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Exponential field dependence of critical current density of underdoped (La1−xSrx)2CuO4 single crystals
PHYSICA ELSEVIER
Physica C 254 (1995) 213-221
Exponential field dependence of critical current density of underdoped (Lal_xSr x) 2CUO4 single crystals T. Kobayashi *'1, T. Kimura, J. Shimoyama, K. Kishio, K. Kitazawa, K. Yamafuji 2 Department of Applied Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan Received 28 February 1995; revised manuscript received 11 September 1995
Abstract
The magnetic-hysteresis curves up to + 9 T of single-crystalline (Lal_xSrx)2CuO 4 (x < 0.061) in the under-doped composition were examined with respect to the dependence of the critical current density J¢ on the internal magnetic-flux distribution B. The exponential model (arc ~x exp(-B/Bo)) could best reproduce the hysteresis curves obtained over a wide field range, while neither the modified Bean model nor the modified Irie-Yamafuji model (J¢(B) = Jco( [ B I/B¢2) v- 1(1 I BI/B¢2) ~) could do so. The exponential dependence of arc on B was discussed in terms of the available models. It was also shown that J~ obtained by fitting to the exponential model did not follow the Kramers scaling law.
1. I n t r o d u c t i o n
There have been many reports describing the exponential field dependence of the critical current density Jc of the high-temperature superconductors (HTSC's). In the case of the polycrystalline sample, this behavior has been attributed to the weak-coupling nature associated with grain boundaries [1]. But a similar behavior has often been reported on single crystals, for example on Y [2] and on La cuprate systems [3], although it has been limited only to the field applied parallel to the c-axis and in the intermediate field range.
* Corresponding author, 1 Present address: Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai 980-77, Japan. 2 Present address: Department of Electronics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812, Japan.
The field dependence of Jc in the bulk material has often been discussed on the basis of the field dependence of the pinning force which is in balance with the Lorentz force acting on a single vortex or on a flux bundle [4-6]. The macroscopic pinningforce density of conventional materials has been frequently represented by Kramers scaling law as Fp ~ bP(1 - - b ) q, (b ~- Bex//Bc2) using the applied external field Bex and the upper critical field Bc2 [7]. However, it is not possible to explain the exponentially decreasing Jc with B ~ in this framework. Hence, the exponential field dependence of Jc of the H T S C ' s suggests a pinning mechanism manifest in these novel superconductors different from the conventional superconductors. The Jc of the H T S C ' s has frequently been determined based on the Bean model [8]. For simplicity, this model averages the change in Jc due to the variation of the magnetic induction over the whole sample. As a result, under a low applied field, where
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Table 1 Parameters describing the field-dependent critical current density according to the exponential model: Jc(B) =,/co, exp(-B/B o) Sample Sr contents Thickness Tc 2K 4.2 K 10 K 20 K # x 2w (K) Jc0 B0 Jc0 B0 J~0 B0 •/cO B0 (mm) (109 m/m 2) (T) 45-80 59-90 61-40 61-94
0.045 0.059 0.061 0.061
0.80 0.90 0.40 0.94
24.8 31.2 32.6 32.6
4.12 -
2.32 -
Jc is observed to change significantly from the surface to the center of the sample, a large error should be involved in the evaluation of J~ [9]. In the previous report using the modified Bean model [2,10] the exponential law appears to hold only in the intermediate field range. It is, therefore, required to reexamine the previous analysis in order to check whether this is only apparent due to the over-simplified approximation of the Bean model or is really intrinsic. In this paper we determined the field dependence of Jc in the following way to avoid the above difficulties. First, we assumed a set of trial functions J~(B) as typical relationship. Then, by calculating the magnetization-hysteresis curves from each of the trial Jc functions, we obtained the best fit to the observed curves by optimizing the parameters in each set of the trial functions to minimize the square sum of the error. The results obtained were compared to those obtained by the Bean model. The distribution of the magnetic induction inside the slab-shaped sample with thickness 2w is represented as Eq. (1), with the aid of the local magnetic induction at the position x ( - w < x _< w), B(x), the external field Box and field-dependent local critical current density Jc(B), considering the symmetry of the sample ( B ( - x ) = B ( x ) ) in the magnetization process. We have w
B ( x ) =Bex -- flxll, XoJc(B(I x I)) d x .
(1)
Then the magnetization is expressed as
M(Be) = -
1
2/z0w
fw{ -
(x)-Box} dx
-1 -
-- [ W
B(x))
dx dx.
(2)
2.44 4.34 9.69 7.86
1.36 1.40 2.67 2.37
. . . . 2.04 0.695 0.404 4.83 1.50 1.52 3.94 1.31 1.07
0.283 0.682 0.608
In this way, the magnetization curve can be calculated provided that the critical current density is given as a function of the field [11]. Although there are many papers calculating the magnetization curves and some are based on an exponentially field-dependent critical current density, none of them referred to (Lal_xSrx)2CuO 4. For example, Ravi et al. calculated the magnetization of a slab [13], Chaddah et al. extended this to a cylinder [14] and Chen et al. showed developments to various forms [15]. In this paper, we examined three models to represent the field dependence of the critical current density in order to reproduce the magnetization curves of under-doped (Lal_xSrx)2CuO 4 single crystals; the modified Bean model, the modified Irie-Yamafuji model [12] and the exponential [3] model [13-15]. (They are defined by Eqs. (3), (4) and (5).) The present La based system has the following advantages compared to other major Y and Bi systems. (1) the demagnetization effect (B II c-axis) can be minimized since large crystals along the c-axis are available; (2) measurements are less subject to the flux-creep effect over a wide temperature range of T / T c because of its lower Tc, and (3) measurements can be made over a wide field range of B/B~z also due to its lower Be2. It has been found that the exponential model can best reproduce the magnetization curves over the whole field range. Therefore, J~ decreases exponentially with field from zero and the deviation from the exponential dependence of Jc which was reported in previous reports should be attributed to artifacts caused by application of the Bean model to samples which are large enough to ignore the variation of the magnetic induction inside the sample at a low field.
