Determination of vortex friction in a rotating type ii superconductor with a self-compensating torsion balance

A torque method is described for investigating pinning and viscous friction of vortices in a type II superconductor which continuously rotates with frequency f in a magnetic field H directed perpendicular to the axis o f rotation. A self-compensating torsion balance was developed to allow the torque D(H, f) to be measured for temperatures in the liquid helium range. The average flux density B(H, f) in the rotating specimen can also be determined. Data obtained from a niobium cylinder can be described by the linear relationship D(B, f) = Do (B) + D'(B) f, where Do and D' are determined by the dynamic pinning force and the viscous friction respectively.

Determination of vortex friction in a rotating type II superconductor with a self-compensating torsion balance M . F u h r m a n s a n d C. H e i d e n

The average pinning force per volume, ~ Pv, has been measured by a variety of methods including direct measurements of critical current density, investigation of flux density profries, or a suitable analysis of the irreversible magnetization behaviour, but Pv can also be determined by mechanical means. For this purpose the maximum torque exerted on the specimen under test just prior to tearing off vortices from pinning sites can be measured. Cylindrical specimens in a transverse magnetic field 2-4, have been used for this together with the geometry of a corbino disc:

and the associated braking torque, exerted on the rotating specimen becomes dD = a × dfv

where a is the vector pointing from the rotational axis to the vortex element. For a cylindrical specimen of radius r and height h (see Fig. 1a) summing over the individual contributions of all vortices gives a total braking torque D 8

These torque measurements can be regarded as analogous to the determination of critical current 1%from the onset of voltage in current-voltage characteristics. 1% must be distinguished from the other critical current Icf, which is obtained by extrapolating the linear part of current-voltage characteristics to zero voltage, and corresponds to the dynamic volume pinning force 6 acting on vortices in motion. In the following we describe a system, which apart from static measurements, can be used to obtain information on the dynamic pinning force per flux thread per unit length, k, by torque measurements under stationary flux flow conditions. The torque due to the viscous friction of vortices can be separated from the pinning torque, thus also providing information on the viscosity coefficient, 7 7, or the related flux flow resistivity, pf = qbo B 7)-~ , where qbo is the flux quantum, and B the flux density averaged over the moving fluxons. In this system samples with axial symmetry are continuously rotated at frequencyfaround their axis in a perpendicular external magnetic field H. Vortices in the mixed state of the sample tend to align themselves with the external magnetic field giving rise to the observed torque.

(2)

D = Do + D' f

(3)

4 Do = ff k B ~o 1 r a h

(4)

~2 D' = -2- r / B ~ ° l r4 h

(5)

H

0 ~~-~

Vz(t)

~i V~ (t)

• H

b

a

If we consider a vortex element ds moving with velocity v± perpendicular to ds, the average force acting on this element is (t} I c B

dfv = V.L ZI ds + v--~-Ikds v±

(1)

The authors are in the Institut for Angewandte Physik der Universit~t MLinster, Roxeler Strasse 70/72, MOnster, Germany. Received 12 January 1976.

CRYOGENICS.

AUGUST

1976

t Fig. 1

Rectified induced voltage of sample rotating at frequency f

plotted against time a -- plan o f rotating sample; b -- location o f pick-up coil

451

A

B

1

II

.. /

28

- - - - 13

o •

..L

19

17

'

