magnetostriction of the model type-ii superconductor nbti

Physica B 319 (2002) 293–302

Magnetostriction of the model type-II superconductor NbTi

A. Gerbera, R. Schleserb,*, P.J.E.M. van der Lindenb, P. Wyderb

aRaymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv,

69978 Tel Aviv, IsraelbGrenoble High Magnetic Field Laboratory, MPI-FKF and CNRS, BP 166, F-38042, Grenoble Cedex 09, France

Received 15 February 2002

Abstract

The magnetostriction technique is shown to be a valuable tool for the study of flux matter. Measurements of a

classical type-II superconductor (NbTi) exposed new data on a range of topics of current interest, including the physics

of the critical state, the peak effect, and memory of the vortex matter both in the vicinity and far away from the peak

effect region. r 2002 Published by Elsevier Science B.V.

PACS: 74.60.�w; 74.60.Ge; 75.80.+q

Keywords: Critical state; Peak effect; Scaling; Thermomagnetic history dependence; Magnetostriction

1. Motivation

The trend to diversify experimental techniquesfor the study of superconductivity has beenstrongly promoted by a group headed by J. Franseat the University of Amsterdam, in particular sincethe discovery of high temperature superconductiv-ity [1,2]. One of the propositions was an applica-tion of dilatometry ([1,3]; for a recent review seeRef. [4]) for the study of flux matter. Already thefirst experiments with strongly anisotropic BSCCOhigh Tc crystals and classical isotropic NbTipolycrystalline samples have demonstrated thefar-reaching abilities of the technique and richnessof the phenomena to be discovered. The present

work is a direct continuation of the researchinitiated at Amsterdam.

2. Magnetostriction in the critical state

The reason behind the dilatometric approach isstraightforward: pinning stabilizes non-homoge-neous flux density profiles in the critical state oftype-II superconductors. At the same time as thepinning force acts on a flux line, an equally largeforce is applied to the pinning center and istransferred to the body of material. As a result, thematerial undergoes a macroscopic mechanicaltransformation. The local stress within a bulk ina magnetic field Ba parallel to its surface isdistributed as

sðxÞ ¼ �B2a � B2ðxÞ2m0

;*Corresponding author. Tel.: +33-476-88-1130; fax: +33-

476-855610.

E-mail address: [email protected] (R. Schleser).

0921-4526/02/$ - see front matter r 2002 Published by Elsevier Science B.V.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 3 2 - 8

For an elastically isotropic infinite slab of width2w the relative transverse dilation is given by

Dw

w¼

1

2m0Cw

Z w

0

½B2ðxÞ � B2a� dx ¼Fp

2C;

where C is an elastic constant and Fp is the totalmacroscopic pinning force per unit length. Itshould be noted that magnetization in the sameconditions is given by

MðBaÞ ¼1

m0w

Z w

0

BðxÞ dx �1

m0Ba:

The local critical current density JcðxÞ and BðxÞ arerelated to each other by

JcðxÞ ¼ �1

m0

qBðxÞqx

:

Therefore, both magnetization and magnetostric-tion can be used to evaluate the critical currentdensity in two independent experiments.The polarity of the flux density gradient and,

therefore, the stress and the resulting dilationexperienced by the sample depends on its magneticprehistory: inward for ascending and outward fordescending fields (in directions perpendicular tothe applied field). For the cylindrical geometryused in our experiments with field parallel to theaxis, the contraction of the diameter underincreasing field corresponds to an expansion ofthe sample’s length (and vice versa for decreasingfield) as

