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VOLUME 82, NUMBER 1 PHYSICAL REVIEW LETTERS 4JANUARY 1999 Pressure Dependence of the Upper Critical Field of the Heavy Fermion Superconductor UBe 13 L. Glémot, 1, * J. P. Brison, 1 J. Flouquet, 2 A. I. Buzdin, 3 I. Sheikin, 2, ² D. Jaccard, 4 C. Thessieu, 2, and F. Thomas 5 1 Centre de Recherches sur les Très Basses Températures, CNRS, BP 166, 38042 Grenoble-Cedex 9, France 2 Département de Recherche Fondamentale sur la Matière Condensée, SPSMS, CEA-CENG, 38054 Grenoble-Cedex 9, France 3 Laboratoire de Physique Théorique, Université de Bordeaux, CNRS URA 764, 33175 Gradignan-Cedex, France 4 Département de Physique de la Matière Condensée, Université de Genève, 24 quai Ansermet, 1211 Genève 4, Switzerland 5 Institut Laue Langevin, BP 156, 38042 Grenoble-Cedex 9, France (Received 12 May 1998) We report measurements under pressure of the upper critical field of the heavy fermion superconductor UBe 13 . An interpretation in the framework of a simple strong coupling model is achieved consistently with only one arbitrary parameter: the strong coupling constant l. We find that UBe 13 is in an extreme strong coupling regime and that the variation of l with pressure explains the pressure dependence of the thermodynamic properties of both the normal and the superconducting phases. It reveals a strong interplay between the mass renormalization and the pairing mechanisms, yielding the first quantitative indication of a nonphonon mediated pairing in a superconductor. [S0031-9007(98)08084-3] PACS numbers: 74.20.Mn, 74.60.Ec, 74.62.Fj, 74.70.Tx A characteristic feature of heavy fermion (HF) inter- metallic compounds is the occurrence at low temperature (l10 K) of a very large renormalization of the mass of the charge carriers: up to several hundred times the mass of a free electron. There is no complete understanding of this huge mass renormalization, but the consensus is that it arises from strong electronic correlations, which also produce different kinds of magnetic excitations. It has also long been suggested that the pairing mechanism responsible for superconductivity in HF may differ from the usual electron-phonon interaction, and is believed to involve the magnetic properties of the normal phase. But as well as for high-T c cuprates, real quantitative results on this point are still lacking. We propose here a new approach to this challenging question, taking advantage of the extreme strong coupling regime met in the HF super- conductor UBe 13 : our measurements and analysis of the upper critical field H c2 of UBe 13 under pressure reveal a direct link between the mechanisms of mass renormaliza- tion and superconductivity in this system, yielding the first quantitative indication of a nonphonon mediated mecha- nism in a superconductor. The normal phase of UBe 13 already presents striking features. The quasiparticles have a record effective mass (renormalization by a factor of 1000 has been suggested [1,2]), and coherence in the lattice occurs only at very low temperature: the resistivity presents a maximum at 2.5 K [1]. Therefore, and as opposed to all other HF super- conductors, UBe 13 has the very unusual feature to be in an ill defined Fermi liquid regime when superconductivity appears at T c l 1 K [1,3]. With regard to the supercon- ducting state, it is clearly in a strong coupling regime, as indicated by the relative jump of the specific heat at T c : of the order of 3, much larger than the BCS weak coupling value of 1.43 [4]. Quite curiously, the specific heat has long been the only superconducting property analyzed in a strong coupling scheme [4,5]. It is only very recently that strong coupling effects were quantitatively discussed on H c2 [6], providing a new but straightforward interpre- tation of its peculiar behavior. The temperature dependence of H c2 in UBe 13 at zero pressure (see the curve P 0 kbar of Fig. 1) is very puz- zling. First, H c2 shows a very strong negative curvature near T c . This observation suggests a strong paramagnetic limitation of the upper critical field (i.e., the effect of the field on the spin of the quasiparticles) [2]. The difficulty for a quantitative fit comes both from the value of H c2 at T 0 in UBe 13 [more than 7 times greater than the Clogston paramagnetic limit: H P sT 0d p 2 DsT 0dygm B l 1.85k B T c , where D is the energy gap, g the gyromagnetic ratio, and m B the Bohr magneton] and from 0 4 8 12 0 0.2 0.4 0.6 0.8 1 P=0 kbar P=4 kbar P=6 kbar P=10 kbar P=20 kbar H c2 (T) T(K) FIG. 1. Upper critical field of UBe 13 under various pressures. Both the strong curvature near T c and the anomalous increase in high field gradually disappear with increasing pressure. Full lines are the best fits of our strong coupling model. 0031-9007y 99 y 82(1) y 169(4)$15.00 © 1998 The American Physical Society 169

