Critical currents of NbTi tapes with differently oriented anisotropic defects

In the paper we present a theory predicting the angular dependence o f the critical current in Nb Ti tapes with elongated defects of different orientation with respect to the tape axis. Corresponding experiments were carried out on samples made from two Nb Ti foils with different critical current densities. The role of the self-field as well as of the small additional perpendicular field on the critical current is also investigated.

Critical currents of NbTi tapes with differently oriented anisotropic defects S. Takacs, M. Polak and I_. Krempasky Key words: superconductors, critical current, niobium titanium, anisotropic defects In NbTi superconductors there are two kinds of defects acting as effective flux pinners x'2 : walls of dislocation ceils (dislocation sub-bands) and a - phase precipitations. The dislocation sub-bands have diameters of typically ~ 500 A and they are elongated in the drawing or rolling direction. It has been shown, that during the heat treatment of the cold worked material the density of a - phase precipitates in the axial direction (along the sub-bands) is higher than in the perpendicular direction. Therefore, both pinning centres are anisotropic. As a result of such pinning centres, NbTi wires exhibit a strong dependence of the volume pinning force on the angle between the transport current and external magnetic field. 3'4'5 The volume pinning force for different field and current configurations occurring in resistivity and magnetization measurements of the wire samples are treated in detail in the literature. 6 It should be noted that in the wire samples, the transport current flows in the wire axis direction so that its direction is identical with the defect direction. Having a NbTi foil of sufficient width, samples with different defect orientations with respect to the sample axis can be prepared. Thus, we can have samples in which the transport current direction is not identical with the defects direction and it may be changed by appropriate cutting of the sample. Concerning the angular dependence of the critical current, a tape sample differs from a wire sample in two ways: one, by changing the external field direction with respect to the wire axis, the angle 7 between the direction of the flux lines and the defect direction is the same as that between the sample axis and the external field ~. In tape samples, the relation between the angles T and ~ois a function of the defect orientation in the sample. Two, the angle between the transport current and the orientation of the defects in the tape samples can be varied, while in the wire samples both directions are identical. Thus, we have additional configurations between defect, field and current direction for testing the theory of pinning on anisotropic defects with experimental results. In this paper, we present a theory predicting the angular dependence of critical currents in NbTi tapes with different defects orientation. Experiments on NbTi 50/50 samples were carried out in order to compare the theoretical predictions with measured angular dependence of

critical currents. In addition, the role of the self field as well as of small additional perpendicular field on the critical current is investigated.

Calculation of the critical current density In general, the calculations are the same, as in our previous papers.S,6 The results of these papers showed that the volume pinning force Fp in NbTi satisfies the result of the elastic theory of pinning, 7 ie

Fp ~ NkZm

(1)

where N is the defect density, and km the maximum elementary interaction force between one defect and one flux line. We investigated one type of defect (although they can be from different origins 1 ), schematically represented by an ellipsoid with mean aspect ratio A = a/l (width/length). By changing the flux line and current direction with respect to the elongated direction d~of the defects, not only kin, changes, but also the effective density of the pinning centres, s'6 Moreover, by changing the angle between current and field, the Lorentz force direction and its value also changes. Therefore, one needs to determine the volume 'pinning force in this (changing) direction of the Lorentz force. We neglect the surface contribution to the volume • pinning force, which can have another type of anisotropy, 6 assuming that its role is less important in superconductors with a stronger volume pinning force. We defined ]co as the critical current density of the superconductor, B perpendicular to the current, flowing in d direction, ie ~ = 0 (Fig. 1 a). Then, for the changing

B±

e

13

._..B ±

b

B±

c

Fig. 1 N b T i tapes w i t h d i f f e r e n t o r i e n t a t i o n o f the elongated defects a -- d e f e c t o r i e n t a t i o n parallel t o t h e tape axis (e = 0 ° ) , b -- defect orientation perpendicular to the tape axis (• = 9 0 ° ) , c -- d e f e c t orientation e

0011-2275/83/030153-07 $03.00 © 1983 Butterworth & Co (Publishers) Ltd CRYOGENICS . MARCH 1983

