Vortex pinning in a rotated YBa2Cu3O7−δ crystal

PHYSICA ELSEVIER

Physica C 291 (1997) 8-16

Vortex pinning in a rotated

YBa2Cu3OT_ crystal

I.M. Obaidat *, H.P. Goeckner, J.S. Kouvel Department of Physics, University of Illinois, Room 2236, 845 W. Taylor Street, Chicago, IL 60607-7059, USA Received 23 June 1997; accepted 31 July 1997

Abstract Magnetization-vector measurements were made on an YBCO (YBa2Cu30 7_ ~) crystal rotated about its c-axis in various fixed fields at different temperatures. The pinning torque per vortex (rp) in the ab-plane is found to have a broad distribution in strength, similar to previous findings for polycrystalline samples of YBCO and BKBO (Bao.57K0.n3BiO3). Simple modeling shows that the rp distribution function rises linearly with increasing ~-p and then drops rapidly to zero. With increasing field at each temperature, the average ~-p decreases gradually after rising to a broad peak. The peak value of ~-p/tZ ( ~ being the vortex moment) defines Hp, a characteristic pinning field per vortex, and it is found to decrease with increasing temperature (T) in a distinctly two-component manner. One component of Hp falls off very rapidly as (1 - T/Tc)~ with /3 = 10, while the other component descends linearly to zero as T approaches To. A similar two-component behavior of Hp versus T is found in polycrystalline YBCO, whereas in polycrystalline BKBO Hp decreases gradually with increasing T at all temperatures. The strongly temperature dependent Hp component in YBCO may derive from vortex pinning by oxygen vacancies, as suggested theoretically. © 1997 Elsevier Science B.V.

1. Introduction The pinning of magnetic vortices in type-II superconductors continues to be studied very actively, largely because of its strategic technological importance but also because of many unresolved questions about the nature of the pinning under different experimental conditions. In this pursuit, the strength of the vortex pinning has commonly been determined from conventional measurements of magnetic hysteresis loops, as interpreted via various critical-state models [1,2]. An alternative and more direct method, first used by LeBlanc et al. [3] and then modified and

* Corresponding author. Tel.: + 1 312 996 3400; fax: + 1 312 996 9016.

applied extensively in our laboratory [4,5], involves the measurement of the sample magnetization as a vector while the sample is rotated in a fixed magnetic field normal to the axis of rotation. From rotational measurements of this kind on a polycrystalline sample of the perovskite compound Ba0.57K0.43BiO 3 (BKBO) at 4.2 K, it was found [6] that in the rotational steady state the vortex flux density vector B bifurcates into two distinct components: a rotational component B R, which turns rigidly with the sample, and a frictional component B v, which stays at a constant angle (O F) relative to the fixed field H and thus turns frictionally relative to the sample. Moreover, with increasing field, the magnitude of B R decreases and eventually vanishes, while B F continues to grow in size. The coexistence

0921-4534/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 ( 9 7 ) 0 1 6 6 0 - 2

LM. Obaidat et al./Physica C 291 (1997) 8-16

of B R and B F over a large range of fields indicates unambiguously that the vortex pinning has a broad distribution in strength. Thus, the frictional angle OF represents an average orientation of the vortices associated with B F and, correspondingly, the quantity H/z sin OF (/x being the vortex moment) represents an average pinning torque on each of these vortices. More recently, Hasan et al. [7] carried out similar rotational measurements on polycrystalline samples of BKBO and the classic YBCO (YBa2Cu307_~), and their results were qualitatively unchanged over a wide range of temperatures, consistently showing a broad distribution in the strength of the vortex pinning torques. Furthermore, in both materials, the average pinning torque per vortex was seen to decrease rapidly with increasing temperature. The grain boundaries in the polycrystalline samples investigated can be expected to contribute to the vortex pinning distribution since they are thought to provide avenues of relatively weak pinning [8]. Hence, we have now focused our rotational magnetic studies on a single crystal of YBCO, which is rotated about its c-axis with the fixed field applied in the ab-plane. As described in this paper, we continue to observe a broad distribution in the size of the vortex pinning torques, which is shown by simple modeling to be a distinctly asymmetric distribution function. Moreover, from the results obtained for the YBCO crystal over a wide temperature range, the average pinning torque per vortex is found to diminish very rapidly at first and then more slowly with increasing temperature, indicating two separate components of variation. This behavior is compared with those of polycrystalline YBCO and BKBO, where the former shows a similar two-component variation whereas in the latter the variation is consistently gradual at all temperatures. A possible implication of this comparison will be discussed.

