n e d by c a l c u l a t i n g t r a n s i t i o n p r o b a b i l i t i e s according t o t h e Golden Rule. P a r t i c l e i n j e c t i o n by

This means, i n p a r t i c u l a r , t h a t T~ d i v e r g e s means of a t u n n e l c o n t a c t and quantum t r a n s i t i o n s

(Tc- T ) - ~ / ~ a t t h e t r a n s i t i o n temperature ( c r i t i c a l induced by r a d i a t i o n g i v e r i s e t o a nonvanishing

slowing down).

source term P.

The d e f i n i t i o n of pS g i v e n p r e v i o u s l y implies A most important p o i n t i s an adequate parame-

t h a t t h e r e is a s t e a d y s u p e r c u r r e n t only between re- t r i z a t i o n of t h e q u a s i p a r t i c l e e n e r g i e s E I t i s

P ' gions having t h e same

is.

I n nonthermal s t a t e s , t h e n e c e s s a r y , f o r i n s t a n c e , t h a t t h e f r e e p a r t i c l 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 pN of t h e q u a s i p a r t i c l e s e n e r g i e s E be reckoned from the energy l e v e l o f t h e

P can be d e f i n e d by means of a d i f f u s i v e e q u i l i b r i u m .

Cooper p a i r s which may be d i f f e r e n t from t h e (un-

However. one h a s t o k e e ~ i n mind t h a t i n c o n t r a s t t o p e r t u r b e d ) Fermi l e v e l EF. Hence (Aronov, Gurevich

thermal e q u i l i b r i u m , t h e r e s u l t depends on t h e na- /2/ 1974)

t u r e of t h e d i f f u s i v e process.

2 The d i s c u s s i o n of conversion o f f e r s an op-

E = J E 2 + &,2 ; E

= L -

P P Zm ( E F - +)

.

p o r t u n i t y t o b r i d g e a gap between s u p e r c o n d u c t i v i t y and s u p e r f l u i d i t y 151. I remind you of Khalatnikov's The p a i r energy

+

i s r e l a t e d t o t h e phase O of t h e

two f l u i d hydrodynamics where f o u r c o e f f i c i e n t s of o r d e r parameter

4

= A exp(-ia) a s f o l l o w s :

v i s c o s i t y appear t o account f o r d i s s i p a t i o n . There

C6-1362

^{JOURNAL D E PHYSIQUE}

is one, called c3, which shows up in a relation of the form

Obviously, we have

In principle, -rQ should control the absorption of the fourth sound in 3 ~ e . There is not yet agreement between experiment and theory.

The finite rate of conversion is responsible for the line width Aw of the nuclear magnetic reso- nance in superf luid 3 ~ e . In the longitudinal

NMR

of 3 ~ e - A, for instance, one finds

TQis result compares favourably with expe- rimental results

;

it is difficult, however, to prove experimentally the divergence

As a second example of a relaxation process, I wish to mention the condensation or the relaxation of the sagnitude of the order parameter. One finds that a deviation from equilibrium decays in time as

-t/TR 6A

a

e

9

where

for ideal conditions near Tc. One should note, howe- ver, that these relaxation times suffer different modifications if pair-breaking, insufficient energy transfer to phonons, etc. have to be taken into ac- count.

Peter and Meissner

/ 6 /

(1973) have investi- gated this relaxation process by measuring the high frequency impedance of a superconductor. In their experiments, the pair breaking effect of a supercur- rent was utilized to manipulate the magnitude of the order parameter. Recently, Schuller and Gray /7/

(1977) observed this relaxation process after having driven the order parameter out of thermal equili- brium by a short laser pulse. It was found in both cases that rR diverges

a

( T c T)-~'~ in the vicinity of the transition temperature.

Due to the anisotropy of the order parameter in 3 ~ e - A, local changes in its magnitude occur at

a

given direction, if the orbital vector +

9,

changes

in time.

On account of the finite relaxation time

T ~ ,

a vis-

cous drag accompanies this orbital motion

:

where the right side is the torque on

+ 9,

represented

by the derivative of the free energy. Specifically,

a result which one has been able to confirm by ex- periments

/8/.

Summarizing the discussion of the collective relaxation processes, I recall that there are two distinct modes

:

(i) Conversion of normal into supercurrent (and vice versa) which is accompanied by the appearance of differences in the two electro- chemical potentials and by density fluctuations.