T. Kobayashi et al./ Physica C 254 (1995) 213-221
215
2. Experimental
3. Results
The samples used were (Lal_xSrx)2CuO 4 single crystals grown by a TSFZ method [3]. They were cut into slab shapes with the longest edge along the c-axis to minimize the demagnetization coefficient, 3( [I c-axis) × 2 × 2w (2w < 0.94) mm 3. The field was applied along the c-axis to obtain the critical current within the CuO 2 plane. Four samples of different Sr contents in the under-doped region and of different thicknesses were employed as shown in the Table 1. The cation compositions were determined by inductively coupled plasma atomic spectrometry on the solution of the neighboring part of the measured sample. The magnetization was measured by a vibrating-sample magnetometer, E G & G PAR model 4500 equipped with a Janis 9T superconducting magnet and cryostat. The field was swept over the range of + 9 T at a rate of 20 m T / s (maximum) which was rather fast in order to minimize the effect of flux creep. The magnetization-hysteresis carves we used for the analysis were taken after a field was applied high enough to achieve the full flux penetration.
We used the following three models to attempt to reproduce the observed magnetization curves: (1) the modified Bean model,
J~(B) =Jc(B:x) = AM(Bex)/W; (2) the modified Irie-Yamafuji model, Jc(B) =Jco([B[/Sc2)~'-l(1 - [Bl/Bc2)~;
(4)
(3) the exponential model, J ~ ( B ) =J~0 e x p ( - I B l/B0).
(5)
It is worthwhile to note that in the model (1) the critical current is determined only by the external field Bex and assumed to be constant through the sample in spite of the change in the local magnetic induction. On the other hand, the critical current is a function of the local magnetic induction in the models (2) and (3). In the modified Bean model, arc is directly obtained from the magnetization curves, following Eq. (3). Then, using Eq. (2), we obtain the magnetization curves for comparison.
I
1.20
[\'---(c)I-Y model 0.80 (a)observed---~1~/ Ay
0.40
~
(3)
A
,~
#59-90
(b)Beanmodel
' ~ ', t '~(d)
exponential model
0.00 -0.40
"~
\\x\\ -0.80
--1.20
"
28
26
24
--'2
IC/ oi Box[T]
Fig. 1. Observedand calculated magnetization-hysteresis-curvesof a (La~_xSrx)2CuO4 slab single crystal (x = 0.059, thickness 2w = 0.9 mm, H [[c, 4.2 K): (a) observed, (b) calculated from the modified Bean model, (c) calculated from Jc(B)=Jco([BI/Bc2)~'-l(1[B]/B~2) s (modified Irie-Yamafuji model with Jco =6.8 × 109 A/m a, Bo2 = 17 T, 7=1.4 and 6= 3.9) and (d) calculated from Jc(B) =arcs exp(--B/Bo) (exponential model with Jco = 4.34 ><109 A/m 2 and Bo = 1.40 T).
216
T. Kobayashi et al. / P h y s i c a C 254 (1995) 2 1 3 - 2 2 1
•59-90 (b)I-Ymodel
(c)e~cponential e~cponentialmodel ~ 21 \ o
4.2K
0
2
4
B~=[T]
1 B~.g'l
6
8
Fig. 2. Critical current density as a function of the magnetic induction, calculated for Fig. 1 according to (a) the modified Bean model, (b) the modified Irie-Yamafuji model (Jo(B) = Jeo(I B I/Bc2)z'- 1(1 - I B I/Be2) ~), and (c) the exponentialmodel (Jc(B) = J~o exp(- B / B o ) ) .
In the modified Irie-Yamafuji model, J¢ is expressed by three parameters J¢0, T and 6. Here, we also treated Bc2 as a fitting parameter. In total there are four fitting parameters in the modified IrieYamafuji model, while in the exponential model only two are included, ./co and B 0. The best fit parameters were determined for the three models by fitting the calculated magnetization-hysteresis curves to the observed ones. The quality of the fitting was evaluated by the square sum of the error. Fig. 1 compares the magnetization-hysteresis curves thus obtained by the above three models and the observed one for sample #59-90 (x = 0.059). The Jc values obtained from each of the models by the fitting are shown in Fig. 2. Comparing the observed and calculated magnetization curves in Fig. 1, it is understood that the modified Bean model raises a significant discrepancy especially in the low-field range. The shift experimentally observed in the position of the magnetization peaks from the zero field cannot be explained by the modified Bean model. The magnetization curve calculated from the modified Bean model is in principle symmetrical against the axes Bex = 0 and M = 0. This symmetry leads to the same magnetization at the same external field, irrespective to whether the field is increasing or decreasing. This is just as expected because the critical current expected for the modified Bean model is a function only of the
external field, as shown in Eq. (3). Thus the modified Bean model approximation fails in the low-field region and hence the J¢ value obtained by this model is subject to a large error in the low-field range. In the high-field range ( > 3 T) in Fig. 1, however, the modified Bean model is shown to approximate the observed magnetization curve very well. Although the modified Irie-Yamafuji model can reproduce the shift in the position of the magnetization peak from the zero field, reflecting a local magnetic induction-dependent critical current density, it cannot reproduce the whole shape of the magnetization curves. This model tends to give a sharp peak in the low-field region and a less steep decrease of the magnetization curve in the high-field region as is seen in Fig. 1. On the other hand, the exponential model can reproduce the observed magnetization curves excellently over the whole experimental field range, and for all the samples at different temperatures. The fitting parameters, J~0 and B o are given in Table 1. Since the exponential model gave the best fit to the observed magnetization it was concluded that Jc of underdoped (Lal_xSrx)2CuO 4 was described by Eq. (5), the exponential model. However, the application of this model should be limited to the low- to intermediate-field ranges. This is because the exponential model assumes the fitted J~ value to remain finite even at the field above Bir r o r Be2 , which is physically unreasonable. In the present fitting procedure, the error is evaluated on the linear scale of the magnetization and hence the error in the largest magnetic field is not apparent because the value is small in the highest field region. From the present systematic study on the underdoped (La 1_ ixSrx)2CuO4 single crystals of different Sr contents, x = 0.045, 0.059 and 0.061, J~ can be concluded to decrease exponentially with the field. From Table 1 it is noticed that both of the two parameters Jc0 and B 0 increase with an increase in the Sr contents. Both Jc0 and B 0 decrease with temperature as expected. On the other hand, in the over-doped region (x_> 0.075) as reported previously by the present authors, there appears a second peak in the magnetization-hysteresis curve [3,16]. Therefore the field dependence of the critical current is complicated in the over-doped region and hence
217
T. Kobayashi et al. /Physica C 254 (1995) 213-221
the magnetization behavior should be examined independently.
3 ==:zZ2Zzz==
4. Discussion
2
/.-@..
(b)
_~.z-"......------.~'~.,~ JG----J -"'---Q~..
