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The influence of concentration gradients on the mixed state critical surface currents of type II superconductors
An investigation o f the critical current o f Pbln alloy strips in which the ahoy composition near the sample surface has been altered by a diffusion process is described. It was found that the mixed state critical current can be increased or decreased by introducing composition gradients of appropriate sign.
The influence of concentration gradients on the mixed state critical surface currents of type II superconductors J. Lowell
It is known that the d c critical current o f type II superconductors is influenced by interactions of the fluxons with both the sample surface and defects in the 'bulk' of the material. The surface contribution to the current-carrying capacity may be of considerable importance in a c applications, because it is probably not subject to inherent a c losses as is the 'bulk' contribution. On the other hand, the usefulness of the surface contribution to the critical current is limited by its rather small magnitude; it is important, therefore, to investigate any possible means of increasing it. This paper reports an investigation of the effect of variations in alloy composition near to the sample surface. The work was prompted by the discovery ~ of Doidge et al, -that superficial concentration gradients can influence the surface current in the 'sheath' regime. As these authors point out, it is reasonable to expect similar effects in the mixed state. In fact, it is probable 2 that such effects have already been detected in NbMo alloys, which exhibited greatly enhanced critical currents after vacuum annealing. It is likely that the increase in critical current was due to composition changes near the surface, induced by preferential evaporation of Mo. In the present work, we have studied Pbgoln~o samples in which the concentration of In near to the surface was increased or decreased. These changes in composition were brought about by evaporating either In or Pb onto the sample surface and then heating to allow diffusion to take place. We find that the critical current is strongly affected by the resulting gradient in impurity concentration. The critical current can be either increased or decreased, according to the sign of the gradient. The author is with "the University of Manchester, Institute of Science and Technology, Sackvilte Street, Manchester M601QD. Received 1 April 1971.
298
Experiments All experiments were performed at 4"2 K on strips (typically 2 mm x 1 cm x 2 cm) cut from rolled foil of an alloy of Pb (913 at %) and In (10 at %). All samples were initially annealed for several days at room temperature. Further annealing for several hours at 100C was not found to influence the critical currents. Each sample was chemically polished in a mixture of glacial acetic acid (8 parts) and 30% aqueous hydrogen peroxide (2 parts). Then, after a few minutes exposure to the atmosphere, the sample was introduced into liquid helium, and its critical current measured in magnetic fields both parallel and perpendicular to the sample surface. The magnetic field was judged to be parallel to the sample plane when the critical current attained a maximum value with respect to field direction; the latter being varied by rotating the external electromagnet. (All samples exhibited the strong angular variation of critical current, characteristic of surface effects.) The critical current was defined as the current giving rise to a voltage drop along the sample o f ] ~V. After measuring the critical current, the sample was withdrawn from the liquid helium and introduced into a vacuum chamber, where a layer about 10 -6 m thick, of either Pb or in, was evaporated onto each of its two larger surfaces. The pressure in the vacuum chamber was of the order of 10 -s torr (1"3 x 10 -3 N m -2) during evaporation. Transfer of the sample from cryostat to vacuum chamber involved exposing the sample surfaces to the atmosphere for about 5 minutes. The critical current of the sample was then remeasured. Subsequently, the sample was heat-treated in a bath of transformer oil, maintained at 102 C, in order that diffusion might occur between the sample and the evaporated layer. The critical current of the sample was re-examined at various stages during the heat treatment. It was verified by CRYOGENICS . AUGUST 1971
experiment that heat treatment of an uncoated sample had no appreciable effect on the critical current. The temperature of heat treatment (102 C) was chosen to give diffusion to a depth ~)t in times of the order of 1 hour. This estimate is however rough because of uncertainty in the diffusion constants of the Pbln system and because o f the possible effect of an oxide layer on the sample surface. It is, of course, to be expected that diffusion will be assisted by grain boundaries running down from the sample surface. We are interested, however, in diffusion lengths (~ 10 -~ m) very much smaller than the grain size (~ 10 -3 m), so the diffused layer will not be affected much by the presence of grain boundaries. On the other hand, the formation of indium-rich (or indium-depleted) 'sheets' at the grain boundaries probably has an important effect on the critical current in fields perpendicular to the treated surface (see Appendix).