DL

L¼

�2nDD

ð1� nÞD;

where L and D are the length and diameter of thecylinder and n is the Poisson ratio.Magnetostriction of NbTi samples was mea-

sured by the capacitance technique [5] in fields upto 14T and temperatures down to 0.4K. A rod ofNb56Ti44 has been supplied by Goodfellow SARLand was machined and cut into samples ofdifferent geometries.Fig. 1 presents a set of typical magnetostriction

curves of a cylindrical sample with diameterD=1.5mm and length L=6.8mm, measured atseveral temperatures as the field parallel to the axiswas swept up and down between zero and 14T.The curves are entirely reproducible after a longenough thermal stabilization of the capacitance

cell. Fig. 1(a) shows the measurement of the lengthof the sample, whereas the measurement of thediameter is shown in Fig. 1(b). Since D5L;the signal-to-noise ratio is significantly smallerand the background signal from the cell becomesmore important in the measurement of D:Two characteristic hysteretic regions can be

identified in low-temperature experiments: a wideellipsoid-like curve up to about 0:8 Bc2 and a sharppeak close to Bc2: Just from the very beginning it isinteresting to compare the Poisson ratio n ex-tracted from the L and D measurements atintermediate and high fields. n ¼ 0:47 is calculatedat B ¼ 4 T at T ¼ 1:5 K: n as high as 0.6 iscalculated in the peak region. This value is

0 2 4 6 8 10 12 14

−2 x10-7

0

−2 x10-7

0

2x10-7

0 2 4 6 8 10 12 14

(b)

(a)

T = 1.5 KT = 4.2 K

T∆D /

D

B (T)

T = 1.5 KT = 2.6 KT = 4.2 KT = 6.7 K

T

∆L /

L

Fig. 1. Normalized change of length L (a) and diameter D (b)

of a cylindrical NbTi polycrystalline sample as a function of

magnetic field measured at different temperatures. The field is

applied parallel to the axis. L ¼ 6:8mm and D ¼ 1:5mm. Thedotted arrow through the curves indicates the direction of

increasing temperature.

A. Gerber et al. / Physica B 319 (2002) 293–302294

unphysical ð> 0:5Þ and surprising given the largeratio L=DE4:5:

3. The peak effect

For several decades, the abrupt and strongincrease of the critical current density in manytype-II superconductors, at fields slightly belowthe upper critical field Bc2; has remained a topicunder vivid discussion both experimentally andtheoretically.Multiple ideas have been put forward about the

origin of this phenomenon. One of the firsttentative explanations, by Pippard [6], assumedthat the rigidity of the flux lattice (FLL) near Bc2would drop more rapidly than the pinning strengthdue to inhomogeneities, leading to a bettermatching of FLL and defects. In contrast to this,Larkin and Ovchinnikov [7,8] put forward the ideaof weak collective pinning, where large amounts ofpinning centers lead to a division of the FLL intosmall regions of volume Vc with short-range order.At fields just below Bc2; the reduction of Vc leadsto a maximization of the pinning force per unitvolume.For the case of superconductors with large spin

susceptibility, Tachiki et al. [9] proposed theoccurrence of a generalized Fulde–Ferrel–Lar-kin–Ovchinnikov (GFFLO) state as the origin ofthe peak effect in these systems, based on work byFulde and Ferrell [10] and by Larkin andOvchinnikov [11,12]. Following Tachiki et al., aspatially modulated order parameter leads toflexible vortices readily pinned through the collec-tive pinning mechanism. Modler et al. [13] claimto have found strong evidence for the occu-rrence of such a state in UPt2Al3 and CeRu2.Neutron diffraction studies of Nb singlecrystals were interpreted in favor of a first-orderphase transition between a stable ordered and afully disordered solid [14] or liquid [15] vortexstate.Fig. 2 shows a detail from Fig. 1(a) for tem-

peratures 1:5 KoTo4:2 K: The peak is defined bya local minimum at its onset Bon for increasingfield, the irreversibility field Birr and an offset fieldBoff ; corresponding to a local minimum in the

descending branch of the curve (all these areindicated by vertical dotted lines at T ¼ 1:5 K).The peak maxima for increasing and decreasingfield Bp;up and Bp;down are also marked.It should be noted that the positions of the

peak maximum for increasing and decreasingfield do not coincide. Bon is clearly higher thanBoff ; i.e. the width of the peak region is signifi-cantly larger for decreasing than for increasingfield. The asymmetry between the increasing anddecreasing field is well illustrated in Fig. 3, where

10 12

-2x10-7

0

2x10-7

B irr

Bp, down

Bp, upBon

Boff

T

T = 1.6 KT = 2.6 KT = 4.2 K

∆ L/

L

B (T)

Fig. 2. Plot of the peak region in magnetostriction of a

cylindrical sample for different temperatures. The solid arrows

indicate the direction of the field sweep. The dotted arrow

through the curves points in the direction of increasing

temperature.