)k~:l I

Io

180 -

25

7a

I

IA

30 mm

I0 8---

Fig. 2

Technical details of the self-compensating torque balance

~) 25mm

452

C R Y O G E N I C S . A U G U S T 1976

This result is derived by assuming that all vortices are parallel to a straight line and that their density is constant throughout the sample. Due to the presence of currents required in the sample to provide the driving Lorentz force, vortices are however more or less bent, and density gradients are likely to occur. The resulting corrections for the relationships (3), (4), and (5) are being studied at present. They should be important if the magnetic field produced by the driving currents near the sample axis becomes comparable to the average vortex field. The influence of the non-uniform demagnetizing field which arises for cylindrical specimens with finite length is strongest at low flux densities and disappears for B approaching Bc~. It can be reduced considerably either by using cylinders with h/r >> 1 or h/r ~ 1, or by choosing rotational ellipsoids as samples. According to (3) a linear relationship between torque and frequency of rotation is expected. From the intercept Do and the slope D', values for k and r~ can be determined if B is known. To measure B a pick-up coil is used which is wound around the sample and rotates with it (Fig. lb). The induced voltage II1(t) is rectified and time averaged leading to a signal I VI (t)[ , which is proportional to the value of B averaged over the sample. In addition to this average value of B information about the distribution of vortices in the sample can be obtained from the shape of the signal I/1 (t), a deviation from the sinusoidal form indicating an inhomogeneous contribution to the vortex density. The voltage V2(t) from a second identical coil, which is placed well above the sample and rotates with it, can be used for compensation purposes so enabling the magnetization B --/1oH of the rotating sample to be measured directly. The signals in the Meissner region (B = 0) are used for calibration.

Apparatus The measuring system is essentially a rotating self-compensating torsion balance. The sample is suspended on a torsion wire and rotates with constant angular velocity between the pole pieces of an electromagnet, whose field is perpendicular to the axis of rotation. The torque acting on the sample is determined by measuring the twisting angle of the torsion wire. Fig. 2 shows details of the apparatus. It consists of two parts, the torsion suspension with the sample mount A and the drive and control system B, to which A is attached. The sub-system A has an overall length of 1050 mm, the lower part of which can be immersed in the helium bath of the surrounding glass cryostat. The sample is mounted in one of two rectangular openings of a plexiglass rod 1, both of which are surrounded by the pick-up coils 2 for magnetization measurements. The plexiglass rod is suspended by means of a steel torsion wire 3 which is fastened to the inner part 4 of a ball bearing fitted into the thin walled German silver tube 5. This tube is the outer mantle of the whole assembly A. Its lower part, which is of a somewhat reduced diameter, has lateral openings to provide access to the sample mount. Another German silver tube 6 is attached to the upper end of 1 and carries a glass mirror 7, which indicates the angular position of the sample with respect to the drive system, B. The lateral guidance of 1 is accomplished by thin circular brass plates 8 and 9 attached to the tubes 5 and 6; each of which has a small central hole serving as a bearing for the pin 10 connected coaxially to 1

CRYOGENICS. AUGUST 1976

and the torsion wire 3. A beneficial effect of the everpresent light vibrations of the rotating system was that friction in these bearings turned out to be negligible even for the smallest observable torques. This construction with a loose lower end of the torsion suspension avoided difficulties caused by contraction of the different parts. An oil damping 11 in the upper part of A, which is at room temperature, counteracts angular oscillations of the sample. This consists of a circular oil tub attached to the inner side of 5 into which a rotor with four blades dips; this rotor is attached to the upper end of 6. Since the torque constant of the torsion wire varies with the fourth power of the wire radius, the sensitivity of the system can be altered most effectively by using torsion wires of different diameter. To achieve this 3 is fastened to 1 and 4 by using set screws. The drive and control system B consists of a circular aluminium plate 12 of 200 mm diameter and 5 mm thickness, which acts as support for all mechanical and electronic parts needed for the automatic torsion balance. This plate is driven by an electromotor 13 using a frictional coupling with a rubber cylinder 14. The angular velocity of 12 is measured by using a light barrier consisting of lamp 15, photo-diode 16, and 100 equidistant holes near the outer circumference of 12. An electric feedback signal acting on 13 is proportional to the difference between the rate of the photodiode pulses and a given rate, and so allows automatic regulation of the frequency of rotation of 12 to a pre-selected value. Slow sweeps o f f are also possible with this system. The torsion balance works in the following way. A small change in the angular position of mirror 7 due to a torque acting on the sample causes an equal and opposite torque to be applied to suspension 4 from electromotor 17 via worm gear 18 + 18a, frictional coupling 19a, axle 19, and connexion 20, 20a. The angular position of 7 with respect to 12 is thus kept practically constant. The resulting twist angle ot torsion wire 3 is proportional to the compensating torque and can be measured by the output voltage of a linear potentiometer 21, which is mechanically connected with 19. The angular position of 7 is controlled by an optical method. After passing lens 22a, light from the filament of an incandescent lamp 23 is reflected by mirrors 7 and 7a and becomes focused on the photo-diodes 24 via lenses 22. Driven by motor 24, a chopper disc 26 with four radial slits rotates between 7a and 22. In the zero position of 7 the image of the filament is divided in two equal parts, each of which covers one lens 22, so that the duration of the light pulses on the two photo-diodes are equal. A deviation of 7 from this position causes a difference in the pulse durations which in the electronic circuit 27, following the photo diodes, is converted into a corresponding proportional output voltage used to control 17. Even under varying cooling conditions for the electronic circuit, which occur when the frequency of rotation is changed or the system operates under different helium gas pressures, a drift of the zero position of 7 and thus of the torque compensation remains negligible due to the feedback technique used. Slip rings 28 and 29 provide the necessary electric connexions of the rotating system to the external power supply, the electronic system for the magnetization measurements, and for the registration of torque. Hole 30 is for the helium transfer tube.

