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Magnetic penetration depth and reversible magnetization in single crystals of Bi2Sr2CaCu2O8d grown by TSFZ method A.I. Rykov a,b, T. Tamegai a,b,* b

a Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan CREST, Japan Science and Technology Corporation (JST), Honcho 4-1-8, Kawaguchi, Saitama 320-0012, Japan

Received 16 October 2000; accepted 6 February 2001

Abstract We analyze the reversible magnetization in Bi2 Sr2 CaCu2 O8d using several theoretical concepts, starting from the London model. The ®eld dependence of the empirical ``super¯uid density'' dM=d ln B is ®tted with using the theory of circular cell. Additionally, we take into account the pinning mechanism associated with dislocation structures naturally existing in the crystals grown by travelling-solvent-¯oating-zone (TSFZ) technique. This allows us to explain: (i) the rapid variation of dM=d ln B at small ®elds; (ii) the reduction of thermal ¯uctuation term with respect to the theoretical prediction for decoupled pancakes. In the light of this reduction we derive the curves Hc1 T and k2 0=k2 T from the reversible magnetization similar to the ones obtained by direct measurements of Hc1 by micro-Hall AC technique, and with microwave surface impedance, respectively. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.25.Ha; 74.60.Ge Keywords: Magnetization; Penetration depth; Lower critical ®eld; Bi-2212

A highly regular dislocation network frequently forms in Bi2 Sr2 CaCu2 O8d , being especially clearly observed in the well oxygenated crystals [1,2]. The defect structures were reported to in¯uence the magnetic hysteresis, in which the peak eect was associated with matching between vortices and pinning centers [2]. The matching eect was also suggested to explain the irreversible magnetization vs ®eld curves in the ion-irradiated Bi2 Sr2 CaCu2 O8d crystals, in which the peak was found at BU =3 [3], where BU is the matching ®eld between

* Corresponding author. Tel.: +81-3-5841-6846; fax: +81-35841-8886. E-mail address: [email protected] (T. Tamegai).

vortices and ion tracks forming the columnar defects (CD). Reversible magnetization is known to be also strongly modi®ed in the presence of CD's [3±5]. On the other hand, possible eects of the dislocation structures on the reversible magnetization in pristine crystals of Bi2 Sr2 CaCu2 O8d have not been investigated. In this work, we have studied the reversible magnetization in a number of crystals of Bi2 Sr2 CaCu2 O8d , and found that the behavior of dM= d ln B changes systematically with doping. We show that the experimental data can be ®tted with the circular cell model and the quality of ®t is improved if the defect structures are additionally taken into account. In our model, the following two eects are taken into account, which are additional to the simple London model:

0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 2 5 3 - 2

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(i) a suppression of the order parameter in the vortex core by the other vortices; (ii) a magnetic interaction between vortices and defect structures. Separately, the eects (i) and (ii) have been already examined in the papers by Koshelev [6] and Hardy et al. [5], respectively. The circular cell model [6] takes into consideration the ®eldinduced suppression of the order parameter in the vortex cores, which was ignored in the London model. Therefore, while the London model predicts for the reversible magnetization the logarithmic mean-®eld part M0 (i.e. without entropy term M1 ), dM0 =d ln B e0 =2U0

1

the circular cell model takes into account the nextorder expansion term with respect to the parameter B=Hc2 and predicts a ®eld dependence slower than logarithmic for M0 : dM0 =d ln B e0 =2U0 1 2B=Hc2 d ln Hc2 =B

2

2

Here e0 U0 =4pk is the vortex line energy and U0 is the ¯ux quantum. Due to the factor 1 2B=Hc2 ln Hc2 =B the curve dM0 =d ln B vs ln B is predicted to show a strong negative curvature at small ®elds. In this work, we observed that the curvature of the experimental derivative of magnetization dM=d ln B vs ln B, if negative, is much smaller than that predicted in the circular cell model. Moreover, some overdoped samples show even positive curvature of dM=d ln B vs ln B. These variations cannot be explained by the entropy term related to the pancake alignment length La . Indeed, this term is also logarithmic in ®eld: dM1 =d ln B

kB T =sU0 s=La

3

We observe in Fig. 1 that the experimental data dM=d ln B vs ln B vary signi®cantly at small ®elds in contrast to the prediction of Eq. (2). This discrepancy at small ®elds could be successfully corrected if one takes into account an additional term, which corresponds to low-®eld change in magnetization due to pinning by the defect structures naturally existing in the unirradiated samples.

