DYNAMICS OF GAPLESS SUPERCONDUCTING WEAK LINKS

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DYNAMICS OF GAPLESS SUPERCONDUCTING

WEAK LINKS

A. Baratoff, L. Kramer

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 8, Tome 39, aoiit 1978, page C6-548

DYNAMICS OF GAPLESS SUPERCONDUCTING WEAK L I N K S

A. Baratoff and L. Kramer +

IBM Zurich Research Laboratory, 8803 RiischZikon ZH, SwitzerZand

R6sum6.- Nous discutons la dynamique couplge du paramstre d'ordre et de la distribution des quasipar- ticules dans un supraconducteur sans gap et prdsentons des solutions num6riques pour un micropont quasi-unidimensionnel alimentdpar courant continu.

Abstract.- We discuss the coupled dynamics of the order parameter and of the quasiparticle distribu- tion in a gapless superconductor, and present numerical solutions for a quasi-onedimensional weak link driven by a dc current.

Consider a superconducting weak link of length

a and small transverse dimensions, carrying a uni-

form current density j supplied by a dc source via massive superconducting electrodes. This ensures that the phase difference 6 and the voltage V are localized within the link/l/, besides efficient heat removal. When Vdc # 0, the increase in I$ is compen- sated by - 2 ~ slips in the local phase

X

of the order parameter A, as the latter periodically vanishes in the center (X = 0). This results in stronlgy space and time-dependent oscillations in A and p (electro- chemical potential). We have extended previous inves- tigations 11-31 by including coupling to nonequili- brium quasiparticles within the theory of Gor'kov and Eliashberg (GE)/4/. This enablesus to-visualize the dependence of the deviation Bf (X, t) of the

E

quasiparticle distribution from the equilibrium Fer-

m i function and to test approximations like those recently proposed/5/ in the much more compli- cated case of a finite energy gap.

The GE equations follow in the limit ( ~ ~ / r ) ~ a

1, from those derived later for a finite gap/6-71. Here,

A

(T) is the equilibrium value of

[ A I

reduced

0

by the finite pair-breaking rate

r.

Inelastic scat- tering by phonons may be included via a conserving relaxation-time approximation as in a recent treat- ment of the local equilibrium limit/8/. Since the contribution 6fE(*) to 6f, which is even in the ener- gy E (referred to the equilibrium chemical poten- tial) relaxes to -vfE(0): we write

OfE =

-

(y+g )f (O) E E

' ,

where f(O) E

'

= -FkB~cosh2

(E/2kBTU ( 1 )

Expressing lengths, times, (as well as eV and gE)

and j in units of

5

(coherence length), rJ (current relaxation time), eVJ = M / ~ T ~ , and j =o V 15, res-

J N J pectively, and defining $ = AIAO we obtain

The characteristic lengths cO,cand times T ~ , T, ~ldepend on the parameters p= MrI(2~rk T) and

B p1 = Mrtr/(4lrkBT), where

rtr

is the elastic scatte- ring rate which determines the normal conductivity

ON'

The first of equations (6) is a consequence of theothersplus current continuity ; it was used as a check on the accuracy of the numerical integra- t'ions. cE/lql characterizesthedistance over which

conversion of quasiparticZes to pairs takes place.

In contrast to the case r<<A0/6-71, this process

is unaffected by phonon scattering unless

/ 1 $ 1

E exceeds the corresponding quasiparticle diffusion length

A.

Typically, A>><O, SO that we ignore it if a< 2A. Like u2, u1 is a weighted integral over the

contribution B£:') odd in E ; ul<O describes heating of the quasiparticles/61.

In the dirty limit (pl>>l), to which we restrict our attention here,^

A

= T. If p>> 1, the relevant energy range is E < kBT<<T, hence

c-2

2.

c-2

=T 1. 12.

... 0

Moreover, ~ 1 ' h ~ % ~ - ~ ; thus one recovers the TDGL equations for $ and y /l-3/. In the opposite limit,

-

T =

c '=

5.79 but

'

;

c

4T1 '2T/(lT2p)>>~.

+ Visiting scientist ; permanent address : Institut

fcr theoretische Physik der ~niversitzt Bayreuth, Assuming summetry of Re $ei'I2, ul and antisym- Postfach 3008, 858 Bayreuth, W. Germany metry of Im $ei'j2, U, g about X = 0, solutions we-

E

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786246

re generated for the boundary conditions.

$ = 1, ul = gE = p = 0 at X = a/2, 2~ = V = g a t x=O (8) Essentially decoupled stationmg solutions locali- zed at the ends result for sufficiently high j ;

the corresponding dc characteristics approach the universal asymptote V = jd

-

0.74911 / as y" and, hence gE and UZ*. StationarythreshoZd soZutions/3/ localized about the normal state are found below a critical current jc which lies above the correspon- ding TDGL value for the same T. A typical periodic steady-state solution is illustrated in the figures for p = 0.2, pl= 100. With this choice, the cou- pling to nonequilibrium quasiparticles is weak, sin- ce -cl = 1.61 is still small compared to T = 6.69 and

CO

= 0.291 is comparable to 5 = 0.387. Accor- ding to figure 1 (c), A% = gE-g, is significantly broader than the corresponding Lorentzian driving term.

Fig. 1 : a) Spatial dependence of

1$1

,U and U l in the right half of a link (length a = 45

,

depairing and impurity parameters : p = 0.2, p = 100) at a

tr

current j = 0.5jJ. Oscillations occur between each

air

of extremals shown ; b) Spatial dependence of

g,

,

A@ and uz under the same conditions ; c) Ener- gy dependence of maximum AgE at X = 1 compared with Lorentzian and derivative of Fermi distribution ;

all curves are normalized to 1 at E = 0. The dis- torted abcissa is used as the integration variable in equations/6-71.

It gives rise to a nonthermal hole in the quasipar- ticle distribution 6f (2). The time average

El

vani-

E

shes, a general consequence of ( 4 ) and of the boun- dary conditions. Moreover, Tul = Ti

($1

'-W)

sin- ce the diffusion term uy is small. The latter domi- nates over in short links at low voltages, how- ever/6/. The validity of such approximations and

Fig. 2 : Time dependence of

l $ )

, p , ul at X = 0 (sa- me conditions as shown in figure l).

References

/I/ Likharev,K.K.and Yakobson,L.A., Sov. Phys. JETP 41 (1975) 570

/2/ Baratoff,A. and Kramer,L., in Superconducting Quantum Devices and Their Applications (Walter de Gruyter, Berlin) 1977,51

/ 3 / Kramer,L. and Baratoff,A., Phys. Rev. Lett.

38

(1977) 518

/4/ Gor'kov,L.P. and Eliashberg,G.M., J. Low Temp. Phys.

2

(1970) 161

151 Golub,A.A., Sov. Phys. JETP

-

44 (1976) 178 /6/ Schmid,A. and Schoen,G., J. Low Temp. Phys.

2

(1975) 207

171 Ovchinnikov,Yu.N., J. Low Temp. Phys.

28

(1977) 43

/81 Kramer,L. and Watts-Tobin,R.J., Phys. Rev. Lett. 40 (1978) xxx