2. Experimental details Our sample was an YBCO crystal platelet, ~ 1.5 mm in diameter in the ab-plane and ~ 0.1 mm thick along the c-axis, which had been well prepared by standard techniques by L. Paulius of Western Michigan University. The crystal was mounted in our vibrating-sample magnetometer (VSM) such that it

9

could be turned about its c-axis while a fixed magnetic field ( H ) was applied parallel to the ab-plane. In our cryogenic VSM system, two sets of pickup coils are arranged in quadrature, allowing us to measure both the longitudinal and transverse components of the sample magnetization vector M relative to H in the ab-plane. For this geometry, demagnetization effects were very small - but routinely corrected for. Typically, the crystal sample was rotated in prescribed steps up to 360 ° - which normally took about 5 min, during which time the flux creep effects were negligibly small compared to the rotational effects of interest. The ab-plane was seen to be magnetically isotropic, indicating that the crystal is probably highly twinned. As our previous rotational magnetic measurements on various superconducting samples have shown, [4-7,9], the total magnetization vector M at fixed H can be decomposed uniquely into a constant diamagnetic component M D and a penetrating vortex-flux component Mp that turns with the sample, rigidly at first for small angles of rotation. Furthermore, our present sample shows that M D = - H/41r, signifying perfect persistent-current shielding. It follows that Mp = B/4"n', such that the flux density B derives exclusively from the penetrating vortices and is therefore the vector quantity into which our data have been routinely converted.

3. Results and discussion

3.1. Rotational magnetic properties In all our measurements at various temperatures, the YBCO crystal before rotation was prepared by having H reduced from + 15 kOe to some positive value (or zero), thus placing it initially on the upper branch of a hysteresis loop. Our rotational data are illustrated in Fig. 1 by those obtained for H = 2 kOe at 4.2 K. In this figure, BL, the longitudinal component of B relative to H, is plotted versus the transverse component B T for various sample rotation angles (0) from zero to 360 °. Starting at 0-- 0 with B x = 0, this polar plot shows that when 0 reaches 180 ° the data points begin to move on a circle synchronously with the sample rotation. Thus, in the rotational steady state for 0 > 180 °, the total B con-

l0

I.M. Obaidat et al. / Physica C 291 (1997) 8 - 1 6

YBCO crystal 4

4.2 K

....... ..

0 °'~

2 kOe

3

~

"., 2

'~

360 °

~./)

2700 0

"'

~.

-0"//

180 ° "1

.

-2

.

.

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-1

.

.

.

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0

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....

1

i

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....

~ ....

3

4

BT(kG)

plotted for the same temperatures in Fig. 3(a), where we see that OF stays fairly constant and then decreases very rapidly with increasing H. We have also examined the quantity H sin OF, which corresponds to the pinning torque per vortex (~-p) divided by the vortex moment (/z) and is directly related to the critical current density (Jc) [10,11]. Fig. 3(b) shows that at each temperature H sin OF rises from zero, reaches a maximum near the field at which B R vanishes, and then diminishes gradually with increasing H. A similar variation of H sin OF with H was previously observed in polycrystalline YBCO, [7] where the gradual decrease beyond the peak was ascribed to the collective pinning process of vortex bundling [12] and fitted to the simple expression, H s i n OF a H -8, where 3 = 0.3 at 4.2 K. The same

Fig. 1. Polar plot of longitudinal versus transverse componentsof B relative to H for increasing angle of sample rotation, marked at intervals of 90°. Center of fitted circle indicated by + sign.

4

....

I ....

I '

= ' ' I ....

I ' , , ,

~

, , ,

YBCO crystal 3

sists of a fixed vector (B E) from the origin to the circle center, lying at a frictional angle (O F) relative to H (along +BE), plus a rotating vector (B R) from the circle center to the periphery of the circle. The simplicity of this polar plot, which also pertains to previous data for polycrystalline samples, [6,7] indicates that the ab-plane of the YBCO crystal is magnetically isotropic, suggesting a high density of twin boundaries. As determined from similar polar plots of all our rotational data for the YBCO crystal, the steady-state magnitudes of B R and B F are plotted versus H for several representative temperatures of measurement in Fig. 2. We see at each temperature that B R decreases from its remanence value at H = 0 and eventually vanishes with increasing H, whereas B F grows monotonically and approaches the B F = H line at high H. Qualitatively, this behavior closely resembles that seen from similar measurements on polycrystalline YBCO and BKBO [6,7], where it was attributed to a distribution in the strength of the vortex pinning, as described earlier. It now appears that a similar vortex-pinning distribution exists in our YBCO crystal despite the absence of grain boundaries. Regarding the changes in the frictional angle OF that accompany the changes in B R and B F, they are

• 4.2K

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1

2

3

4

5

6

H (kOe)

Fig. 2. (a) Rotational component, and (b) frictional componentof B versus H at various temperatures. BF = H line in (b) shown for reference.

