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Angular dependence of critical currents and grain misalignment in Ag clad (Bi,Pb)2Sr2Ca2Cu3O10 tapes
PHYSICA ELSEVIER
PhysicaC252 (1995) 211-220
Angular dependence of critical currents and grain misalignment in Ag clad (Bi,Pb)2Sr2Cazfu3Oao tapes Q.Y. Hu a,b,1, H.W. Weber a, H.K. Liu b, S.X. Dou b, H.W. Neumiiller ° a Atominstitut der Osterreichischen Universittiten, A-1020 Wien, Austria b Centre for Superconducting and Electronic Materials, University of Wollongong, Wollongong, NSW 2552, Australia e Siemens AG, Zentralabteilung Forschung Entwicklung, D-91050 Erlangen, Germany Received 10 May 1995
Abstract
Two-dimensional behaviour of the critical current in highly textured silver clad (Bi,Pb)2Sr2Ca2CU3Olo tapes was observed and analysed by introducing an effective grain misalignment angle, ~eff. This angle was found by SEM to be identical to the average crystallographic grain misalignment angle in the superconducting core. Furthermore, after fast neutron irradiation, which is isotropic, the Jc of the tapes were modified by the introduction of artificial defects, but the %ff remained the same. This proves that the crystallographic misalignment of the grains determines the critical current anisotropy of the tape in a magnetic field, i.e, Ic(B ]ltape plane) scales with I~(B ± tape plane) with the factor sin ~o~ff. Therefore, %ff is an intrinsic property of the tape. ~oeff is found to be around 10° in different samples. We propose that this typical value is determined by intrinsic mechanical properties of the (Bi,Pb)2Sr2Ca2CU3Olo compound and is a result of the mechanical deformation of the tape during the tape fabrication. From a comparison of different samples, we propose that the pinning ability is the determining factor of Jr when ~ff is small enough.
1. I n t r o d u c t i o n
According to Lawrence and Doniach [1], a dimensional crossover from 3D to 2D appears in layered superconductors under the condition that the coherence length ¢c perpendicular to the superconducting layers is smaller than the distance between these layers. In the case o f Bi2Sr2CaCu20 s, Kes et al. [2] proposed (1) that in an external magnetic field only the field c o m p o n e n t perpendicular to the layers gives rise to dissipative behaviour and (2) that for the external magnetic field B aligned along the CuO z planes, B should not influence superconductivity as 1 Present address: University of Wollongong, Australia.
long as the temperature is below the crossover temperature TO from 3D to 2D behaviour. This has been confmned by Raffy et al. [3] and Schmitt et al. [4] in their experiments on Bi : 2212 thin films. We have demonstrated that the model is also applicable to the (Bi,Pb)zSrzCa2CU3Olo (Bi-2223) compound, i.e. the critical current behaviour of the compound is 2D at 77.3 K [5]. Because single crystals o f this compound do not exist and epitaxial thin film have become available only recently [6], we used Ag clad Bi-2223 tapes for our investigations. Because the tape is a polycrystalline material, the two-dimensional behaviour o f the Bi-2223 compound was revealed by introducing an effective misalignment angle tpeff [5] for the characterisation of
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212
Q.Y. Hu et al./ Physica C 252 (1995) 211-220
Table 1 Critical currents at 77.3 K and 0 T and effective misalignment angles Sample
Je ( 108 A/rn2)
%ff (deg)
#92152 #AB 143 #AB135 #AB136 #93824
2.77 2.00 1.73 1.67 1.00
10-12 11 7 7 11
tapes. This kind of analysis has been used intensively ever since [7-9] for the characterisation of tapes. In the present work, we report on further studies of the effective misalignment angle, relate it to the crystallographic misalignment angle, and discuss its role for the critical current density of Bi-2223 tapes.
following. In Fig. 3, the critical currents of Figs. l(a) and l(b) are replotted as a function of BIle(B, 0) and for comparison, the magnetic field dependence of the critical current measured with B II c is also plotted in the same figure. From this figure, we can see that Ic(Bii ¢(B, 0)) coincides with Ic(B II c) in the high field region. This demonstrates that the dissipative behaviour of the compound Bi-2223 at 77.3 K results from BII c, i.e. the tape behaves as a two-dimensional superconductor. The 2D behaviour of HTSC can be briefly described as follows: (1) I
11
|
i
%k
( a ) 10
I
#92152
~°ko~
--o--- 22rnT ok_ --o-- 55mT %0\ ~ ~ 77mT ~oo,kk'~ '~' O,.o..,__z~__108.4mT
9. 8.
2. Experimental
i
~ o7 6-
The tapes were fabricated b y the well established powder in tube technique (PIT), the details of which can be found elsewhere [10,11]. The critical current densities of the samples at 77.3 K and zero field are listed in Table 1. The critical current Ic was measured as a function of the angle between the broad face of the tape (tape plane for short) and the magnetic field direction in different magnetic fields and as a function of field for B IItape plane and B _Ltape plane, keeping the magnetic field and the current perpendicular to each other. An electrical field criterion of 10 - 6 V / c m was used for the definition of I¢. The angular resolution of the rotation is better than 1°. The microstructure of the tape (cross-sectional direction) was determined with a Leica Stereoscan $440 scanning electron microscope (SEM).
s-
o
4-
o
a()
Fig. 1 shows the results on the critical currents of two tapes as a function of the angle between the magnetic field, B, and the tape plane. According to Fig. 2, B sin 0 is the B component in the direction perpendicular to the tape plane (c-direction of the tape for short), which is written as nil c(B, 0) in the
4'0
6'0
8'0
I00
e (o)
(b)
9
o/~'~,
#93824
il .....
~,
5 4:
32 3. 2D behaviour of critical currents and effective misalignment angles
2'0
2-'
1 -" 0
-2o
C ' ~,O~ ~o\ O0~°~-kz~~
o
o,,~. %Ooo
"v,
--o-- 47.5mT "--o-- 59.2mT --~-- 83.0mT ---0--108.4mT 360.0roT
o..~z~,~_~
v~.
- ~--.o__.___ o
"v"v'-v-~v~_v , o o' o'16o e (o)
12o
Fig. 1. Critical current as a function of the angle between the magnetic field and the tape plane of sample #92152 (a) and #93824 (b).