153

angle ~0between the current and field, we have for infinitely long defects s

have to make the substitution 3' = + (~0- a). Therefore

Jc = /co sin-2 ¢

(2)

(~ not too small, where self-field effects can prevail). As an intermediate step to the general case, we investigate the case from Fig. 1 b (current perpendicular to the defect direction). The maximum elementary interaction force km will be proportional to the length at which the flux lines cross the defects (/eft), and the effective pinning density inversely proportional to left 6 :

/ l + t f 3'\1/2 le,, = a ~A~ ; V g q )

(3)

1/2

/c =/co

.

sin-' ~0

[ 1 +tg 2 (~0-_~]

(5)

This is our basic formula for comparing experimental results. The results are given in Fig. 2 for A = 0.4 and A = 0.25 as examples. The curves represent the dependence of/J/co on the angle ~ofor different defect directions ot as a parameter. One can see immediately that for A -~ 0 (infinitely long defects) (5) goes into the form

where 7 is the angle between the field and d. Therefore, the volume pinning force will be in this case

Jc

= /cO

sin (~o- a) sin ~o

1/2 FP = FP° (A ]1 +tg2 ) + t g2 77

(4)

and the critical current density

1/2

/c

Sin-1

1 + tg 2 7 ) A 2 + tg2 7

/co

1 + tg 2 ¢ t 1 +A~ttg2-~ ] sin-I ~

= Jco

which for the current density in 3~direction (a = 0) leads to (2). The most difficult theoretical problem connected with our task is including the effect of the self-field on the critical current, mainly in the geometry used in critical current measurements of tapes. Nevertheless, we are able to calculate within the framework of the critical state model, the critical current density in purely longitudinal fields. Then, only the self-fidd is contributing to the Lorentz force. The induced field for a wire with radius r on the surface is

For determiningjc in the general case (ie different angles a between the current direction and ~Fig. 1 c), we simply

(6)

BO ) = Uo 27rrj

4n

12

'/

II

II

--

I0

I0

--

9

9

--

8

8 --

7

7 --

A= 0.4

l

a=50

~0

a=O

"7 .,,,o 6 -

"-~ 6 ...,~

a=45 ~

5

5

--

4

4

--

a=60 a

;

7

~

a=90

3

5

--

2

p

"_

I

I

0

0

b 90

Fig. 2

154

The

80

70

60

50

ratiOJe/JeO as a function

40

30

20

10

0

90

80

70

60

50

40

o f the angle ~ for different defect orientation calculated f r o m (5) with: a - A

50

20

= 0.25; b - A

IO

= 0.4

CRYOGENICS. MARCH 1983

and the maximum field for the plate is given approximately when 2 r = d (d - thickness of the plate). For B >> B (i) (which is well satisfied for our applied fields of some Tesla), the critical state is given by the equation

]c(Bsin~p+B(Jc)) =]c (Bsin~,O+]c ~ f ) 1 + tg 2 (tp' - o0

]

= [1 + A 2 t ~ - - ( ~ ' - - Z a ) l

t/2 Fpo

(7) where Foo =/'co B, when B is perpendicular to the current and tg~o' ~ tg~o +

Samples and the experimental set-up As a starting material for preparing the samples, two kinds o f NbTi foils were used. The first one was 18.5/am thick, 80 mm wide NbTi 50/50 foil stabilized on both sides with "-" 1.5/am copper layer. The critical current density in a superconductor in a 4 T field perpendicular to the current density and parallel to the foil surface is "" 3.5 x 104 A cm-2 for the sample with the axis parallel to the rolling direction (a = 0). The second starting foil was NbTi 50/50 9/am thick 120 m m wide with one side copper stabilization. The 20/am copper stabilizing tape was indium - soldered to the NbTi foil covered previously by indium using ultrasound soldering apparatus. The critical current density under conditions mentioned above was considerably higher, ~ 1.05 × 104 A cm-2 .