4.1. Application of the modified Bean model The results and fitting analysis as given above have shown: (1) the exponential model can best describe the magnetization curves over the entire experimental range, (2) the modified Bean model can only be applicable in the high field range, and (3) the modified Irie-Yamafuji model, which is known to be applicable to the field dependence of the critical current in many of the conventional superconductors, cannot reproduce the magnetization curve of the HTSC such as in the present case. As seen in Fig. 2, Jo obtained from the modified Bean and the exponential models agree at higher fields. As already reported, the Bean model ignores the change of the critical current density caused by the change of the local magnetic induction in the sample. Hence it causes a large error at lower fields [9]. The approximation by the modified Bean model seems to become better as the field increases because the spatial change in the critical current inside the sample becomes smaller. However, it will be understood that the agreement of the two sets of the estimated Jc value, is just apparent and fortuitous. The following argument can be deduced based on the calculation of the magnetic-induction distribution inside the sample using Eqs. (3), (5) and (1). Fig. 3 shows the magnetic-induction distribution curves calculated inside the sample in the cyclic field according to the exponential model (solid line) and the modified Bean model (dotted line). In this figure the gradient of the B(x) curve represents Jc and the broken lines indicate the tangent of the solid curves, respectively, at the sample edges x - w and x = - w, corresponding to Jc(Bex) according to the exponential model. The areas surrounded by the two B(x) curves in the figure are taken to be the same and should be equal to the width of the magnetization curve AM at Bex in Fig. 1 for the two models. The Jc value for the modified Bean model which is
0
1
,
\\ "x
J
~
[
" ~ /
-w
0 x
model......
//J #59-90
w
Fig. 3. Distribution of the magnetic induction inside the sample of Fig. 1, calculated from the exponential model (solid line) and modified Bean model (dotted line) at the external field of (a) 0, (b) 1.3, (c) 3 T. The broken line indicates the tangent at the sample surface for the curve of the exponential model.
constant in the sample (dotted line) is therefore equal to the averaged value of Jc for the exponential model over the entire sample. Fig. 3(a) is for the case when the external field is zero in the cyclic field process. For the exponential model Jc decreases monotonically from surface to center because of the increasing remanent magnetization toward the center. In the modified Bean model arc, being equal to the average of that of exponential model, is lower at the surface. This explains why the modified Bean model underestimates the Jc value near Bex -- 0. Fig. 3(b) is for the case when the external field is elevated but there remains still a region where B ( x ) = 0 near the center of the sample. In this case Jc is higher inside than at the surface of the sample. Then Jc of the modified Bean model, the averaged Jc, may give a higher value than that of the exponential
218
T. Kobayashi et aL / P h y s i c a C 254 (1995) 213-221
model due to the higher J~ inside the sample, explaining the intermediate field region in Fig. 2 where the modified Bean model curve overshoots the exponential model curve. Fig. 3(c) is for the case when the external field is high enough so that the magnetic induction at the center is also high. For increasing field cycle, B(__+w) > B(0) and hence Jc(___w) < J~(0). For decreasing field cycle on the contrary, B(__+w) < B(0) and hence J~( _ w) > Jc(0). In the application of the modified Bean model one uses the difference in the magnetization AM between the increasing- and decreasing-field cycles. Hence, this procedure cancels the error caused by the averaging as can be understood from Fig. 3(b) or (c). At high fields there still remains some variation of arc in the sample, but it is understood that the errors fortunately cancel in the increasing and decreasing processes of the field for the modified Bean method by taking the difference of magnetization, AM. Therefore, when one applies the modified Bean model to the magnetization curve which possess the peaking structure near zero field, it should be noticed that the error in the estimation of J~ is amplified in the low-field region by the procedure taking the difference AM of the magnetization between the field increasing and decreasing process. Because this procedure has been widely adopted in the HTSC's as well as in the conventional superconductors, we should emphasize here the necessity of reexaminations on the studies performed so far based on the analysis of the magnetization data of HTSC. As one of the examples, it has been regarded that J~ seems not to be following the exponential law in a low field. But it is now clear that this has been due to the misuse of the modified Bean analysis in this range; in fact the exponential dependence extends essentially to zero field.
multi-fine filament (NbTi) embedded in non-superconducting matrix (CuNi) [18]. The origin of the exponential decay of Jc with magnetic field is still controversial but there are two types of models explaining this phenomenon: one is to consider the field-dependent flux pinning such as a change of the pinning potential or softening of the flux line, c44 , and the other attributing the phenomenon to the involvement of SNS junction. Nojima et al. reported the exponentially decaying Jc of c-axis oriented thin films of YBa2Cu30 7 and Bi2Sr2CaCu20 8 at rather high temperatures and fields well below the irreversibility field [10]. They attributed the exponential field dependence of arc to the spatial dependences of the pinning potential U on field B and on current density J as reported by Zeldov et al. from the transport measurement on thin films [19]. They expressed the temperature dependence of B 0 in Eq. (5) to be Bo(T) = a B i r r ( T ) , where Bitr is the field when Jc becomes zero and a is a constant. From the present study, it was not possible to determine Bir r with high enough precision so as to discuss the temperature dependence of a = Bo(T)/Birr(T). But we have observed the same current and field dependences [20] as that of Zeldov et al. by the magnetization-relaxation method on the same samples used in this study. Hence, this model should be one of the potential candidates to explain the exponential decay of Jc, although it is a phenomenological approach. Yamafuji et al. presented a microscopic model to explain the exponential decay of Jc for a quasi-2D system in which the correlation length of the vortex b e n d i n g 144 is confined to the order of the penetration length A owing to the layered structure [21]. In this theory, B 0 is approximately expressed as
Bo(T ) ~ TK 2x/-2~r 4.2. Mechanism for the exponential dependence of J~ on the field An exponential decrease of J¢ with field has been reported not only in poly- and single crystals of HTSC but also in the Y / Y + Pr superstructure thin film which is composed of superconductors with two different T~'s [17]. It has also been reported in a
df
t c/1
cl 1 lP]" ] '
where the numerical factor y is &Ldp/OtLd i and is expected to be in the order of 10 -1 for the weak-pinning case. Taking the G - L parameter K = 85 ~ 119, the coherence length ~ab(4.2 K) -~ tab(O) = 3.2 ~ 3.8 nm as well as B~2 = 22 ~ 31 T [22], Ac = 104 nm [23], d e = 0.66 nm (c-axis lattice spacing), B 0 = 2.4 T in Table 1 for x = 0.061, the value 3' becomes of the order of 10 -3 ~ 1 0 - 2 if ~xc ///pin ~ 4 4 ~ 1 is assumed.