Q.Y. Hu et aL/Physica C 252 (1995) 211-220 Ic(B IJab) is constant, when B changes and (2) I~(B, O)=I~(B sin 0), where 0 is the angle between the a,b-plane and B. However, it can be also seen in Fig. 3 that Io(BII c(B, 0)) deviates from I¢(B II c) at low fields. This does not indicate a failure of the 2D description, but is caused by the grain misalignment of the tape. During its fabrication, the tape is repeatedly pressed by applying a perpendicular uniaxial force to the broad face of the tape. Considering this process, it is reasonable to assume that the crystallographic misalignment angles (CMAs) of the grains (referring to their a,b-planes) in the tape are distributed symmetrically about the texture plane, which is also the tape plane, i.e. N(q~i) = N ( - ~ p i ) , where N(~0i) represents the number of the grains with a CMA of q~i, which is the angle between the a,b-plane of each grain and the tape plane. Let us examine those two planes which enclose the largest CMA, i.e. ~max and -~Pm~x- According to the model proposed by Kes et al. [2], the superconductive dissipation in the 2D regime is only caused by the B component in c-direction, i.e. BII c- Referring to Fig. 4 (inset), these field components are given by BII c = B s i n ( 0 + ~Pmax),
(1)
and by BII~ = B s i n ( 0 - ~Omax).
(2)
213
Therefore, when 0 ° < 0 < 90 °, BII c( 0 + ~0max) > BII c ( O - ~Omax) and when 90 ° < 0 < 0 °, BII c(0 + ~Pmax)< BIIc( 0 ~°max)- Only when 0 = 0 °, both are equal, resulting in the minimum dissipation and therefore the maximum critical current. We define the tape plane by 0 = 0 ° and therefore 0 = 90 ° corresponds to the c-direction of the tape. Because we are not able to align B along the a,b-planes for all of the grains at the same time, we cannot observe that n(o II ab) is independent of I c, which is found for high quality thin films [3,4,6]. Actually, the a,b-plane of a misaligned grain does not only rotate around the y-axis (its normal rotates in the x-z-plane around the coordinate origin), as discussed above, but also around the x-axis (its normal rotates in the y-z-plane around the coordinate origin), i.e. the misalignment angle is a spatial angle (Fig. 5). This is because the tape was pressed by applying a force along its c-direction during fabrication, which does not make any difference in the grain misalignment with regard to the x- and y-axis. However, according to Appendix A, we can see that the spatial misalignment has a complicated solution, but BIIc has an upper limit, which is given by a 2D misalignment, similar to the misalignment around only one axis. For simplification, we discuss only grains, which are misaligned by rotating around the y-axis (Fig. 2). -
-
Z
~( tape plane X
TAPE PLANE
~x
Fig. 2. Schematicillustration of a misaligned grain, B rotates (as a result of the tape rotation)in the transversecross-sectionalplane, n denotes the tape normal, c denotesthe c-directionof a misalignedgrain.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220
214
Because of the existence of the misalignment angle, BII c(O) is not equal to B sin 0 any more. Instead
BIle(B, O) = B sin
(3)
0eff
q~max"According to Fig. 4, Btl c is averaged over the angular spread 2 ~Om.x. With the definition
I
I
I
I
BIle(B, O) = c t g 0 dO.
(5)
14
(b) 12-
~,
#92152 - - ' - - B//c --A-- 108.4mT
~
--0--
•I i
~k 104.
dBllc(B,O)
I
II
E
(4)
we get, according to Appendix B:
has to be used, where O~ff denotes the effective rotation angle, which is not identical to O because of the grain misalignment and 0 - q~max~< 0~ff~< 0 +
(a)
O)-BIle(B, Oeff),
dBiic(B, 0 ) = B I l e ( B ,
#93824
I0-
~
~.
8-
--v-- 47.5mT 59.2mT 83mT ---o-- 300mT
o
77mT
°
o 4-
o
2
6
s'o
1~o
0 -50
1~o 260
6 5'0 16o1~o'26o'2~o'36o'3~o
B'sin0 (mT) I
I
I
I
I
B*sine(mT) 6
I
I
'
I
•
i
•
i
•
I
•
I
'
(d)
(•0)
5
AB143
6
~.
E
.,-...
<
\
4
~
v
2
TM
-'~.~o%.
108.4mT
--o-- B//C
"°-e.e.ee~
'o ~o ~o ~o ~o lbo 120 B'sin0 (mT)
Bflc
--o--- 77mT ,\
o
AB135
•
~
108.4mT
.
6
2'o 4b 6'o d016o 120 B'sin0 (mT)
Fig. 3. Critical currents of sample #92152 (a), #93824 (b), ABI35 (its inset is from Ref. [14]) (c) and AB143 (d) as a function of BII c(B, O) and B II c. The background field, in which the rotation measurements were carried out, were 55, 59.2, 77, 83, 108.4, and 300 mT.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220 '
i
•
!
•
I
'
i
0.8 m
-
0.6
) 0,: 0.2 Grain----:'i':" ------"~~ 0.0
I
2~Pm~ '
-go'- o
6 o
3'o
6'0
go
(°)
Fig. 4. Schematicillustrationof the grain misalignmentof a tape and averagingeffect of B over the angularspread 2 ~max"
The difference between 0~ff and 0 is caused by the averaging. Therefore d 0 in Eq. (5) is substituted by 2 ~max, i.e. dB, c(B, 0) = ctg 0 2~x"
0)
(6)
From Eq. (6), w e see that d BII c/eu c is proportional to ctg 0. Therefore, the smaller the angle 0, the larger is the left-hand side of Eq. (6). This means that the averaging effect becomes remarkable when
the angle decreases. Furthermore, the low field region corresponds to low angles and the high field region to high angles. In the high field region, differences between lc(B II c) and/o(BII c(B, 0)) may not be detected even though the averaging effect exists, because they are small and could be within the resolution of our measurement (Fig. 3). However, when 0 decreases, i.e. in the low field region~ dBiic/Bii c becomes larger and finally leads to a deviation of I~(BIIc(B, 0)) from I~(B II c). Because the dissipation of the samples results only from BII c, the same I c in the curves Ie(B II c) and Ic(BIc(B, 0)) corresponds to the sanae BIIc. Since Ie(Bii c(B, 0)) is reduced compared to I¢(B c) in the low field region, the real BII c must be larger than the calculated one, i.e. BII c(B, 0eff) > BIIc(B, 0). We define an angle ~oeff, which is the effective grain misalignment angle of the tape and ~eff ~--- 10eft- 0 l, The value of BII c(B, 0 e l f ) i n the curve le(B c(B, 0)) corresponds to the field of the curve I¢(B c), at the same Ie level. At 0 = 0 °, BII c(B, 0~ff) = BII c(B, 0 + q~ff) = BII c(B, q~ff), therefore ~eff can b e calculated from sin ~ff = BIIc(B, 0 = 0°)/B. The calculated results are listed for several samples in Table 1. The critical current is determined from the voltage, which is related to dissipation in the tape. The supercurrent path follows a number of grains, which have different crystallographic grain misalignment angles, q~. There is no doubt that the amount of grains, N, with the same q~ value has a statistical distribution N=f(q~),
........
¢ ~
e-direction of a grain
B *°'°'"
L"
Fig. 5. S c h e m a t i c illustration o f t h e g r a i n m i s a l i g n m e n t in 3 D
space, the tape plane is in the x- y-plane.
