Specific heat of high temperature superconductors in high fields at Tc: from BCS to the Bose–Einstein condensation

Physica C 317–318 Ž1999. 333–344

Specific heat of high temperature superconductors in high fields at Tc : from BCS to the Bose–Einstein condensation Alain Junod ) , Andreas Erb, Christophe Renner Departement de Physique de la Matiere ´ ` Condensee, ´ UniÕersite´ de GeneÕe, ` 24, quai Ernest-Ansermet, CH-1211 GeneÕa 4, Switzerland

Abstract We consider the experimental trends in a database of specific heat measurements near Tc in high magnetic fields for type-II superconductors with a large value of k s lrj , including mostly high-temperature superconductors. Whereas the BCS limiting case is well established in low-Tc superconductors, the exact 3D-XY behavior illustrated by the l-transition of 4 He applies only in the particular case of optimally doped YBa 2 Cu 3 O x . Otherwise, transitions are intermediate either between the BCS and the 3D-XY models Že.g., YBa 2 Cu 3 O 7.00 ., or between the 3D-XY model and the Bose–Einstein condensation ŽBEC. Že.g., Bi 2 Sr2 CaCu 2 O 8 .. The key parameter in ordering this sequence appears to be the product k F j of the Fermi wave number by the coherence length, as evaluated from tunneling spectra in the vortex cores. Such a trend, which is consistent with theoretical descriptions of the strong coupling limit, is visible in the thermodynamics of the phase transition. Implications on the effective mass, the density of pairs just above Tc , the pseudo-gap behavior, etc., are discussed. q 1999 Elsevier Science B.V. All rights reserved. Keywords: High temperature superconductors; Specific heat at Tc ; Critical behavior; Bose–Einstein condensation

1. Introduction It has long been known, at least theoretically, that real-space pairing of electrons followed by a Bose– Einstein condensation ŽBEC. leads to superconductivity w1x. The wave function of the Bose–Einstein condensed state is a limiting case of the BCS wave function w2x. Following the seminal work of Leggett w3,4x, Nozieres ` and Schmitt-Rink w5x established that the evolution from BCS to BEC upon increasing the coupling strength is continuous. The BCS case is the mean-field limit where pairs strongly overlap, i.e., if )

Corresponding author. Tel.: q41-22-702-6204; Fax: q41-22702-6869; E-mail: [email protected]

n is the spatial density and a 0 the pair size, na03 4 1, whereas the BEC limit applies to nonoverlapping pairs, na30 < 1, which behave as a hard-core Bose gas. The monotonous dependence of the critical temperature Tc on the carriers density in underdoped HTS w6,7x is unexplained in the BCS framework, but can be considered as a typical BEC property, since the 3D condensation temperature is given by k B Tc ( 3.3" 2 n 2r3 b rm b , where n b and m b are the density and the mass of the bosonic pairs, or Žomitting a logarithmic correction term. by k B Tc ( 2p " 2 n b drm b in quasi-2D systems with periodicity d w8x. The compatibility of measured data with the BEC scenario has been addressed by many groups, see, e.g., Ref. w9x, sometimes controversially w10x. Monte-Carlo simula-

0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 0 7 7 - 5

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A. Junod et al.r Physica C 317–318 (1999) 333–344

tions of the lattice 2D attractive Žor negative U . Hubbard model were performed recently over a wide range of coupling strengths w11,12x and showed that for a given density, the Tcrt curve Žwhere t is the bandwidth. plotted as a function of Urt Žwhere U is the coupling. tends asymptotically towards the BCS curve for low coupling, then goes through a maximum, and finally decreases towards the BEC asymptote for large Urt. This very strong coupling regime, where Tc decreases proportionally to trU because of the increasing effective mass, is characterized by the progressive buildup of incoherent pairs as the transition temperature is approached from above. The number of remaining fermions that determine the magnetic susceptibility decreases, giving rise to a pseudo-gap behavior above Tc . The natural variable that establishes the crossover from BCS to BEC was identified as k F j w13x, where k F is the Fermi electron wave number and j the coherence length. The chemical potential at T s 0 takes the constant and negative value characteristic of a boson gas when k F j - 1rp , whereas it approaches the positive Fermi level of a fermion gas for k F j ) 2p . The smooth crossover occurs in the region k F j f 1, which corresponds to a superconducting gap ratio 2 Drk B Tc f 5 w14x. According to recent tunneling measurements, YBa 2 Cu 3 O6.92 ŽY123. with 2 Drk B Tc ( 6 w15x should be quite close to this crossover, whereas Bi 2 Sr2 CaCu 2 O 8 ŽBi-2212. with 2 Drk B Tc f 8–12 w16x should rather lie beyond. A striking observation is that the crossover condition, k F j f 1, is equivalent to the condition for the first localized state in the vortex core, E1 f Drk F j w17,18x, to coincide with the value of the gap D and therefore to vanish. Indeed, the fact that one localized state is observable in the vortex core of Y-123 w15x whereas none is visible in Bi-2212 w16x has been a puzzling fact that, in the present context, signifies that the former is just below the crossover whereas the latter has passed it. A true BEC would require k F j < 1rp w13x and it is doubtful that Bi-2212 is already in this limit, as we shall show. Our purpose is to demonstrate that essential features of the BEC are already found experimentally in the thermodynamics of many layered HTS, and that deviations from the BCS or 3D-XY models can be used as an empirical way to order compounds on the k F j axis. We adopt the point of view that the

