Critical currents and rapid measurements of magnetic relaxation in superconducting thin films

PHYSICA

PhysicaC 192 (1992) 85-94 North-Holland

Critical currents and rapid measurements of magnetic relaxation in superconducting thin films S.W. G o o d y e a r , J.S. Satchell, R . G . H u m p h r e y s , N.G. C h e w a n d J.A. E d w a r d s DRA. Electronics Division, RSRE, Malvern, Worcs WR14 3PS, UK

Received 14 October 1991 Revised manuscriptreceived 3 December 1991

A fast DC magnetisationtechniqueusinga Hall effectprobe is described. It is used for measuringthin film criticalcurrentsand their relaxationas functions of temperature. At fixed temperatureit has been used to take current relaxation data over6 decades of timescale in under 3 h, much fasterthan standard magnetisationtechniques. Usinga thin film ring geometrywe deduce E-J characteristicsextendinq down to 10- t4 V/cm. Magneticrelaxationmeasurementsmadeon over 300 films consistentlyreveal a sharp divergencenear to Tc.At lowertemperaturesthe relaxationrate is found to varywithfilmgrowthtemperatureand annealing conditions.The effectof field line curvature has been measuredand its significancefor our resultsis discussed.

1. Introduction The critical current density is one of the most widely used figures of merit for high temperature superconductor thin films, as it reflects both the crystallographic and electronic quality of the material. For routine characterisation of material in device research a technique must be fast and non-destructive, and therefore should not require patterning and contacting. Magnetic measurements are widely used for this purpose. However, standard DC magnetisation techniques such as SQUID magnetometry or vibrating sample magnetometry start measuring only after seconds or tens of seconds, and so are rather slow if they are used over a wide range o f temperatures. In this paper we present a technique for measuring critical current density (Jc) and magnetic relaxation rate (S) as functions of temperature on a timescale down to milliseconds, thus achieving the rapid turn round needed. The basis of our technique is a Hall effect sensor and low noise electronics which have been used in an earlier study [ 1 ]. The principle is to apply a short magnetic field pulse sufficiently large to place the film in the critical state. After the pulse has been switched off, a persistent current circulates in tb.c film. The magnetic field due to this current is measured with

the Hall effect sensor as a function of time, and the magnitude of the critical current and its decay rate are deduced. Related contactless techniques have recently been published [2-5] but we believe this is the first account of rapid measurement of the persistent current decay as a function of temperature. Inclusion o f the early ms timescale decay allows measurements very near to the transition temperature, or over six decades in time without excessively long data acquismon.

2. Experimental The apparatus has been designed to maximise the speed with which measurements can be made. This leads to a construction rather different to conventional magnetometers. A schematic set up is shown in fig. I. The film is positioned face down on a pancake coil and put into the critical state by the field generated using a current pulse through the coil. This configuration of cod is chosen because it approximately mimics the desired current distribution in the film: the film behaves rather like a ground plane to the coil. This arrangement minimises both the coil inductance (and hence the field rise and decay times) and the energy dissipated in the coil.

0921-4534/92/$05.00 © 1992 ElsevierSciencePubhshers B.V. All rights reserved.

& B: Goodyear et al. / Magnetic relaxation in superconducting thin films

86

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Fig. 1. Schematic of the magnetometer head.

Since the field pulse will induce transient currents in normal metals, large conducting structures and pure metals (whose resistivity is very low at low t e m p e r m u ~ s ) have been avoided. The magnetometer probe head has been constructed from thin alloy or insulator, to promote the fast decay o f any unwanted transients, and all magnetic materials were avoided• It was, however, discovered that at certain positions in the measurement dewar, the field pulse did induce a response without a sample, believed to be associated with a weld line in the dewar neck. This effect was overcome by adding an outer concentric coil in series with the field coil, wound in opposition. This was balanced to cancel the magnetic moment of the inner coil at the radius of the dewar neck, but has httle effect on the field experienced by the sample. The pulse generator is a current source with very sharp turn o n / o f f properties and good zero current stability, as any noise or offset would be detected as a spurious signal by the Hall effect sensor. It produces pulses up to 10 A giving a maximum applied field size of order 0.0~ T on axis, in the plane o f the sample. A typical high current pulse lasts 3.5 ms, having a falling decay time of under 1 ms. When smaller (0-0.1 A) pulses are required such as when measuring a ver)" thin ring sample, a programmable Keithtey current source is used. As soon as the field pulse has decayed, the field due to the reduced persistent current m the sample ~s measured using the Hall effect sensor. Th~s ts a clover leaf of bulk lnSb. It has a carrier mobility of 5 . 3 × 1 0 s cm2/Vs and a sheet carrier density of 7 X 10 t° c m -2 with little temperature dependence below 100 K. The Hall sensor was cahbrated at hq-