-90°~< ~p~< +90 °
(7)
although the exact type of the distribution is not known [12]. Considering this fact, the net voltage of the tape is
v=
....
215
(8)
where N(q~i) is the number of grains with, misalignment angle q~i, and V(q~i) is the voltage generated in the grains with the misalignment angle q~i. Therefore, the detected voltage is contributed by all grains forming the current paths. The grains forming the current paths are connected by grain boundaries, which could be weak links or strong links. At 77 K, they are considered to be strong links [15], i.e. the connected grains can be treated as one grain, there-
216
Q.Y. Hu et a L / Physica C 252 (1995) 211-220 i
fore we do not discuss the role of the boundaries in the following text. The net effect is the same as if.a group of grains with q~eff would produce the voltage of the tape. It is understandable that the effective misalignment angle measured i n this way is not necessarily the same as q~max, again because of the averaging effect (Fig. 4), instead we have 0 < q~ff <
i
i
i
~
10,
I
i
#92152 I'%.
t
--.--B.abJ -o- H0 t
~0nla x •
103,
4. Effective and crystallographic grain misalignment angles
8 ' 2;o"
It has been proposed [13] that for a rotation of the tape in a field B (also see Fig. 6) from its c-direction (0 = 90 °) to its tape plane direction (0 = 0°), Ic(B, O) will go along I~(B II c)from point b to point c and be cut off by I~(B Iltape plane) (point a). We can see that point c corresponds to Ie(Biic(B, 0 = 0°)) and point b corresponds to Ic(B II c), cf. Fig. 3. Using the results of Fig. 6, we calculate ¢p~ff of the tape over the whole field range investigated. We note that although I~ changes with BIIc and BIItape plane, ~Peff varies very little, i.e. by less than 2 °, as shown in Fig. 7. This indicates that q~ff is caused by the crystallographic grain misalignment of the tape. Because such misalignment is an intrinsic property of the tape, q~ does not change with field, neither does ~Oeff. Recently, we found that tpeff remained the same even following neutron radiation, although the critical current dependence on the magnetic field had been changed considerably (Fig. 3(c)). Since neutron irradiation is not anisotropic and the induced defects are randomly distributed in the tape, this fact again proves that ~eee depends only on the crystallographic grain rnisalignment [14]. We investigated the microstructure of sample #93824 by means of SEM. We found that the crystallographic misalignment angle of the majority of grains, i.e. the average crystallographic misaiignment angle, q~,v, is around 10° (Fig. 8) in excellent agreement with the value of tp~ff. This fact proves in turn, that our 2D description of the tape is correct. Wilhelm et al. [11] have also observed a typical grain misalignment angle of about 10° along the longitudinal section of a Bi-2223 tape. However, the misalignment angle that we observed refers to the plane
8o' 7;o
1
ldooli o'lsoo
B (mT) Fig. 6. Critical current of tape #92152 as a function of B for B [Itapeplane and B _Ltape plane.
of the transverse cross section of the tapes, in which the field rotates. Bulaevskii et al. [7] fitted their Ic-Biic(B, O) curve and obtained a q~av value of 16°, which they claimed to be in quite good agreement with the ~&ff value obtained by Hu et al. [5]. It is interesting to find out what determines the typical q~eff values of around 10° observed in Refs. [5,7-9,11]. We propose that q~av, and therefore tpeff, is governed by the mechanical properties of the Bi-2223 compound. In the process of tape fabrication, a uniaxial force is applied to press the tape, which can be decomposed into a principal force, P], and a sheafing force, P2, directed along [001] and along the basal plane of the grain, respectively (Fig. 9). After several deformations of the grain, the dimensions of the grain tend to a certain ratio. According to theory, the maximum principal stress and the sheafing stress are
½df t / 2 P]l-l x dx 3Pll offaax = l sd 3 = 4sd2 _[0011
(9)
and ,/.(001)_
P2
-max -- Is
(10)
217
Q.Y. Hu et al. /Physica C 252 (1995) 211-220 12
i
,
i
i
•
|
respectively, where d, l, and s are the thickness, the length and the width o f the grain. In a stable mechanical state, we should have
t
o o 11
o o
[001----'~ =
6-"
O'r~ax
v
O ¢:D
O O
10
--
( 1 1)
~
3 P1
Referring to Fig. 9, w e have P 2 / P 1 = tan ~o and d / l = tan ~p. Therefore
O O
#92152
3 ~.(001) "max tan q (tan ~o) 2 = ~ ,r [001"--'~-"-,
(12)
~IDaX
9 •
" 16o' 1
o'2
o'2
o'3oo
or
B//c (mT) Fig. 7. Texture misalignment angle of #92152 calculated from Fig. 6.
~3 tan q~ =
' r (001) "max
4 n.[OOl] "
(13)
-max
Fig. 8. SEM micrograph of sample #93824 (transverse cross section). The vertical grids are parallel to the tape plane, ~'av is estimated to be
10°.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220
218
p2~-
TAPEPLANE Fig. 9. Schematic illustration of the formation mechanism of the misalignment angle by deformation.
Thus, we have g, = arctan -4 rr [°°~1
(14)
~n'~x
In general, = ¢{~.(001> -[0011~ ~0av
v k ' m a x , Un~ax ]"
(15)
Thus, we relate q~avand therefore q~effto the mechanical properties of the grain or compound. During the fabrication of the tape, the grain dimension may be modified by the sintering process, i.e. some grains may be connected together again, but ~oav should stay unchanged. The typical grain dimension ratio of a Bi-2223 tape (after sintering) is d/l-- 1/20, but this does not correspond to the typical q~efe value of 10° (Fig. 9).
5. Grain misalignment and critical current density Table 1 lists the q~ef values of the samples. From the table, we note that there is no direct relationship between J~ and q~¢ff. Samples AB135 and AB136 have the same Je and q~eff values, whereas samples #92152 and #93824 have almost the same ~¢ff, but quite different Je values. If we compare AB136 with #92152, we find that its Jc is lower, although the former has a smaller q~ef, which indicates better grain alignment in the tape. This fact, namely that a smaller q~¢ffdoes not guarantee samples with higher Je and that samples with the same goeff may have different Jc, suggests that some other factors, besides the grain misalignment, may be of the same
importance for Jc. They could be the mass density of the superconducting core, the connectivity of the grains, etc. We believe, however, that the pinning capability for BIIc is a decisive factor for the enhancement of Je at high temperatures, if q~ff is small enough. We suggest, that the efforts to increase J~ of the Bi-2223 tapes should not only be directed towards improving the grain alignment, but also towards the introduction of more artificial pinning centres, which has been proved to be an effective way for the enhancement of Je(Bu c) [14].