‘preformed pairs’ behavior that manifests itself as a tunneling gap persisting above Tc w16x is a consequence of very strong coupling of unspecified origin, but we do not address the problem of the mechanism that leads to this coupling. A recent and extensive survey of the BEC, in particular in the context of superconductivity, is found in Ref. w19x.

2. A corpus of specific heat data near Tc We first illustrate the various shapes that the superconducting transition can take in the specific heat curves. Our choice was guided by the availability of pure samples with sharp transitions, preferably single crystals, in order to avoid spurious effects due to broadening by inhomogeneity. High resolution is required. Recall that whereas the specific heat jump at Tc for vanadium, for example, is larger than the specific heat in the normal state, that of Y-123, which ranks highest among HTS, does not exceed 5% of the total specific heat. The situation for Bi2212 is about three times worse. Therefore to map the details of the Žsmall. superconducting transition, it is necessary to subtract a Žlarge. background. In order to circumvent possible criticisms of arbitrariness concerning the choice of this background, we show the experimentally measured difference C ŽT, B . y C ŽT, Bmax ., where Bmax is on the order of 7 to 14 T. This field is generally sufficient to wash out the singular part of the transition; furthermore, the lattice contribution is insensitive to the field, so that C ŽT, B . y C ŽT, Bmax . gives the essential features of the electronic component CeŽT, B ., in particular for B s 0 Žof course, it does so exactly for low-Tc superconductors if Bc2 Ž0. F Bmax .. We have arranged the samples in the sequence where the BCSlike jump component is less and less apparent at the transition. Correspondingly, the anomaly becomes more and more symmetrical about Tc . With one exception, the samples considered in Fig. 1 are measured adiabatically. Ža. Nb 77 Zr23 , a cubic Žisotropic. polycrystal with Tc s 10.8 K, Ginzburg–Landau ŽGL. parameter k ( ˚ The coherence volume contains more 22, j ( 64 A.

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335

Fig. 1. Specific heat at the transition for various superconductors. The curves measured in the largest available magnetic field are used as a baseline and subtracted Žsee text for details.. Ža. Nb 77 Zr23 , Žb. YBa 2 Cu 3 O 7.00 , Žc. YBa 2 Cu 3 O6.92 , Žd. YBa 2 Cu 3 O6.6 w20x, Že. Bi 2.12 Sr1.9 Ca 1.06 Cu 1.96 O x f 8 , Žf. Bi 1.84 Pb 0.34 Sr1.91Ca 2.03 Cu 3.06 O 10.1.

than 10 4 atoms, therefore superconducting fluctuations are not observable w21x.

Žb. Overdoped Y-123, i.e., YBa 2 Cu 3 O 7.00 equilibrated in 100 bar oxygen at 3008C, a high-purity