From a formal point of view, this mode is characte- rized by changes of the phase of the complex order parameter and by the fact, that the non-equilibrium increment 6n of the quasiparticle distribution

P

function changes sign under a particle-hole trans- formation (branch imbalance). (ii) A mode where the magnitude of the order parameter changes and which

I wish to call condensation. It is accompanied with local changes in the superfluid and in the energy density. In this mode, the non-equilibrium increment 6n is invariant under a particle-hole transforma-

P

tion (e.g. recombination and generation of quasi- particles).

3. TIME DEPENDENT GINZBURG-LANDAU EQUATION.- The theory presented above may provide us with a direct understanding of the two basic processes. From this theory we may learn (i) that the properties of the qhasiparticles depend, through the BCS coherence factors, strongly on the energy

;

and hence (ii) that the inelastic collisions, which change the energy of the quasiparticles, play an important role in the Boltzmann equation.

There are, however, limitations to a quan- titative application of this theory to superconduc- tors. We encounter frequently, materials of short mean free path (impurity collisions)

II <<

c0 ^(dirty

limit). Furthermore, the pair-braking energy rwhich is caused by magnetic fields, spacial gradients of

4, spin flip scattering, etc., is not always small as compared with A. If this is the case, there no longer exists a definite relation between momentum and energy of the quasiparticles and the quasipar- ticle picture in its naive sense breaks down.

Even if this were true, Prange and Kadanoff

/ 9 / (1964) have shown i n t h e c a s e of a normal m e t a l , t h a t i t i s p o s s i b l e t o c o n s t r u c t a Boltzmann equa- t i o n . I n terms of t h e Green f u n c t i o n s t h e s t a n d a r d c o n s t r u c t i o n of a Boltzmann e q u a t i o n r e l i e s on t h e i d e n t i f i c a t i o n

(which r e q u i r e s w e l l d e f i n e d q u a s i p a r t i c l e s ) whereas a l t e r n a t i v e l y , t h e d i s t r i b u t i o n f u n c t i o n f E ,

(5

:

rP d i r e c t i o n of t h e momentum), s p e c i f i c f o r d e g e n e r a t e Fermi systems, i s d e f i n e d a s f o l l o w s :

Such a program can a l s o be c a r r i e d through f o r a superconductor ( E l i a s h b e r g / l o / 1971 ; Schmid and SchZn /11/ 1975 ; Larkin and Ovchinnikov 1121 1977).

However, one should be aware of a d i f f e r e n c e i n mea- n i n g s i n t h e d i s t r i b u t i o n f u n c t i o n s n+ and f

P E,F*

Whereas t h e former ( e x c i t a t i o n p i c t u r e ) r e f e r s t o q u a s i p a r t i c l e s having o n l y p o s i t i v e e n e r g i e s ( i n - deed, E > A), t h e l a t t e r one ( p a r t i c l e p i c t u r e )

P

-

d e s c r i b e s q u a s i p a r t i c l e s having n e g a t i v e and posi- t i v e e n e r g i e s . The p a r t i c l e p i c t u r e i s more o r l e s s e q u i v a l e n t t o t h e semiconductor model, t h e most con- v e n i e n t model t o d e s c r i b e t u n n e l i n g between super-

conductors.

There e x i s t s a formal r e l a t i o n s h i p between t h e in- crements of t h e d i s t r i b u t i o n f u n c t i o n s :

f o r conversion ;

wgere E =

I E I .

Note t h a t i n t h e c o n v e r s i o n mode, P

p a r t i c l e s a r e added symmetrically t o t h e up and down band, whereas, i n t h e condensation mode, a p a r t i c l e - h o l e p a i r i s c r e a t e d o r s h i f t e d symmetri- c a l l y . The f a c t o r ^E /E d e r i v e s i t s e x i s t e n c e from

P P t h e BCS coherence f a c t o r s .

It i s n o t n e c e s s a r y h e r e t o go i n t o t h e de- t a i l s of t h e p a r t i c u l a r s t r u c t u r e of t h e Boltzmann e q u a t i o n f o r f i n a superconductor. It seems

E,G

t h a t i n most c a s e s a form l i n e a r i n 6f i s s a t i s - E , h

f a c t o r y where o n l y some non-linear terms i n t h e electron-phonon c o l l i s i o n i n t e g r a l a r e n e g l e c t e d .

