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Experimental evidence for a dimensional crossoverof the vortex ensemble in BSCCO

(Bi1.8Pb0.4Sr1.8Ba0.2Ca2Cu3Ox) by multi-harmonicac susceptibility measurements

V. Mihalache a,*, S. Popa a, D. Di Gioacchino b, P. Tripodi b,c, J.D. Vinko c

a National Institute of Materials Physics, Atomistilor 105 bis, Judet Ilfov, P.O. Box MG 7, 077125 Magurele, Romaniab INFN-LNF, Via E. Fermi 40, 00044 Frascati, Italy

c H.E.R.A., Corso della Repubblica 448, 00049 Velletri, Italy

Received 11 October 2005; accepted 9 November 2005

Abstract

We have studied a portion of H – T phase diagram of the anisotropic Bi2223 bulk system using multi-harmonic acsusceptibility measurements (vn). Based on the behavior of the third harmonic modulus, jv3j, the irreversibility line(IL) was obtained from the onset of jv3j and an anomalous peak effect (PE) was observed. It has been shown thatthe anomalous peak is due to the crossover between a three-dimensional (3D) and a quasi-two dimensional (quasi-2D) peculiarity of vortex dynamics, the crossover magnetic field being H cr % 0.1 T. The obtained portion of B irr(T )is well described by the melting line of the flux lattice. The 3D flux fluctuation part (low field and high temperature)is described by the nearly parabolic temperature dependence B = B 0(T c/T À 1)n with n = 1.58 (sample 1) andn = 1.48 (sample 2). The other region (high field and low temperature) is well described by the temperature dependenceof the quasi-2D flux fluctuations, T mð H Þ ¼ T

2Dm ½1 þ b=ðln B= BcrÞ1=m corresponding to the weak interaction between pan-

cake vortices from adjacent planes. The 2D limit temperature was determined as T 2Dm ¼ 36:7 K (sample 1) and

T 2Dm ¼ 27:3 K (sample 2).

Ó 2005 Elsevier B.V. All rights reserved.

PACS: 74.25.Dw; 74.25.Ha; 74.25.Nf; 74.72.Hs

Keywords: Bi-based cuprates; Magnetic properties; Superconductivity phase diagrams

0921-4534/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physc.2005.11.005

* Corresponding author. Tel.: +40 21 4930047x193/+40 0745460105; fax: +40 21 4930267.E-mail address: [email protected] (V. Mihalache).

Physica C 433 (2006) 225–233

www.elsevier.com/locate/physc

mailto:[email protected]

mailto:[email protected]



1. Introduction

The magnetic phase diagram of high-tem-

perature superconductors (HTSC) exhibits anextraordinarily rich variety of phases [1]. Stillcontroversial is the presence of a boundary, calledthe irreversibility line (IL), which separates a mag-netically irreversible state, with a non-linear dissi-pative behavior, from a reversible region with alinear dissipative property. Some of the suggestionsare based on a vortex glass formation [2] oronafluxlattice melting [3]. More recent studies investigatingthe irreversibility line have confirmed the presenceof a predicted crossover characteristic value H cr[4,5] in the dimensionality of the vortex fluctuations[6,7]. The dimensional crossover 3D/2D has beenobserved experimentally in various HTSC systemsincluding Tl2Ba2Cu3O10 (Tl2223) films [8], YBa2-Cu3O7+d thin films [9], Bi2Sr2CaCu2O8 (Bi-2212)single crystals and Bi2Sr2CaCu2O8+d thin films [6].In layered superconductors [4,6] the increase of the magnetic field causes a decoupling between thepancake-like vortices in adjacent CuO2 planes (3Dto quasi-2D vortex lattice transition). Recently,numerical analysis confirmed a transition fromordered straight vortices to disordered decoupled

vortices in layered superconducting material [10].This transition has also been used to explain thepresence of the second peak in the magnetizationcycles [10–12]. In fact the increase of the magneticfield was accompanied by a sharp increase of thecritical current density J c (or a sharp increase of the pinning force) for a little span of the magneticfield. The increase of J c is due to the decoupling of the 2D pancake vortices that can easily adjust theirpositions on the defects, such as the dislocation net-works [12], in the CuO2 planes. Hence there is a

