Persistent currents induced in the superconducting surface sheath

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Abstract
The magnetization curves of type 2 superconductors, and type 1 super-conductors, for which H c3>H c, show a hysteretic tail in a range of field below H c3 which corresponds to a persistent current induced in the super-conducting surface layer (the sheath). Existing theories of this magnetization M are based on the hypotheses (a) that departures of the current distribution from uniformity are confined to the ends and may be neglected for long enough cylinders, and (b) that at the persistent (‘critical’) current, the free energy has the same value as it does in the normal state. These are the essential assumptions of what we call the infinite cylinder model. The apparent differences between the results of calculations based on this model made by Fink et al. and those made by Park, are shown to reside in a parameter η which Fink et al. assume to be ∼1. Values of η for a number of values of field and Ginzburg-Landau parameter κ are given. In a certain range, η ∼ 1 but η deviates markedly from 1 when κ is large (η∝ κ1/2 then), and at low fields when κ is small. It is shown that according to the model the tail should be symmetrical about the M=0 axis but that in order to show that it should be so, from free energy arguments, one must take into account the magnetization m of the sheath itself (due to its internal currents) even though m makes a negligible contribution to the total magnetic moment. The theory is compared with experiment, and significant discrepancies are found, particularly in the field dependence of M near H c3, which cannot be explained away as being the result of imperfections in specimen quality. The problem is then reconsidered on the basis of less arbitrary assumptions. First we consider the thermodynamic stability of the sheath. It is necessary to discuss the problem in terms of the current (per unit length, always) rather than magnetization M, so that we may distinguish between current in some particular region of the cylinder from the current averaged over its whole length. The average current J is directly related to the average magnetization M. Adopting the sign convention that a positive current produces a diamagnetic moment, M = −J. The continuous sheath will become unstable when the free energy in the presence of a break in the continuity over a small patch of surface is reduced by an increase in the area of the patch. That is our basic assumption. Unlike the homogeneous systems normally considered in thermodynamics, the sheath becomes unstable at an average current J which depends on the shape and position of the patch. If the patch lies in the central region of the cylinder the associated critical current J 1>>j c, where j c is the critical current calculated according to the infinite cylinder model. If it forms a ring of constant width on one end the sheath becomes unstable at J 2=j c. Thus the free energy criterion used to calculate j c corresponds to the assumption that J 2 is the lowest current at which the sheath becomes unstable. It is shown that the sheath may become unstable to penetration from the end along a large number of narrow slots at a current J 3⩽-j c. Whether or not J 3<j c, however, the sheath would always begin to break up at the end if the current reached a value J u before it became unstable. J u is the average current at which the current on the rim of the continuous sheath would reach j u. j u is the ‘ultimate’ current, the maximum current the sheath can carry under any circumstances. We show that for a high κ superconductor J u<j c, and assume the same inequality to hold for low κ superconductors. Thus at a certain current J p the sheath begins to break up at the end, and at this point, we argue, vortices or ‘flux spots’, of the type whose existence has been proposed by Hart and Swartz, enter the sheath from the ends. J p equals either J u or J 3, whichever is smaller. Only if J p equals J 3 does it correspond to a thermodynamic instability. In any case, since J u<j c, J p<j c, so that j c does not correspond to any observable phenomena, though it is a function with some heuristic value. Unless there is no pinning of the flux spots at all it will not be until a higher (average) current Jk that the spots move continuously through the sheath, entering it at one end and leaving it by annihilation with spots of opposite sign in the middle. It is this current Jk to which the magnetization tail corresponds. Above J p there are observable effects due to the non-uniform distribution of current over a large fraction of the cylinder length, which might serve to determine J p. It is shown, adapting an argument of Hart and Swartz, that Jk should be independent of the size of the cylinder provided the ratio of length to radius is kept constant. Some of the discrepancies between theory and experiment are shown to be resolved. Cylindrical systems such as a toroid, in which the effects of ends have been removed, are briefly considered.