uid nitrogen temperatures using a coil with known geometrical factor. In the measurement apparatus it is positioned 6 mm from the superconducting layer. This distance is chosen as a compromise between maximising the sensitivity to the currents flowing at the edges o f the film without making the measurement too dependent on the current density near its centre, as this is more dependent on the magnitude o f the field pulse. The calibration o f the magnetometer is fairly sensitive to the spacing between film and sensor, and the design is such as to minimise errors in this spacing. For brevity, we shall refer to the signal detected as M, although strictly it includes small contributions from higher order terms than the dipole one. The Hall voltage is sensed with a specially designed detector circuit [6 ] developed from the work of Daniil and Cohen [ 7 ]. This combination of electronics and sensor gives a field sensitivity of 5 nTHz-m2, which allows the detection of critical currents of l03 A/cm2 for a typical 0.35 gtm thickness film on a l 0 X 10 mm 2 substrate. A Si diode temperature sensor is mounted on an aluminium alloy plate in close contact with the back of the substrate. The whole apparatus is enclosed in a thin brass can to provide a stable thermal environment and eliminate any condensation on or near the sample during warming after measurement. To obtain data as a function of temperature this assembly is lowered into a stainless steel He dewar using a linear drive. The cooling rate during measurements is stabilised at 0.02 K / s - t from 95 K down to 80 K and at 0.04 K / s for lower temperatures. This is maintained by raising or lowering the linear drive under computer control using feedback from the temperature sensor. No observable effect of the cooling rate on the data is found. Measurements taken whilst warming snow the same results as those from a cooling run, with hysteresis in Jc corresponding to less than 0.5 K. The decay of the persistent current set up in the sample is routinely measured between !5 and 500 ms after the field pulse has decayed. A positive pulse is followed by an equal negative one, the decay of the persistent currents being measured after each individual pulse. The difference between positive and negative data is then used to deduce the average remanent field over the meamrement time and this value used to calculate the average critical current

S. W. Goodyear et aL / Magnetic relaxation m superconducting thin films

over the sample. This procedure eliminates the effect o f the earth's magnetic field and the sensor offset. Because of the short timescale of the measurement, data can be collected at a spacing of 0.25 K in a routine characterisation experiment, and the overall time to collect a complete set o f data and return the sample to room temperature is about one hour. For recording persistent current decay over much longer times immersion in liquid nitrogen or argon is used to obtain very stable temperatures. The samples used in this work were YBa2Cu307 thin films grown by co-evaporation of the metals onto polished (001) oriented MgO substrates in the presence o f atomic oxygen [ 8 ]. They were epitaxially coriented ( < 1% a-oriented) and had critical current densities up to 4 × 10 6 A / c m 2 at 77 K [9]. The standard film thickness was 0.35 am, and most were l cm 2 in area.

3. Results

In order to determine the current circulating in the film the approximation is made that the sample sustains a uniform current density J over its whole surface. For a ring shaped sample of inner radius a, outer radius b, the current density is deduced from the magnetic field B measured at the Hall sensor using

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where h is the distance of the Hall sensor above the ring and d the film thickness. For disc shapcd samples the appropriately simplified equation with a = 0 is used. For standard square shaped samples a numerical solution is used, calculated by taking concentric square loops. The correction between the field at a distance h above a disc shaped sample and a square shaped one of side equal to the disc diameter is approximately 12% for our standard film size and measurement geometry. Data typical of the persistent current decay we observe are shown in fig. 2, for a number of temperatures. Over this time period the decay transients show a characteristic logarithmic time dependence