11 12 130

1x10-7

2x10-7

3x10-7 T = 1.6 K

(∆L/L)up

-(∆L/L)down

∆L

/L

B (T)

Fig. 3. Comparison of up- ðdB=dt > 0Þ and down-sweep

ðdB=dto0Þ in the peak effect region in the magnetostrictionof a small cylinder. The down-sweep branch is multiplied by

(�1).

A. Gerber et al. / Physica B 319 (2002) 293–302 295

the part of the curve measured for decreasing fieldwas multiplied by (�1) to compare the increasingand decreasing branch in the peak region.The difference in peak positions is not com-

monly observed in magnetization measurementsof the peak effect. However, a similar inequalityof Bon and Boff has been previously found inmagnetization and transport measurements ofhigh quality single crystals of superconductingUPt2Al3, CeRu2 [13], 2H-NbSe2 [16] as well as inamorphous Nb3Ge and Mo3Si films [17]. Despitethe asymmetry we define the total irreversiblemagnetostriction as

DL

L

� �irr

¼1

2

DL

L

� �up

�DL

L

� �down

!;

where up and down subscripts indicate ascendingand descending field, respectively. Fig. 4 presentsthe temperature dependence of the total irrever-sible signal measured at the peak and at anarbitrarily chosen low-field value, defined here at0:2 Bp: The temperature dependence is evidentlydifferent, the peak amplitude drops much quickerwith increasing temperature than the low-fieldsignal, and seems to approach zero far below Tc:Disappearance of the peak in CeRu2 at tempera-tures significantly lower than Tc and the non-applicability of scaling laws to the peak region (tobe discussed later), have been argued to be anevidence for the appearance of an ‘‘anomalous’’

peak effect in weak pinning superconductors.Following the same guidelines, we plot in Fig. 5the normalized peak value of the irreversiblemagnetostriction as a function of a normalizedtemperature t ¼ T=Tc: To test the universality ofthe behavior, we include the magnetostriction datameasured on both cylindrical and cubic NbTisamples together with the results of measurementson a single crystalline 2H-NbSe2 sample. More-over, we include the peak amplitude from magne-tization measurements of CeRu2, measured byKadowaki et al. [18]. All results seem to collapseclose to a single curve.The amplitude of the peak is defined at fields

close to Bc2: The invisibility of the peak attemperatures higher than TEð0:770:1ÞTc mightindicate the presence of a temperature scaledifferent from Tc; but could also be explained bya strong power law dependence ðTc � TÞ�a; ab1;in the vicinity of the upper critical field.

4. Scaling

In order to connect phenomenological models ofthe critical state to a microscopic understanding ofthe pinning mechanisms, Fietz and Webb [19] have

0 2 4 6 81x10

-8

1x10-7

1x10-6

Tc

( ∆L

up-

∆Ld

ow

n)/

L

T (K)

B = Bpeak

B = 0.20 Bpeak

2 4 6 80

5x10-7

1x10-6

T (K)

(∆L

up -∆

Ld

ow

n )/L

Fig. 4. Temperature dependence of the irreversible magnetos-

triction measured at the peak field peak Bp and at a low field

B ¼ 0:2Bp: The y-axis is logarithmic in the main graph and

linear in the inset. The right border of the x-axis corresponds to

TcE9:5K:

0.0 0.2 0.4 0.6 0.8 1.00

1

2

Magnetization: CeRu2 , B = Bp

(Kadowaki et al.)

Magnetostriction: NbTi cylinder, B = B p

NbTi cube, B = Bp

NbSe2, B = Bp

a

T/T c

Fig. 5. Temperature dependence of the peak amplitude, as

observed by magnetostriction, as a function of the normalized

temperature t � T=Tc; in two samples of NbTi with cylindricaland cubic geometry, and a single crystalline 2H-NbSe2 sample.