453

ing. By using suitable transmissions, it should be possible to extend this range on both sides by at least one order of magnitude. The whole system was found to work satisfactorily in a temperature range between 4.2 K and 1.7 K, which was obtained by lowering the helium vapour pressure. a

NI-N2

b Fig. 3 a - Block diagram of the control circuit for the angular b -- electronic set-up used in magnetization measurements

velocity;

System B is encapsulated by a steel cylinder with closed top and vacuum tight electric feedthroughs, the cylinder bottom being sealed with an O-ring to the nitrogen cooled glass cryostar surrounding sub-system A. The temperature of the helium bath can be reduced by pumping. Fig. 3 shows block diagrams of the different electronic setups used with the rotating torque balance. In the control circuit for the angular velocity (Fig. 3a) pulses from the photo-diode of the light barrier are first shaped by a Schmitt-trigger and then transformed by a frequency voltage converter to a voltage Vf proportional to the frequency of revolution f. The difference between Vf and a given reference voltage Uf after amplification is used to control the drive motor 13 for the disc 12. The magnetization B -- tzoH of the rotating specimen is determined according to the principle in Fig. 1b, but with a digital technique (Fig. 3b) allowing high accuracy. For a given number of sample revolutions, which can be selected using a pre-set counter, another (reversible) counter registers the difference N~ -- N2 of the number of pulses from fast voltage frequency converters (VFC) whose inputs are connected to the two pick-up coils in the sample holder, and whose output pulse rates are proportional to the absolute value of their input voltages. Using a potentiometer in the compensating network, N~ --N2 is set to zero for fields above the upper critical field He2 of the sample. N~ --N2 then is proportional to laoH--B.

The accuracy of the torque measurements is limited mainly by the threshold voltage of motor 17, the accuracy to which the torsion constant is known, and the linearity of the potentiometer 21 ; all of which sums up to + 1.5% of measured torque -+ 0.5% of maximum torque for the chosen wire. Except for variations of B, which eventually are introduced by the sample itself, the rotational magnetization B - #oH with the present set-up was found to be reproducible to within -+ 1.5 x 10-ST. This high accuracy is possible since the thermal emfs, which often produce serious drift problems when measuring the magnetization by electronic integration, have a negligible effect in this case. For a sinusoidal signal Vx(t) = a sin 2 rrft and a superimposed thermal emf b ~ a, a condition which can be easily realized experimentally, the relative error of B is given by AB/B = b2]2 a 2.

Results The following results were obtained from a cylindrical specimen of 99.9% pure polycrystalline niobium. The sample was outgassed for three hours in a vacuum of p < 10-9 torr at a temperature of 2200°C, leading to a resistivity ratio of 40.8 (see Fig. 6). Fig. 4 shows the magnetization curves. The solid line obtained from the sample at rest by electronic integration of the voltages induced in the pick-up coils, when the field was swept, exhibits the hysteretic behaviour usually found on similarly treated specimens. No hysteresis is found within the measurement accuracy when determining the magnetization from the rotating cylinder by the method described

0.10 0108

454

/

Nb(I) r =4.2 K

{ I

HC = 2 l ~ x ~0~ A Cm. I ....