B(i)/B

NbTi foil J J

We assume furthermore that the value of the applied field does not change, only its direction. First, we solve (7) for a = 0, and ~o= 0. Then: f

]2 /aor

--(I+(B//B)2) ½ 2 _ 1 +A2 (Bi/B)2ice B

(l+(/aor/c/2B)2) ½ i +(A/ao c/28) ,

=

J

' ~

ct

Rolling direclion

Fig. 3 Preparation of the sample with defect orientation a by cutting from NbTi foil

()2

1 1 - A 2 /aor 2 A2B 2 -~- Ic or

jc2 [

1 + 1-A 2 ice B 4A 2 B 2

/aor] 2

_

2 /ao r

and forjco ~ 3 . 4 x 109 A m -2 at3.5 T a n d r ~ 10/am (see below)]c ~ 4.4 × 10 x° A rn-2 . In the case ~o= 0, we have a >> ~ofor a ~ 20 °, therefore ½

/c

.tf

2

~1 +A 2 tg 2 (~

t

-a)] ½

]coB

(1 1 + t g 2 a

+.42 tg2 ot )

(8)

The calculated self-field critical currents for different values of a are marked in Fig. 2. As the general calculation ofjc near ~ = 0 are very difficult, we can only estimate the expectedjc corrections, as illustrated in Fig. 2 with dashed lines.

CRYOGENICS.

M A R C H 1983

I

I I Fig. 4

Rotating sample holder used for the measurements

155

Table 1.

The geometric dimensions of the samples studied

Sample

~

Width w, mm

thickness, /Jm

Measured in fields B, T

1.3

18.5

4,3,5,3

1

0

2

30

1.65

18.5

4,3,5,2

3

60

1.35

18.5

4,3,5,2

4

75

1.6

18.5

3,5,3,2

5

90

1.3

18.5

2,3,5

I

<

l i. I//i I/ D'tl Y

I; I'll

.o.°"

oO ,"

,"

Experimental results

7 'fI,1<:

0.1

10

If cutting-off a sample from the foil at an angle ot with respect to the rolling direction of the NbTi foil (Fig. 3), than by turning this sample in the magnetic field B starting with perpendicular geometry (~ = 90°), the angle 3' between fluxoids and defect orientation depends on ~ as (see Fig. 4) 3' = + (¢ - a) The'measured samples and their geometric dimensions are listed in Table 1. The sample was mounted on the circular support, as shown in Fig. 4. Current contacts were made by soldering the sample with In to the copper blocks fixed on the support. Potential taps P1, P2 were soldered at a distance AL = 1 cm. To prevent mechanical movement under the Lorentz force, the sample was glued to the support with GE 7031 varnish. The support controlled by a mechanical turning system was placed in the 60 mm bore superconducting solenoid generating magnetic fields up to 4.8 T. The angle ¢ between the sample axis and field direction was adjusted with a precision better than + 1.5 °. The voltage U between the potential taps P1, P2 was amplified and traced on the Y axis of an XY recorder versus sample current. The maximum Y-axis sensitivity was 0.8/JV cm-1 .

20

Perpendicular field. At first, the current -voltage characteristics of each sample in a perpendicular field (¢ = 90 °) were measured as a function of the magnetic field B. As an example, we show the family of current-voltage characteristics of sample 4 in the log U = f(I) representation (Fig. 5). They have an initial exponential part with the slope depending on B up to U ~ 1/~V. For U - 1/.iV this dependence tends to become linear. Further, using U = 0.1 #V (E = 0.1/.iV crn-1 ) as the criterion for the critical current Ic,

I 30

/,A Fig. 5 Semilogarithmic representation of the current-voltage characteristics of sample 4 in perpendicular magnetic fields

exlo4

15

7xlO4 a = 6 0 e Sample 3

I0

/ a = 7 5

=

Sample4

/ / . = 9 0

°

Sample 5

- 6xlO4

^

-

~

4,,10

Eo

3x 104

2xlO4

ixlO4

I

L,

I

I

0

I

2

3

B,T Fig. 6

156

Critical currents versus magnetic field for samples 1,2,3,4 and 5 in perpendicular geometry (~0 = 90 ~)