The data given in Figures I and 2 are typical of several Pb coated samples studied. All lead-coated samples behaved in a qualitatively similar way, but the magnitude of the changes in critical current, and the time scale of the changes, varied from one sample to another. These variations are probably connected with differences in oxide films present on the various samples. In one sample, (that of Figure 2), the critical current did not completely return to its initial value. We suppose that this is due to slight damage to the surface at some stage during heat treatment.
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This section reports the effect of heat-treatment of Pb and In coated samples on the parallel-field critical current. Critical current data taken with the field perpendicular to the sample plane are discussed in the Appendix. Before evaporation, the critical current per unit width was the same within ~ 30% for all samples investigated. The critical current always exhibited a strong dependence on the angle between the field direction and the sample plane, with a sharp maximum in the parallel position. This indicates that the critical current is dominated by the sample surface. With the field parallel to the sample plane, the critical current curve showed a sharp kink at about 0"23 T. The perpendicular field critical current curve dropped rapidly towards zero at about the same field. We identify this field (0"23 T) with Hc2. Evaporation of either Pb or In onto the sample surface had in itself no effect on the critical currents below Hc2. This was usually true also for the critical currents in fields above Hc2 , though a slight decrease was detectable in one or two samples, probably because the sample became rather warm during evaporation. The absence o f large changes in the critical currents on coating with In or Pb suggests the presence of an oxide layer on the sample surface. If good metallic contact between the sample and the evaporated metal had been achieved, a a significant decrease in the critical currents would be expected for fields greater than
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Hc 2 • Figures 1 and 2 show the effect of heat treatment on the critical current of lead-coated samples. We consider only the critical current for fields parallel to the plane of the sample. In the region of field H < Hc2, the critical current is initially unchanged after evaporation of lead, but is strongly reduced after only a few minutes heat treatment. Prolonged heat treatment gradually restores the critical current to its initial value (Figure 2). The mixed state critical current (H~-.Hc2) behaves in a rather similar way, decreasing during the initial stages of heat treatment but returning eventually to its initial value. The changes for HHc2, as is clear in Figure 2.
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Figure 2. Critical current, at t w o different fields, of a lead-coated alloy of Pb/10 at % In as a function o f time o f heat treatment at 100 C. This sample is not the same as that o f Figure 1
299
Figure 3 shows the critical currents of an indium-coated sample after heat-treating for various times at 102 C. In the region H>Hc2 the critical current falls during the initial stages of heat treatment. Prolonged heat treatment causes the critical current to rise again to its initial value. This behaviour resembles that found for lead-coated samples. Below Hc2, there is a slight but significant rise in the critical current during the initial stages of heat treatment. It should be noted that a critical current measurement detects the weakest pinning along the length of the sample. For this reason, the critical current is probably a rather poor indication of the effect of heat treatment on In-coated samples. A sizeable increase in pinning force over most of the sample would be masked if some small regions remained for some reason relatively unaffected. We have investigated this possibility by measuring 11/1characteristics (see Figure 4). It can be seen that the linear portion shifts to higher currents in the initial stages of heat treatment and returns towards the initial curve if heat treatment is prolonged. But it is also evident that, after heat treatment, the 1I/1 characteristics develop a pronounced low current 'tail'. This is characteristic of inhomogeneous pinning, 4 and suggests that the diffusion of indium into the sample affects some regions much less than others. Thus, the critical current
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measurements of Figure 2 probably underestimate the effect of indium diffusion. The absence of any significant change in mixed-state critical current in two of the indium-coated samples studied is also understandable on this basis. A number of experiments were performed to determine the effect of changing the heat-treatment temperature. It was found, as expected, that prolonged periods (several days) at room temperature had no effect on the critical current. Figure 5 shows that high-temperature heat treatment of a Pb-coated sample reduces the critical current and shifts the critical current curve to lower fields whereas high-temperature heat treatment of an indium-coated sample has exactly opposite effects. The effect of hightemperature heat treatment seems quite different from that of the 102 C treatment. The shift in the critical current curves apparent in Figure 5 strongly suggests that the composition of the sample in the surface region is altered sufficiently to change Hc2 by a significant amount. All the experiments described above were performed on samples having two surfaces parallel to the field. We have also performed experiments on samples with a single effective surface, using the triangular geometry described by Swartz and Hart. s The behaviour of these samples was not significantly different from that of the strip samples described above. We had hoped to investigate the effect of indium diffusion on the polarity effects described by Swartz and Hart, but polarity effects were not found in our samples, either before or after coating and heat treatment.