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Page 1: Pressure Dependence of the Upper Critical Field of the Heavy Fermion Superconductor

VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999

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erland

Pressure Dependence of the Upper Critical Field of the Heavy Fermion SuperconductorUBe13

L. Glémot,1,* J. P. Brison,1 J. Flouquet,2 A. I. Buzdin,3 I. Sheikin,2,† D. Jaccard,4 C. Thessieu,2,‡ and F. Thomas51Centre de Recherches sur les Très Basses Températures, CNRS, BP 166, 38042 Grenoble-Cedex 9, France

2Département de Recherche Fondamentale sur la Matière Condensée, SPSMS, CEA-CENG, 38054 Grenoble-Cedex 9,3Laboratoire de Physique Théorique, Université de Bordeaux, CNRS URA 764, 33175 Gradignan-Cedex, France

4Département de Physique de la Matière Condensée, Université de Genève, 24 quai Ansermet, 1211 Genève 4, Switz5Institut Laue Langevin, BP 156, 38042 Grenoble-Cedex 9, France

(Received 12 May 1998)

We report measurements under pressure of the upper critical field of the heavy fermionsuperconductor UBe13. An interpretation in the framework of a simple strong coupling model isachieved consistently with only one arbitrary parameter: the strong coupling constantl. We find thatUBe13 is in an extreme strong coupling regime and that the variation ofl with pressure explainsthe pressure dependence of the thermodynamic properties of both the normal and the superconductingphases. It reveals a strong interplay between the mass renormalization and the pairing mechanisms,yielding the first quantitative indication of a nonphonon mediated pairing in a superconductor.[S0031-9007(98)08084-3]

PACS numbers: 74.20.Mn, 74.60.Ec, 74.62.Fj, 74.70.Tx

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A characteristic feature of heavy fermion (HF) intermetallic compounds is the occurrence at low temperatu(ø10 K) of a very large renormalization of the mass othe charge carriers: up to several hundred times the mof a free electron. There is no complete understandinof this huge mass renormalization, but the consensusthat it arises from strong electronic correlations, whicalso produce different kinds of magnetic excitations.has also long been suggested that the pairing mechanresponsible for superconductivity in HF may differ fromthe usual electron-phonon interaction, and is believedinvolve the magnetic properties of the normal phase. Bas well as for high-Tc cuprates, real quantitative resultson this point are still lacking. We propose here a neapproach to this challenging question, taking advantagethe extreme strong coupling regime met in the HF supeconductor UBe13: our measurements and analysis of thupper critical fieldHc2 of UBe13 under pressure reveal adirect link between the mechanisms of mass renormaliztion and superconductivity in this system, yielding the firsquantitative indication of a nonphonon mediated mechnism in a superconductor.

The normal phase of UBe13 already presents strikingfeatures. The quasiparticles have a record effective ma(renormalization by a factor of 1000 has been suggest[1,2]), and coherence in the lattice occurs only at very lotemperature: the resistivity presents a maximum at 2.5[1]. Therefore, and as opposed to all other HF supeconductors, UBe13 has the very unusual feature to be inan ill defined Fermi liquid regime when superconductivitappears atTc ø 1 K [1,3]. With regard to the supercon-ducting state, it is clearly in a strong coupling regime, aindicated by the relative jump of the specific heat atTc: ofthe order of 3, much larger than the BCS weak couplinvalue of 1.43 [4]. Quite curiously, the specific heat halong been the only superconducting property analyzed

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a strong coupling scheme [4,5]. It is only very recentthat strong coupling effects were quantitatively discusson Hc2 [6], providing a new but straightforward interpretation of its peculiar behavior.