maximum in pinning [10], with respect to the previ-ous weakly pinned 3D vortex line. In other words,the 3D–2D transition lowers the linear dissipationlosses and increases the non-linear (irreversible)dynamic processes. In highly anisotropic supercon-ductors, such as BSCCO single crystals, a peakeffect in which J c shows a sharp increase is fre-quently observed as a function of increasingmagnetic field [11–13]. The experimental resultson the crossover transition were reported fortextured c-axis oriented (Bi, Pb)2223-Ag tapes [7]

and for Bi1.7Pb0.3Sr2Ca2Cu3O [14], by means of magnetic measurements in the vicinity of the irre-versibility line and dumping of mechanical oscilla-

tions, respectively.In this work, we have investigated the irrevers-ibility line, the crossover transition and the peakeffect in bulk Bi2223, using an ac multi-harmonicsusceptibility apparatus [15]. This investigationwas performed measuring the temperature depen-dence of the third harmonic susceptibility for dif-ferent DC magnetic fields. The measurement wasperformed on bulk samples and evidenced differentnon-linear flux dynamics as well as a markedanomalous peak effect.

In fact, a superconductor is characterized by anon-linear I – V , so that its magnetization loop isstrongly deformed (non-harmonic), comparativelyto the elliptical magnetization loop shown by nor-mal conductors with linear I – V characteristics.The non-harmonic signal can be analyzed bymeans of Fourier series containing high-harmoniccomponents. The measurement of higher harmon-ics, in particular the third harmonics, is an efficientmethod to depict the deformation of the magneti-zation loop and, as a consequence, a good toolfor the analysis of the phenomena connected to

the pinning processes, including the peak effect.In the case of PE, the magnetic loop has a steepdeformation in a narrow range of dc magnetic field,and is well characterized by the variation of the v3.

The random orientation of the grains in bulkBi2223 samples represents an apparent disadvantage.However, as will be shown in Section 2 paragraph,the susceptibility signal in our conditions of measure-ments mainly arises from the platelets oriented per-pendicularly to the applied field. Moreover, anadvantage of our bulk samples is the clear separation

between intra- and inter-granular vortex dynamics.Therefore, the study of the transition crossoverdynamics was performed on a system consisting of decoupled Bi2223 plate-like crystals in bulk sam-ples at relatively low fields.

2. Experimental

Two samples with the same chemical composi-tion Bi1.8Pb0.4Sr1.8Ba0.2Ca0.2Cu3Ox were preparedby using the standard solid-state reaction method.

226 V. Mihalache et al. / Physica C 433 (2006) 225–233



For sample 1, the Bi2O3, PbO, BaCO3, CaCO3

and CuO powders were mixed in the ratioBi:Pb:Sr:Ba:Ca:Cu = 1.8:0.4:1.8:0.2:2:3.

For sample 2 the following procedure of pow-der preparation for calcinations was used. High-purity SrCO3, BaCO3, CaCO3, and CuO powdersmixed in the ratio Sr:Ba:Ca:Cu = 1.8:0.2:2:3 werepressed in 9 · 9 · 3 mm3 pellets at 0.75 GPa whichwere treated at 947 °C, this procedure beingrepeated and followed by a thermal treatment at950 °C. Bi2O3 and PbO powders were added tothe obtained precursor, thus yielding the followingratio between the components Bi:Pb:Sr:Ba:Ca:Cu = 1.8:0.4:1.8:0.2:2:3.

Samples 1 and 2 were calcinated in an aluminacrucible at 815 °C for 20 h in air, reground andpressed in 3 · 3 · 10 mm3 pellets at 0.75 GPa.The sintering thermal treatment of the pelletswas performed in air at 848 °C for 330 h and320 h for samples 1 and 2, respectively. Every110 h, sample 1 was quenched in air down to roomtemperature and then submitted again to the ther-mal treatment (848 °C).

According to XRD measurements sample 1contains approximately 97% Bi-2223 phase, whilesample 2 is single Bi-2223 phase without any trace

of secondary or non-superconducting phases. Thedimension of the samples was 5 · 3 · 2 mm3, themean grain size being 5–8 lm for sample 1 and10–15 lm for sample 2.

