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DETERMINATION OF THE THERMAL
RESISTANCE OF A NORMAL-SUPERCONDUCTOR
INTERPHASE BOUNDARY IN HIGH-PURITY LEAD
P. Bonnard, P. Laeng, J. Suter, L. Rinderer
To cite this version:
DETERMINATION OF THE THERMAL RESISTANCE OF A NORMAL-SUPERCONDUCTOR INTERPHASE BOUNDARY
IN HIGH-PURITY LEAD
Abstract.- The thermal conductivity of a regular, optically-controlled intermediate-state structure has been measured in high-purity lead, the heat current being (nearly) perpendicular to the (static) interphase boundaries. The thermal resistance of a n-s boundary is calculated from the measurements with a good accuracy. The results agree fairly with a theoreticalcalculation by Andreev in the case of electronic heat conduction, but at lower temperatures the measured values are smaller than pre-dicted by Andreev; in the latter case, the limitation of phonon mean free path in the s-lamellas has to be taken into account.
As is well known since the pioneering work by Mendelssohn and Olsen /l/,the thermal resistivity of a type-I superconductor can be larger in the in-termediate state than in both of the superconducting and the normal states. This is due to the reduction of the heat carrier mean free path according to a special mechanism for each type of carriers (elec-trons, phonons). This interpretation was first pro-posed by Cornish and Olsen 111, and was then.develo-ped by Laredo and Pippard /3/ for phononic and by Andreev /4/ for electronic heat conduction. A num-ber of papers on the heat transport in the interme-diate-state structure was ignored. We reconsidered the problem and choosed an experimental situation approaching as much as possible the simple theore-tical model of inter-spaced n and s - lamellas per-pendicular to the heat current. A small sample cur-rent is used to align the interphase boundaries per-pendicular to the electric field E in the n -lamel-las (see figure 1, and References 16,71). The sam-ple is given an U-form in order to prevent the Joule's heat developed in the electric contacts from
flowing through the sample. The intermediate-state structure is studied in a separate experiment in function of the intensities of the heat current q
- > •
and of the electric curren j , using a high resolu-tion magneto-optical technique.
Fig.l : Magneto-optical view of a 0.5 mm thick lead plate, at T = 2.6 K and h = 0.900; superconduction lamellas are dark. The direction of the heat cur-rent is either parallel or antiparallel to the e-lectric-current density j , while the n-s bounda-ries are roughly parallel to the vector E A H . (3 the angle between the heat current and the normal to n-s boundaries, is here equal to the Hall angle (between j and E)
Present address : Laboratoire Suisse de Recherches Horlogeres, 2000 Neuchatel.
Present address: Eidgenossisches Institut fur Reaktorforschung, 5303 Wurenlingen.
JOURNAL D E PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août\91i, page C6-670
P. Bonnard, P.Laeng , J.M. Suter and L. Rinderer
Institut de Physique Expérimentale de l'Université de Lausanne, Ch-lOlS Lausanne, Switzerland.
Résumé.- Nous avons mesuré la conductivitë thermique d'une structure lamellaire statique d'état intermédiaire, contrôlée optiquement, dans^le cas où le courant de chaleur est (presque) perpen-diculaire aux frontières de phase. Nous en avons déduit la résistance thermique d'une interface Normal-Supraconducteur avec une bonne précision. Ces résultats sont en bon accord avec les calculs d'Andreev dans le cas où la conduction thermique est principalement due aux électrons. A des tem-pératures plus basses, nous obtenons des valeurs inférieures à celles d'Andreev, et nous devons tenir compte de la valeur plus faible du libre parcours moyen des phonons dans les lamelles supra-conductrices.
Three steps are required to obtain the resis- tance of an interphase boundary : (i) The thermal conductivity, Ks, is measured as a function of tem- perature.(ii) A magnetic field equal to the criti- cal field is applied perpendicular to the heat cur- rent, and the thermal conductivity Kn(Hc,
l)
,
in the normal state is determined, including the magne- toresistance.(iii) The change of the thermal resis- tance in the intermediate state, when the field is varied slowly enough, is recorded at constant heat current and with an electrical current through the sample, as shown in figure 1. In these conditions the following relation holds between the thermal resistivities and the structure parameters :W,Wn, and W are the measured thermal resistivities of the sample in the intermediate, normal and super- conducting state, respectively. Wn is I / K ~ ( H ~ , ~ ) .
The first member of Eq.(l) is the derivative, with respect to the reduced field h, of the difference between the actual (i.e. with boundary resistance) and the calculated (without boundary resistance) resistivities of the lamellar structure represented in Figure
1.
The demagnetizing coefficient is set to 1, i.e. the concentration of the normal phase is assumed to be equal to h. In the second member of the Eq.(l), Wns is the quantity we are interes- ted in, the thermal resistance of a n-s interphase boundary, defined as the temperature jump at the boundary divided by the heat current density per- pendicular to the boundary,q
cosB
( B2140, see Figure 1). Finally, n is the experimentally de-ns
termined number of interphase boundaries ; in the range 0.85 < h < 0.99
,
nnski]_7
=I560 (1-h) for our 1-nun thick sample, independent of T in thedn
range 2.1 K < T <4.2 K.
2
is more accurately de- dhtermined than n itself, and the same applies to ns
d
-
(W-Wn) calculated from the recorded resistance d hcycles. Therefore, Eq (1) was written using deriva-
Fig. 2.:Thermal resistance of a n-s interphase boundary
tes systematically from the data points at lower temperatures; the dashed curve was calculated from
Andreev'sexpression : 2 ~ ~ 1 2 ~
w
=- 1 A(T) ns f o p n(T)+2
)
Tv:
exp (-) T 2 with : fo adjusted at 2.23, po = cm A(T)the gap
A
(T) = 15.46(rn) BBCS (Kelvin) The discrepancy between theory and experi-' ment is of cours due to the occurence of phonon conductionin
the s-lamellas which is the dominant conduction mechanism below 1.5 K.ow ever;
in cor- recting Eq. (1) for phonon conduction, the reduc- tion of the phonon m.f.p. in a s-lamella (with res- pect to the bulk s-state) has to be taken into account; this reduction which is estimated to amount to a factor 15 at 1.5 K in our sample is due to the presence of two adjacent n-lamellaswhere the phonons have a much shorter m.f.p.
tives with respect to h.
In
figure 2 the experimentally determined n-s boundary thermal resistance W is plotted versusns
References
/ I / Mendelssohn,
K.
and Olsen,J.L., Phys.Rev.80
(1950) 859/2/ Cornish,F.H.J. and Olsen, J.L., Helv.Phys.Acta 26 (1953) 369
-
131 Laredo,S. J. and Pippard,A.B., Proc.Cambridge Philos.Soc.
2
(1955) 368/4/Andreev,A.F., Sov.Phys. JETP