LM. Obaidat et al. / Physica C 291 (1997) 8-16 50

nomenological vortex pinning model will now show explicitly.

,0

YBCO crystal

40-,¢, ', • ~oo

~,

/

o 16K o 35K

I

20

o

10 a

3.2. Rotational uortex pinning model

* 4.2K

,~i ,,

o

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~,..

(a)

0 ~

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•

•

,

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6

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12

14

H (kOe) Fig. 3. (a) Frictional angle OF and (b) H sin OF versus H at various temperatures. Solid curve in (b) corresponds to H s i n OF cxH - I at 4.2 K.

type of fit to our YBCO crystal results at 4.2 K is represented by the solid curve in Fig. 3(b), which corresponds to 6 = 1, indicating a much larger effect due to vortex bundling a n d / o r a related mechanism recently proposed for the effective pinning of vortices that are surrounded by vortices in direct interaction with pinning centers [13]. In any case, since there is a distribution in the strength of the vortex pinning, our experimental values for 0 F and H sin 0 F represent averages over the distribution, as a phe-

The effects of a distribution in the strength of the vortex pinning on the rotational magnetic properties of an isotropic superconducting sample will be explored here for some simple forms of the distribution function. But first, as a standard of comparison, we consider the simplistic case where every vortex in the sample has exactly the same strength of pinning. If the vortices are turned by angle 0 from the direction of a uniform applied magnetic field H, the magnetic torque on each vortex is given by: z H = H/z sin 0, where /z is the quantized vortex moment. Hence, if the vortex pinning torque (Zp) exceeds H/x (the maximum possible value of ZH), the vortices will all rotate rigidly with the sample through all values of 0. However, at higher fields such that H/z > ~-~, the vortices will rotates rigidly up to some angle OF, where ~'H comes into balance with ~-p, and for further sample rotation the vortices will be pulled to neighboring pinning sites (assumedly of the same strength) such as to maintain the same frictional angle OF relative to H. The value of OF will clearly satisfy: sin OF = zp/Htz, so that OF will decrease monotonically from 90 ° as H is raised above Zp/i.L. Consider now that the vortex pinning torque has a normalized distribution function P(~'p) from zero Zp to some maximum ~-p value of zm. Then, if H is greater than Zm/~ each vortex will again turn with the sample up some frictional angle 0~ relative to H and remain there while the sample continues to turn. However, 0 v will have a spectrum of values for the different vortices, as determined by the frictional balance between z H and the distributed ~-p, which gives sin OF = ~'p/Hl.~

( 1)

for a vortex with pinning torque Tp. Hence, the frictional magnetic moment m F will involve integrations over the entire distribution function P ( ' r p ) . Specifically, the longitudinal and transverse compo-

LM. Obaidat et al. / Physica C 291 (1997) 8 - 1 6

12

nents of the normalized m F relative to H are expressed respectively as follows: m~ = f0 "mP(~-p) cos OF d~-p

~

(2)

2/'c=1

1/4: m

and m~ = fO~mp(ze) sin 0~ d , p ,

(3)

't

'tm P

where 0~ is determined from Eq. (1). The magnitude of m F and its macroscopic frictional angle OF relative to H are given respectively by rnF = {(mFL ) 2 + (mT)2)l/2

(5)

If H is smaller than ~'m//X, only those vortices with pinning torques (rp) weaker than H/z will turn frictionally relative to the rotating sample, each maintaining its angle 0~ relative to H, as prescribed by Eq. (1). Thus, the expressions for rn~ and mrF have the forms of Eqs. (2) and (3) but where the upper integration limits are H/x rather than ~'m" The remainder of the vortices, whose pinning torques are stronger than Ht~, will rotate rigidly with the sample, and the normalized rotational moment produced by these vortices is given by mR

=S.;.

( r p ) d'rr,.

rnL = zrh/4 and m~ = h/2,

(7)

from which Eqs. (4) and (5) give m F = 0.931h and OF = 32.5 ° .

