Specific heat anomalies in high temperature superconductors

Physica C 174 (1991) 208-214 North-Holland

Specific heat anomalies in high temperature superconductors Critical behaviour

or gaussian

fluctuations?

A.I. S o k o l o v Department of Physical Electronics, Leningrad Electrical Engineering Institute, Leningrad, 197022, USSR Received 19 November 1990

The width of critical region and 2D-~ 3D crossover temperature for high-T¢superconductors of yttrium, bismuth and thallium families are estimated. For Bi- and Tl-based compounds the Ginzburg number Gi turns out to be 0.02-0.05 while for Y-123 Gi ~ 0.003. In Bi-2212 and TI-2212 phases superconducting fluctuations appear to be two-dimensional up to I T - Tc[ - 0.3-0. l K. The results of recent high resolution measurements of the specific heat of the above superconductors [ 1-3 ] are discussed in detail. The experimentally accessible range is shown to cover the region of gaussian fluctuations and, for Bi and Tl compounds, the crossover region. It is argued that the logarithmic temperature dependence of fluctuation specific heat obtained would not be considered as an evidence of true 3D critical behaviour since it is found outside 3D critical region and the critical amplitude ratios measured [2,3 ] turn out to be essentially non-universal.

1. Introduction Recently specific heat a n o m a l i e s in high-T¢ superconductors ( H T S C ) o f yttrium [1 ], bismuth [2] a n d thallium [ 3 ] families were thoroughly investigated, and fluctuations o f the superconducting o r d e r par a m e t e r p r o v e d to give rise to these anomalies. D. Wohlleben a n d co-workers were able to extract the fluctuation c o n t r i b u t i o n to the specific heat from their high precision d a t a which was f o u n d to be appreciable at I z l -~ 0.1, r = ( T - T¢) / T¢, a n d large enough in close vicinity o f the phase transition p o i n t To. A t t e m p t s to fit e x p e r i m e n t a l d a t a o b t a i n e d with p r o p e r theoretical curves led, however, to unexpected results. It t u r n e d out that the t e m p e r a t u r e dependence o f fluctuation specific heat in YBa2Cu307_ m a y be equally well described both with logarithmic a n d potential functions within the same t e m p e r a t u r e range 0.01 < Izl <0.1; inverse square root fit holds good actually up to I vl - 0 . 3 [ 1 ]. Similar contradictory situation takes place in Bi2Sr2Ca2Cu30,o+x [ 2 ] where the fluctuation specific heat manifests logarithmic t e m p e r a t u r e dependence, i.e. d e m o n s t r a t e s 3D critical b e h a v i o u r for Ivl 4 0 . 1 while the resistivity for z>~0.03 obeys 2D A s l a m a z o v - L a r k i n the-

ory which is known to be valid only within the region o f weak (gaussian) fluctuations. The logarithmic character o f specific heat a n o m a l y s p a n n e d over surprisingly wide temperature intervals was revealed also in thallium c o m p o u n d s [ 3 ]. In this paper, I shall e m p l o y available experimental d a t a on coherence lengths, specific heat j u m p and effective thickness o f superconducting layers to est i m a t e the width o f critical region and the temperature o f 2 D ~ 3 D crossover for H T S C s o f Y, Bi and T1 families. W i t h these estimates in h a n d I shall discuss results o f experiments [ 1-3 ] to clear up whether true critical behaviour has been observed in the above c o m p o u n d s or only gaussian fluctuations have been really seen. It is well know that the specific heat o f systems belonging to the universality class o f the 3D X Y m o d e l is governed, within the critical region, by an extremely small critical exponent a - ~ - 0 . 0 1 , a n d this t e m p e r a t u r e d e p e n d e n c e is practically undistinguishable from the logarithmic one. The observation o f such a t e m p e r a t u r e d e p e n d e n c e itself, however, can be considered as an evidence o f true critical beh a v i o u r only if one is sure that i) the o r d e r p a r a m eter has two real components, ii) the superconductor