Discussion I O-I
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Figure 3. Critical current versus field f o r an indium-coated alloy (Pbg01n]0) at several stages o f heat treatment at 102 C. Curves f o r field parallel and perpendicular t o the sample plane are s h o w n
300
We draw the following conclusions from the experimental results. 1. Evaporation of metal onto the sample surface does not in itself influence the critical current. This indicates 3 that the sample surface is protected from metallic contact by an oxide layer. C R Y O G E N I C S . A U G U S T 1971
2. During heat treatment at 102 C, the sample composition remains close to PbgoIn~0, even near to the surface. This is implied by the absence of any shift in the critical current curves towards higher or lower fields (see Figures 1 and 3) and also by the fact that prolonged heat treatment restores the critical currents to their initial values. The absence of marked changes in composition of the surface region is probably due to diffusion through the oxide layer occurring much more slowly than diffusion through the alloy. In contrast, heat treatment at significantly higher temperatures (see Figure 5) does appear to alter the sample composition, at least near to the surface.
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3. The critical current above Hc2 is always reduced in the initial stages of heat treatment. Below Hcz , the initial stages of heat treatment lead to afall in the critical current of lead-coated samples but to a rise in the critical current of indium-coated samples. Prolonged heat treatment always tends to restore the critical current to its initial value. Conclusion 2 shows that we need not consider complications arising from the two-phase region (aroundabout 70% In) in InPb alloys. The return of the critical currents to their initial value after prolonged heat treatment cannot be due to complete depletion of the evaporated layer. For, in the first place, the critical current below Hc2 may still be decreasing after the critical current above Hc2 has almost returned to its initial value. Secondly (conclusion 2), it is apparent that only a small amount of material diffuses out of or into the evaporated layer during heat treatment at 102 C, for the composition of the surface region of the alloy does not appear to change significantly (the experiments at higher temperatures (Figure 5) show that it is possible to change the near-surface composition very considerably with the amount of evaporated metal available). Pinning associated with the sample surface is thought to be due to the Bean-Livingston barrier. 6-a We shall discuss our experimental results in terms of a simple model whereby (for applied fields small compared to Hc2 ) the force, F, on unit length of a fluxon distance x from the metal surface can be written s
F=ae-2X/X-[3Be-X/h+F(B,x)
...(1)
where H is the applied field, B the flux density in the superconductor, and X the penetration depth; a and 13 are approximately constant. The terms on the RHS represent the forces due to interaction of the fluxon with its 'image' in the surface, to interaction with the shielding current, and to interaction with fluxons in the bulk of the material. The absence of rectification effects s in our samples is not compatible with this model and it seems likely that the real situation is more complicated than that described by (1). We shall ignore this difficulty in the absence of a more satisfactory model. We propose that the heat treatment of the coated samples alters the mixed-state critical current because it establishes a gradient in alloy composition ~ near to the surface. We therefore consider how (1) must be modified if C R Y O G E N I C S . A U G U S T 1971
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the concentration C(x) of (say) indium varies with distance x from the sample surface. It is apparent that this equation must be modified in two ways.
