Z. Phys. B 102, 331–336 (1997)
ZEITSCHRIFT
FÜR PHYSIK B
c Springer-Verlag 1997
Specimen geometry effects on the irreversible magnetization
in the low field regime for specimens of bulk Nb3Sn
M. Däumling1,? , W. Goldacker2
1
2
Département de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland
Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany
Received: 18 October 1996
Abstract. Irreversible magnetization curves in particular in
self field conditions were investigated for specimens of bulk
Nb3 Sn as an isotropic and non-granular alternative to high
Tc materials. The specimens were cut into squares with side
length to thickness ratios of about 0.8 to almost 20, leading to diamagnetic slopes |χd | ranging from slightly more
than 1 (slab in parallel field) to 13 (plate in perpendicular
field). The peak in the low field magnetization (caused by
the field dependence of jc ) moves from negative field for
specimens with small |χd | (where it should be according
to the standard Bean model) to zero or even positive fields
for specimens with large |χd |. Finite element self consistent
calculations with field dependent and thus spacially varying
jc were used to fit the measured hysteresis curves. A strong
current peak in the center plane of the specimens – caused
by the demagnetization field – probably causes the shift of
the magnetization peak to positive field. Best agreement between the experimental and calculated magnetizations have
been obtained for the extreme geometries: either thin disk
in perpendicular (χd < −4) or thin slab in parallel magnetic
field (χd > −1.2).
PACS: 74.25.Ha; 74.60.Jg; 74.70.Ad
1. Introduction
The irreversible magnetization of superconductors is usually
connected with the critical current density jc by the critical
state model [1, 2]. The current density values are limited by
flux pinning, and determine the macroscopic magnetic field
profile in the specimens according to (neglecting contributions from surface currents)
jc = curlH.
(1)
Here jc and H are vectorial quantities. If the specimen has a
cylindrical geometry with the symmetry axis z then we can
write (j in circumferential direction)
? Current address: Forschungszentrum Karlsruhe, D-76021 Karlsruhe,
Germany
jc =
dHz
dHr
−
dz
dr
(2)
The original critical state model was conceived for cylindrical or slab geometries without demagnetization effects.
Thus the first term could be neglected. As a result of the disk
or plate like geometry of single crystals of high temperature
superconductors [3–11] lately much interest has been concentrated on how the critical state would behave in geometries that show considerable demagnetization fields. These
magnetic fields are generated in the specimen and are usually not parallel to the applied magnetic field. Thus flux line
bending occurs and the critical current is not only determined by the flux line gradient, but also the curvature of the
flux lines. In this case the second term becomes important,
and even dominant in the case of thin tapes or disks in a perpendicular field. However, most work has been carried out
considered flux penetration for field independent jc in the
zero thickness limit, applicable for thin strips or disks. Recently the field dependence of jc has been taken into account
([8] and references therein), and attempts to model flux penetration [12] and distribution [13, 14] in three dimensional
objects have been made.
One particular aspect that was measured in plate shaped
specimens of the high Tc materials was that the peak in
the irreversible magnetization that usually occurs in negative field when decreasing the external magnetic field was
positioned at a positive field [15–17]. This peak is caused
by the field dependence of jc or flux creep [18] and must be
– in the classical critical state model – always located in a
negative magnetic field. This is the case because a negative
applied field is necessary to create some minimum effective field – and thus maximum shielding current – in the
specimen interior.
In this article we will concentrate on exactly this problem: the irreversible magnetization in the vicinity of zero
magnetic field, the self-field regime. Here the magnetic field
generated by the specimen itself is of the same order of
magnitude or larger than the external field. Significant field
variations exist across the specimen. In addition, the demagnetization field twists the applied magnetic field away from
its original orientation.
332
The goal of this article is to examine this phenomenon
using specimens of bulk polycrystalline Nb3 Sn. These specimens have the advantage that they are – contrary to the high
Tc materials – isotropic, have rather high critical current
densities and can be cut fairly easily into different shapes.
