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Proximity-effected magnetization in short NbTi multifilamentary wires
Cryogenics 35 (1995) 637-643 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved
WORTH EINEMANN
001 l-2275/95/$10.00
Proximity-effected magnetization NbTi multifilamentary wires N. Sakamoto
and T. Akune
Department of Electrical Engineering, Fukuoka 813, Japan Received
in short
2 February
1995; revised
Kyushu Sangyo University,
11 April
Higashi-Ku,
1995
Magnetization measurements have been performed as a function of wire length L for two series of NbTi multifilamentary wires of interfilamentary spacing 0.20 and 0.39 pm. The magnetization of the multifilamentary wire is made up of two components; one due to the superconducting filaments, /kfi, and the other arising from the interfilamentary proximity couplings, IMP. The filaments are so fine that the end effect can be neglected with respect to the magnetization /Lfi. Then the dependence of the magnetization on wire length mainly results from the proximity-induced magnetization I&, which can be determined as the twist-pitch dependent part of the magnetization. For wires which are relatively short compared with their diameters, flux line movement along the wire axis through the weak superconducting matrix exerts a significant influence on the field distribution and the proximity-effected magnetization A$, will be reduced considerably in comparison to that of the long wires. In this paper the two dimensional critical state model including the effect of low f3,, is developed. Characteristic penetration fields of the proximity-effected magnetization A$, in the direction of the wire axis are estimated by fitting with the two-dimensional critical state model and are shown to have a linear dependence on wire length. The shielding current densities both parallel and perpendicular to the wire axis are estimated and compared with those derived from the continuum twist-pitch theory. Keywords: multifilamentary lengths
wires;
end
In the high temperature oxide superconductors, a proximity effect exists between superconducting grains or the superconducting Cu-0 planes’. The anisotropic current resulting from structural features gives noticeable characteristics to the magnetization ‘J . It has been reported that high temperature superconductors exhibit a surface effect at low magnetic field and the onset of magnetic reversibility, i.e. the inability to carry a transport current at high fields near BcZ4. To obtain a clear understanding of the magnetic properties of the high temperature superconductors, further examination on each of these effects should be carried out. Among the investigations to give us further insight into the proximity effect and the anisotropic effect, the multifilamentary wire is considered as an appropriate model because the anisotropy induced from the proximity effect between the filaments has been extensively studied and is well understood5-8. The normal matrix in the multifilamentary wire shows weak superconductivity due to proximity couplings when the interfilamentary spacing is of the order of the normal coherence length &.+ The coupling current generated by the proximity effect gives rise to an additional magnetization. Twisting of the wire decreases the longitudinal distances
effect;
proximity
effect;
twist-pitch;
wire
over which the transverse coupling currents can flow. In this case, the proximity-effected magnetization was shown to be determined by a vector sum of the proximity current between the filaments and the transport current through the filaments. The magnetization currents show anisotropic characteristics depending on the relative direction to the filament. The current densities JcP of proximity-induced superconducting matrices in NbTi multifilamentary wires have been estimated from the twist-pitch (I,) dependence of the wire magnetization. The magnetization Mi of these wires, which is independent of l,, has been shown to arise mainly from the superconducting filament@. In wires with interfilamentary spacing dN comparable to &,, the superconductivity induced in the Cu matrix was shown to behave like that of a type-II superconductor, and the upper critical field BcZp was estimated in an earlier paper7 by applying the scaling law to the proximity currents. In the magnetization process of multifilamentary wires whose lengths L are not much larger than the wire diameter reduction in magnetization was R./Y an appreciable reported9,ro. Even in short multifilamentaty wires, the filament diameter df is usually much smaller than the wire
Cryogenics
1995 Volume
35, Number
10
637
Proximity-effected
magnetization:
N. Sakamoto
and T. Akune
length L, and in such cases the end effect of the wires should be considered to result from the matrix region. In this paper, the length dependence of the proximity-induced magnetization of the multifilamentary wire is measured in wires with different proximity strengths (realized by changing the filament spacing). The reduction of magnetization with decreasing wire length is examined for the first time by means of quantitatively estimating the proximity current density in the twisted wires. We shall report the magnetization measurements made on short multifilamentary wires with various lengths and extract the proximity-induced magnetization from the twist-pitch dependent part of the magnetization. The two-dimensional critical state model based on the Irie-Yamafuji theory” is developed to give a quantitative explanation of the length dependence of the proximity-induced magnetization. The anisotropy of the shielding currents is estimated and compared with results from the continuum twist-pitch theo$.
current
densities
in multifilamentary
(4) where S = 7rL@4, h, denotes the volume fraction of the superconducting matrix in the multifilamentary region and h, is the volume fraction of the multifilamentary region in the wire. If the critical current density of the filament, J&B,), is constant and independent of B,, as in the case of the Bean model”, the magnetization width AM, resulting from the current JCf through the filaments is given by5,‘j (5)
where h is the volume fraction of the superconducting filaments in the wire and is equal to h = A,( 1 - hp).
The net shielding current density J, in a short multifilamentary wire flows perpendicular to the applied magnetic field B,. The magnetic field component which enters or exits perpendicular to the axis B,(r) is determined by the shielding current density flowing parallel to the wire axis JS, and another component which moves along the wire axis B,(z) due to the shielding current density flowing perpendicular to the axis J,,. As shown in Figure 1 the superconducting filaments in the twisted wire make an angle cpto the direction of the wire axis and each shielding current density is the vector sum of two current components; the proximity current density JcP inside the matrix and the transport current density Jf through the superconducting filaments. Then the current densities J,, and jS, are given by J,, = J&in
rJ,,( r)costI dS S
hrM, = 4/&ddcd3T
Theory Shielding wire
In a long multifilamentary wire, the shielding current is composed only of J,,, and the proximity-effected magnetization MP assisted by JcP is expressed by the usual formula to give the magnetization in the cylindrical co-ordinate systern6
p = JCp+-(l+(Y)2)112
(1)
Magnetization
of slab: two-dimensional
case
Calculations are carried out for an infinite slab of width 21, and thickness 21, which will be applied to discuss the twodimensional magnetization of a short cylindrical wire with a length L and a diameter 2r. The magnetic field is applied along the z direction, as shown in Figure 2. The pinning force density Fp is given as CUP,where the parameter y represents the magnetic field dependence of F,, ‘I . The field component which penetrates along the x-axis, Bdx), and another component for the y axis, J&(y), in the initial stage of the magnetization, are expressed, respectively, by Bx(x)2-’
= BZ-Y
_ B;,Y
f
(6) x
J,, = J&OS cp= JCp
(2)
B,,fy)2-Y
= B,2-Y _
B;;Y;
(7)
Y
As is clear from the above equations, the magnetic fields reach the centre at the characteristic field values Bps and Bptin the respective directions. The fields B,, and B,,, which
B,
*.