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(a) A force - p ~ must be added to the RHS, where p is the rate of chang6 of the free energy of unit length of fluxon with respect to concentration. (b) Since the positions of the fluxons within the sample will be modified by the concentration gradient, the term F(B, x) must be adjusted. In the case where OC/ax varies slowly over the barrier region, the contributions (a) and (b) should approximately cancel each other - j u s t as in the interior of the superconductor there is no resultant force on a fluxon due to a slowly varying concentration gradient. If, on the other hand, C varies so rapidly with x that OC/~x is large fo.r x -~ 0 but negligible for x > X, then the fluxons within the sample are unaffected, and only (a) is applicable. The unbalanced effect of (a) will alter the effective barrier height and modify the critical current. 301
Thus, we expect the critical current to be influenced by a concentration gradient near to the surface, providing that the concentration gradient is larger for x ~" 0 than for x~>k. If the diffusion into (or out of) the alloy is limited by the oxide layer, the concentration gradient in the metal at distance x into the surface takes the form x
dC(x)/dx = constant erfc 2 ~
For times t ~ 2 / D the concentration gradient is appreciable near to x = 0 but small for x>~X. For times t >~X2/D, the concentration gradient is approximately constant over a distance X from the surface of the metal. We would therefore expect a change in critical current during the initial stages of diffusion, due to the concentration gradient for x < X. As diffusion proceeds, however, the concentration gradient becomes approximately uniform over a distance X and then has little or no effect on the critical current. This is the behaviour observed in our experiments, and it is feasible that the observed effects are due to concentration gradient near to the surface. One would in addition expect changes in critical current due to the change in composition of the material near to the surface, which results in a change in He2 near to the surface. In fact, if the alloy composition changes so that the material changes from normal to superconducting over a length ~X, then the pinning associated with the surface is entirely removed. 9 Critical current changes observed in the present experiments appear to be due to composition gradients; the change in composition near the surface appears to be too small to affect the critical current appreciably. It seems probable that much larger concentration gradients could be established if the oxide layer were absent. Much larger critical current changes should then result, though the complicating effects of He2 changes would also be present. Conclusion
In our lead alloy, the current-carrying capacity associated with a surface parallel to the applied field can be enhanced by diffusing indium into the surface or reduced by diffusing indium out of the surface. The dependence of critical current on the length of time for which diffusion is allowed to proceed suggests that a gradient in the concentration of impurities is responsible.
302
I am grateful to Prof A. C. Rose-Innes for useful suggestions and criticisms and to the Ministry of Technology for support. APPENDIX Critical current for fields perpendicular to the sample surfaces
The critical current of a lead-coated sample for magnetic fields perpendicular to the sample surfaces is shown in Figure 1. It is apparent that the critical current is considerably enhanced by the diffusion of indium out of the sample surface. Other lead-coated samples were found to behave in a similar way. The critical current of indium-coated samples was also found to increase as a result of heat treatment. The dependence of critical current on heat-treatment time is quite different from that for the parallel-field case. The perpendicular field critical current of both lead-coated and indium-coated samples increased rapidly during the first few minutes of heat treatment, then continued to increase slightly as heat treatment was continued. There was no sign of any decrease in critical current even after many hours heat treatment. We believe the increase in critical current to be an example of pinning by 'surface irregularities', as proposed by Swartz and Hart (1967). Heat treatment of an indiumcoated sample will cause indium to migrate very rapidly down the grain boundaries whence it will diffuse a short distance into the surrounding material. The resulting 'sheets' of indium-rich material will act as barriers to fluxons attempting to move past them. Similarly, heat treatment of lead-coated samples will create regions of abnormally low indium content. REFERENCES
1. DOIDGE, P. R., KWAN SIK-HUNG, and TILLEY, D. R. PhilMag 13, 795 (1966) 2. FRENCH, R.A., and LOWELL, J. Phys Rev 173 504 (1968) 3. SAINT-JAMES, D., and DE GENNES, P. G. Phys Lett 7, 306 (1963) 4. JONES, R. G., RHODERICK, E. H., and ROSE-INNES, A. C. Phys Lett 24A, 318 (1967) 5. SWARTZ, P. S., and HART, H. R. Phys Rev 137, A818 (1965);PhysRev 156,412 (1967) 6. BEAN,C. P., and LIVINGSTON, J. D. Phys Rev Lett 12, 14 (1964) 7. CAMPBELL, A. M., EVETTS, J. E., and DEW HUGHES, D. PhilMag 18, 313 (1968) 8. LOWELL, J. J Phys. C (Solid St Phys) 2,372 (1969) 9. EVETTS, J. E. PhysRev B2, 95 (1970)
CRYOGENICS
. A U G U S T 1971
The influence of concentration gradients on the mixed state critical surface currents of type II superconductors J. Lowell
It is known that the d c critical current o f type II superconductors is influenced by interactions of the fluxons with both the sample surface and defects in the 'bulk' of the material. The surface contribution to the current-carrying capacity may be of considerable importance in a c applications, because it is probably not subject to inherent a c losses as is the 'bulk' contribution. On the other hand, the usefulness of the surface contribution to the critical current is limited by its rather small magnitude; it is important, therefore, to investigate any possible means of increasing it. This paper reports an investigation of the effect of variations in alloy composition near to the sample surface. The work was prompted by the discovery ~ of Doidge et al, -that superficial concentration gradients can influence the surface current in the 'sheath' regime. As these authors point out, it is reasonable to expect similar effects in the mixed state. In fact, it is probable 2 that such effects have already been detected in NbMo alloys, which exhibited greatly enhanced critical currents after vacuum annealing. It is likely that the increase in critical current was due to composition changes near the surface, induced by preferential evaporation of Mo. In the present work, we have studied Pbgoln~o samples in which the concentration of In near to the surface was increased or decreased. These changes in composition were brought about by evaporating either In or Pb onto the sample surface and then heating to allow diffusion to take place. We find that the critical current is strongly affected by the resulting gradient in impurity concentration. The critical current can be either increased or decreased, according to the sign of the gradient. The author is with "the University of Manchester, Institute of Science and Technology, Sackvilte Street, Manchester M601QD. Received 1 April 1971.