Fig. 1. Semilogarithmic plots of the logarithmic derivative of magnetization dM=d ln B in the as-grown optimally doped (a) and overdoped crystals (b). The crystals were grown in the same batch and had initially the value of Tc 90 K. The overdoped crystal (Tc 80 K) was obtained by annealing the as-grown crystal at 450°C under oxygen pressure of 6.6 MPa. The temperature step for all data presented is 5 K. Note the upward curvature of dM=d ln B vs ln B in overdoped sample and in the low-temperature regime of the optimally doped sample.

Therefore, we introduce an additional term, similar to the one introduced by Hardy et al. [5] to ®t the magnetization in irradiated samples containing the CD's. dM2 =d ln B

2

e0 =2U0 B/ =B exp aB/ =B pR2 b4 Hc2 =16U0 1 ln pR2 Hc2 =U0

2

4

where R is an eective radius of sample inhomogeneities, B/ is the matching ®eld, b 0:77, and a is a function of R and Hc2 only [4,5]. Contrarily to the case of CD's [4,5], the Eq. (4) ignores a change in superconductivity volume fractions induced by defects.

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In Fig. 2, we show a typical ®t of experimental data dM=d ln B with the theoretical expression given by the sum of Eqs. (2)±(4) dM0 =d ln B dM1 =d ln B dM2 =d ln B. The alignment length La cannot be established from such a ®t, because the ®eld-independent term dM1 =d ln B is fully correlated with k. We show also in Fig. 2 the decomposition of the ®tted function into two terms. The ®rst term is given by the circular cell model dM0 =d ln B dM1 =d ln B and the second one is given by Eq. (4). Thus, the additional pinning term allows to improve the quality of ®t signi®cantly even for experimental data dM=d ln B with downward curvature vs ln B. The values of Hc2 extracted from the circular cell model are independent of temperature between 0.3 Tc and 0.8 Tc [7], in accordance with the previous estimates [8,9]. Both Hc2 and k show only small change when adding the

Fig. 2. A typical ®t of the logarithmic derivative of magnetization in optimally doped sample at 42 K with using circular cell model (Eqs. (2) and (3)) and the circular cell model corrected for pinning (Eqs. (2)±(4)) (top). Two contribution of the theoretical function ®tted for the data shown at the bottom plot are the circular cell term and defect structure pinning term.

pinning term. The typical values of the parameters B/ and R extracted from the pinning term (Fig. 3) closely correspond to the parameters of mesoscopic models of dislocation structures and stacking faults found in microscopic observations [1,2]. Finally, we show in Fig. 3 that the reduction of the ¯uctuation term dM1 =d ln B by a factor La =s consistent with our pinning model allows to obtain the curves k2 0=k2 T and Hc1 T which are almost linear with (1 T =Tc ) in some rather broad temperature range near Tc (80 < T < 90 K). This is true for the circular cell model with or without pinning. Such linearity is consistent with recent measurements of Hc1 by Mrowka et al. [10] with using micro-Hall AC technique. It is also shown in Fig. 3 that the full ¯uctuation term leads to a large negative curvature of the k2 0=k2 T and Hcl T curves near Tc . For the London model such a result has been obtained previously [11] and here we obtained similar result for the circular cell model. We would note, however, that the microwave surface impedance technique [12] gives the curve of k2 0=k2 T vs T close to our curve for the reduced entropy term. Therefore, the interlayer correlations of pancakes, probably caused by the inter-

Fig. 3. Lower critical ®eld in the optimally doped sample calculated from the ®tted values of penetration depth using formula Hcl U0 =4pk2 ln j 0:5 and j 70. Two curves correspond to two series of calculations with dierent entropy term. In the ®rst series, the full entropy term was allowed with dM1 =d ln B kB T =sU0 . The logarithmic derivative dM1 = d ln B kB T =sU0 s=La with La 10s was taken for the reduced entropy term in the second series.

A.I. Rykov, T. Tamegai / Physica C 357±360 (2001) 284±287

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