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crystal grown in BaZrO 3 ŽBZO. w22–24x with Tc s 88 ˚ and anisotropy K, k a b f 100–150, j a b f 10–15 A ratio g ' j a brj c s 5.3 " 0.5 w25x. Žc. Optimally doped Y-123, i.e., YBa 2 Cu 3 O6.92 equilibrated in 1 bar oxygen at 5108C, a high-purity crystal grown in BZO with Tc s 93 K and g s 7.0 " 0.5 w25x. Žd. Underdoped Y-123, i.e., YBa 2 Cu 3 O6.6 , a crystal with Tc s 62 K w20x. Note that here Bmax s 7 T and the specific heat, measured by an oscillating temperature method, is given in relative units. Že. Bi-2212, i.e., Bi 2.12 Sr1.9 Ca 1.06 Cu 1.96 O x , a crystal obtained by floating zone melting and annealed in 1 bar oxygen at 8608C, with Tc s 85 K w26x. Žf. Bi-2223, more precisely Bi 1.84 Pb 0.34 Sr1.91Ca 2.03 Cu 3.06 O 10.1 , a ceramics with Tc s 108 K w27x. Due to the large anisotropy of Bi-based compounds, the averaging effect of the random grain orientation merely halves the amplitude of the variation of the specific heat at Tc with the field, whereas the position and shape remain unchanged. In addition to these samples, we shall also refer to data from our laboratory on the following crystals grown in BZO: DyBa 2 Cu 3 O 7.00 w28x, YBa 2 Cu 3 O x with x s 6.94 and 6.96 w29x Žsame sample as that of Fig. 1b, with different anneals., EuBa 2 Cu 3 O 7.00 w30x, and to other data sets obtained with one YBa 2 Cu 3 O x crystal grown by traveling solvent floating zone melting ŽTSFZM. with x s 6.82, 6.87, 6.90, 6.95, 6.97 and 7.00 w31,32x. All these 123-phase crystals show characteristic specific heat anomalies on the melting line of the vortex lattice when they are in the overdoped state. The Bi 2.12 Sr1.9 Ca 1.06 Cu 1.96 O x crystal of Fig. 1e was also overdoped in 10 bar oxygen at 5008C, resulting in Tc s 76 K w33x. Additional data are available for Tl-2201 w34x, Hg-1201 w35x and Hg-1223 w35x. The transitions can be classified into three groups: ŽI. low temperature superconductors ŽLTS. ŽII. M-123 with M s Y, Dy, Eu . . . ŽIII. ŽBi, Tl, Hg.-cuprates In group ŽI., the transition in C ŽT, B . –C ŽT, Bmax . appears as a step which is shifted towards T s 0 with little smearing when the field increases. The superconducting curve in H s 0 crosses the normal-state curve slightly above Tcr2 ŽFig. 1a.. A relatively large anisotropy does not change qualitatively this

BCS-like result, see, e.g., the layered compound 2H-NbSe 2 w36x. In group ŽII., the shape depends strongly on the exact oxygen stoichiometry. For fully oxidized samples ŽFig. 1b., the step shape is still obvious, but the crossing point occurs in the region close to 85–90% of Tc . The broadening of the transition with the field is of the same order as its shift, so that the transition onset seems to remain locked to the zero-field Tc whereas the top of the specific heat peak clearly moves to lower temperatures. For optimally doped samples ŽFig. 1c., these characteristics are enhanced and the l-shape of the transition becomes obvious. The transitions of group ŽIII. ŽFig. 1e–f. show two strikingly unusual characteristics: the transition becomes symmetrical, and the top of the specific heat anomaly no longer shifts with the field, as if the upper critical field Bc2 had an infinite slope. We stress that whereas there was some doubt about the effect of the homogeneity in early reports, the shape of the transitions shown here has been reproduced now by several laboratories and confirmed independently by thermal expansion measurements w37x. There is no way to broaden an asymmetrical transition such as that of Y-123 to obtain a symmetrical transition typical of Bi-2212. Group ŽIII. is generally characterized by a large anisotropy ratio g . The shape of the specific heat at the transition is unexplained by conventional ŽBCS. theory, even taking into account strong fluctuations; we shall argue that this qualitative change with respect to Y-123 is due to the crossover towards very strong coupling that ends with the limiting case of the Bose condensation. Strongly underdoped Y-123 Ž60-K phase. represents an intermediate case, both from the point of view of the shape of the transition and that of the anisotropy ŽFig. 1d..

3. Phenomenological description of the transition We first recall a few facts about the specific heat at the BCS, 3D-XY and BEC transitions. The ideal BCS case ŽFig. 2a. is the best known example of a second-order transition with a discontinuity in the second-order derivatives of the free energy F at Tc , e.g., in the specific heat CrT s E 2 FrET 2 or in the

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Deviations from the mean-field model are caused by thermal fluctuations of the order parameter in the vicinity of Tc . The field and temperature dependence of the fluctuation specific heat can be described by the critical exponents a and n , e.g., in B s 0: C" s T

A"

C

ž / T

q

a

background

< t
t'

T y Tc Tc

.