6. Conclusion Based on the 2D behaviour of Ag clad Bi-2223 tapes with regard to their critical currents, the effective grain misalignment angle, q~eff,has been studied. The data of It(B, O) measurements on different samples show, that this value is around 10°, which is identical to the average crystallographic grain misalignment angle, q~av,of the tape. Preliminary studies reveal, that the average grain misalignment of the tape is related to the mechanical properties of the Bi-2223 compound. We suggest that the pinning capability for BII c is an important factor for J¢ in tapes with small misalignment angles.
Acknowledgements One of the authors (QYH) wishes to acknowledge fmancial support by the Austrian Exchange Service (OAD), Vienna during his one year stay at the Atomic Institute. We wish to thank R.M. Schalk for
Q.Y. Hu et al. / Physica C 252 (1995) 211-220 useful discussions and H. Niedermaier for technical assistance. This work was supported in part by Fonds zur F~Srderung der Wissenschaftlichen Forschung, Vienna, under contract No. 8849. A l l the authors from WoUongong wish to acknowledge financial support by Metal Manufacturers Ltd. and by the University o f Wollongong.
References [1] W.E. Lawrence and S. Doniaeh, Proc. 12th Int. Conf. on Low Temperature Physics, Kyoto, 1970, Keigaku, Tokyo (1971) p. 361. [2] P.H. Kes, J. Aarts, V.M. Vinokur and C.J. van der Beck, Phys. Rev. Lett. 64 (1990) 1063. [3] H. Raffy, S. Labdi, O. Laborde and P. Moncean, Phys. Rev. Lett. 66 (1991) 2515. [4] P. Schmitt, P. Kummeth, L. Schultz and G. SaemannIschenko, Phys. Rev. Lett. 67 (1991) 267. [5] Q.Y. Hu, H.W. Weber, S.X. Dou, H.K. Liu and H.W. Neumiiller, Proc. 1992 E-MRS Fall Meeting, Nov. 3-6, 1992, Strasbourg, published in J. Alloys Comp. 195 (1993) 515. [6] R.M. Schalk, G. Samadi Hosseinali, H.W. Weber, S. Griindorfer, D. Biiuerle, S. Proyer, P. Wagner, U. Frey and H. Adrian, Physica C 235-240 (1994) 2625. [7] L.N. Bulaevsldi, L.L. Daemen, M.P. Maley and Y. Coulter, Phys. Rev. B 48 (1993) 13798. [8] B. Hensel, J.-C. Grivel, A. Jeremie, A. Perin, A. Pollini and R. Fl'tikiger, Physica C 205 (1993) 329. [9] T. Kaneko, Y. Torii, H. Takei, K. Tada and K. Sato, Advances in Superconductivity VI, Vol. 2, eds. T. Fujita and Y. Shiohara (Springer, Berlin, 1994) p. 683. [10] S.X. Dou, H.K. Liu and Y.C. Gno, Appl. Phys. Lett. 60 (1992) 2929. [11] M. Wilhelm, H.W. Neumiiller and G. Ries, Physica C 185189 (1991) 2399. [12] Bulaevskii et al. [7] assumed a Gaussian distribution for ~. The application of this distribution to fit the experimental data has proved to be successful to some extent. [13] Q.Y. Hu, H.K. Liu and S.X. Don, Cryogenics 32 (i992) 1038. [14] Q.Y. Hu, H.W. Weber, F.M. Sauerzopf, G.W. Schulz, R.M. Schalk, H.W. Neumiiller and S.X. Dou, Appl. Phys. Lett. 65 (1994) 3008. [15] Q.Y. Hu, R.M. Schalk, H.W. Weber, H.K. Liu, R.K. Wang, C. Czurda and S.X. Dou, J. Appl. Phys. 78, in press.
Appendix A A s s u m i n g that a magnetic field is applied in the x - z - p l a n e in a 3D coordinate system, it intersects the
219
x-, y- and z-axes at the angles ce = 0, /3 = 90 °, and 3~= 90 ° - 0, respectively (Fig. 5). Then the field vector is given by B = ( B cos a . i + B = ( B cos O . i + B
cos / 3 . j + B
cos ~ / - k )
sin O . k ) .
(A.1)
The a - b - p l a n e s o f the grains in a tape are misaligned by tilting off the texture plane, or by tilting not only to the y-axis but also to the x-axis (Fig. 5). W e consider the normals o f those grains, which enclose an angle ~Omax with the z-axis. The entirety o f these normals forms a cone with 3' = ~Pr~x, which encloses all the normals of misaligned grains. A s a result, the a,b-planes o f these grains, enclose an angle ~Pmax with the Z = 0 plane. Therefore, the unit vector o f the grain along the c-direction is (see Fig. 5) n = (cos a i + cos 13 j + cos ~Omax k ) .
(A.2)
Therefore, the B component along the c-direction o f the grain is BIIc=B .n--B(cos
0 cos oL+ sin 0 cos
~Pmax)" (A.3)
Since cos2a + cos2/3 + cosZy = 1, we have / lt c = B . n = B ( c o s
0V/1 - cos2/3 - c o s 2
+ s i n 0 cos ~Pm~x),
max
(A.4)
or BII c = B . ,
= B ( c o s 0 ~sin2CPmax- cos2/3 + s i n 0 cos ~Pm~x)•
(A.5)
W e note that the spatial misalignment depends on BII c in a complicated way with regard to the angles. But, according to Eq. (A.5), we have BIIc
s i n ( 0 + ~Omax).
(A.6)
W e note that the right-hand side is the same as Eq. (1) in the text. Therefore, we see that the 2D misalignment (grain normals lie in the x - z - p l a n e ) represents an upper limit o f BII c for the 3D misalignment case (grain normals are not in one plane). Thus, the solution o f the three-dimensional problem o f grain misalignment can be approximated by a two-dimensional model.
220
Q.Y. Hu et aL / Physica C 252 (1995) 211-220
Appendix C. Summary of notations for the angles
Appendix B In an external magnetic field, the c-component of the field is given by BII c = B sin 0.
(B.1)
Differentiation of Eq. (B.1) leads to dBii ~ = B cos 0 dO.
(B.2)
Substituting B from Eq. (B.2) into Eq. (B.1) results in
dBLI c = BII c ctg 0 dO,
(B.3)
or
dBii c/Bn c = ctg 0 d0.
(B.4)
Eq. (13.4), therefore, represents the relative change of Btl c with 0.