298
Experiments All experiments were performed at 4"2 K on strips (typically 2 mm x 1 cm x 2 cm) cut from rolled foil of an alloy of Pb (913 at %) and In (10 at %). All samples were initially annealed for several days at room temperature. Further annealing for several hours at 100C was not found to influence the critical currents. Each sample was chemically polished in a mixture of glacial acetic acid (8 parts) and 30% aqueous hydrogen peroxide (2 parts). Then, after a few minutes exposure to the atmosphere, the sample was introduced into liquid helium, and its critical current measured in magnetic fields both parallel and perpendicular to the sample surface. The magnetic field was judged to be parallel to the sample plane when the critical current attained a maximum value with respect to field direction; the latter being varied by rotating the external electromagnet. (All samples exhibited the strong angular variation of critical current, characteristic of surface effects.) The critical current was defined as the current giving rise to a voltage drop along the sample o f ] ~V. After measuring the critical current, the sample was withdrawn from the liquid helium and introduced into a vacuum chamber, where a layer about 10 -6 m thick, of either Pb or in, was evaporated onto each of its two larger surfaces. The pressure in the vacuum chamber was of the order of 10 -s torr (1"3 x 10 -3 N m -2) during evaporation. Transfer of the sample from cryostat to vacuum chamber involved exposing the sample surfaces to the atmosphere for about 5 minutes. The critical current of the sample was then remeasured. Subsequently, the sample was heat-treated in a bath of transformer oil, maintained at 102 C, in order that diffusion might occur between the sample and the evaporated layer. The critical current of the sample was re-examined at various stages during the heat treatment. It was verified by CRYOGENICS . AUGUST 1971
experiment that heat treatment of an uncoated sample had no appreciable effect on the critical current. The temperature of heat treatment (102 C) was chosen to give diffusion to a depth ~)t in times of the order of 1 hour. This estimate is however rough because of uncertainty in the diffusion constants of the Pbln system and because o f the possible effect of an oxide layer on the sample surface. It is, of course, to be expected that diffusion will be assisted by grain boundaries running down from the sample surface. We are interested, however, in diffusion lengths (~ 10 -~ m) very much smaller than the grain size (~ 10 -3 m), so the diffused layer will not be affected much by the presence of grain boundaries. On the other hand, the formation of indium-rich (or indium-depleted) 'sheets' at the grain boundaries probably has an important effect on the critical current in fields perpendicular to the treated surface (see Appendix).
The data given in Figures I and 2 are typical of several Pb coated samples studied. All lead-coated samples behaved in a qualitatively similar way, but the magnitude of the changes in critical current, and the time scale of the changes, varied from one sample to another. These variations are probably connected with differences in oxide films present on the various samples. In one sample, (that of Figure 2), the critical current did not completely return to its initial value. We suppose that this is due to slight damage to the surface at some stage during heat treatment.
