Low Temperature Physics - LT14 [2]

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Citation preview

y

) \

I

'rVol ~ 'Volume 2

/

E1ac-Lachlan, R. Mailfert, B. Sou f fach e , and J. Bli!'ger

40

THE MAGNETIZATICN AND UPPER CIUTICAL FIELDS OF PALLADIUM

PRINTED IN HELSINKI, 1975, BY T:MI

P.

AHONEN

HYDRIDES.

D.S. McLachlan, T.B. Doy l e ,

and J. Burg e r

44

v

iv I'

\. THIN FILMS. J. Igalson, L. n Sniadower, and A.J. Pindor ......•..••.. .. ...•....•..

I

SUPERCONDUCTIVITY IN PdH

"I 48

TUNNELING EXPERIMENTS ON SUPERCONDUCTING PALLADIUMDEUTERIUM ALLOYS.

A. Eiehler, H. Wtihl, and B. Stritz-

ker . . . . . . . . . . . . . .. ... .• . . . . . . . . . . . . . . . . • . . . . . • . . . • . . PREPARATION AND SUPERCONDUCTING PROPERTIES OF WIRES.

/)

~1oN

L. Sehultz, H.C. Frey-

hardt, R. Bormann, and B.L. Mordike . . . . . . . . . . . . . . . . .

59 63

CAN THE 're OF SUPERCONDUCTING Be BE ENHANCED? A. Leger, and S. de Cheveigne '"

67 71

MINE SPIN SUSCEPTIBILITY.

P.M. Tedrow and R. Meservey

H. Balster and J . Wi tUg . . . . . . . . . . D.U. Gubser and A.W. Webb . . . . . ,.

1 09 113

THE EFFECT OF UNIAX IAL STRESS ON THE SUPERCONDUCTING TRANB.R. Roth-

berg-Bibby, D.S. MeLaehlan, and F.R.N. Nabarro .. . '.,

117

SITION IN DILUTE INDIUM- TIN ALLOY SINGLE CRYSTALS. M. Skove and H.Ra Ott . .. ............. .. .... ........ .......... . . . . ........ ........ ..

MEASUREMENT OF He 11 IN SUPERCONDUCTING VANADIUM TO DETERON THE PROPERTIES OF AMORPHOUS VANADIUM ALLOYS.

LOW TEMPERATURE.

PRESSURE EFFE CTS ON THE SUPERCONDUCTING TRANSITION TEMPER-

'ANISOTROPIC LENGTH CHANGES AT THE SUPERCONDUCTING TRAN-

J. Klein,

................... .

105

PRESSURE-INDUCED h~TTICE I NSTABILITY IN FCC LANTHANUM AT

SITION OF TIN WHISKERS AND OF BULK TIN.

SUPERCONDUCTING PROPERTIES OF HOMOGENEOUS AND HETEROGENEOUS LEAD-SODIU11 ALLOYS. H. Freyhardt and H. Culber t

1:

ATURE OFALUMINUM.

RESISTIVITIES OF zrxNbl_2xMOx ALLOYS. R . Fltikiger , M. Ishikawa, and R.L. Cappelletti .... .. ... .... .... , ... .

R. L. Fil l er, P. Lindenfeld, an d

G. Deutseher .. .. .... . . . . . . . . . . . . . . . . . .. .. . . . . . . .... .

PRESSURE AND VOLUME EFFECTS 55

SUPERCONDUCTIVITY IN POWDERMETALLURGICALLY PRODUCED COMPOSITES OF CuNb, CuV AND CUSnNb.

UMAND THEIR RELATION ci'O THE SUPERCONDUCTING TRANS 1TION 'rEMPERATURE .

52

AND MoC

H. Bauer, E. Saur, and D. Seheehinger

HEAT CAPACITY AND THERMAL CONDUC'l'ION IN GRANULAR ALUMIN-

121

75

J. Hass e

and K. Weber

79

INTERMEDIATE AND MIXED STATES

81

CRITICAL MAGNETIC FIELDS IN FIRST KIND SUPERCONDUCTING

EVIDENCE FOR THE ISOTOPIC VOLUME EFFECT IN SUPERCONDUCTING MOLYBDENUM.

T. Nakajima, O. Terasaki, and S. Hosoya

FILMS.

STUDY OF A SERIES OF SUPERCONDUCTING V(l_x)Pt x COMPOUNDS BY NUCLEAR MAGNETIC RESONANCE . L.A.G.M. Wulffers, N. J. Poulis, H.R. Khan, and Ch.J. Raub . . • . . . . . . . . . . . . . ANOMALOUS TEMPERATURE DEPENDENCES OF He2 AND

~

TRANSPORT PROPERTIES IN THE INTERMEDIATE STATE. 85

89

R.H. Dee

. ................... ............ .... .... .

129

TURE.

J . M. Suter, L . Rinde r er , and'A.R. Sweed1er . ..

133

STATE IN SUPERCONDUCTORS. H.D . Wiederick, B.K. Mukherjee, and D.e. Baird .. . . . . . . . . . . . . . ... ..... .. .... .

SUPERCONDUCTIVITY AND PEIERLS INSTABILITY IN COUPLED LIN SPECIFIC HEAT AND SUPERCONDUCTIVITY OF Th H and Th D . 4 15 4 15 C.B. Satterthwaite and J.F. Miller

~

THE a.e. RESISTANCE OF 7HE CURRENT-INDUCED INTEffi1EDIATE

LAYER SUPERCONDUCTOR, 2H-NbSe . N. Kobayashi, K. 2 Noto, and Y. 11u to . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . ' B. Horovit z and A . Bl n bolm .... . .

Guerlault..... .. ........ .. ........

TORS: SIZE EFFECT IN A LONGI'rUDINAL LAMELLAR STRUC-

SPECIFIC HEAT AND STRONG ELECTRON-PHONON INTERACTION IN

EAR CHAIN SYSTEMS.

and A.M.

125

INTERMEDIATE STATE THEffi1AL CONDUCTIVITY OF SUPERCONDUC-

IN LAYERED

SUPERCONDUCTOR, 2H-NbSe 2 . Y. Muto, N. Toyot a , H. Nakatsuji, N. Kobayashi, and K. Noto . . . . . . . . . . . . . . . .

A. Rogani and E. Tabet • . . . . . . . . . ..... .. . . .. .

137

KINETICS OF THE CURRENT INDUCED DESTRUCTI ON OF SUPERCON" 97

DUCTIVITY IN A TYPE-II SIJPEP.CONDUCTOR.

B. Makiej, S.

GoXab , A. Sikora, E. Trojnar , and W. Zaeharko . . . . . . . 101

141

INVESTIGATION OF THE SOURCE S OF ASYMMETRY EFFECTS IN THE VELOCITY OF DESTRUCTION OF SUPERCONDUCTIVITY BY CURRENT IN TYPE 11 Ta WIRES.

W.C. Overton, Jr .

.. . . . . .

145

Vii

yi

M. Sheikholeslam, M.R. Esfand ia ri , A. Gilbert, J.P .

OBSERVATIONS OF ASYMl:!ETRY . IN '1'lg:', VELOCg;¥,: QF ,[m pTRUCpON; CU,R,RE~~

OF SUfERCONDUCTIVITY . BY W.C~

THE

Romagnan, J . P . Laheurte , and J . C. NOi L'ay ... . . . . . . . .

IN ,TYPHII; ~T,il. WIRES ~ ~

Overton, Jr., H. Weif\stock" and A" F ,., S ,c h)lch •• ', ' " • . •. . . . ' . r . • • ,. .,."

~ARAMAGNETIC

I

EFFE;~T

,IN_ ,T~P.E-. I

Rothen and C. Lievre •

0

0

0

;

_.'. _ _

SUPERCONDUCTO~p.

••••

0

•

•••

0

••

•••

0

••••••

•

149

THE SUPERHEATING LIl1IT.

F.

·r.

••••

153

MEASURE~ffiNT

INDUCED TRANSITION IN LONGITUDINAL MAGNETIC FIELD. M. Sllgahara ..•••

0

••••••••••

••

• ' • • •; .

' ,

'

"

0

••••••••••

P. Laeng, P ,.

Z~eiacker"

'

; '

, ' "

.1.5 6

1 95

J-P . Girard, E. Paumier, a n d A. llairie . , ....

199

OF THE LONDON PENETRATION DEPTH IN Cd , Sn

AND Zn CRYSTALS .

E.R. Dobbs and M.J. Lea . . . . . . . . . .

203

or:~E

FLUX FLOW VELOCITY AND RES,ISTIVITY MEASUREME,NTS IN INTERMEDIATE STATE.

\

H. Parr . . . . . . . . •. ... . ... ..

STUDY OF THE MAGNETIZATION OF TYPE I SUPERCONDUCTI NG BARS.

A NEW MIXED STATE OF SUPERCONDUCTORS _I~,:,THE CURRENT-

191

FIELD DEPENDENCE OF THE PENETRATION DEPTH IN TIN UP TO

an,~

L,' , 160

THIN FILMS AND PROXIMITY EFFECT

~

164

SUPERCONDUCTING TUNNELING INTO F ILMS WITH VOLTAGE STEPS.

W. Rodewald •.

167

K.E. Gray .... .. .... . . .. .. .. .. .. . · · · · · · ··· · · · · · ·· ··· DYNAM ICS OF THE CURRENT-INDUCED RESISTANCE IN THIN- FILM

Rinderer •••... , • . . • ,.... . .•••....•. •.. " .. . . . •.. . .... •" ., . TWO-DIMENSIONAL MIXED STATE OF , TYPE-I SUPERCONDUCTORS. I.L. Landau

.. .. .. .... .... .. .. . . . . . . .. . . . . .. _~ .. . , . . . . . . .. . .

,e. .... .. ..

t.

..

!>. " .... f

... ,

~

DIRECT OBSERVATION OF THE TWO-DIMENSIONAL M,IXED STATE eN THIN FILMS OF TYPE I SUPERCONDUCTORS.

TYPE -I SUPERCONDUCTORS.

BOUNDARY ENERGIES OF SUPERCONDpCTING TANTALUM AND NI,OBI,l!M. U. Essmann ••••••••••••..•••.•••••.•.•.••• •. •••.••.•. • . f .~

!

.

.

171

R.J. Watts-Tobin

Pb FILMS WITH QUAN TIZ ED RESISTANCES.

and S. Imai .•.•...••.•... •••••• ••.••.••...•••..••... 175 ~.-:~-;.\r~: :i : ;\1'\ j:I;', . ',~, ·c!':'· ~ :'ii!., OBSERVATIONS OF CRITICAL CURRENTS INDUCING MOTION IN ., LANDAU DOMAIN STRgCTUR.¥!S.. T. Miyazaki

............................

~,

Aok).,

..

.e ,. , S.1~i!l~ .za~i,

f 3d.

-.-................................................ . . ']

.t·.-(t.".

'

.'

;:

,

R . P.

Huebener . . . . . . . .. ..... . . . . . . . ... . . .... .... . . . . . . . . .

211

EFFECT OF ELECTROMAGNETIC RADIATION ON SUPERCONDUCTING

CALCULATIONS OF THE THERMAL CONDUCTIVITY IN THE MIXED STATE OF A DIRTY SUPERCONDUCTOR.

D . E . Chimenti and

207

-,

179

215

MAGNETIC-FIELD-INDUCED RESISTANCE IN SUPERC ONDUC TORS. M. C. Leung . . ... • . . . . . . . . . . . ..... .. . .. .....

. ..........

219

MAGNETIC FIELD PENETRATION I NTO SUPERCONDUCTING TI N FILM CYLINDERS.

. !

L . M. Geppert,

R.L. Thomas, and J .T. Chen .. . . . .. .. .. ... . • . . . . . . . . .

E.G. Wi1son and W.M. Fairbank . . . . . . . . . .

223

ON WEAKENING OF SUPERCONDUCTIVITY BY RESONANCE SCATTERING

.

NUCLEATION PROBLEMS

AT SURFACES.

J. Ha1bri tter .. . . . . . . . . . .. .. .. . . . . . . .

227

THE EFFECT OF ELECTRON REFLEXION AT THE ,INTERFACE ON THE DYNAMIC STUDY OF THE TRANSITION OF A SINGLE SUPERHEATED SUPERCONDUCTING GRANULE TO THE NORMAL STATE.

SUPE RCONDUCTI NG TRANSITION TEMPERATURE OF SUPERIM-

C. Va-

lette and G. Waysand . . . . . . . . . . . . . . . . • . . . . . . . • . • . . . . .

POSEiJ FILMS. 183

AN EXACT STUDY OF MAGNETIC SUPERHEATING OF THIN CYLINDERS IN AN AXIAL FIELD FOR SMALL ari and H.J. Fink

1(-

VALUES.

"

SUPERCONDUCTING NUCLEATION FIELDS OF A NORMAL-SUPERCONDUCTING DOUBLE LAYER ' OF \.

'"",

:t '''(.'

F~~~TE tt'

'TH'i hNES'S : " H.J. Fink, ,",

.'t.l"'

.

I"'::,

INTERPHASE SURFACE ENERGY PARAMETER IN LEAD FILMS: PERATURE P.ND THICKNESS DEPENDENCE .

M. R. Esfandi-

......................................................................,

J. R. Hook . . . . . . . . . . . . . . . . . . . . ... .. .. .

Sninozaki, and R. Aok i 187

231

'!'E M-

T. MJyaz ak i, b .

. .. . . . . . . . . .. . . . . . .. . . . . . . . . .

235

CRIT I CAL CURRENTS OF SUPERCONDUCTING THIN Ir>iDIUM FILMS AND EFFECTS OF MACROSCOPIC INHOMOGENEIT IES. N.Y a. Fogel, A.A. Moshp.nskii, A .M. Glukhov, L . P . Ti shchen ko, Ya . M. Fogel and I . M. Dmi tren ko .. . . . . . . . . . . . . . . .

23 9

viii ix

OBSERVATION OF A LOCALLY INDUCED FLUX FLo\v PHENOMENON IN THIN TYPE I SUPERCONDUCTING FILM CROSSINGS. I. Iguchi

TWO-DIMENSIONAL SOLUTION OF THE GORKOV EQUATIONS FOR

243

CLEAN TYPE-II SUPERCONDUCTORS FOR ARBITRARY T AND B.

AC QUANTUM INTERFERENCE IN SUPERCONDUCTING FILMS WITH PERIODICALLY MODULATED THICKNESS.

E.H. Brandt ........ ...... ....... . ......... . . . . . . .

P. Martinoli, O.

Daldini, C. Leemann, and E. Stocker .........•... ..• •

247

TYPE-II SUPERCONDUCTORS.

I-V CHARACTERISTICS OF A SUPERCONDUCTIVE FILM I'iTITH A LOCALLY APPLIED MAGNETIC FIELD.

DEFECTS AND FLUX LINES.

251

FAR INFRARED ABSORPTION OF THIN LEAD FILMS IN A PERPENDICULAR MAGNETIC FIELD AND UNDER PROXIMITY-EFFECT CONDITIONS.

J.\v. Hendriks, H.R.Ott, and P. Wyder ...•

SUPERCONDUCTING PROXIMITY AND THE TOMASCH EFFECT.

255

Pb/Sn SYSTEM.

J.R. Hook and J.A. Battilana ....•....

259

DIUM. 261

IN TITANIUM-VANADIUM ALLOYS.

J.C. Ho and E. Collings

R.J. Hembach, F.K. Mullen, and R.W. Genberg ..

H. Teichler, and

w.

313

H. Kiessig, U.Essmann,

Wiethaup . . . . . . . . . . . . . . . . . . ..... .

265

BEHAVIOUR OF A NEW PHASE TRANSITION WITHIN SUPERCONDUC-

269

TIME OF FLIGHT

TIVITY.

CALORIMETRIC STUDIES OF SUPERCONDUCTIVE PROXIMITY EFFECTS

309

OF THE MAGNET IZATION IN SUPERCONDUCTING VANA-

TYPE-II - TYPE-I TRANSITION.

J.D. Lejeune and

D.G. Naugle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

305

ANISOTROPY OF Hc2 IN SUPERCONDUCTING PbTl ALLOYS SHOWING

Z. Ovadyahu,

baum ......•....•.•••.••.•.•....•..••......•..•..•••.

301

K. Takanaka and A . Hubert ..... ......... . . . . . . . . . . . . .

M. Dayan, G. Deutscher, R. Rosenbaum, and E. GrunPROXIMITY EFFECT IN AMORPHOUS BISMUTH.

S . Takacs . . . . . . . . . . . ...... .

FLUX FLOW OF TILTED VORTICES I N TYPE 11 SUPERCONDUCTORS R . S. Thompson ..... .. ..... ...... ... ... ........ ..... . .

fu~ISOTROPY

CHECK OF PROXIMITY EFFECT THEORIES BY EXPERIMENTS ON THE COOPER LIMIT MEASUREMENTS ON Pb/Cu FILMS.

297

BETh~EN

ANISOTROPY OF LOWER CRITICAL FIELD OF CUBIC MATERIALS.

P.

Nedellec., L. Durnoulin, and E. Guyon . . . . . . . . . . . . . . . . .

C.-R. Hu . . . . . . . . . . . . • . . . . .

PINNING IN SUPERCONDUCTORS WITH WEAK INTERACTION

J.A. Pals and L.H.J.

Graat • . . . . . . • . .. . .. ••............ ....• ••.•.••.....•

293

RE-EXAMINATION OF THERMOMAGNETIC TRANSPORT THEORY IN DIRTY

I

ARRAY.

I. Kirschner ..... . . . . . . . . . . . . . . . . . . . . . . . . . . ~lliASUREMENTS

317

321

ON DEFECTS IN A MOVING FLUXON

C. Heiden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •

325

THERMAL CONDUCTIVITY OF HIGH PURITY NIOBIUM IN SUPERCON-

273

j

DUCTING STATE.

A. Oota, T. Mamiya, and Y. Masuda ...

BETWEEN TWO TYPE-II SUPERCONDUCTORS NEAR H ' M.C. c2 Leung . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TYPE 11 SUPERCONDUCTORS

329

INTERFERENCE EFFECTS IN THE DYNAMIC JOSEPHSON TUNNELING 333

1

DENSITY OF STATES IN TYPE-II SUPERCONDUCTORS IN HIGH MAG -

ORIGIN OF HALF ORDER TERMS (T-T )n+T IN THE FREE ENERGY c OF INHOMOGENEOUS SUPERCONDUCTORS. G. Eilenberger and A.E. Jacobs . . . . . . . . . . . . . . . ......... . ...• . ......... . .

NETIC FIELDS. 277

SUPERCONDUCTORS BY NEUTRON DIFFRACTION.

CORE STRUCTURE AND LOW-ENERGY SPECTRUM OF VORTICES IN CLEAN SUPERCONDUCTORS AT T«T ' L. Kramer : . . ...... . c PHENOMENOLOGICAL THEORY OF THE LOCAL MA GNETIC FIELD IN TYPE-II SUPERCONDUCTORS .

J.R. Cl e m . .. .. .. .• ........

281 285

I

341

NEUTRON DIFFRACTION, HAGNETIZATION, AND SHIELDING CURRENTS OF TYPE-I! SUPERCONDUCTORS.

Y. Wada .. . .. . ..... .. .. .

MECHANICAL EFFECTS IN FLUX MOTION.

345

D.G. Pinatti and N .

Boboshko .... . .•... . . . . . . . . . . . . . .... . . . .... . ......... 289

337

J. Schel ten

and G. Lippmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •

I

REDUCTION OF CORE SIZE OF VORTICES IN CLEAN SUPERCONDUCTING LEAD FILMS AT T « T ' W.T. Band and G.B. Donaldc son ..•..•.......... • . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . .

W. Pesch . .. .. .... . ... . .. . .. . ...... . . .

DIRECT MEASUREMENT OF FLUX FLON VELOCITIES IN TYPE-I!

THE BEHAVIOUR OF DYNAMIC CRITICAL CURRENT IN THE VICINITY

349

x

OF

Hci.

xi

G. Fricsovszky , I.N. Goncharov, and 1.5.

Khukhareva .... • • . . . • . . . . . . . . . . . . . . . . . . . . . . ......•..

ELECTRON-PHONON COUPLING 353

' ( CALCULATIONS OF THE ELECTRON-PHONON INTERACTION AND SUPE RCONDUC'rIVITY IN THE PALLADIUM- HYDROGEN SYSTEM.

FLUCTUA TI ONS

B.M.

Klein and D.A. Papaconstantopou1os . . . . . . . . . ... .....• THE CONTRIBUTION OF THE HIGH FREQUENCY PHONONS TO THE SU-

PHENOMENOLOGICAL APPROACH TO NONSTATIONARY NONLINEAR PHENOMENA IN SUPERCONDUCTORS.

PERCONDUCTIVITY OF THE HYDROGEN-RICH COMPOUND Th 4 H15 . -'-G. Ries and H. Winter . . . . . . . . . • . . . . . . . . . . . . . . . . . . . .

M.Yu. Kupriyanov and K.K.

Likharev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X-Y MODEL DESCRIPTION OF A GRANUAL SUPERCONDUCTOR.

357

SPECIFIC HEAT AND TUNNELING MEASUREMENTS ON AMORPHOUS IN-

361

COMPARISON OF THE ELIASHBERG FUNCTION a

J.

Rosenblatt, A. Raboutou, and P. Pellan . . . . . . . . . . . . .

DIU!-1 FILMS. S.Ewert , A.Comberg, W.Sander, and H.WUhl

CURRENT NOISE BY FLUCTUATIONS eN SUPERCONDUCTING MICROBRIDGES.

S. Dottinger and W. Eisenmenger

Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ABOVE SUPERCOND UCTING TRANSITION TEMPERATURE. A. K.

ELECTRON-PHONON 368

GIliJE.

NUCLEAR SPIN-LATTICE RELAXATION IN SMALL SUPERCONDUCTOR IN MAGNETIC FIELD.

B. Keck and A. Schmid . . . . . . . . . . . . . . . . .

376

Dayan and G. Deutscher . . . . . . . . • . . . . . . . . . . . . . . . . . . . . .

379

S. Takacs

IN DILUTE INDIUM-TIN ALLOYS.

Rapp

387

PHONON THERMAL CONDUCTIVITY OF SUPERCONDUCTING LEAD -T HAL -

ONE-DIMENS I ONAL SYSTEMS. LIUM ALLOYS.

425

391

1.0 . Kulik . . . . . . . . . . . . . . . .

429 433

J.L. Ho, C.K. Chau, H. Weinstock, and 1'1.

C . Over ton, Jr.

. ... . ..... . . . . . . . . . . . . . ..... .. . . . . . . .

437

PHONON SPECTROSCOPY ON THi': EXCITATION SPECTRut1 ARISING FROM S-N BOUNDARIES IN THE INTERl1EDIATE STATE.

D.

A11ender and A. Houghton . . • . . . . . . . . . . . . •. . . . . . . . . . .

o.

SUPi':RCONDl1CTING AND STRUCTURAL TRl,NSFOR!1ATIONS IN QUASI-

ONE DIMENSIONAL FLUCTUATION CONTRIBUTION TO ULTRASONIC ATTENUATION IN CLEAN TYPE 11 SUPERCONDUCTORS.

R. Fo ge lholm and

383

O. Entin-

Wohlman and R. Orbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

421

RESISTIVE MEASUREMENTS OF THE ELECTRON-PHONON INTERACTION

EFFECT OF PROXIMITY EFFECT BETWEEN MAGNETIC AND SUPERCONDUCTING FILHS ON FLUCTUATIONS ABOVE Tc.

417

J. Bolz and

F. Pobe11 . . . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . . . . . . . .

ORDER PARAMETER PERTURBATIONS NEAR THE SURFACE OF SUPERCONDUCTORS.

J.A. Waynert and

MOSSBAUER EFFECT OF 119Sn IN AMORPHOUS TIN.

A.M. Go1dman and R.V.

Car1son . . . . . . . . . .... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

415

TUNNELING MEASUREMENTS ON GRANULAR AND CLEAN Al FILMS. M.

DYNAMICS OF THE FLUCTUATIONS OF THE TWO COMPONENT SUPERCONDUCTING ORDER PARAMETER.

Top1icar and D. K. Finnemore . . . . . . . . . . . . .

M. Levy .... . . .. . . . . . . . ... ..... .. . . . . . . . . . . . . .... ... .

A.M .

Goldman and J.C. Solinsky . . . . . . . . . . . . . . . . . . . . . . . . . .

411

IN THE PROXIHITY EFFECT RE-

VANAD IUM SAMPLES OF VARYING PURITY. 372

THE FLUCTUATION ENHANCED CONDUCTIVITY IN AMORP HOUS SUPERCONDUCTORS.

J. R.

INTER~CTION

COMPARATIVE STUDY OF ELECTRON-PHONON INTERACTION IN THREE

S. Kobayas hi, T. Takahashi, and

HEAT CAPACITY OF A SUPERCONDUCTING FILM NEAR Tc .

J.

Geerk, W. G1 aser , F. Gompf, W. Reichardt, and E.

FLUCTUATIONS-INDUCED PAIR-CONDUCTIVITY IN THALLIUM FILMS

W. Sasaki .. • . . . . . .

407

( w)F( w) AND THE

PHONON DENSITY OF STATES F( w) OF NIOBIUH CARBIDE. 365

Saxena and A.K. Bhatnagar . . . . . . . . . . . . . . . . . . . . . . . . . .

2

403

J. 395

I r I

F.J.

Lin and J. R. Leibowi tz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

441

xiii

xii

SEPARATION OF RESISTIVE AND MAGNETiC CONTRIBUTIONS TO

SPECIAL TOPICS ON TYPE 1I SUPERCONDUCTORS

THE HYSTERESIS OF TYPE II SUPERCONDUCTORS IN TIME DEPENDENT FIELDS.

INFLUENCE OF A DC COMPONENT ON THE AC LOSSES OF FILA-

bertse, and L.C. van der Marel . . . . . . . . . . . . . . . . . . . • . .

MENTARY SUPERCONDUCTORS AND SUPERCONDUCTING COILS. K. Kwasnitza ..•••..•.........•...••.••.....•.•.••.•• MICROWAVE RESPONSE OF MIXED STATE OF PURE NIOBIUM.

L.J.M. van de K1undert, E.A. Gijs501

445

W.L.

McLean and H .R. Segal . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . .

449

OTHER TOPICS

453

THERMOELECTRIC EFFECTS IN SUPERCONDUCTORS AND THERMOME-

THE TEMPERATURE DEPENDENCE OF FLUX JUMPING FIELDS IN Nb25%Zr.

L.S. Wright, J.P. Pendrys, and R.B. Harrison

CHANICAL CIRCULATORY EFFECT IN SUPERFLUIDS.

HYSTERETIC CRITICAL CURRENTS IN Pb-Sn LAMELLAR EUTECTIC. R.A. Brand, J . M. Dupart, and J. Baixeras ..•......•..

457

DUCTORS.

S.M. Khanna, J.R. C1em, and

M.A.R. LeB1anc ...........•..•..... • . . . . . . . . . . . • . . . . .

461

EVIDENCE OF THE EFFECT OF PARTICLE DISTRIBUTION ON FLUX

FOILS BY MEANS OF ALTERNATING TRANSPORT CURRENTS. 469

FLUX-FLOW RESISTANCE OF PARAMAGNETICALLY-LIMITED TYPE-II SUPERCONDUCTORS.

K.S. Kim and Y.B. Kim .••...•....•.

473

H.C. Freyhardt, B.A. Loomis, and A. Tay10r

PINNING IN THE PARAMAGNETIC LIMIT.

R.A. Brand ......... .

477

POTENTIAL.

P.B. Pipes and D.H. Darling ...... .

RELAXATION IN SUPERCONDUCTORS.

521

V.G. Bar'yakhtar and V.P. 525

QUASIPARTICLE RECOMBINATION AND 26-PHONON-TRAPPING IN 1'1. Eisenmenger,

K. Lassmann, H.J. Trumpp, and R. Krauss . . . . . . . . . . . . .

529

ANOMALOUS INFLUENCE OF Ce-IMPURITIES ON THE SUPERCONDUCM.H. TING TRANSITION TEMPERATURES IN Laln 3 _ xSn x ' v an Maaren and W. van Haeringen . . . . . . . . . . . . . . . . . . . . .

533

489

THERMODYNAMIC CRITICAL FIELD OF THE SINGLET GROUND STATE

493

SYSTEM (Lapr)Sn . R.W. McCallum, C. A . Luengo, M.B. 3 Maple, and A.R. Sweedler . . . . . . . . . . . ~ ... . . . . . . . . . . . . .

537

EFFECT OF PRESSURE ON THE SUPERCONDUCTING TRANSITION TEM-

P.H. Kes, G.P. van der Meij, and D. de

K1erk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . ....... .

TEMPERATURE DEPENDENCE OF CONTACT POTENTIAL IN SUPERCON-

485

INHOMOGENEOUS TYPE-II SUPERCONDUCTOR AT HI GH FIELDS AND SUPERCONDUCTING NUCLEI ABOVE H ' E.H.Brandt ... . . . c2 EXPERIMENTAL INDICATIONS FOR ANHARMONICITY OF THE PINNING

517

SUPERCONDUCTING TUNNELING JUNCTIONS.

H.W. We b e r and I. Adakty-

10s ...... . .... ... . . . . . . . . . .. .... . . ..... . . . . . . . . . . . . .

THERMOELECTROSTATIC EFFECTS IN SUPE RCONDUCTORS. N.K. We1ker and F.D. Bedard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481

BASIC PINNING INTERACTION BETWEEN FLUX LINES AND SUPER·· CONDUCTING PRECIPITATES.

513

Semino zhenko . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . , ....... .

FLUX PINNING IN HEAVY ION IRRADIATED TYPE-II SUPERCONDUCTORS.

C.M. Pegrum, A.M. Guenau1t, and G.R. Pickett ... , ....

DUCTING NIOBIUM.

FLUX-FLOW RESISTANCE MINIMA IN TYPE- II SUPERCONDUCTORS. T. Akachi, D.F. Kim, arid Y.B. Kim . . . . . . . . . . . . . . . . . . •

50 9

ERATION OF FLUX IN A SUPERCONDUCTING BIMETALLIC LOOP. 465

INVESTIGATION OF PINNING SITES IN SUPERCONDUCTING Pbln L. Wauters, L. Reynders, and Y. Bruynseraede

Yu.M. Ga1'perin, V.L. Gurevich, and V.I.

Kozub . . . . . . . . . . . . . . . • . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . AN EXPERIMENTAL INVESTIGATION OF THE THERMOELECTRIC GEN-

PINNING OF VORTICES IN SUPERCONDUCTING Al-13 at.% Ag A. Nemoz and J. Assayrenc ..•.•....•.. • •............•

505

ACOUSTOELECTRIC AND THE~10ELECTRIC PHENOMENA IN SUPERCON-

GUIDED FLUX MOTION AND FARADAY INDUCTION IN HOLLOW SUPERCONDUCTING CYLINDERS.

