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PHYSICAL REVIEW B, VOLUME 65, 132508

Evidence for vortex pinning induced by fluctuations in the transition temperatureof MgB2 superconductors

M. J. Qin, X. L. Wang, H. K. Liu, and S. X. DouInstitute for Superconducting and Electronic Materials, University of Wollongong, Wollongong, New South Wales 2522, Austra

~Received 5 October 2001; published 19 March 2002!

The field-dependent critical current densityj c(B) of a MgB2 bulk sample has been obtained using magneticmeasurements. Thej c(B) curves at different temperatures demonstrate a crossover from single vortex pinningto small-bundle vortex pinning, when the field is larger than the crossover fieldBsb. The temperature depen-dence ofBsb(T) is in agreement with a model of randomly distributed weak pinning centers via the spatialfluctuations of the transition temperature (dTc pinning!, while pinning due to the mean-free-path fluctuations(d l pinning! is not observed.

DOI: 10.1103/PhysRevB.65.132508 PACS number~s!: 74.70.Ad, 74.25.Dw, 74.25.Ha, 74.60.2w

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The recent discovery of superconductivity in MgB2 ~Ref.1! has led to intensive experimental and theoretiactivities,2–15 with the purpose of understanding the bamechanism of superconductivity and the vortex-pinnmechanism governing the critical current densityj c in thisnew superconductor. Althoughj c has been improved greatlsince its discovery, the underlying pinning mechanism is sunder investigation.

In type-II superconductors, the most important elementinteractions between vortices and pinning centers aremagnetic interaction and the core interaction.16–22The mag-netic interaction arises from the interaction of surfacestween superconducting and nonsuperconducting matparallel to the applied field, which is usually very smalltype-II superconductors with a high Ginzburg-Landau~GL!parameterk. The core interaction arises from the couplingthe locally distorted superconducting properties with theriodic variation of the superconducting order paramewhich is usually more effective in technical type-II supercoductors due to the highk value. Two mechanisms of corpinning are predominant in type-II superconductors, i.e.,dTcandd l pinning. WhereasdTc pinning is caused by the spatiavariation of the GL coefficienta associated with disorder inthe transition temperatureTc , variations in the charge-carriemean free pathl near lattice defects are the main cause ofd lpinning.

It has been reported by Griessenet al.16 that thed l pin-ning is dominant in both YBa2Cu3O7 and YBa2Cu4O8 thinfilms. A high k value of 26 has been reported10 for MgB2,indicating that the magnetic interaction is negligible, andcore interaction is more important. However, it has not beexperimentally determined whether thed l pinning or thedTcpinning is the dominant mechanism in MgB2. The purpose ofthis paper is to report measurements ofj c of this new mate-rial to achieve an understanding of the vortex-pinnimechanism and to demonstrate that in MgB2 governed bybulk pinning,dTc is the only important pinning mechanism

All measurements have been performed using quandesign PPMS on a MgB2 bulk sample withTc538.6 K anddimensions of 2.1832.7631.88 mm3, which was preparedby conventional solid-state reaction.23,24

Figure 1 shows the hysteresis loops of the MgB2 sample

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every 2 K in the 14–36 Krange. The results at lower temperatures, which have large flux jumping,23 are not shownhere. The symmetric hysteresis loops with respect tomagnetic field indicate the dominance of bulk pinning uptemperatures nearTc , rather than surface barrier. As a comparison, we show in the inset of Fig. 1 the hysteresis loopa pressed MgB2 sample at 5 K. This sample is fabricated bpressing the MgB2 powder into a pellet without sinteringThe loop is highly asymmetric, showing a large reversibmagnetization resulting from the surface current. The surfbarrier has also been observed by Takanoet al.13 in theirpowder sample and bulk sample sintered at low temperat

From these M (H) loops, we calculatej c5DM /a(12a/3b), with a, b the width and length of the sample pependicular to the applied field, respectively. The resultj c(B) curves at various temperatures are shown in a doulogarithmic plot in Fig. 2 as different symbols. As can bseen from the plateau at low field,j c initially has a weakdependence on the field. When the field is increased beya crossover field, it begins to decrease quickly. The crossofield decreases with increasing temperature. Further incring the field results in a faster drop inj c near the irrevers-

FIG. 1. Hysteresis loops of the MgB2 sample taken every 2 K inthe 14–36 K range. Results at lower temperatures are not shbecause of large flux jumping. Inset shows the hysteresis looppressed MgB2 sample at 5 K, showing large reversible magnetiztion from the surface current.