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This section reports the effect of heat-treatment of Pb and In coated samples on the parallel-field critical current. Critical current data taken with the field perpendicular to the sample plane are discussed in the Appendix. Before evaporation, the critical current per unit width was the same within ~ 30% for all samples investigated. The critical current always exhibited a strong dependence on the angle between the field direction and the sample plane, with a sharp maximum in the parallel position. This indicates that the critical current is dominated by the sample surface. With the field parallel to the sample plane, the critical current curve showed a sharp kink at about 0"23 T. The perpendicular field critical current curve dropped rapidly towards zero at about the same field. We identify this field (0"23 T) with Hc2. Evaporation of either Pb or In onto the sample surface had in itself no effect on the critical currents below Hc2. This was usually true also for the critical currents in fields above Hc2 , though a slight decrease was detectable in one or two samples, probably because the sample became rather warm during evaporation. The absence o f large changes in the critical currents on coating with In or Pb suggests the presence of an oxide layer on the sample surface. If good metallic contact between the sample and the evaporated metal had been achieved, a a significant decrease in the critical currents would be expected for fields greater than
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Figure 1. Critical current versus field for a lead-coated alloy of Pb/10 at % In at several stages of diffusion at 102 C. The t w o sets o f curves correspond to magnetic fields parallel and perpendicular to the surfaces, as indicated
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Hc 2 • Figures 1 and 2 show the effect of heat treatment on the critical current of lead-coated samples. We consider only the critical current for fields parallel to the plane of the sample. In the region of field H < Hc2, the critical current is initially unchanged after evaporation of lead, but is strongly reduced after only a few minutes heat treatment. Prolonged heat treatment gradually restores the critical current to its initial value (Figure 2). The mixed state critical current (H~-.Hc2) behaves in a rather similar way, decreasing during the initial stages of heat treatment but returning eventually to its initial value. The changes for H
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Figure 2. Critical current, at t w o different fields, of a lead-coated alloy of Pb/10 at % In as a function o f time o f heat treatment at 100 C. This sample is not the same as that o f Figure 1
299
Figure 3 shows the critical currents of an indium-coated sample after heat-treating for various times at 102 C. In the region H>Hc2 the critical current falls during the initial stages of heat treatment. Prolonged heat treatment causes the critical current to rise again to its initial value. This behaviour resembles that found for lead-coated samples. Below Hc2, there is a slight but significant rise in the critical current during the initial stages of heat treatment. It should be noted that a critical current measurement detects the weakest pinning along the length of the sample. For this reason, the critical current is probably a rather poor indication of the effect of heat treatment on In-coated samples. A sizeable increase in pinning force over most of the sample would be masked if some small regions remained for some reason relatively unaffected. We have investigated this possibility by measuring 11/1characteristics (see Figure 4). It can be seen that the linear portion shifts to higher currents in the initial stages of heat treatment and returns towards the initial curve if heat treatment is prolonged. But it is also evident that, after heat treatment, the 1I/1 characteristics develop a pronounced low current 'tail'. This is characteristic of inhomogeneous pinning, 4 and suggests that the diffusion of indium into the sample affects some regions much less than others. Thus, the critical current
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measurements of Figure 2 probably underestimate the effect of indium diffusion. The absence of any significant change in mixed-state critical current in two of the indium-coated samples studied is also understandable on this basis. A number of experiments were performed to determine the effect of changing the heat-treatment temperature. It was found, as expected, that prolonged periods (several days) at room temperature had no effect on the critical current. Figure 5 shows that high-temperature heat treatment of a Pb-coated sample reduces the critical current and shifts the critical current curve to lower fields whereas high-temperature heat treatment of an indium-coated sample has exactly opposite effects. The effect of hightemperature heat treatment seems quite different from that of the 102 C treatment. The shift in the critical current curves apparent in Figure 5 strongly suggests that the composition of the sample in the surface region is altered sufficiently to change Hc2 by a significant amount. All the experiments described above were performed on samples having two surfaces parallel to the field. We have also performed experiments on samples with a single effective surface, using the triangular geometry described by Swartz and Hart. s The behaviour of these samples was not significantly different from that of the strip samples described above. We had hoped to investigate the effect of indium diffusion on the polarity effects described by Swartz and Hart, but polarity effects were not found in our samples, either before or after coating and heat treatment.