V.L.

Ginzburg and G.F. Zharkov . . . . . . . . . . . . . . . . • . . . . . . . . . .

(~~Pr)Sn3 AND (LaGd)Sn . L.E. DeLong, 3 R.W. McCa11um, and M.B. Maple . . . . . . . . . . . . . . . . . . . . . . .

PERATURE OF 497

541

xiv

AUTHOR INDEX OF VOLUME

2

... .......................... . ~

1

545

S.OOl

IDENTIFICATION OF ORDER EFFECTS IN V Ga 3

R. Flukiger, J.L. Stauderunann, A. Treyvaud, Departement de Physique de la Matiere Condensee, Univer.site de Geneve, 32 , bd d'Yvoy, 1211 Geneve 4, Suisse, and

P. Fischer, Eidg. Institut fur Reaktorforschung, Wurenlingen, Suisse In spite of the efforts which were made in the last years in establishing a correlation between the long range atomic order (LRO) and the superconducting properties of A15 type compounds, the proof of such a correlation was only furnished for cornpounds A T, where both components are transition metals, such 3 1,2 ) d . as V Au , or MO)Os • For compoun s A)B conta1ning a non3 transition element, however, there is much less· data and there are some cases for which the correlation between the highest LRO parameter and the highest Tc seems not to be valid. As an exam-. pIe, V Ge and V Si are known to have the same Tc after different 3 3 heat treatments, while for V Ga it was reported that the order 3 parameter, S, remained unchanged when Tc varied from 13.6 to o 4 15.1 K • This intriguing situation was the starting point of the present work on the V Ga system. 3 We have reinvestigated the V Ga system exposing the samples 3 to extremely different heat treatments in order to induce a mao ximum change of S. The highest T value (T = 15.7 K) was found c c o after a final annealing of 3 months at 620 C, whi l e the lowest o value (T = 13.4 K) was obtained after an Argon jet quenching c o from 1260 C. The effects of different heat treatments on T a re c summarized in fig. 1. The high value of Tc was confirme d by a specific heat measurement. All our measurements were done on pieces of the same maste r sample having a starting weight of 60 g , which had the highest Tc from a series of identically arc-melted samples . It had the analyzed composition V.752Ga.248. All measurements were performed after a prior homogenization anneal of the master sample. It was

2

3

c a n be decr e a sed to l es s than 1 %, wh i ch is d u e e s s ent ia lly t o the ve ry di f f e re n t

• Ar jet quench A15~

7x=092

I

I

a s de termined by both methods , a r e S

·A2

=

o 1 4.9, a n d 1 3 .8 K,

0.9 4 ± 0 .06 , a nd S

=

I

0. 86 ± 0 .0 6 fo r t he X-r a y diffra c t i o n and S

1 1

S = 0. 94 ± 0 . 0 1 for th e n eutron diffr a ct i o n. I n spi t e o f t he

r

q

s cat t~ri ng fac tors f or V and Ga .

The o rde r p aramete r s corre s p ondi ng t o Tc

5=094 n' . { 5j086

0 .9 7 ± 0. 0 1 , and

I 1

15

I

4.5 •

•

11

0.841

I

600

800

1000

1200

4.0 ........

1400 T (C)

Tc=14.9 K • VGa 3 o V3 Ga Tc=13.8 K

• • o •

x

"0

A

en

Fig. 1. Effects of the annealing temperature and

:::J

of the cooling speed on Tc.

"1

E ClI

.0

a" ...

B

0

first treated for 12h at 1450 C (in the BCC region), followed

-K

0

by 24h at 1260 C in orde r to re t ransform the sample into the

3.0

A15 phase . During this anneal, the pre ssure in the furnace was

V Au 3 V3 Au

+

held at 4 atm. of Argon, in orde r to redu ce the weight losse s .

,p

.

•

+0 +

..

• •

+

+

'. '. ,

• ++~

•

••• +

"

0

+.

••

0· •• •

+ .0 0

+ +.

2 .5'-----~---.L..-----'

o

100

Nblt

5

••

+~o

' » ,

vely 2% of i n c l usion s whos e natur e was n o t determined.

~ I .24~_.

Mo Ir .76

Whil e no e x tra lines could be d e tected by X- rays, the microsc o pic observ ation rev ealed a t t h e grai n boundaries approx imati-

10

•

• 0 • +0

a,

l = 2.9 K l= 1.8 K

...... ~ I=u

200 T(K ) 300

o

L..-~_-'--_~-...L

0.5 0.6

Because of the small difference in atomic numbers between

___

0.7 0.8 0.9

to

- 5 ) Saor Sb

V and Ga, the uncertainty in evaluating the order parameter S by X-r a y diffraction is relatively high (6%). Nevertheless, it was possible t o dete ct a syste matic d ec rease of S with the increa se of the annealing temperature or with the increase of the cooling speed (fig . 1). For the determination of the abso -

Fig. 2 . Ef f ect s o f t he

lute v alues for the LRO parameter, the neutron diffraction is a

order ing o n th e mag n eti c

Tc and the order parameter

more appropriate method for V Ga. In fact, the uncertainty in S 3

s u scep t ibil i t y of V Ga 3 and V Au. 3

S(Sa or Sb for non-stoichio-

!

Fig. 3. Correlation between

metric

compounds.

..

5

4

ANOMALOUS BEHAVIOURS OF LONGITUDINAL ULTRASOUND

slight systematic difference between X-ray and neutron measure-

8.002

ments, we consider that the agreement is satisfactory and that

VELOCITY AND ATTENUATION NEAR Tc IN V3Si SINGLE CRYSTAL . N. Toyota, T. Fukase and Y. Mu to The Research Institute for Iron, Steel and Other Metals, Tohoku University, Katahira 2- 1- 1, Sendai, 980 Japan

definitely the change in Tc i s correla t ed with the change of the LRO parameter. In fig. 2, it is shown that this is also the cas e for the magnetic susceptibility. From the same figure, it is seen . that the ordering effects in V Ga are at least of the same 3 order as for VjAu . This is not surprising if we take into account that V3 Ga has the highest electronic density of states among all AlS type compounds. Qualitatively, the behavior is that described by the theory of Labbe, et a 1 • 5 The ordering effects in V Ga are more difficult to detect 3 than for compounds of the type A T. This is due to the tendency 3 of this . compound to form in a nearly perfect ordered state up to the close vicinity of the structural phase transformation. This fact seems to be even more pronounced for V Ge and V Si and 3 3 3 may well be a characteristic of "typicarAlS type compounds 3 in contrast to "atypical" AlS type compounds, such as C~72os.28 ' which can be very disordered. A summary of the dependence of Tc on the order parameter for d i fferent AlS type compounds is given in fig . 3. It follows that for this structure, independently of the components, the highest Tc is always associated with the ·highest ordering state. References : 1. E.C. Van Reuth, R.M . Waterstrat, R.D . Blaugher, R . A. Hein and J.E. Cox, Proc. Int. Conf. Low Temp . Phys., 10th MOscow, Vol. lIB, Viniti, 1967, p . 137. 2 . R. Flukiger, Ch. Susz, F. Heiniger and J. Muller, J. LessCommon Metals, 40, 103 (1975) . 3. R. Flukiger, A. Paoli and J. Muller, Solid State Comm., 14, 443 (1974). 4 . C . C. Koch, J. Phys. Chem. Sol. 34, 1445 (1973). 5. J. Labbe, E.C. Van Reuth, Phys. Rev. Letters, 24, 1232 (1970).

The discontinuous velocity change and the anomalous attenuation of the longitudinal sound waves propagating along [001] of V3Si single crystal were observed in the close v i c inity of the superconducting transition tempera t ure under zero and finite magnetic f i eld. The pre s ence of a s t r uctural transforma t ion was not revea l ed by t he resistivi t y and susceptibility meas urements for our samp l e wi th RRR =17 and Tc =16.7 K determined by the resi st i vity method . Ve l ocity chan ge of t he 15 MH sound waves z was measured by the pUlse - superposi t ion method with p r ecision o f 10 - 7 an d t he rela t ive a t tenuation chan ge by the pulse - echo me thod using 375 MH z sound waves . No correction was made fo r the l en gth change wit h t emp erature an d field. I n Fi g . I-Ca) is s hown the tempera t u r e depen den ce of the s ound veloci t y un der seve r al f i e l ds para l lel t o [0 01] . In the abs ence o f field , the drastic chan ge occurs in t he close vic in i ty o f TcC H= O) . The t empera tu r e a t wh ich sound veloci t y go e s through a dr as ti c chan ge is obs e r ved to de c rease as an e ffect of app l ica t io n of magn etic fi e ld . Our val u es at va r ious fie l ds almos t li e on Pulver ' s l )plot of T versus Hc2 for HP [OO l] . Such behaviours on Cll mode indicate that th e superconductive ordering arrests the lattice softening and that the drastic discontinuous veloci t y change occurs in the close vicinity of TcCH). It must also b e noted that the softening of Cll mode is gradually suppr essed with increas ing fields in the normal s tate near TcC H). As in Fig. I-C a), this trend is reflected by fa l l in the value of temper ature derivative of the sound velocity with incr easing fiel d . 2 ) In Table I, we compare our result s for the elastic constant C ll in the vicinity o f TcCH=O) wit h those of T estardi~ ) The

7

6

15.0

(0)

16.0

155

I\J

16.5

~

V3Si -it I WOll/lq //H

@~

L.A.mode 15 MHz

~~

H= 0 (kOe) • 20.48

59.15

(l)

U)

(I'

v

t)!....---

,

ct---L

5.62

13.5

13.0

V3Si -ttl

(b)

\

• Ours ( from Vs) o Ours ( from o() -------:.Pulver

(\J r0~

~ CO~

NI- 0.1 MeV). In situ Jc(H) measurements were made up to ~3

T, stronger fields being prohibited by space limitations.

To extend the study to higher fields, a method was devised to mount the samples in a

four-termi~al

without ever exceeding 77 K.

probe in lIT's 12 T magnet,

Preliminary studies appear to

indicate that Jc(H) of radiation damaged samples is unaffected

References: l. 2.

3. 4.

M. Pulver, Z. Phys. 257, 22 (1972) T. Fukase, K. Uema and Y. Muto, Phys. Lett. ~, 129 (1974) and in detail to be published. L. R. Testardi, Physical Acoustics, !, (1974) M. Weger and I. Goldberg, Solid State Physics, ~, 2 (1973)

by annealing to this temperature. Jc(H) was determined by passing a current through a sample wire at a given H and displaying the amplified V(I) char a cteristics on an x-y recorder.

Ic(H) was then determined by extra-

polating that portion of V(I) with slope 125 V

=

0 axis.

~V/A

back to the

Transitions from small to large slope are rather

sharp and occur (by design) well below the linear flux fl ow region. radiated,

In Fig. 1, Jc(H) is shown for three cases: (B)

(A) unir-

irradiated, and (Cl irradiated and annealed to

300 K. At 3.2 T, Jc(H) for the irradiated sample (B) is increased

11

10

6 18% above the already high unirradiated value of' 1.32 10 3 A/cm. 2 and agrees with comparable in situ measurements. with 0

.

.-

increasing field, Jc(H)B/Jc(H)A becomeslarger until, at 10 T, is enhanced 50%. (This contrasts with studies on Nb 3 sn irc radiated at "reactor-ambient" temperature ('l.60°C).4 There, J c

I

I'!

J

(!) (J)

tJ>

I'!

..-I

0 W

I~

C

U1

C"1

.0

Z

cs ~

0:: 0::

0

Z

CO

~

~ ~

--

~

::l

qf

I

I

I

ljoRD«:

u"

~oI:.'lEt>

f

c.,

It is clear that Bc2 in

these materials is not understood.

blunting of the density of states peak. unfortunately though one can guess

0·3

J1 b1

-swEn,H

0. )

M RR.:IN 0.1

an e::jIlivalent degree of disorder to give a better fit to the data (especial-

nodels for Bc2 can be developed which

0..1

ly for low C values) no clear way of estinating the degree of inter-chain

are not dependant on the integrity of

0.

scattering is yet known.

the linear chains in A15 naterials.

--

THeoRy

o.s

At present few systematic neasurarents of Bc2 have been nade on disordered naterial;

SSW have produced

:inens, 9Needler

7

SClle

and Martin et a1

8

results on their I\Dst irradiated specI would like to thank J .E . Evetts for

have both nade studies in NbjU and

[ffiny

rreasurerents are being nade on Bett' s Nb Sn spec.inens. The LFvR theo:ry 3 predicts a I\Dre rapid fall in Bc2 with increasing disorder than may be ex-

1. R. E.Sarekh

pected from a lIDdel with a Fenni velocity independent of band structure ~

2 . A.R.SWeed1er, D.G.Schweitzer, G. w.Webb

In

References . J.Phys. F ;

2

713 (1975).

the table we show a carparison of B (0) obtained by SSW =Pared to estic2

3. R. Bett, C:ryogenics, 14, 361 (1974) .

nates using the ideas of LFvR.

4. R. Bett and

Sparks

6 . Yu. A. Izyurrov , 6.0

0.057

0.8

13.0

0.168

9.2

0.055

1. 3

16 .0

0.154

0.8

0.06

0.5

11.0

The rrean free path, m.f.p., associated with scattering along the chains only is taken into a=unt in the last =1=.

(The m. f.p. for the purposes

Phys. Rev. Lett. 33, 168 (1974 ).

to be published.

5 . A. R.Sweed1er and D.E.Cox,

0.i95

useful discussions.

Phys. Rev.

to be published.

Fiz. r-Et. Metall, 35 687 (197 3).

7 . A. R. Sweedler (private ccmnunication ). 8 . P .J. Martin, A.M. Carrpbell, J . E.Evetts,

Se1. Stat . Comn. 11 123 (1972) .

i,1 21

20 S.006

paramagn e t ic limi t a ti on.

SOME PROPERTIES OF GRANULAR NbN AND Nb Ge THIN FILMS 3

In o r d e r t o u nderstand some of the f u nd ame nt a l me cha n i sms which

,

H. Jones,

~.

D~partement

Fischer and G. Bongi de Physique de la Matiere Condensee

Universite de Geneve, 1211 Geneve, Switzerland Niobium Nitride in bulk form is a high field superconductor with a critical field H =150kG., and a critical temperature c2 T =16 K. This compound is one of the only examples of a c superconductor with a Tc above 15 K which does not have the

l'

a r e importan t in this sys t em, we meas ur ed t he r esi stivity a nd

J j;

the s u percond ucting energy gap of NbN. The resis t i vity showed

t

t h a t t he NbN f ilms do not behave as o rdi na r y metal s becaus e t he r esistan ce increases r athe r than d ecr eases a s th e temp e ra t ure is lowe r ed. Th i s is the kind o f be h av i o r o n e would expe c t f o r an array o f me t allic grain s with hop ping con d u c t ion be t we en the gra i ns. Al so, t here i s a cor r elati o n b etween Tc and the rate o f increase of t he r es i stiv ity a s the temp e r a ture i s lowe r ed

A-IS crystal structure. What i s espe c i a l ly interesting i s th a t

3

it can be easily made in thin film form by sputtering. Th e t unne l ef f ec t measu r ed on NbN - oxide - Pb t u nne l j un ction s shows t hat the energy gap of NbN at T = 0 is near l y ins ensi tive

The starting point of this work was a g r a ph in a p ubl i c a ti o n

t o the t r a n s i tion temperatu re of t h e di f f e r e nt films . This i s

by Yamashita et al. 1 • They showed that Hc2 increas e d with

i n con t ras t to the prediction o f the BCS t heory wh i ch states

resistivity for low resistivity NbN samples, as expected, but

t h at the energy gap should be proporti o n a l to Tc acco r ding t o

then saturated at H =250kG., which was very unex pected. The c2 reason why this saturation was unexpected is that the theory of Werthamer, Helfand and Hohenberg

2

shows that the orbital

* is in fact proportional to the r esis t i v i t y critical field Hc2 according to the formula:

* Hc2

the rela t ion :

J,

to

o

=

1. 76kT

c

Fr om these experiments and their compar i son with othe r wo r k done on NbN we have concluded that NbN is a granul ar thin fi l m'.

=

30.4 PYT c (kG)

The difference between the films Is essential l y the barr iers

where P is the resistivity and y is the electronic coefficient

between the g r ains rather than the gr a ins themse l ves. A re c ent

of specific heat. If one applies this formula to NbN, then the

theory by Deutscher et al.

orbital critical field should be about one megagauss for the

superconducting array that Tc is proportional to the coupling

samples with the highest resistivity. The interpretation of

b e tween the grains which explains our correlation between the

Yamashita et al. 1

for the saturation of the critical field

is that H (O) is limited by the paramagnetic effects of the c2 applied magnetic field on the conduction electrons .

4

has shown that in a granu lar

resistivity and Tc. On the other hand it is logical to suppose that the superconducting properties at T the ratio

=

0 wou l d depend on

of the s uperconducting coherence lengt h, ;, and the

diameter of the grains . Making this hypothesis, it is possible The thin films of NbN made in our laboratory had Tc'S between

to explain in a cohere nt fashion the superconducti n g properties

4 K and 14 K, but with H (O) nearly constant and equal to c2 about 250 kG, independant of T 3 as shown in figure 1. The c fact that Hc2 is independant of the resistivity and the

of NbN as well as recent measurements on granular thin films

critical temperature puts in d oubt the interpretation of

During this work we noticed that the resistive transitions

of Nb Ge which were made in our laboratory. 3

22

23

from the normal to the superconducting state .in thin films of NbN and Nb Ge are rounded instead of being sharp. This is exp3 lained by superconducting fluctuations above Tc By making tunneling measurements in the temperature range where the fluctuations exist, we were able to observe fluctuation effects in the current-voltage characteristics of several junctions. It should be remarked that in several cases the effects observed by tunneling are much larger than in the resistive transitions. We believe that this is the first time that fluctuation effects have been observed by tunneling. Finally, we have measured the critical fields of granular Nb Ge thin films. Critical field measurements by Foner et al. 5 3 on Nb Ge films with Tc = 23 K gave H (0) = 375kG. The Nb Ge c2 3 3 thin films in our labqratory were granular (after their resistive behavior) and had Tc'S well below 23 K. Even so, the critical fields H 2(0) are nearly identical to the samples c 5 measured by Foner et al. These measurements confirm the conclusions drawn from the measurements on NbN and point out that it is not always necessary to have a high Tc in order to have good superconducting properties at low temperatures. We wish to thank M. Weger and T. Yamashita for many discussions.

ts

•

ts ts 0.65, it appears that most of the d character has disappeared (N(O) de1 creases by a factor of "'6) and Pd-H behaves more or less like a noble type x metal (like Ag) with about one condu:::-:::ion electron per atan. For x approaching 1, supercooductivity has been observed2 ,3,4,5 and it is the aim of

phonons (described by a Debye spectrum) and the optical phcnons (described

this work to shine sane light on it's origin, superconductivity being generally not observed in metals with so feN electrons. Preparation: Pd-H with x = 0.995 has been prepared by lcw -t:enperature x (~2000K) electrolysis . Details concerning this preparation, the stability of the specimen and the electrical measuranent will be published elSeNhere

6

of the resistivity, we put it in the form P = Pr(x) + p(T,x) , the last term (it is found 6 that the resi-

representing the part due to phonon scattering x~l).

- e

210 K ± 20 K

ac

(lop

p(T,x '" 1) is given on figure 1 to-

gether with the superconducting Tc observed for this specimen. On figure 2 we show the difference p (T ,x=O) - p (T ,x=l) between pure Pd and PdH. The most

important features of our results can be discussed on this last figure: - p (T ,x=l) is always smaller than p (T ,x=0) despite the fact one has introduced new scattering centers. - as is apparent on fig. 2, the difference p (T ,x=0) - p (T ,x=l) goes throu:rh

550 K ± 50 K (for an Einstein spectrun)

8 1000 ·K ± 100 K for a Debye spectrum. 0p - the intens ity of the scattering can be drawn fron 1:,'1e linear relationship which exists between

Normal state phonon resistivity: To analyze the tanperature dependent part

dual resistivity Pr(x) is lcw for

by an Einstein s pectrum or a second Debye spectrum) add their contribution TT to pi.. e. pIT, x = 1) = Pact -e-) + pop(-e-)' The main results of this ac . op analysis are

P and T, i. . e. P = AT for high enou:rh tenperatu-

res (in practice ~ > 0,3). The result is that Aop ~ 3 Aac which means that the optical phonons are the most efficient scatterer; for electrons. This last result has far reaching oonsequences for the occurence of supercondUCtivity. The fact that i t is difficult to decide between an Einstein spectrum and a Debye spectrum for the optical p.'1onons may be due to the la:rge

width observed for this spectrum by neutron diffraction

7

Superconductivity : 1_ where The superconducting Tc is given by Tc = 6 exp _ _ AX eff N(O) 12 x x x A x U A ~--Aeff = A - U ,A = 1+1.. U = ---t:.--::'r::-- ~ 0,1 M 62 F 1+U log "8 (I is an electron phonon interaction matrix element, M = ionic mass,

a maximum near 140 K and it is probable that it will change sign above roan

e=

-t:enperature. This rather strange behaviour can be explained if one amnits

ween the occurence of a Tc and the high temperature normal state resisti-

that the phonon scattering in PdH is due to two types of phonons : acoustical and optical.

Debye temperature). It is well known that there is a correlation bet-

vity

P

= AT

i. e. between Tc and

A, a high value of A favouring

superconductivity (one can show that A one takes this as guaranteed, one canes

*permanent address: Physics Departrrent, University of the Witwatersrand, Johannesburg, South Africa

~L

~6~

is prq:ortional to

A). I f

conclusion that Acp

~

3Aac

and that most of the Tc cares from the optical monons . The expression of

Tc' in terms of the b.o type of phonons has then the following foun

42

43

AX x exp __1_, AX ; Ax + __--'a:::c~-_)l.:=a::::c'___ __::__ OP Ax eff op eac X eff 1 - (A -)lX) lcq e ac ac op X To obtain a fit with tbe observed T of 'V 9°K, one must take A 'V 0.343 c eff (with e ; 550 K) which means A 'V 0. 44 and A 'V 0.15. We can nON make op op ~ sane further rerrarks

T ; c

e

- i f the optical phonons would not be present tbe T would be determined x x ~ only by Aac and (A eff ) ac would be equal to Aac -)l '" 0.04 and tbe Tc would be negligeably small. This is indeed what is observed in pure Ag where pro" tY e ff ects8 , X1.IlU. assl.gn to Ag a BCS f actor AX eff between 0 and O.l.

- it appears that

eac

is a bit softened corrpared to the Debye teroperature

of Pure Pd which is 270 o K. This is probably due to tbe 4% dilatation of Pd-H canpared to Pd. This nay help to increase Tc but we think it is not a decisive factor. - it has been observed

90 BoNo Ganguly, Zeitso flir Phys. 265, 433 (1973)

P(T,x=O)-pn x=1) 10

9

8 7

3

that deuterium increasesT above the hydrogen value c which means a positive isotope effect. This can be most probably explained

6

by stronger anharrronic effects for H than D as i t was proposed by Ganguly9

5

- Tc decreases as the concentration of hydrcqen decreases belON x ; 1 and

"

increases i f one replaces gradually Pd by increasing arrounts of Ag or Cu.

J.1ohmcm

p(T,x:o.995) ()1Qxcml

2

:[J: 7

•

i

10

• Experimental ·

-.-

3

It is difficult at the m:rnent to see the exact reason for this but one must have in mind that the optical phonons are very sensitive to the local envi-

2

ronment of the H atoms. To =nclude, we think that the nonnal state reSistivity ShONS that optical phonons are ltOre efficient in scattering electrons than

300

~oustical

phonons. This means also that the superconducting Tc is mainly determined by these same optical phonons . References 1. C. A. Mackliet, A.I. Schindler, Phys. Rev. 146, 463 (1966) 2. T. Skoskiewics, Phys. St. Sol.

!l K, 123 (1972)

Fig. 1 : Phonon resistivity p(T,x;I) of Pd-H . An analysis

3. B. Stritzker, W. Buckel, Zeits. flir Phys. 257, 1 (1972)

a=ustical phonoos (P

4. J.M.E. Harper, Phys. Lett. 47A, 69 (1974)

tical phonons (p

5 . R. J. Miller, C.B. Satterthwaite, Phys. Rev. Lett. 34, 144 (1975) 6. to be published 7. J . M. RONe, J.J. Rush, H.G. Snith, M. YDstoller, H.E. FlotON, Phys. Rev. Lett. 33, 1297 (1974) 8. C. Valette, Sol. St. Camt.

2"

895 (1971)

OP

(T»

jn

ac

terms of (T»

and cp-

is sh= also.

Fig.2 : Difference in phcnon resistivity between prre Pd (x;Q) and Pd-H (x;l).

45

44

S.012 THE MAGNETIZATION AND UPPER CRITICAL FIELD OF PALLADIUM HYDRIDES D.S. McLachlan and T.B. Doyle University of the Witwatersrand, Johannesburg, South Africa and J.P. Burger Universit~ Paris-Sud, Orsay 91405, France

The magnetization curves of the following palladium hydrides (PdH l , PdH. 983 , PdH. 958 and PdH. ) have been meas933 ured by means of a Foner type magnetometer designed and built by one of us (TBD). The samples were prepared by the method described in reference I, i.e. by electrolysis at 196K. The resistance of a foil (approx 8mm x 3mm x 50mm) was monitored during the final electrolysis and the change in resistivity at 196K" and the resistivity at 80K were used to obtain the hydrogen concentration l . After being stored under liquid nitrogen for a minimum of 48 hours, a specimen 6 x 3mm was, while still under liquid nitrogen,cut out of the foil and placed in the magnetometer specimen holder. The specimen was then transferred in less than four seconds to helium gas at 4K in the magnetometer. Before each magnetization curve was taken the specimens were heated above Tc in zero field. The temperature was measured using a germanium thermometer. During the hydrogenation the specimens undergo a a-a-a phase trans2 ition and are therefore heavily cold worked . The presence of the resistance anOmaly3, at about 80K for heavily charged palladium hydrides,indicates at least short range ordering at this temperature and it may be assumed that the hydrogen is inhomogeneously distributed, probably precipitated at the dislocations 4 , after the ~geing at 80K. Therefore the magnetization curves of the specimens should exhibit multi-component behaviour and strong hysteresis due to fluxon pinning. Figure 1 shows typical magnetization curves obtained for the PdH and PdH. specimens. HCl cannot be determined 958 as the specimen is multi-component, pins heavily and is not

1 jt

.,.

II li

fr

~I

FI G. I : Magne tization curves for thePdH and PdH O. spec i958 mens at va rious temperat ures showing the procedure f or deter min ing ¥C2L and HC2U '

x

•

=

0

I

t

t j,

I t·

)

~. )

:/ J

1 \'

J~,

plo ts f or th e PdH and PdH o . 958 fo r de termini ng HC2 (O) a nd Te . The insert shows t he de penden ce on the H/ Pd r at io of T . c

I'

tl

r

47

46

aligned with the field to better than 3

0

•

Evidence for multi-

component behaviour is (i) the anomalous increase in the ratio

in HC2 is due to the decrease in HC which must almost certainly drop with decreasing Tc. It is apparent from the magnetization curves of PdH at

of HC2 to the peak of the magnetization with temperature (e.g . figs la

& b) (ii) the magnetization in fig. Id is characteris-

tic of small (of order A(T» specimens therefore implying small superconducting regions in a normal matrix.

low temperatures that the G - L parameter K must be of the order unity. Experiments are planned where (i) the initial 70% of

This is obtained from the steepest descent of the magnetizat -

the H2 is added above 300 0 C to eliminate the cold work due to the a-S phase transition 2 and (ii) the effects of various

ion as shown in fig. 1.

ageing and quenchings on the hydrogen distribution will be

For PdH only one characteristic field is id e ntifi ed . The high field tail in fig. l a is

attributed to surface superconductivity.

For th e ot h er thr ee

specimens the point where the magnetizatjon va nish es is id e ntified in the usual way with H . In order t o charact e ri z C2U the H (T) behaviour of the "weaker" compone n ts th e e xtr p olC2 ation of the steepest descent of the magnetization i s u ed a nd labelled HC2L as illustrated in fig. 1. Th ese r esult f o r t h e PdH and PdH. specimens are shown in fig. 2 a n d i n t h t a bl e 958 the results for H (0) and Tc are given. For th e sam hy dr ogC2 en concentration the values of Tc obtained by thi s met h od lie between the extreme values obtained by Miller a nd Satt rthwaite 5 , using a susceptibility method. Sample PdH l PdH. PdH.

983 958

References: 1.

J.P. Burger, D.S.McLachlan, R. Mailfert and B. Souffache (to be published)

2.

F. Lewis, The Palladium-Hydrogen System, Acad . Press

3.

A.W. Szafranski, Phys. St. Sol. 19, 459 (1973)

4.

A.H. Cotterell and B.A. Bilby, Proc. Phys. Soc. A62,

5.

R.J. Miller and C . B. Satterthwaite, Phys. Rev. Lett.

(1967)

49 (1949)

2!!.,

144 (1974)

H (0) C2L 980 oe

TcL 9.62K

H (0) C2U

Tc U

1110 oe

7.91K

900 oe

8.6 6 K

Trebro, and V.M. Zakosarenko, Zh ETF Pis. Red.

1270 oe

6.20K

1600 oe

7. 21 K

676 (1974)

6.

1270 oe 800 oe 5.5K 5 . 95 K 933 Magnetization and susceptibility measurements of Tc a nd HC 2 PdH.

are considered to be superior to resistivity measurement s since the latter are not characteristic of the bulk in inh omo geneous materials e.g. the size enhanced HC2 of thin filam e nt s can be measured, which can account for the much higher value s 6 observed by Alakseeskii et al. The initial increase of HC2 with decreasing hydrogen concentration (see table) is probably due to the increase in K

studied.

with increasing resistivity7 (p

n

increases by .85~ncm per

1% decrease in hydrogen concentration l ).

The subsequent drop

7.

N.E. Alskseevskii, Yu.A. Samarskii, H. Wolf, V.I.

B.B. Goodman, Phys. Rev. Lett.

~,

597 (1961)

12,

48

S.013

49

SUPERCONDUCTIVITY IN PdHn THIN FIUIS.