The ac susceptibilities including the higher har-monics were measured with a susceptometer basedon double pick-up coils surrounded by a drivencoil [15]. The sample was placed on a sapphireholder inserted in the pick-up coils. The tempera-ture was measured with a platinum thermometer(PT100) in a good thermal contact with the sam-

ple. The whole assembly was cooled in zero mag-netic field (ZFC), in a thermally controlled Hegas flow cryostat provided with an 8 T super-conducting magnet. The temperature rate was0.3 K/min, up to a temperature greater than theT c value at zero field, i.e., between 10 K and120 K. The ac driving magnetic field had an ampli-tude of 6 Oe and a frequency of f = 1070 Hz. Thedc magnetic field was swept from 0 to 1.5 T. Theinduced signal has been measured with a multi-harmonic EG&G lock-in amplifier. Both ac and

dc fields were applied parallel to the longest sideof the sample.

We avoided to grind the samples in order to

study the isolated grains, because the grinding pro-cess may introduce defects into the grains (fissures,etc.) which can modify the pinning, and in this waycan modify or even remove the anomaly (it isknown that PE is a characteristic of the order-dis-order transition in the vortex matter). Moreoverthe grinding produces non-homogeneity of thegrain size.

To prove that the signal in our measurementscomes mainly from the platelets oriented perpen-dicular to the applied field we measured a sampleprepared in the following way. An as-grown sam-ple, similar to sample 2 and prepared in the sameconditions, was ground and the obtained powderwas pressed at 3 GPa to diminish as much as pos-sible the grain misalignment. The square shapedplatelets were cut from the pressed material andsuperposed to make a cubic shaped sample. InFig. 1, the v00(T ) dependence is shown for H kab

and H ?ab measured in H dc = 0.0128 T (H ac =1 Oe). It is evident that the response in H ?ab ismore significant than in H kab. Therefore, for theconditions of the measurements used in this work

on the bulk samples, our approximation in consid-ering that the signal origin from the platelets

0

100

200

300

80 90 100 110 120

H⊥ab

H//ab

Hdc

= 0.0128 T

Hac

= 1 Oe

f = 5500 Hz

T (K)

χ 1 "

( a . u . )

Fig. 1. v00(T ) dependence for H kab and H ?ab measured inH dc = 0.0128 T (H ac = 1 Oe). It is evident that the response inH ?ab is more significant than in H kab.

V. Mihalache et al. / Physica C 433 (2006) 225–233 227



oriented perpendicular to the applied field shouldnot affect our results.

3. Results and discussion

In Figs. 2A and 2B, the temperature depen-dence of the imaginary part of the first harmonicsðv00

1Þ measured in the static magnetic field in therange 0 T > H dc > 1.5 T is shown for samples 1and 2, respectively. The clear separation of theinter- and intra-grain dissipation peaks due tothe low values of the inter-grain critical currentdensity (weak coupling) allowed us to scan also aportion of the irreversibility line for the grains.

The v001 peak amplitude in respect to the applied

dc field has a non-monotonic variation. A rapiddrop around 0.07 T for sample 1 and around0.1 T for sample 2 is observed. The decrease of the hysteretic cycle area for these field values canbe due to a contribution of both non-linear effects(pinning) and linear losses. To distinguish thesetwo processes, an investigation of the higher har-monics is compulsory.

In Figs. 3A and 3B, the temperature depen-dence of the third harmonics module (jv3j) fortwo samples is shown. Increasing the magnetic

field, for sample 1, the amplitude of the third har-monics module has an unusual rise (jv3jmax) atH dc = 0.1 T (Fig. 2A) while, for sample 2, aroundthe same H dc value, a significant plateau is presentbefore an abrupt decrease (Fig. 3B). (For higherfields jv3j is much smaller and, as a rule, is embed-ded in the inter-grain signal.)

From Figs. 2 and 3 it is clear that some changein the vortex matter occurs, which could be due toa transition from an ordered to a disordered state[10] (like a 3D/2D transition [16]).

To sustain the anomalous effects mentionedabove, the peak amplitude of jv3jmax versus tem-perature has been plotted as a function of H dcfor both samples (Fig. 4). One can remark a peakfor sample 1, and a plateau for sample 2, in thejv3jmax (H dc) dependence. The peak as well as theplateau takes place in a very narrow H dc range,indicating that around B dc % 0.1 T a 3D/2D phasetransition occurs which can be associated with themaximum of J c (peak effect). At low fields the 3D

-1.0×10-5

0.0×100

1.0×10-5

2.0×10-5

3.0×10-5

4.0×10-5

5.0×10-5

6.0×10-5

50 60 70 80 90 100 110

χ " 1 ( a r b . u n i t )

Temperature (K)

0.5T0.3T

0.1T

0.07T

0.03T

0.002T

Sample 1H

ac= 6E-4T

f = 1070 Hz

Fig. 2A. v001 versus T for different dc fields (sample 1). Around H dc = 0.1 T a rapid variation of the signal occurs.