Ž 1. This form can be derived from the following form for the singular free energy density in the critical region Žsee Refs. w38,39x for a generalization to the anisotropic case.: Fs s

Fig. 2. Comparison of representative theoretical and experimental specific heat curves near Tc . Ža. Ideal BCS curve, together with data for vanadium. Žb. 3D-XY specific heat of 4 He Žline. scaled to that of YBa 2 Cu 3 O6.92 Ždata.. Žc. Specific heat at the Bose–Einstein condensation.

slope of the magnetization EMrET s E 2 FrEHET or EMrEH s E 2 FrEH 2 .

Q "k B Tc

jD

f

Bj 2

ž / F0

j Ž t . s j Ž 0 . < t
,

Ž 2.

where the q and y superscripts correspond to t ) 0 and t - 0, respectively, Q " is a pair of universal numbers, F 0 the flux quantum, and f Ž x . is a universal function normalized to f Ž0. s 1 w38,39x. ‘Universal’ means constant for a class of transitions characterized by the same number of components in the order parameter and the same dimension D of the system. The derivation of Eq. Ž1. from Eq. Ž2. verifies the hyperscaling relation a q Dn s 2. Critical fluctuations in the 3D-XY model Ž D s 3, twocomponent complex order parameter. are characterized by AqrAys 1.054 " 0.001, a s y0.01285 " 0.00038 and n ( 0.6710 " 0.0001 w40x. In this case, the specific heat anomaly at Tc takes the form of an asymmetric quasi-logarithmic divergence, although rigorously it is a finite cusp ŽFig. 2b.. The 3D-XY model accounts well for the thermodynamics of the superfluid transition of 4 He at 2.18 K. In the limit where fluctuations of the vector potential may be neglected, i.e., when k 4 1, type-II superconductors should belong to the 3D-XY universality class. The size of the critical region is estimated using the Ginzburg criterion Ž DC ' specific heat jump at Tc .: tG s

1 32p 2

ž

kB

1

DC j a j b j c

2

/

A Ž number of pairs in the coherence volume.

y2

.

Ž 3.

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Finally, the ideal BEC condensation, the final point of the evolution towards very strong coupling, gives rise to a specific heat which is known analytically over the whole temperature range ŽFig. 2c. w41x. In 3D, C s 1.925k B ŽTrTc . 3r2 per boson at T F Tc . This is followed by a discontinuity in the third derivative of the free energy at Tc , i.e., C does not show any divergence or jump but rather a triangular peak. At T ) Tc , the specific heat decreases smoothly from C ŽTc . s 1.925k B to the Dulong Petit value C Ž`. s 1.5k B . The critical exponents of the BEC are a s y1, n s 1 Žin 3D, and assuming a dispersion relation E A k 2 .; the universality class is that of the spherical model of ferromagnetism w42x.

4. Classification of the data Transitions of group ŽI. ŽFig. 1a. are of the mean-field type. This is due to the large coherence volume; the critical region is too small to be observed. The large value of the GL parameter k of Nb 77 Zr23 and the large anisotropy g of 2H-NbSe 2 do not change this result. Transitions of group ŽII. ŽFig. 1b–c and other R.E.-123 compounds. are critical and look more or less like the superfluid transition of 4 He. Owing to the huge increase of the factor g 2rj a6b , the critical region approaches 10% of Tc for Y-123 w43x Žcuriously enough, the factor DCrk B ( 3.4 = 10 21 cmy3 for overdoped Y-123 is not very different from the corresponding value for Nb.. Therefore it is expected that the 3D-XY model should hold for the superconducting transition of Y-123. This is indeed confirmed by many groups Že.g., Ref. w44x., but we shall point out that this is an accidental coincidence, valid for only one oxygen concentration close to optimal doping. Fig. 2b shows the total specific heat of a high purity single crystal of YBa 2 Cu 3 O6.92 , superposed with the specific heat of 4 He at the superfluid transition. The latter has been scaled both in temperature and in amplitude, and a very smooth background given by an Einstein lattice specific heat curve has been added. The coincidence is striking. The transition specific heat of 1 mol Ž104 cm3 . of YBa 2Cu 3 O6.92 at Tc is equivalent to that of 0.17 mol of 4 He Ž Vmol s 27.4 cm3 . at Tl . The density of liquid 4 He being 2.2 = 10 22 atoms cmy3 , the transition of