J. IgalsoD, L. ~niadower, and A.J. Pindor Institute of Physics, Polish Acade~ of Sciences, Al. Lotnik6w J2/46, 02-668 Warsaw, Poland Since the discovery of superconducti vHy in Pd"n in 1972 a substantial amount of information about the macro scopic proper ties of this material has been gathered. 1 ,2 Just recently two independent tunneling observations of energy gap in this material, prepared by different hydrogen charging techniques, have been reported. J ' 4 Tc of Pd"n depends strongly on hydrogen concentration n = H/Pd and hence it seems important, for quantitative interpretations of experimental results, to know a hydrogen content of a specimen. In particular this problem may be essential in the case of thin film tunneling experiments where the determination of the hydrogen concentration is not trivial. All our samples, both s epa r at e Pd filll8 and junctions, were charged with hydrogen using high R.-----------------------~ pressure. Pd films were prepared ~ by standard vacuua deposition technique at pressure lower than 10-6 Tr. Fig. 1 shows typical re- o~ sistance transitions of four of PdHD films of varying thickness 0,6 which we have made, charged with hydrogen at two different pressures. The transition of both 2000 04 film" is reasonably sharp, very , IlUch like the transi tion of bult samples and does not depend too 0,2 1.5 kbarj 2000 A IlUch on the charging pressure. We have also found that all our Pd"n i filas of about 2000 /those made 2 3 4 T[Kl separately and those forming one Fig. 1. Transitions of PdHn side of tunneling junctions/ fit films. Tc is taken from half well on curve Tc vs. RolRo deterof a normal resistance.

A

I

I

I

A

mined f er t ..llk samples , as shown Ar4t2::.-____1J~O=--------.:::Oj:::5~--..,O-.:-'!JRrJR~ in Fig. 2; contrary is true for :TdK I 9 I thinner f~lmst l~ke the two shown o~ in Fig. 1. Concentrahon scale i s 8 x - "-'2000 A films I obta~ned f ret!) a r elat:i.Gi . between . _ } bulk 0/00 7 x I RnfR Q and n , l Our desorption mea'f surements of the hydrogen concen.. 6 ,I I tratioc have shown that the hydro/ 5 I gen content ln a thin film /e.g. X /, X I x,I JOO A/ is much lower than in a 4 x/x thick film /e.g. 2000 AI charged " I 3 ~ /xo a·, the same pressure. Independent I measurements have shown a disabl- 2 l" I I llty t o sor b hydrogen and form / / hydride phase by finely dispers ed // Pd films IW. Palczewska - private 0,9 0.8 1,0 HIPd communication/. This is consistent Fig. 2. Comparison of Tc vs. with the above result in view of nand RnlR o for bulk 1 and the fact that crystalite sizes in thln film samples Ithis paper!. our JOO A films , determined from electron micrographs , are in the range of 100 A, whereas for 2000 f i lms the crys talite si zes are around 600 On the basi~ of the above evide.n ce it seems safe to assume that the films of the thickness in the ~ooo Arange land thicker/, when charged with hydrogen. do not differ much in their superconducting properti es from bulk samples. Consequent ly , the hydrogen concentration in the films may be determined on the basis of their residual resistivity . ' From a variety of tunnel~ng junctions of different types tested, junchons Nb-AI-AIOx-Pd re tained satisfactory characteristics after hydrogen chargjng. Before charging, junctions were S I-N type already at 4.2 K due to proximity effect in aluminium layer. 5 The th~cknessof the Nb film was about 1000 A and of the Al interlayer about 300 A. Junctions had areas in a r ange of 0.5 - 1.0 . .2 and the normal res ist ance ~n a range of 10 - 20 mn. Figs. 3 and 4 show I(V) and dV/dI characteristics of one of

I

/

0

0

I

A

1.

51

50

---- TC1.

0.001 !--.L.;;"'::----;~--;'';._

o

1.5

%~-~0~ . 5---~----~1.~5----

never risen above 10.5°K. These resu l ts are disappointing and somewhat surprising, therefore the question is raised as to whether previous attempts were unsu ccessful in increasi ng A o r if the above discussion is simply not correct. A possible explanation might be that y Be is, in fact , a strong coupling material. This would mean that is far smal2 ler than expected, for instance bec ause the function a FCw)

d2V d2 \

Al-y

....

overwe ights low energy modes. This hypothesis can be c hecked by

Be junction

(a) normal state

-.. -.. .-... ...~

tunneling experiments. The first test cons ist s in measuring the 2~/kTc ratio; the exp eriment has b een performed 5,6,7 and values between 3.45 and 3.7 are found, pointing to weak coupling. A second possibility is reported her e : the phonon

•c

15

Fig. 2 show s second derivative characteristics of an

10

or in the superconducting s tate Cb) . A peak corresponding to

from 1 . 2°K to

6.5°K(3~

35°K, a challenge !

the predicted crit ical t e mp era ture is

~

.. .... c

'"

inelas tic tunneling via longitudinal phonons of Be can be

same relative change of as in the case of Al whose Tc goes

L. PHDlOI OF Be

~

attempt is made to deduce the value o f A.

* If the (optimist) hypothesis is made that one can make the

l . Pit OlO1 Of AI

~

~

structure in tunneling characteristics is studied and an

AI-(y Be) junct io n, the Be being either in the n ormal state (a)

(b) superconducting state

~

5

o

20

40

60

80

100 mV

75

74

observed at about 75 mV on both curves (it corresponds to an inc r ease of a), but the important point is that strong coupling effects do not appear when the Be becomes superconducting 7. To be more precise, if we deduced curve (b) from curve (a) using the BCS density of states, the deviation between the predicted and the experimental curves, which includes conductance ch anges due to strong coupling renormalisation, was less than 1x10- 3 . Since the theory gives ; (00/0)

str.c.

~ ~2/ 2

(2)

we can deduce limitations for and then f or A (Eq. 1) > 54 meV

A < 0.6

Consequently, y Be appears in fact to be a weak coupling superconductor but, nevertheless, no increase of its critical temperature has yet been observed. We think that cryogenic temperature condensation already gives an almost "optimized" A and that the difficulty is in achieving a further increase. Whether or not there are other possibilities to raise the A of y Be is an open question which is worth being worked on ... References 1. 2. 3. 4. 5. 6. 7. 8.

W.L. Mc Millan, Phys. Rev . .!.§2, 331 (196'6). K.H. Bennemann, J. de Phys. ~, C 4-305 (1974), A. Fontaine and F. Meunier, Phys. Kond. Mat. li, 119 (1972). N.E. Alekseevskii et ai, J.E.T.P. Letters ll, 174 (1971). A. Comb erg and S. Ewert, Z. Phys. B~, 165 (1975). N.E. Alekseevskii et ai, J. of Low Temp. Phys. !, 679 (1971). C.G. Granqvist and Claeson, Z. Phys. B ~, 13 (1975). S.B. Woods et ai, proceeding of LT 10 cbnf., p. 303.

S.020 MEASUREMENT OF Hc 11 IN SUPERCONDUCTING VANADIUM TO DETERMINE SPIN SUSCEPTIBILITY P.M. Tedrow and R. Meservey Francis Bi tter National Magnet Laboratory*, Massachusetts Institute of Technology Cambridge, Massachusetts 02139, U.S.A. Since the introduction of the BC~ theory there has been controversy as to whether it adequately describes superconductivi ty in vanadium. The theory implies 2 that below Tc the spin susceptibility Xs + 0 exponentially as T + O. An indirect way to determine Xs is to measure the NMR Knight shift, K. Below Tc the BCS theory predicts that the Knight shift in the superconducting state Ks(T) dro!,s from its normal value KN and as T + 0, Ks(O) -+ O. However, in measurements of H9,3 Sn,4 Al,5 and V,6 Ks(O) was found not to equal zero. Many suggestions were made to account for these discrepancies. For Hg ard Sn for which there was some decrease in Ks(T) below Tc , the proposa1 7 ,8 that spin-orbit interactions made Xs finite at T= 0 has generally been accepted. 9 For Al, later NMR measurements showed that Xs + 0 for T + O. In addition critical field and tunneling measurements showed that Al behaved in detail as an almost perfectly spin-paired BCS superconductor with a small amount of spin-orbit scattering, b·'" 1\ / 3TSO~= 0.07. The small spin-orbit scattering is consistent with the small atomi c number (A= 13) of Al. Tunneling measurements lO of Ga (A~ 31) have also shown Zeeman splitting of the quasiparticle states which is characteristic of spin pairing but with a higher value of spin-orbit scattering b ~ 0.6. The results in Al and Ga confirm the validity of the BCS theory as modified to include spin-orbit scattering for si mple metals. From t he above results we would expect for vanadium (A= 23) that Ks(O ) ::: O. Howe~'er , i n NMR measurements of V by Noer and Knight,6 Ks did not change bel ow Tc. This result wa s parti cularly interesting because V i s a transition metal and the presence of the 3d electrons complicates the pict ure of superconductivity and also the analys i s of the Knight shift . Much analysis ll has been devoted to understanding the Knight shift results. It seems established that the main contribution to K is from orbital (Van VIed) paramagnetism which is temperature independent and should not change in the

76

superconducting state. There are two main contributions to the temperature dependence of Ks(T): the direct nuclear contact term of unpaired s electrons and the indirect contact term of the unpaired d electrons acting thrOUghl~he exchange interaction with s electrons in filled shells. Calculations have shown that it is probable that these two terms of opposite sign are equal in mag ni tude and cance l sufficiently to give the experimental result. Although this pi cture is consistent, the method of obtaining Xs is indirect. In the present experi ment we measure Hc II vs T for vanadi urn f i 1ms thin enough so that the critical f ield is determi ned almost entirely by Paul i paramagnetlsm and from the result deri ve i nformation about the temperature dependence of Xs. The films were prepared by el ectron beam deposition onto a heated (200 0 C) glass substrate onto which had previously been deposited 20 ~ of yttrium to serve as an oxygen getter. 12 The V films were 170 ~ thick as determined by a quartz crystal monitor. The samples were cooled in a He3 cryostat which could be rotated to align the film with the field produced by a horizontal Bitter solenoid with transverse access and a maximum field Of.8.5 T. The critical fields Hcll (T) and HC.L(T) were measured resistively uSlng a dc four-terminal method with a bias current of 1 vA. The transition temperatures Tc were'" 2 K. The midpoint of the transition was chosen as determining the critical field. The resistivity of the films was" 25 Q/o and Hc 10) " 1.6 T. Transition widths were" 0.2 T. Temperature was measured using a calibrated 10 Q carbon resistor. In the absence of Pauli paramagnetism, Hc~1 for a very thin film (d « A) is well described by Hcll (T)= Hcll (0) [(1 - t )/(1 + t 2)] 1/2, the two-fluid temperature dependence. Figure 1 shows the measured values of Hcll (T) plotted vs [( 1 - t 2)/(1 + t 2)]1/2; the straight line is drawn through the points near Tc. The sharp deviation of the data at low temperature from the straight line predicted by the above equation is characteristic of paramagnetic limiting of Hcll . By fitting [dH (T)/dT]T= Tc with the Maki-Fulde 9 theory we calculate cll a value of the spin-orbit parameter b= 0.325. The lowest temperature point on the figure marks the predicted change of the transition from second to first order due to the Pauli paramagnetism. Thus both the magnitude of dH (T)/dT near Tc and the temperature dependence of Hcll at lower tempera cl1 tures are well described by the theory with a reasonably small value of b.

77

4

•• •

3

Fig. 1. Hcll vs. the two-fluid temperature dependence. The solid line shows the theoretical result for T near Tc for b= 0.325 and C= 0. 325. The lowest temperature point marks the point at which the transition should change from second to first order.

We also obtain from fitting the data a value for the parameter I- 2 C= ~vFe2d2~/18hv2 related to the importance of orbital effects i n finite thickness films. Here ~ is the electron transport mean free path, vF is the Fermi velocity, d is the film thickness, ~ is the energy gap at T= 0, and V is the magnetic moment of the electron , We can then calculate what the slope of the straight line in Fig. 1 would be if there .4 ~ ~ I.O were no spin pairing ('so= 0). (' _ t 2 )112 This slope is given by (4kTc~/ 2 1/2 1- 1+12 TIV C) = 8.4 T with the measured value of C= 0.325. The line in Fig. 1 has a slope of 5.8 T. We conclude that this measurement shows rather directly that the susceptibility Xs ~ 0 for T= 0 and that vanadium is a nearly completely spinpaired superconductor as described by the BCS theory generalized to include spin-orbit scattering.

°

References:

* Supported by the National Science Foundation 1.

J. Bardaen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. (1957).

~,

1175,

78

79 8.021

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

K. Yo~ida, Phys. Rev. llQ., 769 (1958). F. Reif. Phys. Rev. 106, 208 (1957). G.M. Androes and W.O. Knight, Phys. Rev . }fl, 779 (1961) . R. H. Hammond and G.M. Kelly, Rev. Mod . Phys . 36, 185 (1964) . R.J. Noer and W.O. Knight, Rev. Mod. Phys. 36, 177 (1964). R.A. Ferrell,Phys. Rev. Lett . ~, 262 (1959). P.W. Anderson, Phys. Rev. Lett. ~. 325 (1959). See P. Fulde, Adv. in Phys. 22, 667 (1973) for a review of recent developments in electron spin effects in superconductors. P.M. Tedrow and R. Meservey, Phys. Lett. 51A. 57 (1975). See B.N. Ganguly, Phys. Rev. B~, 1055 (1973) for references to earlier work. This technique was suggested by Prof. B. Matthias.

ON THE PROPERTIES OF AMORPHOUS VANADIUM ALLOYS

J. Hasse, K. Weber, Physikalisches Institut der Universit~t Karlsruhe, D-75 Karlsruhe, West Germany

More information have become available in the last years on amorphous transition metals. Most of them have been obtained in form of thin films 1 ,2; some recent experiments are reported on bulk amorphous Nb- and Pd-alloys3. The most comprehensive investigation was made by Collver and Hammond 4 • They find striking deviations of the transition temperatures of 4d- and 5d-alloys in the amorphous state from the .well known Matthias-Rule. Not much is known about the behaviour of amorphous 3d-alloys. The occurrence of magnetism in this period and the superconductivity of Cr published recently5 makes it an attractive object of investigation. We have measured the transition temperatures of V-alloys in the range from Ti 20 V80 to V70 Cr . Most of the 30 samples were evaporated at pressures typically 10- 8 Torr by an electron beam source on quartz substrates held at about 10K. The films were in the thickness range from about 100 to 800 A. The composition of the alloys was determined by microprobe analysis, with an error of : 3 %. All quench condensed films grow in a metastable structure with a high residual resistivity. Warming to 20 to 50K leads to a steep irreversible decrease in the resistance combined with considerable changes in the transition temperature. The transition temperature is at maximum of 4K close· to pure V in the amorphous state and decreases below 1,5K with both 20-30 at% Cr or Ti. This behaviour is dif ferent fr om that which might be expected in analo gy to amorphous 4d and 5d . ' . meta ls 4 . The suppressl0n 0 fT c wlth growlng Cr-concentration may be due to the antiferromagnetism of Cr. If this is so, the

80

amorphous, s ,t ate does not affect the magnetie properties, other than in sputtered Cr-films, where superconductivity was found 5 , To learn more about the structure of amorphous V we started electron diffraction measurements at low temperatures. Only two broad rings were observed immediately after condensation. Parallel to the resistance drop during "crystallisation" several sharp rings appeared. A comparison of the ring radii in both states shows no difference for the position of the first two rings. This indicates the same average coordination number for "amorphous" and crystalline Vanadium other than in most of the amorphous simple metals. An estimat e for the a verage grain size in the "amorphous" films taken from the width of the diffraction rings gave 14 A. Presently we try to obtain information about the electron-phonon coupling in the"amorphous" state from measurements of the ratio 2fi o / kTc by mic r owave transmi s s ion .

References: 1. 2. 3. 4. 5.

J.E. Crow, et. al. Phys. Letters 30A, 161 (1969) W. Felsch, Proc. Low Temp. Conf. Lt 13, Plenum,New ·York(1974) W.L. Johnson, S.J. Poon, IEEE-Transact. MAG-11, 189 , (1974) M.M. Collver, R.H. Hammond, Phys. Rev.Letters 30, 92 (1973) P.H. Schmidt et al., Phys. Letters 41A, 367 (1972)

81 S.022 EVIDENCE FOR THE ISOTOr:C VOLUME EFFECT IN SUPERCONDUCTING MOLYBDENUM. T. Nakajima, O. Ter a saki*, and S. Hosoya** The Research Inst i tute for Iron, Steel and Other Metal s , Tohoku University, Sendai 980, J a pan *Faculty of Science, Tohoku University, Sendai 980, Japan **Ins t i tute for Solid State Phys ics, University of Tokyo, Roppong i, Minato-ku, Tokyo 106, Japan The i sotope ef fect in s up e rconductivity experimentally found directly showed that the superconductivity appears being due to the electr on-phonon inter a ction. This was the mo s t import a nt c lue to the succe s s in constructing a mi croscopic theor y . 1 I n studi es in the pa s t , however, only the mass effect of i s otop e has been taken into ac count but not its volume eff ec t: t he latter ha s been n eg l ected, bec a u se it was a priori con sidere d to be s mall. Bein g b ase d on th e rmodynamical cons ider a tion, one of the author s has recentl y examined the importanc e o f the volume eff ect in add i t i on to th e mass effe ct.2 Accor di n g to his r es ult , th e is ot ope eff ect c an be ex pre s s ed as dInTe dInT dInT dlnV d lnM = ~ + 3ln/ . dlnM ' (1 ) whe r e Tc is the s uperconduc ting tr ansit i on t emperatur e , M is ma ss and V is volume . The firs t t e rm is th e ma ss eff ect a lready consid e r ed, whil e the s econd i s th e new term. Accor din g t o th e BCS the ory, 1 the tran s ition temp era ture is expr esse d by Tc = 0. 8 5 e exp( -l/ g). I f th e i s otope effec t i s assume d to be expressed by Tc~ M-a, th e eq . ( l) is exp ressed in a compact form such as a = (1 - r;)/ 2 + (Y G -

< have never been reported. Since the discovery of intercal a t ed lay r s uperconductors, several theories have been proposed on t he ba s i s of the model in which the superconducting order pa r amet e r in adjace nt layers is phase-coupled by the Josephson tunn eli ng . We ha ve s t ud ie d the applicability of such theorie s to t he nn n intercala t ed l ay er compound, ZH-NbSe ' According to Klemm e t a l ? , Z for the lower field li mi t ml 1 -1 Hc2 (T)=g 4S 2 (nnD)2(T-T*) 2 for the higher field l i mit,

o *~ ..co

-) 1

(1)

Nbs (0)= 3y /2rr2kB 2 (1+:\)

( 2)

Assuming ).1*=0.13 and using Tc ' OD and y values, :\ and Nbs{O) are estimated to be 0.81 and 2 . 1 (states/eV spin Nb atom), re-

96 spectively. These values are much larger than other ordinary superconductors. Mattheiss 8 calculated the band structure of layered chalcogenides and obtained Nbs (0)=3.0 for NbSe z and showed that it is in good agreement with the value estimated from Bachmann's value 9 bf A. Our value of Nbs(O) is in reasonable agreement with this theoretical one. The estimated value of A=O.Bl is nearly same value as Nb and such large value is the first report so far for NbSe Z' This must be one of the most important characteristics in layer superconductors. By using Maki's formula 10 on the specific heat jump from the mixed to the normal state, 6Cm/T=(dHcz/dT)z/{4rr B' (2~~(t)-1)}, we obtained, furthermore, KZ~( t ) va lu e s i n term s of 6C IT and temperature dependence of H 21T).1 Tho se va lu es m I c are 1.Z-1.5 times as large as Kl~(t) in the temperature range, 0.6 + Q(2n+1) _ Th2(2n+1)2 2n+l

1) 0

n,m

+ Q(n-m)+Q(n+m+l) (1- 0 ) 1(2n+l )(2m+1) n,m (7)

_w/o L _ _ _-.L _ __ _

~

Tc is the solution for det(A) = o. Note that in a Id system diverges as T~Tp and would induce a solution with Tc>T. p Such a conclusion is wrong since if is too large MF in the form of (7) is not valid . Therefore, we use the result for in the range (2)(5)

_ _ ____'

Pz -

Fig. 1. Fermi surface following (1) for n = 0. 5 in the plane Py = rr/2a. Full line corresponds to £(pz) which has electron-hole symmetry, broken line corresponds to a free electron dispersion for £(pz). Note that all vectors qo = (rr/a, rr /a,2PF) connect opposite side s of the full line. Let us now investigate the temperature Tc within the range (2). Tc is the temperature where a solution i s obtained for the gap function ~n(p) (6) (4)

where wm = rrT(2m+l) and D(p-p' ,n-m) is the phonon Green' s fun ction. The Eliashberg equation s (6) include in (4) the electron self energy correction s ; however, these can be shown to have a small effect on the re sults in the region (2), as should be fo r a MF calculation .

(8)

The solutions for Tc are shown in Fig. 2 for n = 0. 1 (the dependence on n is weak in the range (2)). The cross between the curves of T and T happens at Tc ~ wo /25 which is the maximal Tc for a given w0 .c Thus ~or (SN) x at T = D.25°K Tc is favoured while for KCP at T = lDDoK Peierls instability is dominant (and is indeed due to qo (1)). For TTF-TCNQ the answer depends upon which Wo we should use . For the acoustic )honon Tp>Tc at 58°K, but for the high frequency C=N bond vibration(7 Tp ~ Tc'

100 '

101

S.027 SPECIFIC HEAT AND SUPERCONDUCTIVITY OF Th H and Th D 4 15 4 15

0.015

a

a b.c.d

/

C. B. Sattert hwaite and J. F . Miller

/

/

/

/

T/TF

Department of Physics, University of Illinois. Urbana, 11. 61801. U.S .A.

/

wo/TF =0.3 Superconductivity was discovered in the stoichiometric compounds

O.OlD

Th H15 and Th D in our research group in 1970 and we have maintained a 4 4 15 continuing effort in the study of these materials and their superconducting behavior.

As part of this program, we have measured the specific

heat of both compounds over the range of temperature from l . 5K to 20K in both normal and superconducting states.

From these measurements we have:

(1) derived the electronic and lattice contr ibuti ons to the norma l specific heat; (2) compared the electronic specific heat with the predictions

0.3

S

of the BCS theory; and (3) found unexpected anomalies in the specific heat

0.4

of both compounds above Tc which suggest lattice transformations.

Fig. 2.

The experiments were carried out using an adiabat.ic, heat pulse

Tp (curve a) and Tc (curves b.c,d) for n = 0.1

method with a calorimeter designed to accommodate , samples of 0.3 to

In conclusion. it is possible to obtain Tc>Tp only at low temperatures, while high temperature superconductivity is eliminated by the lattice i nstabil ity. We thank Prof s. M. Weger and H. Gutfreund for many useful discussions .

40kOe.

Normal sta t e measurements were made in a magnetic field of Temperatures were measured using a germanium thermometer calibra-

ted in zero magnetic field and a calibration at 40kOe was obtained using a field insensitive capacit ance thermometer for inter-comparison.

The

precision of the individual measurements was about 1%. The samples were prepared by synthesis at high pressures of hydrogen

References: 1.

2.0 gms.

B. Renker, L. Pintschovious, W. Glaser, H. Rietschel, R. Comes. L. Liebert and W. Drexel, Phys. Rev. Lett. 32.836 (1974).

and high temperatures with the pressure and temperature pr ogrammed t o yield solid, stoichiometric materials of near theoretical density . samples were determined to be st oichiometric to within

± i%

The

by therma l

R.L. Greene. G.B . Street and L. J. Suter, Phy s. Rev. Lett. 34. 577 (1975).

by X-ray diffraction .

3.

D. Allender . J.W. Bray and J. Bardeen, Phys. Rev. B9. 119 (1974).

breadth of the t ransition were determined by measur i ng the diamagnetic

4.

A. Bychkov, L.P . Gor'kov and I .E. Dzyaloshinskii, Soviet Physics. JETP 23, 489 (1966) .

perature for both the hydride and the deuteride are shown in Fig. 1 .

5.

B. Horovitz, H. Gutfreund and M. Weger, Phys. Rev. , to be published.

RESULTS AND DISCUSSION (1) From the normal state (40kOe) data, shown in Fig. 2, the follow-

6.

C.S. Owen and D. J. Scalapino,

7.

H. Gutfreund, B. Horovitz and M. Weger, Solid State Comm. (1974); J. Pl\Ys. CL. 383 (1974).

2.

Physica~.

691 (1971). ~,

849

decomposition of a representative portion and were further characterized The superconducting transition temperature and the

susceptibility by an ac method.

The susceptibility as a function of tem-

ing values for the Sommerfeld y and the Debye temperature 9p were obtained: 2 for Th H , y 7.9 mJ/gm atom Th_K and aD = l77K 4 15 2 for Th D , y 8.1 mJ/gm atom Th_K and aD = 222K. 4 15

102

103

160 r--_.=..:8·T4°-,-K~_ _...:.:13:;:::.8~o~K_ _ _ _.'..':::18'.;.!..7....!°K~

1.0

~ c ::J

Th 4 HI5

;:-

~

Th 4 DI5

.80

.0

E 120 .8

.s >f-

o

~ .60

m

80

t=

0.. W A ) .The thermoelectric g gN 4 power S in this regime is of considerable interest since

(c)

the electronic component Se should disappear rapidly below Hc (as most of the temperature gradient will appear across the S

1.6

laminae);

however the phonon drag thermoelectric power Sg should

remain almost at its normal state value, since the phonon

1.2

diffusion current generated in the S regions will predominantly scatter against electrons in the N regions.

0.8 .0 Fig. 2

10.0 30 dxl0 A

15.0

This should enable

a firm differentiation between Sand S • ~ e g 2. Nearer Tc (say T T / 3), the anomalously high thermal c 2 resistivities ,3 can be interpreted in terms of an additional electron scattering process at the N-S interface 5 ,6. This Andreev reflection process involves an electron quasiparticle incident from the N region giving a transmitted bound pair in the S region and a reflected hole quasiparticle into the

f i g.2 _ Experimental and theoretical thi ckness dependence of HcF/H c for tin films at various reduced temperature s.

N region;

and it appears that this process is effective much

130

131

more in destroying a thermal current than in causing electrical resistance (i.e. charge is transmitted, but energy is reflected). The specimens, vertical cylinders about 2mm diameter and

Transport Properties of the Intermediate State in Indium

60mm long, are cooled to 1.2K in a conventional helium cryostat

T =1·53K.

and temperature gradients measured using carbon resistance thermometers with an ASL cryobridge.

The intermediate state is

set up using a rota,table electromagnet, following the procedure of walton 3 and others.

/''>< 1
0; the result

in which A and 9 are given by Goodman.

6

We treat A as a parameter to be

adjusted in computer calcul ations while 9

is

I~onl

-1

can be evaluated from known data

a

(~o

[16 A

~(O)(dH

l/dT)T (3n)1/2 c

- flux quantum, n

=

t

3 for a

triangular fluxoid lattice).

, ,

Fig. 2.

I

and curve b (antiparallel case)

I

60 0

,

To :

,- 0 .0127 c m

3 .10 K

,

•

H

/

,I

.. 50 0

/

I

~'OO

with anomaly parameter A

= O.

200

5 6 CURRENT (A )

only the leading term.

We expand exp(-Bz) for z =

+~i.

We next set

Thus, we can evaluate the argu-

Curve c (parallel case) with ~ l l 1.64 X 10-1 watt cm- K- and

ments of the logarithmic terms in (7).

The velocity v is then obtained

A

A, and nt.

=

3.2 X 10-4 T •

Curve d (anti-2

~, H ' S We obtain T from the relation T (1 - III )1/2 where I s critt c o o ical current at 0 K. With this preparation, we can now investigate the 8 sources of the experimentally-observed asymmetry effects in Ta wires.

by computer solutions of (7) for any chosen set of parameters W, U,

parallel case), ~ = 6.7 X 10 4 watt cm- l K- l and A = 2.6 X 103 Tt. All curves: R = 2.0 X 102 4 watt cm- K- • Solid circles

parallel case (current carriers, electrons, flow in same direction as in-

and crosses are experimental

face motien) has intercept Ii

100

10

T and Ti are the interface boundary temperaf We expand exp(az) for z = -~f of Fig. 1 and retain

nt X (coherence length).

t

," ,,

tures shown in Fig. 1. ~ f = ~i = ~ =

=

, ,,

~

Curve a (parallel case)

data.

See Refs. 4, 8.

We consider first the case v =

=

O.

The experimental curve for the

4.33 A (see Fig. 2); that for the anti-

149

148

parallel case (carrier flow opposite to interface motion) has Ii = 4.78 A.

8.039 OBSERVATIONS OF ASYMMETRY EFFECTS IN THE VELOCITY OF DESTRUCTION

We assume KS(T;I > I ) = UT 3 because current in S-region is confined into i a layer of thickness A(T). This necessitates W(parallel case)T>W(I=O)T

OF SUPERCONDUCTIVITY BY CURRENT IN TYPE 11 Ta WIRES.* W. C. Overton, Jr., H. weinstock,t and A. F. Schuch

by 65% and W(antiparallel case)T I , v > 0, we solve (7) by self-consistent computer calculai tions for assumed A and n'. When n' = one coherence length, the curves 8 have the wrong shape and cannot be fitted to the experimental data. With

mentally by a number of authors. 2 -

n'

locity of thermal destruction of superconductivity are given in several

10 and suitable adjustments in A we fit the experimental data well.

We obtain A(parallel case) > A(antiparallel case) by 20%. shown by curves c and d of Fig. 2, respectively.

The results are

When A = 0 we obtain curves

a and b of Fig. 2 which give v too large by a factor of ten. To summarize:

the observed asymmetry8 in v is explained tentatively

by ~(I>Ii; parallel case) > ~(I=O); ~(r>Ii; antiparallel case) < ~(I=O) and propagating interface specific anomaly parameter A(parallel) > A(antiparallel) and the width of propagating interface = 2n' ~ (T) with n' = 10 .

The phenomenon of thermal propagation in current-carrying superconductors, first observed by Bremer and Newhouse,l has been investigated experi-

papers. 3-9

6

Phenomenological theories for the ve-

We report here the results of experiments on Type 11 Ta wires

immersed in a liquid helium bath at a temperature TB between 2 K and Tc in which velocities were measured for both forward and reverse current directions and new effects were observed. In the typical experiment a current I < Ic(T ) (le is the critical curB rent at bath temperature TB) is supplied to the superconductor, usually via heavy superconducting leads.

Application of a heat pulse to a short section

of the sample increases local T to, say, T

s

*Work performed under the auspices of the U.S.E.R . D.A.

References: 1.

2. 3.

(1960) • W. H. Cherry and J. I. Gittleman, Solid State Electron. 10., 287 (1965). 783 Phys. 36, J. Appl. c. N. Whetstone and C. E. Roos, W. c. Overton, Jr. , J. Low Temperature Phys. ~, 397 (1971) •

such that I becomes> I (T ). c

layer and thence into the immersing medium.

In this case the N-S interface

5.

W. C. Overton, Jr. , Phys. Rev. Letters, in press. I. M. Khalatnikov, Zh. Eksperim. i Teor. Fiz. E, 689 (1952).

propagates at a constant velocity.

6.

B. B. Goodman, Physics Letters g, 6 (1964).

tivity heals at the initial spot after cessation of the heat i ng pulse.

7. 8.