Rotation angle between the magnetic field and the tape plane. q~: Crystallographic grain misalignment angle. qOmax : Maximum ~0 which encloses all the misalignment angles. Effective misalignment angle, defined by the ~Peff: critical current measurement, i.e. sin ~t~eff Bit c(B, 0 = O°)/B. 0 < %ff<~ qgmax . Effective rotation angle, 0~tf = 0 + %ff. 0eft: Average crystallographic grain misalignment qOav: angle. For instance, assuming that 9 has a Gaussian distribution, i.e. p(q~) = ( 1 / v ~ - tr) ×exp(qo2/2o-2), then qOav is or. As mentioned in the text, q~av= q°etf" 0:
=
PhysicaC252 (1995) 211-220
Angular dependence of critical currents and grain misalignment in Ag clad (Bi,Pb)2Sr2Cazfu3Oao tapes Q.Y. Hu a,b,1, H.W. Weber a, H.K. Liu b, S.X. Dou b, H.W. Neumiiller ° a Atominstitut der Osterreichischen Universittiten, A-1020 Wien, Austria b Centre for Superconducting and Electronic Materials, University of Wollongong, Wollongong, NSW 2552, Australia e Siemens AG, Zentralabteilung Forschung Entwicklung, D-91050 Erlangen, Germany Received 10 May 1995
Abstract
Two-dimensional behaviour of the critical current in highly textured silver clad (Bi,Pb)2Sr2Ca2CU3Olo tapes was observed and analysed by introducing an effective grain misalignment angle, ~eff. This angle was found by SEM to be identical to the average crystallographic grain misalignment angle in the superconducting core. Furthermore, after fast neutron irradiation, which is isotropic, the Jc of the tapes were modified by the introduction of artificial defects, but the %ff remained the same. This proves that the crystallographic misalignment of the grains determines the critical current anisotropy of the tape in a magnetic field, i.e, Ic(B ]ltape plane) scales with I~(B ± tape plane) with the factor sin ~o~ff. Therefore, %ff is an intrinsic property of the tape. ~oeff is found to be around 10° in different samples. We propose that this typical value is determined by intrinsic mechanical properties of the (Bi,Pb)2Sr2Ca2CU3Olo compound and is a result of the mechanical deformation of the tape during the tape fabrication. From a comparison of different samples, we propose that the pinning ability is the determining factor of Jr when ~ff is small enough.
1. I n t r o d u c t i o n
According to Lawrence and Doniach [1], a dimensional crossover from 3D to 2D appears in layered superconductors under the condition that the coherence length ¢c perpendicular to the superconducting layers is smaller than the distance between these layers. In the case o f Bi2Sr2CaCu20 s, Kes et al. [2] proposed (1) that in an external magnetic field only the field c o m p o n e n t perpendicular to the layers gives rise to dissipative behaviour and (2) that for the external magnetic field B aligned along the CuO z planes, B should not influence superconductivity as 1 Present address: University of Wollongong, Australia.
long as the temperature is below the crossover temperature TO from 3D to 2D behaviour. This has been confmned by Raffy et al. [3] and Schmitt et al. [4] in their experiments on Bi : 2212 thin films. We have demonstrated that the model is also applicable to the (Bi,Pb)zSrzCa2CU3Olo (Bi-2223) compound, i.e. the critical current behaviour of the compound is 2D at 77.3 K [5]. Because single crystals o f this compound do not exist and epitaxial thin film have become available only recently [6], we used Ag clad Bi-2223 tapes for our investigations. Because the tape is a polycrystalline material, the two-dimensional behaviour o f the Bi-2223 compound was revealed by introducing an effective misalignment angle tpeff [5] for the characterisation of
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 4 7 4 - 2
212
Q.Y. Hu et al./ Physica C 252 (1995) 211-220
Table 1 Critical currents at 77.3 K and 0 T and effective misalignment angles Sample
Je ( 108 A/rn2)
%ff (deg)
#92152 #AB 143 #AB135 #AB136 #93824
2.77 2.00 1.73 1.67 1.00
10-12 11 7 7 11
tapes. This kind of analysis has been used intensively ever since [7-9] for the characterisation of tapes. In the present work, we report on further studies of the effective misalignment angle, relate it to the crystallographic misalignment angle, and discuss its role for the critical current density of Bi-2223 tapes.
following. In Fig. 3, the critical currents of Figs. l(a) and l(b) are replotted as a function of BIle(B, 0) and for comparison, the magnetic field dependence of the critical current measured with B II c is also plotted in the same figure. From this figure, we can see that Ic(Bii ¢(B, 0)) coincides with Ic(B II c) in the high field region. This demonstrates that the dissipative behaviour of the compound Bi-2223 at 77.3 K results from BII c, i.e. the tape behaves as a two-dimensional superconductor. The 2D behaviour of HTSC can be briefly described as follows: (1) I
11
|
i
%k
( a ) 10
I
#92152
~°ko~
--o--- 22rnT ok_ --o-- 55mT %0\ ~ ~ 77mT ~oo,kk'~ '~' O,.o..,__z~__108.4mT
9. 8.
2. Experimental
i
~ o7 6-
The tapes were fabricated b y the well established powder in tube technique (PIT), the details of which can be found elsewhere [10,11]. The critical current densities of the samples at 77.3 K and zero field are listed in Table 1. The critical current Ic was measured as a function of the angle between the broad face of the tape (tape plane for short) and the magnetic field direction in different magnetic fields and as a function of field for B IItape plane and B _Ltape plane, keeping the magnetic field and the current perpendicular to each other. An electrical field criterion of 10 - 6 V / c m was used for the definition of I¢. The angular resolution of the rotation is better than 1°. The microstructure of the tape (cross-sectional direction) was determined with a Leica Stereoscan $440 scanning electron microscope (SEM).
s-
o
4-
o
a()
Fig. 1 shows the results on the critical currents of two tapes as a function of the angle between the magnetic field, B, and the tape plane. According to Fig. 2, B sin 0 is the B component in the direction perpendicular to the tape plane (c-direction of the tape for short), which is written as nil c(B, 0) in the
4'0
6'0
8'0
I00
e (o)
(b)
9
o/~'~,
#93824
il .....
~,
5 4:
32 3. 2D behaviour of critical currents and effective misalignment angles
2'0
2-'
1 -" 0
-2o
C ' ~,O~ ~o\ O0~°~-kz~~
o
o,,~. %Ooo
"v,
--o-- 47.5mT "--o-- 59.2mT --~-- 83.0mT ---0--108.4mT 360.0roT
o..~z~,~_~
v~.
- ~--.o__.___ o
"v"v'-v-~v~_v , o o' o'16o e (o)
12o
Fig. 1. Critical current as a function of the angle between the magnetic field and the tape plane of sample #92152 (a) and #93824 (b).
Q.Y. Hu et aL/Physica C 252 (1995) 211-220 Ic(B IJab) is constant, when B changes and (2) I~(B, O)=I~(B sin 0), where 0 is the angle between the a,b-plane and B. However, it can be also seen in Fig. 3 that Io(BII c(B, 0)) deviates from I¢(B II c) at low fields. This does not indicate a failure of the 2D description, but is caused by the grain misalignment of the tape. During its fabrication, the tape is repeatedly pressed by applying a perpendicular uniaxial force to the broad face of the tape. Considering this process, it is reasonable to assume that the crystallographic misalignment angles (CMAs) of the grains (referring to their a,b-planes) in the tape are distributed symmetrically about the texture plane, which is also the tape plane, i.e. N(q~i) = N ( - ~ p i ) , where N(~0i) represents the number of the grains with a CMA of q~i, which is the angle between the a,b-plane of each grain and the tape plane. Let us examine those two planes which enclose the largest CMA, i.e. ~max and -~Pm~x- According to the model proposed by Kes et al. [2], the superconductive dissipation in the 2D regime is only caused by the B component in c-direction, i.e. BII c- Referring to Fig. 4 (inset), these field components are given by BII c = B s i n ( 0 + ~Pmax),
(1)
and by BII~ = B s i n ( 0 - ~Omax).