In addition, flux creep effects [18] are negligible. 3D model
calculations using magnetic field dependent – and thus spatially varying – jc values were carried out in order to fit the
measured magnetization curves.
z
z
r
x
H
H
I(r,z)
I(x)
φ
Fig. 1a,b. Definitions of the geometry for magnetic field and current calculations
2. Experimental
The Nb3 Sn specimens are polycrystalline, non-porous, and
were prepared powder-metallurgically [19] in a high pressure/high temperature reaction between liquid tin and niobium. The specimens were sparc eroded first into a long
square rod of cross section (3.72x3.72 mm2 ) From this rod
slices with thicknesses of 4.25, 1.7, 0.71, 0.33 and 0.2 mm
were eroded. The onset critical temperature (measured with
a SQUID magnetometer) is 18.1K. From the data of Orlando
et al. [20] we can estimate that the lower critical field Hc1
is of order 30mT (10K) or 50mT (5K).
The microstructure of the specimens was investigated
with optical microscopy. For this purpose a slice – broken
into two pieces – was embedded into epoxy resin with the
two pieces perpendicular to each other. The surfaces were
then polished and investigated with the aid of an optical
microscope. This way it should have been possible to detect anisotropies in the grain structure. However, none was
detected; the grains and grain structure were isotropic. The
grain size is approximately 10 µm. X-ray scattering shows
the specimen to be essentially single phase Nb3 Sn with some
traces of unreacted Nb and no texture of the A15 phase.
The magnetization was measured with a vibrating sample magnetometer (VSM) in a superconducting high field
solenoid. The specimen temperature was controlled by admitting helium gas into the specimen chamber through a
needle valve. Thus temperatures between 2 and 120K can
be achieved. The specimen stay in He gas (p < 10mbar)
all the time at all temperatures. In order to obtain as many
different configurations as possible the plate like specimens
were mounted both perpendicular and parallel to the magnetic field. Hysteresis curves were measured at 10 and 15K.
At 5K most specimens showed flux jumping, in particular
the thicker ones.
3. Calculations
3.1. Cylindrical geometry
The squares used in the experiment were approximated by
disks of the same diameter. As typical shielding fields in
our specimens are significantly larger than Hc1 we have neglected contributions from Meissner currents. We will see
(below) that in a more sophisticated calculation these effects will have to be included. Calculations of the critical
state were carried out for a disk of height d and radius r0
in cylindrical coordinates (see Fig. 1a). The field Happl was
applied in the z direction. From the cylindrical symmetry
it follows that currents only flow in the circumferential direction. The disk was divided into n x m ring segments (n
along the radius and m along the height). The radial and
axial fields (hr and hz , respectively) at arbitrary coordinates
r and z created by a current loop of radius a with the current
I flowing are given by [21]
hr (r, z) =
hz (r, z) =
I
z
[−K(k)
2π r[(a + r)2 + z 2 ]1/2
a2 + r 2 + z 2
E(k)]
+
(a − r)2 + z 2
I
1
[K(k)
2
2π [(a + r) + z 2 ]1/2
a2 − r 2 − z 2
E(k)]
+
(a − r)2 + z 2
(3)
(4)
with k 2 = 4ar[(a + r)2 + z 2 ]−1 . The center of the loop is
located at r = z = 0. K and E are complete elliptic integrals of the first and second kind. The magnetic field at the
coordinates of each segment is then the sum of the fields
generated by all the other segments, plus the applied field.