C. J. Gorter, Physica li, 220 (1967) • W. C. Overton, Jr., H. Weinstock, and A. F. Schuch, Phys. Rev. Letters,

4.

in press.

s

This causes the spot to undergo the transition from the superconducting (S) 2 state to the normal (N) state after which the Joule heating P J (J = curN rent density in N region, P = N-state resistivity) may cause expansion of N the N region. When P J 2 is sufficiently large the J oule heating can overN come the transverse heat conductance loss through the Kapitza resistance He can thus define a threshold current

Ii such that when I > Ii propagation occurs and when I < Ii superconducIi

is an important parameter in the analysis and interpr etation of experimental data.

When I is in the range Ii < I < Ic(T ) the velocity of propagation B may have a dynamic r ange from about 10 cm/sec (smallest measurable with 6 typical medium bandwidth e lectronics ) to more than 10 cm/sec, e.g., as reported in Ref. 2. In our experiments voltage pick-off probes are attached to the sample wire at several points located at suitable distances away from the heater. The arrival of the N-S interface at a probe is signa led by the development of a voltage ramp.

Measurement of the time interva l between successive

151 150 ramps then allows an accurate determination of the velocity. Fig. 1.

4 K in Type 11 Ta wire.

. ...... . .. To ;

60 0

• 00

40 0

r

3.954

~

0 .0121 cm

I(

We have performed experiments on Ta wire samples with diameters of

Velocity vs current at Circles,

squares, diamonds (parallel ar-

helium or embedded in epoxy.

rows):

ments on samples of 0.0254 cm diameter that were immersed in liquid helium.

electron flow in current

900

~300 ~

000

~200 w ~

--'--+--,".00 ---:,'O; ., -

100

----j700

We report here only the results of measure-

Figure 1 shows velocity versus current with the sample initially at TB =

parallel to interface motion • 1000

0.0127 cm, 0.0178 cm, and 0.0254 cm; these were either immersed in liquid

Crosses (antiparallel arrows):

3.954 K at which temperature any possible directional effect is probably

electron flow opposite.

concealed within experimental error. · However, as TB is lowered to about

curve: Inset:

Solid

common fit to all data.

3.5 K the velocity asymmetry becomes evident.

v range from 600 to 1,000

temperature to 3.1 K we find the dramatic asymmetry shown in the velocity

Upon further lowering the

versus current plots of Fig. 2.

cm/sec. Asymmetry in v at 4 K is regarded as being within ex-

We define the parallel case velocity as that in which the direction of the electron flow in the current is the same as the interface motion.

perimental error.

The antiparallel case is that in which the current carrier flow is opposite to the N-S interface motion.

Note two effects in Fig. 2.

First, the curve

fitted to the parallel case data has its zero-velocity intercept at Ii = 4.33 A while the antiparallel case data has a higher intercept at Ii = 4.78 A.

Second, the parallel case veloci ty is significantly greater than the

antiparallel case velocity at all I > Ii and the difference increases as the current increases.

For example, when I = 6 A, v(parallel)

- v(antiparallel) is about 35 cm/sec and this difference increases to Fig. 2.

Velocity vs current at

3.1 K. 600

To '

..

Crosses:

,. 0 ,0121 c'"

3.10 K

• 00

•

If

400

? >00

200 /'

/ /'

/

/

,/

:;.-/

i i

(antiparallel case)

= 4.78

A.

At 3.1 K velocity asym-

I 10

=6

= 10

A.

A to 148 cm/see at

=

253 K.

From these we calculate coherence length

10-6(1_t)-1/2; penetration depth A(T)

metry increases from 35 cm/sec at I

Measured properties of the Ta wire are as follows:

Tc = 4.495 K, P =

N

1.9 X 10- 6 ohm-cm, assay of impurities indicates 0.9996 to 0.9998, electronic specific heat y = 5.22 x 10- 4 J_cm- 3_K- 2 (from bulk specific heat lO ), and en

(parallel case) • 4.33 A.

Note:

/

, 6 CURRENT ( It)

antiparallel case.

tinctly different intercepts. I

138 cm/sec for I = 10 A.

parallel case.

Curves fitted to data have dis-

I

/

/0'" 100

Circles:

thermodynamic) K

2

=

788(1-t ) Oe, Hcl (T)

=

~

(T)

=

4.0 X

7.3 X 10-6(1_t)-1/2 cm, H (T ; 2 c 207(1-t ) Oe, and the GL parameter

= 1.8. Weinstock, et alii have measured the S-state thermal conductivity KS

(T) of a Ta wire sample from the same batch that supplied our velocity samples . Initial analysis showed KS(T;I = 0) could be fitted by the form aT 3 2 + bT 3 , where bT ~ aT. Later analysis showed the form aT + bT is probably better for T > 3 K.

+ BT2, where AT ~

Meanwhile,

~(T;· I.

0) for impure Ta has the form AT

the phonon-electron term BT2 (see Ref. 12).

It is there-

152

153

fore of interest to seek a possible correlation between KS(TB;I

=

0) and the observed velocity asymmetry.

~(TB;I =

0) minus

This correlation is only

crudely qualitative.

5.040

THE PARAMAGNETIC EFFECT IN TYPE-I SUPERCONDUCTORS

F. Ro the n,

C. Lievre.Universite de Lausanne,Inst.P hys.Exp.,

CH-I01S La usanne-Dorigny, Sw itzerland.

In a recent analysis 9 of the asymmetry it is assumed KS(T;I > I ) = i KS(T;I = 0) because I is confined, in the S-state adj acent to the N-S interface, into a s heath of thickness A(T). that it is

This then leads to the conclusion

~(T;I

> I i ) that must be asymmetric. The analysis of the 3.1 K data of Fig. 2 indicates ~(T;I > I ; parallel case) > ~(T;I = 0) by about i 65% and ~(T;I > I i ; antiparallel case) < ~(T;I = 0) by about 33%. Meanwhile, the propagating N-S interface is itself in a Type 11 mixed state and thus exhibits a specific heat anomaly.

The analysis 9 indicates that

The paramagnetic effect in superconduct or s Was first observed by Steiner and Schoneck in 1943 1 . This effect t ak es place in a cylindr i cal wire if superconductivity is destroyed by a current J in the pre s ence of a magnetic field He p arallel t o the axis: one notices that the average longitud inal magnetic induction inside the wire can g r eatly exceed

the anomaly coefficients are also current-direction dependent.

He.

*Work

performed under the auspices of the U.S.E.R.D.A.

t LASL Summer Staff Member, Permanent Address:

Among the models proposed for an explan a tion o f

Department of Physics,

Illinois Institute of Technology, Chicago, IL, USA

this

e ffect , Gorter's model 2 of the destruction of s upercond u ctiv ity by a curren t

in a wire was the most inter est ing : it

c o nsists of concentric norm al (n-) and superco nduct in g (s-)

References:

1,

1.

J. W. Bremer and V. L. Newhouse, Phys. Rev. Letters

2.

R. F. Broom and E. H. Rhoderick, Brit. J. Appl. Phys. 11, 78 (1960).

3.

W. H. Cherry and J. I. Gittleman, Solid State Electron,

282 (1958).

1,

287 (1960).

cy linders moving toward the axis of the wire. Thi s mode l g ives a mechanism for l ong itudinal mag ne ti c fl ux entering the wire; however it does not e xplain the tra pping ef th e flux inside the wire.

4.

C. N. Whetstone and C. E. RODS, J. Appl. Phys. 36, 783 (1965).

5.

J. F. Bussiere and M. A. R. Le Blanc, Proc. LT-13 , Vol. 3, Plenum Press, New York, (1974), p. 221

In a previous article, one of uS showed 3 that t he 4 Lond on-An d reev mac ro scopi c equations of t he in t erme d iate

6.

W. C. Overton, Jr., H. Weinstock, and A. F. Schuch, Phys. Rev. Letters,

sta te lead to : he sa me effect a s

In Press, 1975.

ind e ne nd ent ly of the g eometr y of th e n - and s -

7. 8. 9.

Gorter ' s cyli n ders, q ui te do mains .

C. Zener and P. G. Klemens, Thermal Destruction of Superconductivity,

Therefor e, in this situation, piling up of flux insi de

Westinghouse Research Labora t ory Report l25-JOOO-$1, 1962 .

of the wire is a co nsequence of the ge neral fe at ur e s o f

W. C. Overton, Jr . , J. Low Temperatur e Phys.

w.

2,

397 (1971).

C. Overton, Jr., Phys. Rev. Letters, In Press, 1975.

t he i n ter mediate state b ut not of the detailed geometry

10.

D. White, C. Chou, and H. L. Johnston, Phys. Rev. 109, 797 (1958).

u nd dy nam ic s of the indi vid ual n- o r s - do mai n s. In th e sa me paper, it was suggested that the pa r amagnetic e ffect

11 .

H. Weinstock, et aI, to be published.

c oul d be of oscillatory nature: pi lin g up of a l ar ge quan-

12.

Y. S. Touloukian, R. W. Powell, C. Y. Ho, and p. G. Klemens, Thermal Conductivity, Plenum Press, New York, 1970., Vol. I., pp. 355-362.

t ity of flux inside the wire leads the whole cyl inder to un de rgo a trans ition into the normal state; at this stage,

155

154

Each region is separately in a stationary state. Only

t he distribution of magnetic induction c a nnot remain unchanged and must leave t h e sample by a diffusion process leading to the appearance of eddy current s . The al t ernance

the TMS boundary r = r (t) is moving. The normal regions 2 satisfy the phenomenological equations

of two opposite processes is then likely to produce an os5 cillatory state. Independently of this work, I.L. Landau discovered such oscillations of the flux and of the resis-

where

tance in a wire displayi ng a paramagnetic effect. His ex-

satisfies the London-Andreev equations 4

~

is the normal conductivity. The intermediate region Computation of

planation of the process was also a modification of Gorter's

the distribution of the fields inside the wire can be achiev-

model : he suggested that dynamic two-dimens i onal mixed

ed using suitable boundary condition. Knowledge of the fields

s t ate (TMS) could be present while magnetic flux leaves the

enables one to determine the corresponding value of ~ :ff)

cylinder. In the paper we present here, we try to compute

the calculated value of

the maximal value of the longitudinal magnetic permeabili-

define ~J.max as (01"'2(0-= r;, The experimental values of

ty of the current-carrying wire, defined as

~~

(t) in this state. final l y, we

ft.:

~~

taken from current lit-

terature show a rather good agreement with the theoretical

cDl."'-' - )< = --

=

f

curves ~J.max J.

'ftI{).){

(J, Ho. )

=

~~max

(J,He). This can be explained by the

fact that the amplitude of the flux oscillations are small

5

However, perfect agreement cannot be expected because thermal effects as well as magneto - electric effects, which have

-r is the magnetic flux through a normal section of ....... .1the wire of radius R. In the oscillatory regime, one ex4)

pects ~~

to be time-dependant, so that ~~ max is an up-

per bound to ~J. average

(t) and to its (generally measured) time-

< /-4 ... >

In order to determine ~~ max' we determine the field distribution inside the wire divided in three different domains (r

1. for 2. for

=

distance to the axis) : the wire is in the normal state.

r>r. r. > r>

r. (t)

r..m >r

,

,

the wire is in the inte rmediate sta-

te.

3. for

not been considered here, may play a large role in experiments. Moreover, the cylindrical symmetric state we consider is not stable against spatial perturbations. It is a pleasure to thank Prof. Y. Sharvin for stimulating discussions. References: 1 . K. Steiner, H. Schoneck, Z.Physik i!, 346 (1943) 2 . C.J. Gorter, Physica 23, 45

(1957)

3 . F. Rothen, J. of Low Temp. Phys •

.2"

4 . A.F. Andreev, Zh. Eksp. Teor.Fiz. English transl. Soviet Phys. JETP

the wire is again in the normal state.

359 (1972)

21, ~,

1510 (1966) 865 (1968)

5 . I.L. Landau, Zh. Eksp. Teor. Fiz. 64, 557 (1973) English transl. Soviet Phys. JETP

11,

285

(1973)

156

S.041

lS7

A NEW MIXED STATE OF SUPERCONDUCTORS IN THE CURRENT-

INDUCED TRANSI TION IN LONGITUDINAL MAGNETIC FIELD M. Sugahara

is the diffu-

Pb th in fiLm

sion constant,

thickness 2 (pm)

I; is the coher-

Faculty of Engineering, Yokohama National University, Ohoka,

Bc> = 522

6

(Go)

ence length,

Yoko hama, Japan

V

~""

T=4.2(IO

(.,,")

and 'l' is the order-para-

Recent experiments have shown that current-voltage (I-V) cha racteristics of superconductive whiskers l and of superconductors in longitudinal magnetic field 2 reveal a number of

8 0 (lil

meter. In the

+: 770

E=(O,O,E ) and O magnetic field

the flux flow is considered to be prohibited by the smallness

13= (0 , 0 , B0)' gauge A=

field.

an experiment at 4.2 K using film specimen of pure lead (nominal purity 99.9999 %) fabricated by vacuum-deposition method. 2 The resistance-quantization effect is observed in specimens with relatively large thickness made at high-grade vacuum. On the other hand, when the film thickness is reduced or when spe-

Fig. 1: Current-induced transition characteristics in longitudinal magnetic field ob-

'l'=1jJl(x,y)X is separated into

':630 9:6'0 .0:601

!CA)

By putting 1jJ 2 ( z , t), Eq.

6:665 7 : 64'

o~~~~~~~~~~~----~ 1 2

and (j>-O is used.

mechanism in superconductors in longitudinal field BO' we made

5: '/00

1-9

2

a

(O,BOX,Az(t»

In order to investigate the current-induced transition

2:.400 3: 9'0

(mV)

presence of

of specimen dimension or the presence of longitudinal magnetic

I: :ruo

V 4

electric field

step structures presenting a striking contrast to the fluxflow state 3 in transverse magnetic field. In the former cases

I

•

Z5~

(1)

served in 2 ~m thick film of lead made at l XlO- S torr. I nsets show the specimen dimension and the theoretical and the observed interference pattern.

cimen is fabricated in low-grade vacuum, the transition becomes

[32/3X2+(3/3Y+i21elBox/n)2+S1-2)1jJl (x,y) = 0

(2)

very oscillatory. Figure 1 shows I-V curves with the oscillatory

[( 3/3z +i21elAz(t)/~)2_D-13/3t+1;2-2)1jJ2(z't) = 0

(3)

instability observed in a relatively thick (2

~m)

film made at

vacuum pressure ~l~lO -S torr. From the dependence of the critical c urrent on BO this specimen is found to be type 11 superconductor with the critical field B =622 x IO- 4 T. Insets in Fig . 1 c2 show the specimen dimension, the theoretically derived interference pattern which is descr ibed below, and the interference pattern observed in the I-V rela tion when

BO~Bc2

-2 -2 -2 _ wLth 1;1 +1;2 =1; . EquatLon (2) gives the Abrikosov's lattice 4 so lution when S12=(B c2 /B ) 2 or 1;2 -2=s -2(1-B / B 2)' In the O O c f presence 0 a MeLssner-current flow of critical value j in the z direction, we solve Eq. (3) putting 1jJ2=I1jJ2(t) use of a vector potential

leXp[ikz~)

with

in going and reA

turning path of the current variation . In the following we discuss the transition characteristics basing on a linearized TDGL equation

z

(4)

where ~O is the magnetic permeability in vacuum, and AL is the London's penetration depth. Within a restriction

(1) (S)

where A is the vector potential, (j> is the scalar potential, D

Eq.

(3)

is rewritten as

159

158

(6)

It is shown from Eq.

(11) that the supercurrent varing in

3

4 which gives an Abrikosov's lattice solution in (z,t) space: 2 2 21elE v F t k(z+i 2V F t) nk ] (7) 2 I lE ~2(z,t) = Cexp[0 ]6 3 [ '

• Concerning n the dc component of the supercurrent in the z direction, Eq.

where C is a constant and v

jdc obtained from Eqs.

I

n

2TI

TI

eO

is the Fermi velocity. The restrF iction (5) is found to be satisfied when EO < Ec2

=

(8)

(B c2 -B O)V F

The supercurrent density

(9) and a correct solution of Eq.

(3) give an additional super-

current of negative value 6jdc~k2 besides the main component (4), (7) and (9). On the other hand,

when the applied current j is larger than jdc' the proportionality

(~Z,~t)~(As'As2/D)

reveals that

k~IEO

is satisfied there.

In a inset of Fig. 1 is schematically shown the I-V relation

3s =(j. sx ,j sy ,j sz.)

and the charge den5 sity P is given by the follow1ng express10ns + 2 /m*) (ilelfJ./m*) ('l'*V'l'-c.c.)-(4e

_ (i I e IhsO/om*D) ('¥*H/3t- c. c.) - (4e s o / erm*D) I '¥

which is expected to appear when the mode v=l in Eq.

(12) is

considered. The envelope of the observed patterns

qualita-

tively agrees with the theoretical curve.

I 'l' 12+A 2

p

(z,t) space is always offset by normal current

(9)

12~

(10)

The observed oscillatory instability is thought to inate

imental condition, the frequency of the charge variation is too

with

high to be followed by our recorder. It is seen from Eqs.

(11) where m* is the effective mass of a Cooper pair, er is the elec. . tric conduct1v1ty, an d EO l'S the vacuum dielectric constant, It is found from Eqs.

orig-

from the electromagnetic resonance effect. In our exper-

(9)-(11), the Ohm's law and the Maxwell equa -

tions that the characteristic length of the variation of (jsz'p) . in (z,t) space is (As,A 2 /D) with As 2 =erDm*/4e 21 '¥ 12 • Assum1ng s that the lattice spacing (6z,6t) of the lattice solution (7) in (z,t) space is proportional to the characteristic length (As' A 2/ D), we find from Eqs. (7) and (10) that the time average s < >t of the charge-induced voltage Vi which is expected to appear across the length L of specimen is given by

(7)

and (10) that the frequency is given by f = 211IeIEo/-fik = 11 [(v F E;2/ As) (2IeIEo/h)]1/2, which amounts to 10 8 _10

10

(11=1,2,3, ... )

Hi. The electromagnetic field induced

from the oscillatory charge may found many resonance frequencies which are determined from the shape of the specimen. References: 1. J. Meyer and G. Minnigerode, Phys. Lett. 38A, 529 (1972). 2. J.T. Chen, L.G. Hayler, and Y.B. Kim, Phys. Rev. Lett. 30, 645 (1973). 3 . Y.B. Kim, C.F. Hempstead, and A.R. Strnad, Phys. Rev. 139, A1l6 3 (1965). 4. A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957).

00

(E

L/4) V:l Qv (VhrA /E;2) sin (\ikL/2)/(vkI./2) s c2

(12)

5 , T.J. Rieger, D.J. Scalapino, and J.E. Mercereau, Phys. Rev. Let t. 27, 1787 (1971).

with Q (w)=(1I1/2v/w)5exp[-w2]. From the w dependence of Qv(w), v

we know that t increases rapidly with the decrease of As /E;2' which is proportional to the square root of electron mean free path. Therefore dirty materials are favorable to observe t'

160

S.042 FLUX FLOW VELOCITY AND RESISTIVITY MEASUREMENT.5 IN THE INTERMEDIATE STATE.

161

res by severa l authors(7,8). The present communication reports detailed measurements of

P. Laeng, P. Zweiacker and L. Rinderer

flux flow velocity - performed by direct observation with the

Institut de Physique Experimentale de l'Universite de Lausanne,

magneto-optical technique -, and e lectrical resistance in high

CH-1015 Lausanne-Dorigny, Switzerland.

purity polycristalline rectangular lead and indi um plates subjected to a transverse magnetic field Ha' These measurements

As is well known, the transport properties of superconduc-

were intended to illustrate flux flow resistance in the inter-

tors in a magnetic field are intimately connected with the mo-

mediate state and to test the basic flux flow concept governed

tion of their magnetic structure. Such a motion was first sug-

by the following relations between the average electric field

gested in 1952 by Shoenberg(l) , who noticed that in the inter-

in the superconductor E, the velocity of the magnetic structure

mediate state of a type I superconductor, carrying an electrical

v and the driving current density j

current, the phase boundaries parallel to the current should be unstable and move sideways. A few years later, Gorter(2) showed

v

E

(1) ,

E

(2)

that the flux flow accompanying this motion should give rise to an electrical field across the superconductor and proposed this mechanism to explain the appearance of resistance in the intermediate state of type I and in the mixed state of type 11 super-

where xn is the normal phase concentration and P the electrical reSistivity of the normal metal. The term xnP is called the fl ux flow resistivity. 0

conductors. The first experimental hint for current induced flux flow

2.0

or domain motion in the resistive state of superconductors was

~.

obtained by Giaever(3)who observed that in ' a system consisting of two type 11 superconducting films separated by a thin insula-

1. 5

tion layer, voltages were induced in one film when a current was passed through the other one, the motion of the flux vortices in both superconductors being coupled by proximity effects. Similar

200

600 A/cm 2

400

,...V V

~0Ha

~

T-4.2K 'P.,=14°

-i

h

0.9 0.8

illj E

~H

i

Pb 99.1199 'Y.

1.0

0.7

results were obtained by Solomon (4) in the intermediate state of 0.6

lead. Further evidence for domain motion in the intermediate state was reported by Sharvin(5)who recorded resistance fluctuations

0 .5

0.5

be.t ween point contacts placed on the surface of a tin slab. How0.4

ever the most direct demonstration for flux flow in superconductors has been obtained with the high resolution magneto-optical method developped by Kirchner(6)which allows continuous observation of the magnetic structure of superconductors in the intermediate state. The phenomenon has been presented in motion pictu-

I

0 .00~~~~~--~ 1 0~======~=-2~0------------L---~

30

A

Fig . 1. Resistance characteristics of a lead plate in the intermediate state.

162

163

'£he typical behaviour of a lea.d specimen (0.5 x 10 x 50 mm ,

s .' l!flples, related to edges effects. Correspondingly the

p

Jp ~ 10 4 ) can serve as an eXample to illustrate the re293' 4.2 s u Its of the experiment. Fig.l shows the resistance character-

character of the electrical resistance of the specimen changes

istics measured at different values of the reduced field h = = Ha/ Hc and T = 4 . 2 o K. The magnetic structures observed for current intensities above which the voltage increases linearly

is evident if one notes that in the configuration used here, xn

wi. th t.he curre nt are schematized on the right of the correspond-

curves.

from ohmic to inductive in nature. Agreement with relation (2) ~

h, and if the effects of pinning forces are neglected by

considering only the slopes of the linear parts of the V-I Fig. 2 show. the results of the velocity measurements in

ing graphs. For h Ii and all distinctions in the properties of the sample and the normal metal are due to the tm state layer. The measurements

,'Jhen the superconducti vi ty of a hollow cylindrical sample is destroyed by a current flowing through it, the magnetic field in the hole is zero and remains small also in the metal itself near internal surface of the sample. For this reason the normal state of the metal there should be unstable. L.D. Landau made an assumption that in this case a thin layer of some macroscopically homogeneous "mixed" state is produced on the internal surface of the sample. In as much as the mixed state of type-I superconductors apparently always takes the form of thin layers, we shall use here the term "two-dimensional mixed state"

(tm state), to

were made on the indium and tin single crystals . ..le have inves3 tigated two indium samples with RI = 2 mm and R~ = 4 mm, the o 3 resistance ratios being p(300 )/p(4.2 o ) ~ 2.10 , and two tin samples with RI = 3.7 mm and R2 = 4 mm, the resistance ratios 4 2 2.10 and - 1.5.10 respectively. The samples were put

being -

in a single-layer solenoid for producing a longitudinal magnetic field.

In some experiments a niobium wire was placed in the cav-

ity of the sample; the current in this wire could flow independently of the current in the sample. Passing a current through this central wire we could destroy the tm state layer, displace it into the interior of the sample or on its external surface.

distinguish it from the mixed state of type-II superconductors.

,/

/

If part of the volume of the cylinder is occupied by the intermediate state, superconducting domains can emerge to the

I

internal surface, whereas the domains of the normal phase must be covered by the tm state layer ...lheI:l the current I

> I.

1.

=

R2 + R2 I 2 I c 2RIR2

(RI and R2 -

internal and external radii of

the sample, Ic - the critical current), the radius of the intermediate-state core becomes smaller than the radius of the hole and the whole sample except a thin layer of the tm state on the inner surface must be normal. The experiments carried out on a single crystal indium sample have shown that the state really exists on the internal surface of the hollow cylindrical samples when the superconductivl ity inside the metal is destroyed by a current. In the case when a wire is placed in the cavity of the sample and currents in this wire and the sample flow in the opposite directions, the tm sta"te layer can be produced not only on the inner surface, but on the external surface and in the interior of the sample as well, depending on the ratio between the currents in the sample 2 and in the central wire.

()

I

H. (HH;1~ .

Pl ot of H~

3.19 0 K, I

1.25I c · +) . e), &) 2.0 BI sample 2, T = K; +) I = 1. 25I ' e) I = 2. OBlc ; &) I c c a nd the current in the wire is equal to -2.3I ' c 0)

sample I, T

=

=

3.27 0

The figure shows the dependence of the longitudinal magnet-

167

.-cC

fi.eld ia the

:nagnetil~

cav .~Lty

of the indium samples H . on the external .~

field He' When the tm state layer is on the internal sur-

face of 1:he sample (curve 1), the magnetic field in the cavity is largel' t.han the externa.l magnetic field. When t.he tm state layer is ,m':he external surface, the longitudinal magnetic field in

S.044 DIRECT OBSERVATION OF THE TWO-DIMENSIONAL MIXED STATE IN THIN FILMS OF TYPE I SUPERCONDUCTORS.

W. Rodewald Optisches Institut der TU Berlin, 1 Berlin 12, StraBe des 17. Juni 135,

West Germany

·the cavity is smaller than the external one. In the case when the

cm state laye:c' is in the interior of the sample, Hi

~

He' We havE:

The distribution of magnetic flux in type I superconductors strongly

2.1so obsex-ved the same effect for both tile tin samples.

depends on the geometry of the sample. In thin films a transition from

:Cn t.he case He « Hc and I I , 1:he value of ]J = Hi/ He was i 2.1 for the indium samples, ~ 2.35 for the pure tin sample and

line arrangement was studied experimentally. Their distribution and the

~

=

2.U7 tor the dirty tin sample. The magnitude of this paramag-

netic effect practically does not depend on the temperature. Besides we have measured the time constant T characterising

the intermediate state to a two-dimensional mixed state occurs!. The flux

interaction of flux lines with the crystalline structure of the films will be demonstrated. Such a tranSition was predicted by Tinkham et al. 2,3 and Maki 4 •

the velocity of penetration of the longitudinal magnetic field

According to this model, the magnetic flux penetrates the film in regularly

into the cavity of the tin samples. It has turned out that the

arranged flux lines. Each of them should carry only one flux quantum. 5 Fetter and Hohenberg der'ved that a tr1anguar ' I I attice should exist. A

presence of the tm state layer on the inner surface of the sample leads to a large in increase in T compared with the time constant of the sample in the normal state/T , and the relation n

~Tn

T (p ~ -" x-esistance of 'che

have the same maqnitude, a mixture of both lattices may occur. Increasing

P 1.

~Tn

4

phase diagram for thin films of type I superconductors was calculated by 6 Lasher . As th . e free energy of a triangular and of a square Lattice nearly

the film thickness, a transition to multiquanta flux bund les is to be expected.

Pr.

1 . 0 , - - - - -_ _ _ _ _ _ _ _ _~

tm state layer for the longitudinal cur-

n - resistance of the sample in the normal state) is ful-

rent, P

filled fox- both the samples in a wide range of currents flowing

Mixed State Triangular or Square Lattice

t.hrough the sample.

Cv· 1)

References: Landau and Yu.V.

1.

LL.

......

(:1.%9)

1.1"

3.

r.:;: ..

S~,arvin,

ZhETF Pis. Red. 10. 192

[JETP Lett. 121 (1969») •

Landau and Yu.V . Sharvir., ZhE'l'F Pis. Red. 15, 8&

(1972)

["ETP Lett. l~, 59

L~ndau,

Z11. Eksp.

(1972)J.

~eor.

FiR .. .ti7,

2:;0 (1974).

Fig. 1. Phase diagram for the transition from the intermediate state to a two-dimensional mixed sta te of thin type I superconductors. :J

'0

o

El

N H

H

H

Z

Z

rt)

H

Z

11

lJ"I

U"\

0

0

+>8 >< E «> :r: N 0\0

~

M

'"

N

\0

0

0

o

o

.::-

\0 M

CX)

" :1 11

,.

\0

N

""

""

.::-

M

\0

0

\0

.,

0

:1:

+> :r:

)

-«>

~,

0,

@ Ul

11

M

CX)

..," ".c

11 ()

rt)

:r:

o

U"\

\0

U"\ M

U"\

.0

Z

.0

Z

.0

Z

174

1

lat tice at Bo' For Nb we obtain EB=10- 5 ~J/m. Theoretical estimates for the wall energy parameter 0 are about 3 times higher than our experimental resu l ts. The GLtheory fo r Ta ( ~0=92 nm B) yields 0GL=1.B9 ~(T) =1 51 nm and Bar deen ' ~ 9 approach 0B=230 nm. A theoretical inve~tigat ion of the boundary energy of the Shubnikov phase a t Bo in low-~ type-II superconductors valid for t+l has been given by Hubert 10 and will be discussed at the Conference. This project is supported by the Deutsche Forschungsge meinschaft. I am indebted to Prof. A.Seeger, A.Bodmer and W. Wiet haup for us eful d is cussions.

175

S.046

CALCULATIONS OF THE THERMAL CONDUCTIVITY IN THE

MIXED STATE OF A DIRTY SUPERCONDUCTOR. R.J. Watts-Tobin and Syozo Imai+ Department of Physics, University of Lancaster, Lancaster, LAl 4YB, England. In a dirty type 11 superconductor it has proved possible to perform self-consistent calculations of a circular cell 1 approximation to the mixed state structure and density of 2 states by numerical integration of the microscopic equations 3 in the form given by Usade1 . Using ·t hese results one can compute transport coefficients for processes in which the mixed state structure does not change with time.

References: 1. U.Essmann, Proc.Internat.Conf. on the Science of Superconductivity, Stanford, U.S.A. 1969, Ed.F.Chilton (NorthHolland Publ.Co.Amsterdam, 1971). 2. H.Trauble and U.Essmann, Jahrbuch der Akademie der Wissenschaften, Gottingen 1967, (p.l7). 3. U.Krageloh, phys.stat.sol. 42, 559 (1 970). 4. Yu.V. Sharvin, Soviet Phys.-JETP ~, 1031 (195B). 5. R.N.G oren a nd M.Tinkham, J.low Temp.Phy s. 2, 465 (1971). 6. L.D.Landau, Zh.eksper.teor.Fiz. I, 371 (1 93 7). 7. A.Bodmer, phys.stat.sol.(a) 19, 513 (1973). B. J. P.McEvoy, D. D.Jones, and J.G.Park, Phys.Rev.Letters ~, 229 (1969). 9. J.Bardeen, Phys.Rev. 94, 554 (1954). 10. A.Hubert, Theorie der Domanenwande in ge ordneten Medien, Springer-Verlag, Berlin, Heidelberg, New York 1974.