The temperature dependence ofHc2 in UBe13 at zeropressure (see the curveP ­ 0 kbar of Fig. 1) is very puz-zling. First, Hc2 shows a very strong negative curvatunearTc. This observation suggests a strong paramagnlimitation of the upper critical field (i.e., the effect of thfield on the spin of the quasiparticles) [2]. The difficultfor a quantitative fit comes both from the value ofHc2at T ­ 0 in UBe13 [more than 7 times greater than thClogston paramagnetic limit:HPsT ­ 0d ­

p2 DsT ­

0dygmB ø 1.85kBTc, whereD is the energy gap,g thegyromagnetic ratio, andmB the Bohr magneton] and from

0

4

8

12

0 0.2 0.4 0.6 0.8 1

P=0 kbarP=4 kbarP=6 kbarP=10 kbarP=20 kbar

Hc2

(T)

T(K)

FIG. 1. Upper critical field of UBe13 under various pressuresBoth the strong curvature nearTc and the anomalous increasin high field gradually disappear with increasing pressure. Flines are the best fits of our strong coupling model.

© 1998 The American Physical Society 169

Page 2: Pressure Dependence of the Upper Critical Field of the Heavy Fermion Superconductor

VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999

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the shape ofHc2sT d, which displays, on the best samplean unusual upward curvature at a temperature arouTcy2. Tentative explanations have relied on two supeconducting order parameters with weak Pauli limiting [7on additional magnetic phase transitions [8,9], or evena field dependence of the normal or superconducting sparameters [10–13]. Also they cannot be completely dmissed, none of these phenomenological interpretatiohave found a firm basis in other measurements or thretical developments. It was shown recently [6] that sua behavior ofHc2 is predicted directly by the Eliashbergequations for an extreme strong coupling regime, withoany additional hypothesis (see also the discussion beloA prime interest of these new pressure measuremeof Hc2 is to probe if this straightforward interpretationcan consistently explainHc2sT d when the strong couplingregime is changed. As we shall see, they also lead tmuch deeper insight into the connection between normphase properties and superconductivity in UBe13.

Experimental procedure and setup.—Hc2sT d has beenmeasured by monitoring the resistive transition. Tdetermine the critical temperature at fixed magnetic fieor the critical magnetic field at fixed temperature,“junction” criterion is used (i.e., the determination othe crossing point between the extrapolated resistivof the normal phase and the extrapolated linear partthe transition). We use a 12 T compensated magnwhich enables reliable thermometry under field. Thpressure is applied in a piston-cylinder cell, withliquid mixture of alcohol as transmitting medium. Iis measured by the known pressure dependence ofTc

in UBe13 [14], which has been checked in our sampagainst the superconducting transition of a tin manomeat P ­ 6 kbar andP ­ 20 kbar. Hydrostaticity of thecell is controlled by measuring the broadening of thtransition: 2.5 mKykbar. For the highest pressures of 1and 20 kbar, the low field transition is blurred by anothtransition of (we guess) uranium filaments in our sampwhose superconducting transition temperature is knoto increase under pressure but is rapidly suppressed bmagnetic field [15]. Apart from this problem, a speciaeffort is made on the accuracy of the determination of tinitial slope atTc, and on the saturating value atT !0. For this purpose, the current density was kept loTemperature or field sweeps were performed, overlappin a range of intermediate fields and temperatures whthe results were tested to be consistent.

Sample quality and results.—The evolution of Hc2under pressure is reported in Fig. 1. The sample quahas been improved since the pioneering measurementRef. [16] (which lacked a little of data for a quantitativanalysis) so that the agreement between the two setsdata is satisfying. Concerning our data, those at zepressure (from [6]) were measured on a high qualsingle crystal. The data under pressure were taken onpolycrystalline sample of [12]. TheTc of this last sample

170

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at zero pressure is slightly higher than that of the singcrystal (0.97 K instead of 0.957 K) but with a transitiowidth DTc ø 28 mK instead of 18 mK, which remains,nevertheless, good for UBe13. Both samples have aresidual resistivity in high field of about10 mV cm,which also attests to their good quality, and we casafely assume that they are in the clean limit [1,12]. Aexpected, in the range of fields where the polycrystal hbeen measured (up to 12 T), theHc2 of the two samplesmatch when the same onset criterion is used (UBe13 iscubic). So at zero pressure, we have reported hereresults on the single crystal which are more precise aextend up to larger fields.