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209

A.L Sokolov / Specific heat anomalies in HTSCs

is effectively three-dimensional within the region searched, and iii) the Ginzburg criterion ITI << Gi is fulfilled. Up to now, there is not an unambiguous proof of exotic pairing in HTSCs. Therefore, these substances seem to meet the first condition. To the contrary, the second condition will be shown to be satisfied only in the Y-123 compound while the third one appears to be broken, as will be argued, for all the superconductors studied in [ 1-3 ]. 2. YSa2Cn307_ a We begin from the yttrium compound. In order to estimate the Ginzburg number Gi and the temperature of dimensional crossover "['2D/3D one has to know the Ginzburg-Landau coherence lengths ~bL (0), ~ L (0), specific heat jump ACGL and effective thickness of superconducting layers d, provided the material possesses the layered structure. During the last two years a large amount of high precision data on fluctuation conductivity and diamagnetism, magnetoresistance, derivatives

(H~5)'--

dH~l = d r It=to 2e¢8~(0)¢~L(0)'

(H~2)' ~

dH~2

CGL(C0)=1.5--2 [71

¢~bL(0)=11.5+0.5,

~GL( c

G L I, ! ~ 13.6_+0.8, Cabt0x

c 0 )=1.23_+0.19 [9] ~OL(

¢~bL(O) = 13,

¢~L(O) =2 [101

¢~bL(O) = 12,

¢~L(O) =2.4 [ 11 ]

¢~bL(O) = 15-+ 2, [12].

¢~L(O) =2 ~,

(3)

for further estimates. The specific heat jump has also been measured by many groups (see, for example, [ 1,13,14]) whose results converge to ACGL--~4 × 104 J / (m 3 K). The effective layer thickness d is a less definite quantity. Obviously, it lies between 3.8 A, the thickness of the Cu-O sheet, and 11.7 A, the lattice constant c of YBa3Cu307_6. Since we are interested in the lowest possible value of Z2D/3D,the largest value of d should be put into the formula [15] ZED/3D= ( 2 ~ L ( 0 ) / d ) 2 •

(4)

It gives Z2D/3D> 0.12,

(5)

which is consistent with other estimates [ 7,16,17 ]. So, the main body of experimental data [ 1 ] may be thought to lie within the region of 3D superconducting fluctuations. The Ginzburg number is given, in 3D case, by the well-known expression [ 15 ]:

ka

~2

Gi3D= 3--~5~2\¢~L(O)¢~L(O)¢~L(O)ACoL]

,

(6)

( 1)

and Little-Parks oscillations of Tc has been obtained for high quality Y-123 crystals and thin films [4-12]. Derivatives ( 1 ) measured in refs. [ 4,5 ] were found to be two or three times larger than those reported previously by many authors. Nevertheless, they lead to the values ~bL(0)= 13.5-14 A, ~ L ( 0 ) = 2 . 5 A, which are strongly supported by results of experimental study of transport and quantum interference phenomena (values in/k): ¢~bL(0)=13--16,

¢~b(o) = 13 ~,,

1 (

h Ir=rc -- 2e [ ¢~bL(0) 12

These numbers are expected to be close to the true ones. Thus, we accept

0)=1.5_+0.5 [81

Using eq. (3) we obtain Gi=0.0033.

(7)

The accuracy of this value is not very high but it undoubtedly can play a role of reliable order-of-magnitude estimate for the Y-123 Ginzburg number [18,191. Since the specific heat in ref. [ 1 ] was measured for [z[ >i 0.01, it is clear that all of the experimental points lie outside the scaling region; they cover the region of gaussian fluctuations and, at most, the edge of the crossover region. That is why the inverse square root fit ACff = C -+Izl-i/2

(8)

turned out to be so good within the whole temperature interval studied. This conclusion may be confirmed on the base of (2)

21 o

A.L Sokolov / Specific heat anomalies in HTSCs

experimental data [ 1 ] themselves. Indeed, since G i = 2 ( C+ / ACGL) 2 ,

ACGL=1.43YBcsTc ,

and [ 1 ] C + = 289 m J / ( m o l K) , YBCS=33mJ/(molK 2),

Tc=91.9K,

(10)

the following value of G i can be readily obtained: Gi=0.0089 .

apparent, i.e. resulting from the unusual temperature dependence of the background (mean field) component of the electronic specific heat. If it were true it would be thought as an evidence of non-BCSlike superconductivity in this compound.

