specific heat up to 14 tesla of a yba2cu3o6.92 single crystal

ELSEVIER Physica C 234 (1994) 269-279

glllgt

Specific heat up to 14 tesla of a YBa2Cu306.92 single crystal

E. J a n o d a,a, A . J u n o d a, K . - Q . W a n g a,2, G . T r i s c o n e a, R . C a l e m c z u k b, j . . y . H e n r y b

a Universit~ de Gen~ve, D~parternent de Physique de la Mati~re Condensde, 24 quai Ernest-Ansermet, CH- 1211 Gen~e 4, Switzerland b CEA/D~partement de Recherche Fondamentale sur la Mati~re Condens~e/SPSMS/LCP,

17 rue des Martyrs, F-38054 Grenoble Cedex 9, France

Received 13 June 1994; revised manuscript received 3 October 1994

Abstract

A large porous YBa2Cu306.92 single crystal was synthesized using a recrystallisation technique. The superconducting transition at Tc=92.4 K could be narrowed down to 0.14 K by quenching. We present high resolution specific heat measurements in magnetic fields from 0 to 14 T applied along the principal crystallographic directions. Thermal fluctuations near the phase transition in zero field are analysed in terms of critical behaviour. The Ganssian approximation is found to be unsuitable inside of a temperature window I T - Tc I ~ 3 K. Measurements in magnetic fields indicate an anisotropy of e = 7 + 1 for Hc2 , a positive curvature of the He2 (T) line near Tc, and a mixed-state linear term 7roT increasing with H. The slope of the upper critical field at T~, p~OH~2,c/OT, is found to be approximately -3 .2 T /K both by direct determinations of Tc(H), and by the analysis of Gaussian fluctuations in H = 0. The scaling behaviour predicted either by the low-field 3D-XY or by the high-field critical ap- proaches are compared.

1. Introduction

In spite o f the large interest for the anisotropic superconducting phase YBa2Cu3Ox (Y-123) since its discovery [ 1 ], only a few specific heat measurements in single crystals under high magnetic fields have been published to date. This is due partly to difficulties in obtaining large single crystals with sharp superconducting transitions, and partly to the high experimental resolution in high fields required to separate the superconducting anomaly f rom the total specific heat. Such measurements are however extremely use- ful to understand basic the rmodynamic propert ies

Present address: CEA/D~l~trtement de Recherche Fondamen- tale sur la Mati~re Condens6e/SPSMS/LCP, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France. 2 Present address: Physics Department, University of Science and Technology of China, Hefei, Anhui 230026, China.

such as e.g. the type of superconducting fluctuations that occur close to To, either Gaussian [ 2 -6 ] or critical [4 ,6 ,7-10] , the scaling behaviour in magnetic fields [ 1 l - 14 ], etc.

In this paper, we present the measurement and the analysis o f the heat capacity in magnetic fields up to 14 T of a large Y-123 single crystal with a superconducting transition 0.15 K wide.

2. Sample preparation and characterisation

A porous YBa2Cu3Ox single crystal with a mass of 150 mg was synthesised using a recrystallisation

technique [ 15 ]. A final t rea tment at 460°C in air during 120 h followed by quenching was per formed in order to homogenise the oxygen concentration; the latter reaches then about x = 6.90-6.94. For this doping, Tc is m a x i m u m and OTdOx vanishes [16,17].

0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD10921-4534 ( 94 ) 02320-4

270 E. Janod et al. / Physica C 234 (1994) 269-279

This results in a narrow distribution of the critical temperature in the case of persistent oxygen inhomogeneities. X-ray diffraction shows the presence of inclusions of the impurity phase Y2BaCuOs (Y-211 ). This green phase concentration, estimated using the Curie-Weiss component of the normal-state susceptibility [ 18 ], is found to be 15 to 18% in mass. Re- suits will be corrected for its presence.

Fig. 1 shows the AC susceptibility measured with an excitation field H^c=0.1 Oer~ at 81 Hz in the configuration H parallel to the ab planes. The mid- point of the superconducting transition is found at T~°*/*=92.4 K; the transition width measured between the 10% and the 90% points i's ATe=0.14 K. Whereas the critical temperature is determined by the selected annealing conditions, oxygen pressure and temperature, the transition sharpness is rather en- hanced by the final quenching process. Note that the critical temperature is 2.5 K lower for a fully oxidised crystal from a similar batch [ 19 ], (Fig. 1 ).