(8)

As shown in Fig. 4, OF remains constant in this low-h regime, while m F rises linearly with increas-

\\\A/~//

20°

0

0O

(6)

We proceed now to consider some specific forms for the distribution function P(~'p) and investigate the properties that they generate. The simplest normalized form of P ( r p ) is that for which it remains at a constant value (1/~-m) between zero ~-p and ~-p = ~'m, as shown in Fig. 4. Henceforth, this will be referred to as Case A. Inserting this form of P(~-p) into Eqs. (2) and (3) with the upper integration limits taken to be H/z, we obtain as a function of the normalized field h (defined as Htx/~"m) that for h
~

mF(B) i// /~mR(A) 0.25 //~// " \ \ ~ mR(B)(b) 0 L..i ...... , .... 0 0.5 ~1 . . . . 1.5 2 h

and OF = t a n - l ( mXF/m~ ).

/

0.75~ 0.5

(4)

.

.

.

.

0.5 .

.

.

.

.

1h .

.

.

.

1.5 .

.

.

2

Fig. 4. (a) Vortex pinning-torque distribution function P(~'p) versus rp for Cases A and B of the model. (b) Reduced rotational moment mR and frictional moment rnF versus reduced field h for Cases A and B. (c) Frictional angle OF versus h for Cases A and

B.

ing h. Moreover, for h < 1, Eq. (6) gives for the rotational component, m R = 1 - h,

(9)

whose linear decrease to zero at h = 1 is also shown in Fig. 4. Note that the sum of m F and m R goes below unity, which reflects the fact that the individual vortex moments that comprise the frictional m F have a spread in orientational angle (relative to H ) from zero to 90 °. Thus, the value of OF in Eq. (8) represents a weighted average. For Case A but at

LM. Obaidat et al. / Physica C 291 (1997) 8-16

higher fields (h > 1), we obtain from Eqs. (2) and (3) that

m~=2(hsin-l(l/h)+(h2-1)l/2/h

}

(lOa)

and mT = l / 2 h ,

(10b)

which, substituted into Eqs. (4) and (5), yield the calculated values of m F and OF plotted versus h in Fig. 4. In this high-h regime, m F rises from the value of 0.931 at h = 1 and approaches unity asymptotically with increasing h, which results from a tightening in the orientational spread of the individual vortex moments. Correspondingly, the average frictional angle OF decreases rapidly as h rises from unity. The identical procedure is followed for Case B, where the vortex pinning distribution function P(~'p) is considered to increase linearly as 2rp/~ -2, reaching 2/~-m at rp = ~'m, as shown in Fig. 4. In the low-field regime (h < 1), we now obtain for the frictional moment component,

m L = m~ = 2h2/3,

(11)

which, from Eqs. (4) and (5), yields m F = 0.943h 2 and OF = 45 °,

(12)

and for the rotational component, m R -- 1 - h 2.

(13)

In the high-field regime (h > 1), where m R = 0, we find that

mL=(2/3)(h2-(h

2 - 1)3/2/h}

(laa)

and

mrF = 2/3h,

(14b)

which, again via Eqs. (4) and (5), yield m F and OF. For both regimes, the calculated values of m F, OF, and m R are plotted versus h in Fig. 4. In Case B, as compared to Case A: m F rises quadratically with increasing h but again approaches unity at very high h, m R descends quadratically from unity but again vanishes at h = 1, and OF has a higher constant value for h < 1 but again decreases rapidly as h exceeds unity. Several simplifying approximations made in this

13

model need to be noted. First, each vortex is implicitly considered to remain straight during any frictional rotation relative to the sample. Secondly, the frictional rotation of each vortex is assumed to involve hopping to a neighboring set of pinning centers of the same average strength (though different for different vortices), which implies that vortices pinned with different strengths may be situated differently in the sample. Nevertheless, the present model is a definite improvement over a more approximate earlier consideration of a vortex pinning distribution [9].