Y-123 has the same amplitude as that of a set of 1.0 = 10 21 bosons cmy3 condensing at Tc . This number is of the same order of magnitude as the carrier concentration deduced from Hall effect measurements above Tc , i.e., 1.5 = 10 21 holes cmy3 w45x. However, it is clear that the ideal BEC would not be an adequate description of Y-123. The observed transition temperature would require a single-particle mass enhancement mUrm ( 6.4 in the quasi-2D case with the measured anisotropy g s 7 of optimally doped Y-123. This would lead to overestimate the penetration depth l by a factor of about two. The transitions of 4 He and YBa 2 Cu 3 O6.92 belong to the same universality class, but this does not necessarily imply the condensation of preexisting bosons in both cases. Based on Ginzburg’s criterion, Eq. Ž3., it would seem that the transitions in group ŽIII., in particular that of Bi-2212, should be even better examples of 3D-XY behavior. This is expected because these specific heat curves do not show any measurable jump component DC, and the coherence length j has decreased with respect to that of Y-123. Direct measurements of j for Bi-2212 are scarce. In particular, estimations based on an empirical definition of Hc2 depend on an arbitrary criterion for Tc Ž H .; remember that the specific heat anomaly does not shift with the field w26x whereas the resistive transition does w46x. A direct but relative estimation of j is given by the diameter of vortex cores. Recently, scanning tunneling spectroscopy ŽSTS. in Y-123 and Bi-2212 has allowed to measure the distance from the vortex center at which the superconducting density-of-states ŽDOS. is recovered. This distance is at least four times smaller for Bi-2212 than for Y-123 w16x. Therefore, because of the considerable decrease in the product Ž DCj a j b j c . 2 Žor its 2D counterpart Ž DCj a j b d . 2 ., Bi-2212 should exhibit ideal 3D-XY behavior at almost any temperature. Experiment definitely shows that this is not the case. Fig. 3a shows the ‘best’ Žnevertheless poor. fit of a 3D-XY curve on Bi-2212 data. Similarly, all transitions of group ŽIII. deviate strongly from 3D-XY: they are symmetrical, the jump component vanishes, and the positive curvature of C on both sides of Tc is smaller than for 4 He. The unconventional shape of the specific heat of Bi-2212 at Tc has been known for some time w48x,

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339

Fig. 3. Fits of the specific heat of Bi-2212 at Tc . The optimally doped Bi 2 Sr2 CaCu 2 O x ceramics was cooled along a constant x-line from 8468C, 0.4 bar O 2 to 4688C, 0.0005 mbar O 2 , thus avoiding critical quenching requirements w47x. Ža. 3D-XY fit: critical exponent and critical amplitude ratio of 4 He, i.e. a ( 0y and J s 4 Žresiduals: 0.18% r.m.s... Žb., a ( 0, free J converging to 0.2 Ž0.12% r.m.s... Žc. J s 0, free a converging to a s y0.67 Ž0.076% r.m.s... Žd. BEC fit, full function w41x Ž0.076% r.m.s... The fitting range is 70–120 K, 550 points, no data excluded in the vicinity of Tc .

but in the early days the homogeneity of the samples could be suspected. Now this shape has been confirmed not only with improved crystals of Bi-2212, but also with Bi-2223 w27x, Tl-2201 w34x, Hg-1201 w35,49x, Hg-1223 w35,49x, etc. At this point, it is important to emphasize that the agreement of a specific heat curve with the 3D-XY model implies matching two universal parameters, a critical exponent a or n and an amplitude ratio AqrAy. It is true that if one relaxes the second condition, i.e., if one allows the model curve to become symmetrical, then a better fit of Bi-2212 data can be obtained ŽFig. 3b. w37x. However this does not imply agreement with the 3D-XY model. This distinction becomes

clear by rewriting Eq. Ž1. in the approximation < a ln < t < < < 1: Cy

C

( T

ž / T

q Aq Ž J y ln < t < . ,

T - Tc ,

background

Ž 4a . Cq

C

( T

ž / T

q Aq Ž yln < t < . ,

T ) Tc ,

background

Ž 4b . where J ' Ž AyrAqy 1.ra ( 3.99 " 0.19 w40x is a universal 3D-XY dimensionless number. Aq depends on the coherence volume. A fit of Bi-2212

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A. Junod et al.r Physica C 317–318 (1999) 333–344

Table 1 Parameters of Eqs. Ž4a. and Ž4b. given by fits of specific heat data for various HTS