3. Specific heat investigations

Specific heat experiments were carried out in an adiabatic, continuous heating type calorimeter using platinum thermometry [ 19 ]. Its accuracy is 0.5% and the resolution is ~< 0.04% for 0.2 g samples. C(T) was measured from 30 to 300 K in zero field and from 50 to 225 K in magnetic fields. The fields, 1-14 T, were applied in either direction Hllab and HIIc while the

~" 0

-0.01

=~e -0.02 -&

-0.03 91

ATe(10-90%)=0.14 K'

~ ' j H n m = 0 . 1 0 e

, ' ~ . . . . I . . . .

91.5 92 92.5 93

Temperature (K)

Fig. 1. Real pa r t (Z ' ) and imag ina ry par t (X ' ) o f the AC suscep-

t ib i l i ty of the YBa2CusO6.92 single crystal as a func t ion of the tempera ture .

sample was in the normal state to ensure maximum penetration and equilibrium conditions. Specific heat is given in J/g at K; 1 g a t= 51.46 g for YBa2Cu306.92.

The discussion of the specific heat data is divided into three parts. In Section 3.2 we analyse the data between 0 and 14 T to obtain informations such as the anisotropy ratio ~ or the shape of the He2 (T) line. In Section 3.3 we investigate the fluctuations in zero field near the superconducting phase transition. In Section 4 we finally discuss the scaling properties of the specific heat in magnetic fields.

3. I. Preliminary specific heat measurements of Y2BaCu05

As previously mentioned, the YBa2Cu306.92 crystal contains about 15-18% of Y-211 inclusions. We have measured the specific heat of a polycrystalline sample of this impurity phase in zero field (Fig. 2) in order to subtract its contribution from the total heat capacity. Y-211 is an antiferromagnetic insulator which does not present any anomaly in the temperature range of the superconducting transition of Y- 123. Its specific heat may be represented to good accuracy using a lattice contribution only. Three Einstein modes are fitted to the C / T vs. T curve from 40 to

0.1

0.08

~ 0.06

~--. 0.04

0.02

• Y2BaCuO5

- - fit 40-225K

0 ' ' ' I . . . . I . . . . I . . . . I . . . . : . . . .

0 50 100 150 200 250 300

Temperature (K)

Fig. 2. Specific heat C/TofY2BaCuOs as a function of the temperature. The endothermic peak at 230-240 K is caused by the grease used to paste the sample in the calorimeter. The excess specific heat in the 20-30 K range corresponds to antiferromag- netie ordering [ 20-23 ].

E. Janod et al. / Physica C 234 (1994) 269-279 271

225 K (Fig. 2). The parameters of the fit are given in Table 1. At TN=30 K, Y-211 orders antiferromag- neticaUy. This temperature is low enough to avoid any interference with the specific heat anomaly of YBa2Cu306.92 near T¢. The specific heat C/T of Y- 211 at 100 K is 13% smaller (per gram-atom)than that of high-quality Y- 123 ceramics [ 6 ]. It presents a positive slope 0 (C /T) /OT of about 0.148 m J / K 3 g at at 100 K, whereas pure and fully oxidised Y-123 has an essentially zero slope at the same temperature. After correction, the presence of Y-211 should not influence the analysis of the specific heat temperature dependence and its derived parameters. Note however that only 82-85% of the full jump at T¢ is expected in the raw uncorrected data.

3.2. Results in 0-14 T

We first comment on the specific heat in zero field (Fig. 3). The bulk transition width ATe, defined as the width at half height of the derivative 8 (C/T)/OT, is equal to 0.40 K. This value includes intrinsic broadening due to fluctuations. The uncorrected amplitude of the specific heat anomaly ( C / T ) m ~ - (C/T)mi. (see Fig. 3) reaches 3.00 m J / K 2 gat. The positive curvature clearly observed below T~ evi- dences the presence of thermal fluctuations. After correcting for the presence of 15% (lower limit) of green phase, the expected horizontal slope 0 (C /T) / 0Tat 100 K is restored and the specific heat anomaly increases to (C/T)'m~x - ( C~ T) rain = 3.70 rnJ/K 2 g at (lower limit).

Figs. 4(a) and (b) show the uncorrected specific heat measurements in both field directions (Hllab and HIIc). The data tend to confirm earlier measurements performed in similar Y-123 single crystals with a slightly higher oxygen content [ 19 ]. Namely, the high-temperature onset of the transition seems to be

Table 1 Specific heat of the green phase Y2BaCuOs: characteristic parameters deduced from the fit of Eq. (2) between 40 and 225 K

Do 0.181356 0o (K) 121.809 Do+D~ 0.601343 01 (K) 320.262 Do+D~ 4"/)2 1.00854 02 (K) 651.164

0.111

0.11

0.109

0.108

0.107 ,,/

0.106

0.105

0.104

0.103

t ~ . ~~)m,~ l C/T)'max

, / / ..........