0.0×10 0

5.0×10 -5

1.0×10 -4

1.5×10 -4

2.0×10 -4

50 60 70 80 90 100 110

χ " 1 ( a r b . u n i t )

Temperature (K)

0T0.03T0.07T0.1T0.15T

sample 2|Hac| = 6e-4T

f = 1070 Hz

Fig. 2B. v001 versus T for different dc fields (sample 2). Around H dc = 0.07 T a rapid variation of the signal occurs.

0.0×100

2.0×10-6

4.0×10-6

6.0×10-6

8.0×10-6

1.0×10-5

1.2×10-5

70 75 80 85 90 95 100 105 110

| X 3

| ( a r b . u n i t )

Temperature (K)

0.002T

0.03T0.07T0.1T0.15T0.3T0.5T

Sample 1H

ac= 6E-4T

f = 1070 Hz

Fig. 3A. jv3j versus T for different dc fields (sample 1). Around H dc = 0.1 T an increasing of the signal occur.




vortex-pin center interactions are dominant andthe vortices are poorly pinned [6]. The pinningstrength decreases with the incrementing of the

magnetic field due to the escalating of the in-planevortex lattice stiffness [17]. For H dc % 0.1 T there isa sharp decoupling transition and the irreversibil-

0.0×100

1.0×10-6

2.0×10-6

3.0×10-6

4.0×10-6

5.0×10-6

70 75 80 85 90 95 100 105 110

| χ3 |

( a r b . u n i t )

Temperature (K)

0.03T0.07T0.1T0.15T 0T

sample 2|Hac| = 6e-4T

f = 1070 Hz

Fig. 3B. jv3j versus T for different dc field (sample 2). Around H dc = 0.1 T a constant value of the signal (plateau) can realize.

1×10-6

1×10-5

5×10-6

1×10-5

2×10-5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

sample 1

sample 2

| χ3

| ( a r b . u n i t ) s a m p l e 1

| χ

3 | ( ar b . uni t ) s am pl e2

Hdc

(T)

II peak

5×10-6

Fig. 4. jv3j versus H dc for samples 1 and 2. For H dc = 0.1 T these curves show a peak for sample 1 and a plateau for sample 2.




ity response increases. At higher fields the effective-ness of the pinning decreases due to the increasingof the in-plane interactions.

The crossover field between 3D and quasi-2Ddimension regimes follows the expression [4,5]:

H cr % U0=c2d

2 ð1Þ

where U0 = 2.07 · 10À15 T mÀ2 is the elementaryquantum flux, d is the spacing between adjacentgroups of CuO2 layers and c is the anisotropyratio, defined as c = nab/nc. The nab and nc are the

superconducting ab-plane and c-axis coherencelengths, respectively. If one considered the inter-layer spacing d = 18 A [18], Eq. (1) yields ananisotropy factor c % 80 for H dc % 0.1 T.

To confirm the validity of 3D/2D transitionobtained from our analysis of the second peakeffect we investigated the irreversibility line (IL)extracted from the onset of the jv3j for thetwo samples and have examined its behaviornear H dc % 0.1 T. The platelets in our polycrys-talline samples are far from being ‘‘clean’’ so

we can assume that the irreversibility line isclose to the melting line in the high tempera-ture-low field region, in this case IL can replacethe ML and we shall also use B m and T m forthe description of our experimentally obtainedIL. In Figs. 5A and 5B, the plot of the log(B irr)versus T [6] is shown for samples 1 and 2,respectively. It is evident that the irreversibilityline follows two different laws with temperature,separated by a crossover field (B cr). For B irr <B cr, the region in which the nearly 3D-like vor-

tex fluctuations occur [3,6,19,20] we obtain aquite good fit of the melting line with para-bolic-like law:

BmðT Þ ¼ B0½ðT c=T Þ À 1Þn ð2Þ

where n = 1.58 for sample 1 and n = 1.48 for sam-ple 2.

