ELECTRICAL RESISTANCE OF A NORMAL-SUPERCONDUCTING BOUNDARY

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ELECTRICAL RESISTANCE OF A

NORMAL-SUPERCONDUCTING BOUNDARY

Y. Krähenbühl, R. Watts-Tobin

To cite this version:

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JOURNAL DE PHYSIQUE _{Colloque C6, supplkment au no 8, Tome 39, aolit 1978, page}

~6-677

E L E C T R I C A L R E S I S T A N C E OF A NORMAL-SUPERCONDUCTING BOUNDARY

Y. .~r'&enbGhl and R. J. Watts-Tobin

Department of Physics, University of Lancaster, Lancaster LA1 4YB, EngZand.

Rdsum6.- Nous avons calculd la rdsistance de surface 1 partir de la thdorie microscopique pour des supraconducteurs impurs et obtenu une ddpendance en tempgrature correspondant bien aux rdsultats ex- pdrimentaux.

Abstract.- We have computed the boundary resistance from microscopic theory for dirty superconductors and get good agreement with the experimental temperature-dependence.

Harding et al./l/ have shown by measurements on superconducting-normal-superconducting sandwi- ches that there is an extra electrical resistance at a NS boundary when a.current flows

through^

it. Schmid and schEn/2/ have shown that this effect can

be analysed by solving the following Boltzmann equation for the change Gf(E,x) in the electronic dis- tribution function where E is the energy and x the distance from the interface :

Here D = 113 v R is the electronic diffusion coef-

F

ficient,

A

is the superconducting order parameter, M = N12+ N~~

,

and N1 = Re(a), N2 = Re(B) where a and 8 are the retarded Green's functions of Usadel /3/in the notation of referencel21. K is the elec- tron-phonon collision integral. The theory is for dirty superconductors and should be valid for some of the samples used in referencell/.

Because the operator K mixes the solutions of the Boltzmann equation for different energies, equation(1) is prohibitively hard to solve dir ctly.

&

At high temperature Ri<<C(T) where Ri = (DT~) is the inelastic diffusion length for electrons, 'rE

being their inelastic collision relaxation time, and

5

is the coherence length. In reference121 there are approximate calculations of the extra resis-

tance and we think they are only valid in the high temperature regime. At Zow temperature the second

term in equation(1) is much larger than the right- hand side. Ovchinnikov/4/ has given formulae valid

in this case but has not solved them. There is a range of intermediate temperatures in which we

find that an approximation to the form of K in a homogeneous superconductor given by Schmid/5/ can be used to solve the problem. The upper bound of

this temperature range is dictated by the condition R.>>C(T) but there is a lower bound to be defined later. The basic problem is to compute the logarith- mic derivative y =

-

in the normal metal : from

N

ax

yNthe extra resistance can be calculated simply/4/. In the intermediate and low temperature regimes yN may be computed by numerical integration of the coZ- ZisionZess Boltzmann equation across the interface,

provided we know the corresponding quantity yS on the superconducting side.

With the approximation to K given in reference/5/, the Boltzmann equation in the homogeneous superconductor is

[DM

-&

-

2AN2

-

6f (E) = I(E)

E

I

where

@

/

In reference151 it is shown that if there is a solution of equations (2) and (3) of the form Gf (E, x) = C (E) exp (-Xox) _{( 4 )} where ho is independent of E, then

J

~E'N~~(E') 1

-DM(E')T~, L~osh' (E' /2T) h' (El )-loL

-

where

We find that it is only possible to obtain

a solution X of equation (5) provided it satisfies

0

the inequality Ao<X(E) for all E. One can approach

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the problem in another way161. If one assumes that I a Ox-'. and then solves equation(2), one obtains a solution of the form (4) only if ho<A(E). When ,this form of solution is valid, we have y

S =

-

XO.

?he condition <X(E) for all E provides the lower

0

bound to the intermediate temperature regime.

The experimentsof reference111 to which this dirty limit theory is most applicable are those on

Pb-Bi alloys. Taking the value .r = 10-"s for Pb

E

quoted in reference121 we find the condition R.=c(T)

1 which separates the intermediate and high temperature regimes occurs at T = 0.96 T

.

By computing solu- Cions of equation(5) we find that the condition Xo<X(E) is satisfied down to T = 0.68 T

.

In the in-

termediate temperature regime our computations of the extra resistance R give R (I-T/T~)-' with p = 0.63. This exponent fits the measurements of Hardingl71 on Pb-3 % Bi and Pb-15 % Bi very well even above 0.96 T as is shown in the figure 1.

Ovchinnikov/4/ took a simplified model of the interface and obtained the exponent p = 0.75 by analytic methods

.

I

A Expermental values for 15% 8 1

+

Exper~rnental values for 3% 0 1

,,

-Dart) 11m1t theory

Fig. 1 : Log-log plots of the extra resistance R(T) against I-T/Tc. The theoretical lines correspond to p = 0.63. The experimental points are due to Harding 1 7 1 .

References

/ 1 / Harding,G .L., Pippard,A.B. and Tomlinson, J.R.,

Proc. R. Soc. London A

340

(1974) 1

/2/ Schmid,A. and ~chEn,~., 3. Low Temp. Phys. (1975) 207

131 Usadel,K.g., Phys.Rev. Lett.

2

(1970) 507 141 Ovchinnikov,Yu.N.,J. Low Temp. Phys.

8

(1977)43 151 Schmid,A., Preprint

161 Kramer,L., private communication

I71 Harding,G.L., Ph. D. dissertation, Cambridge

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