Apart from the known variation ofTc under pres-sure, the striking feature seen on Fig. 1 is the geneevolution of the shape ofHc2sT d: the peculiar behaviorobserved at zero pressure gradually disappears withcreasing pressure. Let us discuss this evolution frompoint of view of a strong coupling scheme. We will noreproduce here the calculations presented in [6] but ratconcentrate on the physical effects and the parametentering the model, which are all directly probed by thpressure measurements. In the absence of a precisescription of the quasiparticles in the normal state and wno knowledge of the type and spectrum of the interactioresponsible for the pairing, we have used a simple Estein spectrum for the density of interactions of the fora2Fsvd ­ slVy2ddsv 2 Vd [17], where l is the pa-rameter giving the strength of the pairing andV corre-sponds to the characteristic frequency of the excitatioresponsible for this pairing. Because large values ofl areused in our model, we have neglected the possible inflence of the Coulomb repulsion (characterized by a singscalar parametermp), an hypothesis that we will justifymore quantitatively later.

From a general point of view, the strong couplinregime essentially adds two effects onHc2 with respectto a weak coupling scenario. The first one is an enhanment of the orbital limitation due to the mass renormaization of the quasiparticles induced by the interactioresponsible for the pairing. This renormalization alreadexists in the normal state and is controlled byl throughthe relation:mpymband ­ l 1 1. mband is the band massof the quasiparticles, which means in our case the mof the quasiparticles renormalized by all the interactiowhich do not participate in the pairing potential. Thstrong coupling effects on the orbital limitation could bmimicked by a weak coupling scenario if the full renormalized valuemp of the mass is used in the HelfandWerthamer expressions instead ofmband [18,19].

The second type of effects are thermal effects, and thare completely specific to the strong coupling regime. Ideed, the fact thatTc can be quite close to the characteristic energyV of the excitations responsible for thepairing leads to a reinforcement of the superconductiproperties at temperatures much lower thanV where the

Page 3: Pressure Dependence of the Upper Critical Field of the Heavy Fermion Superconductor

VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999

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pair-breaking thermal excitations have disappeared. It afects the orbital limitation but has the most dramatic efects on the Pauli limit [17]. The latter is enhanced alow temperature, compared to the BCS prediction, duethe increase of the ratioDsT ­ 0dykBTc. There is an-other factor which raises further the Pauli limitation in thstrong coupling regime: in a clean Pauli limited superconductor, an elaborate calculation ofHc2 shows that at lowtemperature, the transition to the superconducting phais made into a modulated phase called the Fulde-FerrLarkin-Ovchinnikov (FFLO) phase. It yields a reinforce-ment of Hc2 compared to the calculation for a uniformstate which is at most5% in the weak coupling regime[20], but is much larger in the strong coupling regimeThe appearance of this phase below a temperatureTFFLOof the order of 0.4 K at zero pressure is calculated without any additional free parameters by optimization ofHc2with respect to the modulation vector, and is responsibfor the upward curvature ofHc2sT d aroundTcy2 [6].

The microscopic calculations [6] ofHc2sT d have beenperformed for a simples-wave state, probably not themost appropriate for UBe13, but actually the symmetryof the pairing has little influence onHc2 (only the ampli-tude of the paramagnetic effect may be strongly affectein case of odd-parity pairing).l is the only truly arbitraryparameter of the fit:V is calculated from the values ofl

andTc, wheremp has been fixed at 0. The gyromagnetiratio g for the paramagnetic limitation has been kept closto the free electron value (g ­ 2) and the measured ini-tial slopes≠Hc2y≠T dT­Tc , essentially proportional tomp2

for a clean superconductor, determines the orbital limittion. The best fits ofHc2 for all pressures are displayedin Fig. 1, and their parameters are reported in Table I, tgether withTFFLO, the temperature of the appearance othe FFLO phase.

At first glance, a remarkable feature of the model is thit explains the evolution of the shape ofHc2sT d with avalue ofg ø 2 (the free electron value) remaining almospressure independent: this supports thatHc2 in UBe13is mainly controlled by the Pauli limitation with strongcoupling effects. In particular, the gradual disappearanof the upturn is naturally explained by the decreaseTFFLO due to the fact that the strong coupling parametis decreased under pressure, and that the orbital limitati

TABLE I. Parameters of the fits of Fig. 1. The gyromagnetiratio g is almost constant which supports our interpretation odominant Pauli limitation; pressure in kbar ands≠Hy≠T dT­Tc

in TyK.