3D-XY model DyBa 2 Cu 3 O 7.00 , overdoped EuBa 2 Cu 3 O 7.00 , overdoped YBa 2 Cu 3 O 7.00 , overdoped YBa 2 Cu 3 O 7.00 , overdoped YBa 2 Cu 3 O6.92 , optimally doped YBa 2 Cu 3 O6.6 , underdoped YBa 2 Cu 3 O6.5 , underdoped YBa 2 Cu 3 O 7.00 YBa 2 Cu 3 O6.97 YBa 2 Cu 3 O6.95 YBa 2 Cu 3 O6.90 YBa 2 Cu 3 O6.87 YBa 2 Cu 3 O6.82 Bi 2 Sr2 CaCu 2 O 8qx , overdoped Bi 2 Sr2 CaCu 2 O 8qx , overdoped Bi 2 Sr2 CaCu 2 O 8qx , optimally doped

Tc ŽK.

Aq ŽmJrK 2 g at.

J

Type of sample

Ref.

90.3 94.4 87.8 88.3 93.0 63.1 53.7 88.3 91.1 92.5 92.9 91.4 88.4 76.3 85.5 92.0

A jy3 0.68 0.78 0.72 0.70 0.80 0.17 0.16 0.53 0.51 0.52 0.53 0.52 0.37 0.79 0.61 0.39

4.0 6.3 5.6 6.1 5.9 3.6 2.7 2.4 6.0 6.1 5.3 4.4 3.1 3.2 0.8 0.3 0.2

BZO BZO BZO BZO BZO BZO BZO TSFZM TSFZM TSFZM TSFZM TSFZM TSFZM TSFZM TSFZM ceramics

w28x w30x w25x w50x w25x w30x w30x w31x w32x w31x w32x w31x w32x w31x w32x w31x w32x w31x w32x w33x w26x w47x

BZO, high purity crystals grown in BaZrO 3 crucibles. TSFZM, crystals grown by traveling solvent floating zone melting. The Y-123 TSFZM data refer to one single sample after various anneals.

data with Eqs. Ž4a. and Ž4b. is possible, although not perfect, and converges to J - 1 ŽFig. 3b., showing that the transition does not belong to the 3D-XY class. More generally, the analytic form of Eqs. Ž4a. and Ž4b. gives a simple means to monitor the deviations from the 3D-XY model as long as < a < < 1. This was done extensively using data from our laboratory; the results are given in Table 1. The surprising information given by Table 1 is that the deviations from 3D-XY are not only present in Bi-2212, but enter gradually when the anisotropy increases. Ideally, a pure mean-field curve analyzed in the same way would yield J s ` Žbut the product AqJ would remain finite and equal to the GL specific heat jump. 1, whereas a pure BEC curve would yield J s 0 because of the absence of jump. In this sense, overdoped Y-123 shows deviations towards mean-field behavior, Bi-2212 shows deviations towards the Bose condensation, and optimally doped

1

The 3D-XY model should remain valid in the BCS region, so that experimental values J ) 4 Žmeaning that the divergence is too small. may be understood as a consequence of our inability to observe the rapidly narrowing critical region with state-of-the-art experiment and samples. Alternatively, deviations J - 4 cannot be explained by a similar reasoning, because the jump component is a robust feature that is not washed out by smearing.

Y-123 appears accidentally as the only ideal 3D-XY superconductor. 1 We shall come back later to the validity of the assumption < a < < 1 for transitions of group ŽIII.. Before this, as the progressive deviations shown in Table 1 are an essential point of this work, we shall discuss their experimental reliability. The uncertainty in the estimations of J was evaluated using simulations. An ideal total specific heat curve with parameters corresponding to those of YBa 2 Cu 3 O6.92 , but J set to exactly 4, was constructed, convoluted with a Gaussian smearing function with half-width s s 0.5 K, and added with 0.2% peak-to-peak noise. Eqs. Ž4a. and Ž4b., together with an Einstein background specific heat, was then fitted to this artificial data set over a w Tc y 15 K, Tc q 15 Kx interval, omitting the points within w Tc y s , Tc q s x. This is the typical range used with real data, except that s is generally smaller. The fitted value of J was found to be 4.1 instead of 4.0, an error much smaller than the variations reported in Table 1. Direct inspection of the curves shows that the actual smearing is generally smaller than for this test. For comparison, the same procedure was applied to an ideal mean-field transition defined by the two-fluid model, again analyzed with Eqs. Ž4a. and Ž4b. which is critical. The best fit for J was then 17.