4
4
Bx(x)
..a
Figure 1 Shielding current densities ./., and J., in twisted multifitamentary wire are induced under the external magnetic flux 6,. Penetrations of magnetic flux density S, along the crossing direction and 6, along the wire axis take place. J_ and Js, are the vector sum of the proximity current density Jcp and the superconducting filament current density Jf
638
Cryogenics
1995 Volume
35, Number
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(.
Bx
0
Figure 2 Penetration of magnetic flux densities in the direction of the y-axis BJyY) and the x-axis Bdx) in the initial magnetization process. At the central region, SA y) = Bx(x) = 0
Proximity-effected
relate to the critical shielding current densities along the yaxis, Jsf, and along the x-axis, J,,, are given, respectively, by B,, = ((2 - y)cyl,)“‘*-y’
= /_&Js,
(8)
Bpt = ((2 - y)aZ,)“‘2-Y’ = p&Jsa
(9)
Then the magnetization M is given by ‘Y o dyBx,,&y) - B,
Am=? Pt
(b:-’ + bl-y - 2b,3_Y)
3-Y +2
b *-’ -! 04
2-Y + __ 5-2y
(j,-Y
. i!!i
_
bt-Y)
(11)
(2b,5-%’
_
b:-zY
_
=b:-&(q<
9
Sample specifications
Sample
0,
(mm) 4 (pm)
I
0.299
II
0.156
dN (pm)
1, (mm)
L (mm)
2.13
0.39
2.6
1.12
0.20
2.8
2.40, 5.25, 7.60, 19.30 3.70, 5.10, 7.45, 10.10
L are listed in Table 1. The volume ratio of Cu to NbTi is 1.71. Both sample wires I and II were prepared using the twisting and cutting method described in a preceding paper7. The mean free path of the matrices is 1.5 pm, which was estimated from the resistivity just above the critical temperature of the NbTi filaments. The Cooper pair penetration length K;;’ of the matrices showed clean limit characteristics of a/T, where the (Yvalues were 1.07 and 0.76 for wires I and II, respectively7. A SQUID magnetometer MPMS2 was used to measure the wire magnetization at a temperature T = 4.5 K14.
Results and discussion
b:-2Y)
92-y
(92
and T. Akune
I
Proximity
where b;-y = b2-y e f 1
Table 1
N. Sakamoto
(10)
The reduced magnetization width Am between the increasing and decreasing field is given, when B, is larger than both B,, and Bpt, by
=-2-Y
magnetization:
1)
(12)
1)
(13)
All magnetic fields are normalized by BP? Detailed calculations will be given in the Appendix. An anisotropic parameter q defined by BpJBpptdetermines the field penetration pattern or expulsion pattern in the two-dimensional case. When q < 1, flux lines move easily along the width of the slab (X direction). For a sufficiently wide slab (q s==I), most flux lines enter and exit in the thickness direction and the end effect can be neglected. As will be shown later, the condition for safely neglecting the end effect should be
current
density
The magnetization of the twisted multifilamentary wires is classified into two components according to the existence of twist-pitch I,-dependence6*7. Equation (4) indicates that the magnetization component which depends linearly on Z, originates from proximity coupling between filaments. For a long multifilamentary wire with diameter D, = 0.299 mm, the same as sample I, the magnetization width LLV was measured7 as a function of lp and is plotted in Figure 3. One can see the linear dependence of Ah4 on lp. With the aid of Equation (4), one can estimate Jcp from the gradient of the line in Figure 3. The critical current density of the proximity-induced superconducting matrix of the wire is 30
, t
”
Be CT)
1
t
0
18
I
0.299mm 4
q = 5.
When the applied field B, is not sufficiently smaller than the upper critical field BcZp,the critical current should be modified by the term (1 - B,JBc2,)13. Then the characteristic fields B,,, and BP,, are reduced to (14) where the subscripts are used in the order of appearance. Calculated results for the whole magnetization curves will be summarized and depicted in the Appendix.
Experimental
details
Superconducting NbTi 8230 tine filaments were embedded in a copper matrix. Sample wires were drawn from the same billet. The wire diameter D,, filament diameter df, interfilamentary spacing dN, twist-pitch 1, and wire length
Figure 3 Magnetization width AM versus twist-pitch I, in long sample wires I’. The critical current density of the proximityinduced superconducting Cu matrix J,, can be estimated from the slope using Equation (4). The intercepts AMi of the lines on the AM axis are used to calculate J,r
Cryogenics
1995 Volume
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639
Proximity-effected
magnetization:
N. Sakamoto
and T. Akune
B, 0’)
0
0.1
0.
0.2
O-1”““““““’
‘11
a 8
B
B F -20 g.0 i
L (mm) 0 2.40
B
‘y”
%
.:j’
-40 -
“,
‘7:;;
.
19.30
’ Sample I I,=2.6(mm) I I I I I I I I I I I L I
I
I
I
and and
Figure 6 Initial magnetization curves of sample wires I. With a decrease in wire length L I= 19.3, 7.60, 5.25 and 2.40 mm), the magnitude of the magnetization also decreases. This fact indicates that magnetic flux penetration along the wire axis takes place
shown in Figure 4. AMi obtained from the intercept of the line on the AM axis represents the &-independent component which comes mainly from the superconducting filaments and is shown in Figure 5. The superconducting filament critical current densities J,r are estimated from the magnetization width AMi using Equation (5) and are plotted in Figure 4, together with the proximity current densities JCP. The JCP is two to three orders of magnitude smaller than JCr. The J,r is 2.2 x 10” A m-* at 0.3 T, comparable to the value for ordinary NbTi superconductors measured by Harada et al. using the four-probe resistive method6.