CRYOGENICS. MARCH 1983

the/el f(n) characteristics of the samples 1,2, 3, 4 and 5 were plotted (Icl = Ic/w is the critical current per 1 mm of width of the NbTi tape) as shown in Fig. 6. The initial part of the characteristics is somewhat surprising: the critical current increases with increasing perpendicular field up to a certain value Bp which depends on the defect orientation a. A similar increase oflc with increasing magnetic field in the longitudinal direction however, was observed in 1966.s For samples A and B made of NbTi foil with higher/c, the lcl = f(B) characteristics are shown in Fig. 7. Because of the limited current capacity of the power supply, they have been measured only in fields exceeding "" 3 T. Each linear part of the log U = f(I) curve can be described by the critical current defined at certain electric fields (0.1 #V crn-1 for example) and the value A/1 defined as the current increase corresponding to the increase of the voltage U by a factor of ten. In Fig. 8 the linear part of each measured curve is presented as a point with co-ordinates A/~, Icl (0.1/~V cm-1 ). Apart from the relatively strong scattering of the points we can say, that A/~ increases linearly with the critical current. For a given critical current, A/I depends on defect orientation. The sample with a = 90 ° exhibits the smallest slope of the measured log U = f(1) characteristics; in this case the fluxoids are parallel to the defect direction. Therefore, in the flux creep region U ~ exp [K(B, a)I/Ic] where K is the smallest for flux lines parallel to the defects. Hence, the flux line lattice dislocation motions are more likely at smaller currents than the tunnelling of flux bundles, which is the effect leading to creep of flux lines when they intersect the defects. On the other hand, once the 'critical' current is reached, the flux lines can cross the defects more easily, whereas in perpendicular geometry, some flux line portions =

@:90" A Sample 5 Sample 4

1.5

a=75 o

--< Sample 3 ~ Sample 2 .

~ J ~ a" ~ J "o .

~

a=60=

~

== 50*

0.5

0

I

I

5

10

/c,,A Fig. 8 Parameter A I 1 as a function of the critical current /cl (0.1 # V c m - 1 ) f o r samples 1,2,3,4 and 5 (perpendicular geometry)

are still strongly pinned; therefore, the creep curve is not so steep. However, it is not clear whether this is an internal property of NbTi samples or a general result true for all superconductors.

Angular dependenceof critical currents Current-voltage characteristics were measured for each sample at angles ~ between 90 ° and 0 ° at intervals of 6 °.

B=2T

~ample I 101

.

9~

@=0=

~=9o

15 20

Sample 4 = = 30*

01

,

90 ° 60 °

I

30 °

0o

= 60* < IO

A I0

& Sample 5

0 0

I I

I 2

I 3

I 4

I 5

0 90 =

I 60 °

I 30 °

I 0°

B,T Fig. 7 Critical currents versus magnetic fields for samples A and B in perpendicular geometry

CRYOGENICS. MARCH 1983

Fig. 9 Angular dependence of the critical current Icl for samples' 1,2,3,4 and 5 in a field B = 2 T

157

The measurements were made at the fields given in Table 1. Because of the very small slopes of the measured characteristics at small angles ¢, we were obliged to use higher voltage criterion for estimating the critical current (1 pV cm-1 ). The evaluated Icl = f(¢) characteristics of the samples with defect orientations ~ = 0, 30, 60, 75 and 90 ° for angles from 90 ° up to 0 ° in constant external fields B = 3.5 T and 2 T are shown in Figs 9 and 10, respectively. The lcl =f(¢) characteristics of samples A and B made of foil with a higher critical current density are shown in Figs 11 and 12, respectively. Comparing the experimental Icl = f(¢) characteristics for samples with different defect orientations with the theoretical curves shown in Fig. 2 we see that they are very similar. The maxima and minima oflcl are more marked in samples with higher critical current densities (comparing the theoretical and experimental results)for samples with smaller Icl ). Unfortunately, at this time we have no information about the defect geometry and dimensions, so that a quantitative comparison of the results cannot be made. During the measurements we observed that the V-A characteristic does not always have the same form for increasing and decreasing current. This behaviour was observed with the sample A for angles ~0= 24 ° and 15 ° at B = 4.25 T (as shown in Fig. 13). When the current reaches lc, the voltage increases abruptly up to a value Ua and then it increases exponentionaUy. Decreasing the current, the voltage decreases exponentionaUy. This behaviour was observed in field - current - defect geometry, where lc (~) characteristics have a local minimum (see Fig. 11).