A. Junod et al.r Physica C 317–318 (1999) 333–344

Deviations from XY behavior Ž a ( 0y, n ( 2r3, J ( 4. have already been noticed before in Y-123 and attributed to an unusual asymmetry of the critical exponents, e.g., aq( 0.5 and ay( y0.3 w51,52x, or nq) 1 and ny( 0.67 w53x. Taken together, these conclusions do not obey the 3D hyperscaling relation a s 2 y Dn . Furthermore, such an asymmetry cannot explain the shape of the Bi-2212 specific heat near Tc . Alternatively, Pasler et al. w43x studied the XY behavior of YBa 2 Cu 3 O x near optimal doping ŽTc s 92 K. using differential thermal expansion along the a and b axes, a quantity that is thermodynamically related to the specific heat, but that is not plagued by a large background. 2 They found on one hand that the critical XY symmetrical exponent aq s ays a ( 0 fits the data much more satisfactorily than the Gaussian exponent a s 1r2 over a temperature range of about "10 K about Tc , and on the other hand that the ratio J is not equal to 4.0, but 5.4. Their result is consistent with our specific heat data for TSFZM-YBa 2 Cu 3 O6.95 ŽTable 1.. For Bi2212, the same group confirmed that the anomaly at Tc is symmetric, i.e., J f 0, and noticed that the critical range extends over about 50 K, which is considerable w37x. These thermal expansivity data completely agree with the present specific heat data, and confirm that the determination of J is robust. The working hypothesis < a < < 1 was a convenient way to quantify the deviations from the 3D-XY model. However, whereas Eqs. Ž4a. and Ž4b. fits the data of group ŽII. ŽY-123-like compounds. within experimental noise, it does not so for group ŽIII. ŽBi-2212-like compounds.. For the latter, it is necessary to return to the more general Eq. Ž1. or Eq. Ž2., allowing a to vary. In fact, we already know that a cannot be small by considering the variation of the specific heat in a magnetic field. From Eq. Ž2., it follows that w54x: E Ž CrT . E ln B

a

s By 2 n g

Bj 2

ž / F0

,

Ž 5.

where g Ž x . is a scaling function. Experiment has shown that the amplitude of EŽ CrT .rEln B, a quan-

2

The parameter G in this article corresponds to 1r J in the present work.

341

tity that does not require any phonon background subtraction, is proportional to B f 0 in Y-123 and B f 1r3 in Bi-2212 w26,55x. Therefore < ar2 n < cannot be small for Bi-2212. This determination is not critically sensitive to the homogeneity of the sample, because the initial width of the transition is negligible with respect to the broadening introduced by a field of a few teslas. On the contrary, a determination of the exponent a based on the zero-field specific heat requires a high homogeneity. The best sample of our database in this respect is a Bi 2Sr2 CaCu 2 O x ceramics that was equilibrated by cooling along a constant x-line, with variable pressure and temperature ŽFig. 3. w47x. The sharpness of the C break at Tc and the absence of diamagnetism at T G 92.5 K indicate that the distortion due to a distribution of Tc is very limited. A fit using Eq. Ž1. yields a s y0.67 " 0.1. Combining this result in zero field with the determination ar2 n s y0.33 " 0.05 in B ) 0, we obtain n s 1.06 " 0.3. Scaling plots of the specific heat and the magnetization in the variable < t
A. Junod et al.r Physica C 317–318 (1999) 333–344

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Table 2 Critical exponents in various models, experimental values for Bi-2212, and corresponding behavior of C A < t
1 0.707 1.054 Ž1. Ž1.

a q1 q0.5 f 0y f y0.7 y1

n

C Ž t ™ 0.

0.5 0.5 f 2r3 f1 1

< t
ECrEB Ž t ™ 0. y2

AB A By1.5 A By1 A By0 .7 A By0 .5

Bc2 Ž t ™ 0. A
When a ™ y1, AqrAy is correlated with the background and becomes meaningless. 3D BEC refers to the charged boson gas w56x.