(2)
213
Therefore, when 0 ° < 0 < 90 °, BII c( 0 + ~0max) > BII c ( O - ~Omax) and when 90 ° < 0 < 0 °, BII c(0 + ~Pmax)< BIIc( 0 ~°max)- Only when 0 = 0 °, both are equal, resulting in the minimum dissipation and therefore the maximum critical current. We define the tape plane by 0 = 0 ° and therefore 0 = 90 ° corresponds to the c-direction of the tape. Because we are not able to align B along the a,b-planes for all of the grains at the same time, we cannot observe that n(o II ab) is independent of I c, which is found for high quality thin films [3,4,6]. Actually, the a,b-plane of a misaligned grain does not only rotate around the y-axis (its normal rotates in the x-z-plane around the coordinate origin), as discussed above, but also around the x-axis (its normal rotates in the y-z-plane around the coordinate origin), i.e. the misalignment angle is a spatial angle (Fig. 5). This is because the tape was pressed by applying a force along its c-direction during fabrication, which does not make any difference in the grain misalignment with regard to the x- and y-axis. However, according to Appendix A, we can see that the spatial misalignment has a complicated solution, but BIIc has an upper limit, which is given by a 2D misalignment, similar to the misalignment around only one axis. For simplification, we discuss only grains, which are misaligned by rotating around the y-axis (Fig. 2). -
-
Z
~( tape plane X
TAPE PLANE
~x
Fig. 2. Schematicillustration of a misaligned grain, B rotates (as a result of the tape rotation)in the transversecross-sectionalplane, n denotes the tape normal, c denotesthe c-directionof a misalignedgrain.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220
214
Because of the existence of the misalignment angle, BII c(O) is not equal to B sin 0 any more. Instead
BIle(B, O) = B sin
(3)
0eff
q~max"According to Fig. 4, Btl c is averaged over the angular spread 2 ~Om.x. With the definition
I
I
I
I
BIle(B, O) = c t g 0 dO.
(5)
14
(b) 12-
~,
#92152 - - ' - - B//c --A-- 108.4mT
~
--0--
•I i
~k 104.
dBllc(B,O)
I
II
E
(4)
we get, according to Appendix B:
has to be used, where O~ff denotes the effective rotation angle, which is not identical to O because of the grain misalignment and 0 - q~max~< 0~ff~< 0 +
(a)
O)-BIle(B, Oeff),
dBiic(B, 0 ) = B I l e ( B ,
#93824
I0-
~
~.
8-
--v-- 47.5mT 59.2mT 83mT ---o-- 300mT
o
77mT
°
o 4-
o
2
6
s'o
1~o
0 -50
1~o 260
6 5'0 16o1~o'26o'2~o'36o'3~o
B'sin0 (mT) I
I
I
I
I
B*sine(mT) 6
I
I
'
I
•
i
•
i
•
I
•
I
'
(d)
(•0)
5
AB143
6
~.
E
.,-...
<
\
4
~
v
2
TM
-'~.~o%.
108.4mT
--o-- B//C
"°-e.e.ee~
'o ~o ~o ~o ~o lbo 120 B'sin0 (mT)
Bflc
--o--- 77mT ,\
o
AB135
•
~
108.4mT
.
6
2'o 4b 6'o d016o 120 B'sin0 (mT)
Fig. 3. Critical currents of sample #92152 (a), #93824 (b), ABI35 (its inset is from Ref. [14]) (c) and AB143 (d) as a function of BII c(B, O) and B II c. The background field, in which the rotation measurements were carried out, were 55, 59.2, 77, 83, 108.4, and 300 mT.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220 '
i
•
!
•
I
'
i
0.8 m
-
0.6
) 0,: 0.2 Grain----:'i':" ------"~~ 0.0
I
2~Pm~ '
-go'- o
6 o
3'o
6'0
go
(°)
Fig. 4. Schematicillustrationof the grain misalignmentof a tape and averagingeffect of B over the angularspread 2 ~max"
The difference between 0~ff and 0 is caused by the averaging. Therefore d 0 in Eq. (5) is substituted by 2 ~max, i.e. dB, c(B, 0) = ctg 0 2~x"
0)
(6)
From Eq. (6), w e see that d BII c/eu c is proportional to ctg 0. Therefore, the smaller the angle 0, the larger is the left-hand side of Eq. (6). This means that the averaging effect becomes remarkable when
the angle decreases. Furthermore, the low field region corresponds to low angles and the high field region to high angles. In the high field region, differences between lc(B II c) and/o(BII c(B, 0)) may not be detected even though the averaging effect exists, because they are small and could be within the resolution of our measurement (Fig. 3). However, when 0 decreases, i.e. in the low field region~ dBiic/Bii c becomes larger and finally leads to a deviation of I~(BIIc(B, 0)) from I~(B II c). Because the dissipation of the samples results only from BII c, the same I c in the curves Ie(B II c) and Ic(BIc(B, 0)) corresponds to the sanae BIIc. Since Ie(Bii c(B, 0)) is reduced compared to I¢(B c) in the low field region, the real BII c must be larger than the calculated one, i.e. BII c(B, 0eff) > BIIc(B, 0). We define an angle ~oeff, which is the effective grain misalignment angle of the tape and ~eff ~--- 10eft- 0 l, The value of BII c(B, 0 e l f ) i n the curve le(B c(B, 0)) corresponds to the field of the curve I¢(B c), at the same Ie level. At 0 = 0 °, BII c(B, 0~ff) = BII c(B, 0 + q~ff) = BII c(B, q~ff), therefore ~eff can b e calculated from sin ~ff = BIIc(B, 0 = 0°)/B. The calculated results are listed for several samples in Table 1. The critical current is determined from the voltage, which is related to dissipation in the tape. The supercurrent path follows a number of grains, which have different crystallographic grain misalignment angles, q~. There is no doubt that the amount of grains, N, with the same q~ value has a statistical distribution N=f(q~),
........
¢ ~
e-direction of a grain
B *°'°'"
L"
Fig. 5. S c h e m a t i c illustration o f t h e g r a i n m i s a l i g n m e n t in 3 D
space, the tape plane is in the x- y-plane.
-90°~< ~p~< +90 °
(7)
although the exact type of the distribution is not known [12]. Considering this fact, the net voltage of the tape is
v=
....