The whole problem i s s t i l l h i g h l y non-linear i n t h e o r d e r parameter. Some s i m p l i f i c a t i o n i n t h e s t r u c t u r e of t h e e q u a t i o n s can be obtained i f one r e s t r i c t s o n e s e l f t o a small temperature r e g i o n c l o -

s e t o t h e t r a n s i t i o n temperature where A << kBTc

.

I n t h i s c a s e , t h e BCS-gap e q u a t i o n ( i n c l u d i n g spa- t i a l and temporal d e r i v a t i v e s ) i s e q u i v a l e n t t o t h e time dependent Ginzburg-Landau e q u a t i o n

T - T

2 i e

"

+ 6(012 (V

-

A ) ~

-

b 1 ~ 1 ~ 1 A ,

-

where b = 75(3)/81r~k;~:. One r e c o g n i z e s t h a t t h e Ginzburg-Landau e q u a t i o n f o r thermal e q u i l i b r i u m amounts t o p u t t i n g t h e r . h . s . e q u a l t o z e r o . Save f o r t h e appearance of t h e unknown q u a n t i t y X, t h i s e q u a t i o n i s r a t h e r of a s t a n d a r d phenomenological type which d e s c r i b e s a p u r e l y r e l a x a t i o n a l motion of t h e o r d e r parameter. R e c e n t l y , such e q u a t i o n s have o f t e n been d i s c u s s e d i n connection w i t h t h e c r i t i c a l dynamics a t a phase t r a n s i t i o n .

The complex valued q u a n t i t y X has played a r a t h e r m y s t e r i o u s r o l e i n t h e p a s t . Gorkov and E l i a s h b e r g

1131

(1968) have shown t h a t i t d e r i v e s from v e r t e x c o r r e c t i o n s ( i n t h e language of t h e Green f u n c t i o n technique) and t h e y c a l l e d X t h e anomalous c o n t r i b u t i o n . However, a mystery e x i s t s only i f one f o r g e t s t h i s one l e s s o n of t h e BCS-the- o r y , namely : t h a t t h e b a s i c v a r i a b l e s of a super- conductor a r e n o t o n l y t h e o r d e r parameter b u t a l s o 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 f u n c t i o n . Indeed, X i s a l i n e a r f u n c t i o n a l of t h e increment 6f of

E , h t h e d i s t r i b u t i o n f u n c t i o n

where 6fE i s t h e angular average. One can show t h a t t h e r a t h e r opaque e q u a t i o n s f o r t h e v e r t e x c o r r e c - t i o n s a r e e q u i v a l e n t t o (some s p e c i a l i z e d form o f ) t h e Boltzmann e q u a t i o n .

I do n o t wish t o comment on X v e r y much i n d e t a i l . However, i t i s n o t d i f f i c u l t t o understand t h a t Rex corresponds t o t h e e x t r a term ( s a v e f o r a f a c t o r A) i n t h e BCS-gap e q u a t i o n which r e s u l t s from t h e non-thermalincrement 6n i n t h e d i s t r i b u - t i o n f u n c t i o n . Hence, P

where N i s t h e BCS d e n s i t y of s t a t e s . A s f a r a s 1

ImX i s concerned, one may perhaps keep i n mind t h a t

JOURNAL DE PHYSIQUE

i t must have t r a n s f o r m a t i o n p r o p e r t i e s such t h a t t h e J a c k e l 1161 (1977) a l o n g a f i l a m e n t i n t h e d i s s i p a - time dependent Ginzburg-Landau e q u a t i o n i s gauge co-

v a r i a n t . Furthermore, we n o t e t h a t ImX i s a s s o c i a - t e d w i t h t h e conversion mode, whereas Rex i s s p e c i - f i c f o r t h e condensation mode. We w i l l s e e l a t e r , t h a t t h e appearance of t h e q u a n t i t y X h a s d r a s t i c consequences on t h e time dependent Ginzburg-Landau e q u a t i o n . The r a t h e r small i n e l a s t i c r e l a x a t i o n r a t e

I / T a l l o w s an e x c e s s i v e accumulation of non-thermal E

q u a s i p a r t i c l e s . Furthermore, t h e s t r o n g dependenceof t h e i r p r o p e r t i e s on energy (expressed here by t h e f u n c t i o n L(E)) a m p l i f i e s d i s t o r t i o n s i n t h e shape of t h e d i s t r i b u t i o n f u n c t i o n i n r e g a r d t o i t s in- f luence on X.

