Solid State Communications, Vol. 11, Pp. 1393—1395,1972. PergamonPress. Printed in GreatBritain

MAGNETIC FIELD DEPENDENCE OF THE SUPERCONDUCTINGFLUX FLOW RESISTANCE MINIMUM*

W.C.H. Joiner andJ. Thompson

Physics Department,University of Cincinnati, Cincinnati, Ohio

(Received 17 August 1972 by P.G. de Gennes)

Measurements havebeenmade in lead alloys of the field dependenceof the temperature,T~10,at which the superconductingflux flowresistivity passesthrough a minimum. When the dataare expressedin termsof the reducedvariables tm~ = Tmjn/Tc and h = H/H02 (0),tmirl is the same linear function of h for all samples.The slopedt,njn/dh = —0.72, indicating a stronger influence of magneticfieldon t~, than was obtained in arecentcalculation by Chow.

IN THEIR early measurementso~flux flow re- normalizedto the resistivity value in the normalsistancein superconductors,Kim et at. observed state, ~ is PF/PN = H/H02. For non-zerovalues

that the flux flow resistivity at fixed field passes of C,througha minimum at a temperature,Tmm,where = _~_[i + CF(T)1

1. (2)To> Trnjn> 0. p~. H

02

Clem2 showedthat the source of this mini- In an earlier paper3we haveshownquali-

mum was a thermal dissipationassociatedwith tatively the correctnessof Clem’s results. Usinga moving fluxoid which was in addition to the equation(2) andthe calculated valuesof the

Joule lossesoriginally describedby Kim. Each coefficient C, the resistanceminimum shouldloss mechanismcontributesto the viscosity disappearfor 1/~~o>1/4, correspondingto C =

coefficient associatedwith a moving fluxoid. 0.5. In the PbT1 alloy systemthis ratio is achievedThe temperaturedependentthermal viscosity for Pbg

5Tl0~,and we showedthat at this compo-wasfound to be sition the resistanceminimum disappeared,while

Thh(T)/~(O)= CF(T) (1) for higherTI concentrationsandshorter 1, theminimum was present.

where ~th(T) is the thermal viscosity coefficientat temperature,T, ~(0) is the total viscosity Certain points of discrepancyremain, how-coefficient at zero temperature,F(T) is a tem- ever. For example,the observeddepth of theperaturedependentfunction calculatedby Clem,

2 minimum wasfound to be significantly largerand C is a coefficient also tabulatedby Clem, thanpredicted. This is not surprisingin that the

and which dependson the ratio of electron mean expressionfor the viscosity due to Joule lossesfree path to coherencelength, ~ As this ratio is probably an over-simplification of the actualdecreases,C increasesin value from 0 to 1. For situation.4A more serious error lies in the factpure materials, I is large, C = 0 andthe resist- that Clem’s model treatsan isolatedcylindricalivity is due to Joulelossesalone. It is then fluxoid, andas a result, the flux flow minimumassumedthat the flux flow resistivity, PF, occurs at a temperaturewhich is independentof

the magneticfield. In practicefluxoids interact,* Supportedby NASA Grant NGR 36-004-054. with the result that the temperatureat which the

1393

1394 SUPERCONDUCTING FLUX FLOW RESISTANCE MINIMUM Vol. 11, No. 10

minimum occurs, Tmjn, dependsupon the rnag- Table 1. Samplecomposition,T~,H02(0) andnetic field. K values

To our knowledgeonly one attempt hasbeen Sample T~ H02(0) K

madeto refine Clem’s independentfluxoid model. composition (°K) (oe)In this calculation

5a field dependenceto Tmm Pb90Tl10 6.96* 2975* 1.9*

is found by consideringthe interaction of fluxoids Pb60Tl~ 5.87* 6600* 5.8*through theelectric fields associatedwith their Pb50In50 6.39~ ‘745O~ 4.6tmotion. Thus if the penetrationof theelectricfields due to thesurfacechargedistribution of * Reference6.the normal core of one flux line into the normal t Reference7.core regions of its neighboringflux lines is takeninto account, the viscosity coefficient is modi- 02$

fied to 0.05:

~(T) = ~(0) ~ — ~ H ± CF(T) (3) 0.26

H02(T) .,

and —~ ~ h~.aso0.24

p~(T) = H 1 — ~__H + CF(T) (4)

p~ H~2(0) H02(T) 0.050 022

where y is a positive quantity dependingon the N

structure of the fluxoid lattice, and is only weaklyfield dependent.Chow

5 estimatesy to be of the ~ 020

order of 10’. The additional term yields a mag- ‘.

netic field dependenceto Tmjn. ‘. I0.045

Unfortunatelythere is insufficient data onthe field dependenceof Trnjr, to make a mean- ringful comparisonwith the above theoreticalresults. In the presentwork we have madea FIG. 1. Normalized flux flow resistivity pp/pN

detailed study of Tmjn as a function of H in as afunction of absolutetemperatureT for the

threealloys, Pb90Tl~0,Pb60T140,and Pb501n50. two reducedfields h = H/H02(0)= 0.053 and