In the

dirty limit the transport coefficients are local, provided the perturbing field changes by only a small fraction over 4

an electronic mean free path

Here we report calculations of the electronic contri-

J

r

bution to the thermal conductivity in two different geometries.

In the longitudinal geometry, the temperature

gradient is parallel to the magnetic flux, and is uniform throughout the specimenS

The conductivity of the specimen

is given by the spatial average of the local conductivity. In the transverse geometry the temperature distribution T(r) is determined by numerical integration of the continuity equation

V.

(K~T)

o

(1)

where. K(r) is the local thermal conductivity.

It was shown

previously4 that K(r) is given by the formula K (X'')

J~

3

~

2n 2 (kT)

3

"'0

_..;;.w_2_ _ _ _ C (w"t) dw

(2)

cosh2 (w/2kT)

where the local coherence factor is C(wr)

=

~e G (wr)] 2

-

[jm F (wrU 2

(3)

in terms of the G and F functions defined by Usadel, after +On leave from Research Institute of Electrical Communication, Tohoku University, Sendai, Japan.

~:

177

176

the continuation

iw~

+

physical frequencies.

distribution experimentally.

w+io from Fermi frequencies to It is assumed that pinning forces

stop the vortices from moving in the temperature gradient. In order to compare the calculations with experiment, allowance must be made for the phonon contribution to the heat conduction.

This is particularly important for dirty

material because the electro?ic conductivity is low even in the normal state.

At low temperature the phonon contri-

portional to liB.

If there remain significant differences

b 2 tween the longitudinal and transverse thermal conductivities after subtracting this contribution, then the electron-phonon scattering length must be long.

If it is

short, the phonons will pull the electrons into a uniform temperature gradient and there will be no difference between the geometries.

bution is always predominant, and experiments by Lowe (private communication) on moderately dirty material at

At low fields where the effect

is strongest the phonon conductivity is expected to be pro-

The calculations presented here are for material with a

~ow

temperature show that after subtracting the electronic contribution, the conductivity is accurately proportional to liB near H ' It follows that the thermal resistance of the cl Meissner state is negligible to phonons compared with the

Ginzburg-Landau parameter

the calculations for K=14

K=4;

were for the longitudinal case.

The method of calculation

in the transverse case is very simple in principle.

Within

the circular cell approximation, in which the conductivity is

resistance due to scattering by electrons in the vortex

a function only of r, the distance from the centre of the

cores.

vortex, we look for a solution of Eq. (I) of the form T(r}

Only phonons whose wavelengths are of the order of

the coherence length E; or smaller will be scattered and short wavelength phonons are rare at low temperature.

The phonon

temperature cannot have signif.icant structure on the scale of the vortex. is sufficiently weak, Eg. (1) will describe the conservation of the electronic heat current alone.

T(r) should be

regarded as a field coupling to the energy density of the the thermal conductivity of the electrons is

connected with the response function to T by an Einstein 6 relation . In order to define T no mechanism for inelastic scattering between electrons is needed:

however, it is

necessary that T should vary slowly on the scale of the electronic mean free path.

a, where (r ,a) are the polar coordinates of the

R(r} cos

point r. d 2R

Ci.?

The equation for R is +

(1r

dR dr

+ldK) Kdr

R

?

=

(4)

0

and it needs to be supplemented by boundary conditions.

Provided the interaction between electrons and phonons

electrons:

=

In order that T(r) should have

the origin, we may take R=O.

At

At the cell boundary we take a

condition analogous to the one giving the Clausius-Mossotti formula for the refractive index of a dense medium. leads to the formula aK (a) dR KT If(a) dr

I

r=a

It

(5)

for the transverse thermal conductivity where a is the cell radius. The computations show that under all conditions the temperature gradient is very uniform in the core where the

th e non-uniform structure predicted by our calculations the

conductivity is high.

mean fre e p a th for s c attering of electrons by phonons must be

the conductivity in the two geometries is shown in the figure.

long compa r e d with the vortex size.

The differences are most pronounced at low field and temp-

This does not conflict

The results of the computations of

wi t h the di r ty limi t approximation, because impurity

erature:

sca tt e ring i s a lmo st e l as tic.

transverse conductivity vanishes at T=O.05 Tc until the cores

It should be possible to test

t h e ass ump t i o n o f a se para t e e lectron and phonon temperature

as predicted by Vinen (private communication) the

of the vortices begin to overlap.

Ne ar Hcl the vortices are

179

17 8

separate d by mat e rial wi th t h e n e gli g i bly small Meissner

S.047

OBSERVATIONS OF CRITICAL CURRENTS INDUCING HOTION IN

s t a t e condu ctivity and in t h e t r a n s v e r se geometry ther e is no

LANDAU DOMAIN STRUCTURES

me c h an ism for e l e c t r o ni c h ea t cond uc t ion . We t h a nk G. Ei lenb erg e r and h i s c ol l eages , with whom we

R. Aoki, B. Shinozaki, T. Miy azaki

h a v e h a d many discussi o ns, f or t hei r ho s pi t a li ty i n the

812, Japan.

Department of Physics, Kyushu University Fukuoka, Hakozaki,

I n stitut fUr Festk5r pe r fors chung of t he Kernfor s chunsan lage JUlich wher e t hes e calcu la t i o n s we r e c ompleted .

We thank L .

Introduction

Krame r and W. Pesch fo r the us e o f t he i r c omput e r progr a ms.

The dynamical character of . flux-flow phenomena contains bas-

This work wa s s uppor ted by a gran t from t h e Scienc e Researcn

ic problems of physical interests, but is quite complicated due

Council.

to the entanglement of fluxon-fluxon and fluxon-crystal lattice Lon gl u dlno i

-

-

-

Transver se

0.75

Figur e . Electroni c contrib ut ion

vestigations using type-I superconductors have been proposed,

to the transverse and

because the macroscopic magnetic structure diminishes the flux-

l ongitud inal t hermal con -

on-fluxon interactions and clarifies the flux-pinning problems.

defect interactions. In order to shed light on this problem in1

ductivitie s f or v ariou s t empe ra tur es a~ functi o n s

050

z

";;C I

/

electron shadow miscroscopy . The specimen in an electron micro-

K= 4 .

scope was cooled down through a thermal linkage with a helium

/

We have investigated the intermediate state of Pb films by

vessel and the temperature was regulated with an electrical

025

,/

Experimental

All graphs of B/ H . c2 a r e f or material with

heater. The specimen was placed in an external magnetic field

/

and in the electron beam as illustrated in Fig. 1 . Periodic shadow patterns corresponding to the Landau domain structure were observed in a certain geometry and the distance 0.25

-

05

0.75

10

of periodicity was analysed in order to sort out the no r mal to

- B /H c2

Re f e renc es: 1. L. Krame r , W. Pe s ch and R. J . Wa tt s -To bin , J . Low Te mp. Ph ys .

superconducting interphase energy.2 In this experiment we suc-

~,

2 9 (19 7 4 ). 2 . R. J . Wa tt s -To bin , L . Kr a me r a n d W. Pe sch , J. Low Temp . Phy s .

from a pin-hole edge located in the centre of the sample as il-

1.2.,

in thickness (d), 1 mm in width (w), and 3 mm in length (L).

71 (1 974 ) . 3 . K. Usadel , Phys . Rev . Le tt.

~,

50 7 (19 70 ).

4 . W. Pes c h, R. J . Wa t ts - Tobi n a nd L . Krame r , Z . Ph ys ik

Phy s . Rev .

~,A

1481 (1964 ).

lustr ated in Fig. 1. The specimens were microstrips of 10 - 50 The samples were made b y chemical etching from single c r ystal

~,

253 (1 974) . 5 . C . Caro l i a n d M. Cyrot , Phys . Kon den s . Mate r ie ! , 285 (1965). 6 . J . M. Lu tti ng e r , P hys . Re v . ~ , A 1 505 (1 96 4 ).

ceeded to observe a similar shadow pattern as shown in Fig. 2(a)

f o ils using starting materials of 99.9999 % pure pb . The pinhole as the measurement probe was fabricated

by

and had nearly ellipsoidal cross section (a - 150 50

~ m).

photo-etching ~m,

and b -

Four electrical lead wires were attached with silver

paint and indium soldering. After the final manipulations the

~m

180

181

o specimen 'was annealed in temperatures up to 300 C. The application of a current I in the specimen in the

super~

conducting intermediate state gives rise to a Lorentz like force 3 acting on the magnetic domains. It was observed that as I in-

This critical current was affected by a temperature change of ~ T ~ 2 x 10- 2 K. Therefore we used an lmage . . . er system lntenslfi for minimizing specimen heating due to electron beam irradiation during observations. In the critical pOint experiments the temperature was in-

creases, the shadow pattern may move during a shor.t time in some parts of the specimen. This happens more frequently and more ex-

creased with a constant current, and the voltage across the spe-

tensively the larger I is. Fig. 2 illustrates the change of a

cimen was simultaneously measured. It was found that at the cri-

pattern by the movement. Plate (a) shows the pre-state of the

tical pOint

i10

abrupt increase in the d.c. voltage due to flux

static periodic structure, which suddenly moves into (b), and in

flow resistivity was observed at least within the detection lim-

(c) the periodic structure is again recovering. This random move-

its of ±0.2

ment begins to occur at a threshold current Ip without noticeable

flux jumps wer e detected once in a while. It means that the ob-

slow down during the usual observation time. The current I could + p clearly be determined within the error limits of -2% with good

served critical point corresponds to the critical state defined as

reproducibility after thermal cycling up to room temperature.

Fp

=

FL

=

~V , only small pulses of noise synchorized to large

5001r-----------------------~

Ip x B

D:

where Fp is the pinning force acting on the magnetic flux in the specimen having magnetic induction B.

__ 400 4{

Hext=36G

0 0

.: 1izxt=55G o· Hzxt=75G .: Hrxt=92G

"

•

E

-a.

using a pin-hole edge (about

in a lead film (40

150

~m

x 50

~m

size) as the

measurement probe.

~m

thick) ob-

served via current induced motion: before (a), just during (b), and after (c) the movement profiles; T

=

6.3 K, h

=

0.49.

0 •

100 o

•• •

c

'b Ill..

50

0

o •

o

0

I

55

of different thicknesses. Here we can

•

•

external fields H ' ext The results are poltted in Fig. 3 for samples

•

200

men temperatures T and conducting magnetic structures

A

a

o

current Ip was meas-

in shadow electron microscopy

• • o o

of Fp' the critical

Fig. 2. Shadow patterns of super-

0 0

In order to inves-

Fig. 1. Geometrical conditions

0

a

• 0

Resul ts

ured in various speci-

0 0

300

tigate the character

•

00

I

6.5 TClO 7.IJ

Fig. 3. The dependence of critical

see a linear depend-

current Ip on temperature T and ap-

ence of Ip on T in the measured temperature

plied magnetic field H in single ext crystal Pb samples of different thick-

range near Tc'

nesses; 23

~m

CA), 14 ~m CB) , and 50 ~m (C)

182

The relatiun j

p

can

183

be expressed as

S. 048 DYNAMIC S'IUDY OF '!HE TRANSITICN CF A SINGLE SUPERHEATED SUPERl:::n/DLCTING GRANUIE 'ID THE NORMAL STATE

= a ·[(l-t)-S(h+y)]

C. Valette and G. Waysand

where t

= TIT , h = B/H (t = 0), and j is the current density c c p lp/ w.d. This e xpression seems to to correspond well to the gen4 eral scaling formulae discussed in type-II superconductors.

Physique des Solides, Universite Paris-Sud, 91405 Orsay, France

All samples were prepared using the same procedure but slight

perccnductor to the nonnal state has been discussed by Faber 1 in the case

The dynamic study of the transiticn of a rretastable superheated su-

differences in lattice defect distributions were observed when

of long cylindrical rods of tin,

checking with the etch-pit method.

normal state propagates fast on a thin crown around a longitudinal por_ tion of the cylinder, reaches the axis in a tirre 7f~(l+p)

It was found that the coefficients 13, related to the intercept of the jp(T) curve

with the abscissa axis, is rather in-

SrI!l

in diarreter. The nucleation of the

T =

a , which mostly determines the magnitude of j

(1) P H -H (ro radius of the cylinder, 0 electrical conductivity p = -2.........£ H , H' 0 applied field, Hc themodynamical critical field) ; in the inilial part of

similar single crystals; e ven as l a rge an e ffect as the thick-

dary (diameter r') diverges as

sensitive from one sample to another and may only reflect the thermodynamic character of the material, while the coefficient (T), is quite p structure sensitive and differ s b y more than a f ac tor o f t e n i n

this prcpagation tcMards the axis, the speed dr' of the interphase boundt ~ because of the initial disoontinuity of

surfac~ of the sarrple ; on Faber's results it

ness dep e ndence may be overwhelmed. These r e sults see m to be

the llBgnetic field at the

likely on the basis of current theories that deal with the char-

.91 is well described by a L dT It dependence as long as r' .\: O. 7 r 0 ; this tirre deperrlence is the one which

acter of flux pinning forces.

can be seen that the rate of flux penetration

would be found for a semi-infinite space of super-conducting material with a plane boundary, showing that the flux penetration is initially control-

References: ~,

L.M. Kano and V.A. Shukman; Zh. ekspr. teor. Fiz.

2.

R. Aoki, U. Krageloh, T. Miyazaki, and B. Shinozaki, Phys.

nitial secticn of the cylinder, the normal phase propagates alcng the axis

Lett. 45A, 89

with a speedof sOle tens of an/s. Faber's model is well confirrred by his

3.

K. Maki,Prog. Theor. Phys.

4.

D. Dew-Hughes, Phil. Mag. 30, 293

(1973), JETP

n,

452

890

led by the speed of the interphase boundary and not by the cylindrical

1.

shape of the sarrple. After canplete transition to normal state of this i-

(1973) .

(1973). ~,

448

(1969).

(1974).

exper~ts within the poSSibilities of his experirrental a::ranganent, Le. T»4.10

s (response tirre of his galvanOleter) and p = HoHHc < 0.03 (only

small superheating can be achieved on such macrosccpic sarrples because of the presence of defects). The aim of our study is to see if Faber' s model remains valid in a canpletel~ different range of scale, i.e. T = SOle 10-7 s and p " theoretical limit s~-Hc.

c To do this study we use a very small Hg sphere of a first kind superconductor (diarreter chosen between 20 and 40IJ) with a very good surface shape, obtained by slowly cooling a drcp of liquid rretal. The dynamic study is done by a very special electronic device recently develcpped2 . A pick-up locp (radius ~=65 IJ ) surroun:ling the superheated sphere is connec-

,~,

"---'-

184

185

bed through a strip transmissi on line to a pulse transformer at roam tem-

beam. The observed signal V(t) does not vary with the inpact position

perature . The secondary of the transformer feeds a law-noise change ampli-

(randan, unknavn) and is the same as the one observed without the beam

f i er with a virtual ("electroni cally cooled") resistance. The output ten-

this is a confirrration that the nucleation process is shorter than the 6ns

sion V i s proportional to

dead time of the amplifier. To go further, one must build statistics on

V (t)

t

'V

1

(2)

OOK. ~n may be interpreted as an imaginary coherence length for T > Tcn.

These equations were solved for the lowes t value of

VI

-0

>,

>, I-

~

Cl

194

195

Fig. 2 shows our results of HO(Cu/Pb)/Hc2 vs. ds/E for one of our specimens.

Hc2 and

~ (bulk)

S. 051

Ref. 1, assuming ds = 1450A (the measured value was 1400A) for a better fit ° within the experimental accu racy. The other data are: dn = 1380A,

~n(film) = 1710A, IEnl=

ET//1 +

3ET7~n with ET = 13740/T A for Cu, Tc = 7.26 K,

o

~s

Since f aJ

of the transition signal S

Sn sphere 15.4fJm

with temperature and field .

T jTc = 0.9961

Experimental details are . gLven elsewh ere 4,2 We use

et:

References:

above Hc'

H ' the weak-field approximation breaks down, because the surface order sh parameter W decreases noticeably. An exact one-dimensional calculation by

Gor'kov pair potential is approximately continuous across the ns boundary

We are grateful to E. Guyon for arrangi ng , tlli s co 11 aborati on.

r~ion,

tend the measurements of o(H) into the metastable

ously, ideal superheating up to the field H h = H / KIT has been observed

we conclude that the

for the Cu-Pb system over a large range of temperatures.

A depen-

dence 0 ~ 1 + a(H/H ) 2 in "weak" fieids is predicted by Ginzburg-Landau

the Orsay Group for K«l gives

6.6 210 500 133

He found 0 in tin

to inc rease quadratically with H, by a total of 1-3% up to Hc'

s

eratures were: TOK

FIELD DEPENDENCE OF THE PENETRATION DEPTH IN TIN UP TO THE SUPER-

HEATING LIMIT

were calculated from Ho(Pb) and Fig. 5 of

1) the breakdown of electromagnetic screening in the normal state gives a direct measurement of 0 for a particular crystal direction, while at low frequencies qo "«

1)

(1) simplifies to: aE,s

(ol/ o1n)

------------------------,

(2)

equation

204

205

10

lO

-0

8

,', 2

6

-10

co 1)

4

Tcf

I

-20 2

o (T - T)(mK) c Figure 1.

Electromagnetic attenuation aE,s of f as t shear waves at

Figure 2. Comparison of the observed temperature dependence of the

60MHz in superconducting cadmium for propagation along at

radiation efficienCY H in superconducting tin along ~l> at 15.3

temperatures near Tc

MHz (experimental points.) fitted to a theoretical curve with the parameter O/AL(O) = 15.74.

=

0.535 ± O.OOSK (experimental points 0), and

the normalized conductivity 02s /oln

=

~2 derived from the

attenuation data (experimental points +, with error bars I). where

approximately i sotropic for two directions o f current flow in the basa l plane, in strong contrast to the screening

0l/oln -=- 1, and near Tc' on B.C.S. theory ,

20

2

length 0 in the normal state.

(T c--T) =

°*2

(3)

directi on in Cd.

O/AL(O) for the

superconducting screening. TABLE I

LONDON PENE TRATION DEPTHS "L (0) IN Cd AND Zn. v(MHz)

S imil ar measurements in the bas a l

planes of Cd a nd Zn over a range of ultrasonic frequencies v

we estimated q~o = 1.6 and at this frequency an enhanced A(O,q) shows the onse t of non-lo cal behaviour in the

This linear dependence on (Tc-T) i s shown in fi gure 1 , the gradient providing a direct meas ur e o f

For 270 MHz in Cd ,

Cd ll 20

gave the resu l ts in Table I. The scree ning lengthso in

O/A L( O)

60

1 8 .6 ± 0.7

270

1 6. 4 ± 0.7

o( ~m)

0.81

AL( O)( nm) 40 ± 2 .5 46 ± 2.5

the normal state a re h i ghly aniso tr opic and determined

CD 1010

146

5.4 ± 0 . 2

0.24

41 ± 2.5

se parately, enabling AL(O) t o be obtained for

Zn ll20

60

23.9 ± 0 . 9

0 . 59

25 ± 2

Zn 1010

60

7.8 ± 0.3

0 .24

120

7.6 ± 0.3

each crystall-

ographic direction. These results show that at low frequencies where

~

(q~o«l,

is the coherence length), AL(O) is ( as expected)

30 ± 2 32 ± 2

206

207

In the second me thod the radi ation efficiency H f or the e l ectromagneti c de t ec tion of ultras oni c shear wa ves in the non-local limi t is me asured in the norma l and super4 The normalised e f ficiency i s : conducting s t ates ne ar Tc '

SUPERCONDUCTING TUNNELING INTO FILMS WITH VOLTAGE STEPS* K. E. Gray Argonne National Laboratory, Argonne, Illinois 60439, USA

(4)

which for low frequencies (qo«l) gives a temperature dependence (figure 2) near Tc in excellent agreement with experiment in a Sn crystal. The only adjustable parameter 4 . f" d is again oIAL(O) and in this case Thomas et al. verl le that AL(O) was isotropic for different polarization directions in the (001) pl ane. PageS and Leibowitz (private communication) have estimated 0= 0.23~m from the observation of screening breakdown in Sn , so that our measurement 25.3 ± 2.5 nm, in good of O/AL(O) = 15.74 yiels AL(O) 2 for this direction . a1 agreement with the 25 nm of Tai et An advantage of this second method is e lec troma gnet ic and so not subject to correction of the attenuation method. We are grateful to D.- P .Almond. E. Sendezera and R.L.Thomas for their

that H(T) is ent i rely the deformation A. M. de Graaf, valuab l e a s sistance.

References : * Work supported by the Science Research Council, London. 1. See R. Meservey and B.B.Schwartz,Superconductivity, Chapter 3, e d. R. D.Parks (M. Dekk er, N.Y., 1969). 2. P . C.L.Tai, M. R. Beasley and M. Tinkham, Phys.Rev. ~ 11, 3.

411-419 (1 97 5). D. P. Almond, M. J .Lea and E . R. Dobb s , P-r oc .Roy . Soc . ~

4. 5.

(1975) (to be publi shed). R.L.Thomas , M.J. Lea , E . Senda zer a and E .R.Dobb s , J. Phys. F: Met a l Phys. ~, L2 1-L 25 (19 75 ) . E. A. Page , Ph.D. Thesis, University of Maryl and (196 9) .

S.O q4

Voltage steps in the I-V curves of superconducting films and whiskers is a common occurrence. Regular step structure has been reported in tin whiskers , tin microbridges and lead constrictions. These are explained by phase slip in the one dimensional tin systems and by magnetic field generation in the two dimensional lead constrictions. Wide lead films in a parallel magnetic field above Hc2 have also shown this structurewith quantized resistances of the steps.l This has been suggested2 as evidence for conduction via quantized quasiparticle states. We report observation of voltage- steps in superconducting films along with ~imultaneous tunneling measurements. These voltage steps are very· similar to the quantized resistances fo_u nd in the wide lead films. The sample_ geometry is shown in the lower inset of figure 1. Both films are low pinning granular alumin~ with an oxide layer forming a tunnel junction between them. If the longer strip is current biassed with a shunt capacitor t -o reduce noise, one obs-erves the voltage steps shown in figure - I. After exceeding the critical current and driving the sample into the normal state, the voltage jumps discontinuously from branch to branch as the current is lowered. These branches extrapolate to zero voltage at I .. '" 0.5 lc' and the differential resistance of the branches inCFeases regularly in units- of 0.01 n. This was temperature independent for our measurements between 0 . 94 Tc and 0.99 Tc' Some branches we~e unstable and could not be easily observed at this part1cUla-r temperature. In the top of figure 1 we show two examples of the differential resistance of the tunnel junction corresponding to *Based on work performed under the auspices of the U. S . Energy Research and Development Administration.

209

208

V45 JUNCTION VOLTAGE (0.2 millivolts per div.)

a; La..

0

15 (,,) en

lLJ

U

Z

...

>-

0

~ V) V)

lLJ

Cl:

.!::

:c ... ..s z

0

....J IU

et

i= z z => lLJ

~

n =1

Cl: ....J La.. Z La.. Z Cl =>

""

lLJ lLJ

I-

"-n = 10

'-n = 21

0

5

0

1000 A Z lLJ 10 ALUMINUM FILM, Cl: Cl: u; RN = 3.4 Cl. => u E 15 ~~:::::---::."':.__ 3 .S!

I-

.n

0... Cl:

l-

V)

N

H

\\\~,

]. 20 "\" "",

.

,,~~~::=::--_

---

TUNNEL -JUNCTION- 5

.......... ......... - - . . . - - -- ---,, - .....................

OVERLAY

:~.~~~~~ o

0.4

0.8 1.2 1.6 2.0 V34 STRIP VOLTAGE (millivolts)

Figure 1: Bottom: Current-voltage curve for the lower strip of tunnel junction (inset). Top: Junction curves correspon ding to points Band C,

2.4

points B and C of the lower film characteristic. In B we see the usual characteristic for tunneling between superconductors (the structure near the origin is due to slightly different gaps in the films) with additional structure displaced about 70 ~v to the right. The interpretation is represented schematically in the side view of the junction (inset) . The small "normal" region in the long strip separates two superconducting junctions. The voltage drop across the "normal" region is due to the current in the strip and corresponds to the step voltage and the displacement voltage in the junction curve. In C we have many "normal" regions in the junction. Some "normal" regions are not under the junction so we find fewer displacements in the junction curve than steps in the strip characteristic. Also, in contrast to the normal regions, the superconducting regions are not all of the same length, since the height of the peaks of differential resistance vary . This is clear evidence for localized vo1tages associated with each step in the characteristic . Note also that ,each "normal" region is shunted through the capacitance of the superconducting regions to the upper film forming the junction. The RC time is estimated to be", 10- 11 seconds, and the Josephson period for 70 ~v would be '" 3 x 10 -11 seconds. Hence . if the voltage across the "normal" regions were oscillating at the Josephson frequency or slower, we expect the macroscopic superconducting regions to follow approximately and smear out the tunneling curve . We therefore conclude that the step voltage between macroscopic superconducting regions is independent of time. 2 The explanation of very similar step structure in lead filmsl involves conduction via quantized quasiparticle states . Since the volt ages are not localized in this model we conclude from this experiment that the model i s inapp licable to our case . Magnetic field penetration (i.e .• two dimensional phase slip) models require a nucleation rate of about one half the gap frequency and vortex velocities the order of the sound ve l oc ity, which seems unlikely. Simple heating or hot spot 3

210

models cannot explain the regular pattern of voltage steps nor the extrapolation to finite current at zero voltage. The width of a fully normal region would have to be about 30 microns . We find no explanat ion of the origin of these steps based on simple conventional ideas. Perhaps magnetic flux penetration should not be entirely ruled out . Since the films are ~ 1 mm wide there would be many vortices across the film and the a.c. component of voltage would be negligible. Measurements on tin 4 and aluminum5 indicate a maximum flux' flow velocity which is roughly 600 m/sec for similar films . The minimum flux flow velocity required to explain the "norma l " regions would be the Josephson frequenc y times the coherence length which gives about 3000 m/sec . We feel that tunneling is an excellent probe of these systems and is likely a weaker perturbation than attaching voltage probes. Similar experiments could help determine the validity of the quantized quasiparticle state model in lead films,l,2 and obtai n more informa tion about phase slip models.

References: J. T. Chen, L. G. Hayler and Y. W. Kim, Phys. Rev . Letters l. 30, 645 (1973). Reiner Kummel, Phys . Rev. B10, 2812 (1974). 2. W. J . Skocpol, M. R. Beasley, and M. Tinkham, J . Appl. 3. Phys. 45, 4054 (1974) . Meissner , J . Low Temp . Phys . ~ , 267 (1970) . Hans 4. K. E. Gray, to be published. 5.

211

S.055 DYNAMICS OF THE CURRENT- INDUCED RESISTANCE IN THIN-FILM TYPE-I SUPERCONDUCTORS* D. E. Chimenti and R. P. Huebener** Argonne National Laboratory, Argonne, Illinois 60439 During current-induced breakdown of superconductivity in a thin-film type-I superconductor in zero applied magnetic field , discrete flux-tubes .are nucleated at the edges and traverse the strip in a train perpendicular to the current . l The dimensions of the strip are assumed to be large compared to the penetration depth. With the configuration illustrated in the inset of Fig. 1 , we have been able to control geometrically the nucleation point of the flux-tube trains responsible for the onset of resistance . The samples were prepared using the following three-step procedure . First, on a glass substrate an underlay film was deposited over a thin fiber mask. Second, a gap of a few ~ m width was produced in the underlay by dissolving the fiber. Third, the gap was bridged with a narrow overlay film . In this way we prepared a type-I superconducting stri p having a reduction in thickness and width over a section of only a few ~m in length. The following results have been obtained with a Pb specimen, the ,dimensions o f whi ch a re shown in Fig. 1. Applying a direct current to the sample, we observe a step structure in the V(I) characteristic (Fig . 1). Through simultaneous magneto-optical observations 2 we have succeeded in correlating details of the V(I) curve with the current-induced magnetic flux structures . We find that the voltage step between A and B of Fig . 1 coincides ex actly with the appearance of the first flux-tube train in the gap region of the Pb overlay strip .

*Based

on work performed under the auspices of the U. S . Energy Research and Development Administration. **Present address : Universit~t TUbingen, Lehrstuhl fUr Experimenta1physik 11 .

213

212

0.8

b" b"

>::l... >

0 .6

Ip.m Pb UNDERLAY

150p.m

0.4

B

b"

E

01

1I

b"b"

QJ

be

Pbc9

eIS

~

M

4.2K RUN 2

·0

M:>

b"

(1) U

b"b"b"b"

.Q

AI~

0.2

0..