I n g e n e r a l , i t i s extremely d i f f i c u l t t o f i n d adequate s o l u t i o n s 6f of t h e Boltzmann e q u a t i o n ,

E , B

which i s from a formal p o i n t of view, an i n t e g r o - d i f f e r e n t i a l e q u a t i o n w i t h c o e f f i c i e n t s which depend on t h e v a r i a b l e s . N e v e r t h e l e s s , t h e r e s t r i c t i o n t o t h e c a s e A << kkTc i s a l s o h e l p f u l i n s o l v i n g t h e Boltzmann equation. Then t h e q u a s i p a r t i c l e r e l a x a - t i o n time rE i s f a i r l y independent of t h e energy i n t h e range of i n t e r e s t and c o n s t i t u t e s t h u s , a cha- r a c t e r i s t i c time of t h e Boltzmann e q u a t i o n . I t f o l - lows t h a t , t h e r e i s a l s o a c h a r a c t e r i s t i c l e n g t h A (Pippard e t a l . 1141 1971) d e f i n e d by

A = (DT,)'/~

,

where D =

-

1 v R i s t h e d i f f u s i o n c o n s t a n t . We c a l l

3 F

A t h e i n e l a s t i c d i f f u s i o n l e n g t h which c o n t r o l s t h e s p a t i a l changes of t h e energy d i s t r i b u t i o n of t h e q u a s i p a r t i c l e s .

4. PHASE SLIP CENTER.- L e t me f i r s t r e c a l l t h e ex- p e r i m e n t a l f a c t t h a t a superconducting f i l a m e n t c a r r y i n g s u f f i c i e n t c u r r e n t jumps i n t o a d i s s i p a t i - v e s t a t e d i s c o n t i n u o u s l y . This behaviour i s r e f l e c - t e d i n t h e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c by t h e appearance of a v o l t a g e s t e p of a h e i g h t c o n s i d e r a - b l y l e s s t h a n what one would expect i f t h e f i l a m e n t would undergo a complete t r a n s i t i o n t o t h e normal s t a t e . Now, a superconductor cannot s u s t a i n a v o l - t a g e d i f f e r e n c e and m a i n t a i n phase coherence f o r any a p p r e c i a b l e amount of time. Consequently, Skocpol e t a1. ./I51 (1974) explained t h i s d i s s i p a t i v e s t a t e by t h e appearance of a l o c a l i z e d phase s l i p c e n t e r where t h e phase 0 of t h e o r d e r parameter s u f f e r s d i s c o n t i n u o u s changes such t h a t t h e time averaged p a i r p o t e n t i a l

Fs

jumps by a s t e p i n h e i g h t :

-

1 - 7

pS =

7

Z.0 = eV.

Recent measurements of

<

^and

^KN

by Dolan and

-

t i v e s t a t e c l e a r l y show t h a t pS i s c o n s t a n t e x c e p t a t t h e p o i n t where t h e phase s l i p t a k e s p l a c e . A s f a r a s t h e space v a r i a t i o n of

- vN

i s concerned, one should keep i n mind t h a t conversion of normal and s u p e r c u r r e n t s i s t h e r e l e v a n t c o n t r o l mechanism.

T h e r e f o r e , one e x p e c t s a c h a r a c t e r i s t i c l e n g t h

which i s a l s o b e i n g observed, indeed.

Though t h e g e n e r a l p i c t u r e of a phase s l i p c e n t e r i s i n t u i t i v e l y c l e a r , i t i s d e s i r a b l e t o have a b e t t e r understanding and t o know under which c i r - cumstances i t may be a c t i v a t e d . It i s my impression t h a t i n t h i s r e s p e c t t h e t h e o r e t i c a l i n v e s t i g a t i o n s of Kramer and B a r a t o f f 1171 (1977) and of Kramer and Watts-Tobin 1181 (1978) a r e most important. They

showed t h a t i n t h e l i m i t i n g c a s e

(which one may o b t a i n under r e a l i s t i c c o n d i t i o n s i f M/kgTc~E

2

i t i s p o s s i b l e t o e l i m i n a t e 6 f f r o m t h e system of e q u a t i o n s w i t h t h e r e s u l t t h a t t h e time dependent Ginzburg-Landau e q u a t i o n appears i n t h e f ~ r m

One may convince o n e s e l f t h a t t h e simple d i f f u s i v e time dependent Ginzburg-Landau e q u a t i o n

X + - ( i r e / 4 k T ) $ i s o b t a i n e d i f T ~ A / M << 1.