Discussion I O-I
I 0.2 8, T
I 0-3
Figure 3. Critical current versus field f o r an indium-coated alloy (Pbg01n]0) at several stages o f heat treatment at 102 C. Curves f o r field parallel and perpendicular t o the sample plane are s h o w n
300
We draw the following conclusions from the experimental results. 1. Evaporation of metal onto the sample surface does not in itself influence the critical current. This indicates 3 that the sample surface is protected from metallic contact by an oxide layer. C R Y O G E N I C S . A U G U S T 1971
2. During heat treatment at 102 C, the sample composition remains close to PbgoIn~0, even near to the surface. This is implied by the absence of any shift in the critical current curves towards higher or lower fields (see Figures 1 and 3) and also by the fact that prolonged heat treatment restores the critical currents to their initial values. The absence of marked changes in composition of the surface region is probably due to diffusion through the oxide layer occurring much more slowly than diffusion through the alloy. In contrast, heat treatment at significantly higher temperatures (see Figure 5) does appear to alter the sample composition, at least near to the surface.
==.oOo
"
o In c o o t e d , before treotment A In cooted, o f t e r treotrnent • Pb cooted, befonz treatment • Pb tooted, after treotment
A
A
8o
A A
1.0
A
oo
Ib
k o"
3. The critical current above Hc2 is always reduced in the initial stages of heat treatment. Below Hcz , the initial stages of heat treatment lead to afall in the critical current of lead-coated samples but to a rise in the critical current of indium-coated samples. Prolonged heat treatment always tends to restore the critical current to its initial value. Conclusion 2 shows that we need not consider complications arising from the two-phase region (aroundabout 70% In) in InPb alloys. The return of the critical currents to their initial value after prolonged heat treatment cannot be due to complete depletion of the evaporated layer. For, in the first place, the critical current below Hc2 may still be decreasing after the critical current above Hc2 has almost returned to its initial value. Secondly (conclusion 2), it is apparent that only a small amount of material diffuses out of or into the evaporated layer during heat treatment at 102 C, for the composition of the surface region of the alloy does not appear to change significantly (the experiments at higher temperatures (Figure 5) show that it is possible to change the near-surface composition very considerably with the amount of evaporated metal available). Pinning associated with the sample surface is thought to be due to the Bean-Livingston barrier. 6-a We shall discuss our experimental results in terms of a simple model whereby (for applied fields small compared to Hc2 ) the force, F, on unit length of a fluxon distance x from the metal surface can be written s
F=ae-2X/X-[3Be-X/h+F(B,x)
...(1)
where H is the applied field, B the flux density in the superconductor, and X the penetration depth; a and 13 are approximately constant. The terms on the RHS represent the forces due to interaction of the fluxon with its 'image' in the surface, to interaction with the shielding current, and to interaction with fluxons in the bulk of the material. The absence of rectification effects s in our samples is not compatible with this model and it seems likely that the real situation is more complicated than that described by (1). We shall ignore this difficulty in the absence of a more satisfactory model. We propose that the heat treatment of the coated samples alters the mixed-state critical current because it establishes a gradient in alloy composition ~ near to the surface. We therefore consider how (1) must be modified if C R Y O G E N I C S . A U G U S T 1971
o• o
~ 0.1
8= 0 • 0 •
o!2
i
o.3
i
o.4
B, T
Figure 5. Critical current of a lead-coated sample before and after heat treatment f o r 10 rain at 150 C, and critical current of an indium-coated sample after heat treatment for 5 h at 1 6 0 C . (Magnetic field parallel to the sample surface in each case.) The critical currents of the indium-coated sample have been multiplied by 2"5 to facilitate comparison with the lead-coated sample
the concentration C(x) of (say) indium varies with distance x from the sample surface. It is apparent that this equation must be modified in two ways.
