Effect of surface barriers and lower critical fields to magnetization of type-II superconductors

Abstract

To determine the effect of the surface barrier Δ H and the lower critical field H c1 to magnetization of type-II superconductors, we have calculated the initial magnetization curves and full hysteresis loops within the framework of the modified Kim–Anderson critical-state model, where Δ H and H c1 are explicitly taken into consideration. A sample is located in an external field H( ωt)= H dc+ H ac cos ( ωt), where H dc (≥0) is a DC bias field and H ac (>0) is an AC field amplitude. Derivations of the magnetization equations were carried out on the assumption that the critical-current density J c is a function of the local internal magnetic-flux density B i, J c( B i)= k/( B 0+| B i|), where k and B 0 are constants. In addition, we used the effective magnetic field inside the specimen H eff for evaluations of B i, where H eff is assumed to have the form H eff= H−( H/| H|) H cl−[(d H/d t)/|d H/d t|]Δ H. We consider an infinitely long cylinder with radius a and the applied field along the cylinder axis. Denoting the maximum and minimum values of H by H A (= H dc+ H ac) and H B (= H dc− H ac), respectively, four types of hysteresis loops appear, depending on the magnitude of H A. Among these, three types are further classified into several cases, depending on the magnitude of H B. To describe completely the descending and ascending branches of the loops for all the cases, 113 stages of H are considered. To verify the present derivations, all the equations were confirmed to be continuous at their end points. Some typical hysteresis loops computed using the appropriate magnetization equations are demonstrated. From the results, we recognize the effect of Δ H and H c1 on the hysteresis loops. Δ H merely expands the M( H) curves up and down with the increase of Δ H, while H c1 introduces a step-like feature into the hysteresis loops. From measurements of the difference in level of the magnetization under the applied field of opposite directions, one can precisely determine H c1.