B c

This e q u a t i o n ( t o g e t h e r w i t h t h e c o n t i n u i t y e q u a t i o n l o r t h e c u r r e n t ) does have a r i c h v a r i e t y of s o l u t i o n s f o r a c u r r e n t c a r r y i n g f i l a m e n t . I t i s known t h a t t h e normal s t a t e i s l o c a l l y s t a b l e ( i . e . s t a b l e a g a i n s t i n f i n i t e s i m a l f l u c t u a t i o n s ) i f t h e cur- r e n t J > O . The same i s t r u e f o r t h e superconducting s t a t e provided t h a t J < J m a x , where J i s t h e maxi-

max

ma1 c u r r e n t i n t h e sense of Ginzburg and Landau.

Though r e l a t i v e l y s t a b l e , a t l e a s t one of t h e s e s t a - t e s i s g l o b a l l y u n s t a b l e ( i . e . u n s t a b l e a g a i n s t f i n i - t e f l u c t u a t i o n s )

.

^{I f} T ~ A / K = 0 (more p r e c i s e l y :

%AIM

< 5.5) t h e r e i s , a s Likhareff 1191 ( 1 9 7 4 ) h a s shown,

-

even t h e p o s s i b i l i t y of a d i s s i p a t i v e f i r s t o r d e r phase t r a n s i t i o n a t J = J k <Jmax. Although t h e mea'ning of a f r e e energy i s n o t obvious i n non-equilibrium, s t a t e s , we may summarize t h e r e s u l t s on t h e s t a b i l i t y of t h e v a r i o u s s t a t e s by a f r e e energy diagram i n analogy w i t h t h e e q u i l i b r i u m t h e o r y a s follows.

Fig. 1 : The gap a s a f u n c t i o n of d i r e c t i o n (Amcose) r e l a t i v e t o t h e o r b i t a l v e c t o r

% .

Shown a r e two

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

F i g . 4 : F r e e energy analogue r e p r e s e n t i n g t h e va- r i o u s degrees of s t a b i l i t y of c u r r e n t c a r r y i n g s t a - t e s of a f i l a m e n t . F u l l l i n e : t h e normal s t a t e i s g l o b a l l y s t a b l e . Dash-dotted l i n e : c o e x i s t e n c e of normal and superconducting s t a t e . Dotted l i n e s : l i m i t of l o c a l s t a b i l i t y of t h e normal r e s p e c t i v e superconducting s t a t e

A

IAl2

E,v; ⁰ _{( i ) (ii 1}

0 PN

Fig. 2 : Change i n 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

f u n c t i o n accompanying ( i ) conversion ; ( i i ) conden- ~ i5 ~: Limit cycle . of t h e square of t h e pa-

s a t i o n . rameter

161

v e r s u s e l e c t r o c h e m i c a l p o t e n t i a l FI of

t h e normal component a t t h e c e n t e r of t h e phase s l i p N

Fig. 3 : Excited s t a t e s i n ( i ) p a r t i c l e p i c t u r e ; ( i i ) e x c i t a t i o n p i c t u r e

Fig. 6 : Graphical c o n s t r u c t i o n of t h e s o l u t i o n s of t h e Ginzburg-Landau e q u a t i o n f o r a r a d i a t i o n stimu- l a t e d superconductor

JOURNAL DE PHYSIQUE

Accordingly, we e x p e c t a f i r s t o r d e r phase t r a n s i t i o n a t J = J k '

Most i n t e r e s t i n g , however, i s t h e appearance of a f u r t h e r s t a b l e s t a t e (not contained i n t h e s i m - p l e diagram above) which i s o s c i l l a t o r y and which resembles a t l e a s t q u a l i t a t i v e l y , an a c t i v a t e d phase s l i p c e n t e r . I n p a r t i c u l a r , i f .rEA/H

2

5.5, t h i s o s c i l l a t o r y s t a t e i s t h e only one which i s g l o b a l l y s t a b l e i n a range of c u r r e n t s below Jmax. Thus, a c u r r e n t c a r r y i n g superconducting f i l a m e n t may jump i n t o a d i s s i p a t i v e s t a t e of o s c i l l a t o r y n a t u r e which i s s t a b l e . Usually, one r e f e r s t o such a p r o c e s s a s a t r a n s i t i o n t o a l i m i t c y c l e .