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[ 10,11 ]. We calculate the current density from the average value of the detected magnetic field over this decay curve. A linear fit is made to the transient data (plotted on a logarithmic timescale) in order to calculate a slope and thus deduce the relaxatio-, rate. To measure a true Jc it is essential to ensure that most of the film has been driven into the critical state [ 12 ]. Figure 3 shows the effect o f the size of the applied field pulse on (a) the transient magnitude and slope and (b) the deduced normalised relaxation rate (-dLog(M)/dLog(t)). The lower curves in (a) show the film in the sub-critical state, with low relaxation rate. Both the magnitude of the signal and the normalised relaxation rate increase with increasing applied field, and saturate when the critical state is reached as in the top few transients. By observing this saturation effect at 77 K the minimum pulse size is found for a film with a given Jc. In measuring other films and at different temperatures, it is assumed that the field pulse required scales linearly with the sheet current density. The energy dissipated during this current pulse in the coil can be as large as 1 J. To check that this did not cause a significant thermal transient in the film on the time scale of the measurement, the power was increased by replacing a single pulse by a triple pulse set with alternating signs. This was found to have no observable effect on measured critical currents and little effect on the relaxation rate data in our measurement temperature range. However, large effects were observed when the measurement range was extended to low temperatures where the heat capacity

S. W. Goodyear et aL / Magnetic relaxation in superconducting thin films

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of the structure drops and the pulse size required is large. This is not the only problem in extending measurements to low temperatures. Below 40 K the field pulse has been found to cause catastrophic failure across the films. Figure 4 (a) shows a photograph of one such sample. The breakdown behaviour is in the form of radial "cracks" which on examination in the electron microscope (fig. 4 ( b ) ) show clear evidence oflocalised melting. They are believed to result from energy dissipated at the weakest point as the field in the sample is swept rapidly round a magnetisation h~steresls loop. The "cracks" are often found to pass through precipitates or other film defects. This has also been observed in ring structures even when the size of the field pulse has been minimtsed. For this reason all of the data presented here are for sample temperatures >/40 K.

Ftg. 4 (a) Optical micrograph of cracking caused by a large field pulse at low temperature. (b) Scanning electron microscope image of the crack shown in (a), suggesting that localised melting has occurred.

At the beginning of this section, we pointed out that it is assumed that the current flowing in the film is uniform to deduce the critical current density from the measured magnetic field. To test the validity of this approximation, some samples have been patterned into rings. It was found that the measurements on intact films are representative of the results obtained in this better defined ring geometry. A comparison of the current decay at fixed temperature is shown in fig. 5. There is a small difference ( ~ 5%) between the Jc deduced for the unpatterned film and the rings. This is larger than errors arising from measurement irreproducibility and is discussed in sectton 4 below.

89

S. W. Goodyear et al. / Magnetic relaxauon m superconductmg thin fiLns

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Fig. 5. Decay of the current density in a 8 mm dmmeter film (lower curve) and the same sample after the 4 mm and 6 mm diameter central regions have been removed by ion milling. These measurements were made with the sample in liquid nitrogen.

Figure 5 also d e m o n s t r a t e s the advantage o f using a fast technique: the m a g n e t i c relaxation has been m e a s u r e d over 6 decades o f time in about 10000 s. T h e large n u m b e r o f d e c a d e s covered, if carried o u t with say a fast S Q U I D system would have t a k e n nearly two weeks to measure. This timescale reveals a d e v i a t i o n from logarithmic time behaviour: it is f o u n d that a nearly straight line ts obtained w h e n l o g ( J ) is plotted against l o g ( t ) , as in fig. 5. It is o f interest to c o m p a r e transport and magnetic m e a s u r e m e n t s on the same sample. Four terminal t r a n s p o r t measurements were made on p a t t e r n e d tracks 60 lam long and 6 lain wide. The tracks were defined by p h o t o l i t h o g r a p h y and patterned by wet etching. The voltage criterion for the transport meas u r e m e n t s was 2 ~tV, as c o m p a r e d ~vith n a n o v o l t s calculated for the magnetisation technique. The g o o d a g r e e m e n t o f the results s h o w n in fig. 6 indicates that

both techniques are measuring essentially the same property. However, such good agreement is not always obtained. The differerce is attributable to patterning damage and film inhomogeneity, to which measurements on narrow tracks are rather sensitive. In some models, the temperature dependence of the relaxation rate is expected to give direct information on the nature of pinning centres. Figure 7 shows a number of relaxation rate (S) olots as a function o f temperature together with the corresponding J¢ versus temperature plots Here we have made a reasonable approximation to S by ( - 1/ Ma)dM/dLog(t), where Ma is the average magnetisation over the time period used. Over three hundred films have been measured and most show this general behaviour. A few degrees below the transition temperature the relaxation rate is fairly insensitive to temperature, varying between 0.015 and 0.035 over the whole temperature range studied, and tending to remain constant or increase as the tern~O.OB 07 0