P l g s≠Hy≠T dT­Tc TFFLOyTc

0 15 2.1 255 0.454 13 2.2 242 0.426 12.5 2.25 232 0.37

10 11 2.2 221 0.2620 6.5 2.2 28.5 0.1

f-f-tto

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becomes more dominant at high pressures (see the decrof the initial slope). It appears that the balance betwethe orbital and paramagnetic limitation is reversed in tpressure range of our experiments. We stress that,each fit,l is the only real free parameter if one considethat the value ofg has to be about 2, and that the initiaslope is controlled by the measurements close toTc: Sol is basically fixed by the value ofHc2s0d (in order toprovide the necessary increase of the Pauli limitatioand agreement of the fit with the experiment in sua range of values ofl and of the initial slope is verylikely meaningful. But further insight can be gained banalyzing the pressure variation of the parameters, amost remarkably that ofl.

Pressure dependence of the parameters.—Popular wis-dom has it that in a classical electron-phonon interation scheme, a value ofl as large as reported in Tablewould be impossible because a lattice instability shouthen occur before superconductivity could appear. Butheavy fermion systems, mass renormalizations of mothan 100 times the free electron mass do exist, mlikely due to magnetic interactions and in any case nto electron-phonon coupling. Obviously in most casethese interactions do not favor superconductivity: few Hare superconducting. UBe13 is also necessarily a speciacase, because all other HF superconductors cannot ba similar extreme strong coupling regime. This is showfor example, by the very conventional behavior of theupper critical field, so that the pairing mechanism usuahas to be different from the mass renormalization mechnism. The singularity of UBe13 is also reflected in thefact that the Fermi liquid regime is so ill defined atTc inthis compound, which is not proof of a strong couplinregime, but is compatible with it.

In this respect, the decrease ofl under pressure is con-sistent with the restoration of a well defined Fermi-liquregime atTc as observed in transport measurements [2More quantitatively,lsPd [mainly determined byHc2s0d]gives the pressure dependence ofmp through the rela-tion: mpsPdymbandsPd ­ lsPd 1 1. If one further as-sumes thatmbandsPd varies little in this pressure rangethenflsPd 1 1gyfls0d 1 1g is a measure ofmpsPdymps0d.It is seen in Fig. 2 (solid squares) that this yields a dcrease ofmp of about50% betweenP ­ 0 and 20 kbar.A second independent estimate ofmpsPdymps0d can be ex-tracted from the fit of the initial slopes≠Hc2y≠T dT­Tc :mp decreases by more than60% in the same pressurerange. This depends little on the model used to fitHc2:it is essentially dictated by the experimental valuess≠Hc2y≠T dT­Tc (proportional tomp2). This is nicely con-firmed by a third measurement ofmp: the specific heatSommerfeld coefficientg, whose variation has been measured up to 9 kbar [22]. The agreement of the thrindependent estimates ofmpsPdymps0d can be seen onFig. 2. The consistency of these results, on top of ging confidence in the validity of our model, also show

171

Page 4: Pressure Dependence of the Upper Critical Field of the Heavy Fermion Superconductor

VOLUME 82, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1999

le

,

,

s,

ty,

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

vF*(0)/v

F*(P)

(1+λ(P))/(1+λ(0))γ(P)/γ(0)Ω(p)/Ω(0)

m*(

P)/

m*(

0),

Ω(P

)/Ω

(0)

P (kbars)

FIG. 2. mpsPdyms0d obtained from lsPd (full squares);s≠Hc2y≠T dT­Tc (full circles); and the Sommerfeld constangsPd [22] (full diamond). Full triangles:VsPdyVs0d. Dashedlines are guides to the eye.

that in UBe13 the mechanism responsible for the pairinin the superconducting state accounts for a large partthe mass renormalization in the normal state: about a fator of 16 at zero pressure. It gives a basis to the extremstrong coupling regime found in this compound and alsyields a first quantitative experimental argument againelectron-phonon coupling, because a mass renormalizatof a factor of 16 by such a mechanism is extremely ulikely, especially with a Debye temperature of the order o640 K [23].