©2002 The American Physical Society08-1

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BRIEF REPORTS PHYSICAL REVIEW B 65 132508

ibility line, which is obtained by using a criterion ofj c5100 A/cm2 and shown as open circles in Fig. 3. The bfitting of the data yields,

Birr~ t !5Birr~0!~12t2!3/2, ~1!

shown as solid line in Fig. 3, witht5T/Tc . For high-temperature superconductors, a (12t)3/2 behavior is usuallyobserved25 and has been explained by means of giant flcreep.26,27 The (12t)3/2 law is actually an approximation othe (12t2)3/2 law as t→1. Also plotted in the figure is theBc2(T) data taken from Takano’s work, showing thatBirr iswell below Bc2. Therefore, giant flux creep also plays aimportant role in the new superconductor MgB2.

j c(B) characteristics very similar to those shown in Fighave been observed by other groups.3,14,15Based on differentphysical assumptions on summation of the elementaryning force f p to obtain the macroscopic pinning forceFp ,different pinning models yield differentj c(B) characteristics.The simplest model is the direct summation off p to have

FIG. 2. j c of the MgB2 sample at various temperatures indicatby different symbols. Solid lines are fitting curves using Eq.~4!.

FIG. 3. B-T phase diagram of the MgB2 sample. Open circlesrepresent the irreversibility line obtained from Fig. 2, and the soline is a fit toBirr55.2(12t2)3/2. Solid circles represent theBc2(T)line from Takano’s data using resistive measurements. The daline is just a guide to the eye.

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Fp5 j cB5npf p , wherenp is the density of pinning centersWhen the influence of the flux-line lattice is taken into acount, one haveFp5 j cB5npf p

2(u0 / f p)d/a02 , where u0 is

the maximum distortion of the flux-line lattice, caused bypoint finning forcef p , d is the range of the pinning forcetypically of the order ofj, anda0 the flux-line lattice con-stant. These two strong pinning models yieldj c}B21 andj c}B20.5, respectively. Due to the large densities of the pnp and smallf p , these two models are not representativemost real pinning systems.28 For randomly distributed weakpinning centers, theFp can be estimated using the basic cocept of collective pinning,29 which has been proved to bvery successful in most real pinning systems,

Fp5 j cB5AW

Vc5A W

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~2!

with the correlation volumeVc5Rc2Lc , the correlation

lengthsRc perpendicular to the field direction andLc alongthe vortex line, and the pinning parameterW5np^ f p

2&. Rc

and Lc depend on the applied field, the dimension of tflux-line lattice@three dimensional~3D! or 2D#, and the elas-ticity or plasticity of the flux-line lattice. For the 3D elastiflux-line lattice, it has been derived by Blatteret al.17 that j cis field independent when the applied field is lower thancrossover fieldBsb ~single vortex pinning!

Bsb5bsb

j sv

j 0Bc2 ~3!

wherebsb'5 is a constant,j 054Bc/3A6m0l the depairingcurrent,Bc5F0/2A2plj the thermodynamic critical fieldBc25m0F0/2pj2 the upper critical field, andj sv the criticalcurrent density in the single vortex-pinning regime. WhB.Bsb ~small-bundle pinning!, j c(B) follows an exponentiallaw,

j c~B!' j c~0!expF2S B

B0D 3/2G . ~4!

WhenB.Blb5b lbBc2( j sv/ j 0)@ ln(k2jsv/ j 0)#2/3 ~whereb lb isa constant'2), this large bundle-pinning regime is governed by a power lawj c(B)}B23. The 2D elastic flux-linelattice shows single pancake pinning at low magnetic fiewith field independentj c , then a 2D collective-pinning region at higher fields withj c}1/B. For fields higher than acrossover fieldBb

3D , three-dimensional pinning is predictedFrom thej c(B) curves shown in Fig. 2, it is expected th

the 3D elastic pinning model may explain the dominant pning mechanism in MgB2. We, therefore, use Eq.~4! to fitthe j c(B) curves, with fitting parametersj c(0) andB0. Thefitting results are shown as solid lines in Fig. 2. At intermdiate fields, Eq.~4! fits the experimental data very welwhile deviations from the fitting curves can be observedboth low and high fields. A clearer plot is shown in Fig.where the j c(B) curve at 24 K is shown in a doublelogarithmic plot of 2 log@ j / j (B50)# vs the applied field,which clearly shows a straight line at intermediate fields. Tdeviation at low fields is denoted asBsb, indicating the