/ 0.102 . . . . I . . . . 1 . . . . I . . . .

80 85 90 95 100

Temperature (K)

Fig. 3. Specific heat C/Tvs. T for the YBCO single crystal before (open squares) and after (filled squares) having subtracted the contribution of Y2BaCuO5 and having renormalised the results. A third curve corresponding to a high-quality ceramics measured in our laboratory with the same oxygen concentration is plotted as a reference (open diamonds). Note that the raw amplitude of the specific heat anomaly (C/T)m~-(C/T)mt. of the crystal, corrected for the green phase content, is close to the value for the ceramics.

field-independent. It has been previously interpreted as the result of a decreasing mean-field critical temperature associated with an increasing broadening due to fluctuations [ 19,24 ].

Another interesting feature is the similarity of the deformation of the specific heat peak in both sets of measurements for Hllab and H[Ic. The same round- ing effect is observed in both directions taking a scale factor for the fields that reflects the anisotropy ratio [25 ]. It suggests that the transition in a magnetic field depends mainly on the parameter H/Hc2(T= O, 0), where 0 is the angle between the field and the c axis, and not on the topology of the vortices. It follows that the anisotropy ratio of the critical field slope (OHc2,ab/0 T[ to) / (OHc2,c/0 T J re) can be estimated directly by superposing the curves in both directions. The curve for B[Ic= 2 T coincides with the curve for Bllab= 14 T, and the curve for Bllc= 1 T inserts itself between the curves for Bllab = 6 T and 10 T. It is con- cluded that the anisotropy ratio of the present YBa2Cu306.92 single crystal is e = 7 _+ l, in good agreement with other determinations in Y-123 with the optimum doping [4,26]. It falls between the values

272 E. Janod et al. / Physica C234 (1994) 269-279

0.108

0.107

0.106

0.105 e~0

.~ 0.104

[- 0.103

0.102

0.101

0.1

0.108

0.107

0.106

-~ 0.105

~ 0.104

E- r,.) 0.103

0.102

0.101

0.1

/ ' ' ' I . . . . I . . . . I . . . . I . . . .

75 80 85 90 95 100

H//ab : : (b)

J

75 80 85 90 95 100

Temperature (K)

Fig. 4. (a) C/T as a function of the temperature for magnetic fields applied along the c-axis,/~t/= 0 (filled squares), 1 T (open squares), 3 T (filled diamonds), 6 T (open diamonds), 10 T ( filled triangles ) and 14 T ( open triangles ). ( b ) C/T as a function of the temperature for various magnetic fields applied along the ab planes. Symbols as in Fig. 4(a).

obtained for a fully oxidised sample (07, T¢= 89.9 K, ~ 5 . 5 + 0 . 5 ) [19,26] and for a reduced sample (O~.TS, T¢=68 K, E~ 12-13) [26]. This comparison confirms that the anisotropy is directly correlated with the oxygen content, rather than with the critical tem-

perature. The same observation is true for the amplitude o f the specific heat jump.

The plot of the difference A C / T = [ C( T, O) - C( T, H ) ] / T (Fig. 5) clearly shows on the low-temperature side the increasing mixed-state term 7m (H) . The present adiabatic method gives accurate absolute values o f the specific heat, and this variation is therefore significant. At 14 T and 50 K, i.e. in a region where fluctuations may be neglected, 7~(14 T) reaches 0.51 + 0.10 m J / K z g at. After a correction taking into account the field dependence o f the Schottky term of the Y-211 impurity ( 29+0 .6 p J / K z ga t ) we obtain 7m(14 T ) = 0 . 4 8 + 0 . 1 0 m J / K z gat. This value is o f the same order o f magnitude as the Sommerfeld constant obtained by Loram et al. [27] , i.e. 1.4 m J / K z g at. Note that entropy considerations further suggest that 7m(H) determined at 50 K is somewhat overes- t imated with respect to its value at T = 0.