of sample I shown in Figure 6 clearly indicate sample length dependence. The &-dependent magnetization width values AMP were obtained using the hMi values of Figure 5 for the long wires, and are shown in Figure 7 for sample wires with lengths of 2.4, 5.25, 7.6 and 19.3 mm. Theoretical curves calculated from Equations ( 1 1 )-( 14) with y= 0.7 and B,, = 14 mT using the value of BcZp= 1.7 T obtained in previous work’ are shown as solid lines in Figure 7. This y value corresponds closely to that of a conventional type-II superconductor”. The characteristic field B,, plotted as a function of L in Figure 8 shows a linear relationship, as predicted in Equation (8). From this equation and the slope of Figure 8, one can estimate the ratio of the shielding currents J,, and J,,
Proximity-induced
Jst/Jsa = (M,$~L)I(B,J2r)
Figure4 Proximity-induced critical current densities & filament critical current densities &. Current densities & Jd are estimated from Equations (4) and (51, respectively
magnetization
Even for short wires, the &-independent magnetization AM, produced mainly by the fine filaments should not depend on wire length L, because L (= 21,) S dp On the other hand, the proximity-effected magnetization width AM, for short wires AM,=AM-AM,
= 0.041
From Equations ( 1) and (2) in the continuum twist-pitch mode16, this ratio Js,/Js, is given as 2m/l,, = 0.36. This difference can be ascribed to the varied configuration of the shielding regions in the wire and in the slab. One can suppose that the total shielding current induced against axial flux motion in the wire is much smaller than the corre-
(15)
will depend strongly on L. The initial magnetization
curves
0.299mm 4
0
0.5
B, l(T)
Figure 5 Applied magnetic field L?. dependence of twist-pitch independent part of the magnetization width AMi, which originates mainly from the superconducting filaments and is used to estimate ./,r in Figure 4
640
Cryogenics
1995 Volume
35, Number
10
rs
Figure7 Applied magnetic field 6, dependence of the twistpitch dependent part of the magnetization width AM,, of sample wires I. AM, is the additional magnetization induced by the proximity currents between superconducting filaments. Solid curves represent the calculated results using Equations (ll)(14). A& values are computed from Equation (15) using the observed AM and the values of AM, in Figure 5
Proximity-effected
Sample F30 a
II,,=14
mT
N. Sakamoto
and T. Akune
I mm
1,=2.6
magnetization:
j
:
$20 ..* ” .’
0
5
1.5
20
L &n) Figure 8 Characteristic penetration field along the wire axis BP, versus wire length L in sample wires I 0,
sponding current in the slab. The adjustable parameter B,, in Figure 7 seems to saturate when B,, > 2B,,. This means that the axial penetration of the magnetic fields can be safely neglected when q > 5. This fact is confirmed by the theoretical curves depicted in Figure A3. Sumption et al9 introduced the concept of the coupling field H,, from the creep measurement of untwisted wires of length 2-71 mm with nearly the same df and dN values as sample wire I. H,,,, was defined as the field above which the coupling between filaments weakens and rapid creep begins. H,,,, versus wire length L showed the same tendency as Figure 8, since the travelling distance of the flux lines along the wire axis increased in the long wire. They defined this distance as the transfer length 4,. In the case of a twisted wire, the flux motion along the wire axis is not governed by the proximity current density JCP but by the proximity-effected shielding current density J,, given by Equation (2), then L, will take the form qlltan cp using Equations (8) and (9). On the other hand, the creep rate of the untwisted and twisted wires measured by Sumption et al. showed field dependence with nearly constant peak fields, the origin of which was not clearly explained. This characteristic can be interpreted by considering that the creep in the twisted wire takes place in directions both parallel and perpendicular to the axis and that the proximityeffected shielding current densities J,, and J,, are also a function of JCP. Then the proximity-effected creep of any twist-pitch wire should be dominated only by JCP itself, which leads to peak fields independent of twisting. The measurements of hysteresis loss of short Nb,Sn multifilamentary wires by Goldfarb” showed a knee-like decrease in the loss on decreasing the wire length to its twist-pitch. This reduction is well described by the reduced magnetization loop due to the above-mentioned flux motion along the wire axis. Magnetization wire
in thin intetfilamentary
spacing
The same analysis was carried out for sample wire series II with small filament spacings (dN < &). The wire lengths are 3.7, 5.1, 7.45 and 10.1 mm. The proximity-induced magnetization and characteristic field B,, are estimated in a similar way to that described above and are shown in Figures 9 and 10. The obtained value of y is -0.4, a negative value, as in the exponent of the proximity current’. This value indicates a sharp magnetic field dependence exceeding the limit of y = 0 in the existing pinning theory. The underlying mechanism not yet clear, but could involve
’
0
-
0.5
1
Be CT) Figure 9 Applied magnetic field B. dependence of the twistpitch dependent part of the magnetization width AM, of sample wires II. The solid curves represent the calculated results using Equations (I 1 b-(14)
I I
300
I
I ,
I
I ,
Sample II 1,=2.8 200 F S -
mm
I
,
I
l
-
mT
-
m
e’ 0
$
100 -
8,,=180
OL
0
10 L5
(mm)
Figure 10 Characteristic penetration field along the wire axis S,, versus wire length L in sample wires II
It can be seen that a proximity-induced superconductivity. noticeable increase in the proximity effect takes place in sample wire II. The value of B,, for sample II is 13 times as large as that of sample I. From Equation (9), B,, is proportional to both the sample diameter (D, = 21,) and the shielding current J,,. The diameter term gives a multiplication factor of 0.52. The shielding current is made up of two components, one from JCP itself and the other from the reminder directional term ( l/sin ~0) in Equation ( 1). The JCP values of wires I and II were studied in detail in previous work’ and give rise to a difference of a factor of 10. The directional component is calculated to be a factor of two. The total factor is about 10, which agrees well with the observed value. On the other hand, using Equations ( 1), (2) and (8), one can obtain the relation B,,IB,, = nUlp and the ratio (LU$,$aL)/( B&J becomes constant. From the observed data, these ratios are approximately equal for both samples, that is 0.35 for wire I and 0.42 for wire II. The magnitude of the constant, however, differs considerably from r. The difference in flow pattern of the shielding current in the wire and in the slab is considered to be so large that a detailed theoretical treatment including the effect of wire shape is necessary.
Cryogenics
1995 Volume
35, Number
10
641
Proximity-effected
magnetization:
N. Sakamoto and T. Akune
Conclusions The magnetization change appearing in short multifilamentary wires is quantitatively discussed using the two-dimensional critical state model with the aid of the continuum twist-pitch theory. Characteristic penetration fields parallel to the wire axis B,, vary linearly with the wire length when B,, 5 2B,,. The proximity current density in the thin interfilamentary spacing wire decreased sharply with increasing field. The value of (J,,lJ,,)l(2r/l,) is constant (= 0.4) in all sample wires. This constancy is just as predicted by the continuum twist-pitch theory, but its magnitude is one order smaller than that calculated from the theory. A detailed theoretical analysis for a short wire configuration will be necessary to explain this discrepancy.