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Fig 7 (a) Normahsed relaxation rate (S) and (b) crmcal current density as funcnons of temperature for a number of samples, chosen to illustrate a range of different types of behaviour. The curves are labelled to indicate whichJ¢ curves correspondto which relaxation data. The • on (a) indicates a fixed temperature measurement in liquid argon of the relaxation rate in the region of the divergence.

S. ~!: Goo@ear et al. / Magneuc relaxation in superconducting thin films

90

perature falls. Similar conclusions have been drawn [ 13 ] from measurements on aligned powders, single crystals and sputtered films. The data of fig. 7 suggest that S does not vary very much from sample to sample and similar observations [ 13 ] have led to speculation that some universal mechanism is involved. However, it is worth noting that the two curves plotted showing the highest relaxation rates were for 0.7 pm thick films whilst the rest are from standard (0.35 lain thick) growth runs, and that most of the Ideas we have studied were prepared in much the same way. More recently, we have begun to examine the effect ofvaLying the film preparation conditions on the electrical properties. Figure 8(a) shows S for three fdms grown at different temperatures. A clear relationship is seen between decreasing growth

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temperature and increasing S for this sample set, demonstrating that details of the relaxation behavtour are dependent on preparation conditions. Figure 8(b) shows a similar plot o f S as a function o f temperature for a single film after annealing at successively higher temperatures. Although the generality of the results o f fig. 8 is not estabfished, it is clear that the behaviour o f these films is far from universal. Similar conclusions were made by Griessen et al. [ 14] for YBa2Cu30~ in general. N o obvious relationship between S and Jc has been found: they appear to behave as more or less independent parameters. For all our samples a divergence in relaxation rate is observed at Tc (see fig. 7). Although the magnetic relaxation is more difficult to measure in this region due to lower magnetisation values and the large temperature dependence of the superconducting parameters near T~, this effect is consistently found in our unpatterned 10 mm square films and in the patterned ring structures. Such sharp behaviour has not been reported previously but we are confident that it is a real effect, which is more readily detected at short times. Measurements at fixed temperature ir.~ liquid argon on a specially chosen film with T ~ 88 K, just above the boiling point o f the argon have confirmed this fast creep rate (the • on fig. 7 indicates this measurement). The signal/noise ratio in this experiment was too poor to allow the functional form of the decay transient in this region to be assessed. This technique measures the J¢ and relaxation rate in the self field. The calculation to deduce a J¢ assurn'es the Bean model, i.e. that the critical current is independent o f the magnetic field due to the persistent current (the self field). Many groups do not measure flux decay in the remanent state on the grounds that the flux lines are strongly curved, and the field pattern is complex [ 15 ]. As yet there has been little experimental evidence on how significant ..; ~r¢~, might be. We have *}'~-~". . . . . ..r.,...~,~a prelimina% measurements in low magnetic fields of the J, and relaxation characteristics of a high quahty film ( J c ~ 4 × I06 A/cm ~ at 77 K in self field). With our equipment, experiments in the presence of external fields can only be carried out up to about 14 mT due to saturation of the Halt voltage detection c~rcmt. Figure 9 shows the effect o f a small magnetic

S. W. Goodyear et aL I Magnetic relaxation zn superconducting thin filrns

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field on our standard measurement of the relaxation parameter. In this case a static field was applied with the pancake coil by passing a D C current of up to 0.3 A through it. Even these weak fields have a dramatic effect on the form of the divergence near To. This is reminiscent of the effect that magnetic fields have on the resistivity of high temperature superconductors near To. The above experiment does not take into account the pattern o f field lines in the sample, as the applied field was in the same direction for both positive and negative field pulses. We have therefore examined the effect of field line curvature by changing the direction of the static field with respect to that of the pulsed field. Figure 10 shows the results of experiments in low fields applied with an external coil. Two cases were measured corresponding to whether the sense of the static field straightened the self field lines ( + ) or increased their curvature ( • ). Little difference between the two cases is observed until the applied field is of order the self field. Above this we nnu that the enect on Jc ,rig. ~uta; I of the flux line curvature is small compared with the reduction in J~ solely due to the applied field. The effect of flux line curvature on rela×ation measurements on whole films is found to be more significant. Figures 10(b) and (c) show the unnormalised decay rate ( G = - d M / d L o g ( t ) ) and the S value as a function of applied field. Again the two different senses of applied field

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are found to diverge when the applied field is approximately equal to the self field. Measurements in approximately twice the self field show S > 20% larger when the applied field increases the curvature of the field lines than when it straightens them.