T. Kobayashi et al. /Physica C 254 (1995) 213-221
For a precise calculation of % the pinning center must be identified. This order of magnitude estimation therefore leaves the Yamafuji et al. model as a candidate to microscopically explain the exponential decay of J~ with field. But the model has not been developed to examine the dependences of U on B or on J. It has been known that (Lal_xSrx)2CklO 4 becomes more strongly anisotropic as x decreases [3], suggesting a weaker coupling between CuO 2 layers. We can regard it as quasi-two-dimensional and suitable for application of the Yamafuji et al. model. On the other hand, Senoussi et al. observing a similar exponential decrease of the magnetically obtained Jc with a YBa2Cu30 7 single crystal, suggested the involvement of a microstructural defect such as dislocations to determine Jc [2]. Grujic et al. presented a model based on the SNS junction consisting of a normal metal of effective thickness d embedded in a type-II superconductor under a field applied perpendicular to the junction to explain the exponential decay of J¢ with field [24]. From the present experiment B o has been known to decrease with temperature. According to their SNS junction model the temperature dependence of B o was attributed to the temperature dependence of d as B 0 = q ~ 0 / 2 v d 2, where d is assumed to increase with temperature in a qualitative agreement with the present experiment. From the results in Table 1, B o of sample #61-94 is 2.37 and 0.609 T at 4.2 and 20 K, respectively, resulting in d = 11.6 and 22.9 rim. Therefore if the SNS model [24] applies to this case, there must be normal regions of thickness several times of the coherence length ~ab present in the single-crystalline (Lal_xSr,)2CuO 4. This seems to require the material to be inhomogeneous in the Sr distribution or in the oxygen content, suggesting the single crystal to possess the granular nature of the superconducting particles. As discussed above, there are essentially two microscopic models which may explain the exponential dependence of J~ on field; one is to assume the softening of the vortex line due to the quasi-two-dimensional structure and the other is to assume the presence of the granular SNS junctions. Both of the models need to be further modified to explain the more detailed behavior in order to identify which of the models is to hold.
219
4.3. Pinning force density and the Kramers scaling law Finally we will examine whether the Kramers scaling law [7] can be applied to the HTSC which was successful in describing the pinning-force density as a function of field for many conventional superconductors and has been conveniently used to classify the pinning mechanism in them. The macroscopic pinning-force density Fp = Jc B is expressed as follows, using the local current density Jc(B) and the magnetic induction B(x):
l~j
Fp=wJo
c ( B ( x ) ) B ( x ) dx.
(6)
On the other hand, applying the Kramers scaling to a slab-like shaped conventional superconductors, although ignoring the change of magnetic induction or the critical current inside the sample, that scaling works well. Here, in comparison with Kramers scaling, we approximate Eq. (6) and use the field dependence of under-doped (La 1 xSrx)2CuO4,represented by Eq. (5) to obtain Eq. (7) (our approximation is justified at a higher field as discussed in Section 4.1; the variation in the critical current inside the sample is relatively smaller):
Fp = Jco B e x p ( - B / B o ) .
(7)
This equation shows that the Kramers scaling law of max O( b 2 ( 1 - b) q should not be able to describe the Fp-B relationship for the present material. But in case if b << 1 then
Fp//gp
Fp/Fp max c~ B / B o e x p ( - B / B o )
(8)
- b(1 - b),
(9)
for b << 1,
it can approximately describe the present case, taking p = 1 and q = 1. However, there have been several attempts made, for Jc of the HTSC's obtained by the modified Bean model plotted to apply the Kramers model and the scaling parameters have been deduced, e.g., p = 1, q = 4 [25] and p = 2, q = 4 [26]. But the arguments made in Section 4.1 have indicated that the modified Bean model gives ate and hence Fp with a larger error as the field is reduced for the HTSC. Therefore it may have been just fortuitous that the Kramers scaling law has been
220
T. Kobayashi et al. /Physica C 254 (1995) 213-221 1.0 0.8
-....
0.6 \
0.4
\\ \\
0.2
\\
2
3
4
5
6
7
8
1.0 (b) 0.8 O.6 0.4 0.2
4 ~o °° /
model smaller. Consequently, we conclude that the pinning-force density Fp does not follow the Kramers scaling law in the HTSC at least in the under-doped region. In summary, it has been shown that the field dependence of the critical current density of underdoped (Lal_xSrx)2CuO 4 single crystal under a field applied parallel to the c-axis is well represented by the exponential model over a wide field range from zero to several T. The exponential model explains the shift in the peaks of the magnetization curve from zero field quantitatively. For materials as in the present case in which the critical current density changes sharply with field, it has been shown that the error associated with the estimated Jc from the modified Bean model becomes greater as the field is reduced. It has also been shown that the Kramers scaling law does not hold for the pinning-force density of the HTSC's in the under-doped region.
/1
/// /
0.2
0.4
0.6
0.8
1.0
References
Box/Bmo~ Fig. 4. Normalized macroscopic pinning-force density F p / F p max of thin (2w = 0.4 ram) and thick (2w = 0.94 mm) slab singlecrystal samples ( x = 0.061) (a) and its magnification of the low-field region (b).
reported to fit the Fp-B curves obtained from the Bean approximation. Fig. 4 shows the Fp-B¢x relation as normalized by the maximum Fp, Fp max for the samples of two different thicknesses. Fig. 4(a) illustrates the entire field range and Fig. 4(b) for the low-field range. Comparing the two figures, it is understood that the tendency of underestimating J~ or Fp by the modified Bean model in a low field happens to contribute to the apparent agreement of the calculated Fp-B relation with the Kramers curve. The degree of this underestimation becomes greater for a thicker sample as is seen in the figure, Although the present results do not seem to fit the Kramers scaling law even by following the modified Bean model, it is clearly understandable that the error by using this model makes the deviation from the Kramers scaling
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[22] Q. Li, M. Suenaga, T. Kimura and K. Kishio, Phys. Rev. B 47 (1993) 2854. [23] T. Shibauchi, H. Kitano, K. Uchinokura, A. Maeda, T. Kimnra and K. Kishio, Phys. Rev. Lett. 72 (1994) 2263. [24] L. Dobrosavljevi6-Gruji6 and Z. Radovi6, Physica C 185-189 (1991) 2313. [25] K. Kishio, Y. Nakayama, T. Kobayashi, T. Noda, T. Kimura, K. Kitazawa and K. Yamafuji, Physica C 185-189 (1991) 2523. [26] J.N. Li, F.R. De Boer, L.W. Roeland, M.J.V. Menken, K. Kadowaki, A.A. Menovsky, J.J.M. Franse and P.H. Kes, Physica C 169 (1990) 81.