b:-y(l) = b:-r - 5
(A2)
b;-‘(t)
(A3)
= b;-y - q*-y(
In the initial magnetization process, the flux lines penetrate inwards from the surfaces and b = 0 at 5 = &, and 5 = lo, as shown in Figure Al; then to = (bJq)*-‘,
co = bz-7
(A4)
The flux densities b, and b, in the two regions (a) and (t) in Figure Al at 5 = &, f = ce coincide with each other and lead to
(A%
#klk= 4y-2 The averaged flux densities 6, and 6, are given by
Acknowledgements The authors would like to express their appreciation to Mr Y. Aoki of NEC Co., then a student for a Masters Degree, for his assistance with this work. Thanks are also given to Mr 0. Miura of the Furukawa Electric Co., Ltd for providing the sample wires. References 1 Dinger, T.R., Worthington, T.K., Gallagher, WJ. and SandStrom, R.L. Phys Rev L.etr (1987) 58 2687 2 Gyorgy, E.M., van Dover, R.B., Jackson, K.A., Schneemeyer, L.F. and Wasxzak, J.V. Appl Phys Lerr (1989) 55 283 3 Sauerzopf, F.M., Wiesinger, H.P. and Weher, H.W. Cryogenics 4
( 1990) 30 650 Cave, J.R. and Critchlow, P.R. IEEE Tram Mugn (1991) MAG-
(A7)
Then the magnetization in the initial process is given by (a) 0 5 b, < 1: m = 6, + &t- b, =-b,+
27 1379 5
Carr, WJ. Jr AC Loss and Macroscopic Theory of Superconductors Gordon and Breach Science Publishers, New York, USA (1983) 91
6
Harada, N., Mawatari, Y., Miura, O., Tanaka, Y. and Yamafuji, K. Cryogenics (1991) 31 183 Akune, T., Sakamoto, N., Miura, O., Tanaka, Y. and Yamafuji, K. J Low Temp Phys (1994) 94 219 Akune, T., Sakamoto, N., Miura, O., Tanaka, Y. and Yamafuji, K. Cryogenics (1995) 35 189
I 8 9
Sumption, 40A 199
M.D. and Colllngs,
E.W. Adv Cryog Eng (Mater) (1993)
Sakamoto, N., Aoki, Y. and Akune, T. Physica C (1994) 235240 2515 11 Irie, F. and Yamafuji, K. J Phys Sot Jpn (1967) 23 255 12 Bean, C.P. Rev Mod Phys ( 1964) 36 3 1 13 Matsushita, T. and Ktipfer, H. J Appl Phys (1988) 63 5048 14 Sakamoto, N., Akune, T. and Matsushita, T. Jpn J Appl Phys (1992) 31 Ll470 1.5 Goldfarh, R.B. and Itoh, K. J Appl Phys ( 1994) 75 2 115 10
Let us consider a slab sample having a thickness 21, along the y direction and a width 21, along the x direction and extending infinitely in the z direction. A magnetic field is applied parallel to the z-axis. Then the magnetic flux densities moving parallel to the y-axis, Bdy), and to the x-axis, l&(x), inside the slab are given by Equations (6) and (7) in the initial magnetization process. Normalization of the quantities x, y, B and M 5 = xll,, 5 = yll,, b, = BJB,, b, = BJBpt, bcz = B&B,,,
b5-2Y
(A81
As the applied magnetic field increases further, the flux front Jo reaches the centre line when q 2 1, or front &,does so when q < 1. Whole magnetizations for each process are similarly calculated and summarized as follows, for q 2 1. (b) 1 % b, 5 bc2, at increasing field: (b:-7 - b3_?y) + 2-y 5 -
m = MIB,
1995 Volume
35, Number
1
2 (b:-2y _ 2y 42-y
b5_;2y)
(Al)
10
--
co
ceI n
b, = BdBpt
leads to normalized forms of the magnetic flux densities using the anisotropic penetration parameter q (= BpalBpt)
Cryogenics
2
c
Appendix
642
(2 - Y12
- (3 - y)(5 - 2y) 42-y e
( a>
4
te
>
Eo
Figure Al Flux distribution in the initial magnetization process. Flux lines enter along the 6 axis at region (a) and along the 6 axis at region (t). At the central region (c) the flux density is still zero
Proximity-effected
magnetization:
N. Sakamoto and T. Akune
where ~!;Yz
(c)
62-Y_ e
1
0 5 b, 5 bE2, at
m=-b,+
(A101
decreasing field:
~{l+--&+2(f$-y](b:;Y-b;“)
2-y 2 -p(b:;+‘_ 5 - 2y 42-7
bs-27) e
(All)
where b:;y = b2-y e + 1
(A12)
Figure A2 Magnetization curves in the two-dimensional case, where bc2 (= &,,/B,) = 10, y= 0.5 and 9 (= f3,,&,,) = 0.2,0.4,0.6, 0.6, 1, 2, 5 and 10
(d) 0 L -lb,] z -1, for the reversed case:
OS-.
m=-b,+2-y ,[rl++(~~-y}!$~y
0.4 -
,
,
,
,
,
bJba \
-(I +&)1bei3-‘] 2-y 2 b&-W _ sy -p5 - 2 y 42-y (
Ib.ls-zy)
(A13) Ol/.
where b,:-y = 1 - lbe12-y
0
(A14)
The magnetization width between the increasing and decreasing field for q 2 1 is obtained from Equations (A9)-(A12) and is reduced to Equation (11). When the applied field b, approaches the upper critical field, the penetration fields are modified as in Equations (14) as a first iteration. When the width is relatively small and/or the critical current density J,, is small (the q < 1 case), the flux front along the 5 direction moves faster than that along the [ direction. The geometric pattern of the flux distribution is similar for both the q 2 1 and q < 1 cases. The magnetizations for q < 1 are given as follows:
(a’) 0 5
b, < q, at the initial magnetization, and magnetization m is given by Equation (A8); (b’) q < b, d bc2, at increasing field; (0 0 5 b, I bc2, at decreasing field, and m values are given by Equations (A9) and (Al I), respectively, where b,, is converted to b,,
1
2
3
4
5
4 Figure A3 Magnetization width Am versus 9 (= B,JB,,),where bc2(= Bc,&BpJ= 10, y= 0.5 and b$b,~=0.25,0.3,0.35,0.4 and 0.5
b$,Y = b,2-y f
(d’)
l/q2-Y
(A151
0 2 -/be1 2 -4, for the reversed case, and m is given by Equation (A13), where b,, is converted to brq b;$Y=
q2-y_
[bej2-Y
(A161
Typical magnetization curves are shown in Figure A2, where b,, = 10, y=OS and q =0.2, 0.4, 0.6, 0.8, 1, 2, 5 and 10. The magnetization width values Am are plotted in Figure A3 as a function of q in the case of bc2 = 10 and bJb,, = 0.25, 0.3, 0.35, 0.4 and 0.5. The rapid decrease in Am when lowering q indicates that the entry and exit of flux along the ,$ axis is superior to that along the 5 axis. Above q = 2, saturation of Am begins and the end effect can be regarded as being smaller.