The iterative calculation proceeds as follows: first a constant current density j=j(Hmax ) is assumed to flow in each
segment. Here Hmax is the maximal magnetic field considered for a magnetization curve on the reverse sweep. The
field generated by this current is determined and added to
the applied field. The current direction was chosen as to
cause a positive (trapped) field in zero applied field. After
this first step the local j values were modified according
to the critical state, fulfilling the condition
j = jc (H) at
p
all points in the disk where H = (Happl + Σhz )2 + Σh2r
is the total (radial plus axial) field at each point. Then the
field was again calculated from the modified current distribution, thus changing the fields (and currents) again, and so
on. The magnetic moment of the sample was computed by
summing up the contributions from the individual current
loops. The criterion for convergence was the magnetic moment. If it changed less than 0.1 % from one iteration to the
next, the calculation proceeded to the next lower external
field. Hereby the current distribution from the previous field
was kept as starting condition for the iteration procedure of
the new field. A self-consistent state was usually reached
within less than five iterations. However, if the field dependence of jc is made very steep numerical instabilities occur,
in particular near the magnetic field at which first negative
magnetic flux enters the rim of the disk. If the segment
number was increased these instabilities sometimes could
be avoided. Practical limits (of order 5000 elements) were,
333
1500
3 //
3⊥
6⊥
2 //
2⊥
4 //
4⊥
5⊥
WG4
10K
5
Bp
)
2
1000
m
A
jcB (A/mm 2 )
0
-3
0
1(
m
-5
χd
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
10K
500
3.0
Field (T)
Fig. 2. Irreversible magnetization curves at 10K of Nb3 Sn platelet 1.7 mm
thick in parallel magnetic field
Table 1. Diamagnetic slopes χd and thicknesses for all Nb3 Sn specimens
used in the magnetization experiment. All samples are square shaped with
a side length of 3.73 mm. A perpendicular orientation means that the field
is aligned perpendicular to the square
specimen
WG3
WG6
WG2
WG4
WG5
thickness [mm]
0.2
0.33
0.71
1.7
4.25
field orientation k
⊥
⊥
k
⊥
k
⊥
⊥
1.18 11.1 5.35 1.22 3.58 1.28 2.12 1.44
−χd
however, given by the calculation speed of the computer
used (DEC 3000-900S).
The field dependence of the current used here was of the
Kim type [22] with an additional Gaussian term
H1
H2
(5)
j(H) = j0 Aexp(− 2 ) +
H + H1
H0
The parameters A, H0 and H1 were used to fit the measured
magnetization.
3.2. Slab geometry
The slab was assumed to extend into infinity for the z and
y direction (see Fig. 1b). The slab width is 2W along the
x direction, zero being located in the center. The external
magnetic field is pointed along the z direction, all currents
flow along the y axis. The equation resolved was
dH(x)z
= j(H(x))
(6)
dx
Typically this equation was solved for 1000 segments along
x. The calculation now proceeded in a similar manner to the
one described above. However, compared to the previous
paragraph the calculation was much simpler and faster, as
the current only varies in one dimension.
j(x) =
4. Experimental results
The magnetization curve at 10K of one specimen is shown
in Fig. 2. Diamagnetic slopes were determined from the first
branch of the curve after zero field cooling. Within measurement accuracy the value of the slope is independent of
sample temperature up to 15K, as is expected for specimens
much larger than the magnetic penetration depth. The maximum shielded field (self field) h∗ is roughly equal to the
0.1
1
Field (T)
Fig. 3. Critical current density of all Nb3 Sn specimens at 10K as a function
of magnetic field
field at which the diamagnetic branch and the envelope curve
join, thus about 0.5T for the specimen shown. The curves
were measured up to high magnetic field (typically 2T), so
that the specimen is always fully penetrated on the reverse
leg. The diamagnetic slopes of all specimens are given in
Table 1.
It should be noted that – contrary to high Tc materials
– flux creep plays a very small role here. Measurements
of dynamic relaxation at 15K show that the magnetization
curves for two different field ramp rates differing by a factor
of 5 fall on top of each other within measuring accuracy.
Thus flux creep effects are negligible in this case, and the
shielding current flowing can be identified with the critical
current denstity jc .
The values of the critical current density jc as a function
of magnetic field are shown in Fig. 3. These were calculated
using a modification of the standard Bean formula [1, 23]
−1
2 a
ad
b−
(7)
jcB = ∆m
2
3
where b is the length of the sample, a its width, d its thickness and ∆m the difference between the upper and lower
branch of the magnetic moment. Evidently this expression
is only correct if everywhere in the sample the same current
j is flowing, which is only true if the magnetic field does not
vary significantly across the specimen, an assumption which
is certainly violated in low applied fields.
As expected, the jcB values for all specimens converge
into a single curve for magnetic fields higher than about 1T.
The remaining differences at higher field stem probably from
the errors in determining the geometry of the specimens.
This shows that in fact bulk pinning is responsible for the
measured hysteresis at high fields, irrespective of specimen
shape. The slabs in parallel field and disks in perpendicular
fields behave identical. The field dependence of jc in high
field above about 1T can be approximated by a Kim type
2
dependence (eq. 5 with A=0) with j0 = 4300A/mm and
µ0 H1 = 57mT. As the material of all specimens is identical
we can now conclude that the true jc (H) behavior is identical
for all specimens, even in low magnetic field.