4. D i s c u s s i o n

The critical current densities deduced from the magnetisaiion experiments described above can only underestimate the actual value. Values which are too 1°_~ ~ma.11 are obtained if the appneu nero is too zow azzu most of the film is not driven into the critical state. If the film is granular and does not support large cxrculating currents, the critical current deduced will be reduced in proportion to the ratio of the grain size to the film diameter. The dominance of macroscopic circulating currents in our experiments is confirmed by the data of fig. 5 where fields corresponding to --

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S. ll; Good~.~ar e~at. ~Magnetic ~laxauon m ~mconducnng thin films

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large current flow are observed even after removal of much o f the film area. It is L~.-.~obvious that magnetic relaxation observed on an unpattemed firm should be independent o f the sample geometry, b u t as~in fig. ~ verifies this to b e the case. The ring geometry used for the experiment o f fig. 5 allows the voltage criterion for the measurement o f J, to be defined [ 1 ]. (Other techniques [4,16 ] achieve a similar effect by imposing a constant field ramp d B J d t . ) Starting from a magnetic field decay transient as measured by the Hall sensor we deduce the rate o f decay o f the current using equation ( 1 ). The correspondence between the rate o f decay o f current and the e m f i n the ring is given by Faraday's law V= - L d i / d t . T h e inductance (L) of a thin ring is given approximately by L = poa (log[ 8a/w] - ~ ) ,

(2)

where a is the mean radius o f the ring and w its width. The data in fig. 11 are plotted as electric field ( E ) against current density ( J ) . assuming that both the dissipation and current are uniformly distributed over the ring. A uniform current density in this geo m e t w is a reasonable assumption but it is probable that the dissipation is concentrated local weak points in the ring. Nevertheless, it is conventional to discuss dissipation in terms of an average electric field. The longest times in our measurement correspond to

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an electric field criterion of order 10 - ' 4 V/cm, and we are able to measure up to 10 -8 V / c m due to the speed of the first measurement point. Over these six orders of measurement tune the estimate for the critical current decreases by nearly 20%. Plotted as log(E) versus log(J) as in fig. 11 it can be seen that the E - J characteristics are much more nearly power law ( E ~ J ' ) F~kethan exponential (log(E- E0) ~ J ) , as would be expected for conventional Kim-Anderson flux creep. Relaxation measurements taken over a small number of decades o f t i m e will not reveal the difference between these two forms. The data o f fig. 11 correspond to n varying from 45 at short times to 60 at long times so the power law is not exact. The form o f this dependence should yield information on the relationship between the current density and the effective pinning energy U ( J ) . The data available at present do not appear to agree closely with published models, but more samples need to be studied to confirm this. n is related to the measured normalised relaxation rate S = ( - d L o g ( M ) / d L o g ( t ) ) by S = 1/ ( n - 1 ) so the measurement of a wide plateau for the S value below T¢ (see fig. 7) is equivalent !o :he statement that n is approximately temperature ~adependent. Similar conclusions were drawn by Sun et al. [3 ]. They showed that this type of behaviour is inconsistent with the Kim-Anderson flux creep model and suggested a model involving a distribution of Jc's across the sample to explain the weak dependence of the n value on temperature. Other models [ 17,18 ] have predicted power law like behaviour in the E - J characteristics by allowing for a distribution o f activation energies. There are many conflicting theories of relaxation behaviour under discussion at the moment and all are still controvemial. Many of these theories are specifically aimed at the high field propertie~ of high T¢ materials, where vortex-vortex interactions are strong. The low field case, where inter-vortex forces are less important has attracted less attention. The

/

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terms of thermally activated flux motion over barnets ~ t h a constant height is [19] S= -dL°g(M)