( 11 )

It means that true scaling behaviour should be observed only for [zl <<0.01. Note, that above estimate is in reasonable agreement with eq. (7). Moreover, the averaged coherence length SX;L(0)= 5.8 jl, found in ref. [ 1 ] is close to that resulting from eq. (3): ~'GL ( 0 ) = 7 •, which in turn coincides with the independent estimate of the Illinois group [ 13 ]. Although the scaling region for YBaECU307_ahas not been reached, the fluctuation anomaly measured in ref. [1] was found to be large enough near To, mCfl ~ ACGL, SO the beginning of the crossover regime has been really seen. Perhaps, further improvement of the samples quality will make it possible to observe well developed crossover or even true critical phenomena in superconductors of the Y-123 family in the near future. It is very interesting that within the gaussian fit of high resolution data the ratio C ÷ / C - = 2.02 was obtained [ 1 ] which is three times larger than the theoretically predicted number 1/x/~ but rather close to the ratio values 2.5 and 2.1 found earlier by two American groups [ 13,14 ]. Just after the first observation of this anomaly [ 13 ] it was attributed to the influence of twin boundaries [20,21 ]. This conjecture was shown to result in a good description of the overall temperature dependence of the specific heat near Tc [ 21 ] as well as the explanation of the anomalous C ÷ / C - value [22] provided twin model parameters are chosen in a proper way. However, to account for the almost universal value of C + / C obtained in several different samples by three independent groups one should assume that all the samples studied, crystalline and ceramic, possess i d e n t i c a l d e f e c t s t r u c t u r e s what is hardly believed in. It seems more natural to consider the anomalous ratio C ÷ / C - as an intrinsic property of the Y-123 superconductor. This anomaly may also happen to be

3. Bismuth and thallium systems Up to now, we have more poor and less reliable experimental information on ~ b ( 0 ) , ~ L ( 0 ) and ACGLin Bi and TI compounds than that for Y-123. The evaluation of the Ginzburg number and the 2D--, 3D crossover temperature, therefore, seems to be a somewhat complicated problem in this case. We shall not try to solve such a problem completely, i.e. to find precise values of G i and Z2D/3D for Bi and T1 systems, since it is impossible today. Instead, for each superconductor we will estimate the largest value of Gi and upper and lower bounds for Z2D/3D, compatible With proper experimental data, and evaluate their values Gi* and r2D/3O* which currently may be thought as most likely ones. These quantities can be obtained with reasonable accuracy from available experimental data and, on the other hand, they will be shown to be sufficient to enable us to answer the question put in the title of the paper. We start with Bi2Sr2CaCu2Oa÷x (Bi-2212). This HTSC has been studied by many experimentalists [23-28 ] and numerous data on the temperature dependence of the upper critical field and its anisotropy have been obtained. We estimate first the inplane coherence length ~bL (0) since it is directly related to (H~2)'. Experimental values of this derivative are not very sensitive to relative orientation of magnetic field and crystallographic axes, and it is measured therefore much more precisely than ( H ~ ) ' in highly anisotropic superconductors. Experiments give (H~2)' =0.75 T / K [23] (H~2)'=0.8

T/K[24]

(H~2)'=0.7

T/K[25]

(12)

( H ~ 2 ) ' = 1.2 T / K [26] . Resistive methods used to obtain these numbers are known to underestimate (H~2)' and ( H ~ ) ' . Sample

A.L Sokolov / Specific heat anomalies in HTSCs imperfections also favour such an underestimation as one can easily see, comparing data on YBa2Cu3OT_6 presented in earlier reviews [ 15,29,30] with those summarized in a recent one [ 18 ]. Hence, we have to accept the largest value from ref. [ 12 ]. This choice is motivated as well by our intention to get the upper bound for Gi. Putting To= 85 K we obtain ~ b ( 0 ) = 18 A. Another coherence length may be found from ~bL (0) ~L(O)-

(H~)' (H~2)'-Yu.