o?,)

20

0(2)

"T E E

IO

o

o/

oZ l,, o(,)/

\°7

I

90*

I

30"

60*

o*

$ B =3.5T

Fig. 11 A n g u l a r d e p e n d e n c e o f t h e critical c u r r e n t / c l s a m p l e A in t h e field 4 . 2 5 T, ~ = 6 0 °

for the

Sample I

Conclusions

2O

.~o 6 -

90*

6 0 * 30*

O*

s

/

~

Ir"pte

IO

0

90*

I

I

60*

50*

I

O*

Fig. 10 Angular dependence o f the critical current Icl for the samples 1,2,3,4 and 5 in a field B = 3.5 T

158

We have shown, that defect orientation in NbTi tape samples with elongated defects has a strong influence on the angular dependence of the critical currents when the samples are rotated in magnetic fields parallel to the sample surface. The local maxima o f I c = f(¢) curves are observed for flux lines parallel to the defect direction. With the information obtained on anisotropy of the volume pinning force, the changing field, current and defect direction, it is possible to explain the results. Defect orientation also influences the slope of the voltage current characteristics, ie the defect orientation determines the flux flow mechanism. A hysteresis in the current-voltage characteristics for one sample, A, has been observed between the increasing and decreasing transport current. An anomalous increase of It with B in a small perpendicular field has been observed on all samples made of NbTi foil with smallerjc. The resemblance of theoretical and experimental results shows, that the angular dependence of the critical currents (including the self-field critical current) can be explained qualitatively within the framework of the critical state model by considering anisotropic defects. However, some other very interesting results: change in slope of I-V curves, hysteresis of the V-A characteristics, and mainly the enhancement of critical current by adding small transverse magnetic fields, as well as the possible correspondence of the maxima of zSJl and o f l c l , require more precise calculations if the effects are to be connected

CRYOGENICS. MARCH 1983

~/

8=4.25T

...... ~

~

/

=

U.

4 =240

2O I

Fig. 13 Hysteretic behaviour of the current-voltage characteristic of the sample A in the field 4.25 T, ~o = 24 °

a(

15 O

4T o

o

flux flow mechanism is more or less 'hopping' (ie tunnelling of smaller or larger flux line bundles), whereas flux lines nearly parallel to the defects can 'by-pass' them. In the latter case, the bending properties of the flux lines and the mobility of flux line dislocations will be more important.

tt

a

IO The authors wish to acknowledge the technical assistance of J. Talapa. Authors

The authors are from Electrotechnical Institute, Centre of Electrophysical Research, Slovak Academy of Sciences, 84239 Bratislava Czechoslovakia. Paper received 1 November 1982. References 0

90 °

I 60 °

I

50 °

I

Oo

Fig. 12 Angular dependence of the critical current Icl for the sample B in the fields 4 and 4.49 T, c~ = 30 °

with dynamic phenomena. The dynamic properties of the flux line lattice are very sensitive to changes in flux-line and Lorentz-force direction with respect to the elongated defects. For example, one could expect in flux-line geometries approximately perpendicular to the defects that the

CRYOGENICS,

MARCH

1983

1 2 3 4 5

6 7 8

Pfeiffer, I., Hillmann, H. Acta Met 16 (1968) 1489; Hillmann, H. Siemens Forsch Entw Ber 3 (1974) 197 Barber,A.C., Neal, D., Richards, P., Woolcock, A. Acta Met 19 (1971) 143 Hl~snik,I., Krempask)~,~.., Pol~k, M. Phys Star Sol (a) 16 (1973) 153 Jungst, K.P. IEEE Trans on Magn MAG-I 1 (1975) 340 Tak~tcs,S.,Pevala, A. PhysStatSol(a) 41 (1977) K 175 Tak~cs,S., Hl~Isnik,I. to be published Labusch,R. Crystal lattice defects 1 (1969) 1 Bergeron,C.J. Appl Phys Letters 3 (1963) 63; Le Blanc, M.A.R. Phys Rev 143 (1966) 220

159