Significant differences in the jcB (= apparent jc ) values – calculated assuming constant current flow across the
334
0.05
2.5
2.0
0.00
WG3
10K
peak position Bp (T)
1.5
-0.05
1.0
)
2
-0.10
0.5
m
A
0.0
-3
-0.15
0
1(
m
10K
15K
calc. 10K
-0.20
-0.25
-0.5
-1.0
data
-1.5
fit
-2.0
-0.30
0
2
4
6
8
10
-2.5
12
-1.0
- diamagnetic slope
-0.5
0.0
0.5
1.0
Field (T)
Fig. 4. Field value at which the peak in magnetization in decreasing magnetic field occurs, The calculated values are shown as dashed lines (10K
only)
0.3
fit
WG3
data
10K
0.2
specimens – exist in low magnetic field. The thin specimens
show much higher apparent jc values than the thick specimens. Thus the self field and thus inhomogeneous current
flow must play some role in suppressing the true field dependence of jc for in particular the thicker specimens, which
create a higher self field with the consequence of lower jcB .
The position of the peak in the magnetization on the upper branch (field ramped down, see arrow in Fig. 2) as a
function of the diamagnetic slope is shown in Fig. 4. The
peak is located at low negative fields for the thin slabs in parallel magnetic fields, and moves to more negative fields for
thicker slabs. For further increasing diamagnetic slope |χd |
the peak moves towards zero, and for the thinnest platelets
in perpendicular field is located at a positive field. This trend
is similar to the one observed by us for some high Tc materials [17, 24] and others for single crystals of YBa2 Cu3 O7−δ
[15, 16].
)
0.1
2
m
A
0.0
-3
0
1(
-0.1
m
-0.2
-0.3
-1.0
Figure 5 shows the result of the calculations of the irreversible magnetization curves for the thinnest platelet in
parallel and perpendicular direction using the jc (H) dependence of (5) with the parameters A=2, j0 =4300 A/mm2 ,
µ0 H1 = 57mT and µ0 H0 = 75mT. We note that the general shape of the peak is reproduced well by the calculation.
The differences between calculated and measured curve in
Fig. 5a are small, and somewhat larger in Fig. 5b. It should
be noted that the reversible magnetization is negligibly small
for all our specimens (of order 10−6 Am2 for the specimens
shown).
However, in Fig. 5a the peak in the calculated curve is
occurring in negative magnetic field, and not in positive field
as in the experimental curve (for better resolution see Fig. 4).
In particular the variable A has some influence on the peak
position. For larger A the peak moves to the right, thus in
the direction of the measured curve. This is illustrated in
Fig. 6 for three values of A. More field dependence of jc
shifts the peak to the right, towards more positive magnetic
field, while at the same time sharpening the peak in the
magnetization somewhat. An even larger A – for example
A=5 – causes large zero field current densities, and thus
large field gradients, rendering the calculation unstable. The
0.0
0.5
1.0
Field (T)
Fig. 5a,b. Magnetic moment vs. field for decreasing branch for specimen
WG3 in perpendicular a and parallel b field orientation together with calculated curve using jc (H) of (5)
10K
2.0
)
2
5. Result of calculations
-0.5
m
A
-3
0
1.5
(1
m
A=2
A=1
A=0
1.0
-0.2
-0.1
0.0
0.1
0.2
Field (T)
Fig. 6. Calculated magnetic moment vs. field for decreasing branch for
0.2mm thick platelet in perpendicular field orientation using jc (H) of (5).
with A=0, 1 and 2 (see text)
same happens if a large screening current at the edge of a
current and field free diamagnetic zone is allowed.
The position of the peak Bp of the calculated magnetization curves (with A=2) are shown in Fig. 4 together with the
experimental data. The agreement with the data of Fig. 4 is
good, but not everywhere quantitative. In particular for the
intermediate diamagnetic slopes the slab calculation overestimates |Bp | in the slab approximation, and underestimates
it in the disk approximation (1.5 < |χd | < 4).