Fig. 6 shows the electronic specific heat vs. T for various fields in the configuration Hllc. This contribution was isolated by subtracting the field-independent lattice part as given in Section 3.3. The anomaly is dearly distinguishable even for /~ / - /= 14 T. In order to map the Hez (T) line, we adopted three practi- cal definitions o f T¢. The first one is given by the maximum of the anomaly, which certainly sets a lower limit for the H~2 (T) line. The second definition uses the inflexion point o f the electronic specific heat on

0.003

"~ ~" 1T 0.002 . 3 T

6 T I ~

0.001 • 10 T gt~..~ "

,-. ~ 14T

- 0 . 0 0 1 . . . . ', . . . . ~ . . . . ', . . . . : . . . .

50 60 70 80 90 100

Temperature (K)

Fig. 5. Difference between the specific heat C/Tin zero field and the specific heat in magnetic fields from 1 to 14 T applied along the c-axis.

E. Janod et al. / Physica C234 (1994) 269-279 273

0.006

0.005

~ .004

~ 0.003

0.002

0.001

m

50 60 70 80 90 100

Temperature (K)

Fig. 6. Sl~eific heat C/T (Hllc), after having subtracted the lattice background, as a function of temperature and magnetic field. Symbols as in Fig. 4(a).

the right side of the anomaly. The third one is de- freed as the maximum of 0 (C /T) ~OH=f(T) , e.g. the temperature at which an infinitesimal change of the field induces the maximum specific heat variation. The results, including three transitions taken in a hy- brid magnet at higher fields [28], are presented in Fig. 7. In any case, a description of the H¢2 (T) data by a straight line does not seem to be appropriate close to Tc where a positive curvature is observed. Let us mention that this curvature, which will be discussed in Section 3.3.2, does not depend upon the choice of the baseline given by the phonon spectrum. Using the second definition (inflexion point ), an average slope ~OH¢2/OTI rc gives - 3 . 2 T / K taking the zero and high-field points [ 28 ] and - 4.1 T / K taking only the high-field points. Both are given by dashed lines in Fig. 7. A slope of - 2 . 5 T / K is found taking the high- field points issued from the third definition. These values are somewhat larger than those found in the literature: - 1.9 T / K [30-32] and - 2 . 3 T / K [33] by linear extrapolation of the magnetisation data M ( H , T) , or - 1 . 9 T / K [13] and - 1 . 8 T / K [19] by high-field scaling. In contrast, they are in better ac- cordance with recent measurements of the reversible magnetisation below T¢ [34,35] in high-quality ceramics. Using the WHH temperature dependence of the upper critical field, we extrapolate

° ", ",~ I 25 , ~,

\ \~ • . :I \ \, ~ mflexlon of(C-Cph) \ \ x

\ ~ ,~ ~ max. of d(C/T)/dHI

~ 105 ' ~

0 75 80 85 90 95

Temperature (K)

Fig. 7. Critical field curves Voile2 (T). The lower set of data (open squares) is deduced from the maximum of the electronic contribution shown in Fig. 6. The upper set is deduced from the upper inflexion of the electronic contribution shown in Fig. 6. The intermediate set (filled diamonds) issues from the maximum of O(C/T) ~OH=f(T). The latter was determined with the finite difference [ C / T ( H2) - C/T(H~ ) ] / (1t2- Ill ). The solid lines are the prediction oct 4Is for the critical regime [29]. The dashed straight lines define average slopes in the Gaussian regime.

/toH¢2.c(0)=165, 212 and 270 T at T = 0 for ltoOHe2.c/0TI r o = - 2.5, - 3.2 and - 4.1 T /K, respectively. The corresponding coherence lengths are ~.~(0) = [#o/2n/toH¢2,c(0) ]1/2= 14.1, 12.4 and 11 A, respectively.

3.3. Thermal fluctuations in zero-field

Owing to the small coherence length and the low dimensionality of the high-To cuprates, thermal fluctuations are distinctly observed in experiments such as resistivity, magnetisation, specific heat, etc. As long as their relative contribution remains small, fluctuations can be in principle described in the Gaussian approximation. This approach has been used in recent Y-123 specific heat experiments [2-6] . We will see in Section 3.3.2 that the Gaussian approximation fails near To, and, as pointed out by several groups [ 4,6,7-10 ], critical fluctuations are in a better agreement with experimental data.