The t h e o r e t i c a l p r e d i c t i o n t h a t a t r a n s i t i o n i n t o t h e normal s t a t e and/or an a c t i v a t i o n of a pha- s e s l i p c e n t e r may occur a t c u r r e n t s below t h e ma- ximal c u r r e n t , seems t o me of c o n s i d e r a b l e i n t e r e s t . There a r i s e s t h e n e x t q u e s t i o n on t h e n a t u r e of f l u c - t u a t i o n s such a s t o a c t i v a t e a c e n t e r .

5. RADIATION STIMULATED SUPERCONDUCTIVITY.- It i s w e l l known t h a t 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

s u p p r e s s s u p e r c o n d u c t i v i t y s i n c e they r e s t r i c t t h e phase space a v a i l a b l e t o Cooper p a i r s

-

t h i s explains t h e phase t r a n s i t i o n a t f i n i t e temperatures. Less known, however, i s t h e f a c t t h a t e x c i t a t i o n s n e a r t h e gap edge a r e more d e s t r u c t i v e l y blocking t h e p a i r s t h a n those which a r e off t h e edge. Recognizing t h i s circumstance, E l i a s h b e r g 1201 (1970 ; I v l e v e t a l . 1201, 1973) p o i n t e d o u t t h a t e l e c t r o m a g n e t i c r a - d i a t i o n t e n d s t o remove, under c e r t a i n c o n d i t i o n s , e x c i t a t i o n s from t h e gap edge w i t h t h e consequence of an enhancement of t h e o r d e r parameter and even of a s t i m u l a t i o n of s u p e r c o n d u c t i v i t y above Tc.

R a d i a t i o n o f , s a y , frequency v induces t r a n s i - t i o n of t h e q u a s i p a r t i c l e s from s t a t e s of energy E t o s t a t e s of energy E +- Nv. I n i t i a l l y , t h e gap edge i s h e a v i l y populated w i t h e x c i t a t i o n s on account of i t s h i g h d e n s i t y of s t a t e s , and consequently, t h e n e t r a t e of t r a n s i t i o n s i s away from t h e gap edge.

I n t h e Boltzmann e q u a t i o n , t h e s e t r a n s i t i o n s a r e r e - p r e s e n t e d by a f i n i t e s o u r c e term P which i s propor- t i o n a l t o t h e a p p l i e d r a d i a t i o n power.

A t f i r s t , we look f o r s p a t i a l l y homogeneous and s t a t i o n a r y s t a t e s . One f i n d s t h a t t h e q u a n t i t y X i s r e a l and only a f u n c t i o n of A/$v,

where B i s a c o e f f i c i e n t p r o p o r t i o n a l t o a p p l i e d power. I n t h e c a s e of e l e c t r o m a g n e t i c r a d i a t i o n , G

i s p o s i t i v e and h a s one maximum i n t h e form of a kink a t Hv = 2A. Obviously, t h e s t a t i o n a r y Ginzburg-Landau e q u a t i o n

~ 2 T - T

BG(-)A = ( b

- -

~ ~

H2v2 ) A

Tc

a l l o w s always t h e normal s t a t e s o l u t i o n A = 0. A g r a p h i c a l c o n s t r u c t i o n shows c l e a r l y , t h a t t h e r e i s one superconducting s t a t e s o l u t i o n A

#

0 f o r T < T w i t h t h e p r o p e r t y t h a t A i n c r e a s e s w i t h i n c r e a s i n g r a d i a t i o n power ^a B. For t e m p e r a t u r e s above T b u t below some maximal temperature T t h e r e e x i s t s

max

'

two superconducting s t a t e s o l u t i o n s .

Turning our a t t e n t i o n t o t h e dynamic behaviour s f t h e s e s o l u t i o n s , one f i n d s i n a l i n e a r s t a b i l i t y a n a l y s i s t h a t f o r T < Tc, t h e normal s t a t e i s l o c a l l y u n s t a b l e , i n c o n t r a s t t o t h e superconducting s t a t e which i s s t a b l e . I n t h e temperature range

Tc < T < Tmax, t h e superconducting s t a t e w i t h t h e s m a l l e r v a l u e of A i s l o c a l l y u n s t a b l e , whereas t h e superconducting s t a t e w i t h t h e l a r g e v a l u e of A a s w e l l a s t h e normal s t a t e a r e a t l e a s t l o c a l l y s t a b l e .