The peak irreversible magnetostriction is normalized at t ¼ 0:2Also shown is the peak’s amplitude in magnetization as

measured by Kadowaki et al. [18] in single crystalline CeRu2.


proposed the following power law dependence ofthe pinning force FpðTÞ on the upper critical fieldHc2ðTÞ:

FpðTÞ ¼ KkðkÞgðB=Hc2Þ½Hc2ðTÞ�p

in which the pinning force density depends onmagnetic field through some universal functiongðB=Hc2Þ and on temperature only throughHc2ðTÞ: k depends only on the Ginzburg–Landauparameter k; and K is a constant. p was found tobe 5/2. Kramer [20] compared measurements ondifferent compounds and distinguished betweentwo types of peaks: a wide one at intermediatefields and a narrow one just below Bc2; which iscurrently known as the ‘‘peak effect’’. For thelatter one, only few data points were presented(taken from Ref. [21]) leading to a scalingexponent pE2:9 > 2:5: It remains unclear, how-ever, if this higher value of the exponent is onlycaused by uncertainties in its determination. Morerecently, Wyder et al. have performed a scalinganalysis of magnetostriction measurements ofNbTi in the peak region [1]. The samples havebeen extra cold worked, which resulted in largepeak amplitude compared to the low-field values.p values as large as 4.5 were found and attributedto an additional field dependence of the elasticconstant cðT ;BÞ ¼ c0ðbÞB�2

c2 ; where b ¼ B=Bc2ðTÞ;and c0ðbÞ is a function depending only on b:Magnetostriction of a cylindrical NbTi sample,

normalized to the irreversibility maximum atintermediate fields is plotted in Fig. 6 as a functionof a reduced magnetic field b ¼ B=Bc2ðTÞ at threetemperatures 4.2, 2.6 and 1.6K. The scaling isremarkable over a wide field range up to the onsetof the peak. The maximum of the irreversiblemagnetostriction in this field range is plotted inFig. 7 as a function of the temperature dependentupper critical field Bc2: The power law dependenceon Bc2 is evident with p ¼ 2:35: This value is veryclose to p ¼ 5=2 derived by Fietz and Webb. It isalso evident from Fig. 6 that the magnetostrictionin the peak region does not follow the same fieldscaling as the rest of the curves. The question is:does the peak scale with the reduced field b at all?The answer is, probably, negative as demonstratedin Fig. 8. Here the peak region magnetostriction,normalized by its maximum value at a given

temperature, is plotted as a function of b at fourtemperatures: 6.7, 4.2, 2.6 and 1.6K. The tem-perature dependent maximum is reached atdifferent reduced fields, which rejects the scalingwith b:Nevertheless, the peak values of the irreversible

magnetostriction can be plotted as a function ofthe temperature dependent upper critical field

0.0 0.2 0.4 0.6 0.8 1.0−1.5

−1.0

_0.5

0.0

0.5

1.0

1.5

T

T = 1.6 KT = 2.6 KT = 4.2 K

∆L/ ∆

Lm

ax,

low

B

B / Bc2

Fig. 6. Plot of magnetostriction of a cylindrical NbTi cylinder,

normalized to the irreversibility maximum at intermediate

fields, vs. a reduced magnetic field B=Bc2: The dotted arrowthrough the curves points in the direction of increasing

temperature.