aC(x)
(a) A force - p ~ must be added to the RHS, where p is the rate of chang6 of the free energy of unit length of fluxon with respect to concentration. (b) Since the positions of the fluxons within the sample will be modified by the concentration gradient, the term F(B, x) must be adjusted. In the case where OC/ax varies slowly over the barrier region, the contributions (a) and (b) should approximately cancel each other - j u s t as in the interior of the superconductor there is no resultant force on a fluxon due to a slowly varying concentration gradient. If, on the other hand, C varies so rapidly with x that OC/~x is large fo.r x -~ 0 but negligible for x > X, then the fluxons within the sample are unaffected, and only (a) is applicable. The unbalanced effect of (a) will alter the effective barrier height and modify the critical current. 301
Thus, we expect the critical current to be influenced by a concentration gradient near to the surface, providing that the concentration gradient is larger for x ~" 0 than for x~>k. If the diffusion into (or out of) the alloy is limited by the oxide layer, the concentration gradient in the metal at distance x into the surface takes the form x
dC(x)/dx = constant erfc 2 ~
For times t ~ 2 / D the concentration gradient is appreciable near to x = 0 but small for x>~X. For times t >~X2/D, the concentration gradient is approximately constant over a distance X from the surface of the metal. We would therefore expect a change in critical current during the initial stages of diffusion, due to the concentration gradient for x < X. As diffusion proceeds, however, the concentration gradient becomes approximately uniform over a distance X and then has little or no effect on the critical current. This is the behaviour observed in our experiments, and it is feasible that the observed effects are due to concentration gradient near to the surface. One would in addition expect changes in critical current due to the change in composition of the material near to the surface, which results in a change in He2 near to the surface. In fact, if the alloy composition changes so that the material changes from normal to superconducting over a length ~X, then the pinning associated with the surface is entirely removed. 9 Critical current changes observed in the present experiments appear to be due to composition gradients; the change in composition near the surface appears to be too small to affect the critical current appreciably. It seems probable that much larger concentration gradients could be established if the oxide layer were absent. Much larger critical current changes should then result, though the complicating effects of He2 changes would also be present. Conclusion
In our lead alloy, the current-carrying capacity associated with a surface parallel to the applied field can be enhanced by diffusing indium into the surface or reduced by diffusing indium out of the surface. The dependence of critical current on the length of time for which diffusion is allowed to proceed suggests that a gradient in the concentration of impurities is responsible.
302
I am grateful to Prof A. C. Rose-Innes for useful suggestions and criticisms and to the Ministry of Technology for support. APPENDIX Critical current for fields perpendicular to the sample surfaces
The critical current of a lead-coated sample for magnetic fields perpendicular to the sample surfaces is shown in Figure 1. It is apparent that the critical current is considerably enhanced by the diffusion of indium out of the sample surface. Other lead-coated samples were found to behave in a similar way. The critical current of indium-coated samples was also found to increase as a result of heat treatment. The dependence of critical current on heat-treatment time is quite different from that for the parallel-field case. The perpendicular field critical current of both lead-coated and indium-coated samples increased rapidly during the first few minutes of heat treatment, then continued to increase slightly as heat treatment was continued. There was no sign of any decrease in critical current even after many hours heat treatment. We believe the increase in critical current to be an example of pinning by 'surface irregularities', as proposed by Swartz and Hart (1967). Heat treatment of an indiumcoated sample will cause indium to migrate very rapidly down the grain boundaries whence it will diffuse a short distance into the surrounding material. The resulting 'sheets' of indium-rich material will act as barriers to fluxons attempting to move past them. Similarly, heat treatment of lead-coated samples will create regions of abnormally low indium content. REFERENCES
1. DOIDGE, P. R., KWAN SIK-HUNG, and TILLEY, D. R. PhilMag 13, 795 (1966) 2. FRENCH, R.A., and LOWELL, J. Phys Rev 173 504 (1968) 3. SAINT-JAMES, D., and DE GENNES, P. G. Phys Lett 7, 306 (1963) 4. JONES, R. G., RHODERICK, E. H., and ROSE-INNES, A. C. Phys Lett 24A, 318 (1967) 5. SWARTZ, P. S., and HART, H. R. Phys Rev 137, A818 (1965);PhysRev 156,412 (1967) 6. BEAN,C. P., and LIVINGSTON, J. D. Phys Rev Lett 12, 14 (1964) 7. CAMPBELL, A. M., EVETTS, J. E., and DEW HUGHES, D. PhilMag 18, 313 (1968) 8. LOWELL, J. J Phys. C (Solid St Phys) 2,372 (1969) 9. EVETTS, J. E. PhysRev B2, 95 (1970)
CRYOGENICS
. A U G U S T 1971