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BRIEF REPORTS PHYSICAL REVIEW B 65 132508

crossover from the single vortex-pinning regime to tsmall-bundle-pinning regime. The point of deviation at hifields was first considered as the crossover field from smbundle pinning to large-bundle pinning. However, whenfit the j c(B) data at high fields to the power lawj c(B)}B2n, n is found to be as large as 20 rather than the thretically predicted value of 3, making it unlikely that thsystem changes to the large-bundle-pinning regime. Ashigh-field deviation is very close to the irreversibility linewhich results from giant flux creep, it is likely that the dviation at high field may result from large thermal fluctutions, which lead to the rapid decrease inj c , and, therefore,is denoted asBth . The inset of Fig. 4 shows thej c(B) of thepressed sample at 5 K in a double-logarithmic plot, whichindicates that when the surface pinning is important, theponential drop inj c(B) @see Eq.~4!# no longer applies, but apower law j c(B)}B21.2 is obvious.

The crossover fieldBsb as a function of temperature ishown in Fig. 5 as open circles. We now compare the expmental data with theoretical predictions to get some insiinto the pinning mechanism in MgB2. Usingl}(12t4)21/2

andj}@(11t2)/(12t2)#1/2, Griessenet al. have found thatfor d l pinning the critical current density in the singvortex-pinning regimej sv}(12t2)5/2(11t2)21/2, while fordTc pinning j sv}(12t2)7/6(11t2)5/6. Inserting all these ex-pressions into Eq.~3!, we have

Bsb5Bsb~0!S 12t2

11t2D 2/3

~5!

for dTc pinning, and

Bsb5Bsb~0!S 12t2

11t2D 2

~6!

FIG. 4. j c at 24 K in a double-logarithmic plot of2 ln@ j/j(B50)# vs the applied field. The solid line is a fit using Eq.~4!. Bsb

indicates the crossover field from single vortex pinning to smbundle pinning, whileBth is the crossover field to the thermal fluctuations dominated regime. Inset shows thej c(B) curve of thepressed MgB2 sample with a large contribution from the surfacurrent, which shows aB21.2 behavior.

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for d l pinning. The lines corresponding to Eqs.~5! and ~6!are indicated asdTc pinning andd l pinning, respectively, inFig. 5. The central result of this paper is the remarkably goagreement found betweenBsb and the correspondingdTc-pinning line in the figure. In sharp contrast, thd l -pinning line shown in Fig. 5 is in total disagreement withe experimental data.

Having derived the crossover fieldsBsb and Bth , we re-construct theB-T phase diagram shown in Fig. 3. The finB-T phase diagram is shown in Fig. 6. The vortex soregion is divided into three smaller regions. Single vortpinning governs the region belowBsb, betweenBsb andBth ,small-bundle pinning becomes dominant, while betweenBthand Birr , thermal fluctuations are more important. Largflux-bundle pinning is not observed in MgB2, but may beconcealed by the thermal fluctuation effects.

-

FIG. 5. Temperature dependence of the crossover fieldBsb. ThedTc-pinning line corresponds to Eq.~5!, which is in agreement withthe experimental data, while Thed l -pinning line corresponds to Eq~6!, which is not in agreement with the experimental data.

FIG. 6. B-T phase diagram of MgB2 . Bsb(T) is the fitting curveof Eq. ~3! to the experimental data~see Fig. 5!. Birr(T) is the fittingcurve of Eq.~1! to the experimental data~see Fig. 3!. Bth is thecrossover field to thermal dominant region~see Fig. 5!. Again theBc2(T) line is taken from Takano’s data using resistive measuments.

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BRIEF REPORTS PHYSICAL REVIEW B 65 132508

In summary, we have found strong evidence fordTc

pinning, i.e., pinning via the spatial fluctuations in thtransition temperature, in MgB2, while d l pinning, i.e.,pinning via the spatial fluctuations of the charge-carrmean free path, is not observed. TheB-T phase diagramof the MgB2 sample has been derived, showing that

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low fields below Bsb the system is dominated by singvortex pinning and changes to smaller bundle pinning whB.Bsb. WhenB.Bth , this region in the vortex solid area idominated by thermal fluctuations.

The authors would like to thank the Australian ReseaCouncil for financial support.

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