4E-8

1E-7

4E-7

(∆L/L)max, low B

Power law fit:

(∆L/L) = 6.16 .10-3 . Bc2

2.35

(∆L

/L) m

ax,

low

B

Bc2

6 10

Fig. 7. Scaling analysis for magnetostriction on a cylindrical

NbTi sample at intermediate fields. Squares represent values

extracted from magnetostriction measurements, the solid line

presents a fit of the form DL=L ¼ aBpc2:


Bc2ðTÞ; as shown in Fig. 9. A power law is metwith p ¼ 5:8; a value much higher than the onesfound at intermediate fields. Notably, the peakmagnitude of a variety of samples not shown herecan be fitted as DL=L ¼ aB

pc2: In a cubic NbTi

sample, p ¼ 7:2 was found, and p ¼ 6:22 in a singlecrystalline 2H-NbSe2 sample. Simultaneously,the maximum irreversible magnetostriction at

intermediate fields can be presented by the samepower law with much lower values of p ¼ 2:35 in acylindrical NbTi sample and p ¼ 3:47 in a singlecrystalline sample of 2H-NbSe2. The origin ofthe high power indices in the peak region isnot known. However, an operational conclusiondrawn from a comparison of indices in the peakregion and at intermediate fields is that the peaksignal drops much quicker with decreasing Bc2 andis screened by the intermediate field irreversibilityat temperatures close to Tc:

5. The asymmetry of the peak effect

As mentioned earlier, one of the points mostvividly discussed nowadays in the physics of vortexmatter is the nature of the transition into the peakregime. In this context, the difference DB ¼ Bon �Boff between the onset and offset of the peakregion was interpreted in terms of a first-ordertransition, in the sense that a disordered ‘‘super-cooled’’ vortex phase coexists with an orderedphase [13,18]. The very existence of the ‘‘super-cooling’’ might mean that the peak effect phenom-enon marks a true thermodynamic phasetransition and it is of first order [16]. To test thishypothesis, we have compared the measurementson three cylindrical samples with different dia-meters: 1.5, 2.8 and 10mm. The field differenceDB ¼ Bon � Boff is plotted in Fig. 10 as a functionof the sample’s diameter. Since only differences ofmagnetic fields are considered, slight fluctuationsin the parameters of the samples should notsignificantly affect the result. DB follows linearlythe transverse dimension of the sample within theaccuracy of the measurement. This geometricalscaling is a strong evidence against the interpreta-tion of DB in terms of a first-order thermodynamictransition. On the other hand, it supports a muchsimpler interpretation of DB; as due to a spatialseparation of two phases in the framework of thecritical state model. A cone-like field profile isgenerated within the bulk of the material when thefield is swept down and out of the peak region. Thehighest value of the local magnetic induction, aswell as its gradient and screening current densityare in the center of the sample. The disordered

0.90 0.95 1.000.0

0.5

1.0

T

T = 1.6 KT = 2.6 KT = 4.2 KT = 6.7 K

∆ L/∆

Lp

ea

k

B / Bc2

Fig. 8. Plot of the irreversible magnetostriction in the peak

region for a cylindrical NbTi sample at different temperatures,

renormalized to the respective peak’s amplitude, vs. a reduced

magnetic field B=Bc2ðTÞ: The dotted arrow through the curvespoints in the direction of increasing temperature.

1010

-8

10-7

10-6

(∆L/L)p

Power law fit:

(∆L/L)p=2.29 .10

-13 . Bc2

5.8

156

(∆L

/L) p

ea

k

Bc2 (T)

Fig. 9. Scaling analysis for magnetostriction on a cylindrical

NbTi sample in the peak region. Squares represent values

extracted from magnetostriction measurements, the solid line

presents a fit of the form DL=L ¼ aBpc2:


high-field phase is preserved in the interior of thesample as long as a local induction is higher thanBon: Boff indicates the field on the surface of thesample; therefore DB scales with the diameter. Thismeans that two phases do not coexist in the samepoint of the B–T phase diagram; their coexistenceat the same applied field is possible due to a spatialvariation of the local magnetic induction, simplyfollowing the basic idea of the critical state.