Physica C 254 (1995) 213-221
Exponential field dependence of critical current density of underdoped (Lal_xSr x) 2CUO4 single crystals T. Kobayashi *'1, T. Kimura, J. Shimoyama, K. Kishio, K. Kitazawa, K. Yamafuji 2 Department of Applied Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan Received 28 February 1995; revised manuscript received 11 September 1995
Abstract
The magnetic-hysteresis curves up to + 9 T of single-crystalline (Lal_xSrx)2CuO 4 (x < 0.061) in the under-doped composition were examined with respect to the dependence of the critical current density J¢ on the internal magnetic-flux distribution B. The exponential model (arc ~x exp(-B/Bo)) could best reproduce the hysteresis curves obtained over a wide field range, while neither the modified Bean model nor the modified Irie-Yamafuji model (J¢(B) = Jco( [ B I/B¢2) v- 1(1 I BI/B¢2) ~) could do so. The exponential dependence of arc on B was discussed in terms of the available models. It was also shown that J~ obtained by fitting to the exponential model did not follow the Kramers scaling law.
1. I n t r o d u c t i o n
There have been many reports describing the exponential field dependence of the critical current density Jc of the high-temperature superconductors (HTSC's). In the case of the polycrystalline sample, this behavior has been attributed to the weak-coupling nature associated with grain boundaries [1]. But a similar behavior has often been reported on single crystals, for example on Y [2] and on La cuprate systems [3], although it has been limited only to the field applied parallel to the c-axis and in the intermediate field range.
* Corresponding author, 1 Present address: Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai 980-77, Japan. 2 Present address: Department of Electronics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812, Japan.
The field dependence of Jc in the bulk material has often been discussed on the basis of the field dependence of the pinning force which is in balance with the Lorentz force acting on a single vortex or on a flux bundle [4-6]. The macroscopic pinningforce density of conventional materials has been frequently represented by Kramers scaling law as Fp ~ bP(1 - - b ) q, (b ~- Bex//Bc2) using the applied external field Bex and the upper critical field Bc2 [7]. However, it is not possible to explain the exponentially decreasing Jc with B ~ in this framework. Hence, the exponential field dependence of Jc of the H T S C ' s suggests a pinning mechanism manifest in these novel superconductors different from the conventional superconductors. The Jc of the H T S C ' s has frequently been determined based on the Bean model [8]. For simplicity, this model averages the change in Jc due to the variation of the magnetic induction over the whole sample. As a result, under a low applied field, where
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T. Kobayashiet al./Physica C 254 (1995)213-221
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Table 1 Parameters describing the field-dependent critical current density according to the exponential model: Jc(B) =,/co, exp(-B/B o) Sample Sr contents Thickness Tc 2K 4.2 K 10 K 20 K # x 2w (K) Jc0 B0 Jc0 B0 J~0 B0 •/cO B0 (mm) (109 m/m 2) (T) 45-80 59-90 61-40 61-94
0.045 0.059 0.061 0.061
0.80 0.90 0.40 0.94
24.8 31.2 32.6 32.6
4.12 -
2.32 -
Jc is observed to change significantly from the surface to the center of the sample, a large error should be involved in the evaluation of J~ [9]. In the previous report using the modified Bean model [2,10] the exponential law appears to hold only in the intermediate field range. It is, therefore, required to reexamine the previous analysis in order to check whether this is only apparent due to the over-simplified approximation of the Bean model or is really intrinsic. In this paper we determined the field dependence of Jc in the following way to avoid the above difficulties. First, we assumed a set of trial functions J~(B) as typical relationship. Then, by calculating the magnetization-hysteresis curves from each of the trial Jc functions, we obtained the best fit to the observed curves by optimizing the parameters in each set of the trial functions to minimize the square sum of the error. The results obtained were compared to those obtained by the Bean model. The distribution of the magnetic induction inside the slab-shaped sample with thickness 2w is represented as Eq. (1), with the aid of the local magnetic induction at the position x ( - w < x _< w), B(x), the external field Box and field-dependent local critical current density Jc(B), considering the symmetry of the sample ( B ( - x ) = B ( x ) ) in the magnetization process. We have w
B ( x ) =Bex -- flxll, XoJc(B(I x I)) d x .
(1)
Then the magnetization is expressed as
M(Be) = -
1
2/z0w
fw{ -
(x)-Box} dx
-1 -
-- [ W
B(x))
dx dx.
(2)
2.44 4.34 9.69 7.86
1.36 1.40 2.67 2.37
. . . . 2.04 0.695 0.404 4.83 1.50 1.52 3.94 1.31 1.07
0.283 0.682 0.608
In this way, the magnetization curve can be calculated provided that the critical current density is given as a function of the field [11]. Although there are many papers calculating the magnetization curves and some are based on an exponentially field-dependent critical current density, none of them referred to (Lal_xSrx)2CuO 4. For example, Ravi et al. calculated the magnetization of a slab [13], Chaddah et al. extended this to a cylinder [14] and Chen et al. showed developments to various forms [15]. In this paper, we examined three models to represent the field dependence of the critical current density in order to reproduce the magnetization curves of under-doped (Lal_xSrx)2CuO 4 single crystals; the modified Bean model, the modified Irie-Yamafuji model [12] and the exponential [3] model [13-15]. (They are defined by Eqs. (3), (4) and (5).) The present La based system has the following advantages compared to other major Y and Bi systems. (1) the demagnetization effect (B II c-axis) can be minimized since large crystals along the c-axis are available; (2) measurements are less subject to the flux-creep effect over a wide temperature range of T / T c because of its lower Tc, and (3) measurements can be made over a wide field range of B/B~z also due to its lower Be2. It has been found that the exponential model can best reproduce the magnetization curves over the whole field range. Therefore, J~ decreases exponentially with field from zero and the deviation from the exponential dependence of Jc which was reported in previous reports should be attributed to artifacts caused by application of the Bean model to samples which are large enough to ignore the variation of the magnetic induction inside the sample at a low field.