274 E. Janod et al. /Physica C 234 (1994) 269-279

3.3.1. Method of analysis Omitting a small temperature window around T~o,

the zero-field specific heat was fitted using a phon- onic term C ph and an electronic term C e~. The latter consists of a mean-field step C mr and a contribution from superconducting fluctuations C a . The total specific heat reads

C = C ph -~- C el = C ph -~ c m f - ] - C fl . ( 1 )

For the lattice part (C ph), we have used a phonon spectrum formed by three Einstein modes with adjustable weights and frequencies [ 19 ]:

cPh 3NAke =DoE( Oo/ T) + D~E( O~/ T)

+D2E(O2/T) , (2)

with

x2e x E(x ) - (eX_ 1 )-----------~ " (3)

ke is the Boltzmann constant and NA is the Avogadro number. A realistic phonon spectrum should have a positive density of states (D~> 0), and the total number of modes per gram-atom (g at) should be equal to 3N, i.e. Do+D~ +D2= 1. For the electronic mean field term C mr, two models are tested.

(i) The two-fluid model:

cmf/T=3y(T/T~o) 2, T<T~o, (4a)

cmf/T=?, T> Tco , (4b)

where y is the Sommeffeld constant and T~o the mean- field critical temperature in zero field. In this model, the specific heat jump at T~o is AC/Tc=2y.

(ii) The BCS specific heat. The latter is interpo- lated in the numerical tabulation of Muhlschlegel [36].

Two models are tested for the fluctuation term C n. (i) Critical type behaviour. Sufficiently close to the

transition, critical behaviour should be observed [ 37 ]. In the 3D-XYuniversality class, the divergence is predicted to be nearly logarithmic [29,38 ]:

1 C~ta°s=-Colnt , i . e . oc~ , witho~-+0, (5)

where t= I 1 - T / T d is the reduced temperature and

Co is an amplitude that depends on the coherence length.

(ii) Gaussian fluctuations. Alternatively, the Gaussian approximation is valid farther from To, more precisely when Cn << AC. We used a model including a smooth dimensionality crossover from 2D far from T¢ to 3D near Tc (2D-3D) [38,39],

kB cn~auss = 8~V~th~/{ [d¢/2( V~)l /3] t} 2+ ( t/a) '

(6)

where d is the interlayer distance (11.6 A [40] ), ~=He2,ab/H¢2,c=2c/2ab is the anisotropy ratio,

Vco h --.~ a=either I for T>T~ or 2 for T<T~, CL (~a~bL)Z ~L is the Ginzburg-Landau coherence volume, and ~L and ~L are the coherence lengths.

The free parameters of the fit are Do, D~, D2, O0, 01 and 02 for CPh; y and Too for cmf; and Co or V~ L for C n. The specific heat data C(T) are weighted by I / T.

3.3.2. Analysis of specific heat fluctuations We first comment on the different fluctuation re-

gimes, i.e. 2D Gaussian far from Too, 3D Gaussian in the intermediate range, and finally critical very close to the superconducting transition. In order to locate approximately the crossover, we have excluded an adjustable temperature window AT* around T~o (i.e., T~o + AT*/2) and fitted the remaining experimental specific heat data, using either one of the following four models:

(a) C ph + C tw°'fluid -{- C Gauss ;

( b ) C ph -{- C tw°-fluid "~- C ~ rit'l°g ;

( c ) cPh"~ - c B C S ' ] - C G . . . . ;

(d) C ph -~ C BcS "~- C ~ rit'lOg .

The crossover can also be estimated using a model where fluctuations go from Gaussian to critical upon approaching Tc [8]. Although this model is strictly 3D, the extension of the critical domain deduced with this method is comparable with our determination (see below). The standard deviation is given in Fig. 8 as a function of the width of the excluded window AT*. Clearly, the quality of the fit obtained with the models including 2D-3D Gaussian fluctuations de- grades rapidly when the excluded region is smaller


0.13

~ 0.1

0.07

0.04 0.1

' , ~ - - 2 ft.+ logarithmic

2 ft.+ gauss. 2D-3D

• BCS+Iog

~ - - BCS+gauss .2D-3D

1 10

AT*=excluded window around Tco (K)

Fi& 8. Standard deviations of various fits (models a, b, c and d, see text) as a function of the width of the excluded temperature window around To. Note that the fitted phonon spectra remain realistic in every case (i.e., positive with a total weight Do+Dr +De close to one).