Microscopically, l is given by l ­ 2R

dv a2Fsvdyv. So a large value ofl indicates an anomalous shift of thedensity of interactions at low frequencies, whereas highfrequencies are more favorable for an optimum valueTc at a given spectral weight [19,24]. Indeed, we finan absolute value ofVs0d ø 1.5 K, which is very smalland close toTc. VsPdyVs0d is shown in Fig. 2. Somp,which is usually of the order of 0.1–0.2, might be mucsmaller in UBe13: mp is some frequency integral of theCoulomb repulsion up to a cutoff proportional toV, andin any case, it is quite negligible compared to the valuof l. We also find thatVsPd ø const. It means thatthe whole variation ofTc stems from that ofl, whereasin ordinary superconductors, a variation of bothlsPd andmpsPd is necessary to account forTcsPd, and V is alsogenerally found to only slightly increase with pressureIn this respect,VsPd cannot be directly related to theresistivity maximum (at 2.5 K atP ­ 0), which is thesmallest energy scale known in UBe13 [1], because thismaximum increases much faster with pressure [21].present, a microscopic identification ofV seems hopelessdue to the lack of neutron studies in this compound.

172

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In conclusion, we have shown that it is possible toaccount for the wholesP, T d dependence ofHc2 inUBe13 with an extreme strong coupling regime and onlyone arbitrary parameter:lsPd. The variation oflsPdis consistent with the thermodynamics of the normaand of the superconducting phases. It shows that thpairing mechanism in UBe13 is also responsible for a largefactor of the mass renormalization of the quasiparticleswhich is too large to be reasonably attributed to theelectron-phonon interaction. To our knowledge, this isthe first quantitative indication of a nonphonon mediatedpairing mechanism in any superconductor, making UBe13a privileged candidate for further microscopic studies.

*Present address: Institut de Physique, Rue Breguet 1CH-2000 Neuchatel, Switzerland.

†Present address: Institute for High Pressure Physic142092, Troitsk, Moscow district, Russia.

‡Present address: Department of Material Physics, Faculof Engineering Science, Osaka University, Osaka 560Japan.

[1] H. R. Ott, Progress in Low Temperature Physics(El-sevier Science Publishers, New York, 1987), Vol. XI,p. 215.

[2] M. B. Maple et al., Phys. Rev. Lett.54, 477 (1985).[3] N. Grewe and F. Steglich,Handbook on the Physics and

Chemistry of Rare Earths(Elsevier Science Publishers,New York, 1991), Vol. 14, p. 343.

[4] H. R. Ott et al., Phys. Rev. Lett.52, 1915 (1984).[5] W. P. Beyermannet al., Phys. Rev. B51, 404 (1995).[6] F. Thomaset al., J. Low Temp. Phys.102, 117 (1996).[7] U. Rauchschwalbeet al., Europhys. Lett.3, 757 (1987).[8] G. Schmiedeshoffet al., Phys. Rev. B45, 10 544 (1992).[9] F. Kromeret al., Chin. J. Phys.36, 157 (1998).

[10] M. Tachiki, Springer Ser. Solid-State Sci.62, 230 (1985).[11] U. Rauchschwalbeet al., Z. Phys. B60, 379 (1985).[12] J. P. Brisonet al., J. Low Temp. Phys.76, 453 (1989).[13] L. E. DeLonget al., Phys. Rev. B36, 7155 (1987).[14] J. W. Chenet al., Proceedings of the 17th International

Conference on Low Temperature Physics,edited byU. Eckern, A. Schmid, W. Weber, and H. Wuehl (North-Holland, Amsterdam, 1984), p. 325.

[15] G. H. Landeret al., Adv. Phys.43, 1 (1994).[16] O. Willis et al.,J. Magn. Magn. Mater.63-64, 461 (1987).[17] L. N. Bulaevskiiet al., Phys. Rev. B38, 11 290 (1988).[18] T. P. Orlandoet al., Phys. Rev. B19, 4545 (1979).[19] J. P. Carbotte, Rev. Mod. Phys.62, 1027 (1990).[20] D. Saint James, G. Sarma, and E. J. Thomas,Type II

Superconductivity(Pergamon Press, Oxford, 1969).[21] J. D. Thompsonet al., Phys. Rev. B35, 48 (1987).[22] N. E. Phillips et al., J. Magn. Magn. Mater.63-64, 332

(1987).[23] R. A. Robinsonet al., Phys. Rev. B33, 6488 (1986).[24] S. Nakamuraet al., J. Phys. Soc. Jpn.65, 4026 (1996).