Ib"b"

·N be

'H I-
TCN ) ·

References

s

le.g.: Yu.A.UspenskiI. G.F.Zharkov. JETP 38 , 727 and 1254 (1974) ze.g.: D.Allender. F.Bray. J.Bardeen. Phys .Rev.7B. 1020 (197 3 ) lW.Rilhl. Z.Phys.186, 190 (1965) 'W.Kessel, W.Rilhl. PTB-Mitteilungen 79. 258 (19 69); M.Strongin, Phys i ca 55 , 155 (1 971) 5D.G.Na ugle, R.E.Glover Ill, W.Moormann. Physi ca 55, 250 (1971) 'W.Fels ch, R.E.Gl over Ill, J.Vac .Sci.Te chnol ·2 , 33 7 (197 2 ) 7K.E.Heusler, P. Sc hlut er. Z.Phys.Chem. 69 , 1 42 ( 1970) 8J.Halbritt e r, Phys.Le tt.4 9A, 37 9 (1 97 4) a nd IEEE Tra ns. MAG 11, No. 2 , 427 (1 975) 9A.B.Kaiser, J.Phys. C.Solid St.Phys.l, 41 0 (1 970); C. F.Ra tto, A.Bla ndin, Phys.Re v.1 5 6 , 513 (196 7 ) IOP.Kofstad, High Tempe rature Ox idat ion o f Me t a l s (J . Wiley , New York , 19 66) IIJ .Ha lbr it ter , s ubmi tted to Phys .Lett . A 12J.Ha lb ri t ter , Phys .Let t.4 3A , 309 (1973) 13 J .Halbr i tter , Ext . Beric ht No . 3/74 - 5 (KFZ,Karl sruhe 1974) I" e .g.: L. Y.L. Shen in Superconductivi t y i n d - and f - band Metals (APS , N. Y., 19 72) , p . 31 IS e . g .: J.R. Hopkins , D.K. Finnemore , Phys .Rev . B9, 108 ( 197 4) 16J.Appe l,

Phys.Rev. 1 39A , 1536 ( 1965 )

N

x=O

x=~

Figure 1. Specimen Geometry In the weak coupling limit the superconducting transition temperature To of the composite system is the highest temperature for which the Gor ' kov1 integral equation ",(x)

f~K(x,x') ",(x')

dx' (1) -DS for the superconducting order parameter "'(x) has a non-zero solution. The many attempts that have been made to calculate To have in general neglected the effect of ref1exion of the electrons at the interface between N and S. In some cases 2 ,3,4 this has happened because a simple JOOde1 has been used in which N and S are identical in their normal state properties and differ only in the strength of the electron-electron interaction V responsible for superconductivity. In other cases S,6,7,S, in which N and S are t aken to have different normal state properties, ref1exion has been explicitly ignored, and this has often been justified by stating that in the dirty limit, to whi ch many of the calculations are restricted, the effect of ref1exion is expected to be negligible. Silvert and CooperS do introduce an adjustable parameter a into their theory to account for '~arrier effects" but the relationship of this quantity to the ref1exion properties of the interface is not obvious. It is the purpose of this paper to show that the effects of reflexion can be expected to be significant in the experiments which have been performed even though films of =

233

232

very short mean free oaths have been used in many cases. Since the ex• 9 . perimental results have often been used to pr edict t he strength of V 1n N some doubt must be cast on the values thus obtained. To include the effects of reflexion we use a theory of Hook and WaldramlO (H-W). Reflexion is introduced in a physically reasonable but non-rigorous fashion which should be adequate for estimating the effect on T. Although the H-W theory is not restricted to dirty metals, becaus~ most of the experimental results are for this limit, we give here the dirty limit form of the theory. Note that this is the limit in which the effect of reflexion has often been assumed to be negligible . The H-W theory when used for calculating To is very similar to the theory of S Werthamer 2, Guyon and de Gennes ll , and de Gennes as modified by Hauser 6 et al. , usually called t he de Gennes-Guyon-Wer thamer (dG-G-W) theory. Both the H-W and dG-G-W theor ies assume that equation (1) for 6(X) may be r eplaced by a second order differential equation: 2 d 6

d7

q 26

(2 )

which is similar t o the linearised Ginzburg-Landau (G-L) equati on but 2. . b with a non-local correction. According to the H-W theory q 1S g1ven y

l/q~ = ~i (~Z/[ 4 In(To/Tci )]

(3)

+ 1)

2 wher e t he subscript i is ei ther N or S and ~i = ~vFi li/(6~kBTo) '

can be seen from fig. 2, t he H-W value for q2 differs only sli ghtly f rom the dG-G-Wvalue which i s given by: 2 2 1 q.f;. 1 tn (Tc/To) HI - ~ 1) - H I ) (4) As

where wis t he digamma function. Indeed for qi f;i > 0 (i.e. To > Tci ) equation (3) represents equation (4) rather better than the approximation 2 to (4) suggested by Werthamer • In order to find To' it is necessary to have boundary condit ions for 6 at the S-N interface. The boundary conditions given by the H-W theory which include the effect of r ef lexi on are N To d6 y' l tn (T) dx is continuous q c and

~y' ([ t ] 3 (1-;;)

N?"s

+

-I

-0 .5

2 2 Figure 2: Values of q ~ • Continuous line: equation (4). Dotted line : approximation ---2 to (4) suggested by Werthamer Dashed line: equation (3).

j

if

it

-I

..y

/1

-1.5

where N is the electronic density of states,Y = 6 tn(To/Tc)/(f;2q2) and p s (PS + P )/2 is the mean electron reflexion coefficient at the interface N averaged over all angles of incidence. In the absence of reflexion these 5 boundary conditions differ from those proposed by de Gennes but in the G-L regime are identical to those .proposed by Za~tsev12 as being appropriate to the G-L equation. For a discussion of this point see ref. 10. Solving (2) with these boundary conditions gives the following equation for To cot(1 qslDs)

~~slqsl

([ t]

N?"s

+

[ t ] )

N?"N

(5)

which in the absence of reflexion is identical to the dG-G-W theory result. Although the dG-G-W theory has been criticised 7,4 because it used a diffusion approximation to the kernel in (1) and also approximates (1) by the differential equation (2), the above calculation should be adequate for estimating the effect of reflexion on To' For the purpose of estimating this effect we consider the experimental data of Hilsch et al. 13 ,14 on Pb/Cu films. In fig. 3 we plot Hilsch's 8 dat a as presented by Clarke for the variation of To with DS for the limit ~ +00 together with the theoretical curves of equation (5) for ;; = 0 and P = 0. 5. It can be seen that, al though She electronic mean free paths in both metals are short (1N = 40~, 1S = 5SA), reflexion has a mar ked effect on To ' Any quantitative deductions from such data , in the absence of informat i on concerning the refl exion factor p, must therefore be r egarded as very uncertain.

~----------------~--------~----------------------------~

235

234

S.061 INTERPHASE SURFACE ENEhGY PARAMETER IN LEAD FILMS: TEMPERATURE AND THICKNESS DEPENDENCE T. Miyazaki, B. Shinozaki, and R. Aoki Department of Physics, Fukuoka, Hakozaki, 812, Japan. 0 .8

0 .7 0 .6

0.5 0.4L-_-!-L.lL-±-_ -±_ _ -'-_~_-!-_---;!;--_;!;----; 9

Figure 3: To/Tcs as a function of DS for Pb/ Cu sandwiches in which ~ +00 • The t heoretical curves are from (5), the experimental data are from Hi 1sch et a1 . 13 , 14 .

i'1l l I,

References: l. L.P . Gor'kov, Soviet Phys. J.E.T.P., ~, 1364 (1959). 2. N.R. Wertharner, Phys. Rev., 132, 2440 (1963). 3. K.M. Hong and A.E . Jacobs, J. Low. Temp. Phys . , 12, 519 (1973). 4. W. Si1vert, Solid State Cammun., 14, 635 (1974). 5. P.G . de Gennes, Rev. Mod. Phys., 36, 225 (1964) 6. J.J. Hauser , H.C. Theuerer and N.R. Wertharner, Phys. Rev., 136,A637 7. 8. 9. 10. 11 . 12. 13. 14.

(1964). A. E. Jacobs, Phys. Rev . , 162, 375 (1967). W. Si1vert and L.N. Cooper, Phys. Rev., 141, 336 (1966). J. C1arke, J. Phys. (Paris) Supp1., 29, C2-3(1968). J.R. Hook and J .R. Wa1drarn, Proc . Roy. Soc., A334, 171 (1973). P.G. de Gennes and E. Guyon, Phys. Letters, ~, 168 (1963). R.O. ZaYtsev, Soviet Phys., J.E.T.P., 23, 702 (1966) . P. Hi1sch, Z. Physik, 167 , 511 (1962). P. Hi1sch, R. Hi1sch and G.V. Minnegerode, Proceedings of the Eighth International Conference on Low Temperature Physics, p . 381 (1962).

Bulk lead (Pb) material belongs to type I but has rather large value of the G-L parameter K(-0.4), Accordingly, when it is formed in a film of thickness d,macroscopic flux spot existence in the range of d-lO~m and single quantized vortex state appearance in d~O.l~m regio~ have been reported 1 . We have investigated the intermediate state of Pb films by electron shadow microscopy, and periodic shadow pattern corresponding to the Landau laminar structure was observed in certain geometry2 The measured periodic distance a was analysed for sorting out the energy parameter 6 using the following equation (1)

whe re d is film thickness, ~(h) is a function depending on reduced ma gnetic field h{=B/H c (T)}, and e', the incident angle of ex t e rnal magnetic field Hextto the film s ur f ace. Temperature dependence of 6(T) As for the temperature de pendence of 6 , several functions have been proposed and they are summarized using a general f unction g(t=T/Tc) and critical exponent a in a form of 6(t)= 6( O)·[g(t)]a. Ginzburg-Landau theory gives a function with certain appr oximation condition as 1.

(2)

I t comes from a following representation containing flux peneter ation depth A and the G-L parameter ~ 6 = 1.89 A(T)/K(T)

(K « l)

an d IZ K=H Z/H =2(2e/~c)'A2(T)'H (T). c

c

c

237

2.36

Besides this theoretical prediction, other functions have been proposed from considerations on experimental results for aluminum 3 , and tin 4 . They are get) = (1 - t), In order to find the best fitting expression for our empirical result 2 of ~(T) among these four number of proposed functions, we plotted the data of ~(T) in a form of log~(t) against log get) for each trial function, where we expect a linear relation at the best fitting as log

~

(t) = log

~

(0)

+

(3)

a- log g (t).

As the result of the trials, it was revealed that the best fit was found, as exhibited in Fig.l, at the theoretically predicted function of eq.(2). The intercept and the gradient of the linear relation yield ~

(0) = 460

±

.

80A, and a = 0.52

±

(4)

0.12.

From anintuitive expression of ~(O) for bulk type I superconductor as ~(O)=~(O)-AL(O), another value of ~(O) is obtainable as ~(O) =460A from the parameters of ~(0)=830A, and AL(O)= 370A for leadS. We find this ~(O) agrees reasonably with our result of eq.(4). 2.

Thickness dependence ~(d) For this investigation the observation condition in shadow el f ctron microscopy was improved in such a way that a pin-hole (about SOllm lSOllm in size) was fabricated at the centre of,the specimen surface in place of specimen bending for periphery 2 shadowing . We succeeded in observation of distinct periodic patterns on the flat shadow image of the pin-hole edge as shown in Fig.2, and little effect of the pin-hole size on the 25

«1.)

0(11) 20

4

10

illl

Pb

5xl03

,,/ ~

6(1) 10

15

3

t

10

10

-I

50

i

He=75G 0 B =39 0 00

••••••

o •

.,po 00

•

0

0 0 0 0 0 0 0

°0 0

( C)

5x 102

5

0

• d-251Jm o d~ 15IJm

( b)

/f 1

Pb

100

(1+ t 2 )o(1-t 2 )

Fig.l Temperature dependence of Interphase energy parameter in Pb films of thickness l3-l4~m (e), and 28~m (0). Data for thicker than SO~m (D) by other investigators (Krageloh, eody etal., and Gasprovic etal. see ref. 2) are also cited.

Fig .2 Periodic shadow pattern of Pb film (d=lSllm, 6=45') inte rmediate state by pin-hole edge method. (a ) pattern of short a (t=0.8S, h=O.2l) (b ) pattern of long a (t=0.9S, h=0.6l) (c) shadow of pin - hole edge (t>l)

0~-t.S5~--~6.~0--~~6~5----~7.0

T(K)

Fig.3 Temperature variation of periodic distance a (T) in two samples of different thicknesses

238

239

measurement was recognized. By this means, periodic distance a(d) of samples in several thicknesses d were measured and in Fig.3 two sample's are illustrated. Nearly temperature insensitive region gave reproducible result of a(d), from which the parameter bed) was carried out with eq.(l), and the results are exhibited in Fig.4. We feel some problem remained in b dependence on incident angle 8 of the external field 6 and it may yield rather smaller b values in Fig.4 (8o{i-t) tC{HkfI-tY ( 1 ) j, /:( J if eh 3 V3 1J" Here A is the numerical factor, ~ the Ginsburg-Landau paraT

near

of. l

,the cut-01'1' on

the ordinate presents the contribution due to the inhomogeneiFig.1. Dependen ces

HC2

and

changes,in connection with

al current due to the inhomogeneities

of inhomogeneities,we ob -

H('1((.H"

size.

~ J\

w

62

~E

E ;. ~

o c

~~

~ ~

.= '

~~

~

210

~ 52

Variations 0 and X with 300 CY

~

e

173

Figure 2.

g 3

~

46

8'

~

E

>"

~

48

i!

~

l

4

~ ~

52 182-

G. Eilenberger IFF, KFA Jlilich, 517 Jlilich, West Germany

~

~

g ~

;;

E

188 -

z.

~

:l!

5

c

e 1.l w

3

~ I

Elapsed Time , hours

REFERENCES C. Ho anj E. W. Collings, Titanium Science anj Technology, ed. by 1. J. ff and H M Burte plenum Press, Vol. 2, 1973, p. 815. R. 1. Ja ee .. , ti nal and E. W. Collings, Proceedings of the 13th Interna 0 2. J. C. Ho , ha W J. Conference on Ii:1N Terperature Physics, ed. by K. D. Tl.IlI1'eI' us, . Harmel, Plenum Press, Vol. 3, 1974, p. 403. O'Sullivan, and E. F. Cornell University, ,,~_"'" MS Thesi s , Materials Science Centre, 3. K. K. '"L '\."CU.JC, Sass, phiL Mag. 23, Ithaca , Ne.v York . See also K . K. l'tCabe and S. L.

4. 5.

957 (1971). M=tal 96, 330 (1968) ; see also B. S. Hickman, B. S. Hickman, J. I nst. J . Mater . Sci. 4, 554 (1969). -J. c. Ho, --" R. 1. Jaffee, Phys. Rev. B 2" 4435 E. W. Collings , "" .... (1972) .

A.E. Jacobs Department of Physics , Uni v ersity of Toronto, Toronto (Ontario) Canada M5S 1A7 We describe a new method for the calculation of the free energy of inhomoge neous superconductors as a functional of pair potential 6 (£) and vector potent i al ~(£). The method is applicable whenever the Bogoliubov equations can be r e duc ed to ordinary differential

equations with boundary conditions describing

specular reflection. We apply the method to resolve the question of the origin of (T-T )n+1/2 terms in the expansion of the free energy near Tc' c which has prompted several investigations recently. 1-5 Fo r the class of potentials ~(£)=O, 6 (£)=A+B tgha x the Bogoliubov equations can be solved analytically in terms o f hypergeometric functions; we consider some special cases to demonstrate our method. In the following, we shall describe the main idea of our' 6 method; the details will be published elsewhere. Let ~(~)=O and 6=6 (x), we write (1 )

In the fo l lowing, we suppress the variable

~.

With the usual ap-

proximations, the eigenvalue e quations for u,v are ordinary differential equations of first order. The essential term in the free energy is a sum over all eigenvalues Ev of this equation,

2 ,-I{ (] C " (J I="Rf - f3'C b-' ~J c,enh.l tr" . . v -- KOj ccnh. 12 '-v

(2)

To achieve convergence, the free energy of a r e ference system - i ndex R -

(free electrons for example) has been subtracted . The

279

278

main difficulty encountered by other authors,

1-5

was the deter-

mination of these eigenvalues (for bound states as well as scat-

it

tering states) and the subsequent execution of the sum (2) with sufficient accuracy. This had to be done for a ~ system O1

1

+2b n _ 1 )}

n

(L~' j=1

0)

J

He/Ho - -

with

- - =>

"'--------- ~--~--~----

SK(z)

1 fOOdS exp{- 2SZ}V (-s cosra, ,-s sinCD) , - 2v sin-& 0 v sin lJ ~

dhc

r-s/2

2e j-d!:. ~+~/2

~(!:.)

+

.!S

.",lG . 1: liagn et ization curves a t 'r / Tc =O. 1

!:}>r-

, t f , - - - - - - - - - - --

J

So(z)=SK(z) for

-

J

•

~=O,

.!::=~mn

( __h " ~a

~ =O

I[

_ - o.!?

B/He]= --O.1r

i!

=0.1°.211 )(. =0.9

11

re c iprocal lattice vectors).

For the e xp licit form of over !

- - - ----:::7'_:: -c_:-:_:---:_=-=_-=___ =~ ------ _ -- - - --~ _=_:··::..·: _ _ _ _ '1

:: -------1

•

wQ,311kT(2t+l). The renormalized wt is given by - -> equation for b comes from the Maxwell equation VxE=

c~ntinuity -+

V -+

-+

•

-+-+

-b: bn=(d/dt) (fi.b)=- .jb

Using the fact that ~j=-:;; . VN for a steady flux-flow with veloci-+

-+

+-+

ty v, he finds jr-tVH from th2 continuity rec;:uirement M+V.jM=O.

with jbn=E x fi + a pure curl. Later n we shall see that the pure curl must be chosen as Vx(-~fi/e)= -Vj.lxfi/e, then

This adds a t0rm +H to a, and predicts: ->-J'b

n

=

e

(5)

x fi.

(4 )

Dividing the equation for Tos by dt, and using the equations of 1 where LD (t)=l+p1/J(2)C . h p=eDH /2 TIT, and t= 2 1: +p)/,,,(I)(i+ 'l' 2" p ) ' w~t c2 T/Tc. Since Ln'UT at very low 'f, the third law of thermodynamics is now obeyed.

-th J

We find that this argument can not be com-

te~m.8 But if the sign of the new contribution to

SD is reversed, SD becomes divergent again as T->-O! On the other hand, the coefficient B in Eq. (1) has been calculated by Taka5 yama and Ebisawa, and they found, that the Onsager relation B=

+E

w = J .

+

(jh/T )

wiT, where

+

j -(~/e)j+Eimijbi

~E

pletely correct, since Eq. (2) usually has a minus sign in front of the last

s+V.

continuity, we obtain

+

-+

~

~

-+

(6)

-+

(7)

.XT+J.X~H·Jb··X

*

e

~

~

...

+

*

mi

...

...

+

*

w~th XT=Tv(I/T), ~=E-Tv(~/eT)=~-(~/e)XT and Xmi=TV(mi/T).

Eq.

(7) suggests that the phenomenological linear-transport equations for type-II superconductors should in general be:

afT is satisfied only if the magnetization contribution to a is not included!

(No magnetization-current correction occurs in B,

since a temperature gradient does not cause a flux-flow. 5) The

(8)

purpose of this paper is to resolve both of these enigmas at the same time.

We find that Eq. (4) remains valid, though with

a somewhat different justification, and a ne,. contribution is found for

S,

with the Onsager's relation retrieved.

Using Eq. (6)

to eliminate jE in favor of jh, and putting vmi=o,

we arrive at Eq. (1), with:

At first it seems that the reversal of sign in the last term of Eg. (2) could result from having the demagnetization constant n=l for a flat geometry.

It is concluded that this can

not be the case since what is wanted is a local relation connecting local densities which can not depend on such global quantities as n.

(9)

It is then realized that the correct thermodynamic

relation that we should use is: 6q

=

(10)

T~s = 6E - ~6n + ;.6~,

(ll)

where s,E,n and m are local densities of entropy, energy, par-

ticle and magnetization.

b

is then the local magnetic induc-

tion that varies with the order parameter .

This equation may

be justified using statistical mechanics, but not Eg. (2), if E is to be identified as the ensemble - aver a aed Ch>. n=p/e we have continuity equations

E+~.jE:j . E

YY. where use has been made that for bulk samples ~ s 22 ' Each of Eqs. (9)-(11) is gis o nly along the field direction.

nd a =L

v n as a sum of two parts, with the first part already calcula-

For E and

and p+V . j=o.

d b fo r e , and with the second part being the new terms preA

dl

' d by the present analysis.

Referring back to Eq. (8), we

301

300

see that to

find+L~3i~we can calculate j in the presence of a

Vm i :ssuming that VT~=O. Then because of the general relation Vxm=j, we easily see that L z ,yx=-1. Onsager relation then 23 . th a t L 32 z , yx =+~, 1 wh2ch . . requ2res also follows s2mply from Eq. (5). [Here we see why it is

e,

not

E,

that should appear in Eq. (5).]

S . 078 PINNING IN SUPERCONDUCTORS WITH WEAK INTERACTION BETWEEN DEFroTS AND FLUX LINES S. Takacs Sl ovak Academy of Sciences, Electrotechnical Institute, 809 32 Bratislava , Czechoslovakia

The last term in Eq. (10) becomes just +m, e x actly what is needed to give Maki's expression, Eq o (4 ) ,

The last term in Eq. (11) is

now +m/T, just what we need to save the Onsager relation B=a/T. We now show that the whole second part in Eq. (9) is zero , so that the prediction for Ks is the same as before: First'~33ij is triv i ally zero having vm

because to get ;bi we need

~~O.

Then because

pushing a heat current in any direction would violate

i the second law of thermodynamics, we must have~

i-(~/eiL 23 i 13 On the other hand, it can be shown that a temperature gra· t d +-+ i d 2en oes not dr2ve a +b- current so -+-+ L31 i -(~/e)L32 =0. O.

In conclusion we

emphasiz~

that our analysis is valid for

all temperatures and fields, and it describes not only the spatially-averaged charge and heat currents, but also the local distributions and back-flows, in a scale large compared with the mean free path , mit M=-(4TI)-lH

'

Finally, we note that in the low field liIt is therefore qU2te challenging to show

cl that So again approaches zero as

T~O

in this low f2eld lim2t .

b o th resu I J

tvS

b -:.l TJo . . ., e 11" ___ ._ ....

P

l.r O~!l

*Supported by National Science Foundation ~,

1.

C. Caroli and K. Maki, Phys. Rev .

2.

K. Maki, J. Low Temp . Phy s . !, 45 (1969).

591

(1967).

3.

C. Caro li and M. Cy rot , Phy s. kondens. Mater.

4.

R. S. Thomp s on, Phy s . Rev . B!, 3 27 (1970) .

i,

285 (1965).

5.

H. Ta k a y ama a nd H. Eb i sawa, P r o g . Theo r . Phy s · ii,1450(1970).

6.

A. Houg hton and K. Maki, Ph ys. Rev. B3, 1625 (1971).

7.

K. Maki, Phy s. Re v . Le tt e rs,

8.

See, for example, P. M. Morse, Thermal Physics

~,

1 7 5 5 (1968) . (Be njamin,

1.... -' ., 1 iO:' ,. 11.0(';, 1.1.r;8 Sr mo(;.e.

,

·~.lmp. J

(3

..

References:

New York, 1969) Eq. (6-5) .

The behaviour of t h e flux line lattice in superconductors uith crystal lattice defects and imperfections represents a complicated statistical problem, as in most cases t he interaction of single flux lines with single defects i s smaller than the interaction between neighb ouring flux l ines. Therefore, the distortion of the flux line n ear the defect is inf uenced by the elastic properties of the flux line lattice, too. The resulting quadratic depend 8nc~ of the volume pinning force Fv on the elementary interaction forc~ Ko between flux line and defect (or the maximum force Km ~Ji th which the defect can "hold" the flux line), as well as the existence of the threshold value for the flux-line d~stortion by the def ect, under which the defect can not a, ~t as pitming centre for t he vor tex l att ice,1 is s?mewhat surl.''':::'lU',. ':'he first re,3"1~ ',1:","; i nte r preted by H aasen ~ f r om the s ta ~i~ ti~al ~ oint of v~ ew, And 4

;-lo-

,,_I P

interac tion fer s e, and by ot hers . In all these works, ~ t; quadI.'at.; , Fv("Krr:) dependenc e of Labusch1 was obtai ned for ver:; str an€: pi nni)"l::; c entres. The devia:; ions from this d ependenc '~ '.J eL' e no t, r ot;ts:i ~p'red very seriousl y , a lthough already ment1.0ned by Lar,t1,v'h i . ",t . in some cases (mainl y in fields near Hc2 ' or for s~a tistica l ly distributed point def ects 5 ) the interaction fo rc e can -:)0 small and. therefore the dependence mentioned above will be of another form . Then , the form and charac t er (repulsive, attractive) of the interac tion potenti al can play an important role for the resulting volume pinning fo r ce, as we shall s ee later. The symmetric cubic interaction potential

E = Eo (1 + g1x2 + g2x' sgn x) ads to very useble quadratic force dependence

303

302

(1)

b)

a)

on the flux line distance x from the defect centre for both attractive (upper sign) and repulsive (lower sign) interaction potential. Km is the maximum force acting on the flux line, Xo the "range" of the force (K=O for x :/'%0). We can now calculate the distortions of the flux line from the equilibrium condition between this force and the elastic action of the flux line lattice, represented by the elastic constant? ~~~-c44--c-6-6-' the only one important in the lattice approximation. 1 The resulting equation6 is a quadratic one for the "real" position x of the flux line near the defect in dependence on the "original" position a of the undeformed flux line. We obtain very illustratively the existence of the threshold value for the pinning: if the deformation of the flux line near the defect is small, the flux line can move continuously through the defect and there is no reason for irreversible effects. For stronger pinning sites, some positions of the deformed flux line can not be solutions6 of the corresponding quadratic equation, therefore the flux line must "jump" over these regions (Fig. 1) at the motion of the vortex lattice (influeneed e. g. by the Lorentz force). This leads to irreversible effects and thus to losses. The threshold value for the maximum pinning force is Kom= 5'xo/4 = E)~2 , both for the attractive and repulsive interaction flux line - defect. This is only half the value for the linear model. 3 For the potential used by LabuSCh,1 2 x2 1 x4 E=Eo ( 1 - - - + - - ) 3 d 2 9 d4

x.c:{3d

(2)

we obtain analogically the threshold value: Kom = ~~/? (attractive potential), (repulsive potential). ~

is the distance from the pinning centre for the attractive and from x={3d for the repulsive potential (as the flux lines are held "on", or "from" the defect, respectively).

1

a Xo

1

-1

-1

Fig. 1. The possible positions x of the deformed flux line near the defect vs the original distance a of the flux line from the defect centre. The curves ( 1) correspond to weak defect with no net pinning, k = KmI s-xm = 1/2, (2) - k = 4 • a) attractive, b) repulsive interaction. The volume pinning force for a random system of line defects (density N) nearly perpendicular to the flux lines can be calculated by integrating the elementary force overall possible coordinates x (we suppose that the flux line lattice is regular at larger distances from defects). The results are K2 F = N -ill (1 - b)3(1 + 223 b) v 26'" f or the attractive quadratic elementary force (1), and for b"/0.1715 , N Km Xo (1 - b)3(1 + 2 2/3) F v- { N K~ (-1 + b)2(1 + ~ b _ !!: b 2 ) for b ..(0.1715 ~ 3 3 for the repul sive quadratic elementary force (1). The corresponding results for the potential (2) are very similar e For strong pinning centres, all potentials lead to the "expect ed" quadratic dependence (Fig. 2). However, there are pprec iable deviations from this dependence, as well as consir bl e differences for smaller interaction forces, mainly bet-

305

304

FLUX FLOW OF' TILTED VORTIC~ IN TYPE II SUPERCONIJUCTORS Richard S. Thompson Department of Physics, University of Southern California, If Loe Angelee, California 90007, U. S. A. and L. D. Landau Institute of Theoretical Physios, Vorobyevekoy. + ShoelSe 2, KolScow 117334, U. S. S. R. S.079

f

1

V

Prev{ous caloulations by Saint-James and Jlaki of the upper critical field and impedance of a flat ISuperconductor in a magnetiC field inclined with respect to the surface are incorrect. Our exact solution for small angles of inclination givelS a coneistent flax-flow characteristic.

:~

o J'

20

40

k

Fig. 2. The functions f ..FyI(NK;;26") in dependence on k-Km/5' xm for quadratic (1) and cubic (2) force with attractive (full line) and repulsive (dashed line) interaction. ween attraotive and repulsive ones (even for potentials of the same form). These differences, together with the different values for the threshold pinning force can have some serious consequences for superconductors with weak pinning centres. However, some other quantities (e. g. the maximum reversible displacement of the lattice, the force on i~ in the reversible regime 3 ) are very insensitive with respect to the special form and character of the interaction force. 6 References: 1. R. Labusch, Crystal Lattice Defects 1, 1 (1969). 2. P. Haasen, Z. Metallkd. §Q, 149 (1969). 3. J. Lowell, J. Phys. F g, 547 (1972). 4. A. M. Campbell and J. E. Evetts, Adv. Phys.~, 199 (1972)~ 5. H. Ullmaier, K. Papastaikoudis, S. Takacs and W. Schilling, phys. state sol. il, 671 (1970); S. Takacs and H. Ullmaier, phys. state sol.(a)!2,K35(1973). 6. S. Takacs, Czech. J. Phys. B ~; (1975), in press.

-

INTRODUCTION The static and dynamic properties of a flat Buperconduct~ ing sample placed in a static magrietic tield H, inclined with respect to the surface at an angie 9, have been previously , 1 2 invest;i.gated theoretically by Saint-James and Malti , respectively. Saint-James attempted a calculation of the dependence of the upper critical field Hu on 9 using perturbation theory and found interesting behavior of the elope (J .H;1 4 Hu/J9 Ig _O as a function of sample thickness d. AUhough Saint-James regarded hill calculation of as exact, he actually neglected an infinite number of terms of the same order as those he kept in the perturbation expansion •. We sum the series and find a revised value for (3 , which actually vanishee at the critical thicknese d-1.812 ~ iIUltead of reaching a finl1;e minimum. (:s is the temperature-dependent coherence length.) We also calculate the exact wave functions for small 9 to evaluate the dynamic responee to an additional time-dependent electromagnetic field, derived from the vector potential A..,e- iw t Maki 2 has previously calculated this impedance but using approximate wave functions. Yaki found a flux-flow characteristic with finite conductivity at zero -+ fre quency for the longitudinal orientation when E",.- d A"" /at -+ i n parallel to the surface component of H. However, for the

I.

rs

~

306

307

..

...

transverse orientation when Ew is perpendicular to H he found an infinite conductivity at zero frequency, Whioh is not consistent with the dissipation expected to result from fllU flow. Our exaot caloulation gives a flux-flow character for both orientationa with an anisotropic conduotivity. II. STATIC PROPERrI:SS The static properties of a dirty type 11 superconductor near Hu are found by solVing tbe Ginzburg-Landau equation: (2eH)-1(iV+2et)2£l. _Kt.. >"'A

(1)

wbere ~ is the order parameter, and Hu is determined by setting tbe ground-state eigenvalue ~OO-Hc~H. HC2 is the perpendioular critioal field, which is calculated from microscopic tbeor". The boundary condition is that the normal component of (i if +2eA)~ vanish at t be sample surfaces. Let the sample lie between z-O ~nd z-d, and let H lie in the x-z plane. The static vector potential may be taken to have a single component A .H(x sin9 - z 00s9). The y-dependence of A. 18 then a tr1V~al exponential t::... A(x,z)e ity • To apply perturbation theory separate K into two parts KO+K 1 •

...