than 4 to 7 K, suggesting that the Gaussian approximation is no longer valid inside this range. Irrespec- tive of the model used for fluctuations, the use of the two-fluid model for the mean-field part leads to somewhat better fits. This is taken as an indication in favour of a strong coupling. Eventually, model (b) agrees best with all data, even very close to T~o. The standard deviation is found to be independent of the width of the excluded window down to AT*< 0.1 K; admittedly, inhomogeneities start to play a substan- tial role in this limit. Fig. 9 and Table 2 compare the results of two fits, the first one including critical fluctuations and the second one Gaussian fluctuations, using the same excluded window AT*=0.73 K. The amplitude of the critical fluctuations Co (45 mJ/g at K=0 .87 mJ/g K) is of the same order of magnitude as the values found in the literature: 1.01 mJ/g K [4], 0.82 m J / g K [7], 1.17 mJ/g K [10], 0.77 mJ/g K [41 ]. The discrepancy may reflect the dependence of critical amplitudes on oxygen content [ 10 ]. The parameters resulting from a Gaussian fit obtained in the temperature range where it is expected to hold, i.e. A T * = 9 K, are additionally given in Table 2.

We conclude that the Gaussian approximation should not be used closer than about 2-4 K around T~o (i.e. A T * = 4 - 8 K). In this region, the critical model is better suited. The same study performed on the data after subtraction of the green phase contribution leads to similar results.

If we admit the above conclusion, then the positive curvature of the upper critical field curve finds a con-

0.2

o

.~ -0.2 ~ g

-0.4 L)

-0.6 40

0.10% • ~ :' . ~'~e~____

-0.10% ,"

a

80 120 160 200

0.108

0.107

0.106

0.105

O.lO4 L)

0.103

0.102

0.101

0.108

0.107

0.106

~ 0.105

~.. 0.104

0.103

0.102

0.101

// b

88 91 94 97

J

J I I

88 91 94 97

Temperature (K)

Fig. 9. (a) Residuals of the Gaussian fit (filled triangles) and the logarithmic fit (open diamonds) for a fixed excluded window AT*=0.73 K in both cases. The full lines included for comparison show +0.1% of the total C/T. (b) C/Tin zero field (open diamonds), and fit including a logarithmic divergence for critical fluctuations (model b, see text). (c) C/Tin zero field (filled triangles), and fit including 2D/3D Gaussian fluctuations (model a, see text).


Table 2 Specific heat of YBa2Cu306.92 in zero field: selected characteristic parameters issued from the fits

Fitted AT*= 0.73 K parameters

Critical fluctuations Gaussian fluctuations

AT*=9 K

Gaussian fluctuations

Do 0.2827 0.2817 0.2788 00 (K) 152.26 151.31 150.94 Do +Dr 0.7023 0.7319 0.7096 0t (K) 355.91 359.05 352.63 Do+Dr +D2 1.0184 1.036 1.024 02 (K) 685.3 740.9 703.2 7 (mJ/K2 g at) 1.199 1.137 1.083 T¢ (K) 92.58 92.56 92.60 Co (J/Kgat) 0.04536 - -

GL Vcoh (A 3) - 349.3 183.7

AT* is the excluded temperature window around To. Gaussian fluctuations refer to the model (a): C= CPh+ C~'a~d + C~'~; critical fluctuations refer to the model ( b ): C= O 'h + ~ a ~ i d + Cgit.loL

sistent explanation. In the critical region, one expects [5,29,38]

H¢2( T) oct 4/3 . (7)

This temperature dependence agrees well with the determination of H¢2(T) given in Fig. 7. It remains true that the upper critical field should vary linearly with T i n the 3D Gaussian regime [29]. This is in- deed also observed. There is therefore a crossover between both regimes, which occurs at 88 + 1 K. This is an independent estimation of the width of the critical region. The result, AT*= 9_+ 2 K, is consistent with the former determination based on least-squares fits. This analysis supports the remark of Fisher et al. [ 37 ] who noted that the standard Ginzburg criterion underestimates the width of the critical temperature region.

The Gaussian fluctuation model used sufficiently far away from T¢ (AT*>8 K, see Table 2), and therefore in a region where it is expected to be valid, yields a Ginzburg-Landau coherence volume

GL /~k 3. GL Vcoh = 184 One can alternatively obtain Vooh from the formula [ ~ L ( 0 ) ] 3 / e where ca%L(0) is determined by the linear extrapolation to T = 0 of the upper critical field slope - 3 . 2 T / K

GL ~a~L ( ~ H c 2 x (0) = 296 T, ( 0 ) = 10.5 A, Section 3.2). The Ginzburg-Landau coherence volume determined by this method, 167-195 A 3 considering e = 6- 7, is in good agreement with the previous result. The

0Hc2.c/0TI rc values usually found in the literature,

i.e. from - 1.8 to - 2.3 T / K [ 13,18,27-30 ] (see Sec- tion 3.2 ), lead to somewhat larger V~ohOL'. 280--580 A 3 using an anisotropy ratio of 5-7. The lower values of IzoOHc2,c/OTI rc in the literature ( =larger OL Vcoh ) may be a consequence of the use of smaller fields. Com- parable He2 (T) slopes would be obtained in this work if fields below 7 T only are considered.