T. Kobayashi et al./ Physica C 254 (1995) 213-221
215
2. Experimental
3. Results
The samples used were (Lal_xSrx)2CuO 4 single crystals grown by a TSFZ method [3]. They were cut into slab shapes with the longest edge along the c-axis to minimize the demagnetization coefficient, 3( [I c-axis) × 2 × 2w (2w < 0.94) mm 3. The field was applied along the c-axis to obtain the critical current within the CuO 2 plane. Four samples of different Sr contents in the under-doped region and of different thicknesses were employed as shown in the Table 1. The cation compositions were determined by inductively coupled plasma atomic spectrometry on the solution of the neighboring part of the measured sample. The magnetization was measured by a vibrating-sample magnetometer, E G & G PAR model 4500 equipped with a Janis 9T superconducting magnet and cryostat. The field was swept over the range of + 9 T at a rate of 20 m T / s (maximum) which was rather fast in order to minimize the effect of flux creep. The magnetization-hysteresis carves we used for the analysis were taken after a field was applied high enough to achieve the full flux penetration.
We used the following three models to attempt to reproduce the observed magnetization curves: (1) the modified Bean model,
J~(B) =Jc(B:x) = AM(Bex)/W; (2) the modified Irie-Yamafuji model, Jc(B) =Jco([B[/Sc2)~'-l(1 - [Bl/Bc2)~;
(4)
(3) the exponential model, J ~ ( B ) =J~0 e x p ( - I B l/B0).
(5)
It is worthwhile to note that in the model (1) the critical current is determined only by the external field Bex and assumed to be constant through the sample in spite of the change in the local magnetic induction. On the other hand, the critical current is a function of the local magnetic induction in the models (2) and (3). In the modified Bean model, arc is directly obtained from the magnetization curves, following Eq. (3). Then, using Eq. (2), we obtain the magnetization curves for comparison.
I
1.20
[\'---(c)I-Y model 0.80 (a)observed---~1~/ Ay
0.40
~
(3)
A
,~
#59-90
(b)Beanmodel
' ~ ', t '~(d)
exponential model
0.00 -0.40
"~
\\x\\ -0.80
--1.20
"
28
26
24
--'2
IC/ oi Box[T]
Fig. 1. Observedand calculated magnetization-hysteresis-curvesof a (La~_xSrx)2CuO4 slab single crystal (x = 0.059, thickness 2w = 0.9 mm, H [[c, 4.2 K): (a) observed, (b) calculated from the modified Bean model, (c) calculated from Jc(B)=Jco([BI/Bc2)~'-l(1[B]/B~2) s (modified Irie-Yamafuji model with Jco =6.8 × 109 A/m a, Bo2 = 17 T, 7=1.4 and 6= 3.9) and (d) calculated from Jc(B) =arcs exp(--B/Bo) (exponential model with Jco = 4.34 ><109 A/m 2 and Bo = 1.40 T).
216
T. Kobayashi et al. / P h y s i c a C 254 (1995) 2 1 3 - 2 2 1
•59-90 (b)I-Ymodel
(c)e~cponential e~cponentialmodel ~ 21 \ o
4.2K
0
2
4
B~=[T]
1 B~.g'l
6
8
Fig. 2. Critical current density as a function of the magnetic induction, calculated for Fig. 1 according to (a) the modified Bean model, (b) the modified Irie-Yamafuji model (Jo(B) = Jeo(I B I/Bc2)z'- 1(1 - I B I/Be2) ~), and (c) the exponentialmodel (Jc(B) = J~o exp(- B / B o ) ) .
In the modified Irie-Yamafuji model, J¢ is expressed by three parameters J¢0, T and 6. Here, we also treated Bc2 as a fitting parameter. In total there are four fitting parameters in the modified IrieYamafuji model, while in the exponential model only two are included, ./co and B 0. The best fit parameters were determined for the three models by fitting the calculated magnetization-hysteresis curves to the observed ones. The quality of the fitting was evaluated by the square sum of the error. Fig. 1 compares the magnetization-hysteresis curves thus obtained by the above three models and the observed one for sample #59-90 (x = 0.059). The Jc values obtained from each of the models by the fitting are shown in Fig. 2. Comparing the observed and calculated magnetization curves in Fig. 1, it is understood that the modified Bean model raises a significant discrepancy especially in the low-field range. The shift experimentally observed in the position of the magnetization peaks from the zero field cannot be explained by the modified Bean model. The magnetization curve calculated from the modified Bean model is in principle symmetrical against the axes Bex = 0 and M = 0. This symmetry leads to the same magnetization at the same external field, irrespective to whether the field is increasing or decreasing. This is just as expected because the critical current expected for the modified Bean model is a function only of the
external field, as shown in Eq. (3). Thus the modified Bean model approximation fails in the low-field region and hence the J¢ value obtained by this model is subject to a large error in the low-field range. In the high-field range ( > 3 T) in Fig. 1, however, the modified Bean model is shown to approximate the observed magnetization curve very well. Although the modified Irie-Yamafuji model can reproduce the shift in the position of the magnetization peak from the zero field, reflecting a local magnetic induction-dependent critical current density, it cannot reproduce the whole shape of the magnetization curves. This model tends to give a sharp peak in the low-field region and a less steep decrease of the magnetization curve in the high-field region as is seen in Fig. 1. On the other hand, the exponential model can reproduce the observed magnetization curves excellently over the whole experimental field range, and for all the samples at different temperatures. The fitting parameters, J~0 and B o are given in Table 1. Since the exponential model gave the best fit to the observed magnetization it was concluded that Jc of underdoped (Lal_xSrx)2CuO 4 was described by Eq. (5), the exponential model. However, the application of this model should be limited to the low- to intermediate-field ranges. This is because the exponential model assumes the fitted J~ value to remain finite even at the field above Bir r o r Be2 , which is physically unreasonable. In the present fitting procedure, the error is evaluated on the linear scale of the magnetization and hence the error in the largest magnetic field is not apparent because the value is small in the highest field region. From the present systematic study on the underdoped (La 1_ ixSrx)2CuO4 single crystals of different Sr contents, x = 0.045, 0.059 and 0.061, J~ can be concluded to decrease exponentially with the field. From Table 1 it is noticed that both of the two parameters Jc0 and B 0 increase with an increase in the Sr contents. Both Jc0 and B 0 decrease with temperature as expected. On the other hand, in the over-doped region (x_> 0.075) as reported previously by the present authors, there appears a second peak in the magnetization-hysteresis curve [3,16]. Therefore the field dependence of the critical current is complicated in the over-doped region and hence
217
T. Kobayashi et al. /Physica C 254 (1995) 213-221
the magnetization behavior should be examined independently.