Cryogenics
1995 Volume
35, Number
10
643
WORTH EINEMANN
001 l-2275/95/$10.00
Proximity-effected magnetization NbTi multifilamentary wires N. Sakamoto
and T. Akune
Department of Electrical Engineering, Fukuoka 813, Japan Received
in short
2 February
1995; revised
Kyushu Sangyo University,
11 April
Higashi-Ku,
1995
Magnetization measurements have been performed as a function of wire length L for two series of NbTi multifilamentary wires of interfilamentary spacing 0.20 and 0.39 pm. The magnetization of the multifilamentary wire is made up of two components; one due to the superconducting filaments, /kfi, and the other arising from the interfilamentary proximity couplings, IMP. The filaments are so fine that the end effect can be neglected with respect to the magnetization /Lfi. Then the dependence of the magnetization on wire length mainly results from the proximity-induced magnetization I&, which can be determined as the twist-pitch dependent part of the magnetization. For wires which are relatively short compared with their diameters, flux line movement along the wire axis through the weak superconducting matrix exerts a significant influence on the field distribution and the proximity-effected magnetization A$, will be reduced considerably in comparison to that of the long wires. In this paper the two dimensional critical state model including the effect of low f3,, is developed. Characteristic penetration fields of the proximity-effected magnetization A$, in the direction of the wire axis are estimated by fitting with the two-dimensional critical state model and are shown to have a linear dependence on wire length. The shielding current densities both parallel and perpendicular to the wire axis are estimated and compared with those derived from the continuum twist-pitch theory. Keywords: multifilamentary lengths
wires;
end
In the high temperature oxide superconductors, a proximity effect exists between superconducting grains or the superconducting Cu-0 planes’. The anisotropic current resulting from structural features gives noticeable characteristics to the magnetization ‘J . It has been reported that high temperature superconductors exhibit a surface effect at low magnetic field and the onset of magnetic reversibility, i.e. the inability to carry a transport current at high fields near BcZ4. To obtain a clear understanding of the magnetic properties of the high temperature superconductors, further examination on each of these effects should be carried out. Among the investigations to give us further insight into the proximity effect and the anisotropic effect, the multifilamentary wire is considered as an appropriate model because the anisotropy induced from the proximity effect between the filaments has been extensively studied and is well understood5-8. The normal matrix in the multifilamentary wire shows weak superconductivity due to proximity couplings when the interfilamentary spacing is of the order of the normal coherence length &.+ The coupling current generated by the proximity effect gives rise to an additional magnetization. Twisting of the wire decreases the longitudinal distances
effect;
proximity
effect;
twist-pitch;
wire
over which the transverse coupling currents can flow. In this case, the proximity-effected magnetization was shown to be determined by a vector sum of the proximity current between the filaments and the transport current through the filaments. The magnetization currents show anisotropic characteristics depending on the relative direction to the filament. The current densities JcP of proximity-induced superconducting matrices in NbTi multifilamentary wires have been estimated from the twist-pitch (I,) dependence of the wire magnetization. The magnetization Mi of these wires, which is independent of l,, has been shown to arise mainly from the superconducting filament@. In wires with interfilamentary spacing dN comparable to &,, the superconductivity induced in the Cu matrix was shown to behave like that of a type-II superconductor, and the upper critical field BcZp was estimated in an earlier paper7 by applying the scaling law to the proximity currents. In the magnetization process of multifilamentary wires whose lengths L are not much larger than the wire diameter reduction in magnetization was R./Y an appreciable reported9,ro. Even in short multifilamentaty wires, the filament diameter df is usually much smaller than the wire
Cryogenics
1995 Volume
35, Number
10
637
Proximity-effected
magnetization:
N. Sakamoto
and T. Akune
length L, and in such cases the end effect of the wires should be considered to result from the matrix region. In this paper, the length dependence of the proximity-induced magnetization of the multifilamentary wire is measured in wires with different proximity strengths (realized by changing the filament spacing). The reduction of magnetization with decreasing wire length is examined for the first time by means of quantitatively estimating the proximity current density in the twisted wires. We shall report the magnetization measurements made on short multifilamentary wires with various lengths and extract the proximity-induced magnetization from the twist-pitch dependent part of the magnetization. The two-dimensional critical state model based on the Irie-Yamafuji theory” is developed to give a quantitative explanation of the length dependence of the proximity-induced magnetization. The anisotropy of the shielding currents is estimated and compared with results from the continuum twist-pitch theo$.
current
densities
in multifilamentary
(4) where S = 7rL@4, h, denotes the volume fraction of the superconducting matrix in the multifilamentary region and h, is the volume fraction of the multifilamentary region in the wire. If the critical current density of the filament, J&B,), is constant and independent of B,, as in the case of the Bean model”, the magnetization width AM, resulting from the current JCf through the filaments is given by5,‘j (5)
where h is the volume fraction of the superconducting filaments in the wire and is equal to h = A,( 1 - hp).
The net shielding current density J, in a short multifilamentary wire flows perpendicular to the applied magnetic field B,. The magnetic field component which enters or exits perpendicular to the axis B,(r) is determined by the shielding current density flowing parallel to the wire axis JS, and another component which moves along the wire axis B,(z) due to the shielding current density flowing perpendicular to the axis J,,. As shown in Figure 1 the superconducting filaments in the twisted wire make an angle cpto the direction of the wire axis and each shielding current density is the vector sum of two current components; the proximity current density JcP inside the matrix and the transport current density Jf through the superconducting filaments. Then the current densities J,, and jS, are given by J,, = J&in
rJ,,( r)costI dS S
hrM, = 4/&ddcd3T
Theory Shielding wire
In a long multifilamentary wire, the shielding current is composed only of J,,, and the proximity-effected magnetization MP assisted by JcP is expressed by the usual formula to give the magnetization in the cylindrical co-ordinate systern6
p = JCp+-(l+(Y)2)112
(1)
Magnetization
of slab: two-dimensional
case
Calculations are carried out for an infinite slab of width 21, and thickness 21, which will be applied to discuss the twodimensional magnetization of a short cylindrical wire with a length L and a diameter 2r. The magnetic field is applied along the z direction, as shown in Figure 2. The pinning force density Fp is given as CUP,where the parameter y represents the magnetic field dependence of F,, ‘I . The field component which penetrates along the x-axis, Bdx), and another component for the y axis, J&(y), in the initial stage of the magnetization, are expressed, respectively, by Bx(x)2-’
= BZ-Y
_ B;,Y
f
(6) x
J,, = J&OS cp= JCp
(2)
B,,fy)2-Y
= B,2-Y _
B;;Y;
(7)
Y
As is clear from the above equations, the magnetic fields reach the centre at the characteristic field values Bps and Bptin the respective directions. The fields B,, and B,,, which
B,
*.