6. Magnetic history dependence effects

An important feature commonly observed in thepeak region is the dependence of the experimen-tally studied variable on the thermomagneticprehistory of the experiment. The critical currentdensity Jc during the decrease of the applied fieldB or temperature T was found to exceed therespective values measured at the same T or B

along the increasing direction. As seen in the lastsection, the magnetic field indicating the onset ofthe peak region along an ascending field sweep issignificantly higher than the field corresponding tothe exit out of the peak region along thedescending sweep. These and other examples ofmemory of the thermomagnetic prehistory wereinterpreted as due to the coexistence of an ordered

low-field/temperature phase and a disorderedhigh-field/temperature phase in a certain range ofthe B–T phase diagram [17,22–26]. Moreover, thevery coexistence of two phases has been noted[13,18] as an indication of the ‘‘supercooled’’ and‘‘superheated’’ vortex matter phases, and as suchbeing an evidence of a true first-order thermo-dynamic transition.We now turn to the memory effects as they are

observed in magnetostriction measurements. Hys-teresis of magnetostriction is related to Jc: Similarto isothermal magnetization measurements, asingle valued JcðBÞ translates into a genericmagnetostriction hysteresis loop such that theforward and reverse branches of the curve definean envelope, within which lie all magnetostrictionvalues that can be measured at the given tempera-ture along various paths with different (thermo)-magnetic histories. When the direction of the fieldsweep is reversed at fields lower than the field ofirreversibility, the superconductor is expected toreverse its magnetostriction and the signal shouldmerge into the envelope curve after a field changeof DB: JcðBÞ is a monotonically decreasingfunction of magnetic field below the peak region.The largest DB is needed at low fields where it isjust 2Bfp;Bfp being the field of full penetration. Forthe given sample, Bfp ¼ 0:085 T at T ¼ 1:5 K; asfound by measurements of magnetization; a valuecorresponding to Jc ¼ 108 A=m2: Two criteriamust be satisfied to confirm a single valued JcðBÞor the absence of additional memory effects: (1)minor loops (loops measured by a reversal of thefield sweep at intermediate fields) should beconfined within the generic envelope, and (2) theseloops should merge into the generic envelopewithin a field change not larger than Bfp:Figs. 11(a) and (b) show a number of minor

loops in the peak effect region, starting from theascending and descending sweeps, respectively. Allthe loops shown in Fig. 11(a) remain within ageneric envelope and merge into it at the same fieldBE11 T; coinciding with Boff ; the local minimumalong the descending branch. Notably, loop 5initiated at the high-field section of the peak regionat B > Bp; where Bp is the field of the peak, alsodeviates from the generic envelope along asignificant part of its path. To the best of our

0 2 4 6 8 100.0

0.2

0.4

0.6

∆ Bo

nse

t/o

ffse

t(T

)

d (mm)

Fig. 10. Magnetic field difference between the onset of the peak

region in ascending and its offset in descending field ðDB ¼Bon � Boff Þ as a function of the sample’s diameter for cylindricalNbTi samples.


knowledge, the only example of ‘‘superheating’’ ofthe ordered low-field phase beyond Bp has beenobserved in a Nb single crystal under very low ACfields. The collapsing at the same Boff of all thesedescending minor loops initiated within the peakregion seems to indicate a well-defined limit of thedisordered phase. Loop 2 initiated in the vicinityof Bon is of special interest: its trajectory is almostsymmetrical to the ascending branch below thepeak onset and might mimic a hysteresis envelopenot polluted by the peak region phase.Fig. 11(b) presents selected minor loops initiated

at the descending branch of the generic envelope inthe peak regime. Loops 2–5 overshoot the virginascending branch, while loops 1 and 6 undershootit along a wide field span. This is consistent withdescending minor loops and confirms Boff as awell-defined exit field out of the peak region. Theasymmetry between the ascending and descendingminor loops in the low-field section of the peak

region is consistent with data found by othertechniques in other materials with predominantlyweak pinning [17,23,25,26].Several minor loops initiated at fields below the