KO. [- V~ .+(z·-zo)2]cos9+[- V;.+x· 2]81n9 K1- -2x'(z'-zb)(sin9 cos9)*

(2)

where x·.(2eHsin9)ix and z·=(2eHcos9)iz • For small 9 we now replace sin9 by 9 and cos9 by 1. The eigenvalues of KO are ~~j.(2i+1)9+~j' where fAj are the surface sbeath eigenvalues of KO when 9..0. Por example )'-0,.0.5901 when d'" 0 0 . The eigenfunctions of KO are products of Abrikosov barmonic oscillstor functions Ai(x') and surface sheath wave functions Sj(z'). To second order in perturbation theory: ~ 2 >'00·1"0+(1- oC. )9, ex. L L

~

exp(-invt),

(6)

n

,

(7)

v = ku.

is the Landau-Ginzburg parameter,n sums over the total number

(N ) of the vortices along the x-axis. Ism is the maximum d.c. x Josephson current in the absence of a magnetic field. v is the f requency of vortices passing a point in the upper lattice and is typically of the order of 10 3 /sec. To obtain Eq. (5), we

-

(L ay

iX)21fj t;2

Su

=0

L

(2)

where -uBx/c,

1 N x

f (vt)

Im{6~(X,Y)6u(X'Y)}

have used the facts that I 16 LI2dr = Afj2(1-H/H

srn

= I Afj2 0

and that

)/1.16(1-1/2K 2 ). fj is the BCS order para-

c2 me ter in the absence of a magnetic field. In Eq. (5), we have

al ready equate the factor, exp ti(YU~/4)2l ' to 1 since yut; is

0,

ve ry small under all practical situations. As clear from Eqs. y = 3/V~T1'

t;-2

2eB/~c

( 5 ) and (6), Is(vt) has a period equal to 2rr/v = 2rr/ku, giving

with T1 denoting relaxation time 'of normal electron state due

a direct determination of u. f(vt) becomes a series of delta

to ordinary impurity scattering and B the magnetic induction. 2 The solutions of Eq. (2) are

f u nctions when N goes to infinity. However, it is now possible x t o make Josephson junction of an area of 1 sq micron. In such

6

U

=

eiq>u

L en n

exp [-

~(X-nkt;2

+ iyut;2/2)2 + ikn(y-utJ (3)

a j unction N is only of the order of 10. In deriving Eq. (5), x we have assumed that the number of vortices in the x- and ydire ction to be much greater than 1 so that the boundary ef-

(4)

fe cts may be neglected and a triangular lattice remains meani ngf ul.

~U

1 4 1 4 For triangular lattice,3 k=3 / rr / /t;, C + 2 =C n and C =iC o ' n 1 and ~L are spatially constant phase of the upper and lower

From Eqs(5) and (6), one observes that the largest peak of t he oscillatory Jospehson current is of the same order as t he maximum Josephson current when both the magnetic field

superconductor respectively. Substituting Eqs . (3)-(4) into Eg. (1) and performing the integrations over y and then x, one

nd t he electric field are absent. We show that, when the numbe r of vortices along the x-axis is, ' for example, of the or-

obtains:

d r o f 100, the half width vto of the largest peak is of the 1

1:16

(5)

rde r 0 .02. Since v~103, t ~2x10-5sec. t is roughly inverseo 0 ly p roportional to the number of vortices along the x-axis, o a re sult of the diffraction effects. We would like to thank G. Eilenberger, J. Clem, J. Harris,

J . Kur k i jarvi and H. Kinder for useful discussions.

336

References: 1.

v.

2.

A. SChmid, Phys. Kondens. Mater. ~, 302 (1966).

3.

W.H. Kleiner, L.M. Roth and S.H. Autler, Phys. Rev. 133, A 1226 (1964).

Ambegaokar and A. Baratoff, Phys . Rev. Lett. 10, 486 (1963) .

337

5 .0 87

.DENSITY OF STATES IN TYPE-II SUPERCONDUCTORS IN HIGH

MAGNETIC FIELDS. W. Pe sch, Institut f. Theor etische Physik, Technische Univers iti'it Hannover, 3000 Hannover, Appelstrasse 1, West Germany . The f i rst step towards understanding transport properties i n type 11 superconductors consists in the determination of the density of states. The analytical investigation becomes feasible a t high magnetic fields. Here the functional form of the order-

'Ill I

I

p ar ame ter A{£} is known to be the Abrikosov solution (the space a v e rage of I Ll2., :: 61. tending to zero at H ) and the magnetic c2 f ie ld can be approximated by its spatial average 4(B(r)~ =B . Usua lly o n e starts from the Gorkov equations by. e xpanding the Gr een 's functions wi th respect to the small parameter ·Ll • Only t he s pace average of the normal Green's function must be dealt with rigorously, in order to circumvent singularities in the c le an case at low frequencies [1,2] . In

cont~ast

to the work described above, we shall start

f r o m the Eilenberger equations

[3J.

Using the same kind of

a ppro ximations as in the Gorkov theory, these equations can be re d uced systematically , to a transcendental equation, which covers t he e ntire range of impurity concentration (s- and p-wave scatter ing included). In the clean limit one obtains a closed exfo~

pre ss ion

the density of states (in contrast to the results

from [1]), which shows a

tlln /j1. type singularity at the Fermi

s urface. Adding impurities we find a strong influence of p-wave c at t e ring. The Eilenberger equations are given by

(3J

:+

{-1 }

~ .t) ~ L (/4, k,

"") rf·(We, k.:t) = 1- f (4Je, kit

(z. )

(3) I

-

C

1

i n this work

339

338

«J

functions. L1(~) is given by the Abrikosov solution for the

wh ich p rodu ce near Hc2 a nonana ly tic al behavior (conj e ctured

Q"'!/-.z. i.e d

hexagonal lattice,

=0

t he .

average B is chosen parallel to the z-axis. cos

if cos fJ,

K

-z:

and

r..

are the scattering times

for s- and p-waves, respectively; combining to ~~ .~4_ ~:

•.

As explained before, we approximate g by its spatial average

,tlcJ,k).

N(t.J=oJl_

= (Cos,,"sincp ,

No

sin"") is a uni t vector in the direction of the

momentum of the electrons.

Then Eq.

(1) can be solved formally and g is obtained

afterwards from Eq.

-integral is expressed by ellipti c integral s ,

before from the Gorkov-theory [ 2 ] ) :

i s the gauge invariant

gradient. The magnetic field, approxi mated by its spati al

I1II

Lr

g and f correspond to Gorkov's normal and anomalous Green's

(2) by performing the spatial average. We

B~ 0

The li mi t

(i\ -'"' _)

y i elds at l e ast formal l y the BCS

dens ity of states; an asympto t ic expansion with respect to L-~ rep roduces an expression (for 2r4:-1 ) deri ved before

[4J

by

a noth e r method. In general the Green's function can be determined numer i ca lly from the Eqs.

end up with:

A -+2 E~tn.J 2.

=

Ll-' o

(4 , 5, 6) without difficulties . Some repre-

sentat ive results are contained in Fig. 1. We find a remarkab le

-- --

where we have used:

r

1. I

...

./

,', ' -

r: [J;r .-q

](w)= }(wj:>

WlZJ

vi" ] (w) ]

"

"f{W,Co1>J"),. W{z.)ce-Z~rfc{-;l.)

4-2fL(VF7:r'l{w}-rA{;;}[2~;=W

s:

f~

f(Wlk) ...

-::c. f]

1.

!~frolrJ [A + ?(w,c.T)J')]-"4 C')

(r -'I=:

z:.,-" "" 0

,. ;;;

=(.J )

n tates _ N0 ) a s a functi o n of or L} III

the dens i ty of states

i3 dete rmined b y anal y tical continuation from Eg.

(4,6). For

2.

3.

Density of states (normalize d to th e normal density of

o

In the clean limi t

•

,,/- Y: 1.,V.=O.2

'g(~) is determined by the following selfconsistency condition:

f{w)

Y·O.S/ ~.OS

,

if) 4- ;;; 11

""I l""

.-

I ,,

..

A(W ) ~ - 3 J\. (IF 7:.. r' [ -1 -

,

----

Lcrs

~ 0.-(.

£VIL}

( J' = L\ il

V;1 )

The impurity concentration is me asured by the para-

'i = AI VF r.,..

Solid c urve s correspond to

and

-z;;'

,/-4 "" .Il.1 VF 7:'", ~ f) .

(7:;';

=

-z;-" - LA-~ )

.

341

340

influence of p-wave scattering (

~~

), keeping in mind, that

thermodynamic properties and so also

Ll

and

~

(determined

) depend mainly on r:~. c2 (4,5,6) provide us with a simple calculation scheme

by the magnetization and B ,,",H Eqs.

for the tunneling density of states, and accordingly for all

5 .0 88

DIRECT MEASUREMENT OF FLUX FLOW VELOCITIES IN TYPE-II

SUPERCONDUCTORS BY NEUTRON DIFFRACTION. J. Schelten and G. Lippmann I ns titut fUr Festkorperforschung der Kernforsch ungsan lage 5 17 J Ulich, BRD

thermodynamic properties. It is valid for arbitrary temperature A transport current flowing perpendicular to the flux

and meanfree path. Using the results from above in connection with a time dependent generalization of the Eilenberger equat-

lines in a type-II superconducting slab causes a flux flow if

ions (see e.g. [5J ), also more complicated transport coeffici-

i t s density j is larger than the critical current density jc'

ents (e.g. ultrasonic attenuation rate) can be calculated. Their

This flux flow is accompanied by an electric field E which is

deri v ation from the Gorkov equations is unsatisfactory so far

fo r a dirty superconductor in direction of the transport cur-

(leading to divergencies in the clean limit, see e. g. [6]).

rent. Kim et aLl) w'h o first measured the voltage drop across a current carrying superconductor introduced the relation

(1) References where ,

I!

l.

Brandt, U., Pesch, W., Tewordt, L.: Z. Phys. 201,209

(1967)

2.

Brandt, U.: Phys. Lett. 27A, 645 (1968)

3.

Eilenberger, G. : Z. Phys .

4.

Takayama, H., Maki, K.: J. Low Temp. Phys.

5.

Larkin, A.I., Ovchinnikov, Y.N.: J. Low Temp. Phys. 10, 407 (1973)

6.

Scharnberg, K.: J. Low Temp. Phys.

~,

195 (1968)

~,

~,

51 (1972)

195 (1973)

~L

is the velocity of the flux lines with density B. This

re lation has not been manifested in a microscopic theory. How2 ver, arguments have been given by Josephson ) and recently 4 3 by Eilenberger ) and Clem ) that Equ. (1) is a fundamental relation as a result of thought experiments. There have been two ttempts to measure V directly while numerous experiments L upport the flux flow concept and yield the vortex velocity o nly via Equ.

(1). In the first attemp t Brown and KingS)

meas ured the depolarization of a polarized atomic beam. Alt h ough several experimental observations could not be explained the re , vortex velocities were determined which are 1000 times 6 large r than calculated from (1). In the other attempt Delrieu ) m as ured the motional NMR line narrowing and found for

~L

an

o rder of magnitude agreement with the velocities from Equ.

(1) .

A more p recise compariso n is beyond the s ensi tivity of this m thod . S ince at present a di r ect measurement of V has not L e n s uccessfu lly carried out we applied neutron diffracti on o meas ure the vortex velocity directly and to compare with

Y.r.

from Eq u.

(1 ) although thi s relation is considered by some

342

343

to be fundamental. This method is based on the following:

curves by. which a sensitive detection of small shifts is

lattice planes which are in Bragg reflection position at rest

enabled with an accuracy independent of the width of the

must be rotated by an angle

6~

in order to keep them reflecting

when the lattice is moving. The angular shift

6~

is given by

(2) N(IP)

where Vn is the velocity of the neutrons. For the relatively high vortex velocity V = 1 m/sec and for Vn = 440 m/sec (corL responding to a neutron wavelength of 9 6~ is about 8 mi-

B= 0.08T 1=0

nutes of arc.

E.=O

R)

In the diffraction experiment a dirty but ideal super-

Ill/I

II

conductor of polycrystalline Nb81~a13 with Hc2

= 2.3

= 0.474

-I r~18'

magnetic field in direction of the neutron beam. The Bragg To change

+,

the coil and the sample and hence, the flux

1

lines were rotated around the axis perpendicular to the

i

scattered plane. 6+ was determined from the shift of these rocking curves. Currents up to 80 A could be applied corresponding to j = 2.7.10 7 A m- 2 which was about 20 times larger than the critical current at B

= 0.08

T where most of the ex-

./ • x", 0'

the

mutual misalignment of the flux line sections. For the flux w~

=

0.75 m/sec the mutual misalignment increases drastically to curl

340'. The broadening is caused by the field distortion

g =

i

200

)

.•

•• •• • • ••

\.

200'

200'

F ig. 1: Rocking curves of the flux line lattice in Nb 87 Ta 13 at rest (left) and moving (right) with a velocity E/B = 0.7 m/sec.

is only 18'

while for the flux lattice moving with a velocity E/B

=

t't

If' (minI

In Fig. 1 rocking curves are shown which represent

W.

•

400

XI,

lOO'

periments had been performed.

lattice at rest the full width at half maximum

•••

• • • ••

rt

diffracted neutrons were recorded as a function of the angle ~.

•

••

~=O.7ml!.

"I

and P

•• • •

B= 0.08 T 1= 60 A j = 2.10 7Aim! c

1

T at

= 2.5·10 Gm was used. The sample was n mounted in the center of a Helmholtz coil which provided a 4.2 K, KGL

I

~!

B

600

since it bends the flux lines; furthermore, the

rocking curves have a trapezoidal shape because the bending is parabolic. For such broad rocking curves i t seems hopeless to measure angular shifts 6+ of a few minutes caused by the flux flow. However, the determination of the vortex velocities becomes feasible because of the steep wings of the rocking

rocki ng curves. In Fig. 2 a typical result i s shown where for B

=

0.08 T the velocities determined by neutron diffraction

fro m Equ.

(2) are plotted versus the ratios E/B obtained from

s imultaneously measured E and B values. From such plots i t is demonstrated that (I) the flux lines move if j

>

jc'

di rection of the flux flow is in accordance with Equ.

(11) the (1) and

( Ill ) within the experimental errors the directly measured velocities agree with those from Equ. the agreement is within

~

10 %.

(1). For E/B

>

0 .5 rr/sec

344

345

S .089

1.2

NEUTRON DIFFRACTION, MAGNETIZATION, AND SHIELDING

CURRENTS OF TYPE-II SUPERCONDUCTORS .

B=0 .08T

Y. Wada

1.0

Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113 , Japan

1

v 0.8

It has been theoretically shown that a periodic lattice

L

(m/sec)

s tr ucture of flux lines exists in the whole region of the mixed

0.6

s ta te, irrespective of field strength, temperature, and impurity concen tration l • M~gnetic field takes the form,

• •• • •

0.4

B(r)

0.2

(1)

wh e re B is flux density, 'hk a two-dimensional reciprocal l a ttice vector, and ro is a point around where a triangular

o

vo r t e x lattice has a hexagonal symmetry.

0.2

0.4

0.6

0.8

1.0

1.2

Bs(r) denotes a modi-

i c a tion of the periodic structure at the surface, due to

E/B (m/sec) _

s hielding currents.

Magnetization M has been obtained with the

h Ip of a derivative of thermodynamic potential with respect to Fig. 2: Velocities VL as determined by neutron diffraction (e) and calculated from Equ. (1) (solid line) versus E/B.

B. It is found out that the differentiation of the potential a n be performed forrnally2 The thermodynamic potential is i ven in terms of order parameter, normal Green function, and m g ne tic field.

The formal differentiation gives an expre ss ion

which involves derivatives of these quantities. We can elimina t e these derivatives completely in an expression for the

References:

!l!,

1.

Y.B. Kim, C.F. Hempstead and A.R. Strnad, Phys. Rev. 2986 (1963)

2. 3. 4.

B.O. Josephson, Phys. Letters G. Eilenberger, priv. corn. J.R. Clem, priv. corn.

5. 6.

T.R. Brown and J.G. King, Phys. Rev. Lett. ~, 969 (1971) J.M. Delrleu, J. Phys. F: Metal 1, 893 (1973)

li,

ma gne tization, using equations for these quantities and the r la tion (1).

The elimination finally gives

242 (1965) M

=

(1 / 2cV) fdV(rxj (r»z'

(2)

whe re j(r) is current density and V is the volume. ' ie ld Be is applied along the z-axis.

An external

This relation, familiar

i n o rdinary diamagnetism, turns out to be va lid in the whole r g io n of the mixed state . The periodic part of the magnetic field can be regarded as s up e rposition of fields due to each vortex. I.··

hk is a function of 'hkiFhk

=

F('hk).

The form factor

Carrying out a process

346

347

of an analytic continuation for F(T

) to define a function F(T) hk of a continuous two-dimensional vector T, we define the field due to a vortex at the origin by

2.7 2.4 1.8

Nb

T=4.2K

( 3)

~o

being flux quantum.

11

1.0

It can be readily seen

Fig. 1.

,

(4)

B(r) = B(l-F(O)) + BI $ (r-r,) + Bs(r),

Graphic ex-

trapolations of form factors to obtain

where r , is the point where the '-th vortex is located.

Making

F(O).

use of the relation j(r) = (c/4n)rotB(r), we substitute (4)

Solid line:

B=560G. Dashed line:

into (2) to obtain

B=970G. Dotted line: B=2200G.

MS being the magnetization due to the shielding currents. According to the results of the GL approximation and those in the dirty limit, close to H ' M coincides with (1 / 4n)BF(O), c2 Recently, Brandt is claiming that this holds near

namely Ms=O.

H 2 for O~T13. R.Peters, and H.Meissner, Phys.RBv.Lett., 30, 965,(1973).

361

S .093 X-y · MODEL DESCRIPTION OF A GRANULAR SUPERCONDUCTOR. J. Rosenblatt, A. Raboutou, P. Pellan I .N . S.A., 35031

Rennes Cedex, France

1 1-5 We have suggested , on the basis of experimental results , that an assembly of superconducting grains of linear dimensions a »

~(T), having

critical temperature Tc' should undergo a phase transition to a fully coherent state at a certain coherence temperature To < Tc' The transition should be brought abou t by the Josephson interaction coupling the set of vec tors in the complex plane which represent the order parameters in di f fe rent grains. Recently the dependenc~ of the critical region on grain s iz e and coupling has been studiecfon the basis of Ginsburg-Landau the ory. We show here that the pair tunneling hamiltonian is the exact 7 ana logue of the X-Y model hamiltonian which therefore provides a basis fo r studying the behaviour of granular superconductors. In par t icular, a " paracoherent" region of temperature To < T < Tc (as distinct from paraconductivity at T > Tc) should exist, corresponding to the paramagnetic re gion of the X-Y model. In fact a straightforward generalisation of the re sults of Wallace and Stavn8 for a single junction to the case of N (bulk) superconducting grains, labelled hereinafter by greek indices, gives for the t unneling hamiltonian : H = - 1. L T 2 aa'

V

aa'

S+ S a a'

+ HC

-1.LV

2 aa'

SS +SS aa' ( 1a 2a' 2a 2a')

(I)

where Anderson's9 pseudospin formalism has been used with S± = Lk Sk± + a a ( IEkl < *~), Ska being pair destruction and creation operators in grain a ; v aa ' = va'a .; 0 only for a, a' nearest neighbours. Numerical subscripts refer to the three axes in pseudospin space. The operators satisfy commutati on relations (2)

wi th S3a = Ek S3ka ' S3kabeing a pai r number operator. Now expression ( I ) is just the same as that of the X-Y model hamiltonian 7 i n zero field. Furthermore, an external circuit providing a constant voltage

E

bias and therefore an average f i eld in the sample gives rise to different chemi cal po t entials ~ a among the grains with a contribution to the hamiltonian :

362

363

(3)

0.2

where ;a is the coordinate of the centre of a grain. Eq. (3), in turn, is

0,3

0.4

10

equivalent to the energy contribution of an external magnetic field along the 3-axis of pseudospin space, taking values proportional to u a in each The validity of the model

grain or having a gradient proportional to

E.

requires that the norms of the vectors resulting from the thermal averages

+

be phase-independent and non-fluctuating. In fact, to first order in a

vaa '

t~e

only phase-dependent contribution to the expectation value of the

«

rd3qq 2g2(q)/(&)

J

(6)

where a decoupling approximation has been used to obtain the last step, g(q) is the spatial Fourier transform of the equal time correlation function of M and w is the characteristic relaxation rate of the order parameter. From stat;c scaling g

q-Y/V G(q/K), with K' the inverse correlation length, and from dynamic scaling l4 (&) = qZ F(q/K). Since Y ~ 4/3 and v = 2/3, =

-l

3

(z -

1)

I t has been recently pointed out lS that z me t er is conserved and z

=

2 + cn

~

=

4 - n ~

4 if the orde r para-

2 if it is not conserved. We have mea-

s ured the conductivity of samples made of Nb and Ta grains (a

~

60 u ) i mbedded

i n e poxy resin. Values of To for each sample were obtained by extrapolating the critical current I

c

(T

+

T-) values 0

a few hundredths of a degree and values to the data were determined. No fit for of a fit for n

=

2 is shown in Fig.l. The agreement is only fair, particularly

for temperatures close to To' We think that this is due to coupling of phase

364

365

and quasiparticle fluctuations, which in the case of a single junctio~16 is known to become important when (in our notation) ~10/2e ~ kBT. This is precisely the condition characterising the neighbourhood of the coherence temperature. Our results support therefore the value z

~

4. This is equivalent in fact

to verifying Eq. (4), which is nothing else than a continuity equation for the total order parameter (M I , M ) or (M, ~, M ). 3 3 It may be argued that the above description is a bold idealisation of

Mz,

real samples assumed as regular lattices of uniformly coupled grains. We expect,however, that as long as the universality hypothesis holds the X-Y model will remain a good description of the coherence transition, at least as far as critical indices are concerned. References : I. a) J. Rosenblatt, H. Cortes and P. Pellan, Phys. Lett. A 33, 143 (1970). b) P. Pellan, G. Dousselin, H. Cortes and J. Rosenblatt, Sol. State Comm.

ll,

427 (1972). c) J. Rosenblatt, Rev. Phys. Appl.

!,

217 (1974).

2. T.D. Clark and D.R. Tilley, Phys. Lett. A 28, 62 (1968). 3. I. Warman, M.T. John and Y. H. Kao, J. Appl. Phys. 42,5194 (1971). 4. P.K. Hansma, Sol. State Comm.

ll,

397 (1973).

5. J. Kirtley, Y. Imryand P.K. Hansma, J. Low Temp. Phys.

lL,

247 (1974).

6. G. Deutscher, Y. lmry and L. Gunther, Phys. Rev.B lQ, 4598 (1974). 7. A thorough treatment of the X-Y model can be found in D.D. Betts, Phase transitions and critical phenomena, Vol.3, page 569, edited by C. Domb and M.S. Green, Academic Press, London, 1974. 8. P.R. Wallace and M.J. Stavn, Can. J. Phys. 43, 411 (1965). 9. P.W. Anderson, Phys. Rev.

~,

1900 (1958).

10. G. Deutscher, Phys. Lett. A 35. 28 (1971). 11. R.H. Parmenter, Phys. Rev.

~,

. 09 4 CURRENT NOISE BY FLUCTUATIONS IN SUPERCONDUCTING MICROBRIDGE S. S . Do ttinger and W. Eisenmenger Physi ka lisches Institut der Universitat Stuttgart, Teilinstit ut 1, 7 Stuttgart, West Germany

In the investigation of fluctuations of the super conductlng order parameter me as urements of the excess conductivity near T in highly disordered thin films1 show good agreement c 2 wi th theory . Evidence for the dynamical nature of these . t s3 f luctuations has been given by Josephson-Tunneling exper1men fr om which the fluctuation time constant in the range from 6 t o 21 GHz have been obtained as a function of 6T = T - Tc in a c cordance with the Ginzburg - Landau (GL) t i me-constant. For longer time-constants or decreasing 6T tunneling measurements become difficult. Instead, the direct observation of current no i se resulting from conductance fluctuations in microbridges may be possible. The spectral analysis of the corre s ponding current or voltage noise in microbridges is e xpected to yield valuable information on the fluctuation dynamics, on the c orrect value of Tc or on a spatial Tc distribution 4 recently s ugge sted in connection with voltage steps5 i n the overcriti ca l current range in microbridges. The basic condition for observing current fluctuations can be derived from a general statistical relation 6 between averages and mean deViations resulting in:

387 (1968).

12. V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. lQ, 486 (1963).

(1)

13. R.A. Ferrel, J. Phys. 32, 85 (1971). 14. B.I. Halperin and P.C. Hohenberg, Phys. Rev.

lZL.

952 (1969).

IS. B.l. Halperin, P.C. Hohenberg and S. Ma, Phys. Rev. B

lQ, 139 (1974).

16. V. Ambegaokar and B.I. Halperin, Phys. Rev. Lett. 22, 1364 (1969).

P oe = Average direct current power supplied to the sample in excess to normal conductance at constant voltage by fluctuating superconducting regions. N = Average total number of superconducting regions in the sample. PNoise Total current noise power For C, a constant of order unity, we obtain the value 0.25

367

366

removing . the film with the exception of the shadow zone of the

using the t ime d epe nden c e of cu r r e nt fluc t ua t ions IS IS

~

t

.

e

s c r a tch. The corresponding microbridge left, had cross di-

-2t r

(2)

wa s 1 mm . A low noise preamplifierand frequencyanalyzer with t he equivalent noise level of 10- 15 W has been used for de-

as foll ows f rom theory 2 1/ r(T) is the temperatu re depe ndent GL time-co nstant

2

numeri-

cally determined by the r elation r( T ) = 3.3. 10

11

me nsions of about 1000 ~ times 2 p. The length of the sample

te ction under OC sample conditions as indicated above. Though the experimental sensitivity exceeded the calculated noise le-

.!J.T (Hz)

(3 )

vel by two orders of magnitude we did not find evidence for t he expected current noise. The resistive transition in the small current limit could

The spectral current noise distribution can be calculated

(2) with a frequency bandwidth

by Fourier analysis fr om Equ.

equal to Equ. (3). Experimentally a possible value of = 10- S K appears realistic, with a noise spectrum extending

!J.T

from 0 to 3.3 MHz .

be fitted by the theoretical resu·lt for the one-dimensional case as to be expected for a homogeneous microbridge . We obt ained one well-defined value of Tc for each sample, which is a necessary condition for realizing a!J. T of 10- 5 K. In the range of the voltage steps under critical current

For high noise powers a low value of N is required which appears mor e easily to be obtained in microbridges. We - assume

c onditions noise with the comparatively small bandwidth of

that N can be estimated f r om

a bout 10 KHz has been observed.

~

, the GL coherence le ngth. For

microbridges with lateral dimensions smal l er than dimensional c onditions , we use N = Calculating

L/~

~

Further current noise detection experiments in the low

, i.e. one-

with L = sample length.

current range under improved conditions are prepared. The authors gratefully acknowledge stimulating discussions

~

from experimental data for Al-microbridges we 4 obtain N z 10 0 f or L = 0.1 cm and!J.T = 10- K. For the aver age 8 power P W (100 o hms sample r es istance and 10 ~A OC oc = 10 -13 curre n t ) we estimate PNoises= 10 W within the experimental fr e quency r ang e !J.v = 3 . 10 Hz. Th is bandwid t h is o n l y a

References:

fraction o f t he noi s e bandwidth !J.v = 3 3 MHz calcu l a t ed from r a t!J.T = 10 - 4 K.

1.

R.E. Glover, Phys. Lett. 25 A (1967), 542

2.

E.A. Abrahams, J . W. F. Woo, Phys. Lett. 27 A (1968), 117

3.

J.T. Anderson, R.V. Carlson, A.M. Goldman, J. Low Temp.

4.

W.J. Skocpol, M.R. Beasley, M. Tinkham, J. Low Temp.

5.

J. Meyer, G.v.Minnigerode, Phys. Lett. 38 A (1972) 529

6.

L.O. Landau, E.M . Lifshitz "Statistical Physics",

Th e estimate d noi s e power exceeds the sensitivi t y limit

with K. LaBmann, R.E. Glover, G.v.Minnigerode, B. Mtihlschlegel and H. Six!.

of c ommo n short wave radio-receivers by about one order o f mag n itud e.

Phys. 8 (1972), 29

For first e xp e rimen t s to determine current no ise we have used microbridges prepared on microscope glass slides by a new t echnique. After making one scratch on t h e surface o f the glass slide with a defini t ely loaded disk recorder diamond we evaporated a n AI - f i lm of about 1000 ~ thickness. Then the 3~~ple

was expos ed to an argon beam u nder oblique incidence

Phys. 16 (1974), 145

(Addison-Wesley Publishing Company , Inc., Reading, Mass., 1958), pp. 1 - 8

36 9

368

S. 095 FlDC'ruATlOOS-INOOCED PAIR-a:NXX::l'IV1'IY IN THAI.LnJ.I FIIMS AfYJVE SUPERa:tIDU::TING TRANsrrlOO TEMPERA'IDRE. Arun

were s imilcir to the ones described' previously. 3

To interpret experi,TleI'ltal

data properly it is necessary that the nonnal state oonductivity

K. Saxena, St. John's University, Jarraica, New York 11439, U.S.A.

00

be

determined as accurately as possible. We observed that (Jo of clean thallium

Anil K. Bhatnagif, Fordham University, Bronx, New York 10458, U.S.A.

films , as det ennined by quenching pairoonductivity by applying large enough magnetic fields at various tanperatures above Tc' was temperatUre dependent.

We report here rreasurements of electrical resistance of clean thallillll films above superconducting transition temperature Tc to study pair-oorrluctivity fio = (J (T) - (Jo ' where

00

is the normal state oonductivity, oontributed

by superoorrlucting fluctuations. fi(J for other superoorrlucting films has been

investigated before but results on

clean alllllinum films have been fourrl to

shCM max:im.nn departure fran the A-L theory. HaNever, these results are fOurD to agree reasonably well with the follCMing theoretical expression:

Therefore, an appropriate ac oount of tenpe:rature dependence of (Jo and SlT\3.ll magneto-resistive effect are used to analyze the experimental data. Experimen tal plot of inverse of nonralized pair -oonductivity (110/ 0 0 ) versus tertq?erat ure 'r for a thallium f ilm of thickness 220 A and Ra = 4. 74 Si is sl1= in Fig. 1. Measurements were taken with external magnetic field applied parallel to the fi lm. The field values were 0 KG (~ 10-3 G), 0.5 KG, 1.0 KG, and 3.0 KG. For o"ly

c larity sake, data"for

(1)

0 and HII = 3 KG is shCMn. The measured

. . trans ~tion

width, using the s traight line method, i s about four times larger than pre-

Here, = (T-Tc)/I'c,'o = 1.52 x 10-~, and 6 is a parameter proportional to pair-breaking interaction oresent in the film, and is chosen suitably to fit Eq. (1) to experimental data. The first term on r.h.s. of Fq. (1) is the A-L

term2 and the seoond tenn, known as the Maki-Thanpson term, was originally found divergent by Maki 2 for thin films but was later regularized by 'l'harpson? The question of divergence of the Maki term was later reexamined by Keller

1\ =

dicted by the A-L theory f or H = O. The slope of oo/fio increases with the aoolied field indicating sharpening o f the transition, a s predicted by Eq. (1), for the increase of magnetic field increases the pair-breaking parameter 6. The curves drawn through data points represent fit of theoretical expression (1) with the data using & and Tc as adjustable parameters . Curves (a) and (b)

&

Korenman2 , and patton 2 and was fOUIrl to be noIHlivergent even in absence of

are fit of Eg. (1) to experimental points for H = 0 in the range 0 T (B) c samples.

observed by Farrant and Gough(6)

in clean niobium

We wish to thank Dr. C. E. Gough and Professor K. Maki for many useful discussions.

n ot e that there are significant quantitative differences between our results in part by the National Science Foundation

under Grant No. NSFGH-41512 and by the Brown University

a nd the non-self-consistent, non-relativistic calculations of Mueller, et al. fo r P d 4 and Switendick for PdH. 5 We also note that since it has been found

Materials Research Program funded by ARPA .

that the lattice constants of PdH and PdD are very nearly the same, 6 withln

1) See, for example, K. Maki, J. Low Temp. Phys.