3 ==:zZ2Zzz==
4. Discussion
2
/.-@..
(b)
_~.z-"......------.~'~.,~ JG----J -"'---Q~..
4.1. Application of the modified Bean model The results and fitting analysis as given above have shown: (1) the exponential model can best describe the magnetization curves over the entire experimental range, (2) the modified Bean model can only be applicable in the high field range, and (3) the modified Irie-Yamafuji model, which is known to be applicable to the field dependence of the critical current in many of the conventional superconductors, cannot reproduce the magnetization curve of the HTSC such as in the present case. As seen in Fig. 2, Jo obtained from the modified Bean and the exponential models agree at higher fields. As already reported, the Bean model ignores the change of the critical current density caused by the change of the local magnetic induction in the sample. Hence it causes a large error at lower fields [9]. The approximation by the modified Bean model seems to become better as the field increases because the spatial change in the critical current inside the sample becomes smaller. However, it will be understood that the agreement of the two sets of the estimated Jc value, is just apparent and fortuitous. The following argument can be deduced based on the calculation of the magnetic-induction distribution inside the sample using Eqs. (3), (5) and (1). Fig. 3 shows the magnetic-induction distribution curves calculated inside the sample in the cyclic field according to the exponential model (solid line) and the modified Bean model (dotted line). In this figure the gradient of the B(x) curve represents Jc and the broken lines indicate the tangent of the solid curves, respectively, at the sample edges x - w and x = - w, corresponding to Jc(Bex) according to the exponential model. The areas surrounded by the two B(x) curves in the figure are taken to be the same and should be equal to the width of the magnetization curve AM at Bex in Fig. 1 for the two models. The Jc value for the modified Bean model which is
0
1
,
\\ "x
J
~
[
" ~ /
-w
0 x
model......
//J #59-90
w
Fig. 3. Distribution of the magnetic induction inside the sample of Fig. 1, calculated from the exponential model (solid line) and modified Bean model (dotted line) at the external field of (a) 0, (b) 1.3, (c) 3 T. The broken line indicates the tangent at the sample surface for the curve of the exponential model.
constant in the sample (dotted line) is therefore equal to the averaged value of Jc for the exponential model over the entire sample. Fig. 3(a) is for the case when the external field is zero in the cyclic field process. For the exponential model Jc decreases monotonically from surface to center because of the increasing remanent magnetization toward the center. In the modified Bean model arc, being equal to the average of that of exponential model, is lower at the surface. This explains why the modified Bean model underestimates the Jc value near Bex -- 0. Fig. 3(b) is for the case when the external field is elevated but there remains still a region where B ( x ) = 0 near the center of the sample. In this case Jc is higher inside than at the surface of the sample. Then Jc of the modified Bean model, the averaged Jc, may give a higher value than that of the exponential
218
T. Kobayashi et aL / P h y s i c a C 254 (1995) 213-221
model due to the higher J~ inside the sample, explaining the intermediate field region in Fig. 2 where the modified Bean model curve overshoots the exponential model curve. Fig. 3(c) is for the case when the external field is high enough so that the magnetic induction at the center is also high. For increasing field cycle, B(__+w) > B(0) and hence Jc(___w) < J~(0). For decreasing field cycle on the contrary, B(__+w) < B(0) and hence J~( _ w) > Jc(0). In the application of the modified Bean model one uses the difference in the magnetization AM between the increasing- and decreasing-field cycles. Hence, this procedure cancels the error caused by the averaging as can be understood from Fig. 3(b) or (c). At high fields there still remains some variation of arc in the sample, but it is understood that the errors fortunately cancel in the increasing and decreasing processes of the field for the modified Bean method by taking the difference of magnetization, AM. Therefore, when one applies the modified Bean model to the magnetization curve which possess the peaking structure near zero field, it should be noticed that the error in the estimation of J~ is amplified in the low-field region by the procedure taking the difference AM of the magnetization between the field increasing and decreasing process. Because this procedure has been widely adopted in the HTSC's as well as in the conventional superconductors, we should emphasize here the necessity of reexaminations on the studies performed so far based on the analysis of the magnetization data of HTSC. As one of the examples, it has been regarded that J~ seems not to be following the exponential law in a low field. But it is now clear that this has been due to the misuse of the modified Bean analysis in this range; in fact the exponential dependence extends essentially to zero field.
multi-fine filament (NbTi) embedded in non-superconducting matrix (CuNi) [18]. The origin of the exponential decay of Jc with magnetic field is still controversial but there are two types of models explaining this phenomenon: one is to consider the field-dependent flux pinning such as a change of the pinning potential or softening of the flux line, c44 , and the other attributing the phenomenon to the involvement of SNS junction. Nojima et al. reported the exponentially decaying Jc of c-axis oriented thin films of YBa2Cu30 7 and Bi2Sr2CaCu20 8 at rather high temperatures and fields well below the irreversibility field [10]. They attributed the exponential field dependence of arc to the spatial dependences of the pinning potential U on field B and on current density J as reported by Zeldov et al. from the transport measurement on thin films [19]. They expressed the temperature dependence of B 0 in Eq. (5) to be Bo(T) = a B i r r ( T ) , where Bitr is the field when Jc becomes zero and a is a constant. From the present study, it was not possible to determine Bir r with high enough precision so as to discuss the temperature dependence of a = Bo(T)/Birr(T). But we have observed the same current and field dependences [20] as that of Zeldov et al. by the magnetization-relaxation method on the same samples used in this study. Hence, this model should be one of the potential candidates to explain the exponential decay of Jc, although it is a phenomenological approach. Yamafuji et al. presented a microscopic model to explain the exponential decay of Jc for a quasi-2D system in which the correlation length of the vortex b e n d i n g 144 is confined to the order of the penetration length A owing to the layered structure [21]. In this theory, B 0 is approximately expressed as
Bo(T ) ~ TK 2x/-2~r 4.2. Mechanism for the exponential dependence of J~ on the field An exponential decrease of J¢ with field has been reported not only in poly- and single crystals of HTSC but also in the Y / Y + Pr superstructure thin film which is composed of superconductors with two different T~'s [17]. It has also been reported in a
df
t c/1
cl 1 lP]" ] '
where the numerical factor y is &Ldp/OtLd i and is expected to be in the order of 10 -1 for the weak-pinning case. Taking the G - L parameter K = 85 ~ 119, the coherence length ~ab(4.2 K) -~ tab(O) = 3.2 ~ 3.8 nm as well as B~2 = 22 ~ 31 T [22], Ac = 104 nm [23], d e = 0.66 nm (c-axis lattice spacing), B 0 = 2.4 T in Table 1 for x = 0.061, the value 3' becomes of the order of 10 -3 ~ 1 0 - 2 if ~xc ///pin ~ 4 4 ~ 1 is assumed.