F ~ I I E - J characteristics for a ~ m m oax,~ d m m c t c ; . L :r.:r. mncr d m m e l c r n n g ~ample dc~c.,cd from the da~ of F;g. 5 at " " K No~c hha~ b~th ~ c,~.cal a n d hon~-onta~ .~¢a!c~ a:c i~gan~hm:c

dLog(~)

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~here :~ ~s the time at v, hzch the first point of the

S. W. Good),ear et al. /Magnettc relaxation m superconducting thin films

relaxation data is taken, ~ is a relaxation time (typically l0 - t ° to l0 -t2 s) and U ( T ) the pinning energy. The pinning energy must tend to zero at T~ and in this region thermally activated flux creep should dominate. It is therefore not surprising that the creep rate diverges at T~. U ( T ) vanishes when approaching either Tc or Bc2(T ), so it is worth considering which is likely to be responsible for the observed divergence. The self field (Bs) is proportional to J ~ ( T ) . Near to To, the plot o f J¢ is curved, i.e. J c ~ ( T ~ - T ) ~ with a > 1. Since Be2 ~ ( T ~ - T),

B~ ~ ( Tc_ T)a_,_., 0

at T--} Tc

(4)

and the self field is not expected to play a role in the divergence. Compared with the data taken in just the self field, the broader experimental relaxation rate curves taken in low applied magnetic field (fig. 9) look much more like tho~e measured by other groups in high fields. Very recently Lensink et al. [20] have observed a divergence approaching Tc for a single crystal only after irradiation with neutrons. Suenaga et al. [ 21 ] have associated similar behaviour with the irreversibility line, S being a measure of the flux line "'mobility". We now discuss the significance of field line curvature for our measurements. The self field of a thin film disc can be expressed as the sum of two components, the radial and the axial fields [22 ]. Assuming a uniform current density and J~ independent o f field, they can be calculated from eq. ( 1 ) of ref. [22]. The radial component, absent in standard cylindrical or slab geometries, leads to a much more complicated field pattern [ 15 ]. This component is zero in the middle of the film and maximum at the surface of the film, varying linearly with height in between. It is approximately independent of position over the film area. For a typical film at 77 K with a thickness of 0.35 pm and de of 4X 10~"A / c m 2, the radial field is 6 mT at the film surface. The magnitude o f the axial component varies roughly linearly with radial po3ition, reaching a maximum at the centre of the film. For the typical film at 77 K the peak value is ~ 100 mT. The axial field changes sign at approximately 0.8 of the film radius. Our exper-

93

iments measure predominantly the current flowing at large radii, where the magnitude o f the perpendicular field is rather less than that of the radial component. It is clear that an externally applied field of order 14 m T (as shown in fig. 10 ) is ample to change the shape o f the field pattern dramatically, in particular to make them nearly straight, and move the point where the axial field changes sign to well outside the film area for one sense o f applied field. We find (see fig. 10) that an applied field of either sign reduces J¢ and increases S. The difference between the two signs gives an indication of the importance of flux line curvature. Although the field curvature has an observable effect on Jc, it is much less than the reduction of J~ due to a magnetic field of either direction. The effects on S are more significant, but small enough to show that measurements of remanent magnetisation decay in thin films are not dominated by self field effects. We conclude that our calculation of Jc is largely unaffected and that S remains a reasonable measure of the relaxa tion behaviour. 5. Conclusions We have presented a novel, fast, non-destructive technique for measuring both critical current and its relaxation as functions of temperature. The speed of the measurement (typically less than 1 h) enables us to characterise routinely every film we grow, and several hundred have already been measured. The results have been given without extensive discussion of their interpretation in terms of flux motion. There are many conflicting theories in this subject, and we believe that progress will best be achieved by clarifying the systematics of the magnetisation decay over a wide range o f samples. The rapidity of the measurement has allowed both measurements over 6 decades of time and observations near Tc where the relaxation rate diverges. Although the value of measurements of remanent magnetisation can be questioned because the field lines are curved, we have shown that this effect is not large. At the lowest level, the temperature dependent magnetisation decay rate S gives a parameter characteristic of a film, which can be regarded as a fingerprint of its defect structure, to be correlated with other parameters such as Jc or microwave surface resistance.

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S. ii: Goodyear et a L / Magnetic relaxatwn m superconducting thin films

Acknowledgements We are grateful to T. Ashley for supplying the Hall sensors, C. Muirhead for useful discussions and we would like to thank the referee for his very constructive comments.

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