i.e., Bm ŽT . A t 1.34 for Y-123 and BmŽT . A t 2 for Bi-2212. Since Bi-2212 is closer to the BEC limit than to the BCS limit, we may obtain qualitatiÕe information from a fit of the ideal BEC specific heat function w41x with the experimental data for Bi-2212. This fit ŽFig. 3d. is nearly as good as that with a free exponent a ŽFig. 3c. which represented an intermediate behavior; in all cases the background is smooth and physical, without any inflection point. This fit yields a crude estimation of the boson density at Tc , n b ( 6 = 10 21 cmy3 . Assuming an anisotropy in the range g f 50–200, and using the quasi-2D formula k B Tc s 2 p " 2 n b drmUb , a b ln Ž 2 k B Tc mUb , c d 2 r" 2 . Žwhere mUb, a b s 2 mUa b , mUb,c s 2 mUc and n b s nr2 are boson quantities. w8x, the mass enhancement is then mUa brm a b f 20-25. With these values, the in-plane penetration depth becomes l a b s w mUb, a b rm 0 n b Ž2 e . 2 x1r2 f 220–240 nm, well in the range of published values w59x. Unlike the case of Y-123, there is here no inconsistency. Angle-resolved photoemission spectra ŽARPES. have shown the presence of a Fermi surface in Bi-2212, an observation that excludes an ideal BEC of preexisting bosons. One probably does not have to use the argument that ARPES characterizes only the first ppm at the surface of the sample, whereas specific heat probes the remaining volume. The critical exponents show that we are not in the BEC limit, so that a composite model consisting of both fermions and bosons condensing at Tc is more appropriate. Furthermore, unlike the case of 4 He, the binding energy of the pairs ŽA D . and the condensation energy ŽA k B Tc . differ by a factor of only 4 to 6 w16x. Therefore the number of fermions issued from broken pairs, proportional to expŽyDrk B T ., is not negligible at the transition temperature.

Finally, one might wonder if the anomalous transition of group ŽIII. superconductors is not merely a consequence of extreme anisotropy. The XY model has been studied numerically over the full range from D s 3 to D s 2 w60x. A characteristic feature in the vicinity of D s 2 is the separation between the Kosterlitz–Thouless temperature TKT at which algebraic order sets in Žand at which the specific heat remains continuous in all its derivatives., and the temperature Tp at which the specific heat shows a broad peak due to vortex unbinding. The ratio is TprTKT ( 1.22 in the 2D limit. In contrast to this, experiment shows that the specific heat peak of Bi-2212 is sharp and coincides within 0.5% with the diamagnetic and resistive transition temperatures. Therefore a dimensionality change alone is not at the origin of the change in the calorimetric transitions. The crossover near k F j f 1 from BCS-like superconductivity for weak coupling to BEC-like superconductivity for strong coupling provides a consistent framework in which one can qualitatively explain numerous observations: the presence of a vortex core level in Y-123 and its absence in Bi-2212 w15,16x, the 4 He-like calorimetric transition at Tc in Y-123 and the BEC-like transition in Bi-2212, the corresponding change in the melting line Bm A < t < n of vortex matter, the apparent contradiction between a calorimetric transition that does not shift with the field and a resistive transition that does so in Bi-2212 w56x, the change in the temperature dependence of the magnetic susceptibility from overdoped to underdoped systems w12x, the persistence of the normalstate gap at higher temperatures above Tc in Bi-2212 than for Y-123 w15,16x, the puzzling stability of d-wave HTS against disordering w61x, the nonlinearity of the plot of the penetration depth ly2 vs. T near Tc in Bi-2212 w62x Žin the BEC limit, ly2

A. Junod et al.r Physica C 317–318 (1999) 333–344

A Ž t ln < t <. 2 w41x., etc. A convincing argument given by Alexandrov et al. w56x is the calculation of the specific heat at Tc vs. B in a weakly interacting charged boson gas, in remarkable qualitative agreement with the data for group ŽIII. HTS. Another puzzling feature of HTS is the quasi-linear in T term that is present in the zero-field specific heat at T < Tc w54x. The ideal Bose gas intrinsically provides low-lying excitations which lead to C A T 3r2 or C A T at low T, depending on the dimension; similar pair excitations of finite momentum Žcalled pairons in Ref. w63x. occur in the intermediate coupling regime according to the recent three-fluid model of Chen, Kosztin and Levin w63x. Finally, it is anticipated that a reexamination of the reversible mixedstate magnetization in the very strong coupling limit, another thermodynamic quantity, could provide new insights into the problem of the BCS–BEC crossover.

Acknowledgements We thank all our present and former collaborators who helped to gather the specific heat data in high fields, mainly M. Roulin, B. Revaz, G. Triscone, E. Walker, A. Mirmelstein and K.Q. Wang. The support of J. Sierro and the fruitful collaboration with C. Marcenat and F. Bouquet from the CEA Grenoble are gratefully acknowledged. This work was supported by the Fonds National Suisse de la Recherche Scientifique.

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