peak region are shown in Fig. 12. Clear memoryeffects are observed in all these loops in theentire field range. Characteristic features ofthe minor loops are different from those observedin the peak region: ascending and descendingloops initiated at the same applied field aresymmetric in respect of over- or under-shooting.Loops started at low fields (curves 1a,b and 2a,b)overshoot the generic envelope. Loops initiatedabove BE5 T ‘‘undershoot’’ the generic envelope.Loop 3 crosses the envelope twice, an overshoot isfollowed by a dip. The general form of all loopsinitiated at intermediate fields is similar: a rapidreversal of the polarity followed by a wide non-monotonic approach toward the generic envelope.The first step of all minor loops, a rapid approachtowards an opposite branch of the envelope, takesplace over a field change of the order of a few tensof mT down to a few mT, values consistent with afield dependent reduction of the macroscopic Jc.The wide field span of the order of 1T (up to about3 T in loop 4), needed for minor loops to mergeinto the envelope, exceeds by three orders ofmagnitude the values expected by the critical statemodels, and is an evidence of the thermomagneticmemory.

11 12 13-4x10

-7

-2x10-7

0

2x10-7

(b)

Boff

7

6543

21

T = 1.5 K

∆ L /

L

B (T)

-4x10-7

-2x10-7

0

2x10-7 (a)

Boff

54321

T = 1.5 K

∆L /

L

Fig. 11. Minor loops in magnetostriction vs. magnetic field in

the peak region for (a) descending and (b) ascending magnetic

field.

0 2 4-2x10-7

-1x10-7

0

1x10-7

54

3

2b

2a

1b

1a

T = 1.5 K

∆ L/

L

B (T)6 8

Fig. 12. Minor loops in magnetostriction vs. magnetic field at

low and intermediate fields for both ascending and descending

magnetic field.


A model of artificially induced disorder invortex lattice has been recently proposed by Paltielet al. [28]. Vortices injected predominantly at theweakest points of the surface may destroy the localorder and form a metastable disordered vortexphase near the edge, which drifts into the samplewith the flow of the entire lattice. A priori,generation of such disorder is not directly relatedto the scenario of the thermodynamic phasetransition associated with the peak effect region.On the other hand, the stability of this extrinsicdisorder depends strongly on the conditions of theexperiment (DC or AC), annealing tools (e.g. thepresence of a transport current) and the exactlocation inside the H–T phase diagram.We can try to identify the two types of memory

effects described earlier with two types of disorderthat can exist in vortex matter: intrinsic andextrinsic. Intrinsic disorder is the volume disor-dered phase dominating the peak region. Extrinsicdisorder is generated in the vicinity of the surfacewhen flux is forced to penetrate or leave thesample. Any weak spot on the surface and inparticular extended defects like microcracks serveas injection channels for the extrinsic disorderwhich can be generated at any applied field by areversal of the sweep direction. The stability of thisdisorder in the conditions of our experiments ishigh; a field change of the order of DBE1 T isrequired to screen its effect. There might be tworeasons for the fact that this type of memory hasnot been found earlier far below the peak region.The first is the use of the magnetostriction tech-nique which is (a) sensitive, (b) static (free ofvibrations or extended movements of the samplethat in the presence of a finite field gradients havethe same effect as the ‘‘shaking’’ by an AC field),and (c) passive (no measuring current, which hasbeen found to anneal the disordered phase [23],is used). The second reason is the use of apolycrystalline sample with multiple surfacedefects. It should be noted that the visibility offlux memory in the entire irreversible range ofthe B–T phase diagram is not entirely new. Jcvalues of the field-cooled state have been shown toexceed that of the zero-field-cooled one far belowthe peak position in 2H-NbSe2 [23,27] and CeRu2[26,29].

7. Conclusion

To summarize, we applied the dilatometrictechnique to measure the pinning force in thecritical state of the classical type-II superconductorNbTi. New insights into a number of questions ofcurrent interest has been obtained. A comparativestudy of scaling in the peak region and atintermediate magnetic fields shows that the non-applicability of scaling laws at high fields does nothave to be considered as a hallmark of an ano-malous peak effect. Geometrical scaling of the fieldrange of coexistence of the stable ordered and thedisordered vortex phases evidence against a firstorder thermodynamic phase transition into thepeak region. A new type of magnetic historydependence at low and intermediate magneticfields has been found and tentatively ascribed tothe propagation of metastable vortex structures.

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magnetostriction of the model type-ii superconductor nbti

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