3.3. Application of the model In applying this phenomenological model to experiment, it must be noted that in its normalized form the model does not reflect any changes in the total number of vortices, including any produced by changes in the external field. Experimentally for the YBCO crystal, the situation is quite different when we consider the sum of the steady-state values of B R and B F shown in Fig. 2. This sum is plotted versus H in Fig. 5(a), where we see that for each temperature B R + B F rises rapidly at sufficiently high H, signifying an increase in the total number of vortices in the rotational steady state. However, at lower H, B R + B F goes through a broad minimum, which resembles the behavior of m R + m F predicted by the model for h rising to unity. Within this low-H region at each temperature, Fig. 5(b) shows that the normalized quantity BF/(B R + B F) undergoes a complete variation from 0 to 1. Moreover, this variation with H seems to resemble the quadratic variation of m v with h predicted in Case B of the model, as shown in Fig. 4(b). For a more appropriate comparison, the BF/(B R + B F) versus H points in Fig. 5(b) were fitted with curves calculated in Case B for the similarly normalized mF/(m R + m F) versus h. In this fit, the only adjustable parameter is the scaling factor ~'m/~ between the experimental H and the reduced h ( = H / x / r m) of the model. For the calculated curves shown solid in Fig. 5(b), the chosen values of ~'m//X were 2.92, 1.67, and 0.75 kOe for T = 4.2, 16, and 35 K, respectively. In each case, the fit to the experimental points is quite good, except for the points that show a more gradual behavior as BF/(B R

14

I.M. Obaidat et al./Physica C 291 (1997) 8-16 4

. . . .

f . . . .

,

,

,

. . . .

YBCO crystal

.y

o"

3

,'"

e"

,

function shown in Fig. 4(a) for Case B. At present, we offer no microscopic justification for a distribution function of this asymmetrical shape.

. . . .

,,,e

,"

,0' t"

2

I

,*"

/

./

L +~g

~

°

i ....

I /

r.,.

3.4. Vortex pinning versus temperature

,' _.it. -°

i ....

i ....

/

i ....

i ....

6

5

6

,16.

7..o., 0.2

0 0

1

2

3 H (kOe)

4

Fig. 5. (a) B R + B F, and (b) B r / ( I B R + BF) versus H at various temperatures, as determined from experimental points in Fig. 2. Solid curves in (b) are model fits for Case B, as described in the text.

+ B F) approaches unity. Thus, Case B of the model may indeed pertain to the YBCO crystal, but where the linear pinning-torque distribution function P(~-p) decreases less abruptly at ~-p = r m than that shown in Fig. 4(a). Also note that for Case B the frictional angle OF for h < 1 (i.e., H < 7m//X) is constant at 45 ° , which is the approximate vicinity of the experimental OF values at low H in Fig. 3(a). Relatedly, for Case B the quantity (H/z/~- m) sin OF equals 0.707 whereas the peak measured values of H sin 0 v in Fig. 3(b) are consistently about 0.6 times the values of ~'m//Z used in the curve fits in Fig. 5(b). This rough quantitative agreement further supports the notion that the vortex pinning in our YBCO crystal conforms fairly closely to the linear distribution

There have been several experimental studies of the temperature dependence of the vortex pinning in metallic type-II superconductors [14-17], but only a few such detailed studies have been made of the ceramic high-T~ materials [18,19]. The latter list may be extended to include the rotational magnetic study of polycrystalline YBCO and BKBO referred to earlier, but although this study evidenced a rapid decrease of the vortex pinning with increasing temperature, its emphasis was on other vortex properties [7]. However, from more recent rotational measurements on polycrystalline Eu,.sCe0.sSr2CuzNbO,0 [20], it was found that the vortex pinning varies as (1 - T/T~) 22, which is similar to the results near Tc of a study of sintered YBCO [18]. To obtain analogous information for the ab-plane of our YBCO crystal, we have analyzed our rotational magnetic data taken at many temperatures ranging nearly up to Tc (90 K). Our results at each temperature, when plotted as H sin OF versus H, gave a curve very similar to those represented in Fig. 3(b). The peaks of the curves continue to vary systematically down in H sin OF and to lower H with increasing temperature. At each temperature, the peak value of H sin OF may be regarded as the average pinning field per vortex (He), corresponding to the normalized pinning torque (Tp//z) determined before the collective pinning processes set in appreciably at higher H, as discussed earlier. The values thus obtained for H e are plotted versus T/T~ in Fig. 6. Clearly, H e decreases very rapidly from it value at the lowest T/T~ - but when its steep initial descent is fitted closely by a curve varying as (1 T/Tc) 6, we see that the curve falls increasingly below the H e points at higher T/T~. In fact, there appears to be an additional component of H e whose decrease with T/T~ is much slower. Hence, we have taken this component to he linear in 1 - T/T~ (represented by the dotted line in Fig. 6) and then added to it a component varying as (1 - T / T c ) ~ with /3

LM. Obaidatet al. / Physica C 291 (1997)8-16 2

I

I"

tive resemblance to Eq. (15). For the BKBO polycrystal, however, Fig. 7 shows that the new set of points in particular is fitted closely by an expression with only one component, namely (in kOe),

I

"

15

Y B C O cryst=l

1.5

Hp = 0,192(1 - T/Tc) ''8.