(13)

For the ratio YHexperiments give results close enough to each other yu=60

[231

y . > 2 0 (50) [24]

(14)

yu~50

[27]

yu=55

[28].

while directly measured values of ( H ~ ) ' are strongly scattered [23,26]. So, the proper estimate for ~ L ( 0 ) appears to be 0.3 A. To evaluate the upper and the lower bounds for r2D/3D we assume d lies between 3.8 A and 15.4/k, since the whole lattice constant c = 30.8 A is hardly believed, because of the crystal structure, to play a role of thickness of superconducting layer. Then eq. (4) gives 0.025>Z2D/3D>0.0015. The analysis of experimental data on fluctuation conductivity available for various HTSCs shows that, as a rule, d turns out to be closer to the thickness of superconducting stack than to that of the single C u - O layer [ 17,3134 ]. Hence, we conjecture that for Bi-2212 "~D/3D is about 0.002-0.003. To estimate properly the Ginzburg number we should realize that, because of the smallness of Z2D/3D, the 2D expression for Gi [ 15 ] may be needed

kB Gi2D = 4 n ~ L (0) ~ L (0) dACoL"

( 15 )

For ACoL we take the lowest reasonable value resuiting from fig. 4 of ref. [ 3 ]: ACGL= 6700 J / ( m 3 K). It is about two times smaller than that found in ref. [35]. Then, for d = 3 . 8 - 1 5 . 4 A eq. (15) gives Gi2D = 0.13-0.03 ,

(16)

211

Gi* is anticipated to be equal to 0.04-0.05. Since Gi is an order of magnitude larger than r2D/3D the crossover from classical to critical behaviour in Bi-2212 should occur, indeed, when superconducting fluctuations are still two-dimensional. In a similar way the parameters of interest are estimated for Bi-2223, T1-2212 and TI-2223 phases. The results obtained together with initial experimental data are listed in table 1. Before discussion, some comments on these numbers should be made. Derivatives (H~2)' and ( H ~ ) ' for Bi-2223 in ref. [ 36] are very likely underestimated: when used for determination of coherence lengths [ 34] they lead to the value of ~-~L(0) which is considerably larger than that found in ref. [2 ]. Therefore, to evaluate ~(0) and ~ L ( 0 ) in this case we assumed ~ L (0) = 7--8 A [ 2 ] and accepted the parameter of anisotropy resulting from ref. [36]. The estimates of d for Bi phases and T1-2212 originate from data on fluctuation conductivity [2,31-34]. Specific heat jumps were extracted from fig. 4 of ref. [3]. They are considerably smaller than those obtained by other groups [ 35,41-43 ] but seem to be more reliable because the presence of superconducting fluctuations has been taken into account when evaluating these numbers. The anisotropy of TI-2223 (and probably of TI-2212) phase is apparently too strong to be measured properly, at least today, so the numbers i n the right column(s) of table 1 are actually rather crude estimates. Let us now discuss the contents of table 1. It provides an evidence that well-developed 3D critical behaviour, i.e. the logarithmic temperature dependence of ACn can be observed in substances involved only for 131 <0.01. In Bi-2212 and T1-2212 the 3D critical region should be still more narrow: ITI < 0.001, while strong 2D superconducting fluctuations are to be seen within the interval 0.01 > I~1 > 0.001. In Bi-2223 and, maybe, in T1-2223 the classical-to-critical and 2D--.3D crossovers should occur almost simultaneously, at about 2-3 K away from To. Hence, only 2D gaussian fluctuations can be observed in all superconductors discussed. This is in a good agreement with results of recent measurements of fluctuation conductivity in Bi-2223 samples [2,32] and TI-2212 thin films [31,44] and with independent estimates for Bi-2212 compound [45 ]. Since Aar~ (T) measured turned out to become

212

A.L Sokolov / Specific heat anomalies in HTSCs

Table 1 Parameters of Bi- and Tl-based compounds. The initial values are taken from the sources quoted in the references.

(H~2)(T/K) ~u ACoL(J/m3K) c/2 (,~) d(.~) ~OL(O) (A) c 0 ) (A) ~GL( Z2D/3O r~O/3 D

Gi3D Gi2D Gi*

Bi-2212

Bi-2223

T1-2212

TI-2223

1.2 [261 55 [28] 6700 [3] 15.4 15 [33] 18 0.3 0.025--0.0015 0.003-0.002 1.4 0.13-0.03 0.05-0.04

(0.36) [36] 20 [36] 8600 [3] 18.5 10-15 [2,32,34] 20 [2,34] 1 [2,34] 0.3--0.01 0.04-0.02 0.05 0.08-0.017 0.03-0.02

1.0 [37] 70-94 [37,38] 9100 [3] 14.7 13 [31] 18 0.3-0.2 0.03-0.0007 0.002-0.001 0.8-0.6 0.10-0.015 0.03-0.02