The functional dependence of the IL is a repre-sentation of the flux lattice free energy [21]. Smallthermal vortex fluctuations render a value for n

smaller than 2 [6].

10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

30 40 50 60 70 80 90 100 110

B i r r

( T )

Temperature (K)

B irr (T) = Bcr

exp{b[Tm

2D /(T-T

m2D

)] ν}

Bcr = 0.1T

b = 2.1, ν=3.6

Tm

2D= 36.7K

Birr

(T) = B0[1-(T/Tc)] n

B0

= 1.8T

n = 1.58

Tc = 110K

sample 1

Fig. 5A. IL line obtained from the onset of jv3j (T ) (sample 1). The curve follows two different behaviors of the flux dynamic which areseparates by the characteristic 3D/2D crossover.




IL line for B irr > B cr, shows the same exponen-tial behavior found in 2D flux lattice melting of coherent regions with weak Josephson interactionbetween adjacent layers [22]. We fitted the lowtemperature data using this melting line approachdescribed by the inverse formula [6] of the asymp-totic deviation of T m(H ) from T

2Dm [23]:

T mð H Þ ¼ T 2Dm ½1 þ b=ðln B= BcrÞ1=m ð3Þ

where b, m are constants and their values togetherwith the formula for B irr(T ) are shown in Figs.

5A and 5B. We obtained the crossover field valueB dc % 0.1 T for both samples.It is remarkable to notice that the value of the

crossover field of our bulk Bi-2223 samples is closeto those obtained in the literature for other Bi-based HTS: 0.3 T [7] and 0.0175–0.1750 T [14],obtained on the textured Bi-2223, as well as0.1 T on Bi-2212 single crystals [6].

In the strong field region (B ) B cr) the behaviorof IL shows an asymptotic temperature [24] withvalue 36.7 K for sample 1 and 27.3 K for sample

2. These results fit well with literature data, whereT

2Dm $ 40 K was found using transport measure-

ments on Bi2223/Ag tapes [25] and T 2Dm $

28 – 30 K for Bi-2212 single crystal [26].

4. Conclusion

In conclusion we have studied the multi-harmonic ac susceptibility of the grains in Bi1.8-Pb0.4Sr1.8Ba0.2Ca0.2Cu3Ox polycrystalline samples.

Qualitative as well as quantitative behaviors of thethird harmonic modulus that probes only the non-linear losses (i.e., the pinning processes) have beenstudied. An unusual variation of jv3j (peak effect)at H dc % 0.1 T was observed and a flux dynamicscrossover on the IL at the same value has beenfound for both samples. The irreversibility linesfollow two different behaviors upon increasingfields due to the thermal fluctuations [1,28,29].We analyzed them according to the generalnotions [1,4,27] which predict a characteristic

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10

1

10 2

30 40 50 60 70 80 90 100

B i r r

( T )

Temperature (K)

Birr

(T) = Bcrexp{b[Tm2D

/(T-Tm2D

)] ν}

B cr = 0.1T

b = 6.2, ν=3

Tm2D

= 27.3K

B irr(T) = B0[1-(T/Tc)] n

B0

= 1.6T

n = 1.47Tc = 107.7K

sample 2

Fig. 5B. IL line obtained from the onset of jv3j (T ) (sample 2). The curve follows two different behaviors of the flux dynamic which areseparates by the characteristic 3D/2D crossover.




crossover that separates regions of 3D and 2Dmelting. Near T c the melting line has beenobtained by fitting the experimental points with

a quasi-parabolic temperature dependence (3Dregion) B irr = B 0(T c/T À 1)n. For low temperaturesT m has been obtained by fitting the data with aquasi-2D behavior, the asymptotic deviation of T m(B ) from T

2Dm has been used.

The behavior of the grain flux dynamics inBSCCO bulk has been measured using the accu-rate multi-harmonic susceptibility technique. Theexistence of the anomalous PE has been verifiedtrough the analysis of the third harmonic modulusamplitude. At the same time the IL has beenobtained using the onset of the same third har-monic modulus. Both these phenomena, PE ano-maly and the behavior of IL, are compatible withthe 3D/2D flux dynamic crossover.

Acknowledgements

This work has been supported by the EuropeanCommunity Transnational Program (TARI)under contract number HPRI-CT-1999-00088and by Romanian Ministry of Education and

Research under ‘‘Nucleu’’ Programme.

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