A s t a b i l i t y a n a l y s i s , which i n c l u d e s s p a t i a l f l u c t u a t i o n s of t h e o r d e r parameter i n l i n e a r o r d e r , reproduces t h e r e s u l t s on l o c a l s t a b i l i t y found abo- ve. It i s now an i n t e r e s t i n g q u e s t i o n whether t h e allowance of l a r g e s p a t i a l v a r i a t i o n s of t h e o r d e r parameter may l e a d t o a s t a t e where a normal and a superconducting r e g i o n c o - e x i s t

-

perhaps i n a manner s i m i l a r t o t h e r e s u l t found p r e v i o u s l y w i t h a cur- r e n t c a r r y i n g f i l a m e n t . I n o r d e r t o i n v e s t i g a t e t h i s p o s s i b i l i t y , we add t o t h e s t a t i o n a r y Ginzburg-Landau e q u a t i o n , given above, t h e u s u a l bending energy ex- p r e s s i o n

c2

(o)v'A. (The i n f luence of t h e q u a s i p a r t i - c l e d i f f u s i o n on t h e s p a t i a l s t r u c t u r e i s weak i f t h e r a d i a t i o n power i s small.)

One can show t h a t t h i s e q u a t i o n p r e d i c t s a d i s s i p a t i v e f i r s t o r d e r phase t r a n s i t i o n a t T = T k ' where T < Tk < Tmax, w i t h t h e p r o p e r t y t h a t t h e su- perconducting (normal) s t a t e i s g l o b a l l y u n s t a b l e f o r T > Tk (T < Tk). The r e s u l t a g r e e s w i t h i n v e s t i - g a t i o n s 1211 on t h e s t a b i l i t y of t h e homogeneous sta- t e i n t h e presence of quantum f l u c t u a t i o n s i n t h e q u a s i p a r t i c l e occupation number. We f i n d t h a t t h e r e e x i s t s a p r o b a b i l i t y d i s t r i b u t i o n

where t h e non-equilibrium f r e e energy diagram i s t h e

1

same a s i n F i g . (4) except f o r t h e f o l l o w i n g r e p l a -

cement References

Eventually, we may calculate the probability of the occurrence of such fluctuations which cause transi- tions from a globally unstable state to a stable one. Unfortunately, the numbers seem to be so small that one should not expect to observe such a tran- sition. In contradiction to this result, one does observe, for instance, a transition from the normal to the superconducting state even above T .

(Klapwijk, van den Bergh and Mooij 1221)

These considerations on radiation stimulated superconducting states and on their stability con- clude my talk on superconductors away from thermal equilibrium. There are interesting problems which I have not mentioned and some of them are rather di- rectly related to the problems I have tried to dis- cuss. For instance, there is a prediction of

Scalapino and Owen 1231 on a first order phase tran- sition of a superconductor at low temperatures as a result of laser irradition. The stability of states produced in this way is an intriguing and open question.

I regret that I could not present all these interesting problems. On the other hand, I had to make a selection, and certainly, this selection re- flects, to some extent, my personal taste.

It seems to me that the possibility of manipu- lating the quasiparticle distribution function by irradiation or by tunnel injection is a fascinating aspect of this field of physics. I think there are more interesting phenomena than in the related field

of semiconductor physics because here the quasipar- ticle distribution does have a decisive influenceon the order parameter, which means on the gap. As the magnitude of the gap strongly controls the proper-

ties of the quasiparticles,enhanced mutual inter- dependence of the basic variables of superconductor does occur. Consequently, we meet a variety of phe- nomena and, among others, dissipative phase transi- tions and even a transition into stable dissipative oscillatory states which are realized in phase slip centers.

/I/ Langenberg, D.N., Nonequilibrium phenomena in Superconductors. Proceedings of LT 14

:

M. Krusius and M. Vuorio, Eds (North-Holland)

1975

/2/ Aronov, A.G. and Gurevich, V.L., Fiz. Tverd.Tela, 16 (1974) 2656 bngl. Transl.

:

Sov. Phys. Solid

- State 16 (1975) 17221

131 Clarke, J., Phys. Rev. Lett. 8 (1972) 1363, Clarke,

J.,

and Paterson, J.L., J. Low Temp.

Phys. 2 (1974) 491

/4/ Tinkham, M., Phys. Rev. (1972) 1747;

TinkhamrM. andClarke,J. , Phys. Rev. Lett. 8

(1972) 1366

/5/ See, for instance, the review article

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