4
4
Bx(x)
..a
Figure 1 Shielding current densities ./., and J., in twisted multifitamentary wire are induced under the external magnetic flux 6,. Penetrations of magnetic flux density S, along the crossing direction and 6, along the wire axis take place. J_ and Js, are the vector sum of the proximity current density Jcp and the superconducting filament current density Jf
638
Cryogenics
1995 Volume
35, Number
10
(.
Bx
0
Figure 2 Penetration of magnetic flux densities in the direction of the y-axis BJyY) and the x-axis Bdx) in the initial magnetization process. At the central region, SA y) = Bx(x) = 0
Proximity-effected
relate to the critical shielding current densities along the yaxis, Jsf, and along the x-axis, J,,, are given, respectively, by B,, = ((2 - y)cyl,)“‘*-y’
= /_&Js,
(8)
Bpt = ((2 - y)aZ,)“‘2-Y’ = p&Jsa
(9)
Then the magnetization M is given by ‘Y o dyBx,,&y) - B,
Am=? Pt
(b:-’ + bl-y - 2b,3_Y)
3-Y +2
b *-’ -! 04
2-Y + __ 5-2y
(j,-Y
. i!!i
_
bt-Y)
(11)
(2b,5-%’
_
b:-zY
_
=b:-&(q<
9
Sample specifications
Sample
0,
(mm) 4 (pm)
I
0.299
II
0.156
dN (pm)
1, (mm)
L (mm)
2.13
0.39
2.6
1.12
0.20
2.8
2.40, 5.25, 7.60, 19.30 3.70, 5.10, 7.45, 10.10
L are listed in Table 1. The volume ratio of Cu to NbTi is 1.71. Both sample wires I and II were prepared using the twisting and cutting method described in a preceding paper7. The mean free path of the matrices is 1.5 pm, which was estimated from the resistivity just above the critical temperature of the NbTi filaments. The Cooper pair penetration length K;;’ of the matrices showed clean limit characteristics of a/T, where the (Yvalues were 1.07 and 0.76 for wires I and II, respectively7. A SQUID magnetometer MPMS2 was used to measure the wire magnetization at a temperature T = 4.5 K14.
Results and discussion
b:-2Y)
92-y
(92
and T. Akune
I
Proximity
where b;-y = b2-y e f 1
Table 1
N. Sakamoto
(10)
The reduced magnetization width Am between the increasing and decreasing field is given, when B, is larger than both B,, and Bpt, by
=-2-Y
magnetization:
1)
(12)
1)
(13)
All magnetic fields are normalized by BP? Detailed calculations will be given in the Appendix. An anisotropic parameter q defined by BpJBpptdetermines the field penetration pattern or expulsion pattern in the two-dimensional case. When q < 1, flux lines move easily along the width of the slab (X direction). For a sufficiently wide slab (q s==I), most flux lines enter and exit in the thickness direction and the end effect can be neglected. As will be shown later, the condition for safely neglecting the end effect should be
current
density
The magnetization of the twisted multifilamentary wires is classified into two components according to the existence of twist-pitch I,-dependence6*7. Equation (4) indicates that the magnetization component which depends linearly on Z, originates from proximity coupling between filaments. For a long multifilamentary wire with diameter D, = 0.299 mm, the same as sample I, the magnetization width LLV was measured7 as a function of lp and is plotted in Figure 3. One can see the linear dependence of Ah4 on lp. With the aid of Equation (4), one can estimate Jcp from the gradient of the line in Figure 3. The critical current density of the proximity-induced superconducting matrix of the wire is 30
, t
”
Be CT)
1
t
0
18
I
0.299mm 4
q = 5.
When the applied field B, is not sufficiently smaller than the upper critical field BcZp,the critical current should be modified by the term (1 - B,JBc2,)13. Then the characteristic fields B,,, and BP,, are reduced to (14) where the subscripts are used in the order of appearance. Calculated results for the whole magnetization curves will be summarized and depicted in the Appendix.
Experimental
details
Superconducting NbTi 8230 tine filaments were embedded in a copper matrix. Sample wires were drawn from the same billet. The wire diameter D,, filament diameter df, interfilamentary spacing dN, twist-pitch 1, and wire length
Figure 3 Magnetization width AM versus twist-pitch I, in long sample wires I’. The critical current density of the proximityinduced superconducting Cu matrix J,, can be estimated from the slope using Equation (4). The intercepts AMi of the lines on the AM axis are used to calculate J,r
Cryogenics
1995 Volume
35, Number
10
639
Proximity-effected
magnetization:
N. Sakamoto
and T. Akune
B, 0’)
0
0.1
0.
0.2
O-1”““““““’
‘11
a 8
B
B F -20 g.0 i
L (mm) 0 2.40
B
‘y”
%
.:j’
-40 -
“,
‘7:;;
.
19.30
’ Sample I I,=2.6(mm) I I I I I I I I I I I L I
I
I
I
and and
Figure 6 Initial magnetization curves of sample wires I. With a decrease in wire length L I= 19.3, 7.60, 5.25 and 2.40 mm), the magnitude of the magnetization also decreases. This fact indicates that magnetic flux penetration along the wire axis takes place
shown in Figure 4. AMi obtained from the intercept of the line on the AM axis represents the &-independent component which comes mainly from the superconducting filaments and is shown in Figure 5. The superconducting filament critical current densities J,r are estimated from the magnetization width AMi using Equation (5) and are plotted in Figure 4, together with the proximity current densities JCP. The JCP is two to three orders of magnitude smaller than JCr. The J,r is 2.2 x 10” A m-* at 0.3 T, comparable to the value for ordinary NbTi superconductors measured by Harada et al. using the four-probe resistive method6.