215
(8)
where N(q~i) is the number of grains with, misalignment angle q~i, and V(q~i) is the voltage generated in the grains with the misalignment angle q~i. Therefore, the detected voltage is contributed by all grains forming the current paths. The grains forming the current paths are connected by grain boundaries, which could be weak links or strong links. At 77 K, they are considered to be strong links [15], i.e. the connected grains can be treated as one grain, there-
216
Q.Y. Hu et a L / Physica C 252 (1995) 211-220 i
fore we do not discuss the role of the boundaries in the following text. The net effect is the same as if.a group of grains with q~eff would produce the voltage of the tape. It is understandable that the effective misalignment angle measured i n this way is not necessarily the same as q~max, again because of the averaging effect (Fig. 4), instead we have 0 < q~ff <
i
i
i
~
10,
I
i
#92152 I'%.
t
--.--B.abJ -o- H0 t
~0nla x •
103,
4. Effective and crystallographic grain misalignment angles
8 ' 2;o"
It has been proposed [13] that for a rotation of the tape in a field B (also see Fig. 6) from its c-direction (0 = 90 °) to its tape plane direction (0 = 0°), Ic(B, O) will go along I~(B II c)from point b to point c and be cut off by I~(B Iltape plane) (point a). We can see that point c corresponds to Ie(Biic(B, 0 = 0°)) and point b corresponds to Ic(B II c), cf. Fig. 3. Using the results of Fig. 6, we calculate ¢p~ff of the tape over the whole field range investigated. We note that although I~ changes with BIIc and BIItape plane, ~Peff varies very little, i.e. by less than 2 °, as shown in Fig. 7. This indicates that q~ff is caused by the crystallographic grain misalignment of the tape. Because such misalignment is an intrinsic property of the tape, q~ does not change with field, neither does ~Oeff. Recently, we found that tpeff remained the same even following neutron radiation, although the critical current dependence on the magnetic field had been changed considerably (Fig. 3(c)). Since neutron irradiation is not anisotropic and the induced defects are randomly distributed in the tape, this fact again proves that ~eee depends only on the crystallographic grain rnisalignment [14]. We investigated the microstructure of sample #93824 by means of SEM. We found that the crystallographic misalignment angle of the majority of grains, i.e. the average crystallographic misaiignment angle, q~,v, is around 10° (Fig. 8) in excellent agreement with the value of tp~ff. This fact proves in turn, that our 2D description of the tape is correct. Wilhelm et al. [11] have also observed a typical grain misalignment angle of about 10° along the longitudinal section of a Bi-2223 tape. However, the misalignment angle that we observed refers to the plane
8o' 7;o
1
ldooli o'lsoo
B (mT) Fig. 6. Critical current of tape #92152 as a function of B for B [Itapeplane and B _Ltape plane.
of the transverse cross section of the tapes, in which the field rotates. Bulaevskii et al. [7] fitted their Ic-Biic(B, O) curve and obtained a q~av value of 16°, which they claimed to be in quite good agreement with the ~&ff value obtained by Hu et al. [5]. It is interesting to find out what determines the typical q~eff values of around 10° observed in Refs. [5,7-9,11]. We propose that q~av, and therefore tpeff, is governed by the mechanical properties of the Bi-2223 compound. In the process of tape fabrication, a uniaxial force is applied to press the tape, which can be decomposed into a principal force, P], and a sheafing force, P2, directed along [001] and along the basal plane of the grain, respectively (Fig. 9). After several deformations of the grain, the dimensions of the grain tend to a certain ratio. According to theory, the maximum principal stress and the sheafing stress are
½df t / 2 P]l-l x dx 3Pll offaax = l sd 3 = 4sd2 _[0011
(9)
and ,/.(001)_
P2
-max -- Is
(10)
217
Q.Y. Hu et al. /Physica C 252 (1995) 211-220 12
i
,
i
i
•
|
respectively, where d, l, and s are the thickness, the length and the width o f the grain. In a stable mechanical state, we should have
t
o o 11
o o
[001----'~ =
6-"
O'r~ax
v
O ¢:D
O O
10
--
( 1 1)
~
3 P1
Referring to Fig. 9, w e have P 2 / P 1 = tan ~o and d / l = tan ~p. Therefore
O O
#92152
3 ~.(001) "max tan q (tan ~o) 2 = ~ ,r [001"--'~-"-,
(12)
~IDaX
9 •
" 16o' 1
o'2
o'2
o'3oo
or
B//c (mT) Fig. 7. Texture misalignment angle of #92152 calculated from Fig. 6.
~3 tan q~ =
' r (001) "max
4 n.[OOl] "
(13)
-max
Fig. 8. SEM micrograph of sample #93824 (transverse cross section). The vertical grids are parallel to the tape plane, ~'av is estimated to be
10°.
Q.Y. Hu et al. / Physica C 252 (1995) 211-220
218
p2~-
TAPEPLANE Fig. 9. Schematic illustration of the formation mechanism of the misalignment angle by deformation.
Thus, we have g, = arctan -4 rr [°°~1
(14)
~n'~x
In general, = ¢{~.(001> -[0011~ ~0av
v k ' m a x , Un~ax ]"
(15)
Thus, we relate q~avand therefore q~effto the mechanical properties of the grain or compound. During the fabrication of the tape, the grain dimension may be modified by the sintering process, i.e. some grains may be connected together again, but ~oav should stay unchanged. The typical grain dimension ratio of a Bi-2223 tape (after sintering) is d/l-- 1/20, but this does not correspond to the typical q~efe value of 10° (Fig. 9).
5. Grain misalignment and critical current density Table 1 lists the q~ef values of the samples. From the table, we note that there is no direct relationship between J~ and q~¢ff. Samples AB135 and AB136 have the same Je and q~eff values, whereas samples #92152 and #93824 have almost the same ~¢ff, but quite different Je values. If we compare AB136 with #92152, we find that its Jc is lower, although the former has a smaller q~ef, which indicates better grain alignment in the tape. This fact, namely that a smaller q~¢ffdoes not guarantee samples with higher Je and that samples with the same goeff may have different Jc, suggests that some other factors, besides the grain misalignment, may be of the same
importance for Jc. They could be the mass density of the superconducting core, the connectivity of the grains, etc. We believe, however, that the pinning capability for BIIc is a decisive factor for the enhancement of Je at high temperatures, if q~ff is small enough. We suggest, that the efforts to increase J~ of the Bi-2223 tapes should not only be directed towards improving the grain alignment, but also towards the introduction of more artificial pinning centres, which has been proved to be an effective way for the enhancement of Je(Bu c) [14].
6. Conclusion Based on the 2D behaviour of Ag clad Bi-2223 tapes with regard to their critical currents, the effective grain misalignment angle, q~eff,has been studied. The data of It(B, O) measurements on different samples show, that this value is around 10°, which is identical to the average crystallographic grain misalignment angle, q~av,of the tape. Preliminary studies reveal, that the average grain misalignment of the tape is related to the mechanical properties of the Bi-2223 compound. We suggest that the pinning capability for BII c is an important factor for J¢ in tapes with small misalignment angles.
Acknowledgements One of the authors (QYH) wishes to acknowledge fmancial support by the Austrian Exchange Service (OAD), Vienna during his one year stay at the Atomic Institute. We wish to thank R.M. Schalk for
Q.Y. Hu et al. / Physica C 252 (1995) 211-220 useful discussions and H. Niedermaier for technical assistance. This work was supported in part by Fonds zur F~Srderung der Wissenschaftlichen Forschung, Vienna, under contract No. 8849. A l l the authors from WoUongong wish to acknowledge financial support by Metal Manufacturers Ltd. and by the University o f Wollongong.
References [1] W.E. Lawrence and S. Doniaeh, Proc. 12th Int. Conf. on Low Temperature Physics, Kyoto, 1970, Keigaku, Tokyo (1971) p. 361. [2] P.H. Kes, J. Aarts, V.M. Vinokur and C.J. van der Beck, Phys. Rev. Lett. 64 (1990) 1063. [3] H. Raffy, S. Labdi, O. Laborde and P. Moncean, Phys. Rev. Lett. 66 (1991) 2515. [4] P. Schmitt, P. Kummeth, L. Schultz and G. SaemannIschenko, Phys. Rev. Lett. 67 (1991) 267. [5] Q.Y. Hu, H.W. Weber, S.X. Dou, H.K. Liu and H.W. Neumiiller, Proc. 1992 E-MRS Fall Meeting, Nov. 3-6, 1992, Strasbourg, published in J. Alloys Comp. 195 (1993) 515. [6] R.M. Schalk, G. Samadi Hosseinali, H.W. Weber, S. Griindorfer, D. Biiuerle, S. Proyer, P. Wagner, U. Frey and H. Adrian, Physica C 235-240 (1994) 2625. [7] L.N. Bulaevsldi, L.L. Daemen, M.P. Maley and Y. Coulter, Phys. Rev. B 48 (1993) 13798. [8] B. Hensel, J.-C. Grivel, A. Jeremie, A. Perin, A. Pollini and R. Fl'tikiger, Physica C 205 (1993) 329. [9] T. Kaneko, Y. Torii, H. Takei, K. Tada and K. Sato, Advances in Superconductivity VI, Vol. 2, eds. T. Fujita and Y. Shiohara (Springer, Berlin, 1994) p. 683. [10] S.X. Dou, H.K. Liu and Y.C. Gno, Appl. Phys. Lett. 60 (1992) 2929. [11] M. Wilhelm, H.W. Neumiiller and G. Ries, Physica C 185189 (1991) 2399. [12] Bulaevskii et al. [7] assumed a Gaussian distribution for ~. The application of this distribution to fit the experimental data has proved to be successful to some extent. [13] Q.Y. Hu, H.K. Liu and S.X. Don, Cryogenics 32 (i992) 1038. [14] Q.Y. Hu, H.W. Weber, F.M. Sauerzopf, G.W. Schulz, R.M. Schalk, H.W. Neumiiller and S.X. Dou, Appl. Phys. Lett. 65 (1994) 3008. [15] Q.Y. Hu, R.M. Schalk, H.W. Weber, H.K. Liu, R.K. Wang, C. Czurda and S.X. Dou, J. Appl. Phys. 78, in press.
Appendix A A s s u m i n g that a magnetic field is applied in the x - z - p l a n e in a 3D coordinate system, it intersects the
219
x-, y- and z-axes at the angles ce = 0, /3 = 90 °, and 3~= 90 ° - 0, respectively (Fig. 5). Then the field vector is given by B = ( B cos a . i + B = ( B cos O . i + B
cos / 3 . j + B
cos ~ / - k )
sin O . k ) .
(A.1)
The a - b - p l a n e s o f the grains in a tape are misaligned by tilting off the texture plane, or by tilting not only to the y-axis but also to the x-axis (Fig. 5). W e consider the normals o f those grains, which enclose an angle ~Omax with the z-axis. The entirety o f these normals forms a cone with 3' = ~Pr~x, which encloses all the normals of misaligned grains. A s a result, the a,b-planes o f these grains, enclose an angle ~Pmax with the Z = 0 plane. Therefore, the unit vector o f the grain along the c-direction is (see Fig. 5) n = (cos a i + cos 13 j + cos ~Omax k ) .
(A.2)
Therefore, the B component along the c-direction o f the grain is BIIc=B .n--B(cos
0 cos oL+ sin 0 cos
~Pmax)" (A.3)
Since cos2a + cos2/3 + cosZy = 1, we have / lt c = B . n = B ( c o s
0V/1 - cos2/3 - c o s 2
+ s i n 0 cos ~Pm~x),
max
(A.4)
or BII c = B . ,
= B ( c o s 0 ~sin2CPmax- cos2/3 + s i n 0 cos ~Pm~x)•
(A.5)
W e note that the spatial misalignment depends on BII c in a complicated way with regard to the angles. But, according to Eq. (A.5), we have BIIc
s i n ( 0 + ~Omax).
(A.6)
W e note that the right-hand side is the same as Eq. (1) in the text. Therefore, we see that the 2D misalignment (grain normals lie in the x - z - p l a n e ) represents an upper limit o f BII c for the 3D misalignment case (grain normals are not in one plane). Thus, the solution o f the three-dimensional problem o f grain misalignment can be approximated by a two-dimensional model.
220
Q.Y. Hu et aL / Physica C 252 (1995) 211-220
Appendix C. Summary of notations for the angles
Appendix B In an external magnetic field, the c-component of the field is given by BII c = B sin 0.
(B.1)
Differentiation of Eq. (B.1) leads to dBii ~ = B cos 0 dO.
(B.2)
Substituting B from Eq. (B.2) into Eq. (B.1) results in
dBLI c = BII c ctg 0 dO,
(B.3)
or
dBii c/Bn c = ctg 0 d0.
(B.4)
Eq. (13.4), therefore, represents the relative change of Btl c with 0.
Rotation angle between the magnetic field and the tape plane. q~: Crystallographic grain misalignment angle. qOmax : Maximum ~0 which encloses all the misalignment angles. Effective misalignment angle, defined by the ~Peff: critical current measurement, i.e. sin ~t~eff Bit c(B, 0 = O°)/B. 0 < %ff<~ qgmax . Effective rotation angle, 0~tf = 0 + %ff. 0eft: Average crystallographic grain misalignment qOav: angle. For instance, assuming that 9 has a Gaussian distribution, i.e. p(q~) = ( 1 / v ~ - tr) ×exp(qo2/2o-2), then qOav is or. As mentioned in the text, q~av= q°etf" 0:
=