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THERMOELECTRIC BRANCH IMBALANCE
CURRENTS IN SUPERCONDUCTORS
D. Heidel, J. Garland
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplt!ment au no 8, Tome
39, aofit
1978, page C6-492THEWOELECTRIC BRANCH IMBALANCE CURRENTS I N SUPERCONDUCTORS
D.F. Heidel and J.C. Garland
Physics Deparhent, The Ohio S t a t e University CoZwnbus, Ohio, U.S.A. 43210
REsum6.- Un gradient de tempsrature appliqu6 B un supraconducteur produitun courant d'excitations de quasiparticules accompagns par un contre-courant superfluide. Les solutions des Equations cinE- tiques pour les quasiparticules en prssence de ce courant superfluide montrent l'existence d'un Etat hors dlEquilibre dans le spectre d'excitation des quasiparticules. Ce dEsEquilibre est de l'ordre de v et donne un terme d'ordre vs2 au courant de quasiparticules.
Abstract.- A temperature gradient applied to a superconductor produces a current of quasiparticle excitationswhich is accompanied by a counter-flow of superfluid current. Solutions of the quasipar- ticle kinetic equation in the pr6sence of this superfluid current show the existence of a branch imbalance in the quasiparticle excitation spectrum which is of order v and which contributes to the quasiparticle current a term of order v
-
A temperature gradient applied to a bulk iso- G =
k
-
FkO tropic superconductor will produce a current of nowwhere
equilibrium quasiparticle excitations
5.
Because-
-
Fko = [exp(8Ek) +
g-'
( 6 )Sn
is always accompanied by a counterflow of super-4
current it is important to allow for the influ- FkO should be distinguished from the equilibrium
S'
ence of condensate motion on the quasiparticle dis- distribution F k given by
tribution function. We have investigated the effects FkO = Lexp(~'k) +
Q-'
(7)of a uniform superfluid flow on the solution of the quasiparticle kinetic equation. Our primary finding is that an imbalance between the electron-like exci- tation spectrum is induced by the superfluid flow. This branch imbalance /l /, denoted by Q ~ , is first order in vs and contributes a term to the normal current of order vs2. This branch current may beco- me important near the superconducting transition
temperature where vs becomes divergent 121. In the presence of a uniform superfluid flow the excitation spectrum is given by 131.
In the presence of a temperature gradient Equation
where the collision operator has been separated into a quasiparticle impurity operator and a qua- siparticle-phonon operator.
The expression for the total current density
+
J (i.e., that carried by the quasiparticles and by the condensate) is + - + + J = e
1
[Uk2~k + vk2 (1-
F-~)J (vk + vs) (9) k-
+ -f Ek = Ek + Mk*vs, (1) where Uk2 = 1where Ek is the equilibrium excitation energy /2(1 +Sk/Ek) ; vk2 ='/~(I-E~/E~). (10) Ek =
kk2
+ n q 1 / 2and where
5,
= ($k12/2m-
1.1(2) Using Equations (5) and (IO), Equation (9) can be written
-+ -+ aFko
- + +
-+E,
(3) J=nevteI k (-1 aEk
(6
k .vs)vk+e GkGk+GS 1e(--)Gk Ek (I I) The steady state kinetic equation for the quasipar- The first two terms of Equation (1 1) correspond to ticle distribution function Fk with allowance for the superfluid current Ss, while the last two+
the motion of the condensate
-
-
/4/ is terms represent the thermoelectric current Jn. The gEk.
VrFk-
VrEk.
%Fk +I C F ~ I
= 0 ( 4 ) third term in Equation (11) resembles a normal me- where I{F } is the operator that describes the col- tal thermoelectric current. In a superconductor,k
lisions of the quasiparticles with impurities and however, this current must be calculated by allo- + phonons. A non-equilibrium distribution function wing for the dependence of Gk on vs. The fourth
X+
Gk can be defined as term in Equation (11) can be written as Q vs where
Q X , the charge-weighted branch imbalance, is defi- ned by /l/
Q= = e
1
(Sk/Ek)Gk. (1 2) kThis term produces a current whenever a branch im- balance exist in the superconductor.
-+
If v is assumed to be small enough to satis- P v
F S
fy the condition (p)l , <<the left hand side of
A -+
Equation (8) can be expanded in powers of vs. The solution to the zeroth order term,previously stu- died by GalTperin et a1.,/5/ can be completely relaxed by elastic scattering.The symmetries of
-+
this solution with respect to k and
Sk
can be seen in figure la), where the distribution function is plotted along the direction defined by the tempera- ture gradient. Gal1perin et al.have shown for T near Tc, the quasiparticle current due tothis dis- tribution is approximately equal to the thermoelec- tric current in the normal state at T = T.
Fig.l : Plot of Gk along axis defined by the tempe- rature gradient : (a) zeroth order solution,(b) G1 s (C) G2.
The solution to Equation (8) that is of order v 2 has been studied by Aronov 161, who has shown that the resulting current cannot be relaxed by impurity scattering. Altohough the current propor- tional to v is smaller than that produced by the zeroth order solution by a factor (v P /A)~(T S F ph /
Timp), this current can still be important since
Where
(14)
In Equation (15),r is the branch imbalance relaxa-
Q
tion time. The symmetries of these components of G are shown in figure Ib) and lc).G1 represent a
k
net decrease in the total number of excitations, equivalent to a small increase in the magnitude of the energy gap 161. G2 produces the branch imbalan- ce current. A numerical integration of G2shows that G represents a net excess in the number of elec-
2
tron-like excitations over the number of hole-like excitations. The ratio of the branch imbalance cur- rent to the zeroth order current is approximately
V s P ~ 'Q
( T ) 2
C
.
Since rQ and v are both divergentimp
near the transition temperature /l/, this current can become important for T": T
C'
References
/l/ Tinkham,M,; Phys. Rev. B
6
(1972) 1747. /2/ Garland,J.C., and Van Harlingen,D.J.,Phys.Lett.47A (1974) 423.
131 de Gennes,P.G.,Superconductivity of Metals and Alloys, (W.A.Benjamin,New York, 1965)
/4/Ga11perin,Yu M.,Gurevich,V.L., and Kozub,V.I., Zh. Fksp Teor .Fiz 65 (1 973) 1045, (Sov.Phys .-JETP
-
38 ( 1 9 7 4 ) w ) F
-
-
The solutions to the first order term in Equation (8) in the relaxation time approximation are
= Gl + G2