0.6 [39] 20-200 [39,40] 11300 [3] 18.1 20 1-0.1 0.3-0.0001 0.02-0.0002 0.06-0.0001

o f the order oftro at T - Tc = 1-3 K experiments mentioned support also our estimates o f Gi. Thus, experimental data [2,3 ] cover actually the region o f 2D gaussian fluctuation and, very likely, the region o f crossover to critical behaviour. The fluctuation specific heat, therefore, should behave within the experimentally accessible range as I z I [ 15 ] rather than as ln lzl, at least for I rl > 0.05-0.03. It would be interesting to try to fit high precision data for ACfl [2,3] with the curve

sidered as an evidence of true 3D critical behaviour. In fact, in extremely anisotropic superconductors Bi2212 and TI-2212 fluctuations should be essentially two-dimensional throughout the whole accessible range while in Bi-2223 and T1-2223 c o m p o u n d s a complicated crossover regime near I z l = 0.01-0.03 would occur.

ACft/ACGL = g +-/ I Z l , g + ~ g - ~ Gi2D

The width o f the critical region and the 2D--.3D crossover temperature for HTSCs o f Y, Bi and T1 families are estimated on the base o f numerous experimental data available. It is found that Gt* ~, 0.003, Z2D/3D> 0.1 for YBa2Cu307_6 and Gi* ~ 0.02-0.05 for Bi and TI compounds, apart may be for TI-2223. In Bi-2223 and T1-2223 phases T~D/3D seems to be of the other of Gt* while in Bi-2212 and TI-2212 ones it is at least the order o f magnitude smaller. The results o f high resolution specific heat measurements [ 1-3] are discussed in detail and it is concluded that the experimentally accessible temperature range spans over the region ofgaussian fluctuations and, in the case o f Bi and T1 compounds, the crossover region. Inverse square root fit is shown to be quite natural for data [ 1 ] on Y-123 leading to the estimate o f G i which is consistent with the most o f recent experimental results. At the same time, the logarithmic temperature dependence o f the fluctuation specific heat obtained for 0.01 < Izl <0.1 in Bi and TI phases [2,3] would not be considered as an

( 17 )

prescribed by the theory. Gaussian contributions would not sink at all in the experimental noise since its magnitude is much smaller than ACGL, particularly in Tl c o m p o u n d s [ 3 ]. Recent measurements o f Bi(Pb)-2223 specific heat [46] which appear to reveal 2D gaussian fluctuations for I~1 > 0.03 seem to support this conclusion. On the other hand, if strong critical fluctuations really existed in Bi and Tl HTSCs within the region 0.01 < l z l <0.1 the specific heat critical amplitude ratio measured would be equal to a certain universal value. For 3D XY-model-like systems it is known to be very close to unity [ 47 ]. Critical amplitude ratios for different phases resulting from fig. 4 o f ref. [ 3 ], however, are considerably larger than l and seem far from being equal to each other. Such a ratio for Bi2223 given as 2.3 by high resolution measurements [2] also appears not to conform to the universal value. Hence, the logarithmic fit o f the data [2,3] within the range 0.01 < Izl <0.1 would not be con-

4. Conclusions

A.L Sokolov / Specific heat anomalies in HTSCs

e v i d e n c e o f t r u e 3 D critical b e h a v i o u r since it is f o u n d o u t s i d e the 3 D critical r e g i o n a n d t h e critical a m p l i t u d e ratios m e a s u r e d t u r n o u t to be essentially n o n - u n i v e r s a l d i f f e r i n g m a r k e d l y f r o m the k n o w n t h e o r e t i c a l value.

Acknowledgements T h e a u t h o r is v e r y grateful to Prof. D. W o h l l e b e n a n d Prof. J.L. T h o l e n c e for s e n d i n g p r e p r i n t s a n d reprints o f t h e i r papers. H e also t h a n k s Dr. B.N. Shal a y e v for d i s c u s s i o n s a n d Prof. M.V. S a d o v s k i i for s o m e initial i n f o r m a t i o n . T h i s w o r k is s u p p o r t e d by the Scientific C o u n c i l o n the P r o b l e m o f H i g h - T c Sup e r c o n d u c t i v i t y a n d is p e r f o r m e d in the f r a m e o f the U S S R State P r o g r a m " H i g h T e m p e r a t u r e S u p e r c o n d u c t i v i t y " u n d e r P r o j e c t N o . 6.

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