of sample I shown in Figure 6 clearly indicate sample length dependence. The &-dependent magnetization width values AMP were obtained using the hMi values of Figure 5 for the long wires, and are shown in Figure 7 for sample wires with lengths of 2.4, 5.25, 7.6 and 19.3 mm. Theoretical curves calculated from Equations ( 1 1 )-( 14) with y= 0.7 and B,, = 14 mT using the value of BcZp= 1.7 T obtained in previous work’ are shown as solid lines in Figure 7. This y value corresponds closely to that of a conventional type-II superconductor”. The characteristic field B,, plotted as a function of L in Figure 8 shows a linear relationship, as predicted in Equation (8). From this equation and the slope of Figure 8, one can estimate the ratio of the shielding currents J,, and J,,
Proximity-induced
Jst/Jsa = (M,$~L)I(B,J2r)
Figure4 Proximity-induced critical current densities & filament critical current densities &. Current densities & Jd are estimated from Equations (4) and (51, respectively
magnetization
Even for short wires, the &-independent magnetization AM, produced mainly by the fine filaments should not depend on wire length L, because L (= 21,) S dp On the other hand, the proximity-effected magnetization width AM, for short wires AM,=AM-AM,
= 0.041
From Equations ( 1) and (2) in the continuum twist-pitch mode16, this ratio Js,/Js, is given as 2m/l,, = 0.36. This difference can be ascribed to the varied configuration of the shielding regions in the wire and in the slab. One can suppose that the total shielding current induced against axial flux motion in the wire is much smaller than the corre-
(15)
will depend strongly on L. The initial magnetization
curves
0.299mm 4
0
0.5
B, l(T)
Figure 5 Applied magnetic field L?. dependence of twist-pitch independent part of the magnetization width AMi, which originates mainly from the superconducting filaments and is used to estimate ./,r in Figure 4
640
Cryogenics
1995 Volume
35, Number
10
rs
Figure7 Applied magnetic field 6, dependence of the twistpitch dependent part of the magnetization width AM,, of sample wires I. AM, is the additional magnetization induced by the proximity currents between superconducting filaments. Solid curves represent the calculated results using Equations (ll)(14). A& values are computed from Equation (15) using the observed AM and the values of AM, in Figure 5
Proximity-effected
Sample F30 a
II,,=14
mT
N. Sakamoto
and T. Akune
I mm
1,=2.6
magnetization:
j
:
$20 ..* ” .’
0
5
1.5
20
L &n) Figure 8 Characteristic penetration field along the wire axis BP, versus wire length L in sample wires I 0,
sponding current in the slab. The adjustable parameter B,, in Figure 7 seems to saturate when B,, > 2B,,. This means that the axial penetration of the magnetic fields can be safely neglected when q > 5. This fact is confirmed by the theoretical curves depicted in Figure A3. Sumption et al9 introduced the concept of the coupling field H,, from the creep measurement of untwisted wires of length 2-71 mm with nearly the same df and dN values as sample wire I. H,,,, was defined as the field above which the coupling between filaments weakens and rapid creep begins. H,,,, versus wire length L showed the same tendency as Figure 8, since the travelling distance of the flux lines along the wire axis increased in the long wire. They defined this distance as the transfer length 4,. In the case of a twisted wire, the flux motion along the wire axis is not governed by the proximity current density JCP but by the proximity-effected shielding current density J,, given by Equation (2), then L, will take the form qlltan cp using Equations (8) and (9). On the other hand, the creep rate of the untwisted and twisted wires measured by Sumption et al. showed field dependence with nearly constant peak fields, the origin of which was not clearly explained. This characteristic can be interpreted by considering that the creep in the twisted wire takes place in directions both parallel and perpendicular to the axis and that the proximityeffected shielding current densities J,, and J,, are also a function of JCP. Then the proximity-effected creep of any twist-pitch wire should be dominated only by JCP itself, which leads to peak fields independent of twisting. The measurements of hysteresis loss of short Nb,Sn multifilamentary wires by Goldfarb” showed a knee-like decrease in the loss on decreasing the wire length to its twist-pitch. This reduction is well described by the reduced magnetization loop due to the above-mentioned flux motion along the wire axis. Magnetization wire
in thin intetfilamentary
spacing
The same analysis was carried out for sample wire series II with small filament spacings (dN < &). The wire lengths are 3.7, 5.1, 7.45 and 10.1 mm. The proximity-induced magnetization and characteristic field B,, are estimated in a similar way to that described above and are shown in Figures 9 and 10. The obtained value of y is -0.4, a negative value, as in the exponent of the proximity current’. This value indicates a sharp magnetic field dependence exceeding the limit of y = 0 in the existing pinning theory. The underlying mechanism not yet clear, but could involve
’
0
-
0.5
1
Be CT) Figure 9 Applied magnetic field B. dependence of the twistpitch dependent part of the magnetization width AM, of sample wires II. The solid curves represent the calculated results using Equations (I 1 b-(14)
I I
300
I
I ,
I
I ,
Sample II 1,=2.8 200 F S -
mm
I
,
I
l
-
mT
-
m
e’ 0
$
100 -
8,,=180
OL
0
10 L5
(mm)
Figure 10 Characteristic penetration field along the wire axis S,, versus wire length L in sample wires II
It can be seen that a proximity-induced superconductivity. noticeable increase in the proximity effect takes place in sample wire II. The value of B,, for sample II is 13 times as large as that of sample I. From Equation (9), B,, is proportional to both the sample diameter (D, = 21,) and the shielding current J,,. The diameter term gives a multiplication factor of 0.52. The shielding current is made up of two components, one from JCP itself and the other from the reminder directional term ( l/sin ~0) in Equation ( 1). The JCP values of wires I and II were studied in detail in previous work’ and give rise to a difference of a factor of 10. The directional component is calculated to be a factor of two. The total factor is about 10, which agrees well with the observed value. On the other hand, using Equations ( 1), (2) and (8), one can obtain the relation B,,IB,, = nUlp and the ratio (LU$,$aL)/( B&J becomes constant. From the observed data, these ratios are approximately equal for both samples, that is 0.35 for wire I and 0.42 for wire II. The magnitude of the constant, however, differs considerably from r. The difference in flow pattern of the shielding current in the wire and in the slab is considered to be so large that a detailed theoretical treatment including the effect of wire shape is necessary.
Cryogenics
1995 Volume
35, Number
10
641
Proximity-effected
magnetization:
N. Sakamoto and T. Akune
Conclusions The magnetization change appearing in short multifilamentary wires is quantitatively discussed using the two-dimensional critical state model with the aid of the continuum twist-pitch theory. Characteristic penetration fields parallel to the wire axis B,, vary linearly with the wire length when B,, 5 2B,,. The proximity current density in the thin interfilamentary spacing wire decreased sharply with increasing field. The value of (J,,lJ,,)l(2r/l,) is constant (= 0.4) in all sample wires. This constancy is just as predicted by the continuum twist-pitch theory, but its magnitude is one order smaller than that calculated from the theory. A detailed theoretical analysis for a short wire configuration will be necessary to explain this discrepancy.