The principal result of this study is than when 0.250. The sample i3 Pb60Tl~.Note the scaleof the resistivity axis is different for the twothe dataarecomparedin terms of the reduced curvesvariables, tmjfl = Tmjn/Tc and h = H/H02(O),~ is the samelinear function of h for allsamples.Moreover the field dependenceis sig- Two typical curves showing the resistancenificantly strongerthan expectedfrom equation minimum in Pb60T140aredisplayed in Fig. 1,(4). where we haveplotted pp(H, T)/PN as a function

of the absolutetemperature,T, for two valuesof

Samplepreparationand measuringtechniques reducedfield, h = 0.053 and h = 0.250. Note thathavebeendiscussedpreviously.~ Two samples the scales on the ordinate are different for theof eachcomposition were studied, and in each two curves. It caneasily be seenthat ~ de-casethe results obtainedfor ~ at a given creasesas h increases.We would also like tofield and temperaturewere identical within our point out that the reducedresistivity PP/PN isexperimentalerror. For reasonsdiscussedin our greaterthan H/1102(0)at our lowest temperature,

previouspaper3the absoluteaccuracyin the ratio and appearsto be increasingrapidly as T de-

p~/p~,is probably no better than 5 per cent, while creases.Thus theexpressionfor the resistivitythe relative accuracybetweentwo points is closer due to Joule losses at low fields and low tern-

to 1 per cent. Values of H02(0), T0, and the peratures,pp/°v = H/H02(0), is not valid for

Ginzburg—LandauparameterK are given in Table 1. thesealloys.

Vol. 11, No. 10 SUPERCONDUCTINGFLUX FLOW RESISTANCE MINIMUM 1395

value of C, which determinesthe strengthof the

thermal dissipation.

The curve drawn through the points in Fig. 2is a leastsquaresfit to the collective data, and

j hasa slopedt.~/dh= —0.72. Choosingy = 1.0,

ten times his original estimateof this constant

0.2 in order to maximize the slope, Chow calculates

dt,0~,,/dh= —0.4. More realistically smallervalues

.~ of y would yield an evenweaker field dependence.h It seemsapparentthat the interaction between

FIG. 2. Values of ~ = Tmjn/T0 asa function fluxoids consideredby Chow, although accountingof h = H/H02(0) for all samples.~ — Pb60T140, qualitatively for the field dependenceof ~ isx — Pb90Tl~0,0 — Pb50Tl50.The solid curve is insufficient in itself to accountquantitatively fora least squares fit to the data and yields dtmjn/ our results. However, as noted above, the assumeddh = 0.72. expression for the Joule losses is also violated

at low temperatures on these samples, indicating

In Fig. 2 we show the compilation of our that further corrections to this term must also beresults of ~ vs. h for all threesamples.With- considered.It would be interestingto know if the

in the accuracyof our measurementsthedata linear relationshipbetween t,~,,and h persistsfrom the three samples superimpose on a single to lower temperatures, as shown in the dashed

curve, indicating that the relationshipbetween portion of the curve in Fig. 2, since according tot0~and h is independent of I, andhenceof the Clem, the thermal dissipation should becomezero

at T=0.

REFERENCES

1. KIM Y.B., HEMPSTEADC.F. and STRNADA.R., Phys. Rev. 139, A1163 (1965).

2. CLEM-JOHNR., Phys. Rev.Lett. 20, 735 (1968).

3. AXT CARL J. and JOINER W.C.H., Phys.Rev.Lett. 21, 1168 (1968).

4. HUCHIA-REN and THOMPSONRICHARDS., Phys.Rev. B6, 110 (1972).

5. CHOWW.S., Phys.Rev. 188, 783 (1969); Phys.Rev. Bi, 2130 (1970).

6. AXT CARL J. and JOINER W.C.H., Phys.Rev. 171, 461 (1968); AXT CARL J., Ph.D. Thesis,University of Cincinnati (1968).

7. FARRELL D.E., CHANDRASEKHARB.S. and CULBERTHARVEYV., Phys. Rev.177, 694 (1969).

Messungender Feldabhãngigkeitvon der Temperatur Tmjn, bei derder Strömungswiderstanddes supraleitendenFlussesem Minimumhat, wurderi an Bleilegierungendurchgeführt. Wenndie ResultatealsFunktionender reduziertenVariablen t~, = Tmm/TC und h = H/H02(0)ausgedrückt werden, dann ist. t,,~ dieselbelineare Funktion von hfür alle Proben.Die Ableitung dtmm/dh hatdenselbenWert, —0.72,fur alle Proben,em Befund,der auf einenstärkerenEinfluss desmagnetischenFeldesauf ~ hindeutet, als die jiingsten Berech-umngen von Chow ergeben haben.