4. Scaling

4. I. Reversibility

Before discussing specific heat scaling properties, which are an expression of thermodynamic laws, it should be verified that the measurements do not depend upon the system history, or in other words, that they are performed in the reversible regime. This regime, which is bordered by the irreversibility line in the (H, T) plane, is generally found to be narrow in Y-123 crystals. Furthermore Y-211 inclusions can act as pinning centers. We have therefore investigated, using magnetic measurements, the reversible temperature domain as a function of the external field [ 18 ]. In particular, Maxwell's relation

OS r~ O(MV) 0-~ = ~ o ~ ,,,p, (8)

where S and M are the entropy and the magnetisation, respectively, may be taken as a test of equilib-

E. Janod et al. I Physica C 234 (1994) 269-279 277

rium conditions. Fig. 10 shows that Eq. (8) is satis- fied by specific heat and magnetisation [18] measurements in the region of interest, i.e. near the H¢2(T) line.

4. 2. Models

4.2.1. High-field scaring The high-field model is based on a fluctuation be-

haviour of the critical type. Its validity is restricted to a situation where the carders are confined to their lowest Landau level (LLL). Scaling laws for thermodynamic functions are predicted [ I 1,12 ], with the particular result for the specific heat:

c (A r - ro(n) T =g' (T--~ ] ' (9)

with n= 1/2 or 2/3 if the system is two- or three-di- mensional, respectively. The scaling function g~ (x) is generally not known. The predicted field-depen- dent crossover between Gaussian and critical fluctuations is

( kaltoH ,~2/3 t,~ = kOo-~YT V] ' ( l O )

0.045

0.035 - ' ' " ' " " ' . • . : : . , - _ ~ " Bo dM/dT (IT)

0.025 " = : - - - ~ = ° ' - " L : : ~ - - - - - ~ = ~ a ' . ~ °" / = ilo d M / d T (2T)

:.i." • ~,L i ° la°dM/dT(4T)

" "'"':".°. ~'.° 1 - dS/dH (0.5T)

o.o15 • ", ""';"':~ -~"'i_ I .... dS/dH (2T) ~= "' '~. %'~ . / ....... dS/dH(4.5T)

" - . .? . ~."

0.005

-0,005 , t , i , i ' i ' I

87 89 91 93 95 97

Temperature (K)

Fig. 10. Verification of Maxwell relation: OSlOHand I~M/OTas a function of the temperature. The specific entropy S(H, T) is obtained by integrating the speOfic heat C(H, T) /T from lO0 K to T. The derivative OS/OHis determined by the finite difference (AS/AH)= [S(T, H2)-S(T, HI)I/(H2--HI)], assuming that the value of Sat 100 K is field independent.

where tn= I1 - [T/Tc(H)]I. With a coherence length ~= 1-3 A and a jump AC/V=3.7X 10 4 J / K m 3 [6], we obtain for the width of the critical region about 4 to 8 K if/to//= 14 T. Figs. 1 la and b show the scaling plots expected in the critical region for both cases (2D and 3D). Above To(H) and for fields higher than 1 T, scaling is confirmed in both cases, a result that does not depend critically on the value of the unique free parameter T¢(H). In other words, satisfactory scaling is found for go(OH¢2x/OT¢)lrc taking any value between - 2 and - 4 T/K. Scaling fails to occur below T¢(H). This is to be related with the presence and the variation with field of the specific heat jump, which is not considered in the derivation of Eq. (9).