Figure 7 shows the calculated current and field distributions for the thin platelet WG3 in perpendicular magnetic
field, Fig. 8 the corresponding data in parallel field using
identical parameters. In the perpendicular constellation the
335
parameter: field // z (mT)
-140
-120
-60.0
-20.0
1.5
-100
-80.0
-40.0
0
20.0
40.0 60.0
80.0
100
120
r (mm)
140
1.0
160
180
200
0.5
220
240
260
280
300
320
0.0
0.10
340
0.05
0.00
-0.05
-0.10
z (mm)
8000
6000
4000
jc (A/mm2)
2000
0.00
0.05
z
1.0
r (m
m)
(m
m
0.5
)
-0.10
0
-0.05
0.0
1.5
2.0
0.10
Fig. 7a,b. Calculated axial field a and current b distributions for thin platelet
(0.2mm) in perpendicular magnetic field at 10K. The field unit is mT.
250
200
150
100
5000
Field (mT)
j (A/mm2)
10000
50
0
0
0.000
0.025
0.050
0.075
0.100
x (mm)
Fig. 8. Calculated field and current distributions for thin platelet (0.2mm)
in parallel magnetic field
current maximum occurs near, but not at the edge. This is
due to the demagnetization field which is already negative
at the specimen edge. The current is flowing mainly in the
center plane, diminishing strongly towards the top and bottom surfaces as a consequence of the increasing radial field.
In the slab case the maximum current is flowing directly at
the specimen edges where the magnetic field is zero.
for specimens with diamagnetic slopes smaller than about -4
to -5. The slab model leads to best results if used for slabs
with |χd | < 1.2. If the slab model is used for higher values
of |χd | the measured |Bp | occurs at smaller values than the
calculated ones, indicating penetration along the specimen
edges. Similarly the thick disks behave a bit more like slabs
than the calculation assumes. Thus for those specimens |Bp |
is underestimated in the disk approximation. However, even
for the very thin disks a peak in a positive field could not
be obtained in the calculation. There may be several reasons
for that. Firstly, an even steeper jc (H) dependence causes
a shift of the peak in the correct direction. However, due
to convergence problems in the calculation we cannot really
verify this. Secondly, we have used a disk approximation to
specimens which in fact have a square shape. This means
that the current can fill out the edges of the specimens, which
changes current patterns. Thirdly, in principle a field free annihilation zone should be present at the H=0 line. In such a
zone the magnetic field would have to be below Hc1 , while
at the edges of it a large shielding current flows. We were
unable to model this zone due to problems with the numerical stability of the calculation caused by large field gradients. However, in order to accurately model the shielding
behavior this numerical problem must be overcome, as the
current approach is insufficient. A solution to the problem
has been outlined by Brandt [12], however at the moment
not yet incorporating the field dependence of jc . The observed crossover of the virgin and the envelope curves (see
for example Fig. 5b) may be reproduced when calculating
complete magnetization loops that use Meissner currents as
well as a field dependent jc in the mixed state.
This comment also applies with respect to other materials, in particular tapes of the (Bi,Pb)2 Sr2 Ca2 Cu3 Ox material
where a magnetization peak at a positive magnetic field has
been also observed [24]. However, the peak in this case
is much more narrow and sharp than the one found here
in Nb3 Sn. It is located at Bp ≈ +h∗ . By contrast, in the
thinnest Nb3 Sn platelet measured here the peak occurs at
about + h∗ / 10, and is much broader. In the calculations we
had difficulties to even reproduce the quite broad peak in the
magnetization of Nb3 Sn, and we can only speculate that an
even stronger field dependence of jc (in addition to a low
field zone) is necessary in order to generate these magnetization curves. A jc anisotropy may also play an additional role,
as well as possibly changing length scales for the shielding
currents due to field dependent granularity in the tapes. Another mechanism leading to a peak in the magnetization was
pointed out by Schnack et al. [18]. Here the influence of
the magnetic field on the flux creep rate is considered, with
the result that a peak in the irreversible magnetic moment
is generated even for jc = const due to a field variable flux
creep rate. This mechanism will certainly be important in
particular for high Tc materials like that (Bi,Pb)2 Sr2 Ca2 Cu3 Ox tape, but not for Nb3 Sn.
