RELAXATION AND COLLECTIVE MOTION IN SUPERCONDUCTORS. A TWO-FLUID DESCRIPTION

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RELAXATION AND COLLECTIVE MOTION IN

SUPERCONDUCTORS. A TWO-FLUID

DESCRIPTION

C. Pethick, H. Smith

To cite this version:

JOURNAL D€ PHYSIQUE Colloque C6, supplPment au no

8,

Tome 39, aoit 1978, page _{~ 6 - 4 8 8}

++

C.J. Pethick and H. Smith

Department of Physics, University of IZZinois at Urbanu-Champaign, Urbana, Illinois 61801, and

++ Nordita, BZegdamsvej 17, DK-2100 Copenhagen 0, Denmark

Fys-kkLaboratoriwn I, H.C. 0rsted Institutet, DK-2100 Copenhagen

0,

Denmark

Rdsum6.- Nous prdsentons une discussion unifide d'un certain nombre de phdnomsnes de basse frlquence

1 partir de l'lquation de Boltzmann de quasiparticules. Nous introduisons un modzle 1 deux fluides pour la charge et donnons les dquations hydrodynamiques 1 deux fluidesque nous utilisons ensuite pour discuter le mode collectif et la diffusion de quasiparticules au voisinage des centres de glissement de phase. Nous considlrons aussi l'dvolution des populations de quasiparticule sur les diffgrentes branches et la relaxation du gap vers 1'6tat d'squilibre.

Abstract.- We give a unified discussion of a number of low-frequency phenomena in superconductors, using the quasiparticle Boltzmann equation. We introduce a two-fluid model for the charge, and formu- late two-fluid hydrodynamic equations which we then use to discuss the collective mode, and quasipar- ticle diffusion near phase-slip centers. We also consider quasiparticle branch imbalance relaxation and gap relaxation.

The two-fluid model for the charge arises na- turally from considering variations in the total charge density

Here f is the quasiparticle distribution function,

k*

and

$

= 1

-

v2 = (1

+

Sk/Ek)/2, where

ck

is the nor- mal state quasiparticle energy measured with res- pect to the chemical potential

vs

of the paired elec- trons and Ek =

Ja2+5:

is the quasiparticle energy.

The change in Q may be written as the sum tot

of a normal part, associated with changes in the quasiparticle distribution, and a superfluid part, associated with changes in the coherence factors : &Qtot = &Qn +

&Q;

,

(2)

where

6Qn = $(U;

-

v;, 6fk* ( 3 ) and

&Qs = C(l

-

2f )6v:

kc

k*

(4)

The separation here is completely analogous to that discussed by Leggett and Takagi/l/ for the spin den- sity in superfluid Fermi systems.

Using equation(4) we now define a susceptibili- ty for the superfluid component by the relation

0

&Qs =

X,

&vs

(5)

0

For A<<k T one finds

X

-2N(0)(a/4)(A/kBTc), where B c

N(0) is the density of states per spin.

Close to T and in the hydrodynamic regime one +~esearch supported in part by U.S. National Science

Foundation Grant NSF DMR 76-24011

can show that the quasiparticle distribution is a Fermi function evaluated for a chemical potential

(# 1-1,) which we interpret as the chemical poten- tial of the normal component. The normal component susceptibility is then defined by

6Qn =

:

X

&vn

One finds that n + = 2N(O).

In the absence of collisions, the charge as- sociated with each component is conserved. When the effects of collisions are taken into account, one finds

and

Here 6QA'e. = Xz(6pn-6vs) is the deviation of 69, from its local equilibrium value, and pn(ps) and

%(,ys)

are the normal and superfluid densities and velocities, respectively. These equations are stric- tly valid if A<<kBTc, and the relaxation time 'C is

q then found to be

where 'C (C=O,T ) is the lifetime of a normal sta- in

te quasiparticle at the Fermi energy at T due to

C'

inelastic scattering. By solving the Boltzmann equa- tion for the gap relaxation time

-cA

,

one finds

'CA =(~'/28~(3))~ to lowest order in A/kBTc, in q

agreement with the result obtained in the dirty li-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786217

mit by Schmid and Sch6n/2/ using Green functions. The remaining two-fluid equations are the ac- celeration equation for the superfluid

and an acceleration for the normal fluid similar to that discussed by Bray and Schmidt/3/. The electric field

R

is determined by Poisson's equation.

We now summarize the results of some applica- tions of our equations.

i) Charge relaxation. In the tunnelling ex- periments pioneered by Clarke/4/ the detected volta- ge is directly proportional to dQnl'e' ; even away from Tc. While in the past the relaxation process has generally been referred to as "branch imbalan- ce relaxation1'/5/ a more appropriate name is char- ge relaxation, since the process involves conver- sion of charge associated with the normal component into chage associated with the superfluid component. This is brought about not only by scattering of a quasiparticle from one branch to another, but also by scattering of a quasiparticle to a state on the same branch. The characteristic relaxation time mea- sured in these experiments is -c

q'

ii) Quasiparticle diffusion. In a spatially inhomogeneous situation such as the one occurring near phase-slip centers, the difference 6pn-6us may be shown to obey a diffusion equation with the de- cay length

AD

given by X: = DT

.

Here D = v 2 ~ . 13

q F 1mp

is the normal state diffusion coefficient where v

F

is the Fermi velocity and the elastic scatte- ring time.

iii) Collective mode. Our results for the frequency of the collective mode first discovered by Carlson and Goldman resemble those of reference

1 3 1 , but with the important difference that our approach gives explicit expressions for the suscep- tibilities of the normal and superfluid components. Close to Tc the velocity c of the collective mode is given by c2 = ps/mXz, which in the dirty limit,

71

where pS/p = T(~2~imp/UkBTc), is seen to be c2 = 2DAlbl in agreement with the Green function calculation of Schmid and ~ch6n/6/.

References

/l/ Leggett,A.J. and Takagi,S., Ann. Phys. (N.Y.) 106 (1977) 79

-

/2/ Schmid,A. and SchEn,G., J. Low Temp. Phys. ) (1975) 207

/3/ Bray,A.J. and Schmidt,H., Solid State Commun. 17 (1975) 1175

-

/41 Clarke,J., Phys. Rev. Lett.

E

(1972) 1363 /S/ Tinkham,M., Phys. Rev. (1972) 1747 161 Schmid,A. and Sch6n,G., Phys. Rev. Lett.

2