~,

513

(1969) . 2)

The band structure calculations were done self-consistently using the a u gmented plane wave method (APW) of Slater modified to include relativistic 2 e ffects self-consistently and the spin-orbit interaction as a perturbation. T h e details of these calculations will be reported e lsewhere, 3 but here we

References:

*Supported

the APW method their band structures will be identical.

3) D. J. Thouless, Phys. Rev. Letts. 3A, 946

~,

1025

(1975) •

S. P. Farrant and C. E. Gough, Phys. Rev. Letts. 943

v

8

.- se i sotope effect, 2 will be due to differences in their phonon spectra. i nce o nly the phonon spectrum of PdD has been measured, 7 we cannot prentl y do reliable quantitative calculations for PdH.

li,

(1975) •

T o s tudy the superconducting properties of Pd and PdD we have calcula t e d the e lectron-phonon coupling constant,

5) For a review see J. P. Gollub, M. R. Beasley, R. Callarotti, and M. Tinkham, Phys. Rev. Bl, 3039 6)

Therefore, any

dif fe re n ces in their observed superconducting properties, such as the in-

P. A. Lee and S. R. Shenoy, Phys. Rev. Letts. (1972) •

4)

To gain an understanding of

S. P. Farrant and C. E. Gough, to be published.

A

(1973) •

n

=---?-

(1)

M

w ith the qu a ntitie s in Eq. (1) as defined in Ref. 8.

The numerator of Eq.

d In Is on the ab initio band structure calculation and can be accurately \' ,j

ulate rl u sin g the theory of Gaspari and Gyorffy. 9 For a binary compound

MD " u "h as Pd D w ith the atomic mass ratio - ~ 0.02, th ere '1S a contn"b u" , Mpd l ion ~ o th t ot a l A o f the form of Eq. (1) from each atom in the unit cell

401

400

TABLE IT

separately, with the phonon averages being taken over the acoustic modes for Pd and over the optic modes for D.

That is, to high accuracy, the

coupling, while that from the light atom (D) is due to electron-optic mode coupling.

TABLE 1

M

(eV /1. 2)

0.47 0.21

Pd metal

3.59

(eV /1.2) 7.55 a

Pd

1.28

5.97 b

0.62

0.86 b

} PdD

D

A

0.72

}

0.93

For Pd metal we have calculated A = 0.47, and therefore many body These

many body effects lead to a large value of IJ.'~ in a . McMillan type of equation for Tc'

We have found that

IJ.* must be greater than 0.28 to account for the

39.1

39.8

31. 5

31.2

32.6

0,10

23.2

24.1

25.6

0.15

16.2

17.8

19.0

0.20

10.5

12.2

13.1

6.0

7.5

We note from Table I that the large A for PdD comes primarily from the D site and is due to coupling of the electrons with the optic modes.

Al-

cRef. 15.

(T (2) ) agree well with the Eliashberg solutions (T (1) ) particularly for the c c 0 s maller values of IJ.*. The measured value of T c for Pd:> is 11 K and corresponds to IJ.~'

';; Q.2. This is higher than the value of IJ."' ~ 0.1 generally

a bly related to the large value of IJ.~' that we have found for Pd metal. is, the spin fluctuations are only partially removed in PdD.

That

We would

therefore expect that if the spin fluctuations in palladium-hydrogen systems

though n < 12> for the D site is relatively small, the very low value of

M for the optic modes results in a relatively large value of A.

aMethod of Ref. 13.

8.1

bRef. 14.

asc ribed to transition metals and transition metal compounds, and is prob-

fact that Pd is non-superconducting.

In

effect, the whole set of optic modes in PdD is "soft" in comparison to typical binary compounds; and this indicates that the D atoms have a large mean squared displacement and that there is significant anharmonicity. 7 In this compound, these low-lying optic phonons lead to a large A and are favorable for superconductivity.

40.8

0.05

We see from Table IT that the values of T c obtained from the McMillan type e quations (T~2) and Td 3 ) ), especially the new AlIen and Dynes form

bRef. 7.

effects, such as paramagnons, 10 must suppress superconductivity.

0.0

0.25

aA. P. Miiller and B . N. Brockhouse, Can. J. Phys. 49, 704 (1971).

c

b

a

Table I shows the re suits of this calculation. n

T (3) c OK

T (2) c OK

contribution to A of the heavy atom (Pd) is due to electron-acoustic mode

Recent tunneling measurements of Dynes

and Garno 11 have s hown that electron-optic mode coupling is indeed the

could be further suppressed (,/ decreased), Tc would be increased.

Pd-

A g- H(D) alloys have T 's of up to 16°K, 16 and it is appealing to attribute , c • the increased T to a decrease in ",'" from the value in PdH(D). c We gratefully acknowledge helpful discussions with many of our o lleagues; especially, L. S. Birks, A. Ehrlich, J. M. Rowe and J. J. Rush .

The work at Ge o rge Mason University was supported by the National

A e ronautics and Space Administration through Subcontract No. 953918

dominant mechanism for super conductivity in PdD.

from the Jet Propulsion Laboratory, California Institute of Technology.

In Table II we show calculations of T for PdD using our results from ,', c Table I, for different values of IJ."' , using three different methods: (1) an

n I.

T. Skoskiewicz, Phys. Stat. Sol. (a)

exact solution of the linearized Eliashbe rg equations 12 following the approach of Leavens 13; (2) a new modification of McMillan's equation 8

2.

L. L. Boyer and B. M. Klein, Int. J. Quantum Chem. 59, 1975, to be

proposed by Allen and Dynes 14 involving a logarithmic phonon average;

3.

D . A. Papaconstantopoulos and B. M. Klein, to be published.

(3) Dynes ,15 modification of the McMillan equation.

4.

F . M. Mueller, A. J. Freeman, J. O. Dirnmock and A. M. Furdyna,

fe rence s:

Q,

K 123 (1972).

published.

hy e . Rev. Bl, 1617 (1970).

403

402

THE CONTRIBUTION OF THE HIGH FREQUENCY PHONONS TO THE SUPERCONDUCTIVITY OF THE HYDROGEN-RICH COMPOUND Th:4H 15 G. Ries, H. Wi nter Ker nforschungszentrum D-75 Karlsruhe, West Germany.

5 . 104 5. 6.

A. C. Switendick, Ber. Bunsenges, Phys. Chem. 76, 535 (1972). B. Baranowski, S. Majchrzak and T. B. Flanagan, J. Phys. F: Metal Phys •

.!,

258 (1971).

7.

J. M. Rowe, J. J. Rush, H. G. Smith, M. Mostoller and H. E.

8.

W. L. McMillan, Phys. Rev • ..!..§l, 331 (1968).

9.

G. D. Gaspari and B. L. Gyorffy, Phys. Rev. Lett. 29, 801 (1972).

Flotow, Phys. Rev. Lett.

10.

ll,

1297 (1974).

N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett .

.!I,

R. C. Dynes and J. P. Garno, Bull. Am. Phys. Soc. ~, 422 (1975).

12.

G. M. Eliashberg, Sov. Phys. JETP

13.

C. R. Leavens, Solid State Commun •

1329 (1974).

14.

P. B. Allen and R. C. Dynes, to be published.

15.

R. C. Dynes, Solid State Commun • .!.Q., 615 (1972).

16.

W. Buckel and B. Stritzker, Phys. Lett. 43A, 403 (1973).

\~ 1)(EF)" . L ~ fdw J(W -

1)

696 (1960); G. Bergman and

D. Rainer, Z. Physik 263, 59 (1973).

..!2.,

c

433 (1966).

11.

.!..!,

The contribution of the phonon branch! entering t he formula for T is 1 .

'

,t

.

{Jq I

i , ~,q.

s)'1;;',A,'" -+

I

('i'-:;.I •

k,

1-;'! ~ v' I't~',i) '-tf) _>i till

'1':

AL

with (.)9 8 (R.~. the phonon frequency (polarization vector) .. A I f(,O . , the Bloch states with Fermi energy,~(fF)the density of states per unit cell. The s um j (i) goes over all atoms ( i n the unit ce ll) . According to 2,3,4 it is favourab l e to expand the Bloch s ~ a~e s ~WIA into e igens t at e s in the (muff in t i n ) potential s v 1(J) an~ to express the resulting matrix e leme nt s by t he phases hifts ~"(fF)obtaining:

l ) ~i =

"OWl

£~t.

,

~

t~1T2~(€F) ~

I.£':" 111!}

"'.J,Cf,J..:-m..

l l '

I

q,~

'l,~

1_1_ (~'Vl>,f 1'W11-w...,1

""'~ -1>\1

{7{'M; ~

A A'

-

1 .

,

t\l1j.-~1

i z,'WIJ (,c :1 'V\'l3 1 """'4-'WI J I..(4-"""LI-) Y" (..e~,d Y1 , (r~ J~) . IJi 1'1') .T. . . ~ i.:::: 1 .i. /

( A

~'"S

'

.f. .MM I 1,+ - 0.(3 J" """~, 11. 'Wot 113 ""'31 12, ""':. V{AAIvl where T is the imaginary part of the scattering matrix for t he ele ctrons which can be obtained by s olving the equation:

..1-1Nl

3)

o.!z -

T ...

J

.(" """1 I

"

T, I 1 A~

I

'WI1.1

J z. -m~

.

'I )

-

JJ~ )

= t J 4

J" 1 •. lz.

1

}z. 'WI~

wi th go t he free electron prop a gator. A band structure calcu_ la tion would yield the matrix elements of T. o Now Th 4H15 is cubic with a lattice constant a =9. 11 A containing o H- and 16 Th - atoms in the elementary cell. Each Th-atom is a ~ a common corner of three tetrahedrons, 12 H- atoms are in ~I ce ntres of the tetrahedra, 48 somewhat displaced from the

405

404

~

!

faces of them 5. Taking 2.4 as an average distance between the Th- and the next neighboured H-atoms, the muffin tin radii of Th(H) were chosen to be 1.6(0.8) ~. The phaseshifts turned out to be: r Th

o2

= -1.19,

J~ = -0.85

at an assumed fF of 0.95 Ry. The higher phaseshifts were negligible. So the H-atom is found to be a rather strong scatterer. In order to get the matrix elements of T we solved eq.3 for the cluster(l) of atoms shown in fig.l. As examples we give the following results:

4-) ,)"", To~ , ~o =

0.18

tlWl tU0

I

~.;) " L ,)'»'1 ~1'~ 1~ ,"" .•••

=

0.3 )""" t 1H

if~"Y T T..t, I~ "r tTJ.. S~ Itrn z."""z,W\= 0.8 1__ 2. So THH is considerably reduced as compared to the single site approximation (T::::: t). We also see, that the matrix elements of T at the Th-sites are nonnegligibly affected by the environment. But whereas the values of T for the H-atom 1 in fig.l are reliable 6 ,7, those of the Th-atoms are not,as we did not choose an environment characteristic for them. Therefore we solved the scattering problem 3. once again for cluster(2) of fig.2.

')

-TA, 1,f,

...... 10 °,o0

=

.,

Cluster (1) and cluster(2)

:3.W1

""",-1. . . (5)

The values of the matrix elements of T for the H-atoms 1,2,3 in cluster (2) equal those for the central H-atom of cluster (1) in good app roximation . The same is true for the Th-atom 2 in cluster (2) as compared to 1 in cluster (1). So we conclude that the cluster (2) is now large enough in order to have a reliable approximation for the matrix elements needed to determine ....\ 9 from eq. 2. For the high freguency phonons we assume that they consist of loca lized hydrogen oscillations of frequency ~ o' So 2. is simpl if ied to the following expression : 6)

..A

.1.TJ.

1/.4.1'»'1Ta

L lrv.:TTJ. T'('

I

)..t.

~ H 1-1

(T

00,00

_

tj./l'y)

~, z. (

E/'

;tZ1T Z 'YI (E

F

"" H

H

)

~

TA'WI~,1"""1. + TH 0 ,11-1 /.j

0

SH

61-1) \"

Mi/ W 2.

L-./.lAM A 0 D A'jckooe",.. ~"" ~

..s \.2

VI

w ?: ~ ..J w a:

0

~ ~ 0

- 20

~

FIRST HARMONIC

ImVrm.

resisti~ity

No .

a t 300 K ( ~ncm)

l ow t emp . for tunne l . measTs

ene r gy gap AC Ts)

I IO'"~RELATIVE

CHANGE "CS

FROM

VI W

20

.,Ii.... ... ,.

w

VI

(b)

0

0

!:: U)

z

- 20

Sample

(a.)

w

0

10

(me v)

20 ENERGY

40 30 (me V)

~O

10

20 ENERGY

30 40 (m.Vi

~

(oK) 1 2

4

120

0 . 6. 1.4

0 . 17 0. 26

Fo r second harmonic meas urements the samples were excited by lmVac an d tes t ed at the low temperatures Ts ' and in the n ormal state at 4.2 0 K. From the superconducting state characteristics we subtracted t he nOTmal state's charac t eristics a n d the BCS contribution parts (which is appreciab l e from l5mev downards). The final result for samp l e 1 is shown in Fig . la . The ove r all fea t ures resemble res ult s obtained for lead by W.L. McMillan and J.M . Rowell (lO~ a similarity which is probab l y due to the fact that t he two metals are fcc. Fig. 2a shows t he electron dens ity of s t ates obtained from la by integration. Deviations from the BCS density of states are in accord wi th the estimation given in reference 10. The results for sample 2 are shown in figures lb and 2b. The comparison show s that the peaks remain the same in number, in energy and in polarity . The only two important differences are: 1. The struc t ure is broadened and smeared. This f eature is expected because of the s h or t range crystallographic or der

Fig ure 1: (a) 2nd harmon ic of c le an Al at 0.6 0 K. (b) 2nd ha rmonic of granular Al at 1. 40K. I n both cases t he ac e xci tation voltage was lmv r . m. s .

Figure 2: Electron density of states. (a) For cl e an AI. (b) For granular AI. One unit in the y-axis corre s ponds to 10-4 relative change from B.C.S.

Iy hic h mean s uncertainty in the phonon momentum and energy . 2 . The a r e a under the curves is about doubled, which me ans t ha t th e ch anges from the BCS den si ty of states are roughly d ub l ed , in a greement wi t h the estimation that these chan ges :Ir e proportional to (~)2 (l0) Not e that 0p remains unchanged, 0.. hut t he gaps are different by a factor 1.5 so that the area s hould incre ase by 2.25 - a number which is within the experilIIe nta l e rror. lJ i sc uss io n

We s e e that except for some sme aring of the structure, Lhe 2nd de rivati ve and hence the electronic de nsity of states rC lIl a in rou ghly the same but are enh anced by a constant factor. I:v e ll be fo re inve rting the Eliashberg gap e quations, one can d duce t hat th e phonon spec t rum is essentially unc h anged

425

424

(except fo r a ce rtain bro adening) wh ich imp lies that the is un changed. The data s hows no ev i dence f or a "sof ten i ng" of the phonon spectrum, in contradiction with models which consider low energy surface phonons as the explanation for the enhancement of T (6,7,8). The only rl~usible expl an ation c c'r the enhancement of the 2nd harmonic is an inc,ease of 2 (w) i ldependen t oE w. ~ert haT! I}ears rI, t th T I ne Ius •• 1 i fl6 in y \1 lue t cm ir L re se in '-he C'lectn t1 .[ t'1 cel '11 nF 00

~"'\"

n the shape jI

tcrenc

'\

0

c; tlnicLov,

w. 4. 5. 6. 7. 8. 9. 10. 11.

r the

)110nO~

spcetrul1.

,

:>

:di~

Q)

L..

....::Jco N

"'"

L..

Q)

Cl.

E Q) t-

I 0 0

~

0

IX)

0 tD

:l '1:1'0

~1Il Q)

o Q) I-< U.-l :l +' .-l.-llll III Eo< I-
(

where F' ([.) is the slope of F ( '-) computed numerically. 3

4

2

o

of course, necessary to have an appropriate probe.

8T!l;'-T(mkl

'T.~-Ttn*1

I n order t o detect and measure the modified g(£) it is,

ou t that the transverse acoustic wave at frequency

It turns MHz

f~lOO

is known from independent studies 2 ,3 of the intermediate state

i s ideally suited, being a "monochromatic" propagating wave

that well ordered laminas are not maintained for anla ~ 0.2.

which can sample the bulk laminar structure and, very near Tc'

Quasiparticle scattering at S-N interfaces leads to a modified

suffering an attenuation due to excited quasiparticles

energy spectrum and,

A (r),

parameter

~ccordingly,

corresponding to the order

a density of states g (e) strongly modified

from that of BCS; here

E !!

E/A , where A is the energy gap.

To determine the new density of states it is reasonable under the experimental conditions of the present work, to consider the order parameter to be a one-dimensional periodic step function

6

(x) such that

ness as' and~(x)

=

A

(x)

= 6.

in the S-laminas, of thick-

0 in the N-laminas, thickness aN'

using

(4ic.J« f:::.

(T) except for ~ T ~ 10-5K) capable of sensitively re-

flecting the new g (e) structure.

~ (~r/~Jlt!(()(r/rxlJ/ ~ 2.

wi thin a few InK of Tc '

- ~ (cr;~ 10;,.)).,. /~c /0;,.,).,. )

-li"")['

If

(.

-t;~j

Co;r/(f,,J)..= 7,w/b. ~, *)a(~+A I +e(t+~)Jt~~.)-~\PAf'£

where

eo

Here, superscript E refers to t he elec tromagneti c part of the

~s and orN are the real parts of the complex conductiv itie s ~ and ~ for the s-

transverse phonon-electron interaction, and

the Bogoliubov equations van Gelder 4 has determined the spec-

and N- states respectively.

trum of elementary excitation for this case, obtaining cos

ing contributi o n to « (ltS!OI'N) E would roughl y co rr e spond very

(K-k ) ~

ne ar Tc to the ratio (

=

F (~ ), where for 0

F(£)=eos A£

< E
2B

p

. The losses have a minium because of the two

counteracting Bs dependencies in (7). That the final increase of Q is due to the transport current is also indicated by the comparison with curve b which shows the magnetization loss of a Te wire sample without transport current (curve b has been fitted to curve a at the smallest Bs value, where F is very small.

449

448

S.116 MICROWAVE RESPONSE OF MIXED STATE OF PURE NIOBIUM* W. L. McLean and H. R. Segal t Physics Department, Rutgers University, New Brunswick New Jersey, 08903, U.S.A.

I

751

Te 1045

I

II

20~ I : I

~ 1st

3

I

E

~ \',

\ V

tra.nsport

current

~~

!

i,

with

I

3

10 •

witt.c.ut

/'

lIansport current

sL---~--~--~--~--~--~--~--~

o

_

1

e,. , T

IS

Fig. 2 - Dependence of the ac losses of a small coil made from TC 1045 superconductor on Bs (B s averaged over the coil volume) for a fixed mean amplitude Ba of the ac component (curve Cl). Curve b shows for comparison the Bs dependence of the losses of a TC wire sample without ~ransport current. We thank Prof. J.L. Olsen for his continuous interest on loss investigations. The work was financially supported by the :Sidgenossis:::he Kommission zur Forderung der WiSS, which was published just after the experiments reported here were completed, there had been little progress in understanding the high-frequency flux flow in pure type 11 superconductors since the first attempts towards a microscopic theory by Caroli and Maki 4 • Even now, because the CHM theory deals only with T=O, with fields close to Hc2 ' and does not take into account surface solutions of the order parameter, there is still much left to be done. Experimentally, because of the practical limitation to niobium -- which has a great affinity for oxygen 5-_ the difficulty has been the preparation of suffici~ntly pure surfaces for experiments that involve essentially the skin effect. In this report, we compare some features of our measurements of the surface resistance of niobium at 10 Ghz in fields up to Hc3 w: t h the theory of CHM. For the remaining features, not yet considered in the theory, we make comparisons with the corresponding behaviour in impure type 11 superconductors. Details of the experiments will be published elsewhere. Orientation dependence: The surface resistance depends strongly on the relative orientation of the extel:'nal strong

4~ O

451

ste ady magnetic field, t he micr owave curre nt , an d the surface . Except when the external field is perpendicula r to the surfac e and especially when it if; parallel to the sur face, the re ar e screening effects due to the vortex-free region near t he s ur face that can exist up te; Hc3' The surfa.ce solution f or the order. parameter is not obta ined by C:HM -- the i r ba s i~ calculation is of the conductivity ry of an infinit e space whir.h is "then substituted into the fo rmu la R a(o) - l/3 for the surfa ce resistance :;'n the e xtreme anomal olls ~imi t of the ski n effe :C1: .

ductor . 'The theory, fitted at He=0.98 Hc2 ' does agree with the measurements at higher H but significant deviations were found e at smaller values of He'

Fie l d de pendence: The variation o f the surface resistance R with exter na l fie ld He in the perpendi~ular orientati on is shown i n Figure 1 . The CHM t heory predi cts tha t R/ RN = [(1+1-1) /(1 +31-1) ]113 , where 1-1 = H ( Hc 2 -He)/ ~, RN is t he value ·:>f R in t he normal s t a·te , ,q, is the e le ctronic mean fre e path in the normal state , an d f i s con stant for ~ given super con-

DiM

THEORY FITTEOHERE

.~ ---.-

0 .99

R RN

~ ----.~

:::::::::::-. P ::50

--........-..

0.98 '.

0.9

~:--------~gg ...............

\ \ \

0.96

"-

I

o.95~

CAROU-MAKI

I

0 .002

0.006

---.~

T= 6.0K PERPENaCULAR ORIENTATION

I .

l

~.

i

I

• CHM THEORY I " -E XPT

_I 0.010

~.

I

0.014

I 0.018

I

- He/ HC2 Figure 1. Dependence 'Jf s;..:rface !,~s istance in perpen dicul a r· or i ent at i on on field dnd residual l'p.sis"ti ·./ ity.

Hean free path depend ence: It can be seen from Figure 1 that at a given value of He /H c2 ' the slopes of the three purest samples (which all had l~>'o' where ~o is the BCS coherence length) were approximately proportional to the residual resistance ratios r and therefore to 1 -- as predicted by the CHM theory. This is in marked contrast to the CM theory, which predicted R/RN to be independent of R. in the limit R.»'o' Temperature dependence: The effect of raising the temperature is to increase the slopes of all the curves in Figure 1. It has been found that for a given sample and for 42. 5 V2315

~

techniqu e , which consisted of monitoring the reflected microwave

tance vs. the

power from a res o nant sample cavity and the flux-flow resistance was deduced from it(7).

While the flux-flow resistance was be-

ing measured as a function of the external field its derivative

reduced tempera-

S

ture.

10

with respect to the field was also me asured util iz ing a modulation fi e ld.

The sample cavity was a right-circular - cylindrical

c av ity operating at 35.5GHz (at room temperature) in the TEOll mode.

The s a mpl e was glued in a recess in the end plate . The reflected power from the cavity was measured by a crystal

d e tector, whose output is fed s i multa ne ous l y t o an XY re corder, a phase-lock detector (f o r deriva tive measurements) a nd to a frequency stabilizer keeping the k l ystr on frequency centered at the

0.5

cavity resonant frequency. The q uantities of inte rest are R/RN and S.

1

t

Here R , RN are

the microwave surface resistance of the mixed state and the norma l state, respectively, and S=(H c2 /RN) (aR/ aH)Hc2' The exp r ess i ons for R/RN and S in terms of various crystal dete ctor vol tages are given by Pede rs en et. a l(8).

4K

S

2

(o) ----~~-----------1.16 (2K/ (t) -1) + n

1

A(t), where

Kl(O) = 1.2K (K=Ginzburg-Landau parameter), n=demagnetization

Ill. Resul ts a nd Discussion From the t raci n g of av/aH as a function of H we determined (9)

S is ca lculated from th is quantity us ing the ex(av/aH)H 2 ' c pression g i ven in Ref . 8. Shown in Fig . 1 is S p l otted against the t emperature. Not i ce that for Ti.7625v.2375 S increase s by more than 120% as t( =T/T co ) decreases from 0 . 4 to 0 . 2. It seems t ha t for t t; is obtained: H p = jl H 2 TTt;2 _ ~ o c2 (2) o

c

1. H.W .We ber, G.P.Westphal and I.Adaktylos : t o be published 2 . R.Rieg l er a nd H.W.Webe r : J.Low Temp.Phys . .J2., 4 31 (1974) 3 . H. C.Freyhardt: Phil.Mag . ~, 369 ( 1971) 4 . L.Schultz and H.C.Freyh ardt: phys.stat . sol. ~, 145 (1972) S. I . Adaktylos, E.Scha ch i nger and H.W.Weber: to be publishe d 6 . R.L abusch : Crysta l Lattice Defects 1 , 1 (1969) 7 . H.C.F reyhard t: private communication B(r)

lmT1

t

PbNa, T=5K fJ :::

nons relax slowly enough which is accounted for by phonon-pbC>non colliSions, thereupon quasi-equilibrium Bose distribution wlth the temperature TpJ. is established in the phonon &yst_. Then, for the time corresponding to the process of phonon em18slon and phonon absorbtlon by a quasi-particle, equilibrium in the system of phonons and electrons (Tpl. - T) is found to be established..' When the givan number of quasi-particles (Jl* - 0) is established and the temperatures of electron and phonon systems have the same values (eT _ 0 ) ,as a result of this an equilibrium energy gap -stepness" of both electron and phonon subsystems disappears 4 • Assuming that jk'* and J T change according to the law ex? i-At] ,we shall obtain the following values for the relaxation constants in the simplest case - that of a spherical band mOdel: ~ .

11. ~

~

12 .,(.2m *11 3 (T):J. {02(T/L\) . :> L=1 Ll

P",i,'$'fj>F

_

{j

)

2 exp{-IJ/T »).= 2

(2)

where ot. - constant of electron-phonon interaction, m*quasi-particle effective mass, jDm - density of substance, ~ _ sound velOCity, Pt: -Fermi momentum. Below are given certain calculated estimations for AI: i f T=o,3~ X/-5.jO-~et!~Ji~1-/ofec; ':.1-.1 -$ 'l-i -y T =O,2T.: > .lLi '" iO sec,./lot -10 s~c. As has been shown by EliashbergI , the nonequilibrium value of the energy gap may be derived from the equation analogous to that of BCS for the gap containing a nonequilibrium distribution function. Thus, the gap tends to reach its equilibrium value with the change of p.j/(t) and 8:r(t) in the following way: M d{j(t )=:.V2f7(tJ./T) ~xf{-IJIT){I/'/.(-t)+(IJ/T)ST{-t)] (3) Analyzing general solutions of the equations (I) it is shown that in the process of establiShing equilibrium ·between quasi-particles and condensed pairs, at the beginning fairly quickly, for the time 'l::J. "" 1/;'1 there occurs only a slight change Of}-'* , then much slower (Jl j » A 2 ) for L;-l/Ji2. there occurs a further decrease of jW~ till zero. Otherwise, time (2 is a characteristic relaxation time of chemical pctential, whereas the temperatures become the same for the characteristic time 2i ,i.e. quicker than the relaxation

528

of chemical potential takes place. The characteristic relaxation time of the energy gap is f' ~ 1/.7J.2 ~ This time coi.nsides with "Co/> by its order. An amplitude change of the gap, as is evident from (3), is inSignificant as compared with the decrease of .?~ .and 2.5. This result is obtained by equating for various choices of transition temperatures T a and 1 the ratios

~'2'75

0.5 0.2

~3.00 O~~--~~--~---L---a-

0.1

0.5

1.0 n lat%C. in La) _

O~~~~~--~~~

0.2

0.5

5

2

10

Tc(K)-

fig. 1. T IT versus 71 for various a aO ,'I;-values in (La,Ce)In _,'I;Sn x ' 3

fig. 2.

p:ll a l2n

versul? Ta'

Table 1: Some parameters for th e syst e m Lal00_nRnln3_,'I;Sn,'l; with R = Pr, Nd or Ce. The values of N (0) and ~x are de2 11: duced from ref. 1. For the meaning of 0max see text.

NII:(O) (3)

Eac h re lati on (3) i s the n viewed as a quadratic equation in ZnTK(,'I;) , the ro ots of wh ich have to be r eal. This leads t o inequalities of the form 0 2 ~ O~ax (one inequa lity for each ,'1;value), where the respective o~ax -valu es are given in ta.ble 1In the Kaiser theory it is predicted that

~ ::::'

1.0

(1)

(eV.at)-l

3.00 2.875 2.75 2.625 2.50 2.375 2.25

0.34 0.31 0.30 0.28 0.26 0.24 0.21

~II:

0.27 0.26 0.25 0.23 0.21 0.20 0.18

0

2 max

1.1 2.4 3.4 6.2 6.2 7.6 11

-(dT , / dn)n=O

Pr

Nd

Ce

1.7 1.2

0.5 0.4 0.4

1.5 3.1 3.8 5.0 5.9 7.7 11

0 ..7 0.3 0.5

536

~n(T IT C

Co

):: -Ani (l-Dn),

(4)

where the constants A and D are A :: (2L+1)N,(O)(l+N,(O)U ff/A

2

537

e

giv~n

5 .138 ' THERMODYNAMIC CRITICAL FIELD OF THE SINGLET GROUND S TA TE SYSTEM (LaPr )Sn

by

R .W.McCallum,

)/(A N (O»and D :: :x;:x; :x;

can easily be deduced from ref. 5. Here L is the ,-electron orbital quantum number (L=3); N,(O) is the local ,-electron density of states at the Fermi level' U . • eff ~s an effective ,-, Coulomb interaction parameter and A = N:x;(O)V is the well known parameter in the BCS theory. :x; In analyzing our experimentally obtained T -values there Cl is no difficulty at all in obtaining .a fit in the form of equation (4). It is then easy to derive values for N (0) and U , eff from the expressions for A and D(values for A:x; and N:x;(O) are given in table 1). It turns out, however, that the obtained N,(O)-values are definitely negative in all cases. except for :x; = 2.875 and :x; :: 3, whereas in the latter cases U f turns out e f to be negative. So the Kaiser theory does not apply either. We therefore consider our experimental results as decisive evidence that Ce-impurities in LaIn -:x; Sn:x; behave neither like 3 (MHZ) Kondo impurities, nor (Kaiser) non-magnetic resonance states. Preliminary measurements of the magnetic susceptibility as a function of temperature reveals that the Ce-impurities show no significant magnetic moment. This is, however, compatible with both the MHZ-and the Kaiser theory. (2L+l)N,(O)U ef /(N:x;(O)A:x;),as

Acknowledgement: We thank mrs. A.A. Haak for general assi$ance .

3 * C.A.Luengo, 1.