(17)

Thus, in the BKBO polycrystal Hp remains moderately temperature dependent down to low T/T~, whereas in the YBCO single crystal and polycrystal an additional component of Hp appears at low T/T~, which decreases much more rapidly with increasing

"I"

"" 0.5

~

e"

"

0

"

.

0

Hp=2.48(1-TIT~ e

.

..... .\

.

i

0.2

~

0.8

.

.

.

.

.

.

0.4

.

.

0.6

.

.

.

.

0.8

...

.

1

TIT ¢

Fig. 6. YBCOcrystal: vortexpinningfield Hp versus T / T c, with experimentaluncertaintiesindicated.Curvesrepresentone-component and two-componentfits. adjustable. In this way, we found the solid curve in the figure, which obeys the expression (in kOe), Hp = 2.02(1 - T/T~) 97 + 0.62(1 - T/Tc),

(15)

and gives a close fit to all the experimental points. This two-component fit to the temperature dependence of the vortex pinning field in the YBCO crystal was quite unexpected. Indeed, it propelled us to reexamine the rotational data obtained previously [7] for polycrystalline YBCO and BKBO and to augment them strategically with new rotational data on the same samples. Again, taking the values of Hp as the maxima of H sin OF from curves similar to those in Fig. 3(b), we have plotted the new and previous sets of values versus T / T c in Fig. 7 (To being 90 K for YBCO and 27 K for BKBO). For the YBCO polycrystal, a fit to both sets of points is obtained, as shown, by the following two-component expression (in kOe): Hp = 0.84(1 - T / T ~ ) 6 + 0 . 1 4 ( 1 - T / T c ) .

(16)

Though weaker in both coefficient and in the large exponent, this expression bears an obvious qualita-

YBCOpolycr~tal

0,6

~'="0.4

(I"TITc)e+O'14(1"T/T¢) °

0.2

o

~

~

..... 0

7,"7 0.2

0.2

0.4

0.6

0.8

1

. , . . . . . , . . . , . . . , . . -

0.15

~

0.05

BKBO polyc~tal

o

0 0

. . . . . . . . . . 0.2 0.4

T,v=

0.6

0.8

1

Fig. 7. (a) YBCO polycrystal,and (b) BKBO polycrystal:vortex pinning field Hp versus T/T~. Closedcirclesfrom presentwork; open circles derived from Ref. [7], Solid curves represent twocomponentfit in (a) and one-componentfit in (b).

16

LM. Obaidat et al, / Physica C 291 (1997) 8-16

temperature. This contrasting behavior suggests that the strongly temperature dependent component of Hp may arise from vortex pinning by the oxygen vacancies that are commonly present in YBCO - but are essentially absent in BKBO. This interesting possibility, which is supported theoretically [21], needs to be tested by further experiments with, for example, an YBCO crystal whose oxygen stoichiometry is varied in stages by appropriate heat treatment [221.

4. Conclusions From rotational magnetic measurements at various fields and temperatures, the pinning torque per vortex (~'p) in the ab-plane of a YBCO crystal is found to have a broad distribution in strength, similar to previous findings for polycrystalline Y B C O and B K B O [6,7]. Simple modeling shows that the 7p distribution function rises linearly with rp and then drops rapidly to zero. With increasing field, the average 7p decreases gradually after rising to a broad peak. The peak value of ~-p//Z (/x being the vortex moment) defines Hp, a characteristic pinning field per vortex, and it is found to decrease with increasing temperature ( T ) in a distinctly two-component manner. One component of Hp falls off very rapidly as (1 - T / T c ) tz with fl ~ 10, while the other component descends linearly to zero as T approaches Tc. A similar two-component behavior of Hp versus T is found in polycrystalline YBCO, whereas in polycrystalline B K B O Hp decreases gradually with increasing T at all temperatures. The strongly temperature dependent Hp component in YBCO may derive from vortex pinning by oxygen vacancies, as suggested theoretically [21].

Acknowledgements W e are grateful to L. Paulius for the YBCO crystal sample of this study and to the National

Science Foundation (DMR) for partial support of this work.

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