0.06004

Rotating specimen

~ p e i c i r n eI n~'~11='~

i ~:o :L 0.02

Performance With a 20 cm long torsion wire of 0.l mm diameter, torques up to 220 dyn cm (2.2 mN cm) can be measured. This maximum value increases with the fourth power of the wire radius. A torsion wire with 0.5 mm diameter was successfully used to give a torque range up to 137 500 dyn cm (1.37 N cm). The frequency of rotation f c a n be varied in the range between 0.2 Hz and 3 Hz, the lower limit given by the threshold voltage which the drive motor needs for start-

~ ~'e~

I I

2

~

I 3 x I03

H, A cm-I -0.02 -0.04-

Fig. 4 Magnetization curves for a n i o b i u m cylinder, the dimensions of which e r e given in Fig. 6. H is perpendicular to the cylinder axis

CRYOGENICS. AUGUST 1976

relationship (1). Near the field of first flux penetration into the sample, the torque is mainly caused by pinning whereas for higher fields in the mixed state a considerable contribution to D is seen to arise from viscous friction. The observed linear variation D ( f ) has the obvious consequence that the coefficient D' in (1) here in fact is independent of frequency. A variation of vortex configuration in the sample with frequency thus has negligible influence on D' in the range o f f used. Also via its effect on the vortex configuration pinning might have some influence on D'. However, except for the

16 14

12 I0 8 6

E z

'2f

o

T:4,2 K

I0

d

R29~ K

z

=40.8 R42 K f = 1 4 Hz

'O 6

0

r =6.33 mm h:3184 mm

4 0 2

E 0

I 2 z

H,

0

I 3 I03 A crn -I

I

1

4

5

z

0 16

k

i 0

I

2

5

4

14

H, 103A cm-I 12 Fig. 5 Dependence o f measured torque o f a niobium magnetic field for some frequencies of revolution

cylinder

z

'; d above. Any trapped flux in parucular is driven out of the sample when the field is lowered to values in the Meissner region. Unhysteretic magnetization curves were obtained earlier 9'1° for cylindrical samples in an axial magnetic field, and it appears that the common feature in all these experiments is flux flow in the samples. No significant change of magnetization for any given field value was found for the sample under test when the frequency of rotation was varied from 0.2 Hz up to I Hz. In Fig. 5 the dependence of the measured torque on field and frequency is shown. Within the measurement sensitivity there is no noticeable torque in the Meissner region. A rather steep rise of torque is observed as soon as magnetic flux enters the specimen, followed after a sharp maximum by a further increase of D until the magnetic field reaches its upper critical value Hc2. Then the torque drops in a step towards a field dependence which is caused by eddy currents in the normal sample. Due to the fact that it is the electric field rather than the current which is imposed on the specimen, losses in the mixed state here are higher than those which would be measured with the same technique in the same field region, if the sample were in the normal state. The measured torque is also plotted against frequency for several values of magnetic field (Fig. 6). The data obviously can be represented with a fair degree of accuracy by a linear

C R Y O G E N I C S . AUGUST 1976

h

on I0

g

e

f

6

m

4 J 2

0

I0

'o

6

d

~ 4 2

0

o

I 0.2

I 0.4

I 06

1 0.8

I I0

I 1.2

I 1.4

I 1.6

I 18

I 2.0

f, Hz Fig. 6 Dependence o f torque on frequency of revolution f o r different magneti.c fields

455

and (3). The dynamic pinning force per unit length per vortex, k(B), shown in Fig. 7, has values of the same order of magnitude as the corresponding static quantity obtained by other measurements s from a similar material.