T. Kobayashi et al. /Physica C 254 (1995) 213-221
For a precise calculation of % the pinning center must be identified. This order of magnitude estimation therefore leaves the Yamafuji et al. model as a candidate to microscopically explain the exponential decay of J~ with field. But the model has not been developed to examine the dependences of U on B or on J. It has been known that (Lal_xSrx)2CklO 4 becomes more strongly anisotropic as x decreases [3], suggesting a weaker coupling between CuO 2 layers. We can regard it as quasi-two-dimensional and suitable for application of the Yamafuji et al. model. On the other hand, Senoussi et al. observing a similar exponential decrease of the magnetically obtained Jc with a YBa2Cu30 7 single crystal, suggested the involvement of a microstructural defect such as dislocations to determine Jc [2]. Grujic et al. presented a model based on the SNS junction consisting of a normal metal of effective thickness d embedded in a type-II superconductor under a field applied perpendicular to the junction to explain the exponential decay of J¢ with field [24]. From the present experiment B o has been known to decrease with temperature. According to their SNS junction model the temperature dependence of B o was attributed to the temperature dependence of d as B 0 = q ~ 0 / 2 v d 2, where d is assumed to increase with temperature in a qualitative agreement with the present experiment. From the results in Table 1, B o of sample #61-94 is 2.37 and 0.609 T at 4.2 and 20 K, respectively, resulting in d = 11.6 and 22.9 rim. Therefore if the SNS model [24] applies to this case, there must be normal regions of thickness several times of the coherence length ~ab present in the single-crystalline (Lal_xSr,)2CuO 4. This seems to require the material to be inhomogeneous in the Sr distribution or in the oxygen content, suggesting the single crystal to possess the granular nature of the superconducting particles. As discussed above, there are essentially two microscopic models which may explain the exponential dependence of J~ on field; one is to assume the softening of the vortex line due to the quasi-two-dimensional structure and the other is to assume the presence of the granular SNS junctions. Both of the models need to be further modified to explain the more detailed behavior in order to identify which of the models is to hold.
219
4.3. Pinning force density and the Kramers scaling law Finally we will examine whether the Kramers scaling law [7] can be applied to the HTSC which was successful in describing the pinning-force density as a function of field for many conventional superconductors and has been conveniently used to classify the pinning mechanism in them. The macroscopic pinning-force density Fp = Jc B is expressed as follows, using the local current density Jc(B) and the magnetic induction B(x):
l~j
Fp=wJo
c ( B ( x ) ) B ( x ) dx.
(6)
On the other hand, applying the Kramers scaling to a slab-like shaped conventional superconductors, although ignoring the change of magnetic induction or the critical current inside the sample, that scaling works well. Here, in comparison with Kramers scaling, we approximate Eq. (6) and use the field dependence of under-doped (La 1 xSrx)2CuO4,represented by Eq. (5) to obtain Eq. (7) (our approximation is justified at a higher field as discussed in Section 4.1; the variation in the critical current inside the sample is relatively smaller):
Fp = Jco B e x p ( - B / B o ) .
(7)
This equation shows that the Kramers scaling law of max O( b 2 ( 1 - b) q should not be able to describe the Fp-B relationship for the present material. But in case if b << 1 then
Fp//gp
Fp/Fp max c~ B / B o e x p ( - B / B o )
(8)
- b(1 - b),
(9)
for b << 1,
it can approximately describe the present case, taking p = 1 and q = 1. However, there have been several attempts made, for Jc of the HTSC's obtained by the modified Bean model plotted to apply the Kramers model and the scaling parameters have been deduced, e.g., p = 1, q = 4 [25] and p = 2, q = 4 [26]. But the arguments made in Section 4.1 have indicated that the modified Bean model gives ate and hence Fp with a larger error as the field is reduced for the HTSC. Therefore it may have been just fortuitous that the Kramers scaling law has been
220
T. Kobayashi et al. /Physica C 254 (1995) 213-221 1.0 0.8
-....
0.6 \
0.4
\\ \\
0.2
\\
2
3
4
5
6
7
8
1.0 (b) 0.8 O.6 0.4 0.2
4 ~o °° /
model smaller. Consequently, we conclude that the pinning-force density Fp does not follow the Kramers scaling law in the HTSC at least in the under-doped region. In summary, it has been shown that the field dependence of the critical current density of underdoped (Lal_xSrx)2CuO 4 single crystal under a field applied parallel to the c-axis is well represented by the exponential model over a wide field range from zero to several T. The exponential model explains the shift in the peaks of the magnetization curve from zero field quantitatively. For materials as in the present case in which the critical current density changes sharply with field, it has been shown that the error associated with the estimated Jc from the modified Bean model becomes greater as the field is reduced. It has also been shown that the Kramers scaling law does not hold for the pinning-force density of the HTSC's in the under-doped region.
/1
/// /
0.2
0.4
0.6
0.8
1.0
References
Box/Bmo~ Fig. 4. Normalized macroscopic pinning-force density F p / F p max of thin (2w = 0.4 ram) and thick (2w = 0.94 mm) slab singlecrystal samples ( x = 0.061) (a) and its magnification of the low-field region (b).
reported to fit the Fp-B curves obtained from the Bean approximation. Fig. 4 shows the Fp-B¢x relation as normalized by the maximum Fp, Fp max for the samples of two different thicknesses. Fig. 4(a) illustrates the entire field range and Fig. 4(b) for the low-field range. Comparing the two figures, it is understood that the tendency of underestimating J~ or Fp by the modified Bean model in a low field happens to contribute to the apparent agreement of the calculated Fp-B relation with the Kramers curve. The degree of this underestimation becomes greater for a thicker sample as is seen in the figure, Although the present results do not seem to fit the Kramers scaling law even by following the modified Bean model, it is clearly understandable that the error by using this model makes the deviation from the Kramers scaling
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