b:-y(l) = b:-r - 5
(A2)
b;-‘(t)
(A3)
= b;-y - q*-y(
In the initial magnetization process, the flux lines penetrate inwards from the surfaces and b = 0 at 5 = &, and 5 = lo, as shown in Figure Al; then to = (bJq)*-‘,
co = bz-7
(A4)
The flux densities b, and b, in the two regions (a) and (t) in Figure Al at 5 = &, f = ce coincide with each other and lead to
(A%
#klk= 4y-2 The averaged flux densities 6, and 6, are given by
Acknowledgements The authors would like to express their appreciation to Mr Y. Aoki of NEC Co., then a student for a Masters Degree, for his assistance with this work. Thanks are also given to Mr 0. Miura of the Furukawa Electric Co., Ltd for providing the sample wires. References 1 Dinger, T.R., Worthington, T.K., Gallagher, WJ. and SandStrom, R.L. Phys Rev L.etr (1987) 58 2687 2 Gyorgy, E.M., van Dover, R.B., Jackson, K.A., Schneemeyer, L.F. and Wasxzak, J.V. Appl Phys Lerr (1989) 55 283 3 Sauerzopf, F.M., Wiesinger, H.P. and Weher, H.W. Cryogenics 4
( 1990) 30 650 Cave, J.R. and Critchlow, P.R. IEEE Tram Mugn (1991) MAG-
(A7)
Then the magnetization in the initial process is given by (a) 0 5 b, < 1: m = 6, + &t- b, =-b,+
27 1379 5
Carr, WJ. Jr AC Loss and Macroscopic Theory of Superconductors Gordon and Breach Science Publishers, New York, USA (1983) 91
6
Harada, N., Mawatari, Y., Miura, O., Tanaka, Y. and Yamafuji, K. Cryogenics (1991) 31 183 Akune, T., Sakamoto, N., Miura, O., Tanaka, Y. and Yamafuji, K. J Low Temp Phys (1994) 94 219 Akune, T., Sakamoto, N., Miura, O., Tanaka, Y. and Yamafuji, K. Cryogenics (1995) 35 189
I 8 9
Sumption, 40A 199
M.D. and Colllngs,
E.W. Adv Cryog Eng (Mater) (1993)
Sakamoto, N., Aoki, Y. and Akune, T. Physica C (1994) 235240 2515 11 Irie, F. and Yamafuji, K. J Phys Sot Jpn (1967) 23 255 12 Bean, C.P. Rev Mod Phys ( 1964) 36 3 1 13 Matsushita, T. and Ktipfer, H. J Appl Phys (1988) 63 5048 14 Sakamoto, N., Akune, T. and Matsushita, T. Jpn J Appl Phys (1992) 31 Ll470 1.5 Goldfarh, R.B. and Itoh, K. J Appl Phys ( 1994) 75 2 115 10
Let us consider a slab sample having a thickness 21, along the y direction and a width 21, along the x direction and extending infinitely in the z direction. A magnetic field is applied parallel to the z-axis. Then the magnetic flux densities moving parallel to the y-axis, Bdy), and to the x-axis, l&(x), inside the slab are given by Equations (6) and (7) in the initial magnetization process. Normalization of the quantities x, y, B and M 5 = xll,, 5 = yll,, b, = BJB,, b, = BJBpt, bcz = B&B,,,
b5-2Y
(A81
As the applied magnetic field increases further, the flux front Jo reaches the centre line when q 2 1, or front &,does so when q < 1. Whole magnetizations for each process are similarly calculated and summarized as follows, for q 2 1. (b) 1 % b, 5 bc2, at increasing field: (b:-7 - b3_?y) + 2-y 5 -
m = MIB,
1995 Volume
35, Number
1
2 (b:-2y _ 2y 42-y
b5_;2y)
(Al)
10
--
co
ceI n
b, = BdBpt
leads to normalized forms of the magnetic flux densities using the anisotropic penetration parameter q (= BpalBpt)
Cryogenics
2
c
Appendix
642
(2 - Y12
- (3 - y)(5 - 2y) 42-y e
( a>
4
te
>
Eo
Figure Al Flux distribution in the initial magnetization process. Flux lines enter along the 6 axis at region (a) and along the 6 axis at region (t). At the central region (c) the flux density is still zero
Proximity-effected
magnetization:
N. Sakamoto and T. Akune
where ~!;Yz
(c)
62-Y_ e
1
0 5 b, 5 bE2, at
m=-b,+
(A101
decreasing field:
~{l+--&+2(f$-y](b:;Y-b;“)
2-y 2 -p(b:;+‘_ 5 - 2y 42-7
bs-27) e
(All)
where b:;y = b2-y e + 1
(A12)
Figure A2 Magnetization curves in the two-dimensional case, where bc2 (= &,,/B,) = 10, y= 0.5 and 9 (= f3,,&,,) = 0.2,0.4,0.6, 0.6, 1, 2, 5 and 10
(d) 0 L -lb,] z -1, for the reversed case:
OS-.
m=-b,+2-y ,[rl++(~~-y}!$~y
0.4 -
,
,
,
,
,
bJba \
-(I +&)1bei3-‘] 2-y 2 b&-W _ sy -p5 - 2 y 42-y (
Ib.ls-zy)
(A13) Ol/.
where b,:-y = 1 - lbe12-y
0
(A14)
The magnetization width between the increasing and decreasing field for q 2 1 is obtained from Equations (A9)-(A12) and is reduced to Equation (11). When the applied field b, approaches the upper critical field, the penetration fields are modified as in Equations (14) as a first iteration. When the width is relatively small and/or the critical current density J,, is small (the q < 1 case), the flux front along the 5 direction moves faster than that along the [ direction. The geometric pattern of the flux distribution is similar for both the q 2 1 and q < 1 cases. The magnetizations for q < 1 are given as follows:
(a’) 0 5
b, < q, at the initial magnetization, and magnetization m is given by Equation (A8); (b’) q < b, d bc2, at increasing field; (0 0 5 b, I bc2, at decreasing field, and m values are given by Equations (A9) and (Al I), respectively, where b,, is converted to b,,
1
2
3
4
5
4 Figure A3 Magnetization width Am versus 9 (= B,JB,,),where bc2(= Bc,&BpJ= 10, y= 0.5 and b$b,~=0.25,0.3,0.35,0.4 and 0.5
b$,Y = b,2-y f
(d’)
l/q2-Y
(A151
0 2 -/be1 2 -4, for the reversed case, and m is given by Equation (A13), where b,, is converted to brq b;$Y=
q2-y_
[bej2-Y
(A161
Typical magnetization curves are shown in Figure A2, where b,, = 10, y=OS and q =0.2, 0.4, 0.6, 0.8, 1, 2, 5 and 10. The magnetization width values Am are plotted in Figure A3 as a function of q in the case of bc2 = 10 and bJb,, = 0.25, 0.3, 0.35, 0.4 and 0.5. The rapid decrease in Am when lowering q indicates that the entry and exit of flux along the ,$ axis is superior to that along the 5 axis. Above q = 2, saturation of Am begins and the end effect can be regarded as being smaller.
Cryogenics
1995 Volume
35, Number
10
643