10-4

lc iO -5

I

Nb(I)

d ~

T= 4.2 K

The torque measured at fields above Hc2 can be used to evaluate the normal state resistivity, H resulting in a value of an = 0 . 3 7 / l ~ cm. This allows comparison of the experimental r/with the theoretical value, ~7= ¢o BcJPn, given for fields near Hc~. Quite good agreement is found as can be seen in Fig. 8. Note the step in 77at Hc~ which results from the fact that the resistivity responsible for viscous losses changes at He2 from a highly anisotropic flux flow resistivity ,of to the practically isotropic normal state value

T

E u

z

p n .14

0

10-6

Conclusion

ec, 10-7 ~ 0

L

0.1

-

0.2

I ~

03

0.4

B, T

Fig. 7 Averagepinning force k per unit lengtn per vortex as function of flux density B in the sample. Error bars are determined mainly by D o. Markings c . . . . . m correspond to those in Fig. 6

Nb (I) T=4.2 K

d Z

C1

0.1

0.2

0.3 B, T

0.4

0.5

0.6

Fig. 8 B dependence of the viscosity coefficient n - theoretical value r / = D 0 Bc2Pn 1 . Error bars are determined mainly by D'

region near the field of first flux penetration (see data for fields a to e, Fig. 6), the linear behaviour of D due to viscous friction is seen to extend over a range which always is larger than the associated value of Do. The actual vortex configuration in the sample resulting from both viscous friction and pinning is therefore considered to produce (except again for the low field values) negligible corrections for (2) and (3). No significant deviations from a sinusoidal shape were observed in the signal 1"1(t) over the whole field range investigated. From measured values of Do and D' and the rotational magnetization, data for k and r/are then obtained using (2)

456

The determination of the dynamic pinning force per unit length per vortex, k using the method discussed appears to be most attractive for samples with comparatively large cross-sections. Thinner wires might also be investigated, although the winding of the pick-up coils for the magnetization in this case is more delicate. If the vortices are so strongly pinned that they rotate rigidly with the specimen, they give rise to a periodic alternating contribution to the torque, which cancels out when taking the time average of D. These vortices however do not contribute to B when the rotational magnetization is measured by the method described above.

4

2-

The results shown above demonstrate the feasibility of studying frictional forces on vortices by torque measurements on steadily rotating specimens at helium temperatures. It is not necessary to use the other possible method, a rotating magnetic field, which would be very awkward for heavy electromagnets. A related technique, developed by E.L. Andronikashvili and coworkers, 4,~s uses torsional oscillations of a cylindrical specimen in a transverse magnetic field instead of continuous rotation. Information about the frictional forces acting on the vortices is obtained from damping of these oscillations.

We would like to thank Dr H. Ullmaier, Kernforschungsanlage Jtilich, who supervised the outgassing of the specimen.

References 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15

A review of experiments on flux pinning was given recently by Freyhardt, H.C. Proe Intern Discussion Meeting on Flux Pinning in Superconductors, Sonnenberg, Germany (1974) p 98, see also

Campbell, A.M., l~vetts,J.E. Adv in Phys 21 (1972) 199 Wipf,S.L.,Westinghouse Sci Paper 64-1JO-280 P1 Heise,B. Rev Mod Phys 36 (1964) 64 Andronikashvili, E.L. et al Cryogenics 9 (1969) 119 Eggendorfer, G. JLow Ternp Phys 10 (1973) 715 Yamafuji,K., lrie, F. Phys Lett 25 A (1967) 387 Kim, Y.B., Hempstead, C.F., Strnad, A.R. Phys Rev A 139 (1965) 1163 Fuhrmans,M. Thesis, Universit;it MUnster (1974) Dubeek,L.W., Aston, D.R., Rothwath F. J Appl Phys 41 (1970) 1593 Walmsley,D.G. J Phys: F. Metal Phys 2 (1972) 510 Seh~er, R., Heiden, C. Appl Phys 9 (1976) 121 Sehmid,A. Phys Cond Mat 5 (1966) 302 Caroli,C., Maid, U. Phys Rev 164 (1967) 591 Heiden,C., Fuhrmans, M., Schafer, R. (forthcoming) Andronikashvili, E.L et al Phys Lett 25 A (1967) 85

CRYOGENICS. AUGUST 1976