4

---~ 1

o -0.2

~ ~4 L ! L

ooOo~ ~ ~"ffi~% u (dH,,/dT)T.=-4 T/K 1 . o _~

°~o ~ 2

" D 0 ~

0 -0,2 0.1 0 0.1 0.2 0.3 0,4 0.5 ~ ! o I T - T , ( H ) ) / (TH) v" (K'2"Vl/:)

P o ( d H c z / d T ) T c = ~ I I I I i I

-0.1 0 0.1 0.2 0.3 0.4 0.5 ( T - T (H)) / (TH)I/2 (Kl/2 T-l/2)

i i , , ,

4 b ,°°.° 4T t

" ~ o o ~ I ©°°*o©°°~°°°~

° ° L) 2 o .

o o O.05 0 0.05 0.I0 0.15 - ~ I ~ i ~ o o o ( T - T . ( H ) ) / (TH)"/ '{ (KI ' ~ T =' ~ )

0 0 0 8 0 [ 3

go(dHcJdT)r =-2 T/K 0 i I I

-0.05 0 0.05 0.10 0.15 (T-To(H)) / (TH) 2/3 (K v3 T-2/3 )

Fig. 11. (a) Sealing plot of the specific heat data according to the 2D high-field model for poH~2---2 T/K. Inset: same for goH~2 = - 4 T/K. Symbols as in Fig. 4(a). (b) Scaling plot of the specific heat according to the 3D high-field model for #oH~a ffi - 2 T/K. Inset: same for/toll '2-- - 4 T/If. Symbols as in Fig. 4(a).


4.2.2. Weak-field scaling In this approach, the superconducting phase tran-

sition is supposed to belong to the universality class 3D-XY [42]. In the region of the (H, T) plane close to (0, To), the free energy density associated with fluctuations can be written as [ 37 ]

A = T~-°g2(n~2/¢o), ( 11 )

where g2 is a generally unknown scaling functional and ¢o is the flux quantum. The specific heat follows from Eq. ( 11 ) [42,43]

[C(t, O) - C ( t , H) ] ×H~/2~=g'2(t/H~/2~) , (12)

where t= T/Tco-1, and ot and v are the critical exponents. The same limitation as for high-field scaling applies here, i.e. only the region T> T¢o can be investigated to avoid the contribution of the specific heat jump. This is equivalent to t /H °'747 > 0, using appropriate critical exponents. Fig. 12 shows a scaling plot of the data using the values a = - 0.007 and v = 0.669 given by the 3D-XYcritical model. The only free pa- rarneter, T~o, is found to be 92.5 +0.2 K. Note that this scaling law was also successfully tested by Sala- mone t al. [ 9 ].

It is rather unexpected that all scaling plots are equally satisfactory. In this situation, no strong conclusion can be drawn about the effective dimensionality of the electronic system, or on the most rel- evant theory for fluctuations. Clearly the presence of

e

0 .35

0.3

0.25

0.2

~" 0.15 O =L

* 0.1 ,-r" ~. 0.05

o

-0 .05 '

-0 .005

, ~ ~ + 1 T ',/I , ~ ~ - - 3 T

~ : - - 6T

: ~ ~ lOT

~. ~ 14T

I , , , , I

0 0.005

t / H 0 747 (T-0.747)

0.01

Fig. 12. Sca l ing p lo t o f the specif ic hea t a c c o r d i n g to the 3D-XY low-f ie ld m o d e l [ 38 ].

the specific heat jump and the uncertainty about its variation limits the information given by the fluctuation specific heat. The presence of a large jump at Tc is a distinctive characteristic of the Y-123 compound among high-temperature superconductors. Similar investigations of scaling in highly anisotropic cuprates based on Bi, T1 or Hg seem to be more prom- ising from this point of view.

5. Conclusion

We have performed new high accuracy adiabatic specific heat measurements in high magnetic fields, using a large single crystal of Y-123 with a sharp superconducting transition at its optimum doping. A mixed state specific heat term Ym(H) Tthat increases with the applied field is observed. The anisotropy of the upper critical field, about 7 for the present sample, is found to increase with decreasing hole doping. The anisotropy is not a function of the critical temperature, since the latter passes by a maximum in the same conditions.

The analysis of the zero-field specific heat deafly requires the inclusion of a fluctuation term. We find that in a region I T - Tel ~ 3 K wide, fluctuations are in the 3D-XY critical regime. Consistently with this finding, a non-linear temperature dependence of the upper critical field is observed at the same temperatures. Outside this region, the Gaussian model de- scribes both the fluctuation specific heat and the more linear critical field curve.

Acknowledgements

Helpful discussions with M. Droz, C. Marcenat and J. Muller are gratefully acknowledged. The authors thank J.C. Vallier/SNCI Grenoble for high-field fa- cilities. One of us (E J) would like to thank all mem- bers of the group of J. Muller from the DPMC/Univ- ersit~ de Genrve who welcomed him during sixteen months. This work was supported by the Fonds Na- tional Suisse de la Recherche Scientifique.


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specific heat up to 14 tesla of a yba2cu3o6.92 single crystal

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