6. Discussion
Best agreement between the experimental curves and the
calculations have been obtained for the extreme geometries:
either thin disk in perpendicular or thin slab in parallel magnetic field. In order to obtain quantitative values for flux
penetration and shielding the thin disk model should be used
7. Conclusions
We have measured the irreversible magnetization with particular emphesis on the self field regimeof specimens of
Nb3 Sn with various aspect ratios, giving diamagnetic slopes
336
of about −1 to −13. The peak in the magnetization shifts
from negative to positive magnetic field as the diamagnetic
slope increases, just like in high Tc superconductors. Model
calculations are used to fit the measured magnetizations,
and derive magnetic field and current distributions. One requirement for the positive peak field appears to be a strong
field dependence of the critical current density in zero magnetic field. This causes strongly inhomogeneous current flow,
qualitativly explaining the measured magnetizations. Practical limits for the usability of slab or disk approximations for
the critical state model are given.
This work has been partly supported by the Swiss National Funds (NFP30).
M.D. would like to thank R.Flükiger for his support, and E.H.Brandt for
making available [12] before publication. We thank F.Liniger for the metallographic work, and R.J.Wijngaarden for a critical review of the manuscript.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
References
1. Bean, C.P.: Magnetization of high field superconductors. Rev. Mod.
Phys. 36, 31 (1964)
2. Campbell, A.M., Evetts, J.E.: Adv. Phys. 21, 199 (1972)
3. Norris, W.T.: J. Phys. D 3,489 (1970)
4. Frankel, David J.: J. Appl. Phys. 50, 5402 (1979)
5. Däumling, M.A., Larbalestier, D.C.: Phys. Rev. B, 40:9350, (1989)
6. Indenbom, M.V., D’Anna, G. Andre, M.-O., Benoit, W., Kronmüller,
H., Li, T.W., Kes, P.H.: Proceedings of 7.th Int. Workshop on Critical
Currents in Superconductors, Alpbach (Austria). January 1994
21.
22.
23.
24.
Brandt, E.H.: Phys.Rev.B 48, 6699 (1993)
McDonald J., John Clem, R.: Phys. Rev. B 53, 8643 (1996)
Brandt, E.H.: Rep.Prog.Phys. 58, 1465 (1995)
Zeldov, E., Clem, J.R., Elfresh, M.Mc, Darwin, M.: Phys.Rev.B 49,
9802 (1994)
Schuster, Th., Indenbom, M.V., Kuhn, H., Brandt, E.H., Koncykowski,
M.: Phys.Rev.Lett. 73, 1424 (1994)
Brandt, E.H.: Phys.Rev.B 54, 4246 (1996)
Bhagwat K.V., Chaddah, P.: Physica C 254, 143 (1995)
Roy, S.B., Pradhan, A.K., Chaddah, P.: Physica C 253, 191 (1995)
Oussena, M., de Groot, P.A.J., Gagnon, R., Taillefer, L.: Phys.Rev.B
49, 9222 (1994)
Yeshurun Y., Malozemoff, A.P.: Phys. Rev. Lett. 60, 2202 (1988)
Däumling, M., Walker, E., Flükiger, R.: Phys. Rev. B 50, 13024
(1994)
Schnack, H.G., Griessen, R., Lensink, J.G., van der Beek, C.J., Kes,
P.H.: Physica C 197, 337 (1992)
Goldacker, W., Ahrens, R., Nindel, M., Obst, B., Meingast, C.: IEEE
Trans. Appl. Superc. 3, 1322 (1993)
Orlando, T.P., McNiff, Jr E.J., Foner, S., Beasley, M.R.: Phys.Rev.B
19, 4545 (1979)
Smythe, W.R.: Static and dynamic electricity. New York: McGraw
Hill 1950
Kim, Y.B., Hempstead, C.F., Strnad, A.R.: Phys. Rev. 129, 528 (1963)
Wiesinger, H.G., Sauerzopf, F.H., Weber, H.: Physica C 203, 121
(1992)
Däumling, M., Walker, E., Grasso, G., Flükiger, R.: Proceedings of
ICMC Topical